Key Concepts, Methods and Machinery - lecture 2 -

Transcription

1 B. Roos V. A. Fok C. C. J. Roothaan W. Hetler Quantum Mechancs W. Paul J. Čížek P.A.M. Drac Key Concepts, Methods and Machnery - lecture - J. A. Pople M. S. Plesset N. Bohr E. Hückel E. Schrödnger and many other heroes W. Kohn J. C. Slater W. Hesenberg M. Born 1

2 I. On quantum mechancal state Sx postulates n QM The state of the system s descrbed by the wavefuncton = r,t, whch depends on the coordnates * of partcle r at tme t. Wavefuncton are n general complex functons of real varables, thus ( r,t) denotes the complex conjugate of ( ) II. ( r, t) ( x, t) dv = ( r, t) dv * P( r, t) = (probablstc nterpretaton) On operator representaton of mechancal quanttes The mechancal quanttes that descrbe the partcle (energy, momentum, angular momentum etc.) are represented by lnear operators actng on a wavefuncton The operator of the potental energy The total energy operator, Hamltonan: Hˆ = Tˆ + Vˆ The operator of the knetc energy Drac notaton: III. * ψ Aˆ φdτ ψ Aˆ φ Matrx element of the operator  On tme evoluton of the state The tme evoluton of the wave functon s gven by the equaton: ψ * φ d τ ψ φ Scalar product of two wavefunctons Hˆ ( r, t) = t ( r, t)

3 Sx postulates n QM IV. V. On nterpretaton of expermental measurements not dscussed here Spn angular momentum (n non-relatvstc formulaton of QM) Sˆ α = s( s + 1) S ˆ α = α z m s S ˆ β = z m s β ; ; α α β 1/ 1/ VI. where the spn magnetc quantum number m s = -s, -s+1,,s On the permutatonal symmetry ( 1,,...,,..., j,..., N ) = ( 1,,..., j,...,,..., N ) ( 1,,...,,..., j,..., N ) = ( 1,,..., j,...,,..., N ) Probablty densty of fndng two dentcal fermons n the same poston and wth the same spn coordnate equals to zero -fermons (electrons, ) non-nteger spn -bosons - nteger spn Ferm correlaton (Ferm hole) Paul excluson prncple 3

4 Quantum mechancs n Chemstry Let the molecular system under study contan atomc nucle (q nucle ), electrons (q electrons ) and possbly external felds. The key equaton n quantum mechancs s the nonrelatvstc Schrödnger equaton: Hˆ ( q, t) ( q, t) = t ( q, t) * The vector q collects the spatal and spn coordnates of all partcles (nucle and electrons) n the molecular system. * Postulate III. 4

5 The electronc Schrödnger equaton Hˆ ( q, t) ( q, t) = t ( q, t) Let the Hamltonan be tme-ndependent E tot ( q, t) = ( q) exp t ; Hˆ ( q) ( q) = E ( q) tot Born-Oppenhemer approxmaton Schrödnger equaton for statonary states ( q) ( ) ( ) q nucle q electrons 5

6 The electronc Schrödnger equaton ( q) ( ) ( ) ( q) q nucle q electrons ( Tˆ + E) ( q ) = E ( q ) Hˆ Hˆ nucle electrons = Tˆ nucle + ( q ) = E( q ) electrons nucle Hˆ electrons tot electrons Nuclear-moton Schrödnger equaton nucle electronc Schrödnger equaton The concept of potental energy hypersurfaces (of dmenson 3N-6): the energy of a molecule as a functon of ts geometry 6

7 Hˆ = The Hamltonan (spn-dependent terms not consdered) m Tˆnucle k k m Tˆ electrons e e Z r k k ˆ Ve n Coulombc potental k + < j ˆ e r Ve e j + k< l ˆ e Vn n Z r k kl Z l k,l nucle,j electrons Hˆ electrons = hˆ Ĥ electrons one electron + < j hˆ two electron Due to ths term analytcal soluton of SE s unkonwn Thus, the numercal soluton of the electronc Schrödnger equaton Hˆ electrons ( q ) = E( q ) electrons electrons through a favorte electronc-structure (quantum-chemcal, QC) method. QC methods are also devsed to optmze to the spatal confguraton of nucle to mnmze E - geometry optmzaton. 7

