Proposal for a modified Møller-Plesset perturbation theory

P oposal o a modi ied Mølle -Plesse pe u ba ion heo y Alejand o Cabo,1F ancisco Cla o,2Edua do Menéndez-P oupin,3,*No ge C uz-He nández,4and Ja ie Fe nández-Sanz4 1G upo de Física Teó ica, Ins i u o de Cibe né ica, Ma ema ica y Física, Calle E, No. 309, Vedado, La Habana, Cuba 2Facul ad de Física, Pon i icia Uni e sidad Ca ólica de Chile, Vicuña Mackenna 4860 San iago, Chile 3Depa amen o de Física, Facul ad de Ciencias, Uni e sidad de Chile, Las Palme as 3425, 780-0024 Ñuñoa, San iago, Chile 4Facul ad de Química, Depa amen o de Química Física, Uni e sidad de Se illa, E-41012 Se illa, Spain 共Recei ed 2 June 2005; published 23 Janua y 2006兲 A modi ied e sion o he Mølle -Plesse app oach o ob aining he co ela ion ene gy associa ed wi h a Ha ee-Fock g ound s a e is p oposed. The me hod is es ed in a model o in e ac ing e mions ha allows o an exac solu ion. Using up o hi d o de e ms imp o ed esul s a e ob ained, e en in he limi o loosely bound pa icles. Tes ed in molecules as well, he modi ied me hod appea s o gi e imp o ed esul s in sym- me ic sys ems. DOI: 10.1103/PhysRe A.73.012510 PACS numbe 共s兲: 31.15.Md, 03.65.Ge, 31.25.⫺ , 02.70.⫺c I. INTRODUCTION The s udy o in e ac ing many-pa icle sys ems is se i- ously cons ained by he la ge dimension o he Hilbe space. Se e al app oxima ion schemes ha e been de ised o e he yea s, among which he Ha ee-Fock 共HF兲me hod is one o he oldes and mos ui ul, no ably in a omic and molecula physics. Because i ea s in e ac ions in a mean ield way pa icle co ela ions a e le ou , howe e , a sho - coming ha can limi se e ely he alidi y o i s esul s. One may imp o e o e HF by ea ing co ela ions as a pe u ba- ion. In he so-called Mølle -Plesse me hod, a Rayleigh- Sch ödinge pe u ba i e expansion ha is na u ally sug- ges ed by he same s uc u e o he HF solu ion 关1–4兴is adop ed. This o malism educes he co ela ion ene gy o an in ini e se ies in he pe u ba ion, o which only he i s ew e ms need o be compu ed in p ac ice. This scheme has been used o a long ime as a good s a ing poin o s udy co e- la ion e ec s in molecula sys ems. Howe e , his me hod is use ul only i he pe u ba ion se ies is apidly con e gen , which is no always ue 关5兴. Failu es o he Mølle -Plesse me hod ha e been documen ed e en o small molecules 关5–7兴. In his pape we p esen a a ia ion o he Mølle - Plesse app oach ha in some cases appea s o gi e accu a e esul s in low-o de pe u ba ion schemes. Conside he Hamil onian o a gene al sys em o iden ical pa icles in e ac ing h ough a pai po en ial H=兺 i h共i兲+兺 i,j⬎i 共i,j兲,共1兲 whe e h共i兲=h共 i兲is he sum o he one-pa icle kine ic en- e gy plus ex e nal po en ial ene gy, and 共i,j兲= 共 i, j兲is he pai -in e ac ion po en ial. The HF app oxima ion o he g ound s a e o a sys em o iden ical e mions leads o a a ia ional wa e unc ion ⌽in he o m o a Sla e de e mi- nan , cons uc ed wi h one-pa icle o bi als ha sa is y he sel -consis en ield 共SCF兲equa ions 再 h共i兲+兺 b 关Fb共i兲−Kb共i兲兴 冎 ␾ n共i兲= ⑀ n ␾ n共i兲,共2兲 whe e Fband Kba e he Coulomb and exchange ope a o s, espec i ely 关4兴. He e