Darwin and mass-velocity relativistic corrections in non-born-oppenheimer variational calculations

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1 THE JOURNAL OF CHEMICAL PHYSICS 5, Darw ad mass-velocty relatvstc correctos o-bor-oppehemer varatoal calculatos Darusz Kedzera Departmet of Chemstry, Ncholas Copercus Uversty, ul. Gagara 7, PL Toruń, Polad Moka Stake Isttute of Physcs, Ncholas Copercus Uversty, ul. Gagara 7, PL Toruń, Polad Sergy Bub a Departmet of Physcs, Uversty of Arzoa, Tucso, Arzoa 857 ad Departmet of Chemstry, Uversty of Arzoa, Tucso, Arzoa 857 Mara Barysz Departmet of Chemstry, Ncholas Copercus Uversty, ul. Gagara 7, PL Toruń, Polad Ludwk Adamowcz Departmet of Chemstry, Uversty of Arzoa, Tucso, Arzoa 857 ad Departmet of Physcs, Uversty of Arzoa, Tucso, Arzoa 857 Receved 6 February 006; accepted 8 Jue 006; publshed ole 3 August 006 The Paul approach to accout for the mass-velocty ad Darw relatvstc correctos has bee appled to the formalsm for quatum mechacal molecular calculatos that does ot assume the Bor-Oppehemer BO approxmato regardg separablty of the electroc ad uclear motos molecular systems. The correctos are determed usg the frst order perturbato theory ad are derved for the o-bo wave fucto of a datomc system expressed terms of explctly correlated Gaussa fuctos wth premultplers the form of eve powers of the teruclear dstace. As a umercal example we used calculatos of the trasto eerges for pure vbratoal states of the HD o. 006 Amerca Isttute of Physcs. DOI: 0.063/.363 I. INTRODUCTION I order to acheve quatum mechacal calculatos of small molecular systems the accuracy matchg that of hgh-resoluto spectral measuremets ot oly does oe eed to be able to accurately descrbe electroc correlatos but also to accout for uclear moto ad couplg betwee uclear ad electroc motos. Thus, t s very desrable to depart from the Bor-Oppehemer BO approxmato regardg the separablty of the electroc ad uclear motos. Moreover, oe also eeds to clude relatvstc correctos as ther cotrbuto to the total eergy becomes qute otceable whe a comparso wth accurate expermetal data s made. I recet years we have bee volved developg a approach to perform quatum mechacal orelatvstc molecular calculatos wthout the BO approxmato. 8 There have also bee works of others ths area see, for example, Refs. 9 ad refereces there. The cetral part of our approach has bee the use of dfferet forms of correlated Gaussa fuctos that are explctly depedet o the dstaces betwee the partcles ucle ad electros formg the system uder cosderato. I partcular, we used correlated Gaussas wth premultplers the form of eve powers of the dstace betwee frst ad secod partcles usually ucle. We have demostrated that wth such fuctos oe ca acheve very hgh accuracy groud- ad excted-state calculatos of datomc systems wth two or more electros. 3 6,0, 4,6 8 The hgh accuracy s facltated by the varatoal optmzato of the wave fucto that volves aalytcal frst dervatves of the eergy wth respect to olear parameters of the Gaussas. I geeral, t s ot possble to take the Drac relatvstc Hamltoa ad separate t to orelatvstc ad relatvstc parts. The smplest ad most tradtoal way to calculate the relatvstc effect atomc ad molecular systems s based o the Paul approxmato. It provdes a framework for descrbg a quatum partcle wth the accuracy of the order of, where s the fe structure costat. To get a more accurate descrpto of a quatum system gog beyod the Paul approxmato oe ca use the Bret-Paul equato, whch explctly cludes operators descrbg the orbt-orbt ad sp-orbt teractos, as well as other twopartcle magetc teractos. However, sce the Bret- Paul equato s ot completely varat wth respect to the Loretz trasformato, a approxmato s troduced the calculato of the relatvstc effects. The Paul approxmato descrbes a state of a quatum partcle represeted by a two-compoet wave fucto, whch s a egefucto of the orelatvstc Hamltoa. I such a approach the relatvstc effects ad ther correspodg operators must be treated as perturbatos ad determed as the frst order correctos to the orelatvstc eergy. Ths s a serous defcecy of the approach based o the Paul approxmato. Ths defcecy s the result of sgulartes that appear the operators represetg the relaa Electroc mal: bub@emal.arzoa.edu /006/58/084303//$3.00 5, Amerca Isttute of Physcs Dowloaded 0 Apr 0 to Redstrbuto subject to AIP lcese or copyrght; see

