Hartree-Fock Theory ISTPC Summer School June 14th - 24th 2015 Aussois Vincent Robert, [email protected] UdS - UMR 7177 Laboratoire de Chimie Quantique 1. N-electron problem 2. Independent electron approximation I Non-interacting electrons I Orbitals (AOs and MOs) I Electron-electron interactions I Self-Consistent Field Method I Slater determinant I Hartree-Fock equations I Roothan equations - atomic basis sets I A featuring system : H2 3. General results : Brillouin,Koopmans' theorems 4. Conclusions N-electron Problem N electrons ! fig light particles M ions ! fAg heavy particles N M N M ^ X 1 2 X 1 2 X X ZA H = − ri − rA − 2 2MA riA i=1 A=1 i=1 A=1 N N M M X X 1 X X ZAZB + + rij RAB i=1 j>i A=1 B>A mi =MA ≈ 1=2000 frozen ions : Born-Oppenheimer approximation 1 ! non-independent particles ! rij approximated solution Non-interacting electron Case \ 1 = 0" rij N N X X H^ = h^(i) ! E0 = i i=1 i=1 EE E(N + 1) = E(N) + N+1 convenient and simple, though approximate! Orbital Approximation \electrons in boxes" Atomic Orbitals ! Molecular Orbitals orbital : eigenfunction of a monoelectronic problem Z h^' = ϕ k'k2 = '∗'dr = h'j'i = 1 Linear Combination of Atomic Orbitals (LCAO) N X MO = ci 'i i=1 example 1 : H¨uckel for conjugated systems f2pz g simple zeta example 2 : extended H¨uckel fAOgi multiple zeta 'i = a1exp{−ζ1rg + a2exp{−ζ2rg how do the electrons see each other ? Electron-Electron Interactions \ 1 6= 0" rij Hartree's assumptions : 1. instantaneous interaction ! average interaction spherical for atoms 2. self-consistency f'i g! Coulomb ! h^'~i = i '~i !f '~i g k'~i − 'i k ! 0 3. Pauli exclusion principle \no two electrons with same spin in the same orbital" ! spherical averaging of the Coulomb potential Self-Consistent Field Method atomic calculations - the electrons move independently one from the other : Ψ(fri g; fθi g; fφi g) = '1(r1; θ1; φ1) ··· 'N (rN ; θN ; φN ) ! Hartree product - let us write as for the hydrogen atom : P(r) m m '(r; θ; φ) = r Yl (θ; φ) Yl (θ; φ) : spherical harmonics 0 (e.g. Y0 (θ; φ) ^=1s ) ^ X 1 2 X Z X 1 X X H = − ri − + = fi + gij 2 ri rij i i j>i i j>i P - electrostatic part j>i gij : Z ∗ ∗ 1 'i (r1)'j (r2) 'i (r1)'j (r2)dr1dr2 =( ii; jj): Coulomb integral r12 Spherical Averaging : Hartree's equations δh jHj^ iav = 0 - averaging the charge distribution around r1 : 2 δQ(r1) = Pj (r1)dr1 δQ(r2) δQ(r2) - energy at r2 : r if r1 > r2 and r if r1 < r2 n 1 2 o 1 R r1 R 1 r1 Y0(r1; j) = δQ(r2)dr2 + δQ(r2) dr2 r1 0 r1 r2 - spherically averaged ith and jth charge distributions : Z 1 Z 1Z 1 1 F0(i; j) = δQ(r1)Y0(r1; j)dr1 = δQ(r1) δQ(r2) 0 0 0 rmax 2 ∗ n d li (li +1) Z o − dr 2 + r 2 − r Pi (r) = i Pi (r) ∗ X Z = Z − ri Y0(ri ; j): screening effect j6=i Antisymmetry of the Wave Function 2-electron system - experimental facts : 1s2s ! antisymmetric 1S and symmetric 3S - triplet/singlet spin parts : symmetric/antisymmetric MS = 0 component : α(1)β(2) ± α(2)β(1) triplet : Slater determinant 1 'i (x1) 'j (x1) Ψ(x1; x2) = p with xi = (ri ;!i ) 2 'i (x2) 'j (x2) Note : 'i;j = spin-orbitals - antisymmetry:Ψ( x1; x2) = −Ψ(x2; x1) - Pauli's principle verified : Ψ(x; x) = 0 (identical lines !) Properties-Generalization '(xi ) = '(ri )α(!i ) or '(ri )β(!i ) - Space parts of the wave-functions : jS = 1; MS = 0i = 'i (r1)'j (r2) − 'i (r2)'j (r1) jS = 0; MS = 0i = 'i (r1)'j (r2) + 'i (r2)'j (r1) r1 6= r2 in the triplet - Coulomb interactions larger in S = 0 than in S = 1 - Analytically, E0 − E1 = 2Kij R ∗ ∗ 1 with Kij = ' (r1)' (r2) 'i (r2)'j (r1)dr1dr2 i j r12 = hij j1=rjji i =( ij; ji): exchange integral ! Hund's rule : Smax is favored E S=0 2Kij S=1 Generalization : Slater determinant Starting from 2-electron functions : Ψ1(x1; x2) = 'i (x1)'j (x2)Ψ2(x1; x2) = 'j (x1)'i (x2) 1 'i (x1) 'j (x1) Ψ(x1; x2) = p : Slater determinant 2 'i (x2) 'j (x2) More generally : ' (x ) ' (x ) ··· ' (x ) i 1 j 1 k 1 1 'i (x2) 'j (x2) ··· 'k (x2) Ψ(fxi g) = p . N! . 