Lecture 4: Hartree-Fock Theory

Transcription

1 Lecture 4: Hartree-Fock Theory One determinant to rule them all, One determinant to find them, One determinant to bring them all and in the darkness bind them

2 Second quantization rehearsal The formalism of second quantization: both observables as well as states are represented as operators Each Slater determinant is represented as an occupation number vector The ON vectors constitute an orthonormal basis in the 2 M - dimensional Fock space E.g. two general vectors (states) in the Fock space c < c k d <  k  k k d k k cd <  < km  * * c kmd cd k m k m k

3 Second quantization rehearsal In SQ, all operators and states are constructed as sequences of elementary creation and annihilation operators Creation operator puts an electron into an unoccupied orbital Annihilation operator removes an electron from an occupied orbital Operators feature an associated amplitude f <Ú f f () x fdx * c PQ P Q g <ÚÚ f ( x) f ( x ) g ( x, x ) f ( x) f ( x ) dxdx * * c PQRS P 1 R Q 1 S 2 1 2

4 Spin in second quantization SQ formalism remains unchanged if spin degree of freedom is treated explicitly, e.g. Now operators can be spin-free, mixed or spin operators Spin-free operators depend on the orbitals but have identical amplitudes for alpha and beta spins Spin operators are independent on the functional form of the operators Mixed operators depend on both

5 Spin in second quantization An example of a spin-free operator is the molecular electronic Hamiltonian that can be written as h One-electron singlet excitation operator Ê 1 Z < f () ˆ Ú r Á,, Â f() rdr * 2 K pq p Á q ÁË 2 K r K g Two-electron singlet excitation operator pqrs p r s q pq rs qr ps e < Âa a aa < E E,dE t u t u 1 <ÚÚ f ( r) f ( r) f( r) f( r) dd rr * * pqrs p 1 r 2 q 1 s r12 tu

6 Expectation values and density matrices The expectation value of a one-electron operator with respect to some reference state is given by   0 ς 0 < ς 0a a 0 < ς D PQ P Q PQ PQ PQ PQ 1 1 ς <  ς <  ς a a a a 0 d PQRS P R S Q PQRS PQRS 2 PQRS 2 PQRS And their spin-adapted versions by     0 ς 0 < ς D < ς 0 E 0 < ς aa pq pq pq pq pq p t q t pq pq pq t ς 0 < Âς d < 0 0 pqrs pqrs Âς e < pqrs pqrs 2 pqrs 2 pqrs 1 < Âς  2 pqrs 0aaaa 0 pqrs pt ru su qt tu

7 Expectation values and density matrices The diagonal elements of the one- and two-electron orbital density matrices w w p < D Œ È0,2 pp ÍÎ < d Œ È 0,2(2, d ) ÍÎ pq ppqq pq Occupation number of a single orbital Simultaneous occupations of two orbitals The eigenvectors U that diagonalize D, D=UNU are the natural orbitals of the system, and on the diagonal matrix N are the natural orbital occupation numbers

8 On the evaluation of matrix elements Approach 1: Permute the operator strings such that they form commutators and anticommutators and employ the basic anticommutator rules Approach 2: Employ the Wick s theorem Approach 2: Employ the Wick s theorem s_theorem

9 Orbital rotations Consider a set of spin orbitals obtained from another set by a unitary transformation f% < ÂfU P Q QP Q The unitary matrix U can be written as U < exp(, k), k <, k And the elementary operators and an arbitrary state generated by the unitary transformation as a% < exp(, k) a exp( k) P P a% < exp(, k) a exp( k) k< Â k aa P P PQ % 0 < exp(, k)0 PQ P Q

10 Hartree-Fock wave function In the Hartree-Fock approximation, the electronic wave function is approximated by a single Slater determinant The HF energy is optimized with respect to variations of these spin-orbitals E HF < min k H k ; k < exp(,k )0 and the Hartree-Fock state is the solution to the nonlinear equations, HF HF < exp(,k )0