and in wha ollows he summa ion o e indices a,b,c un o e all occupied s a es, while n uns o e all possible s a es. The eigen alues sa is y he ela ion ⑀ n=具n兩h兩n典+兺 b 具nb储nb典,共3兲 whe e 具n兩h兩n典=兰 ␾ n共1兲*h共1兲 ␾ n共1兲d1 a e he ma ix elemen o one-pa icle ope a o hin he basis 兵 ␾ n其, and 具mn储mn典=具mn兩mn典−具mn兩nm典,共4兲 具mn兩pq典= 冕 ␾ m共1兲* ␾ n共2兲* 共1,2兲 ␾ p共1兲 ␾ q共2兲d1d2共5兲 a e he ma ix elemen s o he pai -in e ac ion ope a o . In he abo e exp ession 1 and 2 ep esen he one-elec on a i- ables o coo dina e and spin. The ene gy o he HF s a e is EHF =兺 a 具a兩h兩a典+1 2兺 a,b 具ab储ab典=兺 a ⑀ a−1 2兺 a,b 具ab储ab典, 共6兲 whe e o ob ain 共6兲Eq. 共3兲is used 关4兴. The many-pa icles ope a o HHF =兺 n ⑀ nc ˆn †c ˆn−1 2兺 a,b 具ab储ab典共7兲 is a na u al choice o an app oxima e independen -pa icles desc ip ion o he in e ac ing sys em. In Eq. 共7兲c ˆn †共c ˆn兲is he c ea ion 共annihila ion兲ope a o o a pa icle in he s a e ␾ n. HHF, known as Ha ee-Fock Hamil onian, is diagonal in he basis o all he Sla e de e minan s composed wi h he HF o bi als, and i s g ound s a e ene gy is he HF ene gy EHF. This Hamil onian ope a o is he s a ing poin o cons uc a pe u ba ion expansion o he co ela ion ene gy using as pe u ba ion VHF=H−HHF 关4兴. Hence, he ene gy o he in- e ac ing elec on sys em is ob ained as a se ies *Elec onic add ess: [email p o ec ed] PHYSICAL REVIEW A 73, 012510 共2006兲 1050-2947/2006/73共1兲/012510共5兲/$23.00 ©2006 The Ame ican Physical Socie y012510-1 E=E共0兲+E共1兲+E共2兲+E共3兲+¯.共8兲 Wi h he choice 共7兲 o he ze o-o de Hamil onian, we ha e ha E共0兲=EHF, and E共1兲=0. A well known ea u e o he HF heo y is ha he many body g ound s a e ene gy EHF is no he ba e sum o ene gies o illed single-pa icle o bi als. In e ac ions a e coun ed wice and his o e es ima ion is co ec ed by sub ac ing he cons an explici in Eq. 共6兲. A second choice o a Hamil- onian is also possible, howe e . Equa ion 共6兲may be w i en in he al e na i e o m, EHF =兺 a 冉 ⑀ a−1 2兺 b 具ab储ab典 冊 ,共9兲 sugges ing as an al e na i e Hamil onian also diagonal in he HF o bi als HMHF =兺 n ⑀ ˜ nc ˆn †c ˆn,共10兲 whe e ⑀ ˜ n= ⑀ n−1 2兺 b 具nb储nb典=1 2共 ⑀ n+具n兩h兩n典兲.共11兲 In 共10兲 he ope a o s c ˆn †and c ˆna e he same as in 共7兲. Hence, HMHF and HHF ha e he same eigen unc ions, and he same g ound s a e ene gy, bu di e en exci a ion ene gies. The co ela ion ene gy can again be desc ibed in e ms o a pe - u ba ion, his ime o he o m VMHF=H−HMHF. Fo in- s ance, he second-o de co ec ion o he HF g ound s a e ene gy has he o m 关4兴 E共2兲=兺 ⌽⬘ 円具⌽兩H兩⌽⬘典円2 EHF −E⌽⬘ =1 4兺 ab s 兩具ab储 s典兩2 EHF −E⌽⬘ .共12兲 This is he i s ini e co ec ion in he Rayleigh-Sch ödinge pe u ba ion expansion since o ei he choice o he Hamil- onian he i s o de e m anishes. Only doubly exci ed s a es ⌽⬘=⌽ab s 共 ␾ a eplaced by ␾ and ␾ b eplaced by ␾ sin he Sla e de e minan ⌽, wi h ⑀ , ⑀ sabo e he Fe mi ene gy兲 con ibu e 关4兴. The choice o he ze o-o de Hamil onian a - ec s he exci a ion ene gies in he denomina o , which ha e he o m EHF −E⌽ab s = 再 ⑀ a+ ⑀ b− ⑀ − ⑀ s,i H0=HHF, ⑀ ˜ a+ ⑀ ˜ b− ⑀ ˜ − ⑀ ˜ s,i H0=HMHF. 冎 共13兲 Replacing in Eq. 