8 Hˆ electrons ( q ) = E( q ) electrons electrons Hˆ electrons = E f =1 hˆ one electron + hˆ, two electron j =, < j E 8

9 The Many Electron Wavefuncton A form for the electronc wavefuncton that satsfes the permutatonal antsymmetry (postulate VI) s the Slater determnant (SD) or a lnear combnaton of SDs. SD for two-electron system 3 SD = 1 ψ 1(1) α(1) ψ () α() 1 spatal component of one-electron wave functon (molecular orbtal, MO) ψ (1) α(1) = ψ () α() 1 spnorbtal χ (1) 1 χ () 1 χ (1) χ () spn component of one-electron wave functon SD for N-electron system SD 1 = N! χ (1) χ () χ ( N) χ (1) χ () χ ( N) χ (1) χ () N χ ( N) N N Symmetry and spn-adapted SD or lnear combnaton of SDs = confguraton state functon (CSF) ˆ S Sˆ z CSF CSF = S( S = M s + 1) CSF CSF & Hˆ electrons CSF = E CSF 9

10 Molecular orbtals, as a buldng elements n SD or CSF, are constructed from atomc orbtals: Bass set ψ = N j = 1 c a ϕa (Lnear combnaton of atomc orbtals, LCAO) Hydrogen-lke (one-electron) AOs are always of the form: ( θ, ϑ) R( r) Y ( θ ϑ) ϕ r =, where R(r) s the radal component that decays exponentally, lm wth ncreasng dstance from the nucleus e -ζr 10

11 The Many Electron Wavefuncton Snce t s mpossble to obtan analytc solutons n systems wth two or more electrons, the exponental behavor of the AOs Slater-type orbtals (STOs) were hence the frst to be used. They are characterzed by an exponental factor n the radal part. ϕ α ( r θ, ϑ) = P( r) e r Y ( θ, ϑ), lm STO Drawback: dffcultes assocate wth evaluatng ntegrals that appear n the soluton of electronc SE. contracted bass functon Lnear combnaton of several GTOs CGTO ϕ p = bapϕa a GTO prmtve Correcton ϕ α ( r, θ, ϑ) = P( r) e r Y ( θ, ϑ) or ϕ( α, l, m, n; x, y, z) = Ne lm αr (Gauss-type orbtal GTO) Drawback: qualtatvely ncorrect behavor at the nucleus and n the asymptotc lmt x l y m z n Segmented contracton scheme: each GTO contrbutes to exactly one CGTO General contracton scheme: each GTO can contrbute to more than one CGTO 11

12 double-, trple-, quadruple n-tuple zeta bass sets DZ, TZ, QZ More STO/GTO/CGTO functons descrbng one AO Balanced bass set - More art than scence Mnmal bass set - (one STO or GTO or CGTO for one core / valence AO) DZ DZP TZ TZP TZPD QZVPD Infnte bass set (deal but not realstc) } Not very flexble Dfferent types of STO/GTO/CGTO functons, e.g., polarzaton functons (P): e.g., for H atom add p functons for Fe atom add f functons dffuse functons (D) (wth small α n exp(-αr ) allowng to descrbe electron densty at larger dstances from nucleus. sutable for anons, soft, large molecules, Rydberg states.. - N electrons n MO t requres AOs orbtals Effectve core potental: f the core electrons (MOs, AOs) are replaced wth an approxmate pseudopotental 1

13 General strateges for solvng the electronc SE ( q electrons ) guess Hˆ electrons = E ( q electrons ) [ optmzed E ] optmzed ε Optmze and obtan E through a varaton guess electrons guess [ ] = E[ ] guess Hˆ guess guess ε(c) opt = opt Hˆ opt electrons ε[ ( c0, c1,..., cp )] ε ( c0, c1,.., cp ) = 0 c opt opt Optmze and obtan E through a perturbaton (0) Hˆ λ = Hˆ + λ ˆ Let λ be a perturbatonal parameter 0 λ 1 We seek the soluton n the form: E ( ) V (0) (1) () ( λ) = + λ + λ +... (0) (1) () ( λ) = E + λe + λ E +... Then, solvng Ĥ ( λ) ( λ) = E( λ) ( λ) c 13

