Hartree-Fock equation definition

The KS equations are obtained by differentiating the energy with respect to the KS molecular orbitals, analogously to the derivation of the Hartree-Fock equations, where differentiation is with respect to wavefunction molecular orbitals (Section 5.2.3.4). We use the fact that the electron density distribution of the reference system, which is by decree exactly the same as that of the ground state of our real system (see the definition at the beginning of the discussion of the Kohn-Sham energy), is given by (reference [9])... 

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. 

The determinantal functions must be linearly independent and eigenfunctions of the spin operators S2 and Sz, and preferably they belong to a specified row of a specified irreducible representation of the symmetry group of the molecule [10, 11]. Definite spin states can be obtained by applying a spin projection operator to the spin-orbital product defining a configuration [12]. Suppose d>0 to be the solution of the Hartree-Fock equation. From functions of the same symmetry as d>0 one can build a wave function d>,... 

Atomic units have been used, and Fi is the integro-differential operator of the Hartree-Fock equations that determines the radial factors P i. The usual way of making stationary an energy functional that includes the orthonormality constraints of these radial factors introduces undetermined multipliers e(nl). As the definition of Fi introduces the same potential for all Pni, there is no need for off-diagonal undetermined multipliers, and the variation of the energy functional with the constraints yields the equation... 

With these definitions of F and S we can now write the integrated Hartree-Fock equation (3.135) as... 

Here, the general index j [i = 1,..., M M = n 2N + l) ]oftheCOshas been replaced by double indexing (for future convenience), where h denotes the band indices and p the levels inside the bands. The subscript 2 at the bracket indicates that integration must be performed over the coordinates of the second electron while Pn is the exchange operator. The detailed form of the relativistic Hartree-Fock equation for a crystal involves the definitions of the Dirac matrices in the form... 

The spin dependency of the UHF Fockian and the corresponding eigenvectors has the unfortunate consequence that the resulting many-electron wave function is usually not a pure spin state, rather a mixture of states of different spin multiplicities. The state of a definite multiplicity can be selected by the appropriate spin-projection operator. The thorough investigation of this problem results in the spin-projected extended Hartree-Fock equations (Mayer 1980). 

When the MM subsystem is described through point charges and permanent dipole moments, the Hamiltonian (7) must be used and the Hartree-Fock equations are simply given by using the following Fock matrix definition ... 

All the early work was concerned with atoms, with Sir William Hartree regarded as the father of the technique. His son, Douglas R. Hartree, published the definitive book, The Calculation of Atomic Structures, in 1957, and in this he derived the atomic HF equations and described numerical algorithms for their solution. Charlotte Froese Fischer was a research student working under the guidance of D. R. Hartree, and she published her own definitive book. The Hartree—Fock Method for Atoms A Numerical Approach in 1977. The Appendix lists a number of freely available atomie structure programs. Most of these can be obtained from the Computer Physics Communications Program Library. 

Over the last years, the basic concepts embedded within the SCRF formalism have undergone some significant improvements, and there are several commonly used variants on this idea. To exemplify the different methods and how their results differ, one recent work from this group [52] considered the sensitivity of results to the particular variant chosen. Due to its dependence upon only the dipole moment of the solute, the older approach is referred to herein as the dipole variant. The dipole method is also crude in the sense that the solute is placed in a spherical cavity within the solute medium, not a very realistic shape in most cases. The polarizable continuum method (PCM) [53,54,55] embeds the solute in a cavity that more accurately mimics the shape of the molecule, created by a series of overlapping spheres. The reaction field is represented by an apparent surface charge approach. The standard PCM approach utilizes an integral equation formulation (IEF) [56,57], A variant of this method is the conductor-polarized continuum model (CPCM) [58] wherein the apparent charges distributed on the cavity surface are such that the total electrostatic potential cancels on the surface. The self-consistent isodensity PCM procedure [59] determines the cavity self-consistently from an isodensity surface. The UAHF (United Atom model for Hartree-Fock/6-31 G ) definition [60] was used for the construction of the solute cavity. 

This is closely analogous to the Hartree equations (Eq. (1.7)). The Kohn-Sham orbitals are separable by definition (the electrons they describe are noninteracting) analogous to the HF MOs. Eq. (1.50) can, therefore, be solved using a similar set of steps as was done in the Hartree-Fock-Roothaan method. 

The Roothaan-Hall equations are not applicable to open-shell systems, which contain one or more unpaired electrons. Radicals are, by definition, open-shell systems as are some ground-state molecules such as NO and 02. Two approaches have been devised to treat open-shell systems. The first of these is spin-restricted Hartree-Fock (RHF) theory, which uses combinations of singly and doubly occupied molecular orbitals. The closed-shell approach that we have developed thus far is a special case of RHF theory. The doubly occupied orbitals use the same spatial functions for electrons of both a and spin. The orbital expansion Equation (2.144) is employed together with the variational method to derive the optimal values of the coefficients. The alternative approach is the spin-unrestricted Hartree-Fock (UHF) theory of Pople and Nesbet [Pople and Nesbet 1954], which uses two distinct sets of molecular orbitals one for electrons of a spin and the other for electrons of / spin. Two Fock matrices are involved, one for each type of spin, with elements as follows ... 

Within the so-called wavefunction-based methods, Eq. (5) is most often solved by first approximating Wg as a single Slater determinant. Thereby correlation effects are by definition ignored. (Parts of) these may, however, be added subsequently either directly or via perturbation theory. The N single-particle functions 0i, 02, , of the Slater determinant are calculated by solving the Hartree-Fock single-particle equations... 

An alternative to the operator approach is to start from the matrix equations (Filatov 2002). Then the elimination the small-component, the construction of the transformation and the transformed Fock matrix are all straightforward. There is no difficulty with interpretation because the inverse of a matrix is well defined. The matrix to be inverted is positive definite so it presents no numerical problems. The drawback of a matrix method is that the basis set for the small component must be used, at least to construct the potentials that appear in the inverse. In that case, the same number of integrals is required as in the full Dirac-Hartree-Fock method, and there is no reduction in the integral work or the construction of the Fock matrix. 

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