PROBLEM SET #2 SOLUTIONS by Robert A. DiStasio Jr.

Transcription

1 PROBLM ST # SOLUTIOS by Robert A. DStso Jr. Q. -prtcle densty mtrces nd dempotency. () A mtrx M s sd to be dempotent f M M. Show from the bsc defnton tht the HF densty mtrx s dempotent when expressed n n orthonorml bss. An element of the HF densty mtrx s gven s (neglectng the fctor of two for the restrcted closed-shell HF densty mtrx): In mtrx form, () s smply equvlent to P * C C μ ν. () P C C () where C s the uped bloc of the MO coeffcent mtrx. To reformulte the nonorthogonl Roothn-Hll equtons FC SCε (3) nto the preferred egenvlue problem ~ ~ ~ FC Cε (4) the AO bss must be orthogonlzed. There re severl wys to orthogonlze the AO bss, but I wll only dscuss symmetrc orthogonlzton, s ths s rgubly the most often used procedure n computtonl quntum chemstry. In the symmetrc orthogonlzton procedure, the MO coeffcent mtrx n (3) s recst s ~ ~ C S C C S C (5) whch cn be substtuted nto (3) yeldng the egenvlue form of the Roothn-Hll equtons n (4) fter the followng lgebrc mnpulton:

2 S ( S FC SCε ~ ~ F( S C) S( S C) ε ~ ~ F( S C) S S( S C) ε. (6) ~ ~ FS ) C ( S SS ) Cε ~~ ~ FC Cε ~ In (6), the dressed Foc mtrx, F ~, ws clerly defned s F S FS nd S SS S ws used. In n orthonorml bss, therefore, the densty mtrx of () tes on the followng form: P C ~ C ~. (7) In order to prove tht the densty mtrx s dempotent n n orthonorml bss, we need to frst prove the followng proposton. Proposton : In n orthonorml bss, C ~ ~. C Proof. In order to prove tht C ~ ~, expnd the uped MO coeffcent mtrces s follows: C ~ C ~ C ~ * ~ ~ * ~ ~ ~ C ~ ~ μ Cμ μ μ μ μ δ (8) ~ μ ~ μ ~ μ whch proves the proposton tht C ~ ~. C QD Wth Proposton, we re now n poston to drectly prove tht the densty mtrx s dempotent n n orthonorml bss. Proposton : The densty mtrx s dempotent n n orthonorml bss wth P P. Proof. From the defnton of the densty mtrx n n orthonorml bss, gven by (7) bove, we hve ~ ~ P C C nd wnt to prove tht P P,.e., the dempotency condton n n orthonorml bss. In order to do so, expnd P s follows

3 P ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ( C C ) ( C C )( C C ) C ( C C ) C C C P (9) ~ n whch the results of Proposton were used n frly strghtforwrd mnner (c.f. (8)). QD (b) Derve the generlzton of the dempotency condton for the cse of nonorthonorml bss, wth overlp mtrx S. In nonorthonorml bss, the results of Proposton must be dusted to derve the generl dempotency condton for the -PDM. In nlogy to Q(), we hve two propostons whch requre proof. Proposton 3: In nonorthonorml bss, C. SC Proof. Usng the results of Proposton, nmely tht C ~ ~ n n orthonorml bss nd C the fct tht C ~ S C from (5), we hve: ~ ~ ( ) ( ) ( )( ) C C S C S C C S S C C S S C C SC () whch completes the proof. QD Wth Proposton 3 t our dsposl, we re now n poston to show tht PSP P s the generlzton of the dempotency condton for the cse of nonorthonorml bss. Proposton 4: The densty mtrx s dempotent n nonorthonorml bss wth PSP P. Proof. From the defnton of the densty mtrx n nonorthonorml bss, gven by () bove, we hve P C C nd we wsh to prove tht PSP P,.e., the dempotency condton n nonorthonorml bss. In order to do so, expnd PSP s follows PSP ( C C ) S( C C ) C ( C SC ) C C C P () whch completes the proof. QD

