Fockian

According to Eq. (63), the 1-RDM and the generalized Fockian F commute at the extremum hence the NSOs are the solutions of the eigenproblem (60) with the nonlocal potential defined by the identity (64). One should note, however, that Eq. (64) does not completely define v e- In fact, the diagonal elements... 

This selection implies that e, = jx for the optimal 1-RDM (compare with the Eq. (54)). The Fockian matrix elements are as follows ... 

We conclude that the problem of finding optimal NOs turns into the iterative diagonalization of Eq. (60) with a Fockian matrix, Eq. (66). The corresponding eigenfunctions are certainly orthonormal and optimize the total energy functional, Eq. (41). 

The one-electron equations (60) offer a new possibility for finding the optimal NSOs by iterative diagonalization of the Fockian, Eq. (66). The main advantage of this method is that the resulting orbitals are automatically orthogonal. The first calculations based on this diagonalization technique has confirmed its practical value [81]. 

Substituting these averaged density matrices into Eqn (36), then into Eqn (35), one obtains the Fockian matrix as a ilinctional of the averaged density matrices ... 

By this the one-electron part of the effective Fockian in the CLS carrier subspace is completely defined. 

In the octahedral geometry, the orbitals of each entering symmetry appear no more than twice. For that reason, the problem of defining variables xai and xtlu (or their equivalents - see below) reduces to diagonalization of the 2 x 2 Fockian blocks corresponding to the respective irreducible representations F ... 

The exact definition of the matrix elements of the Fockian for an HFR-treated group of electrons in the presence of other groups is given in [13] and [33] (and above). 

The construction of the LD theory of the ligand influence evolves in terms of two key objects the electron-vibration (vibronic) interaction operator and the substitution operator. The vibronic interaction in the present context is the formal expression for the effect of the system Hamiltonian (Fockian) dependence on the molecular geometry taken in the lower - linear approximation with respect to geometry variations. It describes coupling between the electronic wave function (or electron density) and molecular geometry. 

Any Fock operator can be represented as a sum of the symmetric one and of a perturbation which includes both the dependence of the matrix elements on nuclear shifts from the equilibrium positions and the transition to a less symmetric environment due to the substitution. To pursue this, we first introduce some notations. Let hi be the supervector of the first derivatives of the matrix of the Fock operator with respect to nuclear shifts Sq counted from a symmetrical equilibrium configuration. By a supervector, we understand here a vector whose components numbered by the nuclear Cartesian shifts are themselves 10 x 10 matrices of the first derivatives of the Fock operator, with respect to the latter. Then the scalar product of the vector of all nuclear shifts 6q j and of the supervector hi yields a 10 x 10 matrix of the corrections to the Fockian linear in the nuclear shifts ... 

A common procedure is to include the potential of a representative set of partial charges of the environment in the Hamilton operator or the Fockian of the prototype molecule [188, 189]. This means that the core Hamiltonian is modified by a set of nuclear attraction integrals with partial point charges ... 

The charges induced in the solvent determine the electric field that has an effect on the electron density of the solute molecule and, consequently, causes its reorganization. One may find the charge distribution of a solute molecule in the polarizable medium by solving modified Hartree-Fock equations in which the Fockian has the form ... 

The best wave function which can be written down as a single Slater determinant is the Hartree-Fock wave function. It is built up from the Hartree-Fock orbitals which are eigenfunctions of the Fockian F ... 

The Fockian can also be introduced into the spatial energy expression of Eq. 

The Hartree-Fock model leads to an effective one-electron Hamiltonian, called the Fockian F. The second quantized representation of the Fockian has that same form as any other one-electron operator. In the basis of orthogonalized spin-orbitals one can write ... 

It is convenient to have the Fockian in terms of spatial orbitals. The situation is simple in the closed-shell case where each orbital is required to be either doubly occupied or empty according to the restricted Hartree-Fock (RHF) method. Than, the spatial density matrix is given by Eq. (9.8), and the transcription of Eq. (10.55) can be performed in the usual manner. We find that the spatial Fock matrix is independent of spin ... 

The situation is more complicated if the restriction of double occupancy is dropped. Here we discuss the unrestricted Hartree-Fock (UHF) method in some detail. In this case one has different density matrices for the a and p electrons, P and pP, respectively. As a consequence, one has also different Fockians for the a and P electrons F and F. The expressions of their matrix elements can be derived from Eq. (10.55) by performing the integration over spin functions ... 

The second quantized form of the Fockian in the UHF case is obtained by performing the summation over spin in Eq. (10.51) in the usual way, but taking into account that the Fock matrix is spin dependent according to Eq. (10.58) ... 

In the UHF method one diagonalizes both F and F self-consistently, that is in the MO basis the Fockian is given by ... 

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). 

Note that this holds provided that the fockian contains only local potential terms, which commute with the... 

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