共12兲bo h o ms clea ly lead o di e en nume ical alues. The analysis o he hi d- and highe -o de e ms in he pe u ba ion expansion yields addi ional modi i- ca ions. In he hi d-o de co ec ion E共3兲, besides he e- placemen s o ⑀ m→ ⑀ ˜ min he s anda d Mølle -Plesse exp es- sion 关4兴, an addi ional e m ⌬E共3兲appea s ⌬E共3兲=−E共2兲−1 4兺 ab s 共h +hss −haa −hbb兲兩具ab储 s典兩2 共 ⑀ ˜ + ⑀ ˜ s− ⑀ ˜ a− ⑀ ˜ b兲2, 共14兲 whe e hnn=具n兩h兩n典. Cu iously, he i s elemen o Eq. 共14兲 cancels he second-o de ene gy co ec ion. Ye , he sum ha ollows con ains he second-o de ene gy as well, wi h a plus sign, so ha ⌬E共3兲is hi d o de in he in e ac ion. TABLE I. G ound s a e ene gies o a sys em o wo ha monically con ined spin-1 2pa icles in e ac ing h ough a ha monic po en ial o s eng h k. Se e al app oxima ions a e included: Ha ee-Fock 共HF兲, Mølle - Plesse pe u ba ion heo y o o de s n=2, 3 共MPn兲, modi ied MPn共MMPn兲, as well as he exac alues. Repulsi e in e ac ions a e ep esen ed by nega i e alues o he elas ic cons an k. kHF MP2 MP3 MMP2 MMP3 Exac −0.25 1.732 1.655 1.836 1.702 1.710 1.707 −0.24 1.744 1.681 1.803 1.717 1.724 1.721 −0.22 1.766 1.725 1.784 1.745 1.750 1.748 −0.20 1.789 1.760 1.791 1.772 1.776 1.775 −0.18 1.811 1.791 1.808 1.798 1.801 1.800 −0.16 1.833 1.819 1.828 1.823 1.825 1.825 −0.09 1.908 1.905 1.906 1.905 1.906 1.906 −0.04 1.960 1.959 1.959 1.959 1.959 1.959 −0.01 1.990 1.990 1.990 1.990 1.990 1.990 0.00 2.000 2.000 2.000 2.000 2.000 2.000 0.04 2.040 2.039 2.039 2.039 2.039 2.039 0.16 2.154 2.150 2.149 2.149 2.149 2.149 0.36 2.332 2.319 2.314 2.316 2.313 2.311 0.64 2.561 2.534 2.522 2.525 2.516 2.510 1.00 2.829 2.784 2.762 2.767 2.749 2.732 CABO e al. PHYSICAL REVIEW A 73, 012510 共2006兲 012510-2 In o de o assess he con enience o ei he o mula ion o ob aining he co ela ion ene gy we ha e applied hem o wo sys ems o known exac solu ions, and o a small num- be o molecules. The ollowing sec ions a e de o ed o his analysis. II. EXACT MODELS Ou i s model sys em is a wo spin-1 2pa icle in a bidi- mensional ha monic po en ial, in e ac ing h ough a ha - monic o ce. This is an in e ac ing sys em in ol ing iden ical e mions ha is sol ed exac ly 关8兴. The Hamil onian is gi en by 共1兲wi h h共i兲=1 2共−ⵜi 2+ i 2兲, 共i,j兲=1 2k共 i− j兲2.共15兲 He e, he coo dina es and he ene gy a e gi en in he oscil- la o uni s o he con inemen po en ial. The exac g ound s a e ene gy is E0=1+冑1+2k.共16兲 The model con ains he pa ame e k ha allows he s udy o a ac i e 共k⬎0兲as well as epulsi e 共k⬍0兲in e ac ions. Equa ion 共16兲shows ha k=−0.5 is he lowes alue o which a bound s a e exis s. To ob ain a nume ical solu ion a he Ha ee-Fock le el, we use a basis o nonin e ac ing ha monic oscilla o s eigen- unc ions ␾ nx,ny共x,y兲= ␸ nx共x兲 ␸ ny共y兲,0艋nx+ny艋5, 共17兲 whe e he ␸ 共x兲a e he usual one-dimensional ha monic os- cilla o eigen unc ions. The se 共17兲is an exac solu ion when k=0. The HF solu ion is ob ained by sol ing he SCF equa ions 共2兲in a spin es ic ed con igu a ion, i.e., con- s aining he spa ial pa o he wa e unc ion o be equal o he wo occupied s a es wi h opposi e spins 关4兴. Table I shows he g ound s a e ene gy o he Hamil onian 共15兲calcula ed using he s anda d Mølle -Plesse pe u ba- ion heo y o o de n=2, 3 