'i (xN ) 'j (xN ) ··· 'k (xN ) notation : Ψ = j'i 'j ··· 'k j f'j g = orthonormal molecular spin-orbital basis set Hartree-Fock Equations E[f'ag] = h jHj i and h j i = 1 (i) single Slater determinant : Ψ = j'1'2 ··· 'a'b ··· 'N j (ii) δE = 0 under the constraints h'aj'bi = δab N N X X L[f'ag] = E[f'ag] − ba(h'aj'bi − δab) a=1 b=1 =) F^(1)'i (1) = i 'i (1) : Hartree-Fock equations (∼ 1930) F^ = Fock operator ! monoelectronic X F^(1) = h^(1) + J^a(1) − K^a(1) a;occupied Z ^ ∗ −1 Ja(1) = dx2'a (2)r12 'a(2) Z ^ ∗ −1 Ka(1)'i (1) = dx2'a (2)r12 'i (2) 'a(1) J^a = Coulomb, local K^a = exchange, non-local Coulomb, Exchange Operators, Hartree-Fock Potential Coulomb J^b Z ^ ∗ ∗ −1 hajJbjai = dx1dx2'a (1)'b(2)r12 'b(2)'a(1)=( aa; bb) = Jab exchange K^b Z ^ ∗ ∗ −1 hajKbjai = dx1dx2'a (1)'b(2)r12 'a(2)'b(1)=( ab; ba) = Kab hαjβi = 0 ) Kab = 0 for electrons of opposite spins P ^ ^ HF potentialv ^HF = occ Jb − Kb X hajv^HF jai = (aa; bb) − (ab; ba) b;occ ! α and β electrons are not correlated Properties-Resolution - α and β electrons form two discernable sets ^ (K operator ! spin polarization) E - closed-shell systems : Restricted Hartree-Fock N=2 X F^(1) = h^(1) + 2J^a(1) − K^a(1) MOs - hajJ^ajai = hajK^ajai cancellation inv ^HF : no self-interaction - F^ = f (f'a; occg) ) iterative resolution until self-consistency : hijFj^ ji = δij i 1 - along the HF procedure ! v^HF : mean field r12 Roothaan Equations introduction of a basis LCAO M X 'i = cµi χµ M ∼ 1000 µ=1 two M × M matrices must be introduced : Z ∗ Sµν = dr1χµ(1)χν (1) : metric Z ∗ Fµν = dr1χµ(1)F(1)χν (1) : Fock matrix X X Fµν cνi = i Sµν cνi µ ν FC = SC : Roothaan equations Roothaan Equations in Practice non-orthogonal atomic orbitals I Slater type (atomic-like) example : double dz´eta,DZ 'i (r) = a1exp{−ζ1rg + a2exp{−ζ2rg I Gaussian type and contractions I minimal basis sets Slater linear combination of Gaussians X 2 'i (r) = ci exp{−αi r g i I contraction : some of the coefficents are frozen example : 6-31G 6 for the core 3 + 1 for the valence ! split valence Unrestricted Hartree-Fock different α and β spatial forms ^ ^ X h ^ ^ i X ^ Fα(1) = h(1) + Ja,α(1) − Ka,α(1) + Ja,β(1) Nα Nβ σσ σσ αβ ^ ^ Jij ; Kij and Jij : coupling between Fα and Fβ ! How to practically solve FC = SC ? y − 1 Find X such as X SX diagonal ! X = S 2 F 0C 0 = C 0 with F 0 = X yFX ! "standard" eigen-value problem Main drawback of UHF method : hS^2i 6= S (S + 1) ! spin contamination issue H2 as an example - minimal basis set : f1sa; 1sbg = fa; bg '1 = ξ1a + ξ2b '2 = ξ2a − ξ1b from group theory, jξ j = jξ j = p1 1 2 2 '1 = g and '2 = u - self-consistency is automatically achieved Fgu = hujFj^ gi = 0 Fgg = hgjFj^ gi = g Fuu = hujFj^ ui = u u = ? EHF = f (g ; u) ? = h + 2J − K g = hgg + Jgg u uu gu gu Hartree-Fock Energy EHF = hΨjHj^ Ψi with Ψ = jaa · · · j from a "natural" inspection : EH2 = 2hgg +Jgg How does this compare to 2g ? P EH2 = 2g −Jgg ) EHF 6= a;occ a ! X 1 X h i E = h + (aa; bb) − (ab; ba) HF aa 2 a;occ a;b X 1 X h i = − (aa; bb) − (ab; ba) a 2 a;occ a;b 1 X h i − (aa; bb) − (ab; ba) 2 a;b avoids double counting of e-e interactions Brillouin and Koopmans Theorems Single Excited Determinants r y jΨai = ar aajΨHF i ^ r hΨHF jHjΨai = 0 What do we learn from the i ? N N+1 eA = −r eA = EHF − Er N−1 N IP = −a IP = Ea − EHF i calculated for the N-electron system ! Pathology of the Hartree-Fock Solution Does ΨHF = jgg¯j survive as R ! 1 ? Since g = p1 (a+b), Ψ = 1 ja¯aj + jbb¯j+jab¯j + jb¯aj 2 HF 2 Ψ = p1 Ψ + Ψ HF 2 ionic neutral incompatible withΨ ≈ Ψneutral as R ! 1 ! ) \suppress" the ionic contributions : Ψ = λjgg¯j − µjuu¯j Hartree-Fock invalidated : beyond mean field is necessary.