2 Kedzera et al. J. Chem. Phys. 5, tvstc correctos. Extedg the applcablty of the Paul approxmato to systems wth more tha oe partcle ca be acheved provded that the Darw cotact term, all Coulombc teractos volvg the partcles the system are cluded. Also the orelatvstc wave fucto used the calculatos must gve fte expectato values for all relatvstc correctos volved the Paul approxmato. I ths work we have calculated relatvstc correctos usg the frst order perturbato theory ad the Paul approxmato. I the calculatos we used the orelatvstc wave fuctos expressed terms of explctly correlated Gaussa fuctos ad obtaed wthout assumg the Bor- Oppehemer approxmato. Such wave fuctos ca be geerated for datomc molecular systems wth more that oe sgma electro wth the approach we have developed. It s mportat to meto that there were prevous calculatos cocerg H by Moss ad Valezao 3 where electroc relatvstc correctos were determed usg wave fuctos obtaed oadabatc calculatos. However, the approach of Moss ad Valezao was restrcted to oe-electro datomcs ad the possblty of ts exteso to systems wth more electros seems ulkely. The eed to perform hghly accurate quatum mechacal calculatos o small molecular systems s motvated by the progress the hgh resoluto gas-phase measuremets of such molecular quattes as rovbratoal ad electroc exctato eerges, electro afftes, ozato potetals, bod dssocato, ad atomzato eerges that acheve the precso exceedg a teth or eve a hudredth of a wave umber. Ths ofte presets a challege to quatum mechacal studes of molecular systems because, order to reach such a accuracy, ot oly orelatvstc wave fucto must be computed wth very hgh accuracy but also, eve for small systems, the relatvstc effects have to be take to accout. As wll be descrbed later ths work, our o-bo approach s based o separatg the ceter-of-mass moto of the system from the teral moto. The separato s acheved by trasformg the laboratory Cartesa coordate system to a ew set of coordates, the frst three of whch are the laboratory ceter-of-mass coordates ad the rest are teral Cartesa coordates defed wth respect to oe of the ucle called the referece partcle. Such a choce does ot restrct the types of the molecular systems that ca be calculated. Molecular systems wth two ad more ucle ca be cosdered ths framework. The approach developed ths work for calculatg the relatvstc correctos to the o-bo eergy wth the Paul approxmato s appled to all vbratoal states of the HD o wth the zero total agular mometum. Such states are usually called vbratoal states, although f the Bor- Oppehemer approxmato s ot assumed, the vbratoal moto s coupled wth the electroc moto ad the vbratoal quatum umber s ot a good quatum umber. H ad ts sotopomers are the smplest model systems that show some terestg o-bo effects whe excted to vbratoal states ear the dssocato threshold. As t had bee kow before ad also show our recet o-bo calculatos of average terpartcle dstaces, 4,6 the hghest two vbratoal levels of HD ad HT, the electro charge desty s strogly polarzed towards the deutero ad the systems ca be descrbed as a proto teractg wth ether a D atom HD or a T atom HT. Ths very strog oadabatc effect dfferetates the behavor of the H o, where the hghest vbratoal states the electroc desty s symmetrcally dstrbuted at the protos, from the asymmetrc HD ad HT os. Due to these dffereces, t was terestg to see how the lack of the symmetry the electroc charge dstrbuto HD the hghest vbratoal states affects the relatvstc cotrbutos to the eergy, partcularly those whch are expected to be sestve to such a effect. II. THE METHOD USED IN THE CALCULATIONS The total orelatvstc Hamltoa for a system wth N partcles ucle ad electros the laboratory Cartesa coordate system has the followg form: N Ĥ tot = = N M R = N j Q Q j R j, wth the masses, charges, ad postos of the partcles formg the system deoted as M, Q, ad R R =R,R,...,R N, where deotes vector trasposto, respectvely a datomc system the frst two partcles are the ucle ad the rest are electros. The laboratory frame Hamltoa cludes the ketc eergy operator for each partcle ad Coulombc teractos betwee each par of the partcles. R j =R j R are terpartcle dstaces. I the frst step we trasform the Hamltoa by separatg the ceter-of-mass moto, thereby reducg the N-partcle problem to a N = pseudopartcle