Quantum Chemical Studies of Protein-Bound Chromophores, UV-Light Induced DNA Damages, and Lignin Formation

Transcription

1 omprehensve Summares of Uppsala Dssertatons from the Faculty of Scence and Technology 1010 Quantum hemcal Studes of Proten-Bound hromophores, UV-Lght Induced DA Damages, and Lgnn Formaton BY B DURBEEJ ATA UIVERSITATIS UPSALIESIS UPPSALA 2004

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3 Do I have any mportant phlosophy for the world? Are you kddng? The world don t need me. hrst, I m only fve feet ten. Bob Dylan

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5 Lst of papers Ths thess s based on the followng papers, whch wll be referred to n the text by ther Roman numerals IIX: I II III IV V VI VII VIII B. Durbee and L. A. Erksson, n the bathochromc shft of the absorpton by astaxanthn n crustacyann: a quantum chemcal study, hem. Phys. Lett., 2003, 375, 30. B. Durbee and L. A. Erksson, onformatonal dependence of the electronc absorpton by astaxanthn and ts mplcatons for the bathochromc shft n crustacyann, Phys. hem. hem. Phys., 2004, 6, B. Durbee,. A. Borg and L. A. Erksson, Phytochromobln 15-Z,syn 15-E,ant somerzaton concerted or stepwse?, Phys. hem. hem. Phys., 2004, submtted. B. Durbee and L. A. Erksson, Thermodynamcs of the photoenzymc repar mechansm studed by densty functonal theory, J. Am. hem. Soc., 2000, 122, B. Durbee and L. A. Erksson, n the formaton of cyclobutane pyrmdne dmers n UV-rradated DA: why are thymnes more reactve?, Photochem. Photobol., 2003, 78, A. Borg, L. A. Erksson and B. Durbee, Electron-transfer nduced repar of 64 photoproducts n DA, 2004, manuscrpt. B. Durbee and L. A. Erksson, A densty functonal theory study of conferyl alcohol ntermonomerc cross lnkages n lgnn threedmensonal structures, stabltes and the thermodynamc control hypothess, olzforschung, 2003, 57, 150. B. Durbee and L. A. Erksson, Spn dstrbuton n dehydrogenated conferyl alcohol and assocated dlgnol radcals, olzforschung, 2003, 57, 59.

6 IX B. Durbee and L. A. Erksson, Formaton of -4 lgnn models a theoretcal study, olzforschung, 2003, 57, 466. Related work: B. Durbee and L. A. Erksson, Reacton mechansm of thymne dmer formaton n DA nduced by UV lght, J. Photochem. Photobol. A: hem., 2002, 152, 95. B. Durbee, Y.-. Wang and L. A. Erksson, Lgnn bosynthess and degradaton a maor challenge for computatonal chemstry, Lect. otes omput. Sc., 2003, 2565, 137. B. Durbee and L. A. Erksson, Photodegradaton of substtuted stlbene compounds what colours agng paper yellow?, 2004, manuscrpt.

7 ontents Introducton Quantum chemstry Molecular orbtal theory The Born-ppenhemer approxmaton The artree-fock approxmaton Electron correlaton Densty functonal theory The ohenberg-kohn theorems The Kohn-Sham equatons Exchange-correlaton functonals Tme-dependent densty functonal theory Proten-bound chromophores Astaxanthn Paper I Paper II Phytochromobln Paper III UV-lght nduced DA damages yclobutane pyrmdne dmers Paper IV Paper V Pyrmdne (64) pyrmdone photoproducts Paper VI Formaton of lgnn Background Paper VII Paper VIII Paper IX oncludng remarks...69 Summary n Swedsh...71 Acknowledgements...74 References...76

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9 Abbrevatons AM A B ASPT2 ASSF SD SD(T) I I IS ISD DFT FI GGA F K L M KS LDA LUM MSF M M-LA MPn MRI PM PES RF RR SF TD UF Adabatc onnecton Method Atomc rbtal Born-ppenhemer [approxmaton] omplete Actve Space second-order Perturbaton Theory [method] omplete Actve Space SF [method] oupled luster [theory] Sngles and Doubles [method] SD perturbatve Trples [method] onfguraton Interacton oncal Intersecton onfguraton Interacton Sngles [method] onfguraton Interacton Sngles and Doubles [method] Densty Functonal Theory Full I Generalzed Gradent Approxmaton artree-fock [approxmaton] ohenberg and Kohn etler and London ghest ccuped M Kohn and Sham Local Densty Approxmaton Lowest Unoccuped M Mult-onfguratonal SF [method] Molecular rbtal Ms as Lnear ombnatons of As [ansatz] nth-order Møller-Plesset perturbaton theory [method] Mult-Reference I [method] Polarzed ontnuum Model Potental Energy Surface Restrcted F [approxmaton] Resonance Raman [spectroscopy] Self-onsstent Feld [approach] Tme-Dependent Unrestrcted F [approxmaton]

10 ZID ZPVE max Zerner s Intermedate eglect of Dfferental verlap [method] Zero-Pont Vbratonal Energy Wavelength of maxmum absorpton

11 Introducton Theoretcal chemstry s the feld devoted to the development and applcaton of theoretcal methods for the study of chemcal systems. These methods can be thought of as technques, each wth ts own merts and dsadvantages, n pretty much the same way as there are dfferent expermental technques more or less well-suted for the elucdaton of a gven chemcal problem. The methods defned n terms of equatons that n the end need to be solved ether analytcally or, as s normally the case, numercally usng computers are developed wthn the framework of an underlyng theory approprate for the problem at hand. In general, applcaton of the theory as such leads nowhere but to a seres of equatons much too complcated even for present-day numercal algorthms and computatonal resources. Instead, computatonally tractable yet accurate methods are developed by ntroducng reasonable approxmatons to the theory. Methods founded on classcal physcs are typcally referred to as force feld methods, and are often employed to smulate the dynamcs of chemcal systems at atomc resoluton. Methods founded on quantum mechancs, n turn, are known as quantum chemcal methods. These are, nherently, more complex than force feld methods, but also account for electronc structure and hence enable a proper descrpton of chemcal reactons. A theoretcal approach to the study of chemstry offers several advantages over conventonal expermental technques. Frst, theoretcal chemsts are not lmted by practcal concerns, as are expermentalsts that perhaps may have to worry whether a chemcal substance s contamnated, or may face sgnfcant dffcultes when settng out to study hghly reactve and short-lved molecular speces. Second, theoretcal chemstry offers a possblty to obtan nformaton not easly retreved from experments. For example, quantum chemcal methods can be used to characterze transton states n chemcal reactons, and molecular dynamcs smulatons can be used to study, e.g., molecular recognton processes of profound mportance n bology. Thrd, t may be argued that theoretcal methods, as stemmng from a seres of physcal and chemcal arguments, consttute a better way of provdng physcal nsght nto a chemcal problem. The Woodward- offmann rules governng the reactvty of percyclc reactons serve as a thereby llustratve example of a mlestone achevement of theoretcal chemstry. Fnally, wth the present avalablty of cheap and hghly effcent 11

