Chapter 7. Ab initio Theory

Transcription

1 Chapter 7. Ab nto Theory Ab nto: from the frst prnples. I. Roothaan-Hall approah Assumng Born-Oppenhemer approxmaton, the eletron Hamltonan: Hˆ Tˆ e + V en + V ee Z r a a a + > r Wavefunton Slater determnant: Ψ χ χ... χ N where χ ψ α, β s the MO spn-orbtals. Substtutng t to the Shrödnger equaton leads to the sngle-eletron Hartree-Fok eq. Fˆ ψ ε ψ where the Fok operator for a lose-shell snglet s F ˆ Z a r a a + N / [ Jˆ Kˆ ] It s dffult to solve beause of ts ntegro-dfferental nature.

2 Roothaan-Hall equatons K ψ LCAO-MO n whh are the atom bass funtons. Substtutng bak to H-F equaton, and multplyng on the left and ntegrate: ˆ F K K ε The overlap matrx S The Fok matrx: K J h K J r Z F F N a a a + + ˆ ˆ ˆ / where the Coulomb and exhange ntegrals: r J λσ σ λ σ λ

3 K λσ λ σ λ σ r Sne the harge densty matrx P λσ / N λ σ The ntegrals an be smplfed as J K P λσ λσ P λσ λσ So, the Fok matrx λσ λ σ F h + K K P λσ λ σ [ λσ λ σ ] The matrx form of R-H eq. FCSCE Roothaan-Hall equaton s nonlnear sne the Fok matrx depends on the LCAO oeffents. In other words, an eletron experene an average feld of other eletrons and nule. Soluton: self-onsstent feld SCF 3

4 ntal guess of sngle eletron feld solve R-H eq. for new LCAO oeffents terate untl onvergene SCF Proedure:. P from ntal guess of LCAO oeffents C.. onstrut Fok matrx from P and ntegrals. onstrut overlap matrx and dagonalze t to obtan S -/ : U SU D D s dagonal S / / UD U / /, so S SS I v. transform R-H eq. to an egenequaton S / / FC S SCE S / FS / S / C S / CE F C C E, where / / / F S FS, C S C v. dagonalzng F for energes and oeffents v. alulate new P from C. v. hek onvergene to determne f to stop or reterate. 4

5 II. Bass sets The atom bass used n LCAO an be of Slater or Gauss-type. Slater-type orbtals STO: STO r n e ζ r Y lm l θ, ϕ Fathful representaton of AOs, but dffult to evaluate ntegrals. Gauss-type orbtals GTO: GTO x a y b z e α r 0 th -order GTO: g s e αr st -order GTOs: αr g x xe, αr g y ye, g z ze αr nd -order GTOs: g 6 xx, g yy, gzz, gxy, g yz, gxz, Lost lear AO haraters, but easy for ntegraton as produt of Gaussans s also a Gaussan. 5

6 Compromse: p d p GTO p The expanson oeffents and Gaussan exponents are hosen to mm the orrespondng STO. These parameters are often fxed n ontraton. Types of bass set. mnmal bass, STO-nG Use n prmtve GTO to ft a STO. Hgher n s not neessarly better.. splt-valene or double-zeta bass: 3-G, 4-3G, 6-3G STO for ore orbtals and STOs for valene orbtals for better desrpton of hemal bonds. 3-G: ore orbtals wth 3 GTOs per STO, valene orbtals wth ontrated GTOs for one STO and GTO for a dffuse STO wth a dfferent ζ.. trple-zeta bass, 6-3G STO for ore orbtals and 3 STOs for valene orbtals. Hgher # of zeta possble 6

7 v. polarzed funtons, 6-3G*, 6-3G** Addng polarzaton p,d,f funtons mproves results, partularly ansotropy. 6-3G*: DZ+P, add polarzaton d funtons to non-h atoms. **: add p to H as well 6-3Gkp, ld, 6-3G** p, d, v. dffuse funtons, 6-3+G, 6-3++G Addng GTO wth very small α mproves desrpton of weak bonds. +: add to non-h atoms ++: add to H as well Example: 6-3G** for H O H: 3 s-type STOs wth 5 3 GTOs, 3 p-type polarzaton funtons. O: s-ore STO wth 6 GTOs 3 s-type STOs wth 5 3 GTOs 9 p-type STOs wth 5 3 GTOs, 5 d-type polarzaton funtons wth 6 prmtve GTOs. total: 30 bass funtons, 48 prmtve GTOs Inreasng bass set leads to more ostly alulatons. 7