problem descrbed by the teral Hamltoa Ĥ. I ths trasformato the laboratory Cartesa coordate system s replaced by a system whose frst three coordates are the laboratory coordates of the ceter of mass r 0 ad the remag 3 coordates are the Cartesa coordates the teral coordate system whose org s placed at the heavest ucleus partcle wth mass M called the referece partcle. The other partcles are referred to the referece partcle usg the Cartesa posto vectors r defed as r =R R. The teral Hamltoa Ĥ s where Ĥ = = q0 q Vr = = r m r = j q q j. j r j M r rj Vr, The separato of the teral Hamltoa ad the Hamltoa of the moto of the ceter of mass s exact. The teral Hamltoa descrbes pseudopartcles wth charges q =Q ad reduced masses m =M M /M M movg the sphercally symmetrc potetal of the charge of the referece partcle. The motos of the pseudopartcles are coupled through the mass polarzato term j /M r rj ad through the Coulombc terac- 3 Dowloaded 0 Apr 0 to Redstrbuto subject to AIP lcese or copyrght; see

3 Relatvstc correctos o-bor-oppehemer J. Chem. Phys. 5, tos depedet o the dstaces of the pseudopartcles from the cetral charge, r =r, ad ther relatve dstaces, r j =R j R =r j r. I the calculato of the mass-velocty MV ad the Darw D relatvstc effects, we start wth respectve Hamltoas the laboratory coordate frame R, Ĥ MV = N 8 = Ĥ D = N N 8 = j 3 M 4 R, 4 M Q Q j R. 5 R j Upo the trasformato of the laboratory coordate system to the teral system, the Darw Hamltoa 5 separates to a term depedet o the posto vector of the ceter of mass the laboratory frame, r 0, ad a term depedet o the teral coordates, r=r,r,...,r, where Ĥ D r,r 0 = Ĥ D r 0 Ĥ D r, Ĥ D r 0 = 4 6 m r0 Vr =0, 7 0 because Vr s depedet of r 0, ad where Ĥ D r = 8 q 0q r = M M r = j r M q q j. r j The Darw correcto ca be calculated ether drectly usg the operator 8, Ĥ D I r=ĥ D r we wll call t here the frst approach, or usg a operator obtaed from 8 by applyg the Posso equatos the secod approach, 8 r = 4r, r r = r rj = 4r j. 9 j r j Ths results the Darw Hamltoa the followg form: Ĥ II D r = q 0 q r M = j = M q q j r j. 0 M I the preset work we used both Darw Hamltoas, Ĥ D I r ad Ĥ D II r, the calculatos. Ths was doe to make sure that the algorthm for calculatg the Darw correctos was correctly mplemeted. Upo the trasformato of the coordate system R r 0,r the mass-velocty Hamltoa ca be represeted as a sum of three terms, Ĥ MV r,r 0 = Ĥ MV r 0 Ĥ MV r Ĥ coupl MV r 0,r, where the term Ĥ MV r relevat to the preset calculatos of the relatvstc cotrbuto to the teral eergy has the form Ĥ MV r = 8 M 3 = r 4 = 3 r M 4. The last term Eq., Ĥ coupl MV r 0,r, descrbes relatvstc couplg betwee the moto of the ceter of mass ad the teral moto. Ths effect s ot cosdered our calculatos as we assume that the system as a whole s at rest,.e., the ceter of mass s ot movg. The calculato of the relatvstc correcto to the eergy of the teral moto of the system s performed for each state usg the frst order perturbato theory as the expectato value of the Hamltoa represetg the teral mass-velocty ad Darw cotrbutos, Ĥr = Ĥ MV r Ĥ D r. 3 I our works cocerg o-bo calculatos o small datomc molecular systems 3 6,0, 4,6 8 we used the explctly correlated Gaussas ECGs volvg fuctos wth preexpoetal multplers cosstg of the teruclear dstace r rased to a o egatve eve power m k, k = r m k exp ra k I 3 r = r m k exp rā k r, 4 where symbol Ā k deotes the Kroecker product Ā k =A k I 3, ad I 3 s 33 detty matrx. The above fucto s a oe-ceter correlated Gaussa wth expoetal coeffcets formg the symmetrc matrx A k. I 3 Eq. 4 s the 3 3 detty matrx. k are rotatoally varat fuctos as requred by the symmetry of the teral groud-state problem descrbed by the Hamltoa. The presece of r k m factor 4 shfts the fucto peak away from the org. Ths shft depeds o the value of m k ad o the expoetal parameters, A k. To descrbe a datomc system, the maxmum of the tral wave fucto terms of r should be aroud the equlbrum teruclear dstace of the system. I a varatoal calculato the maxma of k s are adjusted by optmzato of m k s ad A k s. More detals o the Hamltoa trasformato ad the selecto of the bass fuctos for datomc calculatos the reader ca obta from our recet revews., The formulas for the matrx elemets volvg Ĥ MV r, Ĥ I D r, ad Ĥ II D r operators ad bass fuctos 4 are preseted the Appedxes. I the preset calculatos we use the varatoal method, ad the eergy ad the wave fucto for each state of HD were obtaed by mmzg the Raylegh quotet, Ec k,m k,a k = m chm k,a k c csm k,a k c, 5 wth respect to the expaso coeffcets of the wave fucto terms of the bass fuctos c k, the bass-fucto expoetal parameters A k, ad the preexpoetal powers m k. The optmzato s doe separately for each state usg a algorthm based o aalytcal dervatves of the eergy, Dowloaded 0 Apr 0 to Redstrbuto subject to AIP lcese or copyrght; see

