CONTENTS. 7. Variational Optimization of the Gaussian Nonlinear Parameters in Atomic and Molecular BO

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1 pubs.acs.org/cr Bor Oppehemer ad No-Bor Oppehemer, Atomc ad Molecular Calculatos wth Explctly Correlated Gaussas Ths paper s part of the 0 Quatum Chemstry thematc ssue. Sergy Bub,*, Mchele Pavaello,*, We-Cheg Tug, Keeper L. Sharey, ad Ludw Adamowcz*,, Departmet of Physcs ad Astroomy, Vaderblt Uversty, Nashvlle, Teessee 3735, Uted States Departmet of Chemstry, Rutgers Uversty Newar, Newar, New Jersey 070, Uted States Departmet of Chemstry ad Bochemstry ad Departmet of Physcs, Uversty of Arzoa, Tucso, Arzoa 857, Uted States CONTENTS. Itroducto 37.. Need for Hgh-Accuracy BO ad No-BO Calculatos 37.. Challeges Hgh-Accuracy Calculatos Very Accurate BO Calculatos of Molecular Potetal Eergy Surfaces (PESs) 39. Formalsm 40.. Norelatvstc Hamltoa the Laboratory Frame ad Separato of the Ceter of Mass Moto 40.. Clamped-Nucle Hamltoa 4.3. The Adabatc Approxmato 4.4. Icludg the Noadabatc Effects by Meas of Perturbato Theory 4.5. The Varatoal Method Choces of Bass Fuctos for Hghly Accurate Varatoal BO ad No-BO Calculatos Competto betwee ECG ad JC Fuctos Explctly Correlated Gaussa Bass Sets Bass Sets for Atomc Calculatos wth Ifte ad Fte Nuclear Mass Bass Sets for No-BO Calculatos o Datomc Molecules Bass Sets for No-BO Calculatos o Systems wth More Tha Two Nucle Bass Sets for No-BO Molecular Calculatos the Presece of Exteral Electrc Feld Bass Sets for Molecular BO Calculatos Use of Premultplers ECG Bass Sets for Molecular BO Calculatos Ioc ad Covalet Bass Fuctos Symmetry of the Wave Fucto Permutatoal Symmetry Permutatoal ad Spatal Symmetry BO Calculatos Spatal Symmetry No-BO Calculatos 5 5. Evaluato of Matrx Elemets vech Operato Gaussa Itegral p Dmesos Evaluato of Itegrals Ivolvg Gaussas wth Agular Preexpoetal Factors Aalytc Gradet of the Eergy Evaluato of Matrx Elemets for Molecular BO Calculatos Varatoal Optmzato of the Gaussa Nolear Parameters Atomc ad Molecular No- BO Calculatos Soluto of the Geeralzed Egevalue Problem Geeratg the Ital Guess for Nolear Parameters Dealg wth Lear Depedeces of the Gaussas durg the Varatoal Eergy Mmzato Varatoal Optmzato of the Gaussa Nolear Parameters Atomc ad Molecular BO Calculatos Optmzato Approach Used the BO Molecular Varatoal Calculatos Buldg the Bass Set Geeratg the BO PES FICI Method ad the Multstep Procedure Used Growg the Bass Set Implemetato Improved Fucto Moblty ad Barrer Tuelg Calculato of the Leadg Relatvstc ad QED Correctos The Relatvstc Hamltoa A System of N Fermos A Fermo Boso System Trasformato of the Relatvstc Operators to the Iteral Coordate System QED Effects Atomc Calculatos 63 Receved: Aprl 3, 0 Publshed: October, 0 0 Amerca Chemcal Socety 36 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

2 Chemcal s 9. Results for Atoms Very Accurate Calculatos for Three- ad Four-Electro Atoms ad Atomc Ios Calculatos for Atomc Systems wth Fve Electros Calculatos for Atomc Systems wth Sx Electros Results for Datomc Molecules Obtaed wthout the Bor Oppehemer Approxmato Datomcs wth Oe ad Two Electros: Charge Asymmetry Iduced by Isotopc Substtuto Datomcs wth Three Electros Datomcs wth More tha Three Electros 69. Hghly Accurate BO Molecular Calculatos 70.. Hydroge Clusters 70.. Molecules wth Oe Atom Other Tha Hydroge 7.3. Dagoal Adabatc Correctos to the BO Eergy for Molecules Cotag up to Three Nucle 73. Calculatg Molecular Propertes wth ECGs: The BO Case 74.. EFG at the Nucle ad the Deuterum Quadrupole Costat Summary 75 Author Iformato 75 Correspodg Author 75 Notes 75 Bographes 75 Acowledgmets 76 Refereces 76. INTRODUCTION Sce the early wor of Hylleraas o the helum atom, t has bee commo owledge that, to accurately accout for the teracto betwee the electros a atom or a molecule, wave fuctos that explctly deped o terelectroc dstaces must be employed. To overcome the algebrac ad computatoal dffcultes assocated wth the use of the Hylleraas fuctos for systems wth more tha two to three electros, 960 Boys ad Sger 3 troduced a smpler format of bass fuctos that explctly deped o the terelectroc dstaces, the so-called explctly correlated (or expoetally correlated) Gaussa fuctos (ECGs). ECGs, due to the smplcty of calculatg the Hamltoa matrx elemets wth those fuctos, have become popular very accurate quatum-mechacal calculatos of small atoms, molecules, ad other quatum systems the past 30 years. 4 7 They have bee successfully appled hgh-accuracy atomc ad molecular calculatos performed wth ad wthout the assumpto of the Bor Oppehemer (BO) approxmato for systems wth three to eght partcles. Those clude very accurate calculatos of the BO potetal eergy surfaces (PESs) of two-, three-, ad four-electro systems. The problem of fdg a effectve ad hghly accurate approxmato to the wave fucto that descrbes electros ad ucle a atom or a molecule, or more geerally a system of partcles teractg wth attractve ad/or repulsve Coulombc forces, s very complex ad requres a careful physcal aalyss ad sght. Ths partcularly apples to the proper descrpto of the terpartcle correlato effects resultg from the repulso of charged equvalet partcles (e.g., electros) subject to the Paul excluso prcple. I calculatos where the BO approxmato s ot assumed, the correlato effects also volve couplg of the motos of partcles wth opposte charges, such as ucle ad electros a molecule. I ths case, the correlato effects clude the electro ucleus correlato resultg from the electros, partcularly the core electros, followg very closely the ucle, as they are strogly attracted to them. Aother ssue that arses descrbg the correlato effects s related to whether the correlato s prmarly radal,.e., whether the shape of the Coulomb hole ( the case of two repellg partcles) s symmetrc or t has some agular asotropy. The agular correlato asotropy appears, for example, excted states of atoms, where two electros may occupy cofguratos where they are ot oly radally separated but they are also separated by havg dfferet agular wave fuctos. Such a stuato occurs, for example, excted Rydberg D states of the lthum atom correspodg to electro cofguratos s d, = 3, 4, 5,..., where, addto to the cotrbuto to the wave fucto from the ma s d cofguratos, there are cotrbutos from cofguratos s p. We wll elaborate o ths ssue later ths revew. The goal of the preset publcato s to revew recet wors that have used all-partcle ECGs very accurate varatoal BO ad o-bo quatum-mechacal calculatos o atoms ad molecules. I addto to provdg a geeral overvew, we wll focus partcular atteto o the ey ssues related to the effectve mplemetato of computatoal algorthms. We wll also descrbe several represetatve examples of BO ad o- BO calculatos of some small atomc ad molecular systems, wth emphass o how well the results of the calculatos compare wth the best avalable expermetal measuremets. Eve though the varatoal method combed wth expadg the wave fucto terms of all-partcle ECGs s oe of the most accurate methods avalable to solve for the groud ad excted states of quatum systems, t suffers from ufavorable N! depedecy o the umber of detcal partcles. Ths lmts the applcablty of the method at preset to small atomc ad molecular systems. It should also be metoed that, eve though orbtal calculatos are usually sgfcatly less accurate ad slower covergg for small systems tha the calculatos wth explctly correlated fuctos, some staces wth proper extrapolatos such calculatos are qute compettve ad capable of provdg very accurate results as well. I Table we summarze the acroyms used ths revew... Need for Hgh-Accuracy BO ad No-BO Calculatos From the very begg of molecular quatum mechacs, the developmet of hghly accurate theoretcal models that produce results agreeg wth the most up-to-date hgh-resoluto spectroscopc measuremets has bee a mportat source of owledge ad formato. It has allowed the valdato of the theoretcal foudatos ad provded better uderstadg of the electroc structures of atoms ad molecules. As the expermetal techques advace ad acheve hgher levels of precso, refemets have to be made theoretcal models to descrbe effects ad teractos eglected or treated more approxmately the prevous models. I recet years the measuremets of such quattes as molecular rovbratoal trasto eerges, ozato potetals, ad electro afftes have reached the precso of cm ad eve 37 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

3 Chemcal s Table. Glossary of Acroyms ad Abbrevatos Used Ths abbrevato BO o BO BP CC CH CN COM CPU D DB DBOC DC ECG EFG FPO FICI FNM GPT GSEP/GHEP ICI INM JC KG MDC MBPT MV NRQED PEC PES QED SE SVM hgher. Obtag a smlar precso theoretcal calculatos (whch usually meas covergg the eergy to the relatve accuracy of or hgher) s a very challegg tas. As recet wors have show, 8 achevg hgh accuracy the calculatos requres ot oly a very accurate descrpto of the correlato effects, but also the cluso of relatvstc, quatum electrodyamcs (QED) ad, possbly, the effects due to the fte sze of the ucle. These types of effects have already bee calculated for two- ad three-electro atomc systems, leadg to theoretcal results whch are very precse ad accurate whe compared to the expermets. 3 Now the challege les extedg these types of calculatos to atoms wth more tha three electros ad to small molecular systems wth three or more ucle. Hgh-accuracy theoretcal results for such systems wll provde ew grouds for the verfcato of the theoretcal models ad for the assessmet of ther lmtatos. It should be oted that based o the comparso of the theoretcal ad expermetal data t s prcple possble to accurately determe the values of fudametal costats, uclear rad, uclear quadrupole momets, ad other quattes. Therefore, hghly accurate calculatos o small atoms ad molecules may become a very valuable tool for the precso measuremet scece. As a example, we ca meto the determato of the proto/electro mass rato 33 ad the uclear charge rad. 4,3,34 36 descrpto Bor Oppehemer o-bor Oppehemer Bret Paul coupled cluster Coulomb Hamltoa clamped ucle ceter of mass cetral processg ut Drac Drac Bret dagoal Bor Oppehemer correcto Drac Coulomb explctly correlated Gaussa electrc feld gradet froze partal optmzato free teratve-complemet teracto fte uclear mass Gaussa product theorem geeralzed symmetrc/hermta egevalue problem teratve-complemet teracto fte uclear mass James Cooldge Kle Gordo matrx dfferetal calculus may-body perturbato theory mass-velocty orelatvstc quatum electrodyamcs potetal eergy curve potetal eergy surface quatum electrodyamcs Schro dger equato stochastc varatoal method The developmet of hgh-level quatum-mechacal methods related to the use of ECGs atomc ad molecular calculatos ca also serve as a mportat source of deas ad techcal solutos for the developmet of other approaches, whch ca be appled to larger systems. As ECGs wll start to replace products of sgle-partcle Gaussa orbtals as bass fuctos expadg the wave fucto hgh-level molecular BO calculatos, these techques may fd ew applcatos. For example, further developmet of such approaches as the R (F) method 37 4 may beeft from utlzg the aalytc gradet. Aother example s the use of the ECGs wth tme-depedet olear parameters (e.g., Gaussa ceters) studyg the dyamcs of chemcal processes such as processes tated by photoexctatos clusters. Techques employg ECGs ca also provde useful tools for the developmet of methods that descrbe the dyamcs of the coupled ucleus electro moto atomc ad molecular systems whch ca ow be studed wth the femto- ad attosecod spectroscopes. Aother mportat reaso for performg accurate calculatos o small atoms ad molecules wth very hgh accuracy s that such bechmar calculatos ca provde valuable referece data for testg of less accurate quatum-chemcal methods. A example s the total orelatvstc eergy of the system, whch s oe of the most commoly computed quattes. The orelatvstc eergy s dffcult to determe very precsely eve f hghly accurate expermetal data for ozato potetals, electro afftes, dssocato eerges, or trasto frequeces are avalable. Ths s because the determato of the total orelatvstc eergy requres the owledge of the bdg eerges of all subsystems ad, more mportatly, the exact cotrbuto of relatvstc ad QED effects. The latter quattes caot be drectly obtaed the expermet. There s also a predctve purpose for carryg out very accurate atomc ad molecular calculatos. Several recet o- BO ad BO ECG calculatos have produced quattes that ether have ot yet bee measured expermetally or have bee measured, but wth sgfcat error bars exceedg the ucertaty of the calculatos. The exstece of very precse theoretcal predctos may spre the developmet of more precse expermetal tools ad stmulate remeasuremet of those quattes. The breadth ad accuracy of the expermetal data have bee creasg rapdly, ad further major mprovemets are expected due to the developmet of ew expermetal methods for UV laser geerato ad frequecy metrology wth phase-loced femtosecod combs. 4 The data collected usg those ew techques are begg to reveal devatos that suggest that the accuracy of the exstg calculatos s o loger adequate. For example, extesve studes of the spectrum of H + 3, by Oa s group at the Uversty of Chcago, 43,44 have goe cosderably beyod the lmts of the exstg theoretcal wor. More accurate laboratory ad feld (cludg terstellar) observatos ad measuremets of spectra of atoms ad molecules requre more accurate theoretcal calculatos for terpretato ad assgmet... Challeges Hgh-Accuracy Calculatos Very accurate BO calculatos of groud ad excted states of atomc ad molecular systems are rare quatum-chemcal studes because they usually requre -house software developmet ad substatal computatoal resources. Eve more scarce are atomc ad molecular calculatos where the BO 38 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

4 Chemcal s approxmato s ot assumed. Hgh accuracy also requres that the calculatos clude relatvstc ad QED correctos. Those are usually calculated wth the perturbato theory. A few groups have developed capabltes to carry out such calculatos for systems wth oe ad two electros. The actual applcatos, however, most cases have bee lmted to oe- or two-electro systems oly. Ths s partcularly true for molecular systems computed wthout usg the BO approxmato. Whle the developmet of methods descrbg the coupled moto of the electros ad the ucle has receved some atteto, 45 5 oe of the wors has reached a level of accuracy smlar to that achevable the calculatos wth ECGs. No-BO calculatos of the groud ad excted states of molecular systems are dffcult for the followg reasos: () Treatg ucle ad electros of a molecular system o equal footg adds complexty to the problem due to the creased umber of the degrees of freedom oe eeds to accurately represet the wave fucto. Addtoal degrees of freedom requre addtoal computatoal effort. () The o-bo wave fucto eeds to very accurately represet the correlated moto of the ucle ad the electros, ad t has to be costructed usg bass fuctos that ca effectvely descrbe the electro electro, ucleus electro, ad ucleus ucleus correlato effects. It may soud somewhat uusual to tal about the ucleus ucleus correlato, as the term correlato s usually used to descrbed the effects pertag to electros, but f ucle ad electros are treated o equal footg, as happes o BO calculatos, the ucleus ucleus correlated moto also eeds to be represeted the wave fuctos a smlar way as the electro electro correlated moto s. I addto, the masses of the ucle beg much larger tha that of the electro may lead to rapd varato of the relatve teruclear wave fucto, whch s hard to represet wth the usual bass fuctos. Moreover, the ucleus ucleus correlato s much stroger tha the electro electro correlato the sese that the ucle are heavy ad they stay separated (.e., ther moto s more correlated) wth the separato dstace varyg much less tha the separato dstace betwee much lghter ad, thus, more delocalzed electros. The ucleus ucleus correlato ca oly be descrbed the wave fucto by cludg correlato factors. () After separatg the traslatoal degrees of freedom, the teral Hamltoa of a atomc or a molecular system solato s sotropc (.e., rotatoally varat) ad ts egefuctos belog to the rreducble represetatos of the group of three-dmesoal (3D) rotatos. It s ecessary that the bass fuctos used the wave fucto expaso reflect ths symmetry. (v) As the vbratoal ad electroc degrees of freedom are coupled, the mafold of the excted states for a o-bo molecular system correspodg to a partcular value of the total rotatoal quatum umber cludes a mxture of vbratoal ad electroc states. Whle the mxture for the lower lyg states s usually strogly domated by a sgle compoet, beg a product of the electroc wave fucto tmes a vbratoal wave fucto, for states lyg close to the dssocato lmt, two or more compoets may provde more sgfcat cotrbutos. Those compoets may have electroc wave fuctos represetg dfferet electroc states ad vbratoal wave fuctos correspodg to dfferet vbratoal quatum umbers. Some of those states may have multple odes (e.g., hgh vbratoal states) ad requre flexble bass fuctos to be descrbed. (v) Icludg agular depedecy the wave fucto (to determe hgher rotatoal states) requres addto of agular factors to the bass fuctos. For such fuctos, the multpartcle Hamltoa tegrals are more complcated tha those for bass fuctos descrbg states wth zero agular mometa. More complcated are also the expressos for the dervatves of the Hamltoa matrx elemets that eed to be calculated to determe the aalytc eergy gradet whose use s crucal the mmzato of the varatoal eergy fuctoal. (v) If accuracy smlar to that of hgh-resoluto expermets s the am of the calculato, the lowest-order relatvstc ad QED effects eed to be accouted for. Matrx elemets volvg operators represetg those effects are more complcated tha the Hamltoa matrx elemets..3. Very Accurate BO Calculatos of Molecular Potetal Eergy Surfaces (PESs) The frst success of very accurate molecular calculatos that utlzed explctly correlated bass fuctos was the wor of Kołos ad Wolewcz cocerg the H molecule. I ther wor publshed they preseted calculatos of the H spectra that agreed wth the expermetal data of Herzberg wth cm. Ther wor also led to some revsos of Herzberg s orgal le assgmet. I spte of the eormous advaces computer hardware, t too the ext 30 years to acheve a comparable level of accuracy the calculatos of rovbratoal spectra of a three-proto, two-electro system, H + 3. However, eve at preset the H + 3 rovbratoal spectrum s well uderstood oly for states lyg below the barrer to learty of ths system, whch s located cm above the groud state level. Precse assgmet of the spectral les above ths barrer stll remas a great challege for both theory ad expermet. Oce the assgmet s made, the H + 3 o wll be the best uderstood three-ucleus system ever studed expermetally ad theoretcally. The ECGs were troduced to quatum-chemcal calculatos by Boys ad Sger.,3 I 964 a mportat paper by Lester ad Krauss o the Hamltoa tegrals wth ECGs for two-electro molecular systems appeared 53 that had gve mometum to several wors cocerg mplemetato of these fuctos the molecular calculatos. I the 970s, Adamowcz ad Sadlej had exteded the Lester ad Krauss approach to calculate the electro correlato eergy for some small datomcs the framewor of the perturbato theory About the same tme Jezors ad Szalewcz employed ECGs very accurate calculatos of the teracto eerges usg the symmetry-adapted perturbato theory. 6,6 The late 970s ad early 980s wtessed developmet of ovaratoal methods for calculatg electroc structures of atoms ad molecules. May-body perturbato theory (MBPT) ad the coupled cluster (CC) methods had bee mplemeted ad started becomg route tools for hgh-level ab to calculatos of small ad medum-sze molecules. 63,64 Motvated by ths developmet, the team of Mohorst, Jezors, Szalewcz, ad Zaboltzy troduced ECGs to the CC method ad ths was acheved by usg the coupled cluster equatos at the par level reformulated as a system of tegro-dfferetal equatos for sp-free par fuctos. These equatos were solved usg two-electro ECGs (also called 39 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

5 Chemcal s Gaussa gemals). The wor resulted a seres of bechmar studes for small atomc ad molecular systems. Aother mportat developmet cocerg the use of ECGs MBPT/CC calculatos of medum-sze molecular systems orgated wth the wor of Kutzelgg, 37,69 who suggested that lear r j correlato factors should be added to orbtal products to mprove the descrpto of the electro correlato. I order to mprove the computatoal effcecy the calculatos, he suggested usg the resoluto of detty to avod explct calculato of tegrals volvg more tha two electros. The approach was termed the R method ad qucly became a mastream techque computatoal chemstry to perform hgh-accuracy atomc ad molecular calculatos. 70,7 Recetly the R method has evolved to a array of methods, most otably the F method where the lear r correlato factor s replaced by a Slater-type gemal, 7,73 exp[ αr ], ad the G method where the explct correlato s gve by a Gaussa gemal,.e., exp[ αr ]. The wors o the mplemetato of ECGs the MBPT/ CC methods have bee paralleled by progress the ECG varatoal calculatos. For example, varatoal ECG calculatos were able to reach a aohartree (subwaveumber) precso level for orelatvstc adabatc calculatos of the helum dmer 74 ad below pcohartree level for the hydroge molecule 75 usg ECG calculatos. Followg the mportat wors of Kołos ad Wolewcz, there have bee several other wors o very accurate calculatos of small molecular systems As ths revew prmarly deals wth the calculatos performed wth the use of ECGs, we should partcularly meto the wors of the group of Rychlews. 4,74,8 87 I recet years several methods have bee developed ad mplemeted for more effcet geerato of PESs of small molecular systems employg ECGs. 7 The ey elemet of ths developmet has bee the use of the aalytcal gradet determed wth respect to the Gaussa olear parameters the varatoal eergy mmzato Oe ca compare the crease of the effcecy assocated wth the use of the gradet ths case wth the effcecy crease the molecular-structure optmzato after the eergy gradet determed wth respect to molecular geometrcal parameters was troduced to the feld. As computatoal resources become creasgly more accessble ad affordable, such calculatos become more feasble. However, as the sze of the molecules creases, the complexty ad the cost of the ECG PES calculatos also crease. Let us cosder two-electro systems wth dfferet umbers of ucle as a example. For a system wth oe ucleus, e.g., the helum atom, oe ca reach 0 9 hartree accuracy of the eergy wth oly 00 Gaussas. 9 I the case of two ad three ucle, H ad H 3 +, 500 ad 000 Gaussas are eeded, respectvely, to acheve smlar accuracy. 93,94 Also, for a four-electro, four-uclear system, (H ), eve 7000 Gaussas oly yeld 0 6 hartree accuracy. 95 I the ECG calculatos of larger molecular systems, besdes the eed for larger bass sets, a problem whch oe ca ecouter ad has to deal wth more ofte s the occurrece of the lear depedecy betwee the bass fuctos. Effectvely dealg wth ths problem requres rather sophstcated approaches. Also, better techques for hadlg the usage of memory, for guessg ew bass fuctos whe the sze of the bass set s beg exteded, ad for the bass set optmzato have to be developed. These measures wll be elaborated o ths revew. The ma purpose of the BO molecular calculatos employg ECGs s to geerate PESs of groud ad excted states that ca be used to perform rovbratoal calculatos. The ECG PESs of small systems are usually capable of delverg a subwaveumber accuracy for the full rage of the vbratoal trastos provded that at least the adabatc correcto s cluded the eergy of each PES pot. We wll show examples of such calculatos ths revew. If the calculatos also provde the correspodg surface of the molecular dpole momet, the rovbratoal trasto momets, ad thus the bad testes, ca be calculated. PESs ca also be used to perform reacto dyamcs calculatos usg ether the classcal trajectory approach or a wave pacet quatum approach. I both approaches oadabatcty, or hoppg betwee PESs of several electroc states, ca be smulated. Such smulatos volve calculatg adabatc ad oadabatc couplg matrx elemets, whch ca be relatvely easly doe (though t has ever bee reported so far) for eergy pots determed usg wave fuctos expaded terms of ECGs. The ECG PES calculatos performed so far have bee lmted to states descrbed wth wave fuctos wth o odes at the ucle. If such odes exst, the Gaussas the bass fuctos eed to be multpled by a coordate or a product of coordates. 96 Such coordate premultplers to ECGs are also used other applcatos (see secto 3). Fally, the developmet ad mplemetato of ew algorthms ad procedures always volves solvg umerous techcal problems at both the algebrac ad computatoal levels. A part of ths revew s devoted to ths. We partcularly emphasze those ssues that have broader mplcatos ad applcatos the areas that are ot lmted to the developmet of methods for very accurate BO ad o-bo calculatos of small atomc ad molecular systems.. FORMALISM.. Norelatvstc Hamltoa the Laboratory Frame ad Separato of the Ceter of Mass Moto Let us cosder a system comprsed of N orelatvstc partcles teractg va Coulomb forces. If R s the posto vector of the th partcle the laboratory Cartesa coordate frame, M s ts mass, ad Q s ts charge, the the Hamltoa of the system has the followg form: N Hlab = R + M N N = = j> QQ R Here R deotes the gradet wth respect to R ad R j = R j R s the dstace betwee the th ad jth partcles. We wll call H lab the laboratory frame Hamltoa. As the prmary goal s solvg for the boud states, the frst step wll be to separate out the traslatoal moto of the system as a whole;.e., we wll elmate the moto of the ceter of mass from further cosderato. There are several possble ways to do that. Perhaps the most atural way s to use the terpartcle coordates. Let us place some partcle at the org of the ew, teral, Cartesa coordate system. Ths partcle s called the referece partcle. The we ca refer the other partcles to the referece partcle usg relatve coordates r = R + R. These coordates, alog wth the three coordates descrbg the posto of the ceter of mass, r 0, are our ew coordates. If we deote the total mass of the j j () 40 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