12 computer hardware, theoretcal studes are a vable approach also from an economcal pont of vew. eedless to say, there are also dsadvantages assocated wth theoretcal chemstry. Most notably, t s often the case that the chemcal problem at hand s far too complcated to allow for even qualtatve assessment by theory. In other cases, seemngly smple chemcal systems are not amenable to calculatons, merely because an underlyng, key approxmaton may lose valdty n that partcular system. In the present thess, theoretcal methods are used to provde a better understandng of some chemcal systems of bologcal nterest. In partcular, quantum chemcal methods are appled to shed new lght on unresolved ssues concernng the photochemstry of two proten-bound chromophores; the photoenzymc repar of UV-lght nduced DA damages; and the formaton of lgnn n plant cell walls. Followng a short ntroducton (hapter 1) to the theory underlyng the quantum chemcal methods used, hapters 2-4 present the background to the dfferent proects, and summarze ntentonally n a rather bref fashon the man results of the respectve papers. For a more complete account of the research, the reader s referred to the full papers. hapter 5, fnally, offers some fnal concludng remarks. 12

13 1 Quantum chemstry In 1927, etler and London (L) 1 ratonalzed the formaton of a chemcal bond between two hydrogen atoms at certan nteratomc dstances usng the newly developed theory of quantum mechancs. Pror to ths tme, all attempts to explan the stablty of the hydrogen molecule rooted n the theory of classcal electrostatc forces had faled. The work by L was therefore regarded as a spectacular achevement of quantum mechancs, and marked the brth of quantum chemstry. Durng the almost eghty years that snce have passed, ths feld has expanded tremendously and quantum chemcal nvestgatons of relevance for essentally all branches of chemstry appear regularly n the lterature today. Ths progress can be attrbuted to the development of ncreasngly accurate quantum chemcal methods and the mplementaton of these nto computatonally effcent algorthms and easyto-use computer codes, as well as the dramatc mprovement n computer technology occurrng over the last twenty years or so. Whle L n ther study of the hydrogen molecule ntroduced a theoretcal framework known as valence bond theory, the success as measured by the number of successful applcatons to chemcal problems of quantum chemstry s from a methodologcal pont of vew almost entrely due to the development of so-called molecular orbtal (M) methods, and n later years densty functonal theory (DFT) methods. The present chapter brefly ntroduces the basc approxmatons of M theory and DFT underlyng the quantum chemcal methods used n ths thess. For thorough accounts of M theory and DFT, the reader s referred to the textbooks by Szabo and stlund 2 and Parr and Yang, 3 respectvely. 1.1 Molecular orbtal theory The startng pont for a quantum chemcal descrpton of a system of nucle and electrons formng a molecule s the tme-dependent Schrödnger equaton, (1.1) t 13

14 where s the amltonan operator for the system defned as soon as the system has been specfed and s the wave functon one wshes to determne. If the amltonan s tme-ndependent, the tme dependence of the wave functon s manfested only through an exponental phase factor ( Et / R, r, t) ( R, r) e, (1.2) where E s the energy of the (statonary) state characterzed by such a wave functon. R and r denote spatal nuclear and electronc coordnates, respectvely. By nsertng Eq. (1.2) nto Eq. (1.1), one obtans the tmendependent Schrödnger equaton ( R, r) E ( R, r). (1.3) Fndng solutons to Eq. (1.3) s a central goal of quantum chemstry. owever, ths requres the ntroducton of a number of approxmatons even for the smplest of molecular systems. ne common approxmaton s the neglect of relatvstc effects, whch facltates the calculatons n that the amltonan takes on a smpler form. Ths s normally a good approxmaton for elements of the three frst rows (Z36) of the perodc table. eaver elements, on the other hand, have core electrons that may well acqure veloctes correspondng to non-neglgble fractons of the speed of lght, whch means that relatvstc correctons need to be accounted for. Snce ths thess deals exclusvely wth non-relatvstc quantum chemcal nvestgatons of molecular systems consstng of atoms wth Z15, methods that ncorporate relatvstc effects are from now on not further dscussed. A more fundamental approxmaton, whch allows for the separaton of Eq. (1.3) nto electronc and nuclear degrees of freedom, was frst presented by Born and ppenhemer 4 n 1927, and can followng Jensen 5 brefly be outlned as follows The Born-ppenhemer approxmaton The amltonan of Eq. (1.3) can be wrtten as T T V V V T, (1.4) n e ee ne nn e n wth T n and T e denotng the knetc energy operators for nucle and electrons, and V ee, V ne, and V nn denotng the potental energy operators for nteractons between electrons, nucle and electrons, and nucle, respectvely. e Te Vee Vne Vnn s the amltonan for the electronc Schrödnger equaton 14