共MPn兲, and ou modi ied o m, MMPn. Figu e 1 shows he exac co ela ion ene gy Eco =EHF−E o di e en alues o k, oge he wi h esul s ob- ained o he wo choices o he HF Hamil onian, in he second and hi d o de o he pe u ba ion heo y. No ice ha MMPnyields be e esul s h oughou . No ice also ha he usual MPn ails poo ly in bo h o de s o app oxima ion when he sys em becomes mo e loosely bound, as kapp oaches he c i ical alue −0.5. In ac , con e gence p oblems p e en sol ing he HF SCF equa ions o kbeyond −0.25. By con- as , he MMPncon inue o be good app oxima ions e en in his ange o k. The simples molecula sys em is he hyd ogen molecule, o which we p esen he po en ial ene gy su ace 共PES兲in bo h spin es ic ed and un es ic ed con igu a ions 关8兴. The TABLE II. Bondleng hs 共in pm兲o dia omic molecules and he e o made in di e en app oxima ions. The basis used is cc-pVTZ. Exp HF MP2 MMP2 B3LYP H274.1 −0.7 −0.4 −0.2 +0.2 HF 91.7 −1.9 +0.03 −1.6 +0.5 OH+102.9 −2.2 −0.5 −1.6 +1.0 NH 103.6 −1.9 −0.9 −1.7 +0.3 NO+106.3 −3.6 +1.5 +2.7 −0.6 BH 123.2 −1.0 −1.5 −0.8 +0.0 FIG. 2. Po en ial ene gy su ace o he H2molecule as a unc- ion o he dis ance H-H. The basis se used is 6-31G**. Top: spin- es ic ed con igu a ion. Bo om: spin-un es ic ed con igu a ion. FIG. 1. Co ela ion ene gy o a sys em o wo ha monically con ined spin-pa icles in e ac ing h ough a ha monic po en ial o s eng h k. The exac esul is included, as well as co ec ions up o second and hi d o de o he Mølle -Plesse 共MP兲and modi ied Mølle -Plesse 共MMP兲choices o ze o h-o de Hamil onian. PROPOSAL FOR A MODIFIED MØLLER-PLESSET …PHYSICAL REVIEW A 73, 012510 共2006兲 012510-3 molecula elec onic Hamil onian in a omic uni s is gi en by 共1兲wi h h共i兲=−1 2ⵜi 2, 共i,j兲=1 兩 i− j兩.共18兲 The HF wa e unc ions a e ob ained by he expansion in s anda d basis se s o Gaussian unc ions 关4兴共e.g., 6-31G**, DZP, e c.兲and applying he SCF me hod as implemen ed in he code HONDO 关9兴. The MMP2 and MMP3 ene gies a e ob ained by a mino modi ica ion o he MP2 and MP3 ou- ines in he same code. The PES o H2has been calcula ed by Kolos and Wolniewicz 关10兴wi h ex eme p ecision and can be ega ded as exac , hus allowing a e e ence o app oxima i e me h- ods. I can be seen in Fig. 2 ha he MMP2 me hod imp o es he co ela ion ene gy o e he s anda d MP2 in he whole ange o dis ances. A well known ailu e o he spin es ic ed HF and de i ed me hods is he inabili y o desc ibe he dis- socia ion o he molecule, as he wo occupied s a es mus ha e he same o bi al wa e unc ion. This p oblem is sol ed by he spin-un es ic ed HF me hod, whe e he es ic ion ha opposi e spin o bi als sha e he o bi al pa is li ed. Again, ou modi ied scheme p o ides a solu ion close o he exac one o second o de . Howe e , i can be app ecia ed ha o un es ic ed calcula ions bo h he MMP2 and MP2 a e wo se han he es ic ed case in he ange 1.2–1.6 Å. This shows ha he p oblem o spin con amina ion a ec s bo h MP2 and MMP2. III. MOLECULAR SYSTEMS In his sec ion, we e alua e he MMP app oxima ion o se e al simple molecules, o which no exac wa e unc ions a e known. Table II shows he e o made a he second o de o app oxima