6 Chemcal s system as M tot = = M, the the coordate trasform loos as follows: r0 = M M MN R + R RN Mtot Mtot Mtot r = R + R r = R + R3 () r = R + R N whle the verse coordate trasformato s gve by M M3 MN R = r0 r r... r M M M M M3 MN R = r0 + r r... r M M M M M3 MN R3 = r0 r + r... r M M M tot tot tot tot M M3 MN RN = r0 r r... + r M M M tot tot tot tot tot tot tot tot (3) Upo the trasformato of the laboratory frame, the Hamltoa H lab eq separates to two operators,.e., the Hamltoa descrbg the moto of the ceter of mass (COM) of the system H CM = r M tot 0 ad the followg teral Hamltoa that represets the relatve moto of the partcles: = H r + + r rj μ m j qq j + r < j = 0 = j qq 0 r where = N, the prme symbol deotes the matrx/vector trasposto, r j = r j r, m = M +, q = Q +, ad μ = m 0 m / (m 0 + m ). The Hamltoa eq 5 descrbes the moto of pseudopartcles wth masses m ad charges q the cetral feld of the referece partcle. The motos of the pseudopartcles are coupled through the mass polarzato terms j (m 0 ) r rj ad through the Coulombc teractos depedet o the dstaces betwee the pseudopartcles ad the org of the teral coordate system, r, ad o the relatve dstaces betwee the pseudopartcles, r j. Due to the separablty of the teral moto ad the moto of the ceter of mass, the soluto of the Schro dger equato (SE) wth the laboratory Hamltoa ca be preseted as the product ψ = exp[ r ] ψ( r,..., r) (4) (5) lab 00 (6) where 0 s the mometum of the system as a whole ad ψ(r,...,r ) s the soluto of the SE wth the teral Hamltoa. The Hamltoa eq 5 ca be coveetly wrtte the matrx form. To do that we combe the coordates of the pseudopartcle postos ad the correspodg gradets to two 3-compoet colum vectors: r r r = r r, = r r r Wth that we have qq 0 H = rm + r + r = < j qq j r Here M = M I 3 s the Kroecer product of the matrx M ad the 3 3 detty matrx, I 3. The dagoal elemets of matrx M are /(μ ), /(μ ),..., /(μ ), whle all offdagoal elemets are equal to /(m 0 )... Clamped-Nucle Hamltoa Fdg the egefuctos of the Coulomb Hamltoa (CH) eq expressed the laboratory frame would volve descrbg traslatoal states of the COM ad therefore the use of plae waves the tral wave fuctos. The resultg states would be completely delocalzed space, havg lttle practcal use from a chemcal prospectve. As poted out secto., the separato of the COM moto from the teral moto geerates a Hamltoa of the type eq 5. Such a Hamltoa the mass polarzato terms mxes dervatves of the electroc ad uclear coordates. The drect use of the Hamltoa eq 5 s far from beg trval ad s j (7) (8) 4 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

7 Chemcal s the subject of the sectos devoted to the o-bo approach. To smplfy the problem, frst Hetler ad Lodo 97 ad the Bor ad Oppehemer 98 decded to cosder the ucle as beg partcles wth fte masses. Ths led to a formalsm, partly used also by Hady ad Lee, 99 volvg the splttg of the CH to two cotrbutos: H = H + T lab el (9) H êl s defed as N e e H = e el r + V({ r }; { r }) me = (0) where {r e } ad {r } are labels of electroc ad uclear coordates, respectvely, ad r e s a Laplaca operator that acts oly o the three Cartesa coordates of the th electro, N e represets the total umber of electros the system. T s the etc eergy of the ucle ad s defed as T = N α= M α rα () where M α s the mass of ucleus α. From eq t follows that f the uclear masses are allowed to reach very large values (at the lmt of fte masses) the oly term to survve eq 9 s H êl, also ow as the clamped-ucle (CN) Hamltoa. I ths revew, we refer to BO calculatos as those that approxmate the egefuctos of the CN Hamltoa, where the ucle are ept at fxed postos space ad the oly varables the wave fucto depeds o are the postos of the electros..3. The Adabatc Approxmato The SE volvg the CN Hamltoa has the followg form: e μ e HelΦ μ( r ; r ) = E ( r ) Φμ ( r ; r ) () The solutos Φ μ (r e ;r ) are also called electroc wave fuctos, ad sometmes the CN Hamltoa s called the electroc Hamltoa. I ths secto we assume that the solutos of eq are odegeerate. The semcolo separatg the electroc ad uclear coordates eq deotes the fact that the uclear coordates are treated as parameters both the Hamltoa ad the wave fucto. Ths meas that the uclear coordates tae part ether the dervatve operatos or the tegrals. I the calculato of the groud state wave fucto, oce Φ 0 (r e ;r ) s avalable, or approxmated to a suffcet degree of accuracy, a tral soluto to the CH ca be attempted as a product of a purely uclear fucto, deoted as χ(r ), wth Φ 0 (r e ;r ), amely 0 e Ψ a( r, r ) = χ( r ) Φ0( r ; r ) e (3) where the subscrpt a stads for adabatc to dcate that the electroc part of the wave fucto s calculated assumg that the electros follow the motos of the ucle adabatcally,.e., wthout trasferrg ay part of ther eergy to the ucle, ad vce versa. Because the electroc wave fucto Φ 0 (r e ;r ) must be depedet of the COM moto, the uclear coordates the tral wave fucto above are a redudat set. I mathematcal terms we ca mpose the depedece of the electroc wave fucto from the COM moto by havg Φ 0 (r e ;r ) beg a costat upo ay dsplacemet of the COM coordates, amely H CMΨ a 0 ( r e, r ) = r Ψ a 0 ( r e, r ) = 0 0 M (4) Wth the above propertes of Ψ a (r e,r ), t ca be used as a tral wave fucto the SE volvg the CH wth the assumpto that the electroc part eeds o further mprovemet ad ca be tegrated out. The fucto χ(r ) s the obtaed by frst tegratg over the electroc coordates wth the Φ 0 (r e ;r ) beg replaced by Φ 0 (r e,r ), where ow the uclear coordates are promoted from parameter to varable status. Carryg out such a procedure, oe obtas the followg SE for the uclear fuctos χ (r ): 99 N μ α + U ( r ) λ χ = μ ( r ) 0 α (5) α= where λ represets the egevalue correspodg to χ (r ) ad the potetal, whch the ucle move, s geerally specfc to the electroc state (μ), amely μ μ e U ( r ) = E ( r ) + Φ μ( r, r ) T Φ μ( r, r ) r e (6) The re stads for tegrato over the electroc coordates. I the rght-had sde of eq 5 terms that volve other solutos to the CN Hamltoa, or electroc excted states, do ot appear because they have ot bee troduced the asatz eq 3. Equato shows that the potetal whch the ucle move s ot just the correspodg egevalue of the electroc wave fucto. A addtoal term eeds to be added, called the adabatc correcto, or the dagoal Bor Oppehemer correcto (DBOC hereafter). Geerally, the DBOC s specfc to the partcular electroc state descrbed by the wave fucto Φ μ (r e,r ), ad has the followg form: E ( r ) = Φ μ( r, r ) α Φ μ( r, r ) r e M μ e e a α α e (7) The cluso of the DBOC of eq 7 corrects the CN eergy by a term of the order of O(m e /M). I secto.3 the methodology of calculatg the DBOC wth floatg ECG s preseted. The above procedure to obta the adabatc correctos should be tae by the reader wth a gra of salt. Here, we am to approxmate the solutos of the CH startg from the solutos of the CN Hamltoa. We are ot tryg to justfy whether the form of the tral wave fucto eq 3 s approprate, such as havg proper permutatoal ad rotatoal symmetry. For a more -depth dscusso of the ls betwee the CH ad the CN Hamltoa, we refer the terested reader elsewhere. 00,0.4. Icludg the Noadabatc Effects by Meas of Perturbato Theory The correcto derved secto.3 refes the CN eergy by a term havg the O(m e /M) magtude. Ths correcto modfes the BO PES by the addto of a term. The corrected PES s the μ μ μ a U ( r ) = E ( r ) + E ( r ) (8) A oadabatc correcto to the groud state eergy caot be cluded the same way as the adabatc correcto. That s because ay oadabatcty wll mx groud ad excted electroc states, as well as groud ad excted rovbratoal states. It s mpossble to dsetagle these two cotrbutos. 4 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

8 Chemcal s Therefore, the correcto to the adabatc eergy must deped o both uclear ad electroc coordates, amely E 0 a (r e,r ), ad t s specfc to the rovbratoal adabatc state cosdered. The oadabatc correcto s derved by cosderg wave fucto correctos orthogoal to the adabatc wave fucto of eq 3, e.g., orthogoal to Φ 0 (r e ;r ) the electroc coordates, ad/or orthogoal to χ(r ) the uclear coordates. A asatz of the wave fucto that goes beyod the adabatc approxmato s commoly wrtte as e e c μ, μ μ, Ψ ( r, r ) = χ ( r ) Φ( r ; r ) (9) =Ψ ( r, r ) +Ψ ( r, r ) (0) a e a where Φ μ are the wave fuctos of electroc excted states obtaed by solvg the CN SE eq, the χ are approxmate uclear wave fuctos, ad the c μ, are some real-valued expaso coeffcets. For the sae of clarty, the label deotg the electroc state has bee omtted by the total wave fuctos eqs 9 ad 0, as well as the equatos that follow. I a termedate ormalzato framewor, the frst term of the seres eq 0 would be detcal to the wave fucto eq 3. The oadabatc correcto ca be derved usg the perturbatve formalsm wth the CH represeted as a sum of adabatc Hamltoa, H a, ad a perturbato, H. The adabatc Hamltoa ca be wrtte a spectral form terms of ts egevalues ad the CN electroc egefuctos as μ H = Φ U Φ a μ μ μ e () ad the perturbato s defed as what s left to mae up the CH: = H H H lab a () I order to umercally solve the problem, t s ot possble to cosder matrx elemets of the H perturbato, as they are ot well-defed. 0 A better approach 03,04 s to start by splttg the wave fucto to adabatc ad oadabatc cotrbutos, as eq 9, ad derve perturbatve-le solutos to the oadabatc part. By applyg eqs ad 6, ad by otcg that the frstorder oadabatc correcto to the adabatc eergy s zero, the frst result ca be derved rght away, amely Φ H Φ e = 0 (3) 0 0 r Thus, the leadg correcto s a secod-order quatty. The frst step computg ths correcto s by fdg a expresso for the frst-order correcto to the wave fucto, that s, the Ψ a term eq 0. I dog so, Pachuc ad Komasa 03,04 derved the followg equato: Ψ =Φ δχ + a 0 0 E H T Ψa ( ( r ) ) el (4) where the operators have the same meag as eq 9; the prme dcates that the referece state, Φ 0, s excluded from the verso. The δχ fucto dcates the oadabatc correcto to the uclear wave fucto. The Pachuc Komasa 03,04 oadabatc correcto to the electroc eergy taes the form E = Ψ H Ψ a a a = Ψ T Ψ a a = Ψ a T T Ψ E 0 a ( ( r ) H ) el (5) whch s of secod order terms of the perturbato eq. We should ote that the adabatc asatz volved eq 5 could be labeled wth a rotatoal ad a vbratoal quatum umber, thus showg that the oadabatc correctos are specfc to each rovbratoal state..5. The Varatoal Method Most of the calculatos cosdered ths revew are performed wth the framewor of the Rtz varatoal method. The ma dea of the varatoal method orelatvstc quatum mechacs s based o the fact that the expectato value of the Hamltoa of the system computed wth a arbtrary wave fucto, ψ(r) (here r deotes the coordates of all actve partcles), whch satsfes the proper symmetry costrats, s always a upper boud to the exact groud state eergy, ψ H ψ / ψ ψ E 0. Ths geeral property of the eergy fuctoal facltates a way to obta very accurate approxmatos to the exact wave fucto by the optmzato of the parameters, both lear ad olear, whch the fucto comprses. Ths optmzato s accomplshed by the eergy mmzato. If the wave fucto s expaded terms of some bass fuctos K ψ() r = c ϕ () r = (6) ad oly the lear coeffcets are optmzed, the the eergy mmzato procedure reduces to solvg the geeralzed egevalue problem Hc = εsc (7) where H ad S are K K symmetrc (or Hermta f the bass fuctos are complex) matrces of the Hamltoa ad overlap H l = ϕ H ϕ l ad S l = ϕϕ l, whle c s a K- compoet vector of the lear coeffcets. Equato 7 has K solutos,.e., K eergy values ad K correspodg wave fuctos. Accordg to the m max theorem, f the eergy values are set a creasg order, the frst oe provdes a upper boud to the exact orelatvstc groud state eergy of the system ad the th oe provdes a upper boud to the exact eergy of the ( )th excted state (detals of the proof ca be foud ref 5). I addto to varyg the lear coeffcets the wave fucto expaso, oe ca also vary olear parameters volved the bass fuctos. The accuracy of atomc ad molecular quatum mechacal calculatos, partcularly those volvg explctly correlated bass fuctos, s prmarly acheved by performg extesve optmzato of the bass fucto olear parameters..6. Choces of Bass Fuctos for Hghly Accurate Varatoal BO ad No-BO Calculatos I atomc calculatos the possble choces of the bass fuctos are lmted. The most crucal lmtato s related to the eed for accurate ad expedtous calculato of the Hamltoa matrx elemets. Also, the bass set has to accurately descrbe the state of the system uder cosderato 43 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

9 Chemcal s Table. BO Eerges ( hartrees) of Hydroge Molecule at R H H =.40 bohr a K eergy ΔE type year authors JC 00 Pachuc ECG 008 Cece ad Szalewcz ECG 008 Cece ad Szalewcz ECG 008 Cece ad Szalewcz ECG 008 Cece ad Szalewcz ECG 008 Cece ad Szalewcz ICI 007 Naatsuj et al JC 006 Sms ad Hagstrom KW 995 Wolewcz 0 a ΔE s the eergy dfferece betwee calculatos at R H H =.40 bohr ad R H H =.4 bohr. ad, partcular, the electro correlato effects the state. As these effects cocer electros avodg each other ther motos aroud the ucleus, the most effectve bass fuctos for descrbg the correlato pheomeo are fuctos explctly depedet o the terelectro dstaces. The most effcet explctly correlated fuctos are those that smultaeously deped o dstaces of all electros the system. The most serous problem the developmet of methods employg explctly correlated fuctos s the dffculty that may arse accurately calculatg the tegrals whch appear the Hamltoa matrx elemets. As the complexty of these tegrals grows wth the creasg umber of electros ad wth the electros occupyg hgher agular mometum states, more complcated expressos for these tegrals appear. Ths may create dffcultes extedg the calculatos to systems wth more electros. Amog the bass fuctos most ofte used quatum calculatos of small atoms wth less tha four electros, there are the Hylleraas-type fuctos 3,05 09 ad the expoetal fuctos (also ow as the Slater-type fuctos). 0 3 For a three-electro atomc system the Hylleraas fucto has the followg form (here we cosder the states wth zero total agular mometum): ϕ( r, r, r3) = r3 r3 r r r r3 exp( α r αr α33 r) (8) where r are electro ucleus dstaces, r j are terelectro dstaces, ad α s are parameters whch are subject to optmzato the varatoal calculato. Oe otces that the Hylleraas fuctos are oly correlated through the preexpoetal polyomals ad there s o r j presece the expoet. I the expoetal fuctos ϕ( r, r, r) = exp( αr α r α r βr β r βr ) (9) the opposte happes: the r j factors are oly preset the expoet. Recetly, t was demostrated that the expoetal fuctos eq 9 are very effectve calculatos of atoms wth three electros 3 ad other four-body Coulomb systems. 4 Clearly, they are capable of descrbg both the short-rage cusp behavor of the wave fucto as defed by the Kato codtos 5 ad the log-rage behavor. However, ether the Hylleraas fuctos or the expoetal fuctos have bee appled to study atomc systems wth more tha three electros. Ths lmtato s due to the lac of algorthms for accurate ad effcet calculato of Hamltoa matrx elemets wth those fuctos for systems wth more tha three electros. However, such algorthms exst for ECGs, whch wll be dscussed later ths revew. Explctly correlated fuctos have also bee employed molecular BO calculatos. The most thoroughly vestgated system has bee the H molecule, whose frst study dates bac to James ad Cooldge. 6 Some of the most mpressve calculatos performed for ths system have bee those of Kołos ad Wolewcz. The wave fuctos ther approach were spred by the wor of James ad Cooldge ad were expaded terms of the followg fuctos (deoted by G below) expressed terms of ellptc coordates of the two electros deoted by the labels ad : 7 0 Λ Λ G (, ) = ( x + y) g (, ) ± ( x + y ) g (, ) v r s r g (, ) = exp( αξ αξ ) ρ ξ η ξ vrsr,,,, s η {exp( βη + βη ) + ( ) s s+ s +Λ+ p (30) exp( βη β η )} (3) The ± eq 3 refers to sglet ad trplet states, respectvely, Λ s the agular mometum projecto quatum umber, ad p = 0, for g ad u symmetres, respectvely. ξ j =(r ja + r jb )/R ad η j =(r ja r jb )/R are the ellptc, ad x j ad y j the Cartesa coordates for the two electros wth the z axs cocdg wth the teruclear axs ( a ad b deote ucle). ρ =r /R, r s the terelectroc dstace, ad c, α, α, β, ad β are varatoal parameters. Thus the bass set s defed by the set of expoets v, r, s, r, ad s. The calculatos reported 995 by Wolewcz 0 wth the bass fuctos eq 3 stll rema some of the most accurate ever performed wth the BO approach volvg the calculato of the potetal eergy surface frst ad the calculatg the rovbratoal eergy levels by solvg the Schro dger equato for the uclear moto. The drawbac of the bass fuctos eq 3 s that they caot be exteded to study molecules wth more tha two ucle. Eve a exteso of the approach to datomcs wth more tha two electros has ot bee accomplshed. Thus, at preset, the oly bass set of explctly correlated fuctos that ca be exteded to systems wth more tha two electros ad/or more tha two ucle are Gaussas. We wll descrbe the varous types of molecular Gaussa fuctos used the BO ad o- BO, atomc ad molecular calculatos secto Competto betwee ECG ad JC Fuctos Very recetly, Pachuc 80 employed aother type of explctly correlated fuctos, the James Cooldge (JC) fuctos, 6 BO varatoal calculatos of the hydroge molecule. I those calculatos he obtaed a lower eergy at the equlbrum 44 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

10 Chemcal s teruclear dstace tha the prevous best result by Cece ad Szalewcz obtaed wth 4800 ECGs. 75 The advatage of usg the JC bass set H calculatos les the fact that ths set comprses may fewer olear parameters that eed to be optmzed tha the ECG bass set. However, the JC bass s aga restrcted to two-electro, two-ucleus molecules ad caot be exteded to larger systems. A comparso of the varatoally lowest eerges ever obtaed calculatos for the hydroge molecule at the equlbrum dstace s show Table. Amog those eerges, there s the result of Kołos ad Wolewcz (KW) obtaed wth geeralzed JC fuctos defed eq 3, whch clude more olear parameters tha the orgal JC fuctos to better descrbe the log-rage asymptotc behavor of the wave fucto. There s also the result obtaed by Naatsuj et al. wth the teratve-complemet-teracto (ICI) method whch essetally geerated a set of bass fuctos (expoets tmes prefactors) depedg o ellptc coordates. Ths result was the best eergy value at the tme t was publshed. I Table we also show the H BO eerges of Cece ad Szalewcz 75 calculated at R H H =.4 bohr wth dfferet umbers of ECGs. The results are ot qute comparable wth the other lterature values because most of those values have bee calculated at R H H =.40 bohr. Based o the ECG bass set Cece ad Szalewcz obtaed for H at R H H =.4 bohr, they geerated bass sets for several other teruclear dstaces usg a procedure that automatcally shfts the Gaussa ceters to adjust them for the chagg teruclear dstace. Ths allowed them to calculate a H PEC wthout reoptmzato of the Gaussas at each PEC pot, whch would be very tmecosumg. Oly the lear expaso coeffcets the wave fucto were reoptmzed by solvg the secular equato problem. They showed that the shftg procedure allows matag the accuracy of the whole PEC at a almost costat level. Ths demostrates that the computatoal tme for ECG calculatos ca be sgfcatly reduced f a effectve approach for guessg ew Gaussas to be added to the bass set ad for optmzg them s developed. Aother example where a effectve approach of ths d was mpelmeted s the ECG calculatos of Cece et al. 4 cocerg the molecular hydroge dmer, (H ). The approach volved cotractg two large H ECG bass sets to form a bass set for the dmer wth tes of thousads of ECGs. The olear parameters ths bass set were ot optmzed, but stead some addtoal ECGs were cluded to better accout for the teracto betwee the hydroge molecules. The olear parameters of these added ECGs were varatoally optmzed the calculatos. We should stress that, eve though the varatoal calculatos volvg all-electro ECGs scale as the factoral of the umber of the electros, they ca be exteded to larger umbers of electros tha two, as the algorthms for calculatg the Hamltoa ad overlap matrx elemets are geeral. Ths s ot the case for the JC fuctos, where the tegrals eed to be rederved (they have ot bee yet) to calculate molecular systems wth more tha two electros. 3. EXPLICITLY CORRELATED GAUSSIAN BASIS SETS As metoed, the ma advatage of usg ECGs atomc ad molecular calculatos s due to the smplcty evaluatg the overlap ad Hamltoa matrx elemets ad easy geeralzato to atoms ad molecules wth a arbtrary umber of electros. Whle ECGs have bee show to form a complete bass set, 5 7 they have mproper short-dstace behavor (uable to satsfy Kato cusp codtos 5 ) ad toofast decayg log-rage behavor. Eve though these defceces ca be effectvely remeded by usg loger expasos, they may cause a sgfcat crease the amout of computatoal effort eeded for a well-coverged calculato. Certa ssues may also arse the calculatos of relatvstc correctos ad other propertes, where proper short-rage behavor s mportat. 3.. Bass Sets for Atomc Calculatos wth Ifte ad Fte Nuclear Mass We wll frst dscuss the bass sets we have used atomc ad molecular calculatos performed wthout assumg the BO approxmato. For atoms, the o-bo calculatos are more ofte called fte-uclear-mass (FNM) calculatos. Eve though such atomc calculatos are oly margally more dffcult (the Hamltoa cludes the mass polarzato term, whch s abset whe the uclear mass s set to zero) ad essetally equally as tme-cosumg as the fte-uclearmass (INM) calculatos, most very accurate atomc calculatos publshed the lterature have bee performed wth the INM approach. However, there have bee also atomc calculatos whch employed the FNM approach. 5 7,8 3 Ths method ot oly allows for drectly calculatg eerges of groud ad exted states of dfferet sotopes, thus eablg determato of sotopc eergy shfts, but t also, by settg the uclear mass to fty, allows geerato of INM results. Whe dfferet sotopes of a partcular elemet are calculated, oe may cosder reoptmzg the olear parameters of the Gaussas the bass set for each of them. However, as the umercal expermets show, the chage the mass of the ucleus ca be effectvely accouted for by readjustg the lear coeffcets of the bass fuctos,.e., by recomputg the Hamltoa matrx ad solvg the egevalue problem wth the same bass set. Sce the chage of the wave fucto remas small whe the mass of the ucleus s vared (true as log as the uclear mass stays much larger tha the mass of electros), such a smplfcato has vrtually o effect o the accuracy of the calculatos. I the calculatos of atoms wth oly s electros the r j depedecy the Gaussa fuctos ca be lmted to the expoetal factor. For a -electro system these fuctos have the followg form: ϕ ( r, r,..., r ) = exp[ r ( A I3 ) r] (3) where r s a 3-compoet vector formed by stacg r, r,..., r o top of each other (r s the dstace betwee electro ad the ucleus), A s a symmetrc matrx, I 3 s a 3 3 detty matrx, s the Kroecer product symbol, ad the prme dcates vector (matrx) traspose. I some of the expressos ths revew we wll use a shorter otato for the Kroecer product of a matrx ad I 3 : A A I 3. I geeral, A does ot have to be a symmetrc matrx. However, oe ca always rearrage ts elemets such a way that t becomes symmetrc wthout chagg the quadratc form r A r. Sce dealg wth symmetrc matrces has certa practcal advatages further cosderatos, we wll always assume the symmetry of A. As the bass fuctos used descrbg boud states must be square tegrable, some restrctos must be mposed o the elemets of A matrces. Each A matrx must be postve 45 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