14 Famly of standard Wave-Functon Theores (WFT) General overvew Welcome to the ZOO Sem-emprcal methods (MNDO, AM1, PM3, etc.) Computatonal Cost Ab nto methods Multconfguratonal HF (MCSCF, CASSCF) perturbatonal herarchy (CASPT, CASPT3) exctaton herarchy (MR-CISD, MR-CCSD) Hartree Fock (HF-SCF) Correlaton Energy (usually <1% of the total energy) perturbatonal herarchy (MP, MP3, MP4, ) exctaton herarchy (CIS, CISD, CISDT, ) (CCS, CCSD, CCSDT,...) Full CI Two contrbutons to correlaton energy : statc and dynamc correlaton 14

15 Hartree Fock (HF-SCF) method the Gate to the realm of WFT Equaton from page 8: [ ] = hˆ one electron + hˆ, two electron, j < j E spnorbtals f 1 Slater determnant E = { χχ j hˆ two electron χχ j χχ j hˆ two electron χ j } 1 χ hˆ one electron χ + χ one-electron ntegrals Condton: χ χ = δ MOs LCAO, j (and E mnmzed trough varatonal approach) j j two-electron Coulomb ntegrals Fock equaton Fˆχ = ε χ = Fˆ two-electron exchange ntegrals 1 hˆ ψ Jˆ Kˆ ψ = one electron + Jˆ Kˆ Fock operator = Fockan Fock matrx n the bass of AOs Workng Roothaan equaton: orbtal energy of j-th MO AO-overlap matrx { F( c) ε S} c = 0 Vector of LCAO coeffcents for j th MO { F ( c) ε 1} c = 0 In fact, F depends on c: see next page thus, equatons has to be solved teratvely -> self-consstent feld 15

16 F Matrx element of the Fock matrx n the bass of AOs explct form (for the restrcted Hartree-Fock method) nucle 1 ϕ Tˆ el ϕq ϕ p Vˆ e n ϕq + Prs ϕ pϕr Vˆ e e ϕqϕs ϕ k pϕr Vˆ, e e ϕsϕq k r, s pq = p AOs h pq densty matrx = occuped c c r s Ths s what s optmzed teratvely to get E mnmzed Program flow: Compute and store all overlap, one-electron and two electron ntegrals Choose a molecular geometry Guess ntal densty matrx P (0) Choose a bass set Construct and solve Roothaan equaton Replace P (n-1) wth P n Construct P from occuped MOs HF converged pq yes ( h F ) AO 1 E HF = Pqp pq + pq Is new P (n) smlar to P (n-1)? for restrcted Hartree-Fock method no 16

17 Hartree Fock (HF-SCF) method Computatonal Remarks Computatonal bottleneck the evaluaton of two-electron (four-center) ntegrals ϕ ˆ pϕr Ve e ϕqϕs Approxmatons of such ntegrals through Cholesky decomposton (CD) or Resoluton of Identty (RI-JK). Restrcted (closed-shell / open-shell HF) unrestrcted HF spn-symmetry broken α α α α { F ε S } c = 0 β β β β { F ε S } c = 0 β α β β α ( c, c ); F ( c c ) β F, S = S( S + 1) RHF RHF S S( S + 1) UHF UHF Spn contamnaton 17

18 Hartree Fock (HF-SCF) method Physcal Remarks Each electron experences the Coulombc repulson of other electrons through ther averaged feld (a mean feld) (the lack of dynamcal correlaton see later) Exchange nteracton among electrons wth the same spn projecton (Ferm correlaton) through the antsymmetrc nature of the Slater determnant. One Slater determnant (SD) = one electronc confguraton ( exact wave functon better expressed as a lnear combnaton of many confguratons - SDs). Only the ground-state wavefuncton and ts energy s solved by HF SCF. (HF not for excted states and ther energes) 18

19 ψ ψ +4 ψ +3 Exact non-realstc soluton wth Full Confguraton Interacton (FCI) n the nfnte bass set Hˆ electrons FCI = E exact FCI = Φ FCI C k SD, k k Correlaton energy: E corr = E exact - E HF Slater determnant ψ + H ˆ = FCI (f = 1 electrons FCI E exact FCI FCI ) ψ +1 kφ k Hˆ electrons k C C Φ = l l l E exact ψ Slater-Condon rules many ntegrals = 0 MOs ψ 1 also Brlloun theorem: Φ a HF SCF Ĥelectrons Φ = 0 19

