Chapter-10. Ab initio methods I (Hartree-Fock Methods)
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1 Chapt- Ab nto mthods I (Hat-Fock Mthods) Ky wods: Ab nto mthods, quantum chmsty, Schodng quaton, atomc obtals, wll bhavd functons, poduct wavfunctons, dtmnantal wavfunctons, Hat mthod, Hat Fock Mthod, Roothan quatons. 9. Intoducton In th fst modul, you hav bn ntoducd to sval lmntay numcal (computatonal) mthods to solv smpl and solatd poblms. Now, w bgn applyng ths mthods to poblms of chmcal ntst. W bgn wth ab nto and smmpcal mthods to solv poblms of chmcal stuctus. W wll stct mostly to non-latvstc cass. Th thotcal bass of quantum chmsty has bn ntoducd to you n Engnng Chmsty I, and th Wb and Vdo couss n quantum chmsty. In ths chapt, w wll consd th computatonal aspcts assocatd wth th poblm. 9. Th Schödng Equaton Th non latvstc Schödng quaton fo hydogn atom s, h - E + + Ψ Ψ= Ψ (.) 8π m x y z W can not solv ths quaton n Catsan coodnats bcaus w can not spaat th potntal ngy tm ( / ) n tms of ndpndnt functons of x, y and z. A mo natual opton s to spaat th Laplacan opato, = / x + / y + / z n sphcal coodnats, θ and φ. snθ = + + (.) snθ θ θ sn θ ϕ Th potntal ngy tm can now b combnd wth th pat of th Laplacan opato to gv th adal quaton and th angula pats can b spaatly combnd to gv th spaat quatons fo th functons R(), Θ(θ) and Φ(φ) and th soluton can b wttn as: m m ± mϕ (, θ, ϕ) = Rn ( ) Yl ( θ, ϕ) = Rnl ( ) Pl ( θ. (.3) Ψ ) Tabl.: Radal and angula pats of hydogn-atom wav-functons. Th quantum numbs n, l, and m l a also ndcatd n th tabl.
2 Radal Pat R nl (); a = Boh adus Angula Pat Y ( θ, ϕ) = m lm P l ( θ ) Φ( ϕ) 3 / Z Z / a Rs a = ; n =, l = Y = ; l =,,m / l = (4π ) 3 / / Z Z Z / a Rs a 3 = a ; n=,l= Y = cosθ ; l =, m l = 4π 5 / / Z Z / a R p 6 3 ϕ = a ; n =, l = Y = snθ ; l = m l = 8π R3s = 3 n = 3, l = R3 p = 7 n = 3, l = R 3d 3 / Z Z Z Z / 3a 3 a 3a 7a + 3 / 8 Z Z Z Z / 3a 6 a a 6a 7 / 4 Z Z / 3a = 8 3 a ; n = 3, l = Y / 3 ϕ = snθ ; l=,m l = - 8π 5 / Y = (3cos θ ) ; l=,m l = 6π / 5 ϕ = snθ cosθ ; Y 8π l =, m l = / 5 ϕ Y = sn θ ; 3π l =,m l = 9.3 Multlcton atoms: Wavfunctons and Obtals In multlcton atoms such as H, L, tc. th Schödng quaton may b wttn as HΨ = EΨ wh H s th opato psntng th kntc and potntal ngs of all th patcls and Ψ and E a th wavfuncton and ngy spctvly. Fo a hlum atom, kpng th nuclus fxd (ths s not a bad appoxmaton snc th nuclus s much hav than th lctons) th lctonc pat of H can b wttn as H = + m m (.4) Th fst two tms a th kntc ngs of lctons and, th thd and fouth a th ntacton ngs of th two lctons wth th H nuclus and th last tm s th lcton lcton pulson. H = + +. Smlaly fo all oth s. x y z