11 Chemcal s defte. Rather tha eforcg the postve defteess of A, whch usually leads to cumbersome costrats, we use the followg Cholesy factored form of A : A = L L, where L s a lower tragular matrx. Wth ths represetato, A s automatcally postve defte for ay values of L ragg from to. Thus, the varatoal eergy mmzato wth respect to the L parameters ca be carred out wthout ay restrctos. It should be oted that the L L represetato of A matrx does ot lmt the flexblty of bass fuctos, because ay symmetrc postve defte matrx ca be represeted a Cholesy factored form. I order to mprove the qualty of the atomc s-type ECGs (eq 3), partcularly terms of provdg a better descrpto of the short- or log-rage behavor, oe ca clude those fuctos preexpoetal factors smlar to those preset the Hylleraas fuctos defed eq 8: j ϕ ( r, r,..., r ) = ( r ) exp[ r ( A I) r] j 3 > j (33) From a practcal pot of vew, t s easer to use eve powers of the terelectro dstaces such factors ( j eve), or just ther squares ( j = ), because the evaluato of the Hamltoa tegrals s the more straghtforward. By cludg the r j prefactors Gaussas, 3 oe obtas the followg expoetally ad preexpoetally, explctly correlated Gaussa fuctos: ϕ ( r, r,..., r ) = ( r ) exp[ r ( A I) r] > j j 3 (34) Eve though, prcple, all terelectro dstaces should be cluded >j r j, a smpler approach wth oly a lmted umber of those dstaces ca also be cosdered. I such a approach the Gaussa bass set would comprse the followg subsets of fuctos: 3 j {{exp[ r ( A I) r]}, { r exp[ r ( A I) r]}, { rj rl exp[ r ( A I3 ) r]},...} (35) I partcular, oe ca cosder a bass set that oly cludes the two frst subsets: {{exp[ r ( A I3) r]}, { rj exp( r ( A I3) r)}} (36) Such a bass set was recetly tested, 3 ad t was show that placg r j factors frot of the Gaussa expoets leads to a otceable mprovemet of the eergy covergece. There s aother ssue that arses atomc calculatos, partcularly those cocerg excted states. It s related to descrbg radal odes ad agular odes the wave fuctos. I dealg wth the radal odes, for example, the calculatos of S excted states of the beryllum atom, the radal flexblty of the Gaussas s a ey factor. For lower lyg states, the stadard correlated Gaussas, eq 3, provde a suffcetly flexble bass set to descrbe the few radal odes. However, for hgher lyg states wth more radal odes the stadard Gaussas may eed to be modfed to facltate more radal flexblty. Ths ca be accomplshed usg the followg complex Gaussas: 8,33,34 ϕ = exp[ rcr ] exp[ r (( A + B ) I) r] 3 (37) 3 where A ad B are symmetrc matrces that represet the real ad magary parts of C, respectvely. Cosderg groud states of atoms wth more tha four electros (for example, the boro atom) or some excted states of eve smaller atomc systems requres that agular factors are placed frot of the expoets of the Gaussa bass fuctos (more detals o the rotatoal symmetry of the bass fuctos wll be gve secto 4.3). For the states correspodg to a domat cofgurato wth just oe p electro, the followg form of Gaussas ca be used: ϕ = z exp[ r ( A I) r] m 3 (38) Here m s a teger that depeds o ad may tae values from to. It s coveet to represet fuctos 38 as ϕ = ( v ) rexp[ r A r] (39) where v s a vector whose compoets are all 0, except the 3m th compoet, whch s set to. For the case of states wth two p electros (for example, the groud s s p state of the carbo atom) or oe d electro, oe ca use Gaussas wth two electro coordates placed frot of the expoet: ϕ = ξ ξ exp[ r ( A I) r] (40) j 3 where ξ ad ξ j ca ether be x, y, orz coordates of electro ad j, respectvely, wth ad j ether equal or ot equal to each other. I partcular, descrbg states wth two p electros (for example, the calculatos of the 3 P o groud ad frst excted states of the carbo atom) the followg Gaussa bass fuctos 35 ϕ = ( xy xy)exp[ r ( A I) r] j j 3 (4) were used. I calculatg D states volvg oe d electro, the Gaussas had the form 38,39 ϕ = ( xx + yy zz)exp[ r ( A I) r] j j j 3 (4) where electro dces ad j are ether equal or ot equal to each other. It s coveet to use a alteratve form of the agular preexpoetal multpler. It volves the geeral quadratc form, r W r, that represets the preexpoetal factor. Ths form allows for a more geeralzed approach dervg the matrx elemets. For example, the Gaussas wth the factor x x j + y y j z z j ) ca be wrtte as ϕ = ( rwr )exp[ r ( A I) r] 3 (43) where W s a sparse 3 3 symmetrc matrx that for = j comprses oly three ozero elemets: W 3( )+,3( )+ =, W 3( )+,3( )+ =, ad W 3( )+3,3( )+3 =, ad for j t comprses sx ozero elemets: W 3( )+,3(j )+ = W 3(j )+,3( )+ = /, W 3( )+,3(j )+ = W 3(j )+,3( )+ = /, ad W 3( )+3,3(j )+3 = W 3(j )+3,3( )+3 =. It should be oted that, geeral, we could have used a osymmetrc matrx W (for j ) wth oly three ozero elemets (yeldg the same quadratc form) sce there are oly three terms eq 4. However, as already metoed, t s much more coveet to deal wth symmetrc matrces practce. The ma reaso for ths s that the dervato of matrx elemets 46 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

12 Chemcal s becomes cosderably smpler. I a smlar maer, oe ca devse forms of Gaussas eve for states wth hgher total orbtal agular mometa (or hgher tha umber of o-s electros). These forms, however, become progressvely more complex. 3.. Bass Sets for No-BO Calculatos o Datomc Molecules After separatg the ceter-of-mass moto from the laboratoryframe Hamltoa of a molecule, the Hamltoa that descrbes the trsc moto of the system, the teral Hamltoa, s sotropc (.e., sphercally symmetrc). Egefuctos of such a Hamltoa form a rreducble represetato of the fully symmetrc group of rotatos. Thus, those fuctos are atom-le fuctos, whch, besdes beg egefuctos of the Hamltoa, are also egefuctos of the square of the total orbtal agular mometum operator ad the operator represetg ts projecto o a selected axs. As such, the Hamltoa matrx calculated wth egefuctos of the square of the total orbtal agular mometum s a bloc-dagoal matrx. Ths allows for separatg the calculatos of states correspodg to dfferet total orbtal agular mometum quatum umbers. I partcular, whe sphercally symmetrc bass fuctos are used the calculatos, the so-called rotatoless states are obtaed. Those states correspod to the groud ad excted vbratoal states of the system. We put vbratoal quotato mars because, f the BO approxmato s ot assumed the calculato, the electroc ad vbratoal degrees of freedom mx ad the wave fucto for a partcular vbratoal state may cota cotrbutos from products of dfferet electroc wave fuctos ad dfferet vbratoal wave fuctos. Due to ths couplg of the vbratoal ad electroc motos the vbratoal quatum umber s o loger a good quatum umber. It should be, perhaps more correctly, regarded as a umber whch umbers cosecutve states the mafold correspodg to the partcular total orbtal agular mometum quatum umber. I the fully o-bo ECG calculatos for datomc molecules performed so far oly rotatoless states,.e., states represeted by sphercally symmetrc wave fuctos, have bee cosdered. 7,40,4 I the calculatos of those states the followg explctly correlated Gaussas multpled by eve powers of the teruclear dstace, r (the powers usually rage from 0 to 50), have bee used: ϕ = r exp[ rar ] (44) p where t s assumed that the ceter of the teral coordate system s placed o the frst ucleus (usually the heavest oe), the frst pseudopartcle represets the secod ucleus, ad the remag pseudopartcles represet the electros. As oe otces, fucto 44 s sphercally symmetrc wth respect to the ceter of the coordate system. All partcles are explctly correlated the Gaussa expoet (.e., the expoet explctly depeds o all terpartcle dstaces). There s also a addtoal correlato factor for the two ucle that depeds o powers of the teruclear dstace, r p. The purpose of ths factor s to effectvely separate the ucle from each other ad to descrbe the oscllatos of the wave fucto resultg from vbratoal exctato. I the vbratoal groud state the wave fucto should have a maxmum for r aroud the vbratoally averaged groud state teruclear dstace. The combato of the r p factors wth dfferet p powers ad the Gaussa expoet ca very effectvely geerate such a maxmum. I the frst vbratoal excted state the wave fucto has a sgle ode ad t ca also be very well descrbed by Gaussas (44). Moreover, as the calculatos of the complete vbratoal spectra for such systems as H + ad ts sotopologues, 4 44 H ad ts sotopologues, HeH +, 50,5 ad larger systems,5,53 have demostrated, the Gaussas ca very effectvely descrbe eve the hghest vbratoal exctatos wth multple odes. The Hamltoa tegrals volvg fucto 44 are more complcated tha tegrals wth fuctos wthout premultplers. Wth that the tegrals tae cosderably loger to compute. Also, we should ote that the powers of the teruclear dstace the bass fuctos used to calculate the groud state are smaller tha those eeded to calculate excted states. I order to descrbe the vbratoal states correspodg to the frst rotatoal excted states, oe eeds to use the followg Gaussas expadg the wave fuctos of those states: ϕ = zr exp[ rar ] (45) p Here we assume that the cotrbuto of bass fuctos where z s replaced by x, where, s umportat. Ths assumpto may be ot be strctly correct for hgher states ear the dssocato threshold, for whch z eq 45 should be replaced wth z m, where m rages from to Bass Sets for No-BO Calculatos o Systems wth More Tha Two Nucle Let us frst cosder a molecule wth three ucle (the smplest such molecule s the H + 3 o). Applyg the same argumets as used to justfy the use of Gaussas eq 44 o-bo datomc calculatos, the approprate bass set of correlated sphercally symmetrc (oe-ceter) Gaussas to descrbe rotatoless states of a tratomc molecule, such as H + 3, should cosst of the followg fuctos: ϕ = r r r exp[ rar ] p q t (46) As oe otces, the preexpoetal multper of the fuctos eq 46 cludes ot oe, as eq 44, but three factors. The factors are powers of all three teruclear dstaces the molecule. The presece of the powers allows for effectvely separatg the ucle ad placg them at relatve dstaces whch are o average equal to the equlbrum dstaces for the state whch s beg calculated. The powers are also mportat geeratg radal odes the wave fucto excted vbratoal states. Ufortuately, the Hamltoa ad overlap matrx elemets wth the bass fuctos gve eq 46 become more complcated. Whle the expressos for them were derved a closed algebrac form, 54 a effcet umercal mplemetato s problematc due to a large umber of summato loops the tegral formulas whch results from the powers of the teruclear dstace the preexpoetal factor. A Gaussa bass set whch, prcple, ca be used for rotatoless states of tratomc molecules ad eve of molecules wth more tha three ucle s the bass of complex Gaussas eq 37. A approprate lear combatos of these Gaussas should geerate se/cose type oscllatos the wave fucto whch are eeded descrbg vbratoal excted states. Future tests wll show how effectve Gaussas (37) are descrbg those oscllatos. 47 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

13 Chemcal s 3.4. Bass Sets for No-BO Molecular Calculatos the Presece of Exteral Electrc Feld Let us cosder a solated datomc molecule wthout the BO approxmato. Such a cosderato has may fudametal aspects ad also provdes a procedure for descrbg asymmetry the charge dstrbuto molecular systems due to sotopc substtuto (such a effect taes place, for example, the HD molecule). As metoed, the groud state (L = 0) wave fucto of the molecule s a sphercally symmetrc fucto. Let us ow expose the molecule to a statc electrc feld. If a calculato performed wth the BO approxmato shows that the molecule has a ozero dpole momet or a asotropy of the polarzablty, the teracto wth the feld wll result the molecule oretg tself space to mmze the teracto eergy. Whe the feld s small ths oretato s oly partal, but as the feld creases the dpole momet axs of the molecule or the axs of ts hghest polarzablty essetally becomes fully alged wth the drecto of the feld. It usually taes a very small feld to acheve ths full algmet. Oe ca call ths effect the oretatoal polarzablty. Whe ths happes, the o-bo wave fucto of the molecule loses ts sphercal symmetry ad acqures a axal symmetry. I addto to affectg the rotatoal state of the molecule ( a way oe ca say that the feld exctes the molecule to a hgh rotatoal state whe the molecular dpole algs wth the drecto of the feld), the feld also affects the vbratoal ad electroc motos. The feld essetally polarzes the molecule vbratoally ad electrocally. To descrbe the feld-duced deformato of a molecule ts groud state, the followg correlated Gaussas wth shfted ceters were used: 55,56 ϕ = exp[ ( r s ) A ( r s )] (47) where s are the shfts of the Gaussa ceters. Fuctos eq 47 are also called explctly correlated shfted Gaussas or floatg ECGs. The shfts allow for deformg the molecular wave fucto to a cyldrcal (oval) shape the presece of the feld. ECG bass eq 47 s, prcple, also capable of descrbg the groud state of the molecule the absece of the feld. It s ot the most optmal bass for such a case, but wth all the Gaussa ceters beg located at the org of the coordate system, t has the rght symmetry of the system wthout the feld. I refs 55 ad 56 the o-bo dpole momets of sotopologues of H ad LH were also evaluated usg a approxmate procedure employg the fte-feld approach. Good agreemet wth the expermetal values was acheved. The reader may also revew the recet paper by Feradez, 57 where the procedure used refs 55 ad 56 ad the results obtaed there were put to questo Bass Sets for Molecular BO Calculatos I molecular BO calculatos, there s o eed to trasform the coordate system to a teral coordate system represetato. That s because the ucle are fxed space ad therefore the ceter of mass does ot move. A cosequece s that the coordates ca be represeted the laboratory frame wthout ay effect o the outcome of the calculato. I BO calculatos ECGs become fuctos of the electroc coordates oly, expressed the laboratory frame. The smplest Gaussas used the BO calculatos are those gve eq 47, wth L, I 3, ad s havg the same meag as eq 47, but wth r cotag oly the electroc coordates the laboratory frame staced smlarly to the defto eq 7. Gaussas 47 ca oly be used to descrbe states whose wave fuctos do ot have odes o the ucle. Equatos sectos. ad.3 show how BO calculatos the uclear postos are formally preset the wave fuctos as parameters. I most of the avalable quatum chemstry software pacages ths parametrcal depedece s explctly accouted for by ceterg the atomc orbtals to the postos of the ucle a molecule. Deformatos of the wave fucto eeded to descrbe the drectoalty of chemcal bods are the obtaed by employg Gaussa orbtals cotag premultplers (usually powers of Cartesa coordates). Agular mometum fuctos wth up to l = 5 are typcally used. The ECG-type fuctos defed eq 47 do ot clude ay agular mometum factors. Employg such agular mometum factors would be mpractcal as the formulas for calculatg matrx elemets of the clamped-ucle Hamltoa sadwched by fuctos of the type eq 47 qucly become complcated whe L. The problem s readly crcumveted by avodg the use of agular mometum fuctos ad mag up for that by allowg the ceters of the ECGs to float. Techcally, that s acheved by ot equatg the s vectors wth the uclear postos. Usg floatg ECGs meas that the bass fuctos do ot dsplay ay explct parametrcal depedece o the uclear postos. Istead ther Gaussa ceters, s, are cosdered to be adjustable parameters the varatoal optmzato. It s mportat to ote that employg a floatg ceters bass set matas a mplct parametrcal depedece wth respect to the uclear postos. Ths ca be uderstood f oe realzes that the varatoally optmzed wave fucto s foud usg a Hamltoa that s explctly depedet o the uclear postos, such as the CN Hamltoa () used the BO calculatos. To summarze, the use of floatg ECGs allows the descrpto of chemcal bods through the deformatos of the electro desty that are descrbed by bass fuctos that float away from a ucleus. Ths choce of bass fuctos eables employg umercally soud overlap ad Hamltoa matrx elemets at the expese of havg to optmze 3N e K olear parameters, addto to the Gaussa expoets (L ) ad the lear expaso coeffcets (c ) eq 6, where N e ad K are the umber of electros ad the umber of bass fuctos employed Use of Premultplers ECG Bass Sets for Molecular BO Calculatos. ECGs wth floatg ceters defed eq 47 perform best whe used the calculatos of states wth odeless wave fuctos at the ucle. A ode the wave fucto ca oly be descrbed wth floatg ECGs by a superposto of Gaussas that have opposte sgs. Depedg o the ature of the ode, such a arthmetcal geerato of the ode ca lead to umercal stabltes. Ths does ot happe whe the ode les, for example, o the magary le tercoectg two ucle, whch s the case of Σ + u excted states He 9,58 ad H. 59 I ths stuato the ode occurs a rego away from where the electro desty peas (at the ucle); therefore, the arthmetc ode geerato s less lely to cur to a umercal stablty. Dfferet s the case of odes occurrg at the posto of a ucleus. Such cases tae place may excted states of small molecules ad atoms, ad some cases also the groud state, as the CH + molecular o. To effectvely descrbe a ode at a ucleus, t s suffcet to apped a proper agular fucto as a premultpler to the ECG bass fuctos, amely 48 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

14 Chemcal s ucl ϕ = Y( r r )exp[ ( r s ) A ( r s )] (48) where Y(r r ucl ) s some agular mometum fucto such as a sphercal harmoc multpled by the (r r ucl ) l factor Ioc ad Covalet Bass Fuctos. Whe atomcetered atomc orbtals (AOs) are used molecular BO calculatos, the wave fucto s expressed as a atsymmetrzed product of molecular orbtals (MOs) ad sp fuctos. The MOs are lear combato of AOs havg proper spatal symmetry. The wave fucto obtaed wth the outled procedure ca the be decomposed to several atsymmetrzed products of atomc orbtals ad sp fuctos. As a example, cosder the groud state of the hydroge molecule the mmal AO bass costtuted by two s orbtals cetered o ether hydroge atom, s A or s B, respectvely. The spatal electroc wave fucto becomes a product of doubly occuped MOs deoted as σ g : Ψ H = σ g (49) The above product expaded terms of the (uormalzed) AOs (σ g =s A +s B, where A ad B refer to the atoms), after some rearragemet, taes the form Ψ = s ( r) s ( r ) oc H A A + s A( r) s B( r) covalet + s B( r) s A( r) covalet + s B( r) s B( r) oc (50) where the electroc coordates the laboratory frame of the two electros are cluded explctly. I eq 50, labelg of oc ad covalet products has bee assged to those products of AOs that volve AOs cetered o the same atom or o dfferet atoms, respectvely. Smlarly to the above case, the applcato of ECGs to the calculato of the electroc wave fucto of the hydroge molecule volves bass fuctos of the type A A r s exp ( r s, r s) A A r s (5) whch also ca be labeled as oc ad covalet depedg o whether the ceters s ad s are the eghborhood of the same or dfferet atoms. I the smplfed case of eq 50 the mmal AO bass set for the H molecule geerated a wave fucto havg 50% oc ad 50% covalet products. Ths hts that whe usg ECGs the rato of oc/covalet fuctos must be optmzed to acheve a better eergy covergece wth respect to the umber of bass fuctos employed. The smple example of eq 50 also shows how the rato of oc/covalet bass fuctos must chage as the teruclear dstaces of a molecule are vared. Cosder the spatal wave fucto of the H molecule ts mmal AO bass whe the molecule s completely dssocated. Because the dssocato lmt cossts of two oteractg hydroge atoms, the spatal wave fucto becomes Ψ H = ( s ( r) s ( r ) + s ( r ) s ( r)) A B A B (5) whch cossts of a 00% covalet AO product ad has o oc compoets. Therefore, whe the H molecule s calculated wth ECGs, as the teruclear dstace stretches, some bass fuctos must covert from oc to covalet. Such a coverso may requre mgrato of some Gaussa ceters by several atomc uts ad usually overcomg a eergy barrer, whch s ulely to occur durg the varatoal eergy mmzato. Ths problem wll be tacled later ths revew whe the detals of the optmzato of the olear parameters BO calculatos are dscussed sectos 7.. ad SYMMETRY OF THE WAVE FUNCTION The Hamltoa of a few-partcle Coulombc system always possesses certa symmetres, or other words, t commutes wth the wave fucto trasformatos that belog to certa groups. Amog these groups there may be cotuous groups such as the group of 3D rotatos, or pot groups, such as the symmetrc group S p. I ths secto we brefly cosder what the possble mplcatos of these symmetres are ad how to perform calculatos that tae them to accout. 4.. Permutatoal Symmetry Practcally ay atomc or molecular system cosstg of more tha two partcles cotas some subsets of detcal partcles t. Most commoly, we th of electros ths regard. However, the detcal partcles could also be ucle ad eve such exotc partcles as postros ad muos. Accordg to the Paul prcple, the total wave fucto (cludg the sp degrees of freedom) of such quatum systems must ether be symmetrc or atsymmetrc wth respect to permutatos of detcal partcles. Ay approxmato to the exact wave fucto, whch aspres to be accurate, should tae ths requremet to accout. I varatoal calculatos ths puts a costrat o the symmetry of the bass fuctos that ca be used. It should be sad that some cases, such as whe we are terested the groud state of a bosoc system, we mght prcple use a bass that does ot possess ay symmetry. If the state of terest s the lowest (or oe of the very lowest) eergy amog the states of ay symmetry, the total tral wave fucto wll evetually coverge to the form correspodg to the proper permutatoal symmetry as the bass sze goes to fty. For ay fte bass set, however, there s a bg lelhood of the presece of some symmetry cotamato. More mportatly, such calculatos are usually far less effcet terms of CPU tme ad memory requremets as the umber of bass fuctos ecessary to acheve the same accuracy as the case of properly symmetrzed bass s sgfcatly larger. Thus, eve those few specal cases t s a good dea to use bass fuctos of the proper symmetry. Whe we deal wth systems cosstg of fermos (.e., electros), eforcg the proper permutatoal symmetry o the bass fuctos s ot oly a matter of the computatoal effcecy, t s essetally a strct requremet. Eve the groud state varatoal calculatos of fermoc systems are ot possble wthout a properly atsymmetrzed bass. I geeral, order to buld properly (at)symmetrc wave fuctos, oe has to deal wth both the spatal ad sp coordates. Due to the fact that the Hamltoa of a orelatvstc Coulomb system does ot deped o the sp of partcles, t s possble to completely elmate the sp varables from cosderato. The correspodg mathematcal formalsm has bee well developed (see, for example, moographs of Hamermesh 60 ad Paucz 6 ). I ths formalsm projecto operators for rreducble represetatos of the symmetrc 49 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

15 Chemcal s group (called Youg operators) are obtaed a straghtforward maer from ther correspodg Youg tableaux. A Youg tableau s created from a Youg dagram (sometmes called Youg frame), whch, for a system of p partcles s a seres of p coected boxes, such as The shape of the Youg dagram correspodg to the desred rreducble represetato of the symmetrc group s determed by the ature of the partcles the system (bosos or fermos, ad ther sp). For a set of fermos wth sp / (.e., electros) the Youg dagram for the spatal wave fucto must cota o more tha two colums. For fermos wth sp 3/ the maxmum umber of colums would be four. The umber of colums for bosos s ot lmted. A Youg tableau s created by fllg a Youg dagram wth umbers from to p so that they crease whe gog from left to rght ad from top to bottom, such as Geerally, there s more tha oe way to wrte a Youg tableau. The umber of ways determes the dmeso of the represetato. I the actual calculatos we may restrct ourselves wth the use of bass fuctos correspodg to ay of the equvalet dagrams (the equvalet oes are those that have the same shape). The shape of the dagram also determes the multplcty of the state. The tableaux eq 54 would correspod to a doublet (a sgle upared electro), whle the followg oes correspod to a sglet ad trplet, respectvely. Oce we have a approprate Youg tableaux, the Youg operator ca be wrtte as Y = SÂ, where A = A, S = S c c r r (56) are the product of atsymmetrzers over each colum ad the product of symmetrzers over each row, respectvely. The symmetrzers ad atsymmetrzers ca be coveetly represeted by compact expressos that volve traspostos (par permutatos of partcles), whch we wll deote P l. For example, f we have two partcles whose umbers are ad, the the symmetrzer s S = + P (57) The atsymmetrzer over partcles ad also has a smple form: A = P (58) For the sae of smplcty we dropped the ormalzato factor both expressos 57 ad 58 as t ot essetal here. I the case of larger tha umber of partcles over whch the symmetrzato or atsymmetrzato eeds to be doe, the correspodg expressos ca be gve a factorzed form. Let us assume that the umber of partcles a subset s ad ther umbers rage from to. The the symmetrzer ad atsymmetrzer would loo as follows: S = + P + P + P + P + + P,..., ( )( 3 3) (..., ) (59) A = P P P P P,..., ( )( 3 3) (..., ) (60) Aga, the ormalzato factors (/!) were dropped for coveece. As a llustrato, let us wrte the Youg operator for a doublet state of fve detcal partcles wth sp / (correspods to the frst Youg tableau eq 54). We symmetrze over rows ad by applyg + P ad + P 34. The we atsymmetrze over colums ad by meas of operators ( P 3 )( P 5 P 35 ) ad P 4. The fal expresso for the Youg operator s the Y = ( P )( P )( P P )( + P )( + P ) (6) The operators whose matrx elemets are eeded the varatoal calculato (such as the Hamltoa, etc.) usually commute wth all the permutato operators volved the projector Y, ad thus, they commute wth Y tself. Moreover, we ca restrct ourselves to the mplemetato of oly those cases where the permutatoal operators are appled to the et, because l l (6) Y ϕ OY ϕ = ϕ OY Y ϕ Here O deotes a operator represetg the quatty of terest. Operator Y Y ca be smplfed so that t cotas oly p! elemetal terms (permutatos), where p s the umber of detcal partcles. Now let us cosder how each of those permutatos of partcles act o prmtve ECG bass fuctos. A permutato of the real partcles (.e., ot the pseudopartcles) volved a permutatoal operator, P, ca be represeted as a lear trasformato of the laboratory-frame coordates, R, of the partcles. Sce the relato betwee the laboratory coordates, R, ad the teral coordates, r, s lear, the trasformato of the teral coordates uder the permutato of the partcles s also lear. Therefore, t ca be descrbed by a permutato matrx, P = P I 3. The applcato of P to the smplest bass fuctos eq 3 gves Pϕ = P exp[ rar] l = exp[ ( Pr) Al( Pr)] = exp[ r ( P AP) r] l l As ca be see from the above expresso, the symmetry trasformato of ϕ l s equvalet to a smlarty trasformato of the matrx of the olear parameters for that bass fucto (A l P A l P). Based o ths, a procedure that mplemets the permutatoal symmetry calculatg matrx elemets wth symmetry-projected bass fuctos ca be developed. Evaluatg these matrx elemets volves a summato of tegrals, whose actual umercal values deped o trasformed matrces A ad A l. 50 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