20 Exact non-realstc soluton wth Full Confguraton Interacton (FCI) n the nfnte bass set n/ number of determnants N n 1 3 n Number of SD s: Exact soluton of electronc Schrödnger equaton (For n electrons n n orbtals)

21 Statc versus dynamcal correlaton? dynamcal Short range effects that arses as r 1 0 Dynamcal correlaton s related to the Coulomb hole. Statc ( non-dynamcal ) from confguratonal near-degeneraces or from defcences n Hartree-Fock orbtals He He Interacton energy E / µe h Hartree -Fock Bass set: 4s3pd Wdmark ANO Dynamcal correlaton CCSD r He-He /Å e.g., wth Φ1 Φ = C 1Φ1 + CΦ C = C =

22 QC also devsed for excted states and ther energes Hartree Fock (HF-SCF) perturbatonal herarchy (MP, MP3, MP4, ) exctaton herarchy (CIS, CISD, CISDT, ) (CCS, CCSD, CCSDT,...) Full CI Sngle-reference post-hf approaches (a porton of dynamcal correlaton ncluded) Møller-Plesset perturbaton theory of n-th order (MPn) Ĥ E ( λ) ( λ) = E( λ) ( λ) MP: Truncated CI methods occ a a ab ab = c0 Φ HF + c Φ + cj Φj +... CIS Coupled-cluster methods (CC) e Tˆ vrt a H (0) (1) () ( λ) = + λ + λ +... (0) (1) () ( λ) = E + λe + λ E +... Truncaton of perturbaton to second-order = E (0 ( λ ) = Hˆ + λvˆ ˆ ) from HF-SCF < j a< b CCSD(T) popular and often used as a golden standard method for sngle-reference systems (T) trple exctatons added as a perturbaton occ 1 [ 1+ ( Tˆ + Tˆ +..)] + ( Tˆ + Tˆ ) + Φ HF Φ HF =..] T ˆ = Tˆ T 1 ˆ e Φ = (1 + Tˆ + Tˆ +..) Φ CCD: HF (0) k HF E MP HF + (0) ( k ) k E0 E0 HF χ Vˆ χ vrt CC CISD = spnorbtals from HF-SCF Tˆ e Φ HF Cluster operator Tˆ = Tˆ + Tˆ ˆ + T

23 Formal scalng behavor of some sngle-reference QC methods N the number of bass functons 3

24 Multconfguratonal HF MCSCF (CASSCF / RASSCF) (a porton of statc correlaton ncluded) MCSCF = k C Φ k k ( C c) MCSCF, Φ k s a CSF arsng from selected exctatons wthn the actve space f all possble exctatons are allowed wthn the actve space -> FCI on a lmted set of orbtals CASSCF more general approach RASSCF (actve space dvded nto subspaces RAS1, RAS, RAS3 wthn RAS1&3 selected exctatons, wthn RAS - FCI Weyl s formula S + 1 n + 1 n + 1 N CSF = n + 1 N / S N / + S + 1 n= number of e - n the actve space N = number of orbtals n the actve space S = molecular spn state Current computatonal lmt for CASSCF actve space ~ 18-n-18 Modern approaches allowng to extent the actve spaces Densty-matrx renormalzaton group technology larger actve spaces wthn DMRG-CASSCF (e.g., 30-n-30) Example for S= N CSF 10-n-8: n-10: n-11: n-13: n-14: n-15:

25 Note on the selecton of an actve space Sometmes trval, sometmes more dffcult, sometmes mpossble B. Roos Selecton cannot be automatzed and depends on the partcular system /problem Chemcal nsght s mportant ngredent In choosng a proper actve space 5

26 Mult-reference wavefuncton approaches (a porton of statc and dynamc correlaton ncluded) CASPT PT on top of CASSCF Popular for spectroscopy RASPT PT on top of RASSCF DMRG-CASPT PT on top of DMRG-CASSCF MRCI(SD) CISD on top of CASSCF MRCC(SD) CCSD on top of CASSCF } Hgly Emergng method for complex electronc structure chemcal transformatons accurate but computatonally extremely demandng Very small molecules 6