4 (c) Derve the egenvlues of n dempotent mtrx (you my ssume n orthonorml bss). Hence wrte n opertor form for t, nd descrbe the functon of ths opertor. If we wrte generl egenvlue equton for the densty opertor, Pˆ, s Pˆ ψ λ ψ, nd cnowledge tht the densty opertor s dempotent,.e., Pˆ Pˆ, we hve the followng two smultneous condtons: Pˆ ψ λ ψ () nd ˆ P ψ Pˆ ψ λ ψ. (3) However, we could wrte (3) s follows: P ˆ ψ Pˆ( Pˆ ψ ) Pˆ( λ ψ ) λ( Pˆ ψ ) λ( λ ψ ) λ ψ. (4) Combnng (3) nd (4), we note tht λ ψ λ ψ (5) whch mples tht λ λ, whch cn only be stsfed f λ or λ. Therefore, the egenvlues of the densty opertor re nd ths should me sense snce the mtrx form of the densty mtrx s dgonl wth s n the uped bloc nd s n the vrtul bloc (neglectng the fctor of two rsng from the restrcted closed-shell tretment). To fnd the egenvectors of the densty opertor, one must consder the consequences of hvng n egenspectrum comprsed of { λ λ,}. In essence, the cton of the densty opertor s to ether return n egenvector unchnged ( λ ) or to nnhlte n egenvector ( λ ). Ths defnes proecton opertor,.e., n opertor of the form: Xˆ p p (6) p

5 whch s nothng more thn the nlog of the unt dydc n functon spce. Therefore, the densty opertor s proecton opertor for the uped spce, O Spn{ ψ } wth dm( O ), wth the followng explct form: P ˆ ψ ψ. (7) When ctng on n uped MO such s ψ, n egenfuncton of Pˆ tht s contned wthn the uped spce, the densty opertor wll return ψ unchnged: ψ ψ ψ P ˆ ψ ψ δ ψ. (8) When ctng on vrtul (or unuped) MO such s ψ, n egenfuncton of Pˆ tht s OT contned wthn the uped spce, the densty opertor wll nnhlte ψ : Pˆ ψ ψ ψ ψ ψ δ (9) snce,. In other words, the vrtul spce, V Spn{ ψ c } wth dm( V ) vrt nd the uped spce re orthogonl spces, correspondng to vrtul spce, nmely, O V vrt. Of course, one could defne proecton opertor Q ˆ ψ ψ () from whch one cn show tht P ˆ Q ˆ ˆ. Why? (d) Suppose mtrx M (n n orthonorml bss) s pproxmtely dempotent, wth devton from dempotency tht s D (.e. M M D wth dempotent). Consder 3 3M M then trnsformed mtrx defned s M. Derve n expresson for ts devton from dempotency n terms of D (to ledng order), nd hence dscuss the cton of ths trnsformton. M

6 Snce M M D, wth M nd D the devton from dempotency, we hve: M M ( M D) M M D DM D () whch clerly llustrtes tht M devtes from dempotency to O(D),.e., lner n order of D. If we now consder the trnsformed mtrx defned s M 3M M 3, then the devton from dempotency of M cn be derved by computng M M n terms of D. Lets frst consder M, n whch we wll need to te successve powers of M. quton () provdes us wth n expresson for M, so we need to explctly consder 3 M now: M 3 ( M M M M D DM M M DM DM DM DM D )( M D D M M D) M D M D M D M D DM D D DM D 3 () where, the term tht ws cubc n D ws omtted before we proceed ny further n the dervton. From () nd (), we hve M s: 3 M 3M M M M D DM 3D M DM D M M D DM D. (3) From (3), we cn compute the devton from dempotency of M, by consderng the quntty M M drectly (eepng n mnd tht only the ledng terms n D must be consdered) M M 3( M D M D M M DM D DM DM D M DM D D ). (4) f the ledng terms n D,.e., O( D ) re retned. Hence, the trnsformton mtrx s now dempotent to frst order n the devton D from dempotency. In ths sense, the trnsformton mtrx hs purfed M nd s ths procedure s referred to s purfcton of n lmost dempotent densty mtrx (For n extremely redble detled revew of the centrl role the densty mtrx plys n quntum chemstry, see R. McWeeny, Phys. Rev. 3, 335 (96)).. Koopmn s theorem for onzton energes nd electron ffntes.