3 Fgu 9. Coodnats n H. Th soluton of ths quaton s th wavfuncton Ψ = Ψ(, ) o Ψ(, ) = Ψ(x, y, z, x, y, z ). Ths s a functon of sx vaabls and s dffcult to handl. A ath smpl way to appoxmat ths s to wt t as Ψ(, ) φ ( ) φ ( ) (.5) whch s a poduct of obtals, φ ( ) dpndng on th coodnat of th fst lcton and φ ( ), whch dpnds on th coodnat of th scond lcton. Wavfunctons f to th solutons of th Schödng quaton whl obtals f to th functons of th coodnats of ndvdual lctons and a th solutons to an appoxmat o an ffctv quaton whch s dscbd blow. 9.4 Hat-Fock Equatons In th quaton fo an obtal, w want to duc th numb of vaabls fom many (n a multlctonc cas) to on (actually on cosponds to th vaabls x,y,z). Ths s don by assumng that ach lcton movs n an avag fld catd by all th oth lctons. Fo xampl, on of th lctons n H, say lcton may b thought of movng n a fld of th nuclus plus th avag fld catd by th scond lcton. [ + V f f( ) ] ϕ( ) = Eϕ( ) (.6) 8π m Th th tms f to th kntc ngy, nucla attacton ngy and th ffctv fld and E s th obtal ngy. Th ffctv fld at s obtand by avagng th potntal ngy btwn lcton at and lcton at by allowng th scond lcton at to cov th whol spac V ( ) = ( ) ( ) dx dy dz (.7) ff * ϕ ϕ all volum
4 By th sam analogy, lcton movs n an avag fld catd by lcton. Ths quatons fo lcton and lcton a solvd tatvly statng wth optmzd bass functons φ ( ) and φ ( ) and th tatons (patd solutons) a stoppd whn th avag fld catd by lcton on lcton s consstnt wth th avag fld catd by lcton on lcton. Ths obtals a calld slf consstnt obtals. Ths mthod s th slf consstnt mthod and foms a vy mpotant mthod fo studyng atomc and molcula stuctu. Th abov quaton s th Hat s quaton fo an atomc obtal. Th Hat quaton as wll as th Hat Fock quaton can b dvd fom a vaatonal pncpl, namly, δ ψ H ψ = (.8) Ths ssntally follows fom th vaaton thom whch stats that, fo th gound stat of a systm, th xpctaton valu obtand fom any (wll-bhavd) tal functon f (satsfyng appopat bounday condtons) s always gat than th tu gound stat ngy E of th systm * f H f = f Hfdτ E (.9) Fo molculs contanng N lctons and M nucl, th hamltonan may b wttn H = m N N, M, Z + N + M K L Z K Z R KL L (.) Th fst tm abov s th opato fo th lcton kntc ngs, th scond tm s fo th lcton nuclus attacton, th thd tm s fo th lcton lcton pulson and th last tm cosponds to th nucla pulson ngs. Th a N lctons and M nucl th chags psntd by Z. It s common pactc to us th Bon-Oppnhm appoxmaton whch consds th nucl as fxd, and thfo, th nucla kntc ngy tms a absnt n Eq. (.). Th coct fnal ngy s obtand fo th lctonc moton by addng th ntnucla pulson ngy. Th a sval appoxmaton mthods avalabl fo solvng th poblm. A common mthod s th slf consstnt fld molcula obtal (SCF-MO) mthod wh on stats wth a tal dtmnantal wav functon (th Hat-Fock mthod) wth bass functons o obtals φ, = to N ψ = φ () φ () φ (3)... φ N ( N ) (.) 3 Ths contans N! tms whos dagonal tm s φ ( ) φ () φ3 (3)... φ N ( N). Tms wth odd pmutatons of,,3, N such as φ ( ) φ () φ3 (3)... φ N ( N) appa n th xpanson of th dtmnant wth a ngatv sgn. In alty, th lcton spn functons also nd to b takn nto account. Th concpt of a spatal obtal s xtndd to that of a spn obtal. Spn obtals a wttn as φ ( ) α (), φ ( ) (), wh th spn functons α and do not dpnd on th spatal coodnat but a ntnsc popts of lctons,, tc. Fo th spn functons, th s no classcal analogu.