4 Kedzera et al. J. Chem. Phys. 5, TABLE I. Total o-bo eerges E o-bo, mass-velocty ad Darw correctos, total eerges that clude relatvstc correctos E o-borel,adv v trasto eerges that accout E o-bo ad do ot accout E o-borel relatvstc correctos. All eerges are gve a.u., whle trasto eerges are cm. v E o-bo Mass-velocty Darw E o-borel E o-bo E o-borel D atom Ec k,m k,a k, wth respect to elemets of A k. I geeral, smultaeous optmzato of the eergy fuctoal 5 wth respect to olear parameters of all bass fuctos represets a dffcult ad very tme cosumg computatoal task whe the umber of bass fuctos exceeds a few hudreds. To acheve the best results the parameter optmzato wth the least computatoal effort, we have mplemeted a hybrd method that combes the gradet based optmzato wth the stochastc selecto method.,3 The strategy s based o alteratg the gradet based ad the stochastc based optmzatos growg the bass set from a relatvely small tal set to a much larger fal set. The small tal bass set s obtaed by meas of smultaeous optmzato of all olear parameters. The bass set for each vbratoal state was geerated a separate calculato. To acheve hgh accuracy we used 500 bass fuctos for all states, except v = 3 state, where the umber of bass fuctos was The rage of the preexpoetal powers m k used was 0 50, ad all the powers were partally optmzed for each state. For all 3 v=0,..., vbratoal states of HD we calculated the expectato values of the relatvstc Hamltoa ad added t to the varatoal eergy of that state. Those values were used to calculate the trasto eerges. The uclear masses used the calculatos were m p = m e ad m d = m e, whch were take from Ref. 5. Here, m e stads for the mass of the electro. The value of the fe structure costat was =/ III. THE RESULTS AND DISCUSSION The trasto eerges for all 3 rotatoless boud vbratoal states of HD are preseted Table I. Both orelatvstc o-bo eerges ad eerges cludg the relatvstc correctos are show. I the table we also clude the values of dvdual Darw ad mass-velocty correctos. The Darw correctos were calculated usg both Ĥ D I ad Ĥ D II ad the results agreed wth the umercal accuracy. The relatvstc electroc correctos for HD were calculated before by Howells ad Keedy 6 usg the frst order perturbato theory ad the BO wave fuctos obtaed for a wde rage of teruclear dstaces. These results were the averaged over vbratoal wave fuctos obtaed by solvg the vbratoal equatos wth the potetal eergy take from the BO calculatos. The comparso of our total relatvstc correcto for each vbratoal state.e., the sum of the Darw ad mass-velocty correctos wth that obtaed by Howells ad Keedy 6 s show Table II. As oe ca otce, the results are ot detcal, but close. I geeral our correctos are cm lower magtude tha the correctos of Howells ad Keedy. The dffereces may be caused by several factors such as the use of the reduced electro mass our calculatos versus the use of the real electro mass ther calculatos, ot assumg the BO approxmato our approach versus assumg ths approach thers, the dffereces the bass fuctos ad ther abltes to descrbe the cotact destes, etc. It s terestg to meto here a comparso of the relatvstc correctos calculated for H by Howells ad Keedy the Dowloaded 0 Apr 0 to Redstrbuto subject to AIP lcese or copyrght; see