16 Chemcal s 4.. Permutatoal ad Spatal Symmetry BO Calculatos Whe employg all-electro bass fuctos, spatal symmetry ad permutatoal symmetry BO wave fuctos are treated o a smlar footg. Geerally, the bass fuctos eq 47 do ot possess the proper permutatoal or spatal symmetry. As metoed above, the geeral case, the approach usually chose s to project away the part of the bass fuctos that does ot have the proper symmetry. Ths s acheved by usg tesor products of Wger-type projectors. The total symmetry operator, P, acts o a ECG fucto as follows: Pϕ = P exp[ ( r s) A( r s)] l l l l = exp[ ( Pr PP sl) Al( P( r PP sl))] = exp[ ( r P sl) ( P AP l )( r P sl )] (63) The symmetry operator cotas a product of operatos belogg to the group of permutatos of partcles, f are the electros the studed molecule, ad elemets of the pot symmetry group the molecule belogs to. The geeral form of the P operator s! Γ = S G P P P = D D S α OO G g! = α G α (64) where stads for tesor product operato, G s the label of a pot group, Γ a specfc rreducble represetato of that pot group, ad g s the umber of elemets the pot group G. The coeffcets D S ad D α G ca be the characters of a specfc rreducble represetato of ether the symmetrc group or the pot group, or a elemet of the matrces that costtute the rreducble represetato. Whe calculatg the molecular electroc groud state, the rreducble represetato to use s fully symmetrc wth every D α G =. Excted states, stead, may have egatve values of D α G, or values magtude dfferet from uty. As a example, cosder the H 3 + molecular o. The pot group symmetry for ths o ts groud state equlbrum geometry s D 3h. Ths pot group cotas sx elemets, ad for the A ad E states, the projecto operators oto the respectve pot group symmetry rreducble represetato ca be wrtte as A D3h 3 3 P = + C + C + σ + σ + σ E 3 (65) P = D C3 C3 + σ σ σ3 3 h (66) respectvely. The operator eq 66 was obtaed after otcg that C 3v s somorphc wth the permutato group of three partcles, S 3, ad by employg the Youg tableau. It s terestg to otce that at the groud state equlbrum geometry of H + 3 there s a cocal tersecto of two degeerate E states (the E represetato s two-dmesoal). For the secod of the two states the symmetry projector ca be geerated usg the Youg tableau Spatal Symmetry No-BO Calculatos I the case whe there are o clamped partcles the system ad o exteral felds are preset, the Hamltoa commutes wth the total orbtal agular mometum operator, L. Therefore, the exact solutos of the correspodg Schro dger equato must also be the egefuctos of L. For ths reaso, bass fuctos for a varatoal calculato of a gve system/state eed to possess certa rotatoal symmetry propertes. It should be sad that some cases the proper rotatoal symmetry of bass fuctos s ot strctly requred. For example, whe oe deals wth the groud state of the system ad does ot mpose ay rotatoal symmetry of the wave fucto the calculato, the total o-bo tral wave fucto should coverge to the groud state provded the bass fuctos have suffcet flexblty ad oe performs a thorough optmzato of the lear ad olear parameters volved the tral fucto. The tral wave fucto wll evetually approach the rght rotatoal symmetry of the groud state the lmt of a complete bass set. Noetheless, eve such a case t s a good dea to use bass fuctos of the correct symmetry as ths yelds a much faster covergece rate. For cosderato of excted states, the use of correct rotatoal symmetry s usually ot optoal. If the symmetry s ot mposed, the tral wave fucto wll smply coverge to a wrog state whe the lear coeffcets ad olear parameters are optmzed. To remedy ths, oe may clude some pealty terms the varatoal eergy fuctoal whch, eve wthout strctly mposg the rght rotatoal symmetry, would force the wave fucto to effectvely assume ths rght symmetry the process of the bass set optmzato. I addto to the Hamltoa commutg wth L, t also commutes wth the projecto of the total orbtal agular mometum operator o a selected axs, L ẑ (aga, assumg o clamped ucle are volved the system). Due to the degeeracy of the eergy levels correspodg to dfferet quatum umbers M (egevalues of L ẑ ), t s ot requred that bass fuctos must correspod to a partcular M value. I prcple, oe ca use ay lear combato of bass states wth dfferet M s (ad the same L ad other quatum umbers). However, for effcet umercal mplemetato, t s desrable that the bass fuctos be real. Ths s automatcally satsfed whe M =0. For states wth L = 0 (eve those that arse from the couplg of the ozero agular mometa of separate partcles) the wave fucto of the system s rotatoally varat. Thus, ay sphercally symmetrc Gaussa exp[ r (A I 3 )r] multpled by a arbtrary fucto of the absolute values of r, r j, or ther dot products, s a sutable bass fucto. The actual choce of the premultpler s dctated by the structural peculartes of the cosdered system ad ts state, so that the covergece of the varatoal expaso s suffcetly fast. For example, for the groud state of a atom wth s electros oly, t s usually suffcet to use the premultplers that are equal to uty. For excted Rydberg states of atoms the calculato may beeft from usg some of the bass fuctos factors of the form r or eve hgher eve powers of the electro ucleus dstaces to better descrbe the radal odes the wave fuctos of these states. For a datomc molecule, where partcles ad are ucle, premultplers eed to be troduced to descrbe the spatal separato of these partcles. As metoed before, these premultplers ca have the form of powers of the teruclear dstace, r m = R m, where m s a teger whch typcal o-bo calculatos may rage from 0 to 00. To obta a proper fuctoal form of the bass fuctos sutable for the calculatos of states wth a gve L ad correspodg to a gve couplg scheme (.e., a certa set of termedate total agular mometa) of the orbtal agular mometa of the costtuet partcles, oe ca use the wellow rules of the addto of the agular mometa. The case 5 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

17 Chemcal s of a L = state, whch all partcles but oe have zero agular mometum (.e., l =,l =0, ; here l j s the orbtal agular mometum quatum umber of partcle j), s trval ad leads to a prefactor the form r Y 0 (r ), where Y lm (r ) deotes sphercal harmocs. Sce Y 0 (r ) s proportoal to z / r, the followg bass fuctos are geerated: ϕ = z exp[ r ( A I) r] 3 (67) I a slghtly more sophstcated case of two partcles wth ozero agular mometa (l, l j ) the expresso for the premultpler s evaluated as the followg sum: l l j l l j mm, j m+ m= M j j r r LM = r r ( LMl m l m) l m l m j j j j j (68) where (L M l m l j m j ) are the Clebsch Gorda coeffcets, ad l,m are shorthad for sphercal harmocs Y l m (r ). The l mathematcal fuctos gve by eq 68 wthout the r l r j j factor are ofte called the bpolar harmocs. 6 Whe may or all partcles the system have ozero orbtal agular mometa, ther couplg to multpolar harmocs, whch may symbolcally be represeted as l l r r [[[ l, m l, m ] L, M l3, m3 ] L, M l, m ] 3 3 L, M (69) becomes progressvely more complcated as the umber of partcles creases. Noetheless, for ay relatvely small umber of partcles, the exact form of the proper Gaussa premultpler ca be easly determed wth the use of moder computer algebra pacages. For very complcated cases oe may also employ the approach proposed by Varga, Suzu, ad Usuura, 5,63 whch avods the couplg of orbtal agular mometa completely. Ths approach ca be partcularly useful calculatos of states wth very hgh L. The spatal bass fuctos ths approach have the followg form: p ϕ = exp[ rar ]( v ) Y ( v) (70) LM where v = u r = u r, p s a oegatve teger parameter, ad u are a set of addtoal olear parameters. Both p ad u are subject to optmzato. Oly the total agular mometum L appears eq 70, whle the couplg scheme of the dvdual agular mometa of the partcles s ot strctly defed. I geeral, t s a lear combato of dfferet couplg schemes correspodg to the same fal L value ad ths lear combato (or the weghts of the dfferet couplg schemes) may chage cotuously durg the eergy mmzato. The eergy mmzato performed wth respect to u amouts to fdg the most sutable agle or a lear combato of agles defed by the v /v ut vector, terms of whch the wave fucto s expaded. 5. EVALUATION OF MATRIX ELEMENTS The dervatos of the Hamltoa ad overlap matrx elemets wth ECGs ca be coveetly carred out wth the use of the formalsm of matrx dfferetal calculus (MDC). Whle ths formalsm s ofte employed the feld of ecoometrcs ad statstcs, t has ot bee well-ow chemstry ad physcs. It has prove to be a very sutable tool to wor wth all types of ECGs. A detaled troducto to the subject of MDC ca be foud ref 64. I ths secto we wll supply the reader wth most mportat deftos ad brefly descrbe the geeral procedures for evaluatg the Hamltoa ad eergy gradet matrx elemets. For a complete descrpto ad explaato of all techcal detals, we refer the reader to refs 7, 9, 8, 34 37, ad vech Operato I some stuatos, such as whe computg dervatves of matrx elemets, t s hady to mae use of the operator vech. It trasforms a matrx to a vector by stacg the colums of a matrx, oe udereath the other, but for each colum oly the elemets located o ad below the dagoal of the matrx are used the stacg. Hece, vech trasforms a matrx to a ( + )/-compoet vector. For example, f X s a 3 3 matrx wth elemets X j, the X X X3 vech X = X X3 X33 (7) The vech operator s partcularly useful the case of symmetrc matrces; the vech X cotas oly depedet elemets of X. 5.. Gaussa Itegral p Dmesos I the evaluato of the Hamltoa ad overlap matrx elemets the followg p-dmesoal Gaussa tegral s used most ofte: + p/ π exp[ xax+ yx ] dx = exp[ ya y] A / (7) where x s a p-compoet vector of varables, A s a symmetrc p p postve defte matrx, ad y s a p-compoet costat vector Evaluato of Itegrals Ivolvg Gaussas wth Agular Preexpoetal Factors Let us start wth the smplest -partcle ECG bass fucto that s used to costruct the bass for calculatg boud states of atoms wth oly s electros: ϕ = exp[ rar ] (73) By drectly applyg eq 7, we obta the expresso for the overlap tegral betwee two bass fuctos gve by eq 73: 3 / π ϕϕ l = A 3/ l (74) where A l = A + A l. I dervg tegrals over Gaussa bass fuctos represetg atomc states wth hgher agular mometa, the so-called geerator fuctos are used. Let us cosder agular Gaussas geerated by multplyg eq 73 by oe or two sgle partcle Cartesa coordates. For the former case the geerator Gaussa fucto s ϕ = exp[ rar + α ( v) r] (75) where α s a parameter ad v s a vector whose compoets are all 0, except the 3m compoet, whch s set to. For the latter case oe ca choose the geerator Gaussa to be ϕ = exp[ rar + α rwr ] (76) 5 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

18 Chemcal s where W (whch wll defe the preexpoetal factor x y j x j y or a smlar oe) s a sparse 3 3 symmetrc matrx comprsg oly four ozero elemets, two of whch have values of / ad the other two have values of /. The / elemets are placed the (3, 3j ) ad (3j, 3 ) postos, whle the / elemets are placed (3j, 3 ) ad (3, 3j ) postos. The geerator Gaussas are used to geerate Gaussa bass sets for expadg wave fuctos wth dfferet agular mometa. For example, the bass fucto z exp[ r A r]s geerated from eq 75 by dfferetatg wth respect to α ad settg α to 0. I ths case the elemets of the v vector are all 0 except for the 3th elemet, whch s set to. I the same maer, order to geerate the bass fucto (x y j x j y ) exp[ r A r], oe eeds to dfferetate the geerator eq 76 wth respect to α ad set α to 0. Ths approach, prcple, ca be exteded to geerate ay agular preexpoetal factor for a Gaussa bass fucto. As a example, let us cosder the overlap tegral betwee fuctos of the followg type: ϕ = ( xy xy)exp[ r Ar] j j It ca be obtaed as ϕϕ = l α = α α l α φ φ l + l α= αl= 0 exp[ r ( A + α W + αw) r] dr l l l α= αl= 0 (77) Ths approach ca be used to evaluate all other types of tegrals that appear the calculatos wth ECGs cotag Cartesa prefactors. I o-bo calculatos of datomc molecules a sutable bass set cossts of ECGs multpled by powers of the teruclear dstace. Eve a smpler case of atomc calculatos oe eeds to deal wth powers of the teruclear dstace whe evaluatg the matrx elemets of the potetal eergy. To obta these matrx elemets t s coveet to use a approach employg the Drac delta fuctos.the expresso for the matrx elemet of the Drac delta fucto, δ(r j ξ), where ξ s some three-dmesoal vector (parameter), allows oe to evaluate the matrx elemet of a arbtrary fucto f(r j ), whch depeds o a sgle pseudopartcle coordate or a lear combato of the coordates. For the case of a smple (o premultplers) sphercal Gaussas we have: ϕ f( r) ϕ = f( r) ϕ δ( r ξ) ϕ dξ j l j j l = ϕϕ l π ξ f tr[ Al J ] j 3/ / e ξ dξ (78) wth J j beg a symmetrc matrx wth the ad jj dagoal elemets ad the j ad j off-dagoal elemets. Whe f depeds oly o the absolute value of the terpseudopartcle dstace, ths formula becomes 4 / ξ ϕ f( r ) ϕ = ϕ ϕ f(tr[ A J ] ξξ ) e dξ j l l l j π 0 (79) The above tegral s easly evaluated aalytcally for may commo forms of f(r j ) (cludg /r j ). I the worst case scearo t ca always be computed wth quadrature formulas Aalytc Gradet of the Eergy A very mportat aspect of the atomc (ad molecular) calculatos wth ECGs s that achevg hgh accuracy s possble oly whe the olear expoetal parameters of Gaussas are extesvely optmzed based o the mmzato of the eergy. Ths process usually taes large amouts of computer tme. To accelerate the bass set optmzato the ECG calculatos, oe ca derve ad mplemet the aalytc gradet of the eergy wth respect to the olear parameters of the Gaussa bass fuctos. The term aalytc here meas that the compoets of the gradet are ot evaluated usg fte dffereces of the eergy (whch s a very costly procedure sce the umber of those compoets may reach may thousads for large bass sets). Istead, they are evaluated umercally usg aalytc expressos. The use of the aalytc gradet has eabled the performace of very accurate BO ad o-bo calculatos of varous atomc ad molecular systems wth accuracy umatched by prevous calculatos. Below we outle the approach used calculatg the gradet. We start wth the dfferetal of the secular equato (eq 7): d(h εs)c = (dh)c (d ε)sc ε(ds)c + (H εs) dc (80) Multplyg ths equato by c from the left, we obta: dε = c(dh ε ds)c (8) To get eq 8, we utlze eq 7 ad assume that the wave fucto s ormalzed,.e., csc=. For geeralty we also assume that the bass fuctos ad ther lear coeffcets may be complex. The relato eq 8 costtutes the well-ow Hellma Feyma theorem. Now let α t be a olear parameter, whch bass fucto φ t depeds o. As the tth row ad tth colum of matrces H ad S deped o α t, the dervatve of ay arbtrary elemet belogg to that row or that colum of ether of the two matrces ca be wrtte as ad Hl = H l ( α α δ t + δ lt δ t δ lt ),, l =,..., K t t Sl = S l ( α α δ t + δ lt δ t δ lt ),, l =,..., K t t (8) (83) Next, applyg relatos 8 83, the dervatve of the total eergy, ε, wth respect to parameter α t s 53 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

19 Chemcal s ε α t = c* H cl α K H = R c* t cl αt l= K S tl + * H ct cl α α K t tl ε l= t t l= tl ε S tl * H cc t t αt αt lt t ε tt ε S α lt t * H cc t t α S tt α t tt t ε S tt α t (84) By calculatg all such dervatves for each α t, the complete eergy gradet s obtaed. The total umber of the gradet compoets s equal to the product of the bass sze ad the umber of olear parameters cotaed each bass fucto. To mae the calculatos effcet, t s best to evaluate all dervatves of ε wth respect to the etre vech L vector (ad other olear parameters f ay) a sgle step rather tha performg separate dfferetatos for dvdual parameters (L ),(L ),..., (L ) because may of the operatos calculatg the dervatves are detcal. Wth that, the calculato of expresso 84 requres owledge of the followg dervatves of the H ad S matrx elemets: Hl (vech L ), Hl (vech L), Sl (vech L ), l Sl (vech Ll ) (85) The explct expressos for these dervatves for dfferet types of ECGs were derved ad preseted several papers. 90,9,8,3,35 37, Evaluato of Matrx Elemets for Molecular BO Calculatos There are may smlartes the dervato of the tegrals used BO calculatos ad the oes used the o-bo calculatos. Ths s partcularly the case the dervato of the overlap ad the etc eergy tegrals. I the dervato of the potetal eergy tegrals,.e., the electro repulso ad the uclear attracto tegral, the followg trasformato ad detty ca be employed. 7 = exp[ μ rj ] dμ / r π 0 j 3/ βμ ( + αμ ) exp + αμ / π / β = erf / β α 0 dμ (86) (87) Here the square of the terelectro dstace ca be represeted as a quadratc form: j r = rjr j (88) Followg Kghor, 65 the gradet of the molecular tegrals wth respect to the olear varatoal parameters (.e., the expoetal parameters A ad the Gaussa ceters s ) are also derved usg the methods of MDC. The gradet of ε ow volves the dervatves of the overlap ad Hamltoa matrx elemets wth respect to ot oly vech L, but also wth respect to the coordates of shft vectors s : Hl (vech L ), Sl (vech L ), H l S, l s s (89) Detals cocerg the dervato of the gradet matrx elemets ad the correspodg algorthms ca be foud refs 7 ad VARIATIONAL OPTIMIZATION OF THE GAUSSIAN NONLINEAR PARAMETERS IN ATOMIC AND MOLECULAR NON-BO CALCULATIONS 6.. Soluto of the Geeralzed Egevalue Problem Usg quc ad stable algorthms for solvg the geeralzed symmetrc/hermta egevalue problem (GSEP/GHEP) gve by eq 7 s very crucal for the overall effcecy of the varatoal calculatos wth ECGs. Ths s partcularly mportat whe the sze of the ECG expaso of the wave fucto s large (thousads of terms). It s worthwhle to ote that, prcple, oe ca vary the lear coeffcets of the bass fucto wthout resortg to the soluto of the geeralzed secular equato at all. It s possble to smply mmze the Raylegh quotet based o some olear optmzato algorthm (wth approprate orthogoalty costrats mposed o the wave fucto excted state calculatos). I practce, however, usg umercal algorthms of lear algebra s much more coveet ad computatoally effcet. The most straghtforward approach to solvg for all or some of the egevalues eq 7 s based o the reducto of the GSEP/GHEP to the stadard (.e., ogeeralzed) egevalue problem. Due to postve-defte ature of S, t ca be factorzed the Cholesy form, S = LL. The, after applyg the verse of L o the left ad the verse of L o the rght, oe obtas the stadard egevalue problem. Ths scheme s mplemeted may umercal lear algebra pacages, such as LAPACK. However, ths geeral algorthm has several drawbacs, oe of whch s a relatvely low speed. For large matrx dmesos, K, the soluto of eq 7 becomes qute expesve, ad for smaller atoms ad molecules, t may eve tae more computer tme tha the evaluato of the S ad H matrx elemets. Ths happes because the soluto of the GSEP/GHEP wth dese matrces requres K 3 arthmetc operatos, whle the evaluato of matrx elemets requres oly K operatos. Although the proportoalty costat s much larger the latter K term tha the K 3 term, as K creases the tme requred to solve eq 7 may start to exceed the tme eeded for the evaluato of matrx elemets. For ths reaso t s very mportat to use a egesolver whch s effcet for large dmesos of the bass. It should also be oted that the calculatos that volve optmzato of olear parameters, eq 7 usually eeds to be solved a very large umber of tmes (thousads, f ot mllos). I addto to that, hgh relatve accuracy of the egevalues/egevectors s desrable the calculatos. 54 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

20 Chemcal s Perhaps the most mportat factor choosg the most sutable algorthm s ts ablty to obta the soluto by performg a quc update of the prevous soluto the case whe just oe or a few bass fuctos have bee chaged,.e. whe oly oe or few rows/colums of matrces S ad H have bee modfed. Oe such computatoally cheap scheme s gve by Varga ad Suzu. 69 I ther wors they perform the Gram Schmdt orthogoalzato, whch reduces the geeralzed egevalue problem wth already dagoalzed (K ) (K ) submatrces (ths dagoalzato eeds to be carred out oly oce f the frst K bass fuctos are ept uchaged) to the covetoal form. Aother opto s to use the verse terato method. 70 I the vast majorty of cases oe s usually terested determg oly a sgle egevalue/egevector ad a good approxmato, ε appr, to the egevalue usually ow. The dea of the method s smple ad cossts performg the followg teratos: ( j+ ) () j (H ε S)c = Sc appr (90) The tal vector to start the terato process, c (0), ca be chose radomly f o better guess s avalable. Aother opto s to tae the soluto obtaed the prevous step of the optmzato procedure ad update t for the chaged matrces S ad H. Such a approach wors partcularly well f the chages matrces S ad H are small ad lmted to oly a few rows/colums of these matrces. The terato process (90) coverges as log as the desred egevalue s closer to ε appr tha ay other egevalue. The rate of covergece depeds o the rato (ε appr ε c )/(ε appr ε ), where ε s the desred egevalue ad ε c s the ext closest (after ε ) egevalue. Typcally, just a few teratos are eeded to obta the desred egevalue ad the correspodg egevector wth suffcetly hgh accuracy. I each terato oe has to perform a matrx vector multplcato ad solve a system of lear equatos wth a symmetrc (Hermta) matrx H εappr S. I the case whe the calculated state s the groud state of the system ad ε appr s chose to be below the actual egevalue, oe ca use the Cholesy method (ote that the matrx factorzato requrg K 3 operatos eeds to be performed oly oce regardless of the umber of teratos). I the geeral case, however, the matrx H εapprs s ot postve defte ad stead of the Cholesy factorzato oe should use aother type of factorzato, such as LU, QR, or LDL T (LDL H s used for complex matrces). The LDL T factorzato s probably the best choce as t taes advatage of the symmetry of the matrx ad requres the least amout of computatoal wor. The above-descrbed algorthm of solvg the geeralzed egevalue problem s rather robust ad accurate despte a apparet problem wth the matrx ( H εappr S), whch may become ll-codtoed whe ε appr les close to the actual egevalue. As dscussed by Partlett, 70 the error whch may occur whe solvg the system of lear equatos wth a early sgular matrx s cocetrated the drecto of the egevector ad therefore does ot lead to a falure of the algorthm. A very mportat feature of the algorthm wth LDL T (LDL H ) factorzato s the fact that, upo chagg the last row ad colum (or a few last rows ad colums) of matrces S ad H, the updated soluto ca be obtaed by performg oly K operatos. Eve the case whe oe must obta the soluto of GSEP/GHEP from scratch, the verse terato scheme gve by eq 90 s several tmes faster tha the usual reducto of GSEP/GHEP to the stadard egevalue problem. Besdes that, the verse terato scheme geerally exhbts better umercal stablty. 6.. Geeratg the Ital Guess for Nolear Parameters The covergece of the wave fucto expaso terms of correlated Gaussas strogly depeds o how oe selects the olear parameters the Gaussa expoetals, as well as other parameters preset the bass fuctos (cludg teger parameters, such as the values of powers preexpoetal polyomals ad the dces referrg to the coordates of partcles volved those polyomals). I order to reach hgh accuracy the calculatos, t s ecessary to perform optmzato of those parameters. The ey compoet here, as was dscussed before, s the use of the aalytc gradet. Aother possble method, whch ca also be very capable, s based o the stochastc selecto of the olear parameters. The dea ad the potetal of ths coceptually smple yet very powerful approach (ofte called SVM, stochastc varatoal method) was frst demostrated by Kuul ad Krasopol sy 7 ad the further developed by Varga ad Suzu. 5,69,7 Some calculatos based o stochastc geerato of the olear parameters were also performed by Alexader et al., 73,74 ad others. It should be oted that eve f oe heavly reles o the drect optmzato wth the use of the aalytcal gradet, t s stll very mportat to be able to geerate a good tal guess, where the optmzato ca start from. As the umercal experece shows, a totally radom guess may lead to a covergece to a very shallow local mmum ad/or result very slow progress of the optmzato. Ufortuately, the hypersurface of the objectve fucto (whch s the total eergy our case) becomes extremely complcated whe the umber of bass fuctos exceeds a few tes. For ths reaso, geeratg a good tal guess of the olear parameters for a bass cosstg of thousads of fuctos s a otrval tas. Usually the followg strategy wors qute well. The bass set s grow cremetally ad the tal values of the olear parameters of ew bass fuctos are selected usg a approach smlar to SVM. That s, the selecto procedure s based o the dstrbuto of the olear parameters the bass fuctos already cluded the bass set. I ts smplest verso, the geerato of each olear parameter of ew radom caddates ca be doe usg a lear combato of ormal dstrbutos cetered at the values defed by the olear parameters of those already cluded bass fuctos. For example, f each bass fucto, ϕ, cotas m cotuous olear parameters α,...,α m, ad the curret bass sze s K, the the th olear parameter of the caddates for the ext bass fucto, ϕ K+, s obtaed from the followg dstrbuto: K α ρ = ( x ) () x exp K πσ ( ) ( σ ) = (9) I the above equato σ are some costats whose magtude s comparable to the magtude of α. Usually a large umber of bass fucto caddates (hudreds f ot thousads) s geerated ad the tested agast the total eergy lowerg. The caddate whch lowers the eergy the most s the cluded the bass as ϕ K+. After that (or after cludg several more bass fuctos), the olear parameters of the ew bass fucto(s) are further optmzed usg the gradet- 55 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