15 e ; ; ( r R) ( r R), (1.5) where ( r ; R) s the electronc wave functon for state wth energy, whch depends explctly on the electronc coordnates and parametrcally on the nuclear postons. If a full (complete) set of solutons to Eq. (1.5) s avalable, whch can be chosen to be orthonormal, the total wave functon of Eq. (1.3) can be expressed n terms of these solutons wth the expanson coeffcents (R) beng functons of the nuclear coordnates n ( R, r) n ( R) ( r; R). (1.6) ow, by nsertng Eq. (1.6) nto Eq. (1.3) and makng use of the facts that e only acts on electronc wave functons, and that the knetc energy operator for the nucle s gven as (where M k s the mass of nucleus k ) T n k n, (1.7) 2 k M k one gets 2 2 { ( n n ) 2( n ) ( n n ) n ( n ) n } E n. Multplyng Eq. (1.8) from the left by yelds (1.8) 2 2 {2 ( ) } E n n n n n n n n (1.9) after ntegraton over electronc coordnates. Apparently, the electronc wave functon has been removed from the frst two terms of the LS of Eq. (1.9). The two dfferent types of matrx elements n the sum correspond to nonadabatc couplng elements, and represent couplng between dfferent electronc states. In the adabatc approxmaton, one consders only one electronc state at a tme ( ),.e., all off-dagonal couplng elements are neglected. It s also assumed that the frst-order dagonal couplng element s neglgble. ence, n n ( ) 2 n 2 n n E. (1.10) n 15

16 Snce the mass of the lghtest nucle (a proton) s larger than that of an electron by a factor of ~1800, t furthermore holds that s sgnfcantly 2 larger than the second-order dagonal couplng element (often referred to as the dagonal correcton). By neglectng ths term as well, one arrves at the Born-ppenhemer (B) approxmaton ( ) E. (1.11) 2 n n n n The B approxmaton ntroduces the concept of a potental energy surface (a soluton to the electronc Schrödnger equaton) upon whch the nucle move, and reduces the complcated problem of solvng Eq. (1.3) nto separate electronc and nuclear problems. Frst, the electronc Schrödnger equaton (1.5) s solved for a set of nuclear confguratons to obtan (R). Then, the nuclear Schrödnger equaton (1.11) s solved wth (R) actng as the effectve potental, whch yelds molecular vbratonal energy levels at zero temperature and n combnaton wth statstcal mechancs thermochemcal propertes. The adabatc and B approxmatons are usually good approxmatons, but are not vald when electronc states le very close n energy. In such cases, t s no longer possble to make use of Eq. (1.11) for the nuclear problem, and solvng the electronc counterpart by means of Eq. (1.5) wll nevtably requre a consderaton of non-dynamcal electron correlaton effects (cf. subsecton 1.1.3). In the followng, focus s on methods for solvng the electronc Schrödnger equaton. For the sake of notatonal smplcty, electronc amltonans, wave functons, and energes are from now on denoted by,, and E, respectvely. Snce the electronc moton s assumed to take place n the feld of fxed nucle, one may exclude the constant V nn term from the electronc amltonan, and add the nuclear-nuclear repulson energy once the electronc problem has been solved. The problem at hand s thus E wth T V e ee V ne (1.12) Unfortunately, exact (.e., analytc) solutons to Eq. (1.12) can only be obtaned for one-electron systems. It s therefore crucal to ntroduce further approxmatons through whch one may obtan accurate numercal solutons. The artree-fock approxmaton, ntroduced by Fock 6 and Slater 7 16

17 and buldng on the work by artree, 8 consttutes the frst step towards more elaborate treatments The artree-fock approxmaton In order to completely descrbe an electron, t s necessary to specfy ts spn state. Snce the electronc amltonan of Eq. (1.12) s spn-ndependent, ths has to be done by mposng a certan form for the wave functon. The procedure s as follows. Frst, two orthonormal spn functons correspondng to spn up, (), and spn down, (), are ntroduced ( s here a generalzed spn coordnate). The four electronc coordnates (three spatal, r; one spn, ) are usually collectvely denoted by x, and the wave functon for an -electron system s wrtten ( x, x,..., x ). Snce electrons are 1 2 fermons,.e., spn ½ partcles, t s now possble to ensure that electron spn s dealt wth n a satsfactory manner by makng use of the fact that a manyelectron wave functon therefore must be antsymmetrc (change sgn) wth respect to the nterchange of the coordnates of any two electrons. Ths requrement, whch s known as the antsymmetry prncple and consttutes a generalzaton of the Paul excluson prncple, s met by formng the manyelectron wave functon as a lnear combnaton of antsymmetrzed products (Slater determnants) of one-electron wave functons (Ms). The Ms, ) (x also referred to as spn orbtals are products of a spatal M, (r), and a spn functon ( r) ( ) ( x ) or. (1.13) ( r) ( ) In the artree-fock (F) approxmaton, the many-electron wave functon s represented by a sngle Slater determnant (gven here for the general case of electrons and spn orbtals) ( x, x 1 2,..., x ( x ) 1( x 2 ) 2 ( x 2 ) ( x 2 ) -1/2 ) (!), ( x ) ( x ) 2 ( x 2 1 ) ( x ) ( x 1 (1.14) ) 17

18 -1/2 where (!) s a normalzaton factor. It s seen that an nterchange of the coordnates of two electrons corresponds to an nterchange of two rows of the determnant, whch changes the sgn of the wave functon and ensures that the antsymmetry prncple s fulflled. Furthermore, any two electrons are prevented from occupyng the same spn orbtal snce ths would make two columns of the determnant equal and the wave functon zero, n accordance wth the Paul excluson prncple. If the wave functon of Eq. (1.14) s normalzed, t s straghtforward to show that the correspondng electronc energy equals E h ( J 1 2, K ), (1.15) where h are one-electron ntegrals descrbng the moton of electrons n the feld of M nucle h M 1 2 Z k ( x1 ) 1 ( x1) dx 2 k R k r1 1 (1.16) and J and K are two-electron ntegrals accountng for repulsve nteractons between electrons J K 1 ( x dx dx (1.17) 1 ) ( x1) ( x 2 ) ( x 2 ) r1 r2 1 ( x dx dx (1.18) 1 ) ( x1) ( x 2 ) ( x 2 ) r1 r J are referred to as oulomb ntegrals and represent classcal electrostatc repulson between two charge dstrbutons. K are known as exchange ntegrals, and have no classcal counterpart. Ther occurrence n Eq. (1.15) s a drect consequence of the determnantal form of the wave functon requred by the antsymmetry prncple. From Eq. (1.15), t s realzed that the exchange energy reduces the classcal oulomb repulson. It can be shown that ths effect s entrely due to nteractons between electrons wth parallel spns. In order to calculate the best possble F wave functon, a partcular set of spn orbtals needs to be determned. Snce the F wave functon constructed from a tral set of spn orbtals by vrtue of the varatonal prncple yelds an energy whch s always larger than or possbly equal to the exact electronc energy, ths set can be determned by 18