7 () Use the Slter-Condon mtrx element expressons to derve the Koopmn s expresson for the energy dfference between n optmzed n-electron HF determnnt, nd the (n-) electron determnnt obtned by removng electron from the th spn-orbtl. Let Ψ χχ χ χ be the optmzed -electron ground stte HF determnnt nd Ψ χχ χ χ χ be the - electron determnnt obtned from removng n electron from spn orbtl χ n the optmzed -electron ground stte HF determnnt. ote: The - electron determnnt hs OT been reoptmzed n the vrtonl sense, s t ws smply obtned by removng n electron from the HF -electron reference. The onzton potentl (IP) for removng n electron from the th spn orbtl s gven s the dfference between the energy of the - electron determnnt obtned from removng n electron from spn orbtl χ n the optmzed -electron ground stte HF determnnt nd the energy of the optmzed - electron ground stte HF determnnt,.e., IP. (5) Usng the Slter-Condon rules for mtrx elements, we need to determne the energes correspondng to both of these wvefunctons. Lets strt wth the optmzed -electron ground stte HF determnnt, n whch: Ψ ˆ H Ψ. (6) As ws dscussed n both lecture nd dscusson, the Hmltonn opertor s the sum of one- nd two-electron opertors ( Oˆ nd & Oˆ ). From the Slter-Condon rules, we hve Ψ Oˆ Ψ h (7)

8 Ψ Oˆ Ψ (8) where the sums re over ll uped spn orbtls. Usng (7) nd (8), we cn mmedtely wrte down the energy of the optmzed -electron HF ground stte determnnt s h (9) nd the energy of the - electron determnnt obtned by removng n electron from χ s h (3) snce the th spn orbtl s now unuped n the - electron determnnt. Substtuton of (9) nd (3) nto (5) yelds the desred expresson for the IP: IP h h h. (3) h Of course, the energy dfference n (3) s nothng more thn the negtve of the orbtl energy ε, thereby yeldng very smple, but nterestng physcl nterpretton of the cnoncl HF orbtl energy egenvlue, nmely, tht IP ε. (3) Ths result s nown s Koopmn s Theorem for onzton potentls. In the next prt, we wll nvestgte Koopmn s Theorem for electron ffntes. (b) Repet for the electron ffnty, ddng n extr electron to the th vrtul orbtl.

9 As bove n (), let χ χ χ χ Ψ be the optmzed -electron ground stte HF determnnt nd Ψ χ χ χ χ χ be the electron determnnt obtned from ddng n electron to spn orbtl χ n the optmzed -electron ground stte HF determnnt. ote: The electron determnnt hs OT been reoptmzed n the vrtonl sense, s t ws smply obtned by ddng n electron to the HF -electron reference. The electron ffnty (A) for ddng n electron to the th spn orbtl s gven s the dfference between the energy of the optmzed -electron ground stte HF determnnt nd the energy of the electron determnnt obtned from ddng n electron to spn orbtl χ n the optmzed -electron ground stte HF determnnt nd the energy of the optmzed -electron ground stte HF determnnt,.e., A. (33) Usng the Slter-Condon rules for mtrx elements gven by (7) nd (8) bove, we cn wrte the energy for the electron determnnt obtned from ddng n electron to spn orbtl n the optmzed -electron ground stte HF determnnt s: χ h h. (34) Usng the energy expresson for the optmzed -electron HF ground stte determnnt gven by (9) nd the energy ust computed (34) n the A expresson gven by (33) yelds: A h h h. (35) h

10 As seen bove n the dervton of Koopmn s Theorem for onzton potentls, the energy expresson n (35) s nothng more thn the negtve of the orbtl energy ε. Therefore, A ε (36) whch s Koopmn s Theorem for electron ffntes. (c) Dscuss how onzton potentls nd electron ffntes computed s the dfference of seprte HF clcultons on the neutrl nd on mght dffer from the Koopmn s estmte. On ths bss, how do you thn the qulty of the Koopmn s IP s nd A s would compre? In () nd (b), we re usng the so-clled frozen orbtl pproxmton,.e., ssumng tht the spn orbtls n the ± electron wvefunctons re the sme s the spn orbtls n the electron wvefuncton. By performng seprte reoptmzton of the ± electron wvefunctons, the vrtonl SCF procedure would yeld lower energes correspondng to set of MO coeffcents unquely tlored to the respectve onc speces. Therefore, neglect of relxton often overestmtes IPs nd underestmtes As computed usng Koopmn s theorem. There re two dstnct resons why estmtes of IPs re consderbly better thn As obtned n ths mnner. Frst, correlton effects (neglected t the HF level) tend to ncrese wth the number of electrons, thereby tendng to cncel the relxton error for IPs whle ddng to the relxton error for As. Second, the egenvlues of HF vrtul orbtls on neutrl molecules re lmost lwys postve, despte the fct tht most neutrl molecules wll form stble non, thereby ddng to the error n predctng As. Q3. Toy SCF clcultons on HeH. Use ether pper/clcultor, or wrte smll progrm, or use mtrx-wre mth envronment le MtLb (or ScLb). The mn nformton (hopefully the only nformton) you ll need s the followng: the overlp mtrx, S, the core Hmltonn mtrx, H (derved from netc energy, nd nucler ttrcton to He nd H), nd the symmetry dstnct non-zero -electron ntegrls n the mnml bss t the equlbrum geometry (R.463.u.). The frst functon s on He nd the nd s on H.