5 Relatvstc correctos o-bor-oppehemer J. Chem. Phys. 5, TABLE II. Comparso of the relatvstc correctos sum of MV ad Darw obtaed ths work wth those of Howells ad Keedy.Ref. 6.All quattes are cm. v Ths work Ref same work where they preseted the HD results wth the results obtaed by Moss ad Valezao 3 ad show the latter paper. Ths comparso shows smlar dffereces betwee the relatvstc correctos for the H vbratoal eerges obtaed by the two teams to the dffereces betwee our correctos ad those obtaed by Moss ad Valezao for HD. Ths seems to dcate that calculatg relatvstc correctos t s dffcult to acheve hgher accuracy tha about cm due to the ature of the operators volved the calculatos.e., hgher dervatves ad Drac delta fuctos. The o-bo eerges wthout the relatvstc correctos show Table I are vrtually detcal to those preseted before Ref. 4. The trasto eerges corrected for the relatvstc effects the lower part of the spectrum are lower by cm tha ther ucorrected couterparts. Ths tred reverses the upper part where the trasto eerges obtaed from the eerges corrected for the relatvstc effects are lower tha those obtaed from ucorrected eerges. Although geeral the relatvstc correctos to the trasto eerges are small, they are ot eglgble ad, for most trastos, they are a lttle larger tha the usual precso of the expermet. Thus, ther cluso should result mproved accuracy of the predcted trasto eerges as was the case for the trasto eerges we recetly calculated for the HeH Ref. 4 o where a drect comparso wth the expermetally determed three lowest vbratoal trastos was possble. There s oe addtoal observato oe ca make upo comparg the relatvstcally corrected trasto eerges wth the ucorrected oes. It cocers the hghest trasto the spectrum betwee the v= ad v= levels whose relatvstcally ucorrected trasto eergy s cm ad the corrected oe s cm. As we determed our prevous work, 4 both v= ad v= states, HD ca be descrbed as a D atom teractg wth a dstat proto. Ths s dfferet tha the lower states, where the degree of the electro charge polarzato s much lower. The trasto s somewhat a aomaly as far as the relatvstc correctos are cocered. Based o the trastos just below the trasto oe would expect to see a over 0.0 cm decrease of the trasto eerges whe the relatvstc effects are cluded. However, the relatvstcally corrected ad ucorrected trasto eerges are almost detcal. We attrbute ths lack of chage to the uusually hgh electroc polarzato of HD the v= ad v= states. IV. SUMMARY I ths work we descrbed the algorthms for calculatg mass-velocty ad Darw relatvstc correctos to the o- Bor-Oppehemer eergy of datomc systems wth electros. Wth ths, for the frst tme a geeral framework for calculatg these two relatvstc effects for systems wth more tha oe electro was preseted ad mplemeted wth a approach that does ot separate the electroc ad uclear motos as t happes whe the BO approxmato s assumed. Thus the calculato we ca descrbe o a equal footg the relatvstc effects due to electros ad ucle, as well as effects due to teractos betwee these two types of partcles. The dervatos of the tegrals volvg explctly correlated Gaussa fuctos for both the Darw ad mass-velocty correctos are legthy but lead to expressos that ca be readly programed. The code for the correctos has bee tegrated to our o-bo datomc computer program that has bee effcetly parallelzed usg message passg terface MPI. As we have demostrated the o-bo calculatos for some datomc systems see, for example, the recetly preseted calculatos for HeH Ref. 4, our approach s capable of producg total ad trasto eerges wth accuracy that matches that of hgh resoluto expermets. I our pursut to develop a predctve method for calculatg datomc rovbratoal spectra wth the accuracy of the stateof-the-art hgh resoluto expermet, we have to accout for the relatvstc effects. Icludg the Darw ad massvelocty effects s the start. Next stage wll be the cluso of magetc sp-sp, sp-orbt, ad orbt-orbt teractos. It s deftely a exctg task to push the theoretcal developmet to ts lmts as descrbed by the orelatvstc ad relatvstc quatum mechacs. Fally, we hope that the relatvstcally corrected trasto eerges determed ths work wll be helpful assstg the expermet. We eed to add that at preset tme the o-bo calculatos such as those for HD ad to a much hgher degree for systems wth more electros requre a lot of computatoal tme. We hope that the progress the computer hardware wll eable calculatos of spectra of sys- Dowloaded 0 Apr 0 to Redstrbuto subject to AIP lcese or copyrght; see

