Hartree exchange operator

The Hartree-Fock equations form a set of pseudo-eigenvalue equations, as the Fock operator depends on all the occupied MOs (via the Coulomb and Exchange operators, eqs. (3.36) and (3.33)). A specific Fock orbital can only be determined if all the other occupied orbitals are known, and iterative methods must therefore be employed for determining the orbitals. A set of functions which is a solution to eq. (3.41) are called Self-Consistent Field (SCF) orbitals. 

If no correlation is introduced (ec = 0), the KS equations reduce to the well known Xa method proposed by Slater22 as a simplification of the Hartree-Fock scheme with a local exchange operator ... 

In many calculations beyond the Hartree-Fock level a first step is the transformation of at least some integrals. For the simplest such calculation, second-order perturbation theory, integrals with two indices transformed into the occupied MO basis axe required. Such integrals appear in many situations, including the MO basis formulation of coupled-perturbed Hartree-Fock theory. We can represent the first phase of this transformation as obtaining Coulomb and exchange operators ... 

In the above formula, HFvx([ J r) is the familiar nonlocal Fock exchange operator, but here built from the one-particle Kohn-Sham orbitals of instead of from the Hartree-Fock orbitals. 

For singlet excitations a = —1 andb = 2 for tritlet excitations a = —, b = 0 h, J, and K are, respectively, the one-electron, the Coulomb, and the exchange operators. They have the same meaning as in the usual Hartree-Fock operator. Here Ff1 3 is the N-electron operator with the hole in the ith occupied MO. Thus, Hunt and Goddard have replaced a single Hartree-Fock operator by a whole set of operators [Eq. (28)] differing in the position of the vacancy. The spectrum of each of these operators is an orthonormal set of MOs ... 

The Hartree-Fock equations read (F — ej)(j)j = 0. Note that the definition of the Fock operator involves all of its eigenfuctions fa through the coulomb and exchange operators, Jt and Kt. 

One guesses at an initial set of wave functions, , and constructs the Hartree-Fock Hamilton S which depends on the through the definitions of the Coulomb and exchange operators, (/ and One then calculates the new set of , and compares it (or the energy or the density matrix) to the input set (or to the energy or density matrix computed from the input set). This procedure is continued until the appropriate self-consistency is obtained. 

With the first of these forms, a single exchange operator, we may write down the unique equation which the optimising orbitals must satisfy in order that there be a turning point in the Hartree-Fock energy functional eqn ( 2.1) ... 

In fact, this is the principle role of the exchange term to cancel the unphysical self-repulsion in the Coulomb sum. It is the difference between the Hartree and the Hartree-Fock methods, and the reason why all the MOs are the eigenfunctions of the same Hartree-Fock operator, while a separate Hartree operator is needed for each MO which excludes the self-repulsion for that MO. 

The Coulomb and exchange operators in the Hartree-Fock Hamiltonian depend on the expansion coefficients and through them on the nuclear coordinates. 

The precursor to Kohn-Sham density-functional theory is Slater theory [12], In the latter theory, the nonlocal exchange operator of Hartree-Fock theory [25] is replaced by the Slater local exchange potential Vf(r) defined in terms of the Fermi hole p,(r, r ) as... 

The one-electron Kohn-Sham operator (1) in (15.123) is the same as the Fock operator (15.82) in the Hartree-Fock equations except that the exchange operators -2"=i in the Fock operator are replaced by (Problem 15.63), which handles the effects of both exchange (antisymmetry) and electron correlation. 

The exchange operator represents a (non-intuitive) result of the antisymmetrization postulate for the total wave function (Chapter 1) and it has no classical interpretation. If the variational wave function were the product of the spinorbitals (Douglas Hartree did this in the beginning of quantum chemistiy),... 

We have seen the same in the Hartree-Fock method for molecules, where the Coulomb and exchange operators depended on the solutions to the Fock equation, (cf. p. 4121. 

The difference between this Fock operator and the Hartree-Fock counterpart in Eq. (2.51) is only the exchange-correlation potential functional, Exc, which substitutes for the exchange operator in the Hartree-Eock operator. That is, in the electron-electron interaction potential, only the exchange operator is replaced with the approximate potential density functionals of the exchange interactions and electron correlations, while the remaining Coulomb operator, Jj, which is represented as the interaction of electron densities, is used as is. The point is that the electron correlations, which are incorporated as the interactions between electron configurations in wavefunction theories (see Sect. 3.3), are simply included... 

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