Electronic Structure Methods

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Morgan Barnett
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1 Electonc Stuctue Methods One-electon models e.g., Huckel theoy Semempcal methods e.g., AM, PM3, MNDO Sngle-efeence based ab nto methods Hatee-Fock Petubaton theoy MP, MP3, MP4 Coupled cluste theoy e.g., CCSD(T) Mult-confguatonal based ab nto methods MCSCF and CASSCF (also GVB) MR-MP and CASPT MR-CI Ab Into Methods fo IPs, EAs, exctaton eneges CI sngles (CIS) TDHF EOM and Geens functon methods CC

, AM, PM3, MNDO Sngle-efeence based ab nto methods Hatee-Fock Petubaton theoy MP, MP3, MP4 Coupled

2 Densty functonal theoy (DFT)- Combne functonals fo exchange and fo coelaton Local densty appoxmaton (LDA) Pedew-Wang 9 (PW9) Becke-Pedew (BP) BeckeLYP(BLYP) Becke3LYP (B3LYP) Tme dependent DFT (TDDFT) (fo excted states) Hybd methods QM/MM Solvaton models

Becke-Pedew (BP) BeckeLYP(BLYP) Becke3LYP (B3LYP) Tme dependent

3 Why so many methods to solve Hψ = Eψ? Electonc Hamltonan, BO appoxmaton H = Z A + + A j j A B Z AZ R / j s what makes t tough (nonsepaable)!! Hatee-Fock method: Wavefuncton antsymmetzed poduct of obtals In geneal, τ efes to both spatal and spn E = Fo the -electon case Ψ = ϕ τ ) ϕ ( τ ) = [ ϕ ( τ ) ϕ ( τ ) ϕ ( τ ) ϕ ( )] AB Enegy mnmzed vaatonal pncple Ψ H Ψ Ψ Ψ h ϕ = ε ϕ obtals and obtal eneges B (n a.u.) Ψ = φ ( τ ) φ ( τ ) Slate detemnant N N accounts fo ndstngushablty of electons ( τ coodnates Vay obtals to mnmze E, keepng obtals othogonal h Z = A A + ( J K ) ϕ() ϕ() K = d j ϕ() ϕ() j j j J = φ ( ) d j

( τ ) ϕ ( τ ) ϕ ( τ ) ϕ ( )] AB Enegy mnmzed vaatonal pncple Ψ H Ψ Ψ Ψ h ϕ = ε ϕ obtals and obtal eneges B (n a.u.

4 Chaactestcs Each e - sees aveage chage dstbuton due to othe e -. (It s a mean feld method.) Electon coelaton mssng In-out Left-ght h depends on the obtals one s tyng to solve. Reques an teatve (self-consstent) soluton. Pattonng of Hamltonan H = F + V F = Σh Fφ = ε φ V = Σ(/ j ) - Σ(J -K ) (weake than Σ/j) E HF = <ψ H ψ > = <ψ F ψ > + <ψ V ψ > 0 th ode st ode Note: E not just a sum of obtal eneges (would double count Coulomb nteactons)

Reques an teatve (self-consstent) soluton.

5 Soluton of the HF poblem Intoduce a bass set (usually atomc obtals) Evaluate - and -electon ntegals Geneate an ntal guess fo the MO s Constuct Fock matx Dagonalze new obtals and obtal eneges Rebuld Fock matx + teate (SCF) Stop when specfed convegence cteon met Obsevatons Bute foce (/8)N 4 -el. ntegals χ a( d d ) χb( ) χc( ) χd ( ) Cleve codes - scale as N 3 o weake Can be multple solutons!! These ae nonlnea equatons, and the soluton can depend on the ntal guess o the teatve pocedue mght not convege.

cteon met Obsevatons Bute foce (/8)N 4 -el.

6 Most QC codes usng GTOs (Gaussans) s e α p ( x, y, z) e α Usually these ae contacted e.g., s = 3 j= c j e α j The contacton coeffs. c j and exponents α j ae fxed and ae dffeent fo each element. The above functon counts as 3 fo the ntegal evaluaton, but only fo the sze of the SCF. Example: 3-G - s (3 pmtves), tght sp ( pmtves), extended s'p' ( pmtve) Conventonal vs. dect SCF Conventonal stoe ntegals Dect compute ntegals on the fly Essental fo bg systems

The above functon counts as 3 fo the ntegal evaluaton, but only fo the sze of the SCF.

7 Some common Gaussan bass sets Bass set H L-Ne Na-A STO-3G s sp 3sp 3-G s 3sp 4s3p 6-3G(d) s 3spd 4s3pd 6-3G(d,p) sp 3spd 4s3pd 6-3G(d,p) 3sp 4s3pd 5s4pd 6-3G(df,pd) 3spd 4s3psf 5s4pdf 6-3++G(d,p) 4sp 5s4pd 6s5pd cc-pvdz sp 3spd 4s3pd cc-pvtz 3spd 4s3pdf 5s4pdf cc-pvqz 4s3pdf 5s4p3dfg 6s5p3dfg aug-cc-pvdz 3sp 4s3pd 5s4pd

6-3G(df,pd) 3spd 4s3psf 5s4pdf 6-3++G(d,p) 4sp 5s4pd 6s5pd cc-pvdz sp 3spd 4s3pd

8 Geneal emaks about atom-centeed bass sets Pople bass sets ( ) polazaton functons + dffuse functons anons, polazablty Dunnng coelaton-consstent bass sets Valence double-zeta, tple-zeta, etc. Aug adds one dffuse functon of each angula momentum type aleady n the bass. Cusp condtons Ψ' dscontnous as 0 Neve acheved n a sngle confguatonal wavefuncton Reques vey hgh angula momentum bass functons Also makes t had to satsfy the val theoem <V>/<T> = - Bass set supeposton eo (BSSE) Consde AB A uses functons on B and B uses functons on A to lowe the eneges atfcal enegy loweng at shot dstances. Countepose coecton Lnea dependency poblems Plane-wave bass sets Commonly employed by physcs/mateals scence communty At pesent only used wth DFT methods

Cusp condtons Ψ' dscontnous as 0 Neve acheved n a sngle confguatonal wavefuncton Reques vey hgh angula momentum bass functons Also makes t had to satsfy the val theoem <V>/<T> = - Bass

9 Hatee-Fock calculatons The HF potental fo H does not dssocate coectly: E(H, R = nf.) *E(H) H + H + H - + H + σ g = ( L + R) = ( L + R ) + ( LR + H + H RL) σ g, σ u obtals degeneate as R nfnty Solve by usng Ψ = C ϕgϕg + C σ u σ u

10 LUMO HOMO Closed-shell, S = 0 Open-shell, snglet (S = 0) o tplet (S = ) Open-shell snglet cannot be descbed by HF method!!!! Dadcal HF also not vald

Excitation energies for molecules by Time-Dependent. based on Effective Exact Exchange Kohn-Sham potential

Excitation energies for molecules by Time-Dependent. based on Effective Exact Exchange Kohn-Sham potential Excitation enegies fo molecules by Time-Dependent Density-Functional Theoy based on Effective Exact Exchange Kohn-Sham potential Fabio Della Sala National Nanotechnology Laboatoies Lecce Italy A. Göling