27 Densty Functonal Theory - DFT The total energy s represented as a functonal of densty: E The realm of DFT methods bult on two fundamental theorems: 1 st Hohenberg-Kohn theorem: shows that electron densty of an arbtrary molecular system (n an electroncally nondegenerate ground state) n the absence of external electromagnetc felds determnes unambguously statc external potental nd Hohenberg-Kohn theorem: proves that the correct ground state electron densty mnmzes the energy E[ρ] [ ρ] = V [ ρ] + T[ ρ] + V [ ρ] = ρ( r) v ( r) dr + T[ ρ] V [ ρ] ne ee v ext ( r) = nucle k = 1 Z 1 ext + k r R k ee nucleus-electron attracton energy knetc energy of (nteractng) electrons electron-electron nteracton energy 7

28 Kohn-Sham Densty Functonal Theory (KS-DFT) [ ρ] = ρ( r) v( r dr + T[ ρ] V [ ρ] E ) + ee Coulomb electron-electron nteracton E [ ρ] = ρ( r) v ( r dr + T [ ρ] + J[ ρ] + ( T[ ρ] T [ ρ] ) + ( V [ ρ] J[ ρ] ) ext ) s s ee Knetc energy of non-nteractng electrons ( r' ) ( r) 1 ρ r ρ r' dr' dr [ ρ] ρ( r) v ( r dr + T [ ρ] + J[ ρ] + E [ ρ] E ) = ext s xc Exchange-Correlaton (XC) Energy 8

29 Workng Kohn-Sham Equaton The dea of consderng the determnantal WF of N non-nteractng electrons n N orbtals, the T s [ρ] s exactly gven as: T s N [ ρ] = = 1 χ χ m e Kohn-Sham spnorbtal & fulfllng condton: ρ = N = 1 χ χ Real electron densty Then, one-electron KS equaton: m e + v eff LCAO ansatz ( r) χ ( r) = ε χ ( r) wth: ( r' ) v (Fock-lke equatons) = vext ( r) + ρ dr' vxc( r) r r' eff + Roothaan-lke equatons Restrcted / Unrestrcted Kohn-Sham equatons - as n HF ρ = ρ α + ρ β Alpha-omega n KS-DFT exact form unknown 9

30 Most common of exchange-correlaton potentals Local densty approxmaton most popular way to do electronc structure calculatons n sold state physcs Generalzed gradent approxmaton (GGA) xc potentals are functonals of electron densty and ts frst spatal dervatves ( gradentcorrected LDA functonals) Meta-GGA approxmaton extenson of GGA. xc potentals are functonals of electron densty, ts frst and second spatal dervatves and knetc energy densty Hybrd exchange functonals a porton of exact exchange from HF theory s ncorporated nto xc potentals. Usually, GGA hybrd and GGA approach are combned. PBE, BP86 TPSS. TPSSH, B3LYP, PBE0. Hybrd exchange and hybrd correlaton (double-hybrd) functonals - essentally extenson of hybrd-gga, whch uses MP correcton to replace part of the sem-local GGA correlaton. BPLYP 30

31 Lmtatons of standard KS DFT methods Lack of long-range correlaton (dsperson) emprcal correctons ~1/R 6 B3LYP+D3 Incorrect long-range exchange behavor e.g. ncorrect energes of charge-transfer exctatons (exchange should decay asymptotcally as r 1-1 ; B3LYP : 0.r 1-1 ) CAM-B3LYP Lack of statc correlaton energy Generally lower senstvty of DFT to multreference character s dependent on the amount of HF exchange ncluded n the functonal Self-nteracton error SIE nterpreted as the nteracton of an electron wth tself. Whle the dagonal exchange terms K cancel exactly self-nteracton Coulomb terms J n HF, t s not vald for standard KS-DFT methods. Lack of systematc mprovablty!!!!! 31

32 Some fnal notes on solvng SE through WFT and DFT methods For a gven geometry wavefuncton optmzaton -> electronc energy E (sngle-pont calculaton) On the other hand: QC methods can be also used to optmze geometry algorthms allowng to evaluate (frst, second) dervatves of E wth respect to the nuclear coordnates and to search crucal ponts on the potental energy surface Mnma & frst-order statonary ponts (transton states) (geometry optmzaton) Thus now, n prncple, you are able to read the followng sentence: GGA-type PBE functonal n combnaton wth RI-J approxmaton and the DZP bass set was used for the geometry optmzaton, whle CASPT(10-n-8) approach combned wth a larger bass set (e.g. TZVP) was employed for the fnal sngle-pont energes. 3