6 Kedzera et al. J. Chem. Phys. 5, tems wth three ad four electros wth a smlar accuracy as t s curretly possble for datomc systems wth oe or two electros. ACKNOWLEDGMENTS Ths work has bee supported part by the Natoal Scece Foudato. We would lke to thak Professor Jacek Karwowsk for very helpful commets cocerg ths work. APPENDIX A: SOME AUXILIARY FORMULAS Two types of fuctos are used the dervatos, f k =exp rā k r, k =r m k exp rā k r=rj r m k/ exp rā k r, where the matrx J =J I 3 s a partal case of matrx J j, whch we wll defe the followg way: J j = j j j j, j, J =, A ad where s the Kroecker symbol. By settg m k =0 oe gets the f k fuctos from the k fuctos. I order to smplfy the otatos we wll be deotg the sum of the powers of k ad l as pm k m l /m kl /. For matrces B =B I 3 ad B we wll use the followg relatos: trb =3trB, B = B 3. Here ad below vertcal bars aroud a matrx deote the determat of the matrx, whle tr stads for the trace of a matrx. To avod ay cofuso, we wll ot assume that the matrces appearg the tegrals below are symmetrc uless explctly stated. The frst ad secod dfferetals of the k fucto have the followg forms: ad k = k m k r J r Ā k r k = k m k m k r 4 rj J r It follows from here that m k r rj Ā k r rā k J r A 4rĀ k Ā k r m k r J Ā k. A3 B k = k m k r B J r B Ā k r, A4 r B r = B k = k m k m k r B m k r rj B Ā k Ā k B J r 4rĀ k B Ā k r 6trA k B. A5 Throughout our dervatos we wll extesvely use the relato exp rārdr = 3/ Ā /, A6 whch holds for postve defte symmetrc matrx A. Accordg to A6 the overlap of f k ad f l s f k f l = 3/ A 3/. A7 Expressg p r p ad r j the followg way: r p = p exp urj r u p A8 u=0 ad = exp t r j 0 rj jrdt, A9 ad usg A6 we ca evaluate the followg useful tegral p=m kl q: q kr r j l p =f kr r jf l = p p 0 u p exp rā kl uj t J jrdrdt u=0. A0 We ca dfferetate A7 wth respect to Ā kl, f k f l f k f l dr, Ā kl = Ā kl A whch yelds f k r r f l = f kf l. Ā kl A Ths result ca be geeralzed as k grr r l = kgr l, A3 Ā kl k grr r r r l = k gr l, A4 Ā kl Ā kl where gr s a arbtrary fucto of r that does ot deped o Ā kl, for example, r or /r j. Calculato of determats ca be hadled usg the followg theorem. Theorem o the verse matrx ad the determat 7. If G ad GH are osgular matrces, Dowloaded 0 Apr 0 to Redstrbuto subject to AIP lcese or copyrght; see

7 Relatvstc correctos o-bor-oppehemer J. Chem. Phys. 5, rak H=r0, H=H H H r, where rak H k =, kr, ad C k =GH H k s osgular for k=,..., r C =G. the, C k =C k v k C k H k C k, where v k =trc k H, kr. GH=v v,...,v r G. Usg the above theorem we ca express determats I ah ad I ah bh as a sum. To do ths we wll set C = I, v =a tr H, C = a tr H H, v =b tr H a tr H ab trh H. The results are ad I ah =a tr H A5 I ah bh =a tr H b tr H abtr H tr H trh H. A6 We wll also be usg the Lebz formula for the dervatve of a product of two fuctos f ad g, x q fxgx = q s=0 s q s fq s g s. q q APPENDIX B: MASS-VELOCITY MV TERM A7 After the trasformato from the laboratory coordate system to the teral coordate system the MV Hamltoa has the followg form s the umber of pseudopartcles, the case of HD =: Ĥ MV r = 8 M 3 = r 4 = 3 r M 4. The matrx elemets that eed to be calculated are k Ĥ MV l = 8 M 3 r J r k r J r l B 3 r J r k r J r l, B = M where we used the matrx J wth o dces, whose elemets are equal to oe: J =. Matrx J s defed A. Oly oe type of tegral appears the expresso for the Ĥ MV matrx elemets: r D r k r D r l, where D s ether J or J. To compute t we ca express t through the followg elemetary tegrals: r D r k r D r l =36trA k DtrA l D k l 4trA k D k rā l D Ā l r l 4trA l D k rā k D Ā k r l 6 k rā k D Ā k rrā l D Ā l r l 6m k m k tra l DD k r l 6m l m l tra k DD k r l 4m k m k D k r rā l D Ā l r l 4m l m l D k r rā k D Ā k r l m k tra l D k r rā k D J J D Ā k r l m l tra k D k r rā l D J J D Ā l r l 8m k k r rā k D J J D Ā k rrā l D Ā l r l 8m l k r rā l D J J D Ā l rrā k D Ā k r l m k m k m l m l D k r 4 l m k m k m l D k r 4 rā l D J J D Ā l r l m k m l m l D k r 4 rā k D J J D Ā k r l 4m k m l k r 4 rā k D J J D Ā k rrā l D J J D Ā l r l. Thus, to carry out the calculatos of the MV correcto oe eeds the followg tegrals: k r q l, k r q rb r l, ad k r q rb rrc r l, where q=0,, ad 4. We preset the expressos for those tegrals below.. Itegral Š k r q l We start wth the tegral k r q l. Accordg to A0 oe ca wrte Dowloaded 0 Apr 0 to Redstrbuto subject to AIP lcese or copyrght; see