19 mnmzng the energy of Eq. (1.15) wth respect to the choce of spn orbtals. Ths mnmzaton, whch has to be carred out n such a way that the spn orbtals reman orthonormal, results n the so-called F equatons f, (1.19) where f M 1 2 Z k ( J K ) 2 R r (1.20) k k s the Fock operator and s the matrx of the Lagrange multplers used n the mnmzaton to enforce the orthonormalty of the spn orbtals. J and K are the oulomb and exchange operators defned by and Apparently, 1 J ( x1) ( x1) ( x 2 ) ( x 2 ) dx 2 ( x1) r1 r2 (1.21) 1 K ( x1) ( x1) ( x 2 ) ( x 2 ) dx 2 ( x1) r1 r2 (1.22). J and K account for nterelectronc repulson only n an average fashon a gven electron nteracts wth the average feld from the other electrons. The Fock operator s therefore an effectve one-electron operator, and the F method s referred to as a mean-feld approxmaton. It s also worthwhle notng that the result of operatng wth K x ) on ( x 1 ) depends on the value of throughout all space. Ths means that K, unlke J, s a non-local operator, and that the calculaton of the exchange energy therefore s a demandng task. s ermtan and Eq. (1.19) can therefore be brought about to standard egenvalue form by means of a untary transformaton of, yeldng the canoncal F equatons ( 1 19

20 f. (1.23) It mmedately follows that f, whch shows that the Lagrange multplers can be nterpreted as M energes. Snce the Fock operator depends on the spn orbtals one wshes to determne, Eq. (1.23) can only be solved teratvely. For computatonal purposes, the canoncal F equatons need to be reformulated n terms of spatal Ms (r) rather spn orbtals (x). Ths can be acheved by ntegratng out the spn functons. If the restrcton s made that every occuped spatal M should contan two electrons wth opposte spns, one obtans f, (1.24) where wth and 20 f M / Z k (2J K ) 2 R r (1.25) k k 1 J ( r1 ) ( r1 ) ( r2 ) ( r2 ) dr2 ( r1 ) (1.26) r1 r2 1 K ( r1 ) ( r1 ) ( r2 ) ( r2 ) dr2 ( r1 ). (1.27) r1 r2 These equatons underle the so-called restrcted F (RF) formalsm often used for closed-shell molecules, and are except for the oulomb operator n Eq. (1.25) occurrng wth a weght of 2 and the sum of oulomb and exchange operators beng over occuped spatal Ms analogous to those nvolvng spn orbtals. If the double-occupancy constrant s released,.e., f dfferent spatal Ms for and electrons are ntroduced, one arrves at two sets of F equatons upon ntegratng out the spn functons; one for the electrons and one for the electrons. Ths formalsm s referred to as unrestrcted F (UF), and s prmarly used for open-shell molecules such as radcals, snce t allows for spn-polarzaton to be taken nto account. As expected, the two sets of F equatons cannot be solved ndependently of each other snce the Fock operators depend on

21 both and Ms. Ths ncreases the computatonal complexty of UF compared to RF. Due to the fact that UF s assocated wth a hgher degree of varatonal freedom, an optmzed UF wave functon wll always yeld an electronc energy whch s lower than or equal to that of the correspondng RF wave functon. owever, for closed-shell molecules at equlbrum geometres, UF wave functons n general do not dffer from those obtaned by RF. Some further aspects of UF theory wll be dscussed n subsecton It s possble to solve Eq. (1.24) by usng a numercal grd to represent the Ms. Such methods are known as numercal F methods. 9 In 1951, however, Roothaan 10 showed that t s possble to transform Eq. (1.24) a set of ndeed complcated dfferental equatons nto a less demandng matrx egenvalue problem by frst expandng the Ms n terms of lnear combnatons of known analytc one-electron functons (bass functons) chosen so as to represent atomc orbtals (As), and then nvokng the varatonal prncple for the M coeffcents. Wth the M-LA (Molecular rbtals as Lnear ombnatons of Atomc rbtals) ansatz, (1.28) one obtans the so-called Roothaan equatons (or Roothaan-all equatons 11 ) F S, (1.29) where F s the Fock matrx, S F F ; S s the overlap matrx, ; s the (dagonal) matrx of M energes; and s the matrx of M expanson coeffcents. Snce the Fock matrx depends on the M expanson coeffcents one wshes to determne, the Roothaan equatons are solved teratvely, typcally by startng off wth an estmate for the densty matrx P uv obtaned from, e.g., a prevous calculaton / 2 P. (1.30) uv 2 u v Ths procedure s known as the self-consstent feld (SF) approach. Whle the Roothaan equatons were derved wthn the context of RF, Pople and esbet 12 shortly thereafter ntroduced the correspondng UF equatons the Pople-esbet equatons allowng effcent calculatons to be performed also for open-shell systems. A F calculaton carred out by means of Roothaan or Pople- esbet equatons that does not ntroduce any approxmaton to the electronc amltonan or the two-electron ntegrals resultng from the choce of bass 21

22 functons s referred to as an ab nto ( from the begnnng ) calculaton. In prncple, such a calculaton employs no other parameters than physcal constants once the set of bass functons s specfed. The computatonal bottleneck n ths procedure s the calculaton of the two-electron ntegrals requred for the constructon of the Fock matrx. Snce the number of twoelectron ntegrals grows as the fourth power of the number of ntroduced bass functons (~sze of the system), t s clear that ab nto F calculatons have lmted applcablty to large molecular systems. In order to crcumvent ths problem, a wde varety of so-called sememprcal methods have been developed. In prncple, these methods dscard certan two-electron ntegrals, and ntroduce emprcal parameters (ftted to expermental data) and/or approprate functonal forms for some or all of the remanng ntegrals. Furthermore, only valence electrons are consdered explctly wth core electrons beng accounted for mplctly by, e.g., reducng the nuclear charge accordngly. In general, a mnmum bass set of Slater-type orbtals s used for the valence electrons. Wth the computatonal resources of today, these approxmatons extend the applcablty of M methods from systems consstng of, say, a few hundreds of atoms to systems wth a few thousands of atoms *. Another advantage of sememprcal methods s that electron correlaton effects (see subsecton 1.1.3), whch are not ncluded n F theory, to some extent are accounted for ndrectly va the parameterzaton (expermental data of course nclude electron correlaton). The fundamental dsadvantage of sememprcal methods s, however, ther lack of transferablty,.e., they perform rather poorly when appled to systems for whch they have not been parameterzed Electron correlaton As noted above, the F method uses the approxmaton that each electron s movng n the average electrostatc feld created by the other electrons. Ths s, however, an dealzed descrpton snce the explct dependence of the moton of each electron on the nstantaneous postons of the other electrons thereby s neglected. Ths defcency s often expressed as F theory falng to properly account for electron correlaton effects. There are two types of electron correlaton. Exchange (Ferm) correlaton, whch concerns electrons wth parallel spns and s related to the Paul excluson prncple, s accounted for n F theory through the determnantal form of the wave functon. oulomb correlaton, whch concerns all electrons and arses from nterelectronc repulson, s on the other hand completely neglected n F theory due to the employed mean-feld approxmaton. * These estmates refer to calculatng the electronc energy for an organc molecule at a sngle geometry. 22