8 The Hartree-Fok lmt: As the bass sze nreases, the eletron energy and geometry onverge to the Hartree-Fok lmt. The H-F lmt: good equlbrum geometry 0.05 Å 0-5% hgher vbratonal frequenes 0.89 salng dssoaton energy and barrer heght ~50% or more. Spn restrted and unrestrted HF RHF: use one set of MOs for both α and β eletrons May lead to problems wth open-shell moleules suh as NO and O. UHF: use two dfferent sets of MO for the α and β eletrons. More general and flexble, but dfferent equaton. More aurate for lose-shell moleules as well, partularly at dssoaton asymptotes. 8

9 III. Eletron orrelaton Beause of the mean feld approxmaton, H-F gnores the nteratons between ndvdual eletrons. The energy dfferene s the orrelaton energy. Correlaton energy mportant for quanttatve results. Correlaton energy an be aptured by ether varatonal or perturbatve methods Confguraton nteraton CI: Example: H, n the -MO approxmaton, two eletrons an be plaed n the followng ombnatons: σ σ, g g σ g σ u, σ σ, u g σ u σ u party: g u u g CI wavefunton: Ψ g σ σ + σ σ g g u u Varyng, lowers energy. 9

10 Confguraton state funtons CSF: For K spn-orbtals, the H-F onfguraton flls the lowest N. Exted onfguratons possble by movng e - to hgher vrtual orbtals. Sngle extatons: eletron s exted. Φ p a χ χ... χ pχb... χ N Double extatons: eletrons are exted. Φ pq ab χ χ... χ pχq... χ N Illustraton: CSF: lnearly ombned determnants that have orret eletron symmetry. CI wavefunton: Ψ CI C p p pq pq 0 Φ + Ca Φa + Cab Φab + a, p a< b p< q... 0

11 The oeffents are varatonally determned to lower E. Numerally, solvng egenequaton HCEC Note: the MO oeffents are already determned n the H-F alulatons and fxed n CI. S/D/T/CI: CI wth sngles/doubles/trples extatons. SDCI preferred. Full CI: nlude all CFSs, very expansve / N! K N Problem: sze nonsstene, energy and energy error do not nrease wth the sze of moleule. CI vs. HF Multonfguraton SCF MCSCF: optmze both MO oeffents and CI oeffents, more aurate. MCSCF-CI: MCSCF followed by CI.

12 CASSCF omplete atve spae SCF: spn-orbtals dvded nto three lasses: natve orbtals: low energy doubly ouped spn-orbtals vrtual orbtals: hgh energy empty spn-orbtals atve orbtals: spn-orbtals n between CFSs n CASSCF arse from all possble ways of dstrbutng eletrons among the atve spn-orbtals. Better for bond formng/breakng proesses nvolvng the atve eletrons Mult-referene CI Generate CFSs not only from the H-F onfguraton, but also exted ones as well. Good for exted states and dssoaton lmts. Coupled-luster CC methods Tˆ + Tˆ +... ΨCC e Φ HF where the operators n the exponent represent sngle, double, trple extatons, et. Better than CI beause t nludes extatons n all orders and s better n apturng more orrelaton energy. CCSD: oupled-luster wth sngle and double extatons.

13 Møller-Plesset MP perturbaton theory H ˆ Hˆ + Hˆ 0 where at H-F lmt they are ˆ 0 H ˆ F Hˆ Hˆ ˆ F 0 Energy orretons to E Φ ˆ 0 H Φ0 HF E st order Φ ˆ 0 H Φ 0 It an be shown that 0 E HF E + E, so MP s not effetve n nludng orrelaton energy. Φ Hˆ Φ 0 J E 0 J E Φ E J J Hˆ Φ 0 nd order MP: nludes double extatons. MP3/4 better, nlude hgher extatons, but muh more expensve. 3

14 Pros and ons of CI and MP approahes: CI: varatonal provde nfo for exted states sze-nonsstene an be qute expansve, and onverge slowly MP: sze-onsstent good for moleular propertes often used for sngle pont alulatons dffult to do hgh-order perturbaton alulatons non-varatonal, energy an be lower than true energy not good for geometry far from equlbrum not applable to exted states Protool for aurate determnaton of mportant propertes suh as atomzaton energy, onzaton energy, eletron and proton affntes et. G3: HF/6-3G* geometry optmzaton MP/6-3G* geometry optmzaton Sngle pont MP4 Corretons 4