8 Kedzera et al. J. Chem. Phys. 5, k r q l = p p a p exp rā kl aj rdr a=0 p A kl 3/ 3/ p a pi aj A kl 3/ a=0 = p A kl 3/ 3/ p a pa trj A kl 3/ a=0 = p f k f l p p 3/A p kl = f k f l m kl q/ 3/A m kl kl q/, B3 where A6, A8, ad A5 were used. After smplfcato we obta k r q l = f k f l m kl q/ 3/A m kl kl q/. B4 I the case of q=0,, ad 4 the correspodg expressos are k l = f k f l m kl 3/A m kl kl /, B5 k r l =m kl A kl k l, k r 4 l =4m kl A kl k l.. Itegral Š k r q rb r l B6 B7 Ths tegral s evaluated usg the followg relato: k r q rb r l = k r q r r l B. Together wth A3 we have k r q r r l = kr q l B8 Ā kl. B9 To determe the above dervatve the followg dettes are used for detals see Ref. 8: Ā kl = Ā kl Ā kl Ā kl ad B0 trj Ā kl = Ā kl J Ā kl. Ā kl After some trasformatos we obta k r q l = k r q l Ā kl m mk q ad tr J Ā kl Ā kl J Ā kl Ā kl B B k r q rb r l = kr q l 3trA kl B m kl q A kl A kl BA kl. B3 3. Itegral Š k r q rb rrc r l Smlarly to B8 we ca wrte k r q rb rrc r l = k r q r r r r l B C, B4 whch, combed wth A4, gves k r q r r r r l = k r q l. B5 Ā kl Ā kl I addto to some expressos derved above we also eed the relato Ā kl J Ā kl Ā kl = Ā kl Ā kl J Ā kl Ā kl J Ā kl Ā kl. We use t to evaluate the followg dervatve B6 k r q l = Ā kl Ā kl 4 kr q l m kl qm kl q tr J Ā kl Ā kl J Ā kl Ā kl J Ā kl m kl qtr J Ā kl Ā kl J Ā kl Ā kl Ā kl J Ā kl Ā kl m kl qtr J Ā kl Ā kl J Ā kl Ā kl Ā kl J Ā kl Ā kl Ā kl Ā kl Ā kl Ā kl. B7 The fal result s Dowloaded 0 Apr 0 to Redstrbuto subject to AIP lcese or copyrght; see

9 Relatvstc correctos o-bor-oppehemer J. Chem. Phys. 5, k r q rb rrc r l = 4 kr q l m kl qm kl q A kl A kl BA kl A kl CA kl 3m kl q A kl A kl BA kl tra kl C A kl CA kl tra kl B m kl qa kl A kl BA kl CA kl A kl BA kl CA kl 9trA kl BtrA kl C 6trA kl BA kl C. B8 APPENDIX C: DARWIN TERM I the frst approach based o the Ĥ D I r Hamltoa the matrx elemet to calculate s k Ĥ D I l = 8 = = j q 0 q M M q q j M r J r k r l k r r J r l J r J r k r j l k r r J r j l J r k r J r l r k r jj r l. C I the secod approach H II D r the followg matrx elemet that eeds to be calculated s k Ĥ II D l = q 0 q k r l M = = j=,j M q q j k r j l. C M APPENDIX D: DARWIN CORRECTION: THE FIRST APPROACH I the expresso for the matrx elemet volvg the Ĥ D I r operator the followg sum of tegrals appears: k Ĥ D I l = r J r k r g l k r g r J r l followg four tegrals: k r g l, kr r g l, k r g rb r l, kr r g rb r l. I the expressos we derve ext we use the followg otato: ā =trj Ā kl, b =trj gā kl, a =trj A kl, b =trj g A kl, D D3 J r k r gj r l, D c =trj Ā kl J gā kl, c =trj A kl J g A kl, D4 where g stads for ether or j. Usg A4 ad A6 ad smplfyg the resultg expresso, we obta r J r k r g l k r g r J r l J r k r gj r l = 6A kl k r g l 4 k r g rā kl J Ā kl r l m kl m kl kr r g l q = 3 q = q q 3/, q / q /, D5 D6 where matrx J g for g= becomes J ad for g=j becomes J j. m kl kr r g rā kl J J Ā kl r l. To complete the above formula we eed to determe the. Itegral Š k r q /r g l From A0 we have Dowloaded 0 Apr 0 to Redstrbuto subject to AIP lcese or copyrght; see

10 Kedzera et al. J. Chem. Phys. 5, q kr r g l = p p 0 u p exp rā kl uj t J grdrdt u=0 = p p u p 0 3/ A kl uj t J g 3/ dt u=0 = p f k f l p u p I uj A kl 0 t J g A kl 3/ dt u=0. D7 Usg A6 ad deftos D D4 oe obtas I uj A kl t J g A kl =ua t b ut ab c D8 ad ua t b ut ab c 3/ dt = au b 0 uab c /. D9 Now usg the Lebz formula A7 oe may evaluate the followg dervatves: ad p s u p sau u=0 = p s p s a p s, s u sb uab c / u=0 s = ss / c / ab s a s, b u sau b uab c / u=0 = p p ap p s=0 Wth the expresso for k r q l B4, D0 D 3 s ab c s. D f kf l a p = kr q l p 3/, D3 we obta the followg fal expresso for the tegral k r q /r g l : kr q r g l = k r q l p. Itegral Š k r q /r g rb r l p b s=0 3 s ab c s. We frst eed to calculate the followg dervatve: k r q /r g l q = kr r r l. Ā kl r g D4 D5 Usg the prevously determed dervatves ad the followg dervatve: c = trj Ā kl J gā kl = Ā kl J Ā kl J g Ā kl Ā kl we have Ā kl J gā kl J, D6 k r q /r g l = q kr Ā kl r g l p ā Ā kl J Ā kl Ā kl b Ā kl J gā kl k r q l 3 3 p p ā b b s= 3 ss āb 3c s āb Ā kl J Ā kl J gā kl āb Ā kl J gā kl J Ā kl āc Ā kl J gā kl b c Ā kl J Ā kl. Usg the above we obta the fal expresso for the k r q /r g rb r l tegral, q kr rb r r g l = q kr r g l m kl q A a kl BA kl 3trA kl B b tra kl J g A kl k r q l m kl q/ a b b m kl q/ s= B 3 ss ab c s aba kl BA kl J g A kl aba kl J g A kl BA kl ac tra kl J g A kl B bca kl BA kl. D7 D8 Dowloaded 0 Apr 0 to Redstrbuto subject to AIP lcese or copyrght; see