23 Wthn the space of a gven bass set, the dfference between the exact, non-relatvstc electronc energy and the F energy s referred to as the correlaton energy 13 E corr E E. (1.31) exact F The correlaton energy, whch takes on negatve values, s thus a measure of the error n the F method arsng from the neglect of oulomb correlaton. It s worthwhle emphaszng that the error s not gven relatve expermental data, but relatve the exact egenvalue of a non-relatvstc, electronc amltonan. In prncple, the correlaton energy can be thought of as havng two components resultng from dynamcal and statc (near-degeneracy) correlaton effects, respectvely. Dynamcal correlaton essentally corresponds to the r nteracton between electrons at short nterelectronc 1 dstances. It s precsely ths type of correlaton that the mean-feld approxmaton fals to capture. Statc correlaton, on the other hand, s a somewhat more subtle phenomenon, and corresponds to the nteracton and mxng of electronc states that le close n energy. The falure of F theory to descrbe statc correlatons can be attrbuted to the F wave functon beng represented by a sngle Slater determnant that cannot n a balanced fashon account for the two (or more) dfferent electronc confguratons that come nto play at near-degeneraces. Whle dynamcal electron correlaton s always present n a molecular system, statc correlaton effects for electronc ground states are manfested prmarly n open-shell molecules (e.g., radcals, bradcals, transton metal complexes) and n closed-shell molecules at non-equlbrum geometres ( 2 at large nternuclear dstances s a classcal example). For a closed-shell system, the use of UF theory generally mproves upon RF n the sense that a porton of the statc correlaton energy can be recovered, as evdenced by, e.g., studes of 2 dssocaton. 2 The correlaton energy s therefore most approprately defned as the error n the RF method. Unfortunately, when UF and RF wave functons are dfferent, the former s not an egenfuncton of the total 2 2 electron spn operator S, S UF S( S 1) UF. Ths means that an UF wave functon for a closed-shell system ( S 0) at a non-equlbrum geometry does not correspond to a pure snglet state, but may contan also contrbutons from hgher spn states. Ths s referred to as the snglet UF wave functon beng spn contamnated, and s an nevtable dsadvantage of UF theory that arses also when one normally would choose to use UF (.e., for open-shell systems). The degree of spn contamnaton can be assessed by consderng the devaton from the deal value of S ( S 1). Even though the F method may provde a qualtatvely correct descrpton of many chemcal phenomena nvolvng closed-shell molecules around ther equlbrum geometres, t s n general necessary to obtan 23

24 24 (parts of) the correlaton energy for achevng quanttatve accuracy n a quantum chemcal calculaton. Ths has stmulated the development of a wde varety of M methods that n dfferent ways account for dynamcal and/or statc electron correlaton effects. These methods are referred to as post-f methods, and often use the F wave functon as a reference wave functon for the treatment of electron correlaton. In the confguraton nteracton (I) method, the many-electron wave functon s expressed as a lnear combnaton of the F wave functon and excted Slater determnants T T T D D D S S S F F I c c c c. (1.32) Subscrpts S, D, T etc. denote sngly, doubly, trply etc. excted determnants, where S s obtaned by replacng one occuped M n the optmzed F wave functon wth one that s unoccuped; D s obtaned by replacng two occuped Ms n the optmzed F wave functon wth two that are unoccuped, etc. The respectve sums n the I expanson then nclude all such determnants that can be formed. In order to calculate the best possble I wave functon, the expanson coeffcents are varatonally optmzed (wthout reoptmzng the F Ms) wth the normalzaton constrant 1 I I, whch yelds the I matrx egenvalue equaton T D S F T D S F c c c c E c c c c T T D T S T F T T D D D S D F D T S D S S S F S T F D F S F F F (1.33) The matrx elements S F and F S are zero by vrtue of Brlloun s theorem. 14 Furthermore, snce contans only one and two-electron operators, all matrx elements of between Slater determnants whch dffer by more than two Ms (such as T F and F T ) are also zero. The lowest egenvalue of the I matrx then gves the I energy and the correspondng egenvector the expanson coeffcents for the I wave functon. The second lowest egenvalue, n turn, gves the I energy for the frst excted state, etc. If the I expanson of Eq. (1.32) ncludes all possble excted determnants of each type, the soluton to Eq. (1.33) consttutes an exact soluton (n the B approxmaton) to the non-relatvstc electronc problem wthn the space of the employed bass set. Ths s referred to as full I