ion in he bondleng h o a se o dia omic mol- ecules, as compa ed wi h he expe imen al leng h. I can be app ecia ed ha he bondleng h is imp o ed by he pe u ba- ion schemes sa ed o BH, whe e he MP2 inc eases he e o . MMP2 gi es be e esul s o H2and BH, while MP2 is mo e accu a e in he o he cases. Fo e e ence, he esul s wi h he popula densi y unc ional me hod B3LYP a e in- cluded in he able. Nex , we u n ou a en ion o he alue o he co ela ion ene gy a he expe imen al geome ies. Tables III and IV show a compa ison o he s anda d MPnene gies wi h he p esen MMPnene gies. Fo e e ence and compa ison, he alues wi h he mos accu a e con igu a ion in e ac ion 共CI兲 quan um chemis y me hod a e included. Fo hese mol- ecules i u ns ou ha he co ela ion ene gy is, in gene al, be e in MPn han in MMPnme hods. Only o he symme - ic pai s he me hods a e compa able, MMPngi ing a sligh ly be e esul o H2. The eason o he a he low co ela ion ene gy in he MMP2 can be explained in e ms o Eqs. 共12兲and 共13兲.I co esponds o an inc ease o he ene gy gap be ween he occupied and unoccupied s a es. This can be app ecia ed in Fig. 3, whe e he Fock eigen alues 共3兲and he modi ied eigen alues 共11兲 o he HF a e shown. As can be no iced, he co ec ion o he Fock eigen alues is g ea e o he 共dou- TABLE III. Co ela ion ene gy 共in a omic uni s兲 o se e al dia omic molecules a he expe imen al bondleng hs. The basis used is cc-pVTZ. EHF MP2 MP3 MMP2 MMP3 CI H2−1.133 −0.032 −0.037 −0.034 −0.038 −0.039 HF −100.058 −0.290 −0.290 −0.228 −0.268 −0.282 C2−75.402 −0.385 −0.347 −0.328 −0.346 −0.407 NH −54.875 −0.172 −0.191 −0.149 −0.179 −0.207 NO+−128.966 −0.439 −0.422 −0.369 −0.408 −0.445 OH+−74.866 −0.182 −0.202 −0.156 −0.189 −0.222 BH −25.130 −0.084 −0.101 −0.077 −0.095 −0.110 TABLE IV. Calcula ed ene gies 共in a omic uni s兲 o se e al dia omic and ia omic molecules a he expe imen al geome ies. Geome ies and basis unc ions a e speci ied as ollows: BH: =2.329 a0, DZP. HF: =1.733 a0,DZP.CH 2: =2.11 a0, ␪ =102.4°, DZP. H2O: =1.88973 a0, ␪ =104.5°, DZP. NH3: =1.911 65 a0, ␪ =106.7°, DZV. a0: Boh adius, DZV: Dunning’s 共9s,5p兲/共3s,2p兲basis se , DZP: DZV plus pola iza ion. EHF MP2 MP3 MMP2 MMP3 CI BH −25.124 −25.183 −25.199 −25.187 −25.198 −25.209 HF −100.048 −100.232 −100.233 −100.186 −100.220 −100.251 CH2−38.885 −38.998 −39.017 −38.992 −39.013 −39.027 H2O −76.041 −76.242 −76.247 −76.200 −76.234 −76.254 NH3−56.176 −56.290 −56.297 −56.268 −56.289 −56.304 CABO e al. PHYSICAL REVIEW A 73, 012510 共2006兲 012510-4 bly兲occupied le els 共le el艋5兲 han o he unoccupied ones. This is ela ed o he alues o he in e ac ion in eg als 具nb储nb典p esen in Eq. 共11兲. These in eg als a e gene ally posi i e and ha e la ge alues o he occupied le els be- cause he co esponding o bi als ␾ nha e mo e o e lap wi h he occupied o bi als 共 ␾ b兲. Fo example, in hyd ogen lou ide he occupied o bi als a e concen a ed a ound he F a om, while he lowes unoccupied molecula o bi al and highe molecula o bi als a e mo e dense a ound he H a om. Fo he C2molecule he e ec is less p onounced hough quali- a i ely simila . The case o H2and he wo- e mion oscilla o is di e en . In his case, he e a e only wo occupied s a es wi h he same ene gy, and he sum o e he bs a es in Eq. 