5 Th dtmnantal fom of th wavfuncton s antsymmtc n th xchang of ows/columns. It nsus that th Paul Excluson Pncpl s natually ncopoatd,.. no two lctons hav dntcal spatal and spn pats. Th xpsson fo ngy n ths Hat-Fock mthod s gvn fo a closd shll confguaton contanng an vn numb ( N ) of lctons by E HF N / N / N / = < H ψ > = H + = = = ψ ( K ) + V (.) Nucla R pulson H = φ () m M Z = φ () (.3) = φ () φ () φ () φ () (.4) K = φ () φ () φ () φ () (.5) Eqs. (9.4) and (9.5) psnt th Coulomb th xchang ntgals. Th xchang ntgal has no classcal countpat. V s th total nucla pulson ngy at th Nucl Rp confguaton of th nucl at whch th lctonc calculatons a cad out (s Eq. 9.). In tms of opato quatons fo ndvdual obtals, ths quatons can b wttn as F ( ) φ () =ε φ () (.6) Co N / F () = H = () K = Z () (.7) Co M = (.8) m = H * φ () φ() d (.9a) () f () = f () K * φ () () () φ f d (.9b) () f () = H, F ˆ () s th Fock opato dfnd n tms of th co lcton opato H (), th Coulomb opato ˆ () and th xchang opato K ˆ () whch a dfnd though Eq. (9.6) to (9.9b). It s a common pactc to tak th functons φ s as lna combnatons ˆ co
6 of appopat bass sts and thn obtan th bst sts of th coffcnts. Ths mthod was dvlopd by Roothaan. φ = B s= c χ s s (.) Th coffcnt of th s th bass functon n th th MO s c s. Th st of quatons to b solvd a calld th Hat-Fock-Roothaan quatons and a gvn as follows. H, B s th numb of bass functons usd n th calculaton. B s= c s ( F s ε S ) = ; =,... B (.) s Th Fock matx lmnt s dnotd by F s, th ovlap ntgal btwn obtals and s s Ssand ε s th ngy of th th obtal. Th atomc obtals w ntally takn as Slat-typ obtals whch a gvn by n + / ( ς / a ) n ς / a m χs( ς,, θφ, ) = Y (, ) / l θϕ (.) ( n!) Slat gav uls to assgn th valus of ς fo dffnt atoms and quantum numbs n, l, and m. You may hav notd that th computatonal poblm nvolvs solvng ntgodffntal quatons. It also nvolvs fndng th oots of th scula quaton to obtan th ngy lvls of th systm aft dagonalsng th Fock matx (Eq. 9.). Th attmpts of solvng th poblm usng Slat obtals could not b xcutd satsfactoly as t s not yt possbl to calculat all th ntgals analytcally and an altnatv had to b found. 9.5 Gaussan Functons A convnnt way was found by wtng th Slat obtals n tms of sutably chosn lna combnaton of Gaussan functons. Fo xampl, th s obtal can b wttn as a lna combnaton of sval Gaussan functons as follows. N χ ( ς, ) = cg( α ) (.3) s = Wh, g( α) a Gaussan functons wth xponnt α. Fo xampl, whn N = 3, th coffcnts and th xponnts ( c, α ) of th th Gaussan functons n th STO-3G cas a gvn by ( ,.988), (.53538,.4577) and (.5439,.766) spctvly. It s nstuctv to plot both functons and s th xtnt to whch thy ag and dsag wth ach oth n dffnt gons of spac, spcally at th ogn. Th nomalzd Gaussan functon cntd at = R A s gvn by
7 (, ) ( / ) A 3/4 g s α RA = α π α R (.4) Th man advantag s that th poduct of two Gaussans at two dffnt cnts s a sngl Gaussan cntd on th ln onng th two Gaussans. Thus, all mult cnt ntgals can b ducd to two cnt ntgals and ths can b valuatd analytcally. W lst som lmntay xampls of ths ntgals. α αxa+ xb ( xa xb) ( )( x ) α( x xa) ( x xb) = (.5) To s ths, stat wth α( x x ) + ( x x ) = αx αxx + αx + x xx + x A B A A B B = ( α + ) x αxx xx + αx + x A B A B α + α + α + + α + α + ( ) x x( xa xb) ( xa xb) α xa α xa α xb xb α xaxb = ( α + ) x x( αx + x ) + α ( ) A B α x + x α x + x + α xx α x + α x α xx = ( α + ) [ x x + + α + ( α + ) ( α + ) ( A B) A B A B A B A B ( αxa + xb) αxa + x B α = + [ + ] + + α + α + α + ( α ) x x ( xa xb xx A B) ( αxa + xb) ( ) α = ( α + ) x + ( xa xb) α + α + In cas of functons cntd at R A and R B, th abov sult cosponds to α α( R R ) ( R R ) ( )( R R ) ( RA RB) A B P = (.6)