25 (FI). owever, n computatonal practce, such calculatons are possble only for very small molecular systems (contanng around 10 electrons at most). It s therefore necessary to truncate the I expanson, typcally by consderng only sngly and doubly excted determnants. Ths method s commonly known as ISD. For systems n whch a sngle-reference F wave functon s qualtatvely accurate, ISD typcally recovers 80-90% of the correlaton energy. owever, f statc correlaton effects need to be consdered, most computatonally tractable sngle-reference I methods are napproprate snce the underlyng F wave functon n such cases s qualtatvely ncorrect. Instead, mult-reference I (MRI) methods are to be preferred. As the name suggests, these methods carry out a I calculaton nvolvng excted determnants derved from a reference wave functon ncludng contrbutons from several electronc confguratons. The requred reference wave functon can be calculated by means of mult-confguratonal selfconsstent feld (MSF) technques. ne of the most appealng features of I methods s that they as beng founded on the varatonal prncple provde an energy whch represents an upper bound to the exact electronc energy. Unfortunately, all I methods but FI are also assocated wth a sgnfcant undesrable feature: the lack of correct scalng of energy wth respect to the sze of the molecular system under study. Ths defcency s often expressed as truncated I methods falng to dsplay sze-extensvty, whch mples that the part of the correlaton energy that can actually be recovered decreases as the sze of the molecular system ncreases. As a consequence, a ISD calculaton on two nfntely separated water molecules wll not gve the same energy as twce the ISD energy of a sngle water molecule (whch wll be lower), a problem whch n turn s referred to as a lack of szeconsstency. omputatonally tractable, sze-extensve electron correlaton M methods have nevertheless been developed wthn the framework of Møller-Plesset (MP) perturbaton theory and coupled cluster () theory. These methods (e.g., MP2, MP4, SD, SD(T)) are non-varatonal and may hence yeld energes below the exact electronc energy. Due to the use of lmted bass sets, ths however rarely happens n computatonal practce. Standard MP and methods are based on a sngle-reference F wave functon. Ths means that they prmarly account for dynamcal correlaton effects. To account for electron correlaton usng ab nto M methods s a computatonally demandng task not only from the pont of vew of PU tme, but also n terms of memory and dsk space requrements. Furthermore, the computatonal cost of correlated ab nto M methods scales (wth respect to bass set sze) ncreasngly unfavourably as one proceeds to use ncreasngly accurate methods. For example, n the large-system lmt, MP2, ISD, SD, and SD(T) dsplay M 5, M 6, M 6, and M 7 scalng, respectvely. As a result, MP2 cannot at present be routnely appled (e.g., 25

26 for geometry optmzatons and frequency calculatons) to organc molecules wth more than a few tens of atoms, whereas SD(T) has a lmtaton of around ten atoms. In order to perform correlated quantum chemcal calculatons on large molecular systems, t s hence crucal to smplfy the treatment of electron correlaton effects. Durng the past twenty years, developments wthn the feld of DFT have led to a plethora of methods that, albet beng assocated wth computatonal requrements comparable to those of F, challenge and n many cases outperform correlated ab nto M methods. In the next secton, the central concepts and approxmatons underlyng the DFT methods used n ths thess wll be presented. 1.2 Densty functonal theory The basc (ndependent) varable of the M methods outlned above s the electronc wave functon. For an -electron system, the wave functon depends on 4 (3 spatal and spn) coordnates. From a general vewpont, the (relatve) complexty of M methods may therefore be regarded as an nevtable consequence. DFT methods, on the other hand, use the electron densty (r) as the basc varable. Ths quantty depends on 3 spatal coordnates, regardless of the number of electrons. Even though the reducton n the number of degrees of freedom as such not necessarly leads to a smplfed formalsm, ths suggests a possblty that quantum chemcal methods founded on DFT wll be less ntrcate than M methods. The theoretcal ustfcaton for the use of (r) to extract molecular propertes n quantum chemcal calculatons was gven by ohenberg and Kohn (K) 15 n The concept of functonals wll occur frequently n the followng subsectons. Loosely, a functonal can be thought of as a functon that transforms another functon nto a number. We wll adopt the notaton F F f ( r) to denote that F s a functonal of f(r) The ohenberg-kohn theorems The problem at hand s, agan, the electronc Schrödnger equaton E wth (1.34) T V e ee V ne, 26

27 and the am s to calculate the electronc energy wthout frst havng to calculate the wave functon. The electronc energy can be wrtten as E r) V ( r) dr T V, (1.35) ( ne e ee where the electron densty s defned by 2 ( ) ( r, r,, r ) dr dr dr (1.36) r and the wave functon s assumed to be normalzed. The electronc amltonan s determned by the number of electrons and the external potental V ne ( r) due to the nucle. Therefore, the wave functon and, consequently, all molecular propertes are determned by and V ne ( r). It s thus possble to express E as a functonal of and V ne ( r) E E,V ne. (1.37) In ther poneerng work, K frst showed that, for non-degenerate electronc ground states, the external potental s unquely determned by the electron densty. Ths s referred to as the frst K theorem. Snce the electron densty also determnes the number of electrons through ( r) dr (whch follows trvally from Eq. (1.36)), the electronc energy may hence be represented as a functonal of the electron densty alone E E. (1.38) It s worthwhle emphaszng that the external potental s not restrcted to nclude the oulomb potental from the nucle only. For example, external electrc and magnetc felds may well be ncluded too. Eq. (1.38) s commonly wrtten as E ( r ) V ( r dr F, (1.39) ne ) K where F K s a functonal accountng for knetc electron energy and electron-electron repulson energy V ee F K T V. (1.40) e ee, n turn, can be decomposed nto classcal and non-classcal parts 27

28 ee J V, (1.41) xc J represents classcal electrostatc repulson and where xc s the exchange-correlaton energy functonal. K moreover showed that t s possble to nvoke the varatonal prncple for the ground-state electronc energy as a functonal of the electron densty. Specfcally, for any tral densty (r) such that (r) dr, t holds that E E 0 exact, (1.42) where E 0 exact s the exact ground-state energy obtaned from the exact ground-state densty. Assumng that E s dfferentable, applcaton of the varatonal prncple to Eq. (1.39), wth the constrant ( r) dr accounted for usng a Lagrange multpler, yelds the Euler-Lagrange equaton FK V ne. (1.43) s the chemcal potental. Eq. (1.43) consttutes the basc workng equaton of Kohn-Sham DFT. 16 Wthn ths framework, the K theorems are used to formulate the computatonal methodology that underles all modern DFT methods The Kohn-Sham equatons For a unform gas of non-nteractng electrons, t can be shown that and the exchange part of xc are gven as T e T e F 5 / 3 ( r) dr T TF F 3 10 (3 ) 2 2 / 3 x / 3 4 / x ( r) dr K D x ( ) (1.44) 28