11 () Whle smple ntl guess s P, form better ntl guess densty mtrx by ssumng tht both electrons re n the He (s) orbtl. Wrte the P mtrx nd verfy tht t s properly dempotent. If the ntl guess ssumes tht both electrons re n the He (s) orbtl, then the uped bloc of the MO coeffcent mtrx tes on the followng form: C (37) from whch we cn compute the restrcted closed-shell HF densty mtrx s P C C ( ) (38) whch clerly corresponds to plcng electrons n the He (s) orbtl. The densty mtrx n (38) s clerly wrtten n nonorthonorml bss, so to demonstrte dempotency, we need to use the results of Proposton 4 bove,.e., tht PSP P. However, ths form must be further modfed for the restrcted closed-shell cse densty mtrx s follows. Snce the restrcted closed-shell HF densty mtrx s defned s P C C n (38), nsted of PSP P for the dempotency condton, we hve: PSP (CC ) S(CC ) 4C ( CSC ) C 4CC P. (39) Therefore, redefnng P P C C yelds P SP ( C C ) S( CC ) C ( CSC ) C CC P (4) whch defnes the generlzton of the dempotency condton for the restrcted closed-shell densty mtrx n nonorthonorml bss. So, usng the overlp mtrx gven n the problem S (4) we cn demonstrte dempotency s:

12 P SP P. (4) (b) Form the Foc mtrx ssocted wth ths ntl guess, nd evlute the totl energy ssocted wth ths ntl guess (don t forget nucler repulson!). Use the fct tht ths clculton s restrcted to do the mnmum clcultons (.e. C α C β so you only need F α, etc.). The Foc mtrx ssocted wth ths ntl guess s restrcted closed-shell Foc mtrx wth the followng form: F core H II P H core G (43) core where nd re the one- nd two-electron prts of the Foc mtrx,.e., H G core nucler H T V nd G II P [( ) ( μσ λν )] P. To compute the Foc mtrx from the guess densty gven n (38), one needs the netc energy mtrx ( ) gven s T T (44) He H the nucler-electron ttrcton potentl energy mtrces ( V nd V ) gven s V He (45) nd V H , (46).66 respectvely, s well s the set of two-centered ntegrls gven below (n Chemst s notton)

13 ( ) ( ) ( ).773 ( ).657 ( ).38. (47) ( ).7746 Usng (43), (44), (45), (46), nd (47), the Foc mtrx cn be computed s: F (48) To evlute the electronc RHF energy, we need the followng formul: RHF core P [ H F ] (49) from whch the totl RHF energy follows s TOT RHF V nn RHF M M α β > α Z α R Z αβ β. (5) From (38), (44), (45), (46), (48), nd (49), RHF s computed s hrtrees. When the nucler-nucler repulson energy, whch s equl to / hrtrees, s ncluded, the totl energy TOT hrtrees. (c) Solve the generlzed egenvlue problem to obtn new uped orbtl nd new densty mtrx. How dfferent re they from your ntl guess? To solve the generlzed egenvlue problem, we need to frst symmetrclly orthogonlze the Foc mtrx usng S. To compute S, frst dgonlze the overlp mtrx U SU.549 s.458 (5) te the nverse squre root of the egenvlues nd then undgonlze to form the desred S mtrx.3493 s (5).83