21 Chemcal s based approach. Ths procedure s repeated utl the sze of the bass reaches the desred value. At the ed of the procedure oe gets relatvely good tal values of the olear parameters. Usg these tal values, t s the possble to use algorthms for drect mmzato of the eergy fuctoal to cotue the optmzato of the olear parameters of bass fuctos utl the desred level of covergece s acheved Dealg wth Lear Depedeces of the Gaussas durg the Varatoal Eergy Mmzato I some cases thorough optmzato of the olear parameters may lead to lear depedeces betwee bass fuctos. The reaso for the appearace of lear depedeces vares from case to case. Ofte the lear depedeces arse as a result of poor ablty of the bass fuctos to effectvely descrbe certa features of the wave fucto of the system uder study. I ay case, the lear depedeces betwee bass fuctos may cause umercal stabltes the calculato, partcularly whe solvg the secular equato. It should be oted, however, that the presece of learly depedet fuctos as such does ot automatcally lead to umercal stabltes the computed egevalues. Some lear depedeces, such as those radomly geerated, mght ot cause ay harm at all, provded a proper algorthm for solvg the geeralzed egevalue problem s chose. O the other had, the lear depedeces that arse durg optmzato of the olear varatoal parameters may cause problems the calculato. Ths happes, for example, whe the lear coeffcets of two (or more) bass fuctos are large, close magtude, but have opposte sgs. Let us assume that bass fuctos ϕ ad ϕ j are almost detcal (.e., almost learly depedet). Rememberg that each egevalue of eq 7 ca be represeted as a smple Raylegh quotet, t s ot dffcult to realze that a cacellato of leadg dgts wll occur the course of subtracto ϕ O ϕ + ϕ j O ϕ j ϕ O ϕ j ϕ j O ϕ, where O s ether the Hamltoa or the detty (overlap) operator. Therefore, the cotrbutos to the fal egevalue due to the learly depedet bass fuctos wll be of reduced accuracy ad, depedg o the overall mportace of those bass fuctos, the last several dgts of the computed eergy egevalue may be accurate. I the case of severe lear depedeces the assocated umercal accuracy may ot oly affect the qualty of the optmzato of the olear parameters (whch s sestve to the accuracy of the computed egevalues), but may eve lead to completely urelable results. Therefore, t s geerally a good practce to avod severe lear depedeces the varatoal calculatos. There are several ways to eep the lear depedecy problem uder cotrol. The most straghtforward approach s to remove learly depedet fuctos from the bass set. Ths may be of partcular use whe ew bass fuctos are geerated durg the process of growg the bass set. The crtero of lear depedecy ca be adopted from the Gram Schmdt orthogoalzato process. Our experece has show that by far most ofte lear depedeces occur as par lear depedeces;.e., oe bass fucto becomes very close to aother oe. Whe ths happes, the magtude of ther ormalzed overlap, S j, becomes very close to uty. I ths case, testg for lear depedecy amouts to checg all curret overlap matrx elemets. Whe a ew bass fucto s added to the bass, the test s reduced to checg a sgle row of the overlap matrx, whch taes very lttle computatoal tme. Rejectg a ew bass fucto whose overlap matrx elemets exceed a certa threshold ca be easly mplemeted ad wors very well whe a stochastc selecto of ew bass fucto caddates s performed. The stuato becomes somewhat more complcated whe oe eeds to optmze the olear parameters of the exstg bass fuctos. Oe possble way to proceed here s to smply dscard ay chages of the olear parameters that result the bass fucto becomg too learly depedet wth aother fucto the bass set. However, ths strategy caot be easly adopted actual calculatos, as t s the optmzato procedure/software that pcs the values of the olear parameters each gve optmzato step. A better approach s to use a pealty fucto, whch adds a certa postve value to the mmzed fucto (the total eergy) wheever the overlap of two or more bass fuctos s larger tha the assumed threshold. Sce oe wats to eep the objectve fucto smooth, t arrows the choce of possble expressos for the pealty fucto. Oe ca, for example, use the followg form of t: = j (9) where the sum s over all motored bass fucto pars ad Sj t β, S > t = j j t 0, Sj t (93) I eq 93, t s the value of the overlap threshold ad β cotrols the magtude (.e., maxmum) of the pealty for each par overlap. The choce of t ad β s usually based o experece ad may dffer depedg o the system ad the sze of the bass. The value t = 0.99 ca be a reasoable choce most cases, whle β should ormally be tae as a small fracto of the total eergy of the system. Excessvely large values of β ted to cause falures the optmzato, because the the objectve fucto exhbts very sharp jumps, whch are cosstet wth the assumpto of a smooth, dfferetable fucto. O the other had, too ty values of β may ot result effcet elmato of par lear depedeces. 7. VARIATIONAL OPTIMIZATION OF THE GAUSSIAN NONLINEAR PARAMETERS IN ATOMIC AND MOLECULAR BO CALCULATIONS 7.. Optmzato Approach Used the BO Molecular Varatoal Calculatos A practcal BO PES calculato volves (at least) two steps: buldg the bass set for the wave fucto expaso at the equlbrum structure of the molecule ad geeratg the PES for dfferet geometrcal structures. Both steps usually requre a sgfcat computatoal effort. The computatoal resources eeded for the calculato crease rapdly wth the umber of electros (! depedecy). I geeral, three factors determe the amout of the computatoal tme eeded for the calculato. The frst factor s the umber of expoetal parameters volved each ECG has, whch s [(( + ))/] +3, where s the umber of electros. The secod factor s the umber of ECGs eeded to reach the adequate level of the eergy covergece. 95 Ths umber creases wth the crease of the umber of electros the system. The tme eeded for the calculato of the Hamltoa ad overlap matrces scales as K wth the umber of ECGs, K. Also, the calculato tme 56 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

22 Chemcal s for each prmtve matrx elemet (before symmetrzato) creases as 3 wth the umber of electros. The thrd factor s related to satsfyg the Paul prcple ad mplemetg the correct permutatoal symmetry of the wave fucto. It volves actg wth a approprate symmetry operator ether o the et or bra bass fucto calculatg each Hamltoa or overlap matrx elemet. As the symmetry operator cludes! terms, the computatoal tme of each matrx elemet scales as!. Ths s the factor whch maes the calculato tme crease most rapdly whe a larger system s cosdered. There are also other factors that fluece the calculato tme. Oe of them s the effcecy of the optmzato of the ECG expoetal parameters. Ths effcecy usually decreases as the umber of electros ad the umber of ECGs crease, because more parameters have to be optmzed. Therefore, the developmet of more effectve optmzato strateges has become partcularly mportat as the molecules cosdered the calculatos become larger. Oe of the goals of mag the optmzato more effectve has bee the reducto of the umber of ECGs by better optmzg them. Optmzato approaches are dscussed sectos 7.. ad Buldg the Bass Set. I buldg a larger bass set, ew ECGs eed to be guessed. At the begg of the bass buldg process, a tal small set of fuctos s usually radomly chose usg Gaussa expoets tae from a stadard orbtal bass set. After ths tal set s optmzed, the calculato proceeds to grow the bass set larger. Oe way to guess ew Gaussas s to employ the free teratve-complemet-teracto (FICI) preoptmzato procedure. 75 More detals of the procedure are descrbed secto 7.. We should ote that the procedure adjusts the postos of the Gaussa ceters more effcetly tha the procedure based o the eergy mmzato where such ceter adjustmet has to ofte overcome sgfcat eergy barrers. After the bass set s elarged ths maer to a certa target umber of fuctos, a varatoal, gradet-based reoptmzato ca be appled to the whole bass set. I the optmzato of the whole bass set, ether oe ca choose to optmze all olear parameters smultaeously,.e., perform the full optmzato, or oe ca choose to optmze oly a part of the olear parameters at a tme. We call the latter approach partal optmzato. Usually the eergy coverges sgfcatly faster ( terms of the bass sze) the full optmzato tha the oe-fucto-at-a-tme optmzato. I Table 3 we show the eergy covergece wth the umber of ECGs BO calculatos of the LH molecule performed at the equlbrum teruclear dstace wth the optmzato procedure employg the aalytcal eergy gradet ad the full optmzato approach. 95 The results are compared wth the results of Cece ad Rychlews 8 obtaed the optmzatos performed wthout the gradet ad employg the oe-fucto-at-a-tme optmzato approach. The comparso shows the advatage of usg the gradet-based full optmzato ths case. At the same bass szes the eerges obtaed by Tug et al. 95 are cosderably lower tha those of Cece ad Rychlews. However, to be far, t eeds to be metoed that the calculatos of Cece ad Rychlews were doe 0 years before Tug et al. s ad certaly the mproved computer hardware has also cotrbuted to the creased accuracy of the results obtaed ref 95. Full optmzato does have certa drawbacs. They are the frequet appearace of lear depedeces betwee the optmzed bass fuctos Table 3. Comparso of the Covergece of the BO Eergy wth the Number of Bass Fuctos, hartrees, for the Groud State of the LH Molecule at R = 3.05 bohr bass sze Tug et al. 95 Cece ad Rychlews estd a a The estmated orelatvstc BO eergy at R = 3.05 bohr made by Cece ad Rychlews 8 usg the BO eergy of atoms, adabatc correctos, ad the expermetal equlbrum dssocato eergy. ad, possbly, large memory demads. These lmt the usefuless of the approach. As lear depedeces frequetly appear the bass set optmzato, partcularly at earler stages of the bass set buldg (whe K s small), they eed to be cotuously elmated the course of the procedure order to mata the umercal stablty of the calculato. The progress made by the bass set elargemet could be sgfcatly hampered or eve put o hold by ths pheomeo. As for the memory demads, they occur because of the use of the aalytcal eergy gradet the optmzato. The gradet comprsed the dervatves of the Hamltoa (ad overlap) matrx elemets, eq 89. The sze of the Hamltoa matrx dervatve s equal to the umber of olear parameters a sgle fucto tmes the square of the sze of the Hamltoa matrx, K ([(( + ))/] + 3). Whe thousads of ECGs are geerated the calculato, the sze of the Hamltoa matrx dervatve may exceed the amout of radom access memory avalable. It should be metoed that prcple t s possble to orgaze calculatos wthout storg the etre matrx dervatve. However, ths comes at a cost of addtoal complexty ad somewhat creased computatoal tme. Though the lear-depedecy problem ca be easly hadled by swtchg to the oe-fucto-at-a-tme partal optmzato approach (called partal optmzato ), the use of the full optmzato s more desrable the calculatos. Besdes the better eergy covergece (as show Table 3), there are two other reasos. Frst, to buld a molecular PES, the bass sets at dfferet geometres of the molecule have to be reoptmzed to esure a smlar accuracy level at all PES pots. Ths ca be accomplshed the full optmzato by motorg the orm of the aalytcal gradet vector ad always covergg the calculato to a value of the orm below a certa assumed threshold. Ths s ot possble the partal optmzato, because a small value of the gradet orm for dvdual ECGs does ot guaratee that at the ed of the optmzato the orm of the total gradet s also small. Secod, as the molecular geometry s deformed from the equlbrum the PES calculato, the olear parameters requre less adjustmet. I such a case, t oly taes a few teratos for the full optmzato approach to coverge. The procedure for hadlg lear depedeces betwee bass fuctos the BO calculatos performed by Pavaello et al. 76 comprse four steps. They are called to detfy, to replace, to avod, ad to bypass. At dfferet stages of the calculato (.e., dfferet szes of the bass set) dfferet procedures are usually used to acheve the best overall 57 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

23 Chemcal s computatoal performace. The four steps volve the followg:. To detfy. 76 At varous stages of the calculato the overlap for each par of bass fuctos s checed. Two fuctos, ϕ ad ϕ l, are cosdered learly depedet f the followg crtero s met: ϕϕ l ϕϕ ϕϕ t l l (94) If the absolute value of the overlap s close to ad hgher tha a certa assumed threshold, t (typcally 0.99 s used for t), ths par of fuctos s mared as learly depedet ad further treatmet s appled to resolve the problem.. To replace. 94 It s suffcet to replace oe of the fuctos the learly depedet par to remove the lear depedecy. Nolear parameters of the replacemet fucto, ϕ, are geerated by maxmzg the overlap betwee the fucto ad the lear combato of the fuctos, ϕ ad ϕ l, of the learly depedet par tae wth the lear coeffcets wth whch the par eters the wave fucto, c ϕϕ + c ϕϕ l l l l l l l ( c ϕϕ + c ϕϕ + c c ϕϕ ) ϕϕ (95) After the optmzato, the learly depedet par s replaced by a par comprsg the ewly geerated fucto ad oe of the old fuctos of the par. After the replacemet, the optmzato of the bass set s restarted. 3. To avod. 7,95 The replace procedure was prove to be effectve the PES calculatos of two-electro systems, (.e., H 3 + ). However, the effcecy drops for four-electro systems. Furthermore, ay bass fucto replacemet requres restartg the optmzato. Also, t usually results some crease of the total eergy. The addtoal tme eeded for the eergy to retur to the value before the replacemet cosderably slows dow the optmzato process. Ths causes the fucto replacemet to be computatoally expesve ad mpractcal whe the sze of the bass set becomes large. Thus, for larger bass sets, stead of replacg fuctos learly depedet pars, a method that prevets the formato of lear depedeces altogether the optmzato process was developed. 7,95 It volves addg the pealty term gve by eq 93 to the varatoal eergy fuctoal. Whe the value of the overlap betwee a par of bass fuctos reaches threshold t, the pealty term for that par, whch was zero (below the threshold), becomes postve ad ts value creases f the par becomes more learly depedet. I the mmzato of the eergy fuctoal, whch cludes the pealty term eq 93 for each par of bass fuctos, the fuctos automatcally stay learly depedet. 4. To bypass. 95 I the case whe lear depedecy occurs too frequetly, ad oe of the above procedures s able to correct the problem, a decoupled approach (partal optmzato) s appled the optmzato. I ths approach, the bass set s parttoed to subsets ad each subset s optmzed separately. Ths lowers the probablty of the lear depedecy occurrece. I extreme cases, each fucto s optmzed separately. Ths actually may mprove the scalablty of parallel calculatos as each processor ths case ca carry out the optmzato of a dfferet fucto ad the amout of terprocessor commucato s sgfcatly reduced. Usually, after elargg the bass set to a certa sze usg the partal optmzato, the approach s swtched bac to the full optmzato to fsh the calculato. As show by Tug et al., 95 the lear depedecy problem becomes much less severe or eve vashes etrely whe the bass set grows to a large sze. At that pot, the bass set s exteded eough to descrbe certa mssg features of the wave fucto ad the appearace of learly depedet fuctos (whch would otherwse descrbe those features) the course of optmzato s o loger favorable. Ths explaato s based o a observato that certa features of the wave fucto ca be represeted ether by pars of almost learly depedet fuctos or, alteratvely, by fuctos whch are ot learly depedet but whose ceters are shfted to the rght places. It s upredctable whch represetato s used the bass set optmzato. If, however, the represetato wth a learly depedet par of fuctos s selected by the optmzato procedure, t ca be always coverted to the other represetato. Ths s what the replace procedure does Geeratg the BO PES. The questo the PES calculato s how to effectvely geerate bass sets for dfferet PES pots. It s clear that, stead of regeeratg a ew bass set for each PES pot from the begg, a more effectve way would be to geerate the bass from the bass set of a earby PES pot, where the optmzato of the bass fucto parameters was already performed, ad the reoptmze t for the ext pot. The hgh qualty of the bass set at each PES pot s crucal for a effectve ad accurate PES calculato. I geeratg the bass set for a PES pot from the bass of a earby pot, t s assumed that the two pots are close eough that the expoetal parameters, L, for the two pots are very smlar. The oly parameters that eed some adjustmet are the Gaussa shfts, s. Two methods have bee developed to hadle ths problem: the sprg model 94 ad the Gaussa product theorem model. 95 Both methods adjust the Gaussa ceters whe the PES calculato moves from oe PES pot to aother. I the sprg model each Gaussa ceter s assumed to be attached to every ucle of the molecule wth sprgle coectos. If the posto of ucleus α chages from R α to R α + ΔR α, the Gaussa ceter, s, where s the dex of the Gaussa, follow the uclear movemet ad chages to s + Δs, where N Δ Rα Δ s = d R α= α N d = R α= α (96) ad R α are the dstaces from a Gaussa ceter s to the ucle. The other method for adjustg the Gaussa ceters s a reformulato of the procedure by Cece ad Kutzelgg. 59 I bref, the procedure volves trasformg each ECG to a product of ECGs wth ceters cocdg wth the postos of the ucle, ϕ = N = φ. A more detaled descrpto of ths method ca be foud secto, where t s used for the calculato of adabatc correctos to the BO eergy. By shftg these ceters alog wth shftg the ucle the potby-pot PES calculato ad the retrasformg the ECG from the product form bac to ts orgal form, the ew bass 58 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

24 Chemcal s set s obtaed. These approaches wor best f the PES pots are close to each other. Though oly the Gaussa ceters are adjusted from oe PES pot to a eghborg pot, the calculatos performed by Pavaello et al. 94 ad Tug et al., 95 the expoetal parameters, L, are tued durg reoptmzato of the bass. It s worth otg that there are as may as [(( + ))/] + 3 olear parameters, where s the umber of electros, each Gaussa ad K([(( + ))/] + 3) s the umber of parameters the bass set of K bass fuctos. Oly the use of the aalytcal gradet allows for a effcet optmzato wth such a large umber of varables. Though the lear depedecy s ot a cocer at ths stage, the varatoal eergy mmzato oe ca ether optmze all the above-metoed parameters smultaeously or optmze them separately to acheve a better computatoal performace. Also decouplg the optmzato of the Gaussa ceters from the optmzato of the L expoetal parameters ca be a helpful approach. The advatage of decouplg s based o the fact that the formulas for the frst dervatves of the total eergy wth respect to the Gaussa ceters are much smpler tha the formulas for the dervatves wth respect to the L expoetal parameters. I the decouplg approach, oe usually performs the optmzato of the Gaussa ceters frst (total of K(3) parameters). Ths s followed by the optmzato of L. 7.. FICI Method ad the Multstep Procedure Used Growg the Bass Set Reachg spectroscopc accuracy wth BO eergy calculatos of molecules cotag more tha two electros wth floatg ECGs s a challege. 9 A mportat ssue the varatoal optmzato of floatg ECGs s how to grow the bass set to acheve a faster covergece of the calculato. I ths secto some deas cocerg ths topc are dscussed. I partcular we descrbe how to tacle ths problem wth a approach based o the free FICI method developed by Naatsuj ad coworers The method descrbed here has bee appled calculatos of three- ad four-electro molecules. 95,75,80 For two-electro molecules, such as H + 3, optmzato of the bass set based solely o the prcple of mmum eergy leads to excellet results. 94,76 The exact BO wave fucto, Ψ, satsfes the Schro dger equato volvg the CN or the electroc Hamltoa defed eq 0: ( H E ) Ψ = 0 el el (97) A tral wave fucto, Φ, s ot exact ad does ot satsfy eq 97. Istead, t gves ( H E Φ ) Φ 0 (98) where, for the sae of clarty, we dropped the subscrpt el ad troduced the subscrpt Φ to deote E Φ = Φ H Φ / Φ Φ. It s easy to otce that the result of (H E Φ ) actg o Φ s a fucto orthogoal to Φ: Φ ( H E Φ ) Φ = 0 (99) Let us call fucto χ =(H E Φ )Φ the Naatsuj fucto assocated wth Φ. If such a orthogoal fucto s added to the wave fucto expaso, the eergy would be lowered. Ths s because χ has a ozero off-dagoal matrx elemet wth the Hamltoa ad Φ, amely Φ H χ = Φ H( H E ) Φ Φ Φ = Φ H Φ E 0 (00) Thus, the H E Φ operator may be used to geerate a correcto to the approxmate wave fucto to brg t closer to the exact wave fucto Ψ. Followg ths dea, Naatsuj 77 used the followg seres to costruct Ψ: Ψ = a ( H E ) Φ (0) where E s the eergy assocated wth the th trucato of eq 0 ad a are determed varatoally. Provded that Φ satsfes certa codtos, 77 eq 97 should mootocally coverge to the exact wave fucto as creases. Eve though t s ot yet clear whether the FICI method should lead to a mprovemet of the covergece of the varatoal calculato, t provdes a systematc way of mprovg the varatoal eergy. I several applcatos of the FICI model to atomc ad molecular systems, Naatsuj ad co-worers 77 showed how expasos eq 0 obtaed wth just few teratos (usually 6) produce orelatvstc BO eerges wth a sub-cm absolute accuracy. Utlzg the orthogoalty of the Naatsuj fucto geerated eq 98 ad the property eq 00, Pavaello et al. 75 devsed a approach where the floatg ECG bass set the varatoal molecular calculato s elarged by guessg ew Gaussas to best resemble the Naatsuj fucto. I the approach the followg procedure based o the frst-order term eq 0 was used. Let us assume that a approxmate wave fucto expaded terms of K 0 floatg ECGs, Φ K0, has already bee fully optmzed. The procedure for growg the bass set to K 0 + K fuctos comprses the followg fve steps:. A set of K ew floatg ECGs s costructed. The A matrces of these fuctos are geerated radomly ad the Gaussa ceters, s, are placed at the ucle. Wth the addto of the ew fuctos, the bass set ow has K 0 + K fuctos.. The lear expaso parameters of the wave fucto are foud by a smultaeous dagoalzato of the Hamltoa ad the overlap matrces. 3. Oly the olear parameters of the ewly added floatg ECGs are optmzed (recall that ths step was termed partal optmzato before) usg a fuctoal that maes the ewly guessed fuctos to best approxmate the Naatsuj fucto for Φ K0. 4. The whole K 0 + K bass set s fully optmzed,.e., the olear parameters of all K 0 + K bass fuctos are subject to varatoal optmzato (recall that ths step was termed f ull optmzato before). I every optmzato cycle the lear parameters are updated. 5. The cremetal elargemet route relabels the K 0 + K set as the ew K 0 set, ad the procedure returs to step. The fuctoal for the partal optmzato of step 3 s desged to best approxmate the = term eq 0 by K floatg ECGs. It volves maxmzg of the followg fuctoal: K = c g H E Φ = K0 K0 FK [ ] K K c g c g = = (0) 59 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

