Fock operators

We quote the following result without proof or further discussion. When applied to the state wavefunction I , the variation principle leads to the Fock equation 

Since F and i/, depend only on the coordinate of a single electron, equation 8.15 may be written as 

The Fock operator is made up of three terms. The first is a one-electron term describing the core potential and the other two are two-electron terms that contain the electron-electron repulsion energies. For an electron fx located in the MO i/fi, the core potential, namely, the kinetic plus nuclear-electron attraction energies, is given by the expectation value of the core-Hamlltonian h(/i,) 

For two electrons /x and v jx v) accommodated in the MOs and xlxp respectively, the two-electron terms are the Coulomb repulsion and the change repulsion energies. 

From these definitions, it follows that Kii=Jg. Also note that the Coulomb repulsion between two electrons is independent of their spins while the exchange repulsion vanishes unless their spins are the same.J,j- is repulsive (i.e., positive) and represents the electrostatic repulsion between electron fj, in orbital jfj and electron v in orbital j. It increases with increasing the overlap between the electron densities and 

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Even Hartree-Fock calculations are diflTicult and expensive to apply to large molecules. As a result, fiirther simplifications are often made. Parts of the Fock operator are ignored or replaced by parameters chosen by some sort of statistical procedure to account, in an average way, for the known properties of selected... 

The HF [31] equations = e.cj). possess solutions for the spin orbitals in T (the occupied spin orbitals) as well as for orbitals not occupied in F (the virtual spin orbitals) because the operator is Flennitian. Only the ( ). occupied in F appear in the Coulomb and exchange potentials of the Fock operator. 

As fonnulated above, the FIF equations yield orbitals that do not guarantee that F has proper spin symmetry. To illustrate, consider an open-shell system such as the lithium atom. If Isa, IsP, and 2sa spin orbitals are chosen to appear in F, the Fock operator will be... 

This method [ ] uses the single-configuration SCF process to detennine a set of orbitals ( ).]. Then, using an unperturbed Flamiltonian equal to the sum of the electrons Fock operators // = 2 perturbation... 

Ihe Fock operator is an effective one-electron Hamiltonian for the electron in the poly-tiectronic system. However, written in this form of Equation (2.130), the Hartree-Fock... 

The Fock matrix elements for a closed-shell system can be expanded as follows by substituting the expression for the Fock operator ... 

Note that Jj r ) and A)(ri) are operators that go to make up the Fock operator. They operate on functions. One often sees the notation... 

Guess the average potential that a specific electron would feel coming from the other electrons that is, you guess at the Fock operator. 

A variety of methodologies have been implemented for the reaction field. The basic equation for the dielectric continuum model is the Poisson-Laplace equation, by which the electrostatic field in a cavity with an arbitrary shape and size is calculated, although some methods do not satisfy the equation. Because the solute s electronic strucmre and the reaction field depend on each other, a nonlinear equation (modified Schrddinger equation) has to be solved in an iterative manner. In practice this is achieved by modifying the electronic Hamiltonian or Fock operator, which is defined through the shape and size of the cavity and the description of the solute s electronic distribution. If one takes a dipole moment approximation for the solute s electronic distribution and a spherical cavity (Onsager s reaction field), the interaction can be derived rather easily and an analytical expression of theFock operator is obtained. However, such an expression is not feasible for an arbitrary electronic distribution in an arbitrary cavity fitted to the molecular shape. In this case the Fock operator is very complicated and has to be prepared by a numerical procedure. 

We recently proposed a new method referred to as RISM-SCF/MCSCF based on the ab initio electronic structure theory and the integral equation theory of molecular liquids (RISM). Ten-no et al. [12,13] proposed the original RISM-SCF method in 1993. The basic idea of the method is to replace the reaction field in the continuum models with a microscopic expression in terms of the site-site radial distribution functions between solute and solvent, which can be calculated from the RISM theory. Exploiting the microscopic reaction field, the Fock operator of a molecule in solution can be expressed by... 

The solvated Fock operator can be naturally derived from the variational principles [14] defining the Helmlioltz free energy of the system (fA) by... 

So far, we ve presented only general perturbation theory results.We U now turn to the particular case of Moller-Plesset perturbation theory. Here, Hg is defined as the sum of the one-electron Fock operators ... 

Since Hq is the sum of Fock operators, then is the sum of the orbital energies ... 

Boys and Cook refer to these properties as primary properties because their electronic contributions can be obtained directly from the electronic wavefunction As a matter of interest, they also classified the electronic energy as a primary property. It can t be calculated as the expectation value of a sum of true one-electron operators, but the Hartree-Fock operator is sometimes written as a sum of pseudo one-electron operators, which include the average effects of the other electrons. 

The third and fifth terms are identical (since the summation is over all i and j), as are the fourth and sixth ternis. They may be collected to cancel the factor of 1/2, and the variation can be written in terms of a Fock operator, F,. 

The Fock operator is an effective one-electron energy operator, describing the kinetic energy of an electron, the attraction to all the nuclei and the repulsion to all the other electrons (via the J and K operators). Note that the Fock operator is associated with the variation of the total energy, not the energy itself. The Hamilton operator (3.23) is not a sum of Fock operators. 

The Lagrange multipliers can be interpreted as MO energies, i.e. they expectation value of the Fock operator in the MO basis (multiply eq. (3.41) by the left and integrate). 

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. 

The orbital energies can be considered as matrix elements of the Fock operator with the MOs (dropping the prime notation and letting 0 be the canonical orbitals). The total energy can be written either as eq. (3.32) or in terms of MO energies (using the definition of F in eqs. (3.36) and (3.42)). 

The S matrix contains the overlap elements between basis functions, and the F matrix contains the Fock matrix elements. Each element contains two parts from the Fock operator (eq. (3.36)), integrals involving the one-electron operators, and a sum over... 

Level Shifting. This technique is perhaps best understood in the formulation of a rotation of the MOs which form the basis for the Fock operator. Section 3.6. At convergence the Fock matrix elements in the MO basis between occupied and virtual orbitals are zero. The iterative procedure involves mixing (making linear... 

This is an occupied-virtual off-diagonal element of the Fock matrix in the MO basis, and is identical to the gradient of the energy with respect to an occupied-virtual mixing parameter (except for a factor of 4), see eq. (3.67). If the determinants are constructed from optimized canonical HF MOs, the gradient is zero, and the matrix element is zero. This may also be realized by noting that the MOs are eigenfunctions of the Fock operator, eq. (3.41). 

So far the theory has been completely general. In order to apply perturbation theory to the calculation of correlation energy, the unperturbed Hamilton operator must be selected. The most common choice is to take this as a sum over Fock operators, leading to Mdller-Plesset (MP) perturbation theory. The sum of Fock operators counts the (average) electron-electron repulsion twice (eq. (3.43)), and the perturbation becomes... 

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