Hartree-Fock exchange operator

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. 

In 2003, Heyd et al. proposed to use a screened Coulomb operator in the Hartree-Fock exchange part of hybrid functionals [41]. The Coulomb operator was split into short-range (SR) and long-range (LR) components. 

Harrison, et al. have reported an efficient, accurate multiresolution solver for the Kohn-Sham and Hartree-Fock self-consistent field methods for general polyatomic molecules. The Hartree-Fock exchange is a nonlocal operator, whose evaluation has been a computational bottleneck for electronic structure calculations, scaling as for small molecules... 

It should be noted that the Hartree-Fock equations F ( )i = 8i ([)] possess solutions for the spin-orbitals which appear in F (the so-called occupied spin-orbitals) as well as for orbitals which are not occupied in F (the so-called virtual spin-orbitals). In fact, the F operator is hermitian, so it possesses a complete set of orthonormal eigenfunctions only those which appear in F appear in the coulomb and exchange potentials of the Foek operator. The physical meaning of the occupied and virtual orbitals will be clarified later in this Chapter (Section VITA)... 

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. 

Note that the operator fKS differs from the Fock operator f that we introduced in Section 1.3 in connection with the Hartree-Fock scheme only in the way the exchange and correlation potentials are treated. In the former, the non-classical contributions are expressed via the - in its exact form unknown - exchange-correlation potential Vxc, the functional derivative of Exc with respect to the charge density. In the latter, correlation is neglected... 

Note that the Kohn-Sham Hamiltonian hKS [Eq. (4.1)] is a local operator, uniquely determined by electron density15. This is the main difference with respect to the Hartree-Fock equations which contain a nonlocal operator, namely the exchange part of the potential operator. In addition, the KS equations incorporate the correlation effects through Vxc whereas they are lacking in the Hartree-Fock SCF scheme. Nevertheless, though the latter model cannot be considered a special case of the KS equations, there are some similarities between the Hartree-Fock and the Kohn-Sham methods, as both lead to a set of one-electron equations allowing to describe an n-electron system. 

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 ... 

The terms on the right-hand side of eq. (11.41) denote the kinetic energy, the electron-nuclear potential energy, the Coulomb (J) and exchange (K) terms respectively. Together J and K describe an effective electron-electron interaction. The prime on the summation in the expression for K exchange term indicates summing only over pairs of electrons of the same spin. The Hartree-Fock equations (11.40) are solved iteratively since the Fock operator / itself depends on the orbitals iff,. 

A special case of this approach is represented by the Hartree-Fock equations, where the effective operator Heff contains the usual kinetic (T), nuclear attraction (U), Coulomb (J), and exchange (K) components such that... 

The sum over coulomb and exchange interactions in the Fock operator runs only over those spin-orbitals that are occupied in the trial VF. Because a unitary transformation among the orbitals that appear in F leaves the determinant unchanged (this is a property of determinants- det (UA) = det (U) det (A) = 1 det (A), if U is a unitary matrix), it is possible to choose such a unitary transformation to make the Ey matrix diagonal. Upon so doing, one is left with the so-called canonical Hartree-Fock equations ... 

Whereas Si and s2 are true one-electron spin operators, Ky is the exchange integral of electrons and in one-electron states i and j (independent particle picture of Hartree-Fock theory assumed). It should be stressed here that in the original work by Van Vleck (80) in 1932 the integral was denoted as Jy but as it is an exchange integral we write it as Ky in order to be in accordance with the notation in quantum chemistry, where Jy denotes a Coulomb integral. 

Hartree-Fock theory as constructed using the Roothaan approach is quite beautiful in the abstract. This is not to say, however, that it does not suffer from certain chemical and practical limitations. Its chief chemical limitation is the one-electron nature of the Fock operators. Other than exchange, all election correlation is ignored. It is, of course, an interesting question to ask just how important such correlation is for various molecular properties, and we will examine that in some detail in following chapters. 

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