29 T and TF K D are referred to as the Thomas-Ferm knetc energy functonal 17,18 and the Drac exchange energy functonal, 19 respectvely. eglectng electron correlaton, the use of these n Eq. (1.39) defnes the socalled Thomas-Ferm-Drac energy functonal E E TFD ( ) Vne ( r dr TTF J K D TFD ) r. (1.45) Ths model has to some extent been successfully appled n sold state physcs (n partcular to metallc systems where the underlyng unform nonnteractng electron-gas assumpton consttutes a reasonable frst approxmaton). For molecular systems, ths approxmaton s however far too crude, resultng prmarly n a poor representaton of the (real) knetc energy. As a consequence, chemcal bonds are not predcted. ne way to mprove the model s to consder a non-unform electron gas by ncludng n the knetc energy functonal terms dependng on dervatves of the densty (.e., non-local terms). A frst-order correcton was frst derved by Wezsacker, 20 and later hgher-order correctons have been obtaned as well. 21,22 Up to fourth order, these correctons gradually mprove upon TFD theory and enable the descrpton of chemcal bonds. evertheless, the resultng models are wth respect to both accuracy and computatonal effcency nferor to M methods. Rather than tryng to explctly derve an approprate expresson for T e, Kohn and Sham (KS) 16 devsed a scheme n whch T e s decomposed nto two terms; one leadng term whch can be calculated exactly and one small correcton term whch can be accounted for ndrectly. The frst step s the ntroducton of a fcttous reference system of nonnteractng electrons movng n an effectve external potental V eff ( r), constructed n such a way that the electron densty of the reference system equals that of the real system. It follows from the frst K theorem that a potental meetng ths requrement actually exsts. The Euler-Lagrange equaton for the reference system s (cf. Eq. 1.43) TS S V eff, (1.46) where T S s the knetc energy functonal for the non-nteractng electrons. V eff ( r) s then constructed so that the chemcal potental of the reference system equals that of the real system, leadng to (after makng use of Eqs. (1.40) and (1.41) ) 29

30 J V eff Vne ( Te TS xc ). (1.47) By defnng the exchange-correlaton functonal E T T and the exchange-correlaton potental xc V r E xc e xc S xc, one has V eff V ne J V xc, (1.48) and for the energy functonal for the real system E ( ) Vne ( r) dr TS J Exc r. (1.49) In ths last expresson, the knetc energy of the non-nteractng electrons T S occurs explctly, whereas the dfference Te TS (the correcton term) s absorbed n E xc. The crucal advantage of the Kohn-Sham approach s now that, by usng a wave functon descrpton for the reference system whose Schrödnger equaton can be separated nto exactly solvable one-electron equatons of the form 1 2 V ), (1.50) 2 ( eff where { ( r)} are referred to as Kohn-Sham orbtals the exact knetc energy of the non-nteractng electrons s straghtforwardly obtaned as 1 2 TS (1.51) 2 once the one-electron equatons have been solved. Furthermore, gven the approprate effectve external potental, the electron densty 2 ( r) ( r) (1.52) equals that of the real system. By nsertng ths densty nto Eq. (1.49), whch also can be wrtten as 30

31 J E E r) V ( r dr, (1.53) xc ( xc ) the electronc energy of the real system s obtaned. Even though the ntroducton of orbtals n a way s nconsstent wth the orgnal am of not havng to consder any other varable than the electron densty, t should be emphaszed that Eq. (1.50) represents a one-electron problem. The Kohn- Sham methodology can be summarzed as follows. 1. Gven an ntal electron densty for the reference system, construct the effectve external potental by means of Eq. (1.48). 2. Solve Eq. (1.50) usng ths potental. 3. Determne the electron densty due to the Kohn-Sham orbtals by means of Eq. (1.52). 4. Use ths densty to construct a new effectve external potental and repeat steps 2, 3, 4, etc. The calculatons are consdered converged when the denstes from two consecutve teratons are equal to wthn a certan tolerance. The resultng densty s then that of the real system. Eqs. (1.48), (1.50), and (1.52) are commonly known as the Kohn-Sham equatons. In order to make use of the above methodology, an expresson for the exchange-correlaton functonal E xc Te TS xc s needed. Unfortunately, the exact analytc form of ths s not known, whch means that approxmate expressons have to be used. It s mportant to note that the theory would be formally exact (but not necessarly amenable to actual calculatons) f the precse form of E xc was known. The key to the success of Kohn-Sham DFT s two-fold. Frst, the knetc energy correcton term of E xc s small, whch means that the problem of constructng an approxmate functonal essentally s a problem of expressng exchange and correlaton energes n terms of electron denstes. Secondly, functonals that perform well at a low computatonal cost can actually be constructed Exchange-correlaton functonals There are essentally two man strateges for obtanng approxmate exchange-correlaton functonals. ne s more emprc n nature and, gven a basc form for the terms nvolvng the electron densty, nvolves the ntroducton and fttng of a number of parameters to expermental or accurately calculated (by M methods) energes or denstes. The other reles more on attempts to make sure that the resultng functonal satsfy 31

32 known exact constrants. For example, t should hold that lmv xc ( r ) 1/ r. r It s of course possble to optmze functonals usng both strateges. The exchange-correlaton functonal s usually decomposed nto separate exchange and correlaton parts Ex Ec ( r) x ( r) dr ( r) c ( r E xc ) dr, (1.54) x and where c are the exchange and correlaton energy densty functonal, respectvely. Snce the procedure by whch the respectve part s obtaned n general does not ensure a well-defned separaton of exchange and correlaton contrbutons, t s clear that ths decomposton s only approxmate. In the local densty approxmaton (LDA), the functonals are based on the unform electron gas model. Ths means that the exchange part derves from the Drac formula Eq. (1.44) E LDA x LDA x x x 1/ 3 4 / 3 ( r ) dr (1.55) x 3 3 ( ) 4 1/ 3 As for the correlaton part, no analytc dervaton has yet been reported for the unform electron gas (the smplest of model systems). owever, by means of quantum Monte arlo methods, numercal correlaton energes have been obtaned for unform electron gases at a number of dfferent denstes. 23 These energes have been used as the bass for the development of two dfferent correlaton functonals commonly referred to as VW and VW5, respectvely. 24 The use of the Drac formula (sometmes wth a slghtly modfed value for x ) n combnaton wth ether VW or VW5 then defnes an LDA calculaton. In computatonal practce, LDA methods rarely outperform F, and have not found wdespread use as a quanttatve tool n quantum chemstry. In order to mprove upon the LDA, functonals that depend on both the densty and the gradent of the densty have been ntroduced. Ths s referred to as the generalzed gradent approxmaton (GGA). Many GGA functonals (for both exchange and correlaton) are constructed by addng a correcton term to the correspondng LDA functonal 32