25 Chemcal s The partal optmzato step 3 wll be called the FICI refemet hereafter. I eq 0 the superscrpt labels ECG fuctos, g, belogg to the K set. It s straghtforward to otce that the maxmzato of the F[K ] fuctoal yelds a approxmato to the Naatsuj fucto (H E K0 )Φ K0. After the maxmzato of F[K ], the ewly refed K set of floatg ECGs s added to the K 0 set ad a partal varatoal optmzato of the K set s performed. The steps 4 ad 5 follow. The cycle of the fve steps s repeated utl satsfactory covergece of the eergy s reached Implemetato. I the actual mplemetato, 75 the maxmzato of the F[K ] fuctoal s replaced by the mmzato of the followg fuctoal: GK [ ] = + FK [ ] (03) Havg F rather tha F the fuctoal smplfes the calculato, because the fuctoal becomes depedet of the phase of the wave fucto. Usg + F stead of F prevets the G fuctoal from reachg a sgularty at F 0, whch ca happe f the tal choce of the K set s poor. I ref 75 the mmzato of G[K ] was carred out usg the trucated Newto optmzato route (TN) of Nash et al. 8 To speed up the covergece, t s also possble to supply the TN route wth the aalytcal gradet of G[K ] determed wth respect to the olear parameters of the fuctos the K set: 75 GK [ ] FK [ ] FK [ ] = K ( + FK [ ] ) K (04) where K represets the partal dervatve wth respect to the olear parameters of the floatg ECGs belogg to the K set Improved Fucto Moblty ad Barrer Tuelg. As poted out secto 3.5, a mportat aspect of the floatg ECG bass set s ts flexblty descrbg dfferet features of the wave fucto cludg ts oc ad covalet compoets. Usually the rato of the umber of oc bass fucto wth respect to the covalet fuctos s mataed at a certa level whch vares depedg o the specfc molecular system. 94 If a certa oc/covalet rato s set for a bass set optmzed usg the varatoal method, t remas essetally uchaged durg the calculato. Ths s because chagg t ad mag some oc floatg ECGs become covalet or vce versa eeds mgrato of ceters of the floatg ECGs betwee atomc ceters, whch requres overcomg eergy barrers. Ths s ulely to happe the varatoal eergy optmzato. A aalyss of how the FICI refemet deals wth adjustg the oc/covalet rato was carred out ref 75. The umercal evdece preseted there showed that FICI allows for a much mproved fucto moblty terms of a more proouced varato of the postos of the Gaussa ceters wth each optmzato cycle. 8. CALCULATION OF THE LEADING RELATIVISTIC AND QED CORRECTIONS Recet theoretcal studes of the helum atom that clude the wors performed by Morto et al., 3 Korobov, 8,83 ad Pachuc 6,9,84 have demostrated that, by systematcally cludg the fte mass correctos, as well as the relatvstc ad QED correctos, to the orelatvstc eerges of the groud ad excted states of ths system, oe ca acheve a accuracy of the predcted ozato ad trasto eerges that some cases exceed the accuracy of the preset-day expermet. The recetly publshed summary of the avalable theoretcal ad expermetal results for the boud statoary states of He by Morto et al. 3 demostrates very well the hgh level agreemet betwee theory ad expermet acheved the calculatos. It also shows that for a few states such as P ad 3 P J there s stll some otceable dsagreemet betwee the theoretcal ad the expermetal values. Hgh-accuracy calculatos o the H molecule 8,85 87 also revealed that to acheve a hgh level of agreemet betwee the expermet ad the theory for electroc ad vbratoal trasto eerges the domat α -depedet terms (where α =/cs the fe structure costat), relatvstc correctos, ad some hgher order correctos have to be cluded the calculatos. The effectve operators for hgher order correctos are derved wth the framewor of orelatvstc quatum electrodyamcs (NRQED). For example, all terms up to α 3 were cluded the calculatos for the He atom by Morto et al. 3 Also QED terms of the order of α 4 were cluded for some lower states the calculatos performed by Korobov ad Yelhovsy, 8,88 by Pachuc, 9 ad by Drae ad Mart. 89 Moreover, there have bee wors where the terms of the order of α 6 for the He atom were calculated. 6,84 These mae the He atom the most accurately descrbed atomc system apart from the hydroge atom. I addto to the relatvstc ad QED correctos, some He calculatos also cluded correctos for fte values of the 3 He ad 4 He uclear charge rad of.9659 ad.67 fm, respectvely, whch were derved from the sotope shft measured by Sher et al. ad from the measuremets of the Lamb shft of the muoc hydroge. 90,9 Aother system for whch the theoretcal calculatos have bee ofte used to mae a comparso wth hghly accurate spectroscopc measuremets s the lthum atom. 9,93 The most accurate calculatos of ths three-electro problem have bee performed usg Hylleraas-type fuctos that are capable of accurately descrbg the asymptotc behavor of the wave fucto at both the electro ad uclear cusps ad at fty. There have bee several wors devoted to very accurate L calculatos. 8,3,09,94 97 They have cluded the relatvstc correctos of the order of α, as well as the QED correctos of the order of α 3 ad estmates of the α 4 correctos. 7,98 00 QED whch descrbes the behavor of quatum partcles a electromagetc feld creates a geeral theoretcal framewor for the aalyss of the relatvstc ad QED effects boud states of atoms ad molecules. However, eve for small atomc ad molecular systems wth a few electros, accurate relatvstc calculatos are very hard ad too expesve to be carred out o preset-day computers. Furthermore, the QED Drac Coulomb (DC) equato s oly fully correct for a sgle electro the Coulombc feld ad approxmatos are troduced whe systems wth more tha oe electro are cosdered. Also, a addtoal problem appears due to the lac of a lower boud for the egatve eergy spectrum the DC equato. Faced wth those dffcultes, a effort has bee made to develop a effectve approach to accout for the relatvstc effects lght atomc ad molecular systems the framewor of the perturbato theory. The zero-order level such a approach s the orelatvstc Schro dger equato. The perturbato Hamltoa represetg the relatvstc effects s the obtaed based o the NRQED theory. 0,0 We should meto that the perturbato approach to accout for the 60 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

26 Chemcal s relatvstc correctos ca also be developed wthout usg NRQED, as show by Bethe ad Salpeter. 03 The relatvstc correctos ca ow be routely computed by some quatum chemstry pacages. I the NRQED theory the relatvstc correctos appear as quattes proportoal to powers of the fe structure costat α /37 ( atomc uts α =/c, where c s the speed of lght a vacuum): E( α) = E + α E + α E + α E +... NR REL 3 QED 4 HQED (05) Ths eables cluso of creasgly hgher order effects a systematc way the calculatos. The leadg terms of the expaso eq 05,.e., the orelatvstc eergy E NR, the relatvstc correcto α E REL, ad the hghest-order radatve correcto α 3 E QED, have bee well-ow sce early wors of Bethe ad Salpeter. 03 The relatvstc ad QED correctos are determed usg the perturbato theory wth the orelatvstc wave fucto as the zero-order fucto. I addto to the NRQED correctos, oe ca also calculate correctos due to the structure of the ucleus ad ts polarzablty. Wth those correctos NRQED s at preset the most accurate theoretcal framewor for calculatg boud state eerges of lght atoms ad molecules. I most of the very accurate atomc ad molecular calculatos the relatvstc correctos are calculated as expectato values of the approprate NRQED operators wth the orelatvstc wave fucto of the state of terest. If the BO (.e., fte-uclear-mass) wave fucto s used, oe ca also clude the calculatos the so-called recol effects, whch are fte-uclear-mass correctos to the relatvstc eergy. I the calculatos performed by Stae et al. 8 0, 4,47,04 a somewhat dfferet approach has bee used. They started wth the relatvstc mass-velocty, Darw, sp sp, sp orbt, ad orbt orbt operators for all partcles volved the system expressed terms of laboratory coordates ad trasformed them to a teral coordate system. The they determed the correctos as expectato values of those operators wth the wave fucto obtaed a o-bo calculato that treats the ucle (or ucleus for a atom) ad the electros o equal footg. I ths way the recol correctos were automatcally cluded the relatvstc eergy. Such a procedure allows, for example, drect calculato of how a sotopc substtuto the system affects the relatvstc eergy. The eergy calculated ths way cludes the relatvstc cotrbutos due to the moto of the ucle, as well as small relatvstc cotrbutos orgatg from the couplg of the electroc ad uclear motos. The trasformato of the relatvstc operators to a teral coordate system s dscussed secto The Relatvstc Hamltoa A complete accout of the teractos betwee elemetary partcles that clude the electrostatc ad magetc forces descrbed by the Loretz-varat teracto potetal ca be obtaed from QED. 05,06 Wth ths model partcles teract by emttg ad absorbg vrtual photos. Relatvstc correctos have a smple form oly for the hydroge atom. For atoms ad molecules wth more electros the relatvstc problem s much more complcated. To accout for the relatvstc effects, a relatvstc multpartcle Hamltoa eeds to be costructed. Such a Hamltoa ca be wrtte for two teractg fermos descrbed by a wave fucto cosstg of 6 spor compoets. However, for a more geeral case of N teractg fermos ad/or bosos, the costructo of the relatvstc Hamltoa caot be doe wthout mag sgfcat approxmatos. Ths ca be best accomplshed wth the QED theory where a system of teractg partcles the relatvstc lmt s descrbed by a sum of sgle-partcle relatvstc Hamltoas (H rel ()) ad two-partcle teracto operators accoutg for the Coulombc teractos (V j = >j /r j ) ad the relatvstc teractos (B j ): H = Hrel + r () > j j > j B j (06) The B j term orgates from the applcato of the QED theory to two teractg partcles ad s derved by tag to accout sgle-photo scatterg ampltude the calculato. As the o-bo approach oe cosders all partcles o equal footg, oe s forced to mae a dstcto betwee the fermos ad bosos the relatvstc Hamltoa. Ths dstcto appears at the level of formulatg the theory ad at the level of the calculatos. The case of the two teractg partcles beg fermos (f f) s well descrbed the lterature. Less dscussed are cases where a fermo teracts wth a boso (f b) or a boso teracts wth aother boso. The dstcto at the relatvstc level betwee the fermos ad the bosos seems, perhaps, somewhat artfcal ad arbtrary, as the dfferece the relatvstc treatmet of the two types of partcles s small. However, a rgorous, very accurate relatvstc treatmet of ther teractos requres such a dstcto. We wll ow dscuss the α relatvstc cotrbutos two cases. The frst case cocers a system where all partcles are fermos ad have sps equal to / (for example, the 3 He atom). I the secod case the system cossts of fermos (electros ad some of the ucle) ad bosos (the other ucle). States of a sgle fermo wth sp s = / (a electro) ad a sgle boso wth sp s = 0 (for example, a α-partcle) are descrbed by the oe-partcle relatvstc Drac (D) ad Kle Gordo (KG) equatos, respectvely. The costructo of a geeral N-partcle quatum relatvstc equato s, however, ot as smple as the case of the orelatvstc Schro dger equato. I the Schro dger equato, t s suffcet to clude the Coulomb operators to accout for the teractos betwee the partcles. I the relatvstc case, apart from the Coulombc forces, there are terpartcle teractos that are related to the magetc propertes of the partcles. Those propertes result from the orbtal ad sp motos of the partcles. Furthermore, sce all the teractos betwee partcles are affected by the fte veloctes of ther motos, the so-called retardato effects appear. 8.. A System of N Fermos A system that cossts of N fermos the absece of a exteral feld ca be descrbed usg the Drac Bret (DB) Hamltoa (H f f ) exteded to a N-fermo case. We cosder oly the Paul approxmato of ths Hamltoa. Usg the geeralzed form of the DB Hamltoa ad expadg each term powers of α, oe gets H f f rel (the Bret Paul (BP) operator) as a sum of the followg terms of the order of α : H f f MV = N 8 = 4 P M 3 (07) 6 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

27 Chemcal s ad H H H H f f D f f OO f f SO SS f f N N QQ j = 8 M = j R R j (08) N N QQ j = PP + j R ( R P) P j j j MM R = > R j j j j j (09) N N QQ j = R P S 3 ( j j ) j = j Mj Rj N N QQ j (( Rj Pj ) S ( Rj P ) S 3 j ) MM R j= > j j j 8π SS δ R 3 ( ) 3 j ( j ) (0) N N QQ j = 3 = > MM R ( SS) ( SR )( SR 3 j j j ) j j j j Rj f f f f rel MV D f f OO f f SO f f SS f f H = H + H + H + H + H () f f f f where H MV s the mass velocty Hamltoa, H D s the f f Darw Hamltoa, H OO s the orbt orbt teracto f f f f Hamltoa, H SO s the sp orbt Hamltoa, ad H SS s the sp sp teracto Hamltoa A Fermo Boso System As derved by Datta ad Msra, 07 the fermo boso Hamltoa the orelatvstc lmt (H f b rel ) has to clude the teracto betwee the boso ad the fermos. For a sgle electro ad a boso these addtoal terms are the same as the orelatvstc-lmt Hamltoa for two fermos wth oe small excepto. The dfferece betwee the two Hamltoas s the absece of the term the Darw operator descrbg the teracto of the boso wth the feld geerated by the fermos. Let us expla ths dfferece usg a two-partcle system. The Darw operator the BP equato for two fermos, (M,Q ) ad (M,Q ), has the form H f f D QQ = + R 8 M R M QQ R R The frst term the above operator 8 M QQ R R descrbes the teracto of fermo wth the feld geerated by fermo, ad the secod term 8 M QQ R R s due to the teracto of fermo wth the feld geerated by fermo. I the case of a fermo boso par ( beg the fermo ad beg the boso), the term descrbg the teracto of the boso wth the feld geerated by the fermo, the term 8 M QQ R R s abset the order of α. However, t wll appear hgher orders α Trasformato of the Relatvstc Operators to the Iteral Coordate System The trasformato of the orelatvstc Hamltoa to the teral coordate system ad the elmato of the COM moto was descrbed secto.. A smlar trasformato eeds to be appled to the relatvstc Hamltoa. Whle a full separato of the laboratory Hamltoa to the Hamltoa descrbg the etc eergy of the COM moto ad the teral Hamltoa ca be exactly performed, the separato of the relatvstc Hamltoa to the teral ad exteral parts s ot exact. I geeral, the BP Hamltoa the ew coordate system ca be wrtte as a sum of three terms, H rel = H CM rel + H t rel + H CM t rel, where H CM rel s the term descrbg the t relatvstc effects of the moto of the ceter of mass, H rel descrbes the teral relatvstc effects, ad H CM t rel descrbes the relatvstc couplg of the teral ad exteral motos. The cotrbutos to the eergy of the system due to H CM rel ad H CM t rel vash f the ceter of mass of the system s at rest. The relatvstc correctos to the teral states of the system are calculated usg H t rel. For states wth the S symmetry, the trasformato of the coordate system leads to the relatvstc operator the followg form: H H H H f f MV f b = = H ( ) MV 3 r 3 r 8 m m 0 = = π = + δ = m m qq () r + δ = = m qq 3 ( r ) j j j, j f f 3 D 0 0 π = δ + m qq 3 () r 0 3 δ ( r ) j D f b = = j=, j f f OO f b m qq j = HOO qq 0 = r + r r r 3 ( r ) r = mm 0 r r qq 0 r + r r rj 3 ( r ) rj = j=, j mm 0 r r qq j + + r ( r ) r rj 3 j j r rj mm j rj rj = j> 6 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

28 Chemcal s π qq f f HSS = ( SS 0 ) δ ( r) 3 = mm 0 8π qq j 3 ( SS j) δ ( rj) 3 mm H f b SS j= > j 8π qq j = ( δ 3 mm SS 3 j) ( r j) j= > j where, for cosstecy of the otato, we used m = M QED Effects Atomc Calculatos j j I ths revew we focus o the calculatos of QED effects atoms wth more tha three electros, because very accurate eerges of such systems ca oly be obtaed wth the use of ECGs. A effectve way for calculatg atomc QED correctos proportoal to α 3 ad α 4 was descrbed the wor of Pachuc et al. 98 Let us cosder a four-electro atom. The leadg QED correcto that accouts for the two-photo exchage, the vacuum polarzato, ad the electro self-eergy effects ca be expressed as 98 E QED = + α Ψ δ r Ψ = > 5 3 l INM 3 ( j ) INM j 4 Ψ INM P Ψ π r 3 INM 3 4 j 4 9 q + lα l( ) 4 0 Ψ δ () r 0 INM 3 = 30 3 Ψ INM () The above expresso does ot clude the recol cotrbutos, whch are usually much smaller tha the leadg cotrbutos. The last term eq s the so-called Ara Sucher dstrbuto. 96,09 Ths cotrbuto s determed as the followg lmt: Ψ P Ψ = lm Ψ* ( r) Ψ ( r) Θ r a r ( ) 3 r 3 a πδ ( r)( γ + l a) dr (3) where Θ s the step fucto ad γ s the Euler costat. To overcome the usually slow covergece of the hghly sgular P(/r 3 j ), oe ca use the so-called expectato value detty approach mplemeted by Pachuc et al. The term volvg the so-called Bethe logarthm, l( 0 ), eq s more dffcult to calculate for a atom wth more tha oe electro. The Bethe logarthm ca be expressed as l( = Ψ 0) ( Ht Eorel)l[( Ht Eorel)] D Ψ where for a four-electro atom ad = 4 = r (4) (5) INM = D = πq Ψ δ ( r) Ψ INM (6) Hgh precso calculatos of l( 0 ) have bee doe for some oe- ad two-electro atoms by Drae 3 ad Korobov ad Korobov, 4 as well as for the three-electro lthum atom by Ya ad Drae 97 ad Pachuc et al. 98 More recetly, values for the Bethe logarthm were also reported for the groud state of Be + ad L ad the groud ad the frst excted states of the eutral Be atom by Pachuc et al. 7,99 The procedure used to evaluate the Bethe logarthm those wors was based o the tegral represetato of l( 0 ) proposed by Schwartz 5 ad refed by Pachuc et al. 99 The α 4 QED correcto s smaller tha the leadg α 3 correcto ad ca be determed approxmately. The domat compoet of the α 4 correcto usually accoutg for about 80% of ts value ca be calculated usg the followg formula: 99 E 39 5 l 4πq + Ψ INM 8 9 HQED 0 4 = 3 δ () r Ψ INM (7) The remag α 4 QED cotrbutos are more dffcult to calculate because they volve some sgular terms. 9,8 9. RESULTS FOR ATOMS The spectra of small atoms are measured wth very hgh accuracy. Thus they provde a excellet testg groud for very accurate quatum mechacal methods. Below we wll provde a few examples of recet calculatos o atomc systems wth three, four, fve, ad sx electros. I ths we wll partcularly emphasze the comparso of the calculated quattes wth the expermetal values. The goal of the comparso wll be also to show how volved the calculatos have to be ad what effects they eed to clude to match or eve approach the accuracy of the state-of-the-art expermet. We wll also commet o the role the calculatos ca play refg the spectral eerges of small atomc systems. 9.. Very Accurate Calculatos for Three- ad Four-Electro Atoms ad Atomc Ios We wll show here three examples of recet atomc calculatos. They cocer the lowest fve S states of the Be atom, 4 the lowest two S states of the C + o, 6 ad the lowest fve D states of the L atom. 38,39 We chose these three examples because they very well represet the terplay betwee the expermet ad theory studyg the atomc spectra. The calculatos of the fve lowest S states of beryllum were performed for the 9 Be sotope. 4 The bass set of ECGs was geerated depedetly for each state a process volvg startg wth a small radomly chose set of fuctos ad gradually addg more fuctos to the set ad optmzg them wth the varatoal method employg the aalytcal eergy gradet determed wth respect to the olear parameters of the Gaussas. I all other examples preseted ths secto the bass set was grow ths way. After the fal bass sets were geerated, the eerges of the states were recalculated for a fte uclear mass. The dfferece betwee the fte- ad fte-uclear-mass eerges for each state gves the mass correcto. 63 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

29 Chemcal s Table 4. Trasto Eerges betwee Adjacet S States of the Be Atom Computed Usg Fte- ad Ifte-Nuclear-Mass Norelatvstc Eerges ad The Corrected by Accoutg for the Leadg Relatvstc ad QED Effects a trasto S 3 S 3 S 4 S 4 S 5 S 5 S 6 S ΔE orel (INM) () (3) (6) (30) ΔE orel (FNM) () (3) (6) (30) ΔE rel (0) (30) (40) (70) ΔE QED (4) (36) (50) (85) ΔE HQED (30) (38) (50) (85) expermet 6, (0) (0) (0) (0) a The results are tae from ref 4. All values are cm. Next, the fte-uclear-mass wave fuctos were used calculate the leadg relatvstc correcto employg the frstorder perturbato theory approach. We also calculated the α 3 ad α 4 QED correctos. Wth the correctos were added to the orelatvstc eerges the trasto eerges betwee the adjacet states;.e., the S 3 S, 3 S 4 S, 4 S 5 S, ad 5 S 6 S trastos, were calculated. The results are show Table 4. For each value the umercal ucertaty s show. They were determed based o the level of the covergece of the partcular value wth the umber of bass fuctos ad o other factors cotrbutg to the umercal ose the calculatos. I Table 4 we also show the expermetal trasto eerges tae from the revew paper of Kramda ad Mart, 6 but orgally measured by Johasso. 7 The accuracy of the expermetal results ca be estmated based o Johasso s statemet, whch ca be foud hs paper, that the error hs trasto eergy measuremet should be less tha ±0.05 cm. As each expermetal trasto cluded Table 4 was determed drectly from two mp S trastos, t s reasoable to assume the expermetal ucertaty s about 0.0 cm (or less). As oe ca see Table 4, the eerges for the S 3 S, 3 S 4 S, 4 S 5 S, ad 5 S 6 S trastos calculated usg the FNM orelatvstc eerges augmeted wth the relatvstc ad QED correctos dffer from the expermetal results by 0., 0.0, 0.05, ad 0.06 cm, respectvely. Ths shows that the accuracy level of the preset calculatos s very hgh. Ths s the frst tme hgher excted states of a fourelectro atom have bee calculated wth such a accuracy. Also, t s clear that cludg the correcto for the fte uclear mass, ad the relatvstc ad QED correctos, s ecessary to acheve ths level of agreemet betwee the theory ad the expermet. The ext example cocers the lowest two states of the C + o. I a o wth a larger postve charge, such as C +, the relatvstc ad QED correctos are sgfcatly larger tha eutral atoms wth the same umber of electros. For ths reaso these correctos have a larger effect o of the trasto eerges. Carbo s also a terestg system because t has three stable sotopes, C, 3 C, ad 4 C, ad thus the uclear mass effects o ts trasto frequeces ca be studed. I presetg the results for C +, we start wth showg Table 5 the covergece of the trasto S 0 3 S 0 eergy as a fucto of the umber of bass fuctos for C +, 3 C +, ad 4 C +, as well as C +. As oe ca see, to acheve the covergece of about cm of the trasto eergy, a fucto bass set was eeded. Ths s a typcal covergece patter for a trasto eergy of a small atom. I Table 6 we show the covergece of the S 0 3 S 0 trasto eergy of the three sotopes of C + as creasgly hgher level of theory s employed the calculatos. As oe ca see, the cluso of the fte mass correcto chages the trasto eergy by about 0 cm, addg the relatvstc correcto has a much larger effect of about 300 cm, ad the QED correcto adds about 0 cm. Whe all of the above correctos are accouted for, the calculated trasto eergy for C + becomes (000) cm, whch s oly less tha a waveumber off the expermetal value of cm. 8 The thrd example cocers the most recet calculatos performed o the fve lowest D Rydberg states of the lthum atom. 38,39 The results preseted Table 7 cocer the eerges of the trastos betwee adjacet states of 7 L. The calculated values are compared wth the expermetal results. 8 Results obtaed for 6 L, whch has ot yet bee measured, as well as the L results, are also show. For 7 L, the results obtaed wth dfferet umbers of ECGs are also preseted Table 7 to show the eergy covergece patter ad demostrate that the trasto eerges obtaed wth 4000 bass fuctos are very well coverged. As the relatvstc ad QED effects ca be expected to be very smlar for all fve D states, the eergy dffereces agree very well wth the expermet. The s 4d s 5d ad s 5d s 6d trastos oly dffer from the expermet by 0.0 cm. However, for s 6d s 7d there s a more otceable dfferece maly caused by a less accurate expermetal eergy value for the s 7d state. The results of the calculatos allow for a refemet of ths eergy. Ths ca be doe by tag the expermetal eergy of the s 6d state of cm ad addg to t the very well coverged s 6d s 7d eergy dfferece of cm. Due to a eglgble cotrbuto of the relatvstc ad QED effects, the eergy value of cm obtaed ths way should be qute accurate. Ths value s slghtly dfferet from the expermetal value of cm. 8 The same procedure ca be appled to determe the eerges of the D states of 6 L, oce the eergy of the s 3d level becomes avalable from the expermet. Fally, we should ote that performg separate varatoal calculatos for the dfferet sotopes of the same elemet s ot the way these types of calculatos are usually performed. Usually, for atoms heaver tha hydroge oe frst computes the wave fucto wth a fte uclear mass ad subsequetly cludes the fte-mass effects by addg the frst-order perturbato correcto for the mass-polarzato operator. There are two reasos for usg the fte-mass approach atomc calculatos. Frst, as metoed earler, the bass fuctos eed to be optmzed for just oe sotope (usually the ma oe) ad the calculatos of other sotopes oly the lear expaso coeffcets the wave fucto are allowed to adjust. The effort volved such a adjustmet (doe by solvg the secular equato) s smlar to the effort 64 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