11 Relatvstc correctos o-bor-oppehemer J. Chem. Phys. 5, APPENDIX E: DARWIN CORRECTION: THE SECOND APPROACH I the expresso for the matrx elemet volvg the Ĥ II D operator the followg tegrals eed to be evaluated: k Ĥ II D l = q 0 q k r l M c = j = M q q j k r j l. E M To evaluate the above formula we eed to determe the followg tegrals: k r l, k r j l.. Itegral Š k r l The tegral has the followg form: m k r l r k m l exp rā kl rr dr. = Sce for a arbtrary fucto Fx, we have f=, Fxxdx = F0, m k r l r kl exp rā kl rr dr = f, 0 Û m kl =0, m k r l r kl exp rā kl rr dr = 0 Û for ay m kl. The matrx elemet of the Drac delta fucto wth smple sphercal Gaussas ca be obtaed usg the Gaussa represetato of the delta fucto, 5 r = lm s s 3/ exp sr. E Wth ths, f k r f l = lm s s 3/ f k exp sr sr s f l. E3 If j s a -compoet vector whose frst elemet s equal to ad the rest are zeros, the f k r f l = lm s s 3/ exp s f k exp srj r sj rf l = lm Ā kl r sj rdr = lm s s 3/ exp s s s 3/ 3/ A kl sj exp s exp rsj exps j Ā kl sj j. E4 I the last expresso we used the relato exp rb r yrdr = B 3/ exp yb y. 4 We ca rewrte the determat A kl sj as A kl sj = A kl I sj A kl = A kl s trj A kl. The, f k r f l = 3/ lm A kl s /s trj A kl exps j Ā kl sj j s. 3/ E5 Sce the lmt of the preexpoetal part of E5 s a fte umber, the lmt of the expoet must be equal to wth beg a fte umber. Otherwse the etre expresso E5 would have bee ether zero or fty, whch s ot the case. Hece, f k r f l = 3/ A kl tra kl J 3/ exp = f k f l 3/ tra kl J 3/ exp. E6 Makg use of the ormalzato codto, Dowloaded 0 Apr 0 to Redstrbuto subject to AIP lcese or copyrght; see

12 Kedzera et al. J. Chem. Phys. 5, f k r f l d = f k f l, we fd that =tra kl J. Thus, f k r f l = f k f l 3/ tra kl J 3/ E7 exp tra E8 kl J. The last relatoshp s ow used to evaluate the matrx elemet k r l. To do that we defe p=m kl / ad accordg to A8 we obta k r l = f k r p r f l = p p u pf k exp urj rr f l u=0 = p p u p 3/ A kl uj 3/ 3/ tra kl uj J 3/ exp tra kl uj. E9 J u=0 Applyg the followg formulas: u A kl uj = A kl sj tra kl sj J, E0 u tra kl uj J = tra kl uj J A kl uj J, E ad usg trxj XJ =trxj =X for a arbtrary matrx X lead to the fal result, k r l = f k f l p 3/ A 3/ kl p A kl exp A kl I the above expresso we used the followg:. k l = f k r p f l = p 3/A kl p f k f l. So f we put =0 ad p=0, we have k r l = f k f l p 3/ E E3 3/. E4 A kl. Itegral Š k r j l The matrx elemets of r j ca be obtaed by straghtforward tegrato. The procedure s very smlar to the evaluato of the overlap tegral ad yelds k r j l = p 3/D p kl 3/ = k l A kl D kl 3/ D kl 3/ D kl A kl p, E5 where D kl s a matrx formed from A kl by addg the jth row to the th row, the addg the jth colum to the th colum, ad the crossg out the jth colum ad the jth row. 3. Itegral Š k r l I the case of k r l we obta the same expresso as E5 but D kl s formed from A kl by crossg out the th colum ad the th row wthout addg aythg. M. Cafero, S. Bub, ad L. Adamowcz, Phys. Chem. Chem. Phys. 5, S. Bub, M. Cafero, ad L. Adamowcz, Adv. Chem. Phys. 3, D. B. Kghor ad L. Adamowcz, J. Chem. Phys. 06, D. B. Kghor ad L. Adamowcz, J. Chem. Phys. 0, D. B. Kghor ad L. Adamowcz, Phys. Rev. Lett. 83, C. E. Scheu, D. B. Kghor, ad L. Adamowcz, J. Phys. Chem. 4, M. Cafero ad L. Adamowcz, Phys. Rev. Lett. 88, M. Cafero ad L. Adamowcz, J. Chem. Phys. 6, M. Cafero ad L. Adamowcz, Phys. Rev. Lett. 89, S. Bub ad L. Adamowcz, J. Chem. Phys. 8, M. Cafero, L. Adamowcz, M. Dura, ad J. M. Lus, J. Mol. Struct.: THEOCHEM 633, S. Bub ad L. Adamowcz, J. Chem. Phys. 0, S. Bub ad L. Adamowcz, J. Chem. Phys., S. Bub, E. Bedarz, ad L. Adamowcz, J. Chem. Phys., S. Bub ad L. Adamowcz, Chem. Phys. Lett. 403, E. Bedarz, S. Bub, ad L. Adamowcz, J. Chem. Phys., M. Pavaello, S. Bub, M. Molsk, ad L. Adamowcz, J. Chem. Phys. 3, S. Bub, L. Adamowcz, ad M. Molsk, J. Chem. Phys. 3, H. Naka, M. Hosho, K. Myamoto, ad S. Hyodo, J. Chem. Phys., A. Reyes, M. Pak, ad S. Hammes-Schffer, J. Chem. Phys. 3, C. D. Sherrll, Au. Rep. Comp. Chem., H. A. Bethe ad E. E. Salpeter, Quatum Mechacs of Oe- ad Two- Electro Atoms Pleum, New York, R. E. Moss ad L. Valezao, Mol. Phys. 0, M. Stake, D. Kȩdzera, M. Barysz, S. Bub, ad L. Adamowcz, Phys. Rev. Lett. 96, See CODATA 00 recommeded values ad NIST Physcal Referece Data at 6 M. H. Howells ad R. A. Keedy, J. Chem. Soc., Faraday Tras. 86, K. S. Mller, Some Eclectc Matrx Theory Robert E. Kreger, Malabar, FL, E. Bedarz, S. Bub, ad L. Adamowcz, Mol. Phys. 03, Dowloaded 0 Apr 0 to Redstrbuto subject to AIP lcese or copyrght; see