Hartree-Fock solution

A Perturbation Theory is developed for treating a system of n electrons in which the Hartree-Fock solution appears as the zero-order approximation. It is shown by this development that the first order correction for the energy and the charge density of the system is zero. The expression for the second order correction for the energy greatly simplifies because of the special property of the zero order solution. It is pointed out that the development of the higher order approximation involves only calculations based on a definite one-body problem. 

Of the many quantum chemical approaches available, density-functional theory (DFT) has over the past decade become a key method, with applications ranging from interstellar space, to the atmosphere, the biosphere and the solid state. The strength of the method is that whereas conventional ah initio theory includes electron correlation by use of a perturbation series expansion, or increasing orders of excited state configurations added to zero-order Hartree-Fock solutions, DFT methods inherently contain a large fraction of the electron correlation already from the start, via the so-called exchange-correlation junctional. 

Although the methods suggested above are not, by any means, completely satisfactory, they are sufficient to describe the main qualitative aspects of the problem. In the remainder of this section, therefore, we shall discuss methods of solving the one-electron equations (12), with h defined by (13). We shall assume that the Coulomb repulsion term can be neglected and, therefore, will not consider Hartree-Fock solutions. Details of the latter can be found in Refs. 25-27. 

Differential Virial Theorem for the Hartree-Fock Solution. 103... 

Shiozaki, T., Hirata, S. Grid-based Hartree-Fock solutions of polyatomic molecules. Phys. Rev. A 2007, 76, 040503(R). 

If the supermatrix A becomes degenerate (at the point where the Hartree-Fock solution for which it is calculated loses its stability i.e. ceases to be a minimum of the energy functional) the inversion is not possible any more, but the Hartree-Fock picture of the electronic structure itself becomes invalid. In this case the above treatment obviously loses any sense. 

In addition, comparison to solutions of the Hubbard and PPP models including electron correlation shows the VB wave function to be a more accurate initial approximation than the Hartree Fock solution at the correlation strengths likely to be encountered in realistic semiempirical models. In spite of the qualitative superiority of the VB wave function, systematic computational approaches to more accurate treatment of correlation are still most readily achieved when starting from the independent particle limit, but the correlated wave functions thus built up are likely to be interpretable in valence bond terms. 

Of paramount importance in this latter category is the Hartree-Fock approximation. The so-called Hartree-Fock limit represents a well-defined plateau, in terms of its methematical and physical properties, in the hierarchy of approximate solutions to Schrodinger s electronic equation. In addition, the Hartree-Fock solution serves as the starting point for many of the presently employed methods whose ultimate goal is to achieve solutions to equation (5) of chemical accuracy. A discussion of the Hartree-Fock method and its associated concept of a self-consistent field thus provides a natural starting point for the discussion of the calculation of potential surfaces. 

The orbital approach to the problem of electronic structure reduces a many-electron problem to a corresponding number of one-electron problems. The Hartree-Fock solution represents the best attainable description of the electronic structure of a many-electron system in terms of the one-electron orbital approach. 

Use of a complete (necessarily infinite) set of basis orbitals in the expansion for the molecular orbitals would insure absolute convergence to the Hartree-Fock limit. In practice, this is both impossible and unnecessary, and only a finite number of basis functions is employed in the expansion. The selection of the finite basis set is, therefore, of crucial importance in determining how closely one approximates the true Hartree-Fock solution. 

The multipllclly of the Hartree-Fock solutions obtained for the BH molecule motivated us to perform a special study to clarify which ones of these solutions correspond to true (local) minima, and which ones correspond to saddle points on the energy hypersurface, and to determine the bifurcation points in which new types of solutions appear as exactly as possible. For that reason we have investigated the Hessians for the RHF, UHF/2 and GHF/1 wave functions discussed above. As is well-known [19,20], the Hessian is defined as a matrix H = (Hij) with the elements... 

There are a number of ways in which one may begin to correct the Hartree-Fock wave function so as to include electron correlation ". The simplest in concept is configuration interaction (Cl) which takes the Hartree-Fock solution as a starting point, or reference con-... 