33 APPENDIX 33

34 Propertes as dervatves of the energy - Bonus Consder a molecule n an external electrc feld ε. E de dε d E dε 1 ( ε ) = E( ε = 0) + ε + ε + ε = 0 ε = 0 Dpole moment (µ) Polarzablty (α) Frst hyperpolarzablty (ß) µ = α = β = de dε d E dε 3 d E 3 dε ε = 0 ε = 0 ε = 0 34

35 α de dε α α d E dε dε β d E dx dε α β 3 d E dε dε dε de dx d E dx dx 3 d E dx dx dx α 3 d E dx dε dε γ 4 d E dx dx dx dx j j j k k β l dpole moment; n a smlar way also multpole moments, electrc feld gradents, etc. polarzablty (frst) hyperpolarzablty forces on nucle harmonc force constants; harmonc vbratonal frequences cubc force constants; anharmonc correctons to dstances and rotatonal constants quartc force constants; anharmonc correctons to vbratonal frequences dpole dervatves; nfrared ntenstes polarzablty dervatves; Raman ntenstes 35

36 d E db db α d E di db α d E di di α d E db dj α d di α d ds de ds α E dj α E ds β β j β β β β magnetzablty nuclear magnetc sheldng tensor; relatve NMR shfts ndrect spn-spn couplng constants rotatonal g-tensor; rotatonal spectra n magnetc feld nuclear spn-rotaton tensor; fne structure n rotatonal spectra spn densty; hyperfne nteracton constants electronc g-tensor and many more... 36

37 Restrcted Hartree Fock (RHF) results for LF Hartree Fock Error r e / Å % ω e / cm % µ e / D % µ e / DÅ % D e / kj mol % For LF, the Hartree- Fock method s qute useful for calculatons around the equlbrum, although the bndng energy s too low by 6%. But the RHF model dssocates ncorrectly nto L + and F. E/E h D e LF ground state Comparson of restrcted Hartree-Fock results wth full CI Bass set: Ahlrchs pvdz r L-F /Å RHF(L + + F ) ROHF(L + F) FCI(L + F) 37

38 Post-Hartree-Fock for qualtatve or quanttatve reasons E/E h LF molecule State-averaged-CASSCF+nternallycontracted-MRCI results Bass set: Ahlrchs pvdz Why do we want to go beyond the Hartree-Fock descrpton? Frst, we may wsh to mprove the accuracy of the computed energy and other propertes. snglet excted state snglet ground state L + + F L + F Second, we are dealng wth a stuaton where the Hartree- Fock model s a very poor zeroth-order approxmaton of the wavefuncton. Near-degeneracy effects Ground-state equlbrum propertes r L-F /Å 38

39 Near-degeneracy problems of perturbaton theory LF ground state Closed-shell coupled-cluster RHF/CCSD(T) results Bass set: Ahlrchs pvdz MP E/E h MP4 The CCSD(T) trples correcton becomes very negatve for dstances > 3 Å. FCI The CCSD(T) method breaks down. The MP4(SDTQ) energy also shows ths behavor, but only at a much longer dstance (> 6 Å). B-CCD(T) r L-F /Å CCSD(T) 39

40 Multreference perturbaton theory appled to LF E/E h LF ground state (,)CASPT results (MOLPRO, g=4) Bass set: Ahlrchs pvdz RHF non-dynamcal correlaton (,)CASSCF dynamcal correlaton (,)CASPT FCI r L-F /Å 40

41 Unrestrcted UCCSD(T) coupled-cluster calculatons of the LF ground state The UCCSD(T) results compare favorably wth the full CI potental energy curve. The expectaton value <S > s zero for the (unprojected) UHF wavefuncton at dstances < 3 Å, but <S > 1.0 at larger dstances (> 3 Å). E/E h LF ground state Unrestrcted coupled-cluster UCCSD(T) results Bass set: Ahlrchs pvdz RHF In ths example, the spncontamnaton represents no real problem for the ground state energy. However, spn-contamnaton may make the UHF-based methods unsutable for the study of a varety of molecular propertes. UCCSD(T) FCI UHF r L-F /Å ROHF(L + F) 41