8 R P αra+ R = B (.7) 9.6 Molcula Intgals Involvng Gaussans In ths scton w wll outln how to calculat som of th ntgals usng Gaussan functons. Th ovlap ntgal s th asst as t nvolvs an lmntay ntgal nvolvng a sngl Gaussan. < g R g R >=òd -α( - A) - ( - B) s( A) s( B) α α ( R R ) R R 4π R R A B A B ( )( RP ) ( ) = d = d (.8) (.9) Th standad ntgals that wll b usful a / / π π α 4 α Usng th abov ntgals, Eq. (9.8) ducs to α x α x dx = ; x dx = 3 (.3) 3/ æ π ö < s( A) s( B) >= ç α - RA- R g R g R (.3) èα + ø Th kntc ngy ntgal can b valuatd to gv 3/ α Ñ α é α ùæ π ö - RA- RB < gs( RA) - gs( RB) >= 3 - RA -RB α + ê α + úç èα + (.3) ë û ø Now th nucla attacton and th lcton lcton pulson ntgals can b valuatd by usng th mthod of Fou tansfoms. Th sults a as follows. Th nucla attacton ntgal s gvn by ( A ) ( -α B ) s A C C s B C C < g ( R ) - Z / g ( R ) >=- Z d / - R ò B (.33) 3/ α æ π ö - RA- RB ç F é ë( α ) RP RC ù û - π ZC = + - α + èα + ø Wh F (x) s latd to th o functon f (x) and s gvn by (.34) x - / y ò (.35) F x x dy x f x -/ - -/ -/ ( ) = = ( )
9 Th a vy ffcnt algothms to valuat th o functon and th complmntay o functon dfnd by x y = ò (.36) f ( x) dy - And fnally, th lcton-lcton pulson ntgal s gvn by < g ( R ) g ( R ) / g ( R ) g ( R ) > s A s B s C s D α γδ - RA- RB - RC- RD + γ + δ α d d = òò - µ - RP -ν - RP (/ ) (.37) H, α,, γ and δ a th xponnts of Gaussans cntd at R A, R B, R C and R D and μ = α + and ν = γ + δ. Th poduct of th two Gaussans on th lft s cntd at R P and th poduct of th two Gaussans on th ght s locatd at R Q. Th valu of th ntgal s α γδ - RA- RB - RC- RD γ + δ s A s B s C s D < g ( R ) g ( R ) / g ( R ) g ( R ) >= C F ( x) (.38) 5/ / C = π /[( α + )( γ + δ)( α + + γ + δ) ]; (.3) x= R - R [{( α )( γ δ) / ( α γ δ)} P Q ] Usng th fomula gvn n ths chapt, t s n pncpl to pocd to comput th qud ngs and wavfunctons fo molculs at th Hat-Fock lvl. Ths s only a statng pont and th mpovmnts and dtals a th subct of ths modul. Calculatons of ths lcton pulson ntgals (ERI) s a tm consumng pocss and ths has to b don fo vy molcula confguaton untl a mnmum ngy confguaton s avd at. Th computatonal tm ncass vy apdly wth ncas n th numb of bass sts usd. Bcaus of ths dffcults, t s stll qut dffcult to do na xact calculatons fo systms wth a lag numb of lctons ( >5 ). 9.7 Summay In ths chapt, w hav outlnd th basc quatons lvant to th poblm of stuctual chmsty govnd by th Schödng quaton (at th Hat-Fock lvl) and dscussd a common appoach to solv th poblm of a multlctonc systm usng Gaussan functons. Th st of th modul wll outln th computatons of vaous molcula popts wth th avalabl computatonal mthods wth llustatv xampls. Sval poblms wll also b gvn solvng whch, you should gt a good fl fo dong calculatons of ntst to you. Mthods of gomty optmzaton of molculs, dtmnaton of molcula popts and computatonal ffot nvolvd n HF calculatons tc. wll b consdd n dtal n th nxt moduls Rfncs McQua, D. A. Quantum Chmsty, Vva Books Pvt Ltd, 3. Lvn, I. N. Quantum Chmsty, 4 th d., Pntc Hall of Inda, 99. Szabo A. and Ostlund, N. S. Modn Quantum Chmsty, Dov Publcatons Inc., 989.
10 In patcula s th fnc of S. F. Boys, Elctonc wavfunctons I. A gnal mthod of calculaton fo statonay stats of any molcula systm, Poc. Roy. Soc. [London], A, (95).
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