30 Chemcal s Table 5. Covergece of the S 0 3 S 0 Norelatvstc ad Relatvstc Trasto Eerges ( cm ) for Dfferet Isotopes of C + wth the Number of Bass Fuctos 6 a bass sze C +, orel C +, rel 3 C +, orel 3 C +, rel 4 C +, orel 4 C +, rel C +, orel C +, rel b (0) (70) (0) (70) (0) (70) (0) (70) expermet a The results were calculated usg up to ECGs. The values paretheses are estmates of the remag ucertaty due to fte sze of the bass. b Several addtoal optmzato cycles were performed. volved calculatg the frst-order perturbato correcto accoutg for the chage of the uclear mass. However, performg the calculatos varatoally automatcally allows for accoutg for hgher-order mass effects. Secod, the massdepedet wave fucto geerated FNM calculato ca subsequetly be used to calculate relatvstc correctos, as well as other expectato values. Thus, these quattes automatcally clude the mass depedecy (for example, the recol effects the case of the relatvstc correctos), whch the covetoal calculato ca oly be accouted for by a laborous perturbato procedure. Fally, havg compared the results for some D states of lthum wth the expermet, we should commet o the accuracy oe expects to acheve varatoal calculatos wth explctly correlated Gaussas comparso to the best calculatos performed wth Hylleraas fuctos. Such a comparso ca be made, for example, for the lowest S ad P states of L because hgh-accuracy results obtaed wth the Gaussas ad Hylleraas fuctos exst for these states. The comparso s show Table 8. As oe ca see, the Hylleraas results are otceably more accurate due to larger umber of bass fuctos used. However, the data Table 8 also demostrate that whe the olear parameters of the Gaussas are thoroughly optmzed oe ca acheve very hgh accuracy that s good eough for most applcatos that requre hgh precso. I fact, the covergece terms of the umber of bass fuctos ca be eve better the case of Gaussas tha the case of the Hylleraas fuctos f suffcet effort s vested the optmzato of the olear parameters of the Gaussas. 9.. Calculatos for Atomc Systems wth Fve Electros Precse calculatos o fve ad more electro systems are of partcular mportace as o calculatos of spectroscopc accuracy for such systems exst at the preset tme. Due to creased complexty ad much larger computatoal demads, most methods exhbt a huge deterorato of the accuracy for such relatvely large systems. At the same tme, these larger systems are of great sgfcace sce they serve as a mportat test for developg ad tug less accurate quatum-chemcal approaches. As a example of the hgh level of accuracy achevable for a fve-electro atomc system wth ECGs, we ca cosder the results obtaed for the groud ad low-lyg P ad S states of the boro atom. 30 Ths example ams to demostrate that, f suffcet computatoal resources are used the calculatos, ECGs are capable of producg results for the groud state ad some lower lyg excted states whch almost match the accuracy acheved before for four-electro systems. I Table 9 we preset the results of the ECG calculatos for the two lowest L = 0 states ad two lowest L = states of the boro atom. 30 The covergece of the total orelatvstc eerges of all four states s show. A quc loo at the covergece patters suggests that the ucertaty of the total orelatvstc eerges does ot exceed 3 μhartrees. I Table 0 the groud state eergy obtaed wth ECGs s compared wth some of the best lterature data. As ca be see, the ECG varatoal upper boud of hartree for the groud state eergy les otceably lower tha the best recet upper boud of hartree obtaed a state-of-the-art CI calculato 0 wth the followg very exteded bass set of atomc orbtals: 65 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

31 Chemcal s Table 6. Covergece of the S 0 3 S 0 Trasto Eerges ( cm ) for C +, 3 C +, ad 4 C + Icludg Icreasgly Hgher Level Correctos (Fte-Mass, Relatvstc, ad QED) to the Eerges of the Two States a cotrbuto cluded ΔE ( C + ) ΔE ( 3 C + ) ΔE ( 4 C + ) E orel (INM) (0) (0) (0) E orel (FNM) (0) (0) (0) α E rel (70) (70) (70) α 3 E QED (70) (70) (70) α 4 E HQED (000) (000) (000) expermet a The values paretheses dcate the remag ucertaty due to fte sze of the bass. I the case of α 4 correcto the ucertaty s due to a approxmate treatmet of that correcto. The results are tae from ref 6. Table 7. Eergy Dffereces ( cm ) betwee Adjacet D States of 7 L, 6 L, ad L a bass s 3d s 4d s 4d s 5d s 5d s 6d s 6d s 7d 7 L expt L L a For 7 L, the covergece of the dffereces wth the bass set sze s show. For 6 L ad L oly the results obtaed wth 4000 Gaussas are show. For 7 L, the results of the calculatos are compared wth the expermetal values. The results are tae from ref 39. 4s3pdf0g9h876l5m43oqr0t9u8v7w6- x5y4z. Aother llustratve example of the calculatos of a fveelectro system wth ECGs s the sgly ozed carbo atom. The results of the recet ECG calculatos for ths system 3 are show Table. Smlar to the boro atom calculatos, the accuracy curretly achevable wth the varatoal method employg ECGs s sgfcatly hgher tha that of the best-todate lterature values obtaed wth other methods. It s terestg to ote that the estmated accuracy for the excted S state of C + s roughly twce lower tha that for the groud P state. I cotrast, calculatos o the B atom yelded the eergy of the frst excted S state that was approxmately 5 tmes more tghtly coverged tha the eergy of the groud state. Such a order of magtude dfferece the accuracy of the frst excted S states of C + ad B does ot result from a lower qualty of the C + calculatos, however. The ma reaso for a somewhat worse covergece of the eergy the case of C + s the fact that the wave fucto of the lowest excted S state of ths system s domated by the s sp cofgurato. Ths Table 8. Comparso of the Norelatvstc Varatoal Eerges of the s s( S), s 3s( S), ad s p( P) States of L Obtaed ECG Calculatos wth the Best Lterature Values Obtaed Calculatos That Used the Hylleraas Bass a state source method bass sze eergy s s( S) Puchals et al. 3 Hylleraas Puchals et al. 3 Hylleraas Puchals et al. 3 Hylleraas (3) Wag et al. 9 Hylleraas Wag et al. 9 Hylleraas Wag et al. 9 Hylleraas (45) Stae et al. 3 ECG Bub 9 ECG s 3s( S) Puchals et al. 93 Hylleraas Puchals et al. 93 Hylleraas Puchals et al. 93 Hylleraas (9) Wag et al. 9 Hylleraas Wag et al. 9 Hylleraas Wag et al. 9 Hylleraas (3) Stae et al. 3 ECG Bub ECG s p( P) Puchals et al. 93 Hylleraas Puchals et al. 93 Hylleraas Puchals et al. 93 Hylleraas (4) Wag et al. 9 Hylleraas Wag et al. 9 Hylleraas Wag et al. 9 Hylleraas (5) Bub 9 ECG a Ifte bass sze stads for a extrapolated value. 66 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

32 Chemcal s Table 9. Covergece of the Total Norelatvstc Eerges ( hartrees) for the Ma Isotope of Boro Atom, B a sotope bass P o (s s p) S (s s 3s) P o (s s 3p) S (s s 4s) B b (50) (50) (50) (50) 0 B 500 b (50) (50) (50) (50) B 500 b (50) (50) (50) (50) a Eerges obtaed for 0 B ad B wth the largest bass set of 500 fuctos are also show. The values paretheses are estmates of the remag ucertaty due to fte bass sze used the calculatos. The results are tae from ref 30. b Bass set was geerated wth a more extesve optmzato of the olear parameters. Table 0. Comparso of the Avalable Lterature Results for the Groud State Eergy ( hartrees) of B Atom eergy method (year) cofgurato teracto + expermetal data (99) cofgurato teracto + expermetal data (993) (3) dffuso quatum Mote Carlo (007) (3) dffuso quatum Mote Carlo (0) ECG, 000 bass fuctos (009) selected cofgurato teracto (00) ECG, 500 bass fuctos (0) estmate of the exact orelatvstc eergy 30 cofgurato s dfferet from the domat cofgurato (s s 3s) the wave fucto of the lowest S state of B. The latter cofgurato s easer ad more effectvely descrbed wth the bass fuctos (3) tha the s sp cofgurato whch results from a couplg of two p electros to a S state. The covergece the calculatos of the states domated by the s sp cofgurato ca probably be mproved f prefactors the form of dot products r r j are cluded some bass fuctos Calculatos for Atomc Systems wth Sx Electros The carbo atom ( C) has bee the largest system cosdered so far calculatos performed wth all-electro correlated Gaussas. 35,36 The calculatos focused o the groud ad frst excted 3 P states of ths sx-electro system. As sx-electro bass fuctos are used these calculatos, there are expoetal parameters each fucto to be optmzed. Also, each Hamltoa or overlap matrx elemet volves tegrato over 8 Cartesa coordates. Eve wth the ad of the aalytc gradet the varatoal optmzato, such calculatos represet a dautg tas. As oly 3 processors were used the calculatos, after vestg a cosderable amout of computer tme, a bass set of oly 000 Gaussas was geerated for each of the two states. Eve though the lowest ever varatoal eerges have bee obtaed for both states wth that may fuctos, ther umber s clearly ot eough to determe the trasto eergy betwee the two states wth a accuracy eve close to what we acheved for smaller atoms. It would tae a dedcated computer system wth hudreds of processors to reach hgh accuracy the carbo calculatos. However, eve wth up to 000 bass fuctos the bass set, t s terestg to exame the covergece of the total eerges of the two states, as well as the covergece of the trasto eergy. The data for such a aalyss are show Table. As oe ca see, oly at best four dgts the total eerges are coverged. The trasto eergy shows the rght covergece tred, but t s stll off from the expermetal value the thrd dgt after the decmal pot. 0. RESULTS FOR DIATOMIC MOLECULES OBTAINED WITHOUT THE BORN OPPENHEIMER APPROXIMATION The o-bo approach utlzg ECGs has bee so far mplemeted oly for rotatoless states (.e., vbratoal states) of datomc molecules. I ths revew we preset examples of those applcatos that best llustrate the accuracy level that the approach ca delver. Table. Total Norelatvstc Eerges ( hartrees) for the Lowest S ad P States of C + Io a sotope bass P o (s s p) S(s sp ) P o (s s 3p) C b (50) (500) (350) 3 C 500 b C 500 b C 500 b DMC, c (4) a The ECG results are tae from ref 3. b Bass set was geerated wth a more extesve optmzato of the olear parameters. c Dffuso Mote Carlo, ref dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

33 Chemcal s Table. Covergece of the Total Norelatvstc Eerges of the Groud ad Frst Excted 3 P States (s p ad s p3p) of C ad the Correspodg Trasto Eergy wth the Number of Bass Fuctos a C C bass sze s p 3 P s p3p 3 P trasto eergy expermet DMC b (6) estmated exact a Total eerges of C ad C are hartrees ad trasto eerges are cm. The results are tae from ref 36. Dffuso Mote Carlo. 0.. Datomcs wth Oe ad Two Electros: Charge Asymmetry Iduced by Isotopc Substtuto Two examples wll be show here. The frst cocers the calculatos of the complete spectrum of vbratoal states of the HD + molecule. 43 As the calculatos were performed wthout vog the BO approxmato ad the ucle them possessed fte masses, t s terestg to aalyze what effect ths has o the wave fucto ad, partcular, o the probablty that the electro s o average closer to oe ed of the molecule tha to the other. The secod example cocers the HD molecule ad ts pure vbratoal spectrum. The HD + pure vbratoal spectrum has bee studed by may researchers, ad very accurate, vrtually exact calculated orelatvstc eerges have bee publshed the lterature. 6,7 Ths cludes the eergy for the hghest vbratoal v = state, whch s oly about cm below the D + H + dssocato lmt. I Table 3 we compare the ECG varatoal eerges for all 3 states 43 wth the values of Hlco et al. 7 As oe ca see, the values agree very well. The agreemet s cosstetly very good for all the states calculated. The wave fuctos for all 3 states were used to calculate the average teruclear dstaces ad the average dstaces betwee the ucle ad the electro. Also, averages of the squares of the dstaces were calculated. The results are show Table 3. As ca be expected, the average teruclear dstace creases wth the rsg level of exctato. Ths crease becomes more promet at the vbratoal levels ear the dssocato threshold. For example, gog from v =to v = the average teruclear dstace creases more tha - fold from.95 to 8.6 au. I the v = state the HD + o s almost dssocated. However, the most strg feature that becomes apparet upo examg the results s a sudde crease of the asymmetry betwee the deutero electro ad proto electro average dstaces above the v = 0 exctato level. I levels up to v = 0 there s some asymmetry of the electro dstrbuto wth the p e dstace beg slghtly loger tha the d e dstace. For example, the v = 0 state the d e Table 3. Total Eerges, Expectato Values of the Deutero Proto Dstace, r d p, the Deutero Electro Dstace, r d e, ad the Proto Electro Dstace, r p e, ad Ther Squares for the Vbratoal Levels of HD + at the Rotatoal Groud State a v E, Bub et al. 43 E, ref 7. r d p r d e r p e r d p r d p r p e D atom b a All quattes are atomc uts. b D atom the groud state. 68 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

34 Chemcal s Table 4. Comparso of All Pure Vbratoal Trasto Eerges ( cm ) of HD Calculated wth the ECG No-BO Approach ad wth (rel) ad wthout (orel) Icludg the α Relatvstc Correctos wth the Trastos Reported by Pachuc ad Komasa (PK) 0 a trasto v+ v E orel E orel E v+ v rel E rel E v+ E v v v + Bub et al. PK Bub et al. PK expermet (4) (4) (7) (5) (5) (50) (5) (5) (49) (5) (5) (6) (0) (0) (7) (0) (0) (36) (0) (0) (0) (0) (30) (30) (30) (30) (50) (50) (50) (50) (50) (50) (70) (70) (70) (70) (70) (70) (70) (70) a The values paretheses dcate the estmated remag ucertaty due to fte bass sze used. The ECG o-bo results are derved from the wor of Bub et al. 49 The last colum lsts several trasto frequeces derved from expermetal data. average dstace s au ad the p e dstace s 5.56 au. The stuato becomes completely dfferet for the v = state. Here the p e dstace for ths state of.9 au s almost equal to the average value of the teruclear dstace, but the d e dstace becomes much smaller ad equals oly.306 au. It s apparet that ths state the electro s essetally localzed at the deutero ad the o becomes hghly polarzed. A aalogous stuato also occurs for the v = state. Here, aga, the p e average dstace s very close to the teruclear dstace whle the d e dstace s close to what t s a solated D atom. Oe ca say that the low vbratoal state the HD + o s covaletly boud, but the hghest two states t becomes oc. The secod example cocers the o-bo calculatos of all 7 rotatoless vbratoal eerges of the HD molecule. 8 For each state ECGs were used those calculatos. The total eerges were used to calculate the v v + trasto eerges, whch are preseted Table 4. I Table 4 the ECG trasto eerges obtaed wth ad wthout the relatvstc correctos of the order of α are compared wth the trastos determed usg the results of Pachuc ad Komasa (PK). 0 They performed ther calculatos usg the stadard approach where the potetal eergy curve was calculated frst ad, after beg corrected for the adabatc ad oadabatc effects, as well as relatvstc effects, t was used to calculate the vbratoal frequeces. The comparso show Table 4 cludes vbratoal frequeces calculated by the two methods wth ad wthout the relatvstc correctos. Frst let us focus o the orelatvstc trastos Table 4. As oe ca see, the ECG o-bo results agree very well wth the PK values. The dfferece for the trastos betwee low-lyg states, as well as for the states the mddle of the spectrum, s cosstetly less tha cm. As oe ca see from the comparso wth the expermetally derved trastos, the dfferece betwee the ECG o-bo results ad those of PK s much smaller tha the expected cotrbuto due the α 3 QED ad hgher order correctos. For the three top trastos the dfferece creases to about 0.0 cm. Whle the exact reaso for ths dscrepacy s ot mmedately clear, t s possble that t arses as a result of the perturbato approach used by PK ad ot accoutg for the fte-ucleus-mass effects for those two states as accurately as t s doe the drect varatoal o-bo approach. 0.. Datomcs wth Three Electros As stated, the applcato of the o-bo ECG approach for datomc molecules s curretly lmted because of the form of the bass set to pure vbratoal states ad to molecules wth σ electros. However, t s ot lmted wth respect to the umber of the electros the molecule has. Hece, ths s the frst accurate o-bo method avalable that allows performg calculatos of molecules wth more tha two electros. A example of o-bo ECG calculatos performed for a molecule wth more tha two electros s the wor o all fve pure vbratoal trastos of the 7 LH + o. 53 Up to ECGs for each state were used the calculatos. Table 5 shows the covergece of the trasto frequeces calculated wth ad wthout the relatvstc correctos wth the umber of the bass fuctos. Oe ca see that the thrd dgt after the decmal pot for all trastos expressed waveumbers s essetally coverged. The sgfcace of these calculatos les the fact that the pure vbratoal spectrum of the 7 LH + o has ot bee measured yet, ad the very accurate predctos of the vbratoal trastos geerated the calculatos ca gude the future expermet where such a measuremet s attempted Datomcs wth More tha Three Electros The major bottleec the o-bo calculatos wth allpartcle ECGs s ther! scalg wth the umber of detcal partcles (electros). As metoed before,! s the umber of permutatos the symmetry operator that eeds to be appled to each bass fucto to eforce the rght permutatoal symmetry of the wave fucto. For example, the 69 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

35 Chemcal s Table 5. Covergece of the Pure Vbratoal Trasto Eerges of 7 LH + Io Determed wth ad wthout the Icluso of the Leadg Relatvstc Correctos a v v bass sze E v NR E v NR E v REL E v REL b b b b b a All values are cm. The results are tae from ref 53. b Results obtaed by performg several addtoal cyclc optmzatos of the olear parameters. calculatos of the BH molecule there are 70 such permutatos. The BH molecule s the largest system ever calculated wth the o-bo ECG approach. 5 The am of the calculatos was to determe the dssocato eergy of ths system. The results are show Table 6 wth the ECG FNM results obtaed for the boro atom also cluded. As ca be expected, the eergy for the B atom coverges sgfcatly faster tha for BH. Wth 000 bass fuctos the orelatvstc o-bo eergy of B s essetally coverged wth fve sgfcat fgures whle the BH eergy s coverged wth four fgures. The better covergece for the B atom tha for BH assures that the BH dssocato eergy calculated as the dfferece betwee the total eergy of BH ad the sum of the eerges of the B ad H atoms s a lower boud to the true dssocato eergy of ths system. We should add that, eve wth oly 000 bass fuctos, the o-bo eerges of the B atom ad the BH molecule show Table 6 are the best varatoal upper bouds ever calculated for these systems. However, 000 bass fuctos s oly eough to esure covergece of two sgfcat fgures the dssocato eergy. Our best result for ths eergy s D 0 = cm (8 074 cm after addg the relatvstc correctos). A approxmate extrapolato to a fte umber of bass fuctos creases the dssocato eergy to D 0 = ± 50 cm. Ths result agrees wth the expermetal value recommeded by Bauschlcher et al. 3 of ± 0 cm. However, t s clear that order to match the expermetal accuracy wth the calculatos the eerges of both B ad BH wll have to be computed wth hgher accuracy ad much larger ECG bass sets approachg or eve exceedg fuctos.. HIGHLY ACCURATE BO MOLECULAR CALCULATIONS.. Hydroge Clusters I ths secto we show that t s possble to reduce the computatoal tme eeded for a ECG BO PES calculato by avodg costly optmzato of the ECG olear parameters at every PES pot ad stll mata hgh accuracy of the results throughout the whole PES. The approach developed for ths purpose wll be descrbed ad llustrated through ts applcato the calculatos of the potetal eergy curve (PEC) of the (H ) dmer the lear geometrcal cofgurato eepg the moomer at the froze teruclear separato of.4 bohr. A essetal part of a very accurate ECG calculato s growg the bass set from a small umber of fuctos to a large oe that assures the desred accuracy of the results. I the process of buldg the bass set s usually tated wth a set of ECGs geerated usg cotracto of bass sets of the moomers. 4 For (H ) ths s ϕ = φ φ,(h ),H j,h (8) where φ,h ad φ j,h are ECGs obtaed for a hydroge molecule. As metoed, the Gaussa ceters of φ,h ad φ j,h are placed at the same moomer or dfferet moomers to produce the so-called oc or covalet bass fucto. Next, the bass set s elarged by addg subsets of ECGs followed by ther partal or global optmzato depedg whch of these two procedures s more computatoally effcet the partcular case. As show Table 7, 95 at R H H = 6 bohr 99.6% of the bdg eergy s recovered wth the 5000 ECG bass set. I the calculatos reported ref 95 the procedure was used to geerate 5000 ECGs for (H ) for varyg termoomer dstaces. Wth that, whe movg to the ext PEC pot, oly the lear expaso parameters had to be recalculated. However, as PEC pots got further separated from the pot where the full bass set optmzato was performed (at the equlbrum), the procedure became creasgly less accurate. To remedy ths ad to reduce the error, 000 addtoal ECGs were geerated at each PES pot wth the FICI method ad added to the bass set. I ths way a 7000 ECG bass set was obtaed for each PEC pot. To verfy f the FICI-optmzed Table 6. Norelatvstc ad Relatvstc (E rel = E orel + α H MV + α H D ) Eerges of B ad BH au a bass sze E orel (B) E rel (B) E orel (BH) E rel (BH) D 0 orel D 0 rel a Values for D 0 are the correspodg dssocato eerges expressed cm. The results are tae from ref 5 wth the excepto of the last colum, D 0 rel, whch cotas corrected data (ref 5 has a error ths quatty). 70 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

36 Chemcal s Table 7. Covergece of the (H ) Iteracto Eergy, elv a bass sze teracto eergy exact b a The teracto eergy s determed wth respect to the exact eergy of two solated H molecules each wth the teruclear dstace equal to R H = bohr (E H = hartree 75 ). The calculatos are performed at the termoomer separato of 6 bohr. 95. b Best lterature value obtaed usg the moomer cotracto method wth a ECG bass set augmeted wth addtoal 400 ECGs ad extrapolated to the complete bass set lmt. 3 The estmated stadard devato of the eergy value s σ = K. addtoal set s capable of effcetly correctg the error, full optmzato of the 5000 ECG bass set was performed at some selected PEC pots. A comparso of the results obtaed those optmzatos 95 wth the eerges obtaed wth the 7000 fucto bass sets s show Table 8. Table 8. Comparso of the Total ad Iteracto Eerges of (H ) Obtaed wth 7000 ECGs (5000 ECGs Geerated wth the Shftg Procedure ad 000 Geerated wth the FICI Method) wth the Eerges Obtaed wth Fully Optmzed 5000 ECGs a FICI FO FO R H H (bohr) E 7000 E 5000 ΔE E a Calculatos have bee performed at sx selected PEC pots cludg the R H H = 6 bohr pot. Total eerges, E 7000 ad E 5000, are hartrees, ad the total eergy dffereces, ΔE, teracto eerges, ad ΔE 5000, are elv. 95. Although the eerges calculated wth the full optmzato procedure are lower tha the eerges obtaed wthout reoptmzg the ECG olear parameters, the eergy dfferece s early costat (about 0.9 K) as the termoomer dstace creases beyod 8 bohr. Therefore, oe ca coclude that the absolute error of the FICI-optmzed PEC s just uder 0.5 K at R H H = 6 bohr ad sub-0.3 K for larger termoomer dstaces. Ths valdates the approach used the (H ) calculatos ad shows that the floatg ECGs are capable of represetg the groud state wave fucto of ths four-electro four-ceter molecule wth hgh accuracy... Molecules wth Oe Atom Other Tha Hydroge I ths secto we descrbe hgh-accuracy, state-of-the-art PEC ECG calculatos of the LH molecule. 80 The PEC cludes fte-mass adabatc correctos added to the BO eergy. The PEC accuracy s better tha 0.3 cm, showg oce aga that the varatoal method wth ECGs complemeted wth the use of the aalytc gradet s a very hgh accuracy approach for geeratg PEC/PESs for small molecules. The ECG calculato of the LH BO PEC of Tug et al. 80 started wth the 400 ECG bass set (see Table 3) bult usg the full optmzato approach at the equlbrum teruclear dstace (R L H = 3.05 bohr). Usg ths bass set, the bass sets for the adjacet PEC pots were geerated by applyg the Gaussa ceter shftg procedure. After shftg, the bass sets were fully reoptmzed usg the full optmzato approach. Ths procedure was repeated utl the target teruclear dstace of R L H = 40 bohr was reached. Smlarly to the (H ) ECG calculatos, a cocer oe may have s related to possble accumulato of error whch may arse at each PEC pot due to ts tal bass set ot beg geerated from scratch, but beg extrapolated from the prevous PEC pot. Such a accumulato would lely produce the largest error for the logest teruclear dstace ( the calculatos t was R L H = 40 bohr). I Table 9 the magtude of ths error s Table 9. Comparso of the Estmated ad Calculated LH BO Eerges at R L H = 3.05 ad R L H = 80 E, estmated 8 E, calculated 80 R L H = a R L H = a Calculated at R L H = 40 bohr. examed by comparg the eergy obtaed at 40 bohr wth the estmated accurate eergy obtaed as a sum of the atomc eerges (the BO eergy Table 9 at R L H = s the sum of the BO eerges of the L ad H atoms). As oe ca see, the error s oly about 0 6 hartree. Thus, o sgfcat accumulato of error had occurred. As metoed, the BO PEC geerated the calculatos of Tug et al. 80 was corrected for the adabatc effects to partally overcome the defcecy of the BO approxmato. At the equlbrum dstace the adabatc correcto lowers the depth of the PEC well by 0.7 cm. Because the correcto vares wth the teruclear dstace, the shape of the corrected PEC dffers slghtly from the PEC wthout the correcto. Ths tur affects the eerges of the vbratoal levels, as ca be see Table 0. I the calculatos of Tug et al. 80 these levels were obtaed usg Le Roy s Level 8.0 program. 33 I Table 0 we also compare the results from ref 80 wth recet orbtalbased calculatos of Hola et al. 34 The frst observato oe ca mae upo examg the results of Hola et al. show Table 0 s that they were uable to obta coverged values for the two hghest trastos ad had to resort to extrapolato to determe them. Secod, the root-mea-square devato (from expermet) for the best set of trastos obtaed by Tug et al., whch were geerated wth the BO ECG PEC corrected for the adabatc effects, s half those by Hola et al. It should be oted that the values by Hola et al. also cluded scalar relatvstc correctos. Noetheless, the comparso betwee the two sets s vald because those correctos have a eglgble effect o the calculated trastos. 7 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