共11兲con ains only one e m, as 具nn储nn典=0. On he o he hand, o he unoccupied s a es he bsum con ains wo e ms and he nega i e shi o hese eigen alues is la ge , hus educing he gap. This e ec is also p esen in la ge molecules, bu i is o e come by he di e ences in he spa ial dis ibu ion o he wa e unc ions, as discussed abo e. IV. CONCLUSIONS We ha e p oposed and analyzed c i ically an al e na i e o mula ion o he Mølle -Plesse pe u ba ion se ies o he g ound s a e o many-pa icle e mion sys ems. The new ze o h-o de Hamil onian has he same wa e unc ions as he Ha ee-Fock Hamil onian, bu wi h a di e en spec um. He e, he co ec ion o ob ain he co ec Ha ee-Fock en- e gy is included in he single-pa icle ene gies, ins ead o h ough a global ene gy shi . This opens a possibili y o including pa o he co ela ion ene gy in he ze o-o de Hamil onian, co ec ing a he same ime o he well known double coun o he in e ac ion ene gy p esen in he o dina y Ha ee-Fock o mula ion. We ha e applied he new me hod o a numbe o di e en sys ems and ha e ound ha he new me hod is as good o be e han he Mølle -Plesse scheme in symme ic sys ems. Howe e , o molecula sys ems wi h mo e han wo elec ons, he s anda d MP se ies seems o p o ide be e accu acy. I u ns ou ha he co ec ion o he one-pa icle ene gy le els is la ge o he occupied le els han o he unoccupied ones, inc easing he gap be ween occupied and unoccupied s a es. This educes he size o he lowes -o de con ibu ions o he co ela ion ene gy, dec eas- ing he a e o con e gence o he pe u ba ion se ies. How- e e , we no e ha in o de o ob ain he HF ene gy in he ze o-o de app oxima ion o he many-pa icles p oblem, Eq. 共11兲needs o be applied only o he occupied HF s a es. We s ess ha his choice is mo i a ed by he a emp o elimi- na e he double coun ing o he mean ield in e ac ion ene - gies in he HF ene gy. The e is no eason o he han simplic- i y o apply he same scheme o he emp y le els. This deg ee o eedom could be exploi ed o imp o e he con e gence o he many-pa icles pe u ba ion se ies. ACKNOWLEDGMENTS Suppo om FONDECYT G an Nos. 1020829, 1050293, 7020829, and he Thi d Wo ld Academy o Sci- ences, is g a e ully acknowledged. N.C-H. hanks he Span- ish Minis e io de Ciencia y Tecnología–Ramón y Cajal p o- g am. We also hank M. Dupuis o p o iding his code HONDO, and P. Fuen ealba and J. Ga za o help ul discus- sions. 关1兴C. Mølle and M. S. Plesse , Phys. Re . 46, 618 共1934兲. 关2兴J. M. Eisenbe g and W. G eine , Mic oscopic Theo y o he Nucleus 共No h-Holland, Ams e dam, 1972兲. 关3兴N. H. Ma ch, Sel -Consis en Fields in A oms 共Pe gamon P ess, Ox o d, 1975兲. 关4兴A. Szabo and N. S. 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Fock eigen alues 共uppe g oup兲and modi ied eigen al- ues 共lowes g oup兲o he molecule HF, calcula ed wi h di e en basis se s: STO-3G, 6-31G, and cc-pVTZ. The lines a e guides o he eyes. The dashed e ical line indica es he highes occupied molecula o bi al. PROPOSAL FOR A MODIFIED MØLLER-PLESSET …PHYSICAL REVIEW A 73, 012510 共2006兲 012510-5