14 We cn now form the trnsformed Foc mtrx s: S Us U. (53) ~ F S FS. (54) The trnsformed Foc mtrx hs egenvlues, nmely, nd , nd to proceed forwrd towrd the ground stte energy, we must choose the lower egenvlue, Wht re we convergng to f we contnuously choose the hgher egenvlue? The egenvector correspondng to the lower egenvlue s the orthogonl MO coeffcent vector for the uped orbtl (usng the normlzton condton on the uped MO coeffcent): ~ C (55).479 Snce ~ ~ C S C C S C, we hve the nonorthonorml uped MO coeffcent vector s ~ C S C (56) whch gves rse to the followng new (nd mproved) densty mtrx: P C C ( ). (57) Clerly, the new uped MO coeffcents (56) nd new densty mtrx (57) re qute dfferent from our ntl guess, n whch both electrons were plced n the He (s) orbtl. There re severl wys to quntfy ths dfference (mxmum or RMS devtons mong elements of the densty mtrx for nstnce), but the mn pont here s tht the electrons re now shred mong the H nd He toms, s expected t equlbrum seprton. (d) vlute the totl energy wth the mproved densty mtrx. If you hve wrtten code or re usng n envronment, feel free to terte the SCF procedure to convergence!

15 Wth the new densty mtrx n (57), the new Foc mtrx s computed s: F (58) from whch the electronc nd totl RHF energes re gven by -4.8 nd hrtrees, respectvely. At convergence, the totl RHF energy s gven by hrtrees. ote tht snce the ntegrls were gven to you wth 4 sgnfcnt fgures, you should only expect to converge the energy to 4 sgnfcnt fgures (t best). 4. The mnmum bss dssocton problem.: () Whle I summrzed the H cse wth restrcted nd unrestrcted orbtls n lecture, t s stll useful exercse to crefully derve the expressons yourself. So, for the UHF tretment of H t dssocton, express the wvefuncton n terms of spn-egensttes, nd dscuss. At dssocton, the UHF wvefuncton s gven by (fter some lgebr): Ψ UHF 3 [ ϕ A() ϕb () ϕ B () ϕ A()] [ α() β () β () α()] 3 [ ϕ A() ϕ B () ϕb () ϕ A()] [ α() β () β () α()] (59) n whch there re equvlent mounts of the snglet (frst term) nd trplet (second term, M s ) egenfunctons. At the dssocton lmt, the wvefuncton s 5% snglet nd 5% trplet n spn chrcter. Of course, ths s not desrble feture of the UHF wvefuncton nd s commonly referred to s spn contmnton. Snce the Hmltonn does not depend on spn, t wll commute wth the Ŝ opertor. Therefore, we should be ble to choose set of egenfunctons tht s common to both of these opertors. It should be stressed tht ust becuse two opertors commute does not men tht n egenfuncton of one opertor s necessrly n egenfuncton of the other opertor. If two opertors commute, t cn be shown tht the ntersecton of ther correspondng egenspces s not null.

16 (b) For the restrcted dssocton of H n mnmum bss, estblsh the expresson for the wvefuncton t dssocton n terms of onc nd covlent confgurtons. At dssocton, the RHF wvefuncton s gven by: Ψ RHF 3 [ ϕ A() ϕ A()] [ α() β () β () α()] 3 [ ϕ A() ϕb () ϕb () ϕ A()] [ α() β () β () α()]. (6) 3 [ ϕ B () ϕ B ()] [ α() β () β () α()] In the RHF cse, the wvefuncton s comprsed of onc terms (terms nd 3) nd covlent terms (contned wthn the composte term gven by term 3). Therefore, the RHF wvefuncton s 5% covlent nd 5% onc t dssocton. Snce the wvefuncton should be % covlent t dssocton, t s the presence of the onc terms tht rse the energy nd led to deleterous behvor t nfnte seprton. However, unle the UHF wvefuncton, the RHF wvefuncton s pure snglet. Hence we hve reched compromse between ccurte energetcs nd ccurte wvefunctons. At the UHF level, the energetcs t dssocton re superb (reltve to the RHF level) due to the lrger degree of vrtonl flexblty, but the wvefuncton s spn contmnted nd s therefore not pure spn egenfuncton. On the other hnd, the RHF wvefuncton, wth less vrtonl flexblty, provdes worse energetcs but s ndeed pure spn egenfuncton. (c) For the HeH cse tht you wored on n q. 3, crefully repet your clculton t nfnte seprton (the dssocton lmt). Wrte down the exct expressons for S, T, V He, V H nd the symmetry-unque -electron ntegrls t dssocton. At dssocton, the ntegrls re gven by:. S(R ). (6)