33 LDA GGA x/c x/c x/c. (1.56) 4 / 3 Rather than beng a functonal of the absolute value of the gradent, the correcton term s a functonal of a dmensonless reduced gradent. Functonals of ths form have most notably been developed by Becke 25,26 and by Perdew and co-workers. 27,28,29,30,31,32 As for exchange, the most popular functonal (commonly known as B88 or B) s due to Becke 26 B88 x x 4 / 3 1/ 3 2 x 1 6xsnh 1 x (1.57) The parameter was determned through a ft to exact exchange energes for sx noble gas atoms. It has been stated 33 that the ntroducton of B88 was responsble for the acceptance of DFT as a valuable tool for computatonal chemstry. In 1988, Lee, Yang, and Parr presented a GGA correlaton functonal (commonly known as LYP) whch does not nclude any LDA component. 34 Ths functonal s based on the work by olle and Salvett, who derved an approxmate correlaton energy formula for helum n terms of densty matrces. 35 Lee, Yang, and Parr then turned ths formula nto a functonal nvolvng not only the densty, but also the gradent and laplacan of the densty. Shortly thereafter, Mehlch et al. were able to remove the laplacan terms whch are cumbersome to calculate by ntegraton by parts. 36 The LYP functonal has four emprcal parameters, all of whch stem from the orgnal formula of olle and Salvett. 35 The prmary advantage of LYP s that t consders the complete correlaton energy wthout makng any reference to the unform electron gas. Gven that F theory deals adequately wth exchange, t seems lke a sound strategy to somehow try to nclude F exchange n the DFT formalsm. The adabatc connecton method (AM) 37 provdes a theoretcal motvaton for the addton of F exchange to a general exchangecorrelaton functonal. The key equaton of the AM makes t possble to express the exchange-correlaton energy n terms of a parameter, 0 1, whose value gves the extent of nterelectronc nteracton rangng from none ( 0 ) to fully nteractng ( 1). Explctly, 33

34 1 E ( ) V ( ) ( d. (1.58) xc xc ) 0 In order to make use of Eq. (1.58), one may as a frst approxmaton crudely assume that V depends lnearly on. Then xc 1 E xc ( (0) Vxc(0) (0) (1) Vxc(1) (1) ). 2 (1.59) The frst term n Eq. (1.59) refers to a system of non-nteractng electrons, for whch the exact wave functon s a Slater determnant of Kohn-Sham orbtals. The exchange energy due to a Slater determnant can be obtaned F F exactly by F theory. ence, ( 0) V xc (0) (0) Ex, where E x s calculated usng Kohn-Sham orbtals. The second term n Eq. (1.59) can, of course, be approxmated usng any densty functonal E. Therefore, DFT xc E 1 DFT xc E xc F ( Ex ). (1.60) 2 Ths result s ndcatve of F exchange beng a natural component of densty functonals. Functonals that ndeed nclude F exchange are referred to as hybrd densty functonals. In 1993, Becke showed that usng DFT LDA Exc Exc yelds a functonal (commonly known as Becke half-and-half or &) of promsng accuracy. 38 In addton to addng F exchange to an LDA functonal, Becke moreover generalzed the & methodology by developng a functonal (commonly referred to as B3PW91) addng a sutable porton of F exchange to a GGA functonal 39 E B3PW91 xc 1 a) LDA F B88 LDA PW91 ( Ex aex bex Ec cec. (1.61) B88 PW91 E x and E c are the correcton terms of B88 and PW91 (the 1991 GGA correlaton functonal due to Perdew and Wang 29 ), respectvely. The a, b, and c parameters were determned through a ft to expermental data. B3PW91 was shortly thereafter modfed wth respect to the correlaton part by ntroducng LYP n place of PW91, yeldng the B3LYP hybrd densty functonal. 40 Snce LYP has no LDA component, B3LYP takes the form E B3LYP LDA F B88 xc 1 a) Ex aex bex (1 c) LDA LYP ( Ec cec. (1.62) 34

35 B3LYP has found wdespread use as a quanttatve tool n quantum chemstry, and has n a number of benchmark studes been shown to provde, e.g., energes (atomzaton energes, onzaton potentals, electron affntes, enthalpes of actvaton, etc.) and geometres whch compare very favourably wth those of (hghly) correlated ab nto methods. 41 Furthermore, as s generally the case n DFT, ts bass set requrements s rather modest Tme-dependent densty functonal theory DFT as developed by Kohn and co-workers s essentally a theory for electronc ground states. The formalsm can, n analogy wth F theory, n prncple be extended to nclude any excted state whch s the lowest-energy state of a gven symmetry and s well-descrbed by a sngle-determnantal wave functon, but does not allow for a general treatment of electroncally excted states. Tme-dependent (TD) DFT, whch ncorporates tmedependent external potentals, consttutes a remedy for ths defcency. TD-DFT s by now a well-establshed theory. The K theorems and the Kohn-Sham methodology have been generalzed to nclude also systems subect to TD external potentals, 42,43 and useful approxmatons to TD exchange-correlaton potentals have been ntroduced. 44,45 The TD Kohn- Sham equatons take the form V eff ( r, t) ( r, t) ( r, t) t V eff ( r, t) V eff ( r, t) V ( t) (1.63) V eff ( r, t) V ne J ( r) ( r, t) V xc ( r, t) where V (t) s the appled feld. ormally, the adabatc approxmaton V xc r, t) V ( r) (1.64) ( xc t s ntroduced. Ths makes t possble to use the exchange-correlaton functonals of statc DFT, and to consder the dependence of the energy on the densty at fxed tme t. The adabatc approxmaton appears to work best for low-lyng excted states. 46 A TD formalsm enables the computaton of the dynamc polarzablty (as well as other frequency-dependent response functons), whch has the mportant property that t dverges (has poles) at electronc 35