As an example of the accuracy of the higher order corrections to Hartree-Fock solutions obtained by perturbation methods and by parametrised density functional methods. Table 3.1 looks at the binding energy (or minus the atomisation energy) for an oxygen molecule, calculated in a number of dif-... 

K. Deguchi, K. Nishikawa, and S. Aono,/. Chem. Phys., 75,4165 (1981). Instabilities of the Hartree-Fock Solution and the Vibrational Mode in a Molecule. 

An important feature of the 1-CSE is that it is exactly satisfied by at least two sets of RDM s [6,7,13,19], the set corresponding to the Hartree-Fock solution and the set corresponding to the FCI solution. In what follows, the set of HF matrices, as well as the corresponding energy will be distinguished from those corresponding to the exact FCI solution by an upper index ( ). Thus the HF form of Eq.(14) is written... 

Just as there exist the so-called Thouless stability conditions on the Hartree-Fock solutions in nuclear physics (Thouless, 1960, 1961 Rowe, 1970) and in quantum chemistry (CiZek and Paldus, 1971), one has stability conditions on the mean field solutions in lattice dynamics problems (Fredkin and Werthamer, 1965). The mean field solutions are obtained from the condition AA . = 0 (see Section IV,A). They are stable i.e., they correspond with a local minimum in the free energy if > 0. Substituting the mean field solution (109) into the equation (107) for AA ., the term with Apf vanishes and we can express the stability condition as... 

In the Fock space coupled cluster method, the Hartree-Fock solution for an iV-electron state, 0), is used as the vacuum. The Fock space is divided into sectors, (m,n), according to how many electrons are added to and removed from 0). Thus, the vacuum is in the (0,0) sector, single ionizations are in the (0,1) sector, one-electron attached states are in (1,0), and (1,1) are single excitations relative to 0). The orbitals are also divided into active, which can change occupation, and inactive, for which the occupation is fixed. All possible occupations of the active orbitals in all possible sectors constitute the multireference space for the system. 

No matter how good the basis set is made by extension toward an infinite set, one encounters the Hartree-Fock limit on the accuracy of molecular energy, because the influence of one electron upon the others has not been fully accounted for in the SCF averaging procedure. The difference between a Hartree-Fock energy and the experimental energy is called the correlation energy. To remedy this fault, correlated models are made up which consist of a linear combination of the Hartree-Fock solution plus singly, doubly, etc. substituted wave functions... 

Transition metals are important materials with intriguing properties and they have been studied with ever improved methods. A major difficulty is posed by the standard one-electron models where the tight-binding model seems appropriate for the narrow, so-called d-bands while near-plane-wave crystal orbitals are adequate for the conduction bands. Canonical Hartree-Fock solutions are awkward starting points for the description of magnetic structures and the use of spin-polarized versions destroys basic symmetry properties. 

Electron-electron interactions determine the detailed state formation for a partially filled shell from nearly degenerate atomic d-orbitals. The emphasis here is to demonstrate the nature of the grand canonical ensemble Hartree-Fock which allows partial occupancy of spin orbitals while maintaining point group or spherical symmetry. The analysis is set in the proper spin orbital basis with spin-orbit coupling as well as octahedral symmetry accounted for in the canonical Hartree-Fock solutions. Explicit expressions for the separable representation of the electron interaction are offered to exemplify the available reductions. 

Apart from practical and computational sides, the SCF model has both intuitive appeal and formal mathematical advantages the Hartree-Fock solution is a well-defined one, with known mathematical properties (Brillouin theorem, Hellmann-Feynman theorem etc.). As an example one may mention the invariance of the coupled perturbed Hartree-Fock solution under a translation of the gauge of an external perturbing magnetic field 78). Such properties give the SCF approach a certain special position. 

We now choose our periodic Hamilton to be in the form of (9). We shall let our expansion point be the Slater determinant 0) corresponding to the Hartree-Fock solution of Jdo- We further assume the periodic perturbation operator P (t) to be given by (20). The quasienergy can then be written as (39). In our notation... 

We have chosen the reference determinant to be the Hartree-Fock solution of Hq. This implies that the gradient e[, is zero, that is... 

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