Fock operator model

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... 

An important consequence of the only approximate treatment of the electron-electron repulsion is that the true wave function of a many electron system is never a single Slater determinant We may ask now if SD is not the exact wave function of N interacting electrons, is there any other (necessarily artificial model) system of which it is the correct wave function The answer is Yes it can easily be shown that a Slater determinant is indeed an eigenfunction of a Hamilton operator defined as the sum of the Fock operators of equation (1-25)... 

In Eq. (2.30), F is the Fock operator and Hcore is the Hamiltonian describing the motion of an electron in the field of the spatially fixed atomic nuclei. The operators and K. are operators that introduce the effects of electrons in the other occupied MOs. Hence, when i = j, J( (= K.) is the potential from the other electron that occupies the same MO, i ff IC is termed the exchange potential and does not have a simple functional form as it describes the effect of wavefunction asymmetry on the correlation of electrons with identical spin. Although simple in form, Eq. (2.29) (which is obtained after relatively complex mathematical analysis) represents a system of differential equations that are impractical to solve for systems of any interest to biochemists. Furthermore, the orbital solutions do not allow a simple association of molecular properties with individual atoms, which is the model most useful to experimental chemists and biochemists. A series of soluble linear equations, however, can be derived by assuming that the MOs can be expressed as a linear combination of atomic orbitals (LCAO)44 ... 

The use of the Hartree-Fock model allows the perturbation-theory equations (1.2)-(1.5) to be conveniently recast in terms of underlying orbitals (,), orbital energies (e,), and orbital occupancies (n,). Such orbital perturbation equations will allow us to treat the complex electronic interactions of the actual many-electron system (described by Fock operator F) in terms of a simpler non-interacting system (described by unperturbed Fock operator We shall make use of such one-electron perturbation expressions throughout this book to elucidate the origin of chemical bonding effects within the Hartree-Fock model (which can be further refined with post-HF perturbative procedures, if desired). 

Following a canonical order to get molecular wavefunctions, we introduce here the Hartree-Fock (HF) level of the two-step approach described above. In this framework we have to define the Fock operator for our model. We adopt here an expansion of this operator over a finite basis set and thus all the operators are given in terms of their matrices in such a basis. The Fock matrix reads ... 

Let us now pass to continuum models. As for the isolated molecule, also here the new Fock operator defined in Equation (1.108) and determining the new solution I hf) is obtained by minimizing an appropriate functional. However, now the kernel of this functional is not the Hamiltonian Heff given in Equation (1.105) but rather HefS — VmJ2 and thus the energy of the system is given by... 

The self-consistent nature of the Fock operator is sometimes modeled in the methods without interaction by the schemes relating the ionization potentials of each AO with the overall electronic population (or effective charge) of a given atom and sometimes with the orbital populations of the AOs centered on the considered atom. This generally leads to the expressions of the form [55] ... 

The parameters used to construct the model Fock operator s matrix elements in the OAOs basis set are as follows ... 

Inserting the expansion eq. (4.40) rewritten in terms of matrices V in the energy expression eq. (4.31) with the perturbed Fock operator eq. (4.39) yields a DMM model of the CC of an arbitrary symmetry since the transition densities V take account of all possible perturbations of the electronic structure, keeping the CLS a separate entity. The series eq. (4.40) in fact appears by expanding the closed expression for the projection operator ... 

As can be seen from Equation 6.33, the one-electron part of the Fock operator (cf. Equation 6.25 and Equation 6.26) is set to be proportional to the overlap between atoms A and B with respect to the two specified AOs. Note that this approach violates the NDDO formalism consequently, the approach was called MNDO, and the successor models AMI, PM3, and PM5 are in fact MNDO-type methods. The one-electron matrix element of Equation 6.33 represents the kinetic and potential energy of the electron in the field of the two nuclei (represented by their core charges), and the P terms are the respective single-atom resonance integrals. 

A last but promising procedure is based on the use of effective Hamiltonians. In such approaches, it is supposed that the valence Hamiltonian that reproduces the Fock operator is a sum of kinetic and various effective atomic potentials for atoms within their characteristic chemical environment. In practical computations, those effective potentials are, for example, chosen in a non-local form of Gaussian projectors with spherical or non-spherical symmetry. Parameterization of these potentials is performed by least square fitting of corresponding valence Fock operators for small model molecules ( ). 

Fock matrix (1.5.1) reaction field (2.7.2) matrix element of the Fock operator between AOs and (I-5.1) electron repulsion operator (1.2.3) ground state of the 3 x 3 model (4.4.1) and of 4)V-electron perimeter (2.2.7) free energy (1.4.3) one-electron operator of kinetic and potential energy (1.2.3)... 

In an independent-particle model, Equation (3.7) defines a Fock operator describing valence electrons moving in a field generated by the electrons in orbitals k. Using this relationship, we define an approximate operator... 

The most straightforward approach to constructing an ECP is to use the Fock operator of a valence orbital

Hartree-Fock potential by a simpler operator VCv using the following identity ... 

The most obvious approximation is to replace the terms in the effective Fock operator F + Pc(e — c) which depend on the core orbitals, by an analytic, local repulsive model potential. One commonly used model potential of this type is the Hellmann potential... 

The model molecules chosen to parameterize carbon and hydrogen atomic potentials were ethane, transbutadiene, and acetylene(6) for sulfur and carbon linked to sulfur, dimethyl sulfide and thiophene( 8) for nitrogen and carbon linked to nitrogen, dimethylamine and pyrrole.(15) For each molecule, the Fock operator is constructed as ... 

The zero-order Hamiltonian H0 corresponds to the Fock operator, whereas the fluctuation potential V represents the difference between the full, instantaneous two-electron potential and the averaged SCF potential of the Hartree-Fock model ... 

This also puts in doubt the nomic character of the ab initio procedure. If, as it were, the final model that is used in a ah initio calculation, i.e., the Hartree Fock operator applied to the LCAO model is not deducible from the primordial model, i.e., the Schrodinger-equation with the original Hamiltonian, then it seems that the law that is de facto operating in the ab initio derivation is contained in the Hartree Fock model. But this simplified model is based on assumptions that in part are contradictory to the assumptions of the original model. These assumptions include ... 

Since the model space for LiH contains an open-shell CSF, we use two different Fock operators, viz. and /, and their corresponding generahzed versions (/ = and /= respectively) in our partitioning strategy. As is evident from... 

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... 

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