37 Chemcal s Table 0. Comparso of Calculated ad Expermetal Trasto Eerges of 7 LH a Hola et al. 34 Tug et al. 80 v v b,h ΔE c,h ΔE d,h ΔE BO e,h ΔE (0.89) f (.3) f (.43) f (.70) f rms g a The calculated trastos clude those obtaed wth ad wthout the adabatc correcto ad are gve cm. ΔE s the dfferece betwee the expermetal ad the calculated trastos. The BO PEC cludes DBOC, MVD (mass-velocty ad Darw), ad oadabatc cotrbutos. 34 c The BO PEC cludes DBOC ad MVD cotrbutos. 34 d The BO PEC. 80 e The BO PEC cludes DBOC cotrbutos. 80 f Values paretheses are extrapolated vbratoal levels. 34 g The root-mea-square calculatos clude the vbratoal trasto up to v =0 v = 9. h Calculated trastos are compared wth the emprcal values derved by Coxo ad Dcso. 35 The ucertaty s smaller tha cm for v =0 5, smaller tha 0.00 cm for v = 6, ad smaller tha 0.0 cm for v 7. Let us ow exame how well Hola et al. s 34 ad Tug et al. s 80 calculatos reproduce the expermetal dssocato eergy. I Hola et al. s calculatos, the eergy at each PEC pot cludes three compoets,.e., the BO eergy, the dagoal adabatc correcto (DBOC), ad the relatvstc (mass-velocty ad Darw) correcto. The cotrbutos of the BO eergy, the DBOC, ad the relatvstc correcto to the BO dssocato eergy are ,.0, ad 0.5 cm, respectvely. Compared wth the results of Tug et al., Table shows that Hola et al. s results overestmate the BO cotrbuto by about 6.6 cm ad the adabatc correcto Table. Comparso of the BO Potetal Depth ad the Cotrbuto of the DBOC Calculated by Hola et al. ad Tug et al. a D e (BO) ΔDBOC Hola et al Tug et al ΔE a The ΔE s the dfferece of two calculatos. by about.3 cm. Eve though these two errors have opposte sgs ad partally cacel each other, there s stll a sgfcat dfferece of about 5.3 cm betwee Hola et al. s ad Tug et al. s results ad the expermetal value. Ths error ca be averaged the calculato of vbratoal levels. The dffereces of eerges from the orbtal calculato are compettve wth the ECG values. It should be oted that the result of Tug et al. s vrtually detcal to the expermetal value, whch has a estmated root-mea-square devato of 0.6 cm ad a maxmum devato of 0.79 cm. As the dssocato eergy s usually a good dcator of the accuracy of the calculatos, we compare the values of ths eergy obtaed wth dfferet methods wth the expermetal eergy Table. The comparso shows that the ECG value obtaed by Tug et al. 80 best matches the expermet. Table. Comparso of Values of the LH Dssocato Eergy Calculated wth Dfferet Methods wth the Expermetal Eergy authors D e (ev) method L ad Paldus CCSD-[4R]/cc-pVQZ Ludsgaard ad.49 full CI/all-electro Rudolph 4 Gadeá ad Leger 4.5 CCSD(T)/all-electro Bade et al FC LSE Hola et al a MR-CISD + Q p /cc-pwcvxz (X = Q, 5, 6) Tug et al b ECG Stwalley ad Zeme c expermet (± ev) a7 LH. Ths value cludes BO, DBOC, ad MVD (mass-velocty ad Darw) cotrbutos. LH. Ths value cludes BO ad DBOC cotrbutos. c Expermetal value for 7 LH. Aother good test of the qualty of the calculatos s to exame the results obtaed at the equlbrum teruclear dstace for the system. A large-scale CI [9s8p6df] performed for LH by Bedazzol et al. 36 gave the eergy of hartree, whle the GFMC calculato of Che et al. 37 yelded (5) hartree. CCSD[T]-R coupled-cluster method wth lear r terms 54 gave E = hartree. Ths eergy s close to the eergy of hartree obtaed wth the teratve-complemet-teracto method by Naatsuj. 38 The frst ECG calculatos performed wth 00 ECGs by Cece et al. 39 gave the eergy of hartree. Ths result, as well as other results obtaed wth larger ECG bass sets by Cece et al. 8 were used to mae a extrapolato to the fte bass set lmt, whch yelded (5) hartree. Ths eergy s cosstet wth ad close to the result of hartree obtaed by Tug et al. 80 wth a bass of 400 ECGs. Whe the ECG results are compared wth the MR- CISD calculatos of Hola et al., 34 oe becomes puzzled at a dscrepacy. Hola et al. s eergy obtaed at R = bohr of hartree s 39 μhartrees below the best ECG value obtaed at R = 3.05 bohr ad about 50 tmes outsde ts estmated ucertaty. The reaso for ths dscrepacy s uclear at the momet. It s possble that t s due to a overestmated value of the sze-cosstecy correcto ad (or) flaws the extrapolato procedure used Hola et al. s calculatos. I cocluso, curretly oly ECG-based approaches ca provde relable data ad at the same tme acheve mcrohartree-level accuracy the LH calculatos. 7 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

38 Chemcal s The relatvstc correcto to the dssocato eergy of 0.5 cm (or ev) calculated by Hola et al. 34 dcates that the accuracy of the calculatos of the vbratoal trastos caot be further mproved by oly cludg the relatvstc (ad oadabatc) correctos to the BO PEC corrected for the adabatc effects. It wll also eed to volve mprovg the qualty of the PEC BO eerges. Oe should be able to acheve the accuracy of at least 0 7 hartree calculatg these BO eerges order for the relatvstc effects to start to matter. Such a crease of the accuracy s certaly wth the reach of ECG calculatos..3. Dagoal Adabatc Correctos to the BO Eergy for Molecules Cotag up to Three Nucle I ths secto we wll ot dscuss specfc calculatos, but rather descrbe how to perform calculatos of the dagoal adabatc correctos (DBOC) whe employg floatg ECGs. Icludg effects beyod the BO approxmato the calculatos volves the calculato of the gradet of the wave fucto wth respect to the uclear coordates. Ufortuately, floatg ECGs do ot explctly deped o the uclear coordates, ad therefore drect methods to retreve that depedecy eed to be devsed. I ths secto a approach whch s capable of approxmately accoutg for the mplct depedecy of the floatg ECG bass fuctos o the uclear coordates calculatg the DBOC s descrbed mathematcal terms. The method was frst troduced by Cece et al. 59 ad t s based o the Gaussa product theorem (GPT). The GPT states that the product of two Gaussas s a Gaussa. I addto, t s possble to relate the olear parameters of the two Gaussas the product to the oes of the Gaussa beg the product of the two. Ufortuately, to our owledge, there s o GPT equvalet for floatg ECGs ad some approxmatos eed to be made to represet a floatg ECG as a product of other floatg ECGs. Applcato of the GPT to datomcs s straghtforward ad volves mmal approxmato. As a testamet of ths, there are above-revewed calculatos of the LH molecule by Tug et al. 80 ad calculatos o other datomcs by Cece et al., 59 whch have prove a hgh accuracy of the GTP-based procedure. More problematc s the exteso of the GPT algorthm to tratomcs ad, more specfcally, ther lear cofguratos. I ths secto, we descrbe a geeral dervato of a GPT-based approach to calculate the DBOC. The approach reduces to the procedure of Cece et al. for the datomc case, but t also allows for the calculato of the DBOC for tratomcs eve ther lear cofguratos. The calculato of the adabatc correctos s performed by usg the Bor Hady formula: E ad = Ψ K M α= α α = 3 Q α Ψ (9) where K s the total umber of ucle the molecule, M α are the uclear masses, ad α deote the Cartesa drectos of the uclear coordate dsplacemets. I the above approach, the partal secod dervatve wth respect to uclear coordate Q α s evaluated umercally by calculatg the groud state BO wave fucto at two uclear cofguratos, where Q α s dsplaced bacward ad forward by a small dstace, Q + α ad Q α, accordg to the followg formula: 59 Ψ Q Ψ = ( S ) Δ Q α (0) where ΔQ = Q + α Q α, ad S = Ψ(Q + α ) Ψ(Q α ). Let us ow descrbe how the olear parameters of the wave fucto at the shfted uclear cofgurato ca be approxmated. As a example let us cosder H + 3. Let us frst troduce three 3- dmesoal vectors, Q, Q, ad Q 3, cotag the coordates of the three ucle of H + 3. Next, we troduce three two-electro oc fuctos, ϕ I, ϕ II, ad ϕ III, that have the followg shfts of the Gaussa ceters: Q s = Q () where s equal to ether,, or 3. I ths dervato the subscrpt deotes the olear parameters of the ewly troduced, atom-cetered fuctos ϕ I, ϕ II, ad ϕ III, whle the subscrpt deotes the olear parameters of the fucto we are shftg. The ϕ I, ϕ II, ad ϕ III fuctos are called oc because eq both Gaussa ceters cocde wth the posto of a ucleus. Wth that, we ca approxmate ay floatg ECG bass fucto (ϕ ) by a product of the three oc fuctos troduced above as ϕ = ϕϕ ϕ = exp[ ( rar + ras sas )] I II III 3 = () where A s A I 3. By equatg le terms eq, oe gets 3 A = A = 3 As = A s = sas = sas 3 = (3) (4) (5) where s s the 3-dmesoal (sx-dmesoal for H 3 + ) Gaussa shft vector. By assumg that A = a A, eqs 3 ad 4 become 3 a = = 3 a s = s = (6) (7) Notce that eq 7 s actually composed of two depedet equatos, oe for the x coordates ad oe for the y coordates. For olear geometres of H 3 + eqs 6 ad 7 are suffcet to predct the floatg-ecg shft vectors for the ew geometrcal cofgurato of the ucle. The procedure volves the followg steps.. For each floatg ECG, the three auxlary fuctos, ϕ I ϕ III, are costructed by usg the H 3 + uclear coordates as the shft vectors as show eq.. Three depedet equatos eqs 6 ad 7 are solved to obta the values of the a, a, ad a 3 parameters. 73 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

39 Chemcal s 3. The ew Gaussa shft vector s s computed drectly from eq 7 for the ew, chaged H 3 + geometry,.e., Q ± (/ )ΔQ. However, eqs 6 ad 7 are ot depedet whe the H 3 + geometry becomes lear. The lear case s dealt wth by mag use of oly those equatos eqs 7 ad 5 whch do ot zero out ths stuato. I addto, eq 5 eeds to be smplfed (approxmated) by decouplg the terms correspodg to dfferet electros order to mae eqs 3 5 specfc to each Gaussa ceter. I the decouplg we assume that the off-dagoal terms A are small compared to the dagoal terms. Ths turs eq 5 to a equato that costras the squares of the x-coordates of the Gaussa ceters to the square of the correspodg x-coordate of the α ucleus: a + a + a3 = (8) ax+ ax+ a3x3 = x α (9) ax + ax + a3x3 = x α (30) where t s assumed that the lear H + 3 les o the x-axs. Wth that, eve for a lear H + 3 cofgurato, the system of equatos, eqs 3 5, s osgular ad ca be solved. We should pot out that, strctly speag, eq 5 (ad tur also eq 30) s a ad hoc codto as t formally eforces the product of the three uclear-cetered ECGs to have the same orm as the orgal Gaussa.. CALCULATING MOLECULAR PROPERTIES WITH ECGS: THE BO CASE I addto to the total eergy, some molecular propertes calculated wth ECGs have also bee a focus of recet studes. Importat propertes, such as the quadrupole momet, 44 dpole ad quadrupole polarzabltes, 45 oe-electro desty, 9 electrc feld gradet at the ucle, 46 ad post-bo correctos to the total eergy 6,03, have bee successfully calculated wth the ECG bass set eve though ECGs are ot capable of satsfyg the Kato cusp codto 5 (see secto.6). The ECGs decayg faster at large terpartcle dstaces tha expected by the asymptotc codtos for the exact solutos of the SE usually have small effects o the eergy, but may more strogly affect the calculatos of some propertes. BO calculatos performed wth ECGs may eed to volve thousads of bass fuctos (from about 000 for two-electro systems 76 to over 5000 for systems havg more tha four partcles 95,80 ), f covergece of 0 sgfcat dgts the eergy s targeted. I ths revew we lmt ourselves to descrbg oly oe example of property calculatos. The example cocers the electrc feld gradet (EFG) at the uclear postos the H molecule, as well as ts sotopologues volvg deuterum... EFG at the Nucle ad the Deuterum Quadrupole Costat The EFG s a mportat quatty for may spectroscopes: molecular beam resoace, uclear magetc resoace, 47 uclear quadrupole resoace, 48 Mo ssbauer spectroscopy, 49 ad electro paramagetc resoace, 50 just to meto a few. These techques explot specfc uclear characterstcs (.e., dstct sotopes, decay of excted uclear states, uclear sp trastos, etc.), whch may volve the couplg betwee the quadrupole momet of a ucleus wth the gradet of the electrc feld due to the electro ad uclear charges at the posto of that ucleus. The couplg costat of the uclear quadrupole/efg teracto s defed by frst-order perturbato theory as γ = (e Q q )/(4πε 0 h), where q s the vbratoally averaged asotropc part of the EFG evaluated at the ucleus whose quadrupole momet s Q, e ad h are the electro charge ad the Plac costat, respectvely, ad 4πε 0 s the usual costat that appears Coulomb s law. γ HD ad D has bee obtaed wth four to fve dgt accuracy by Ramsey ad coworers from molecular-beam magetc resoace measuremets. 5,5 The fudametal costat Q ca be estmated by combg the measured γ wth a calculated value of q. It s mportat to stress that the accuracy of the calculatos of q determes the accuracy of the determato of the hyperfe quadrupole teracto. The ECG calculato of q does ot come wthout complcatos, as EFG matrx elemets over bass fuctos of the type show eq 48 cota the secod dervatve of the error fucto. 46 I Fgure a plot of the dfferece betwee the EFG calculated at the ucle of the H molecule obtaed by Fgure. Dfferece betwee the feld gradet calculated wth ECGs ref 46 ad the oe calculated wth the Kołos Wolewcz bass fuctos ref 53. Pavaello et al. 46 ad the prevously best lterature results 53 for q as a fucto of the H H teruclear dstace s show. There are two mportat observatos to be made about Fgure. The frst oe s that the U-shaped tred of the curve seems to suggest that the Kołos Wolewcz-type bass set used by Red et al. s lely ot large eough. I addto, the curve shows that the values calculated by Red et al. are subject to some umercal stablty. Namely, there are some q values whch seem too hgh ad some whch seem too low. I Table 3 the values of the deutero quadrupole costat Q are derved from the vbratoally averaged q value ad the expermetally measured quadrupole splttg. The value of Q of (30) fm derved ref 46 s so far the most accurate to date. The ucertaty ths value s almost etrely due to the expermetal ucertaty of the quadrupole teracto costat, γ (ad perhaps also to the oadabatc effects). 74 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

40 Chemcal s Table 3. Calculated Deutero s Quadrupole ( fm) from D ad HD the State v = 0 ad J =, a ECG46 Bshop Cheug76 Kołos Wolewcz (30) (30) 0.86(5) 0.86(7)b 0.860(5) 0.860(7)b (40) J Bographes D HD 0.875(6)b a All values are calculated usg the expermetal value eqq/h = ad Hz for J = ad J =, respectvely, for D ad Hz for HD the J = state.5,53 Maxmum ucertates ( paretheses) are estmated based o the expermetal stadard devatos. bucertates corrected by addg the cotrbuto from expermetal radom error. Sergy Bub receved hs B.S. ad M.S. degrees physcs from Taras 3. SUMMARY Shevcheo Natoal Uversty of Kyv, Urae. He was awarded hs I ths wor we have attempted to revew the curret state of efforts to use ECGs calculatos of small atomc ad molecular systems wth very hgh accuracy. Such hgh accuracy ca be acheved ether by performg very precse BO calculatos ad correctg the results for adabatc ad oadabatc effects or by treatg the ucle (ucleus for a atom) ad electros o equal footg ad explctly cludg ther motos the Hamltoa ad the wave fucto. For hgh accuracy, the results have to be also corrected for the relatvstc ad QED effects. It s show that by usg large bass sets of ECGs ad by varatoally optmzg ther olear parameters wth a method based o the aalytcal eergy gradet t s possble to determe the eerges of groud ad excted states of these systems wth a accuracy approachg the accuracy of the most precse expermetal measuremets. The ECGs seem to be capable of very well descrbg the oscllatory ature of the wave fucto at ay exctato level. I movg forward wth the developmet of ECG techques for very accurate BO ad o-bo atomc ad molecular calculatos, several drectos eed to be cosdered. Future wor eeds to clude extedg the o-bo approach preseted ths revew to datomc states wth rotatoal quatum umbers hgher tha zero. It wll also eed to clude extedg the o-bo approach to molecules wth more tha two ucle. Furthermore, the wor, whch s curretly already progress, eeds to cocer the developmet of BO ad obo tools to calculate the leadg QED effects groud ad excted molecular states. Wth that, t s hoped that the remag dffereces betwee the results of the theoretcal calculatos ad the data acqured expermets wll be further arrowed. Ph.D. physcs at the Uversty of Arzoa 006, carryg out research uder the supervso of Ludw Adamowcz. After spedg two years as a postdoc at Arzoa ad oe year at Quatum Chemstry Research Isttute (Kyoto, Japa), he the wet to Vaderblt Uversty to wor wth Prof. Kalma Varga. Hs research terests clude explctly correlated methods for hghly accurate treatmet of varous few-body systems, o-bor Oppehemer quatum chemstry, quatum dyamcs molecules ad aostructures, ad the teracto of strog laser fields wth matter. Mchele Pavaello obtaed the Laurea degree 004 at the Uversty of Psa the PCM group worg o NMR propertes of solutes lqud crystals uder the supervso of Beedetta Meucc. From 004 to 00 he was a graduate studet at the Uversty of Arzoa the group of Ludw Adamowcz. There he came cotact wth explctly correlated methods quatum AUTHOR INFORMATION chemstry. A Mare Cure fellowshp too hm to the Neugebauer Correspodg Author *E-mal: sergy.bub@vaderblt.edu (S.B.); m.pavaello@ rutgers.edu (M.P.); ludw@u.arzoa.edu (L.A). group Lede (Netherlads) from 00 to 0, explorg desty Notes September 0, he has bee a assstat professor at Rutgers The authors declare o competg fiacal terest. Uversty, Newar. parttog techques ad charge trasfer pheomea. Sce 75 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79

41 Chemcal s admtted to a Ph.D. program at UW ad the Isttute of the Orgac Chemstry of the Polsh Academy of Sceces (PAN), he fially made gaful employmet the Calormetry Departmet of the Isttute of Physcal Chemstry (PAN) as a research assstat. I hs free tme at PAN, he wored o hs PhD Quatum Chemstry uder supervso of Prof. Adrzej Sadlej (977). I 980 he joed the group of Prof. Ed McCullough at Utah State Uversty as a postdoctoral fellow worg o the umercal-orbtal MCSCF method for datomcs. I 983 he moved to the Uversty of Florda as a postdoctoral fellow wth Prof. Rod Bartlett, where he wored o the aalytcal gradets for the CC method, o the CC method wth umercal orbtals, ad o the optmal vrtual orbtal space (OVOS) method. I 987 we was appoted a assstat professor the Chemstry Departmet at the Uversty of Arzoa (UA). He s curretly a professor the Departmet of Chemstry ad Bochemstry ad the Departmet of Physcs at UA. He has publshed over 500 papers whch ecompass hs curret research terests of the developmet of very accurate methods for atomc ad molecular quatum-mechacal calculatos wth ad wthout assumg the Bor Oppehemer approxmato usg explctly correlated methods ad multreferece coupled cluster methods. He also develops methods for electro ad eergy trasport bomolecules. Applcato studes performed hs lab clude wors o aos, fullerees, ad aotubes, molecular complexes volvg uclec acd bases, etc. Most of the applcato wors have bee performed collaborato wth expermetal groups across the globe. We-Cheg Tug receved hs B.S. degree chemstry from Natoal Tawa Uversty 00. He s curretly a graduate studet uder the drecto of Prof. Ludw Adamowcz at the Uversty of Arzoa. Hs research terests focus o very accurate varatoal Bor Oppehemer quatum-mechacal calculato of molecules usg explctly correlated Gaussa bass fucto wth floatg ceters. ACKNOWLEDGMENTS The authors tha the Natoal Scece Foudato for partal support of ths wor. M.P. acowledges support from the start-up fudg by the Departmet of Chemstry ad the office of the Dea of FASN of Rutgers Uversty, Newar. The wor of S.B. was supported part by NSF Grat CMMI REFERENCES Keeper Laye Sharey, curret doctoral caddate at the Uversty of Arzoa, graduated from the Uversty of Arzoa 009 wth a udergraduate degree chemstry ad mathematcs. She s a member of the Adamowcz group ad has partcpated a umber of academc publcatos wth the group. Her doctoral studes ceter o chemcal physcs. She s the recepet of a NSF graduate research fellowshp (0). Her wor cludes o-bor Oppehemer calculatos of atoms ad molecules. () Hylleraas, E. A. Z. Phys. 99, 54, 347. () Boys, S. F. Proc. R. Soc. Lodo, Ser. A: Math. Phys. Sc. 960, 58, 40. (3) Sger, K. Proc. R. Soc. Lodo, Ser. A: Math. Phys. Sc. 960, 58, 4. (4) Rychlews, J. Explctly Correlated Fuctos Molecular Quatum Chemstry. I Advaces Quatum Chemstry; Sab, J. R., Zerer, M. C., Bra das, E., Wlso, S., Marua, J., Smeyers, Y., Grout, P., McWeey, R., Eds.; Elsever: New Yor, 998; Vol. 3, p 73. (5) Suzu, Y.; Varga, K. Stochastc Varatoal Approach to QuatumMechacal Few-Body Problems; Lecture Notes Physcs; Sprger: Berl, 998. (6) Rychlews, J. Explctly Correlated Wave Fuctos Chemstry ad Physcs: Theory ad Applcatos; Progress Theoretcal Chemstry ad Physcs; Kluwer: Dordrecht, The Netherlads, 003. (7) Bub, S.; Cafero, M.; Adamowcz, L. Adv. Chem. Phys. 005, 3, 377. (8) Stae, M.; Ke dzera, D.; Mols, M.; Bub, S.; Barysz, M.; Adamowcz, L. Phys. Rev. Lett. 006, 96, (9) Stae, M.; Ke dzera, D.; Bub, S.; Adamowcz, L. Phys. Rev. A 007, 75, (0) Stae, M.; Ke dzera, D.; Bub, S.; Adamowcz, L. Phys. Rev. Lett. 007, 99, () Stae, M.; Ke dzera, D.; Bub, S.; Adamowcz, L. J. Chem. Phys. 007, 7, () Stae, M.; Komasa, J.; Ke dzera, D.; Bub, S.; Adamowcz, L. Phys. Rev. A 008, 77, (3) Stae, M.; Komasa, J.; Ke dzera, D.; Bub, S.; Adamowcz, L. Phys. Rev. A 008, 78, For Ludw Adamowcz t all started 973 wth a M.S. degree Quatum Chemstry from Warsaw Uversty (UW) obtaed uder the supervso of Profs. Kolos ad Pela followed by a year as a assstat researcher uder the supervso of Prof. Joaa Sadlej. After several moths of uemploymet ad three faled attempts to be 76 dx.do.org/0.0/cr0049d Chem. Rev. 03, 3, 36 79