17 .643 T (R ).76 (6) V He (R ) (63) V H (R ).66 (64) nd ( ).37. (65) ( ).7746 vlute the Foc mtrx nd energy for the ntl guess wth the electrons on He, nd perform one SCF step gn. Interpret the chnges tht you see. Gven the sme ntl guess densty mtrx s n Q3(), the Foc mtrx s now gven s: F (66) wth correspondng electronc RHF energy of hrtrees whch of course s completely equvlent to the totl RHF energy snce R nd therefore V. The totl RHF energy αβ nn t dssocton s clerly hgher thn the totl RHF energy t equlbrum (-.865 hrtrees). Snce the Foc mtrx s lredy dgonl, we re done,.e., we hve found bss n whch the Foc mtrx s dgonl nd no further tertons wll chnge the densty mtrx. Therefore, our fnl densty mtrx t dssocton s equl to the guess densty n whch both electrons re plced n the He (s) orbtl: P(R ). (67)

18 Ths should me sense to you, snce the off-dgonl, or couplng elements, of the ntegrl mtrces vnsh t the dssocton lmt. Therefore, the electrons on He relly hve no wy of feelng the presence of the H (s) orbtl t s n fct nfntely fr wy Dscuss your result n terms of the nture of the products, the onzton potentls of H nd He, nd the queston of whether RHF s pproprte for ths cse. From the RHF totl energetcs t equlbrum (-.865) nd t dssocton ( hrtrees), we cn compute the bndng energy for HeH s pproxmtely.6 hrtrees 5.9 ev. For reference, the correct vlue s only.4 ev, wth the error mnly due to the mnml bss set used nd the HF pproxmton. From the ntegrls gven, we cn lso determne the energes of H, He, nd He wthn the mnml STO-3G bss. The energy for the H tom s: H T V H.4666 hrtrees. (68) The energy for the He tom s: He ( T V ) ( ) * ( ).37 He.6438 hrtrees. (69) Fnlly, the energy for the He cton s: He He T V ( ) hrtrees. (7) Wth ths set of energetcs, we cn now compre dssocton energes for the followng two competng processes: HeH He H HeH He H Δ ( ) (-.865) hrtrees.67 hrtrees Δ ( ) (-.865) hrtrees.484 hrtrees from whch we fnd tht HF clcultons wthn mnml bss correctly predct tht HeH wll dssocte nto He H (whch re both closed-shell speces). Unle the dssocton for H, the

19 products of the dssocton of HeH re closed shell speces. Ths n fct ws confrmed nlytclly bove n whch we found tht the totl RHF energy t dssocton ws hrtrees n cse you hve not notced, ths s ust the energy of the He tom (69). From n onzton potentl pont of vew, the IP for H H e - s much smller thn for He He e -, thereby supportng our fndngs. So, s RHF pproprte for ths dssocton problem? Let s revst ths queston fter we consder the UHF method below (d) To further clrfy, evlute the energy of sngle determnnt UHF wvefuncton for HeH t dssocton wth one electron on ech tom, nd compre gnst (c). In the prt, we wll consder the UHF tretment of HeH dssocton wth the products beng open-shells: He nd H. In ths cse, the ntegrls t R, wll be the sme s gven n (6), (6), (63), (64), nd (65). In generl, the UHF energy s gven by UHF α β core α α [( P P ) H P F P β β ] F (7) where the lph nd bet Foc mtrx elements re F α H core α β α [( P P )( ) P ( μσ λν )] (7) nd F β H core α β β [( P P )( ) P ( μσ λν )]. (73) Assumng (WLOG) tht the electron n the He (s) orbtl s of lph spn nd the electron n the H (s) orbtl s of bet spn, the lph nd bet densty mtrces re gven s: α P (74) nd

20 β P. (75) From (6), (63), nd (64), we hve: core H (76).4666 Therefore, α F (77) nd β F (78).4666 Usng (7), (74), (75), (77), nd (78), we cn compute the UHF energy t dssocton s -.44 hrtrees. Ths supports our erler fndngs n tht HeH would prefer to dssocte nto closedshell frgments He nd H rther thn open-shell frgments He nd H. Therefore, the RHF method s completely pplcble n ths cse. In the cse of H dssocton, however, n whch bond s cleved homolytclly, extr cre must be ten when comprng the restrcted nd unrestrcted solutons to the HF equtons.