Hartree-Fock approximation wave function

The only atomic wave-functions that do not have a node at the nucleus are s-functions. The isotropic coupling constant is thus a measure of the s-character of the wave-function of the unpaired electron at the nucleus in question. The coupling constant for an atomic s-electron can be either measured experimentally or calculated from Hartree-Fock atomic wave-functions so that, to a first approximation, the s-electron density may be calculated from the ratio of the experimental and atomic coupling constants. Should the first-order s-character of the wave-function of the unpaired electron be zero, as for example in the planar methyl radical, then a small isotropic coupling usually arises from second-order spin-polarization effects. The ESR spectra of solutions show only isotropic hyperfine coupling. 

We have used the terms SCF wave function and Hartree-Fock wave function interchangeably. In practice, the term SCF wave function is applied to any wave function obtained by iterative solution of the Roothaan equations, whether or not the basis set is large enough to give a really accurate approximation to the Hartree-Fock SCF wave function. There is only one true Hartree-Fock SCF wave function, which is the best possible wave function that can be written as a Slater determinant of spin-orbitals. Some of the extended-basis-set calculations approach the true Hartree-Fock wave... 

Other calculations tested using this molecule include two-dimensional, fully numerical solutions of the molecular Dirac equation and LCAO Hartree-Fock-Slater wave functions [6,7] local density approximations to the moment of momentum with Hartree-Fock-Roothaan wave functions [8] and the effect on bond formation in momentum space [9]. Also available are the effects of information theory basis set quality on LCAO-SCF-MO calculations [10,11] density function theory applied to Hartree-Fock wave functions [11] higher-order energies in... 

We note that the virial theorem is automatically fulfilled in the Hartree-Fock approximation. This result follows from the fact that the single Slater determinant (Eq. 11.38) built up from the Hartree-Fock functions pk x) satisfying Eq. 11.46 is the optimum wave function of this particular form, and, since this wave function cannot be further improved by scaling, the virial theorem must be fulfilled from the very beginning. If we consider a stationary state with the nuclei in their equilibrium positions, we have particularly Thf = — Fhf, and for the correlation terms follows consequently that... 

Just as in the unrestricted Hartree-Fock variant, the Slater determinant constructed from the KS orbitals originating from a spin unrestricted exchange-correlation functional is not a spin eigenfunction. Frequently, the resulting (S2) expectation value is used as a probe for the quality of the UKS scheme, similar to what is usually done within UHF. However, we must be careful not to overstress the apparent parallelism between unrestricted Kohn-Sham and Hartree-Fock in the latter, the Slater determinant is in fact the approximate wave function used. The stronger its spin contamination, the more questionable it certainly gets. In... 

In order to find a good approximate wave function, one uses the Hartree-Fock procedure. Indeed, the main reason the Schrodinger equation is not solvable analytically is the presence of interelectronic repulsion of the form e2/r. — r.. In the absence of this term, the equation for an atom with n electrons could be separated into n hydrogen-like equations. The Hartree-Fock method, also called the Self-Consistent-Field method, regards all electrons except one (called, for instance, electron 1), as forming a cloud of electric charge... 

Mean-field approximation of quasi-free electrons (the Hartree-Fock approximation). The total wave function is described, in this case, by a single Slater determinant. 

Two different correlation effects can be distinguished. The first one, called dynamical electron correlation, comes from the fact that in the Hartree-Fock approximation the instantaneous electron repulsion is not taken into account. The nondynamical electron correlation arises when several electron configurations are nearly degenerate and are strongly mixed in the wave function. 

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

Considering the different calculated values for an individual complex in Table 11, it seems appropriate to comment on the accuracy achievable within the Hartree-Fock approximation, with respect to both the limitations inherent in the theory itself and also to the expense one is willing to invest into basis sets. Clearly the Hartree-Fock-Roothaan expectation values have a uniquely defined meaning only as long as a complete set of basis functions is used. In practice, however, one is forced to truncate the expansion of the wave function at a point demanded by the computing facilities available. Some sources of error introduced thereby, namely ghost effects and the inaccurate description of ligand properties, have already been discussed in Chapter II. Here we concentrate on the... 

Establishing a hierarchy of rapidly converging, generally applicable, systematic approximations of exact electronic wave functions is the holy grail of electronic structure theory [1]. The basis of these approximations is the Hartree-Fock (HF) method, which defines a simple noncorrelated reference wave function consisting of a single Slater determinant (an antisymmetrized product of orbitals). To introduce electron correlation into the description, the wave function is expanded as a combination of the reference and excited Slater determinants obtained by promotion of one, two, or more electrons into vacant virtual orbitals. The approximate wave functions thus defined are characterized by the manner of the expansion (linear, nonlinear), the maximum excitation rank, and by the size of one-electron basis used to represent the orbitals. 

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