11 Fock operator The optimal determinant may be found by solving a set of effective one-electron Schrödinger equations for spin-orbitals the Hartree-Fock equations The associated effective Hamiltonian is the Fock operator The eigenvectors of f are the canonical spin-orbitals and the eigenvalues the orbital energies f< h V Effective oneelectron Fock potential Full one-electron Hamiltonian V < (2 g, g ) E ÂÂ pq iœocc pqii piiq pq

12 Self-consistent field method Since the Fock matrix is defined in terms of its own eigenvectors, an iterative procedure the selfconsistent field (SCF) method is needed Initial orbitals {f} Construct the Fock matrix with {f} Diagonalize fu < Ue New orbitals f% < fu No E HF converged? Yes SCF>

13 Fock operator The spin-orbitals of the optimized HF wave function are eigenfunctions of the Fock operator Eigenvalues (orbital energies) are given by the expression Ê e f ˆ f f f f f, 1 2 Z K 1 < () () 2 () () ()() p Ú r p 1 Á,, Â rdr dd p 1 1 p 1 p 1 i 2 i K r ÂÚÚ r r r r rr ÁË ik iœocc r12 ÂÚÚ 1 f( r) f( r) f( r) f( r) dd rr r p 1 i 1 i 2 p iœocc 12 Exchange interaction Coulombic repulsion Therefore, in the HF picture an electron experiences the Coulomb potential generated by the nuclear framework and by the charge distribution of N-1 remaining electrons as well as Fermi correlation

14 Fock operator The occupied and virtual orbital energies are e e   < h (2 g, g ) g i ii iijj ijji iiii j i < h (2 g, g ) a aa aajj ajja j The HF closed-shell wave function corresponds to N/2 lowest orbitals doubly occupied HF Ê < Á aa ˆ i a i b ÁË i vac The HF wave function is an eigenstate of the Fock operator with an eigenvalue equal to the sum of occupied orbital energies (not to the HF energy!) f HF 2 e HF < i i

15 Hartree-Fock energy The HF energy is the expectation value of the true electronic Hamiltonian 1 E < HF H HF < HF ÂD h d g pq pq  2 pq pqrs pqrs pqrs For a closed-shell HF state the HF energy becomes <  Â, E 2 h (2 g g ) HF i ii iijj ijji ij

16 Koopmans theorem Consider removal of an electron from the HF state When describing the ionized system with a single determinant ai s HF we can define the ionization potential as N IP < HF a Ha HF, E <, f i is is HF ii In the special case of canonical orbitals, we may connect the negative orbital energy with the IP, i.e. E N, 1 N < E, e i HF i Identification of the orbital energies with negative IP s is known as Koopmans theorem (1934)

17 Restricted and unrestricted HF wave functions In the restricted Hartree-Fock (RHF) approximation the energy is optimized subject to the condition that the wave function is an eigenfunction of the total and projected spin operators Accomplished by requiring the alpha and beta spin orbitals have the same spatial parts In the unrestricted Hartree-Fock (UHF) this condition is not imposed Different spatial orbitals are used for different spins

18 Restricted and unrestricted HF wave Consider the H 2 molecule functions RHF UHF < < a 1sa 1sb g a a a fa fb 1 2 g vac vac

19 Roothaan-Hall equations Expand MOs as a linear combination of AOs (LCAO), using the expansion coefficients C as variational parameters The variational conditions for optimized energy become AO f C < SCe  1 ( ) AO AO f < h D g, g mn mn rs mnrs msrn rs 2 AO D < 2ÂCC rs ri si i These are solved by applying the SCF method

20 Things to think about & homework Could the Koopmans theorem be employed to the other direction, i.e. adding electrons to the N- electron system as N 1 < HF? aa t Derive the Roothan-Hall equations Study the Chapter 3 Do the preparatory exercise for Friday at