Fock operator, matrix elements

In general, a mean-field approximation is defined by any set of occupation numbers rt/t by means of a corresponding Fock operator matrix element, and the dependence of the results on the specific set of occupation numbers turned out to be very weak in practical calculations. This approximation has also been developed independently by Beming etal (2000). 

Analysis of Green s functions can be useful in seeking to establish model hamil-tonians with the property of giving approximately correct propagators, when put in the equations of motion. In this section, we explore a particularly simple model in order to familiarize the reader with various molecular orbital concepts using the terminology of Green s function theory. We employ the Hartree-Fock approximation and seek the molecular Fock operator matrix elements... 

When one evaluates the expressions for the (E )jj one simply obtains the usual Hartree-Fock operator matrix elements since and F become identical to the usual SCF Fock operators when the valence orbitals become doubly occupied (ie... 

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

When the Coulomb and exchange operators are expressed in terms of the basis functions and the orbital expansion is substituted for xu then their contributions to the Fock matrix element take the following form ... 

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). 

At the HF level of theory, the operator corresponds to addition of an extra term to the Fock matrix elements (Section 3.5). 

Fock Space Representation of Operators.—Let F be some operator that neither creates nor destroys particles, and is a known function in configuration space for N particles. In symbols such an operator must by definition have the following matrix elements in Fock space ... 

In addition, partial cross Fock operators are also to be defined for evaluating the matrix elements in which the orthogonal orbitals are involved ... 

The NAO matrix elements of the operators employed in this example can be obtained by standard keywords of the NBO program (FNAO for the Fock operator, KNAO for the kinetic-energy operator, etc.) see note 43. 

There is another widely used method of obtaining the Fock operator, namely to obtain its matrix elements F lv as the derivative of the energy functional with respect to the density. In our case that yields... 

Here, Flffl are matrix elements of a zeroth-order Hamiltonian, which is chosen as a one-electron operator in the spirit of MP2. is an overlap matrix The excited CFs are not in general orthogonal to each other. Finally, Vf)(i represents the interaction between the excited function and the CAS reference function. The difference between Eq. [2] and ordinary MP2 is the more complicated structure of the matrix elements of the zeroth-order Hamiltonian in MP2 it is a simple sum of orbital energies. Here H is a complex expression involving matrix elements of a generalized Fock operator F combined with up to fourth-order density matrices of the CAS wave function. Additional details are given in the original papers by Andersson and coworkers.17 18 We here mention only the basic principles. The zeroth-order Hamiltonian is written as a sum of projections of F onto the reference function 0)... 

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