Hartree-Fock wave functions, general

Drowicz F W and W A Goddard IB 1977. The Self-Consistent Field Equations for Generalized Valence Bond and Open-Shell Hartree-Fock Wave Functions. In Schaeffer H F III (Editor). Modem Theoretical Chemistry III, New York, Plenum, pp. 79-127. 

If an excited state is concerned, this is done under the restriction that the function should be orthogonal to all of the lower-energy states. We may specify these as the uni-configurational Hartree-Fock wave functions . The "best orbitals constructing the determinants in these wave functions are in general not orthogonal to each other. 

We see the the Hartree-Fock wave functions, 4>o and i, do not, in general, satisfy orthogonality constraints analogous to those obeyed by the exact wave functions. However, we may impose constraints upon the Hartree-Fock function without loss of generality so that, for example. 

Pu JZ, JL Gao, DG Truhlar (2004b) Generalized hybrid orbital (GHO) method for combining ab initio Hartree-Fock wave functions with molecular mechanics. J. Phys. Chem. A 108 (4) 632-650... 

F. W. Bobrowicz and W. A. Goddard, The self-consistent field equations for generalized valence bond and open-shell Hartree-Fock wave functions, in Methods of Electronic Structure Theory, edited by H. F. Schaefer, pages 79-127. Plenum Press, New York, 1977. 

The latter expression clearly shows that Hartree-Fock wave functions are not properly correlated they allow two electrons of opposite spin to simultaneously occupy a same elementary volume of an atomic or molecular space. Consequently, two-electron properties which are completely determined by the second-order density matrix cannot be correctly evaluated at the Hartree-Fock level and, a fortiori, from approximate SCF wave functions. On the contrary, satisfactory values of one-electron properties may be generally provided by those functions, at least in the case of closed-shell systems. However, due to the large contribution of pair correlation, the energy changes associated with the so-called isodes-mic processes (Hehre et al., 1970) can be reasonably well predicted at the Hartree-Fock level and also using SCF wave functions. Indeed, in that case, correlation errors approximately balance each other. 

The MP2 energy is always lower than the Hartree-Fock energy and usually represents a rather good approximation to the total electronic energy. The MP2 model usually works well whenever the Hartree-Fock wave function is a reasonable one, typically recovering about 90% of the total correlation energy, at a cost that scales formally as K5. Still, it is less robust and somewhat less generally applicable than the CCSD model. 

Now consider the basis functions used. Generally, each MO is written as a linear combination of one-electron functions (orbitals) centered on each atom. For diatomic molecules, one can use Slater functions [Eq. (11.14)] for the AOs. To have a complete set of AO basis functions, an infinite number of Slater orbitals are needed, but the true molecular Hartree-Fock wave function can be closely approximated with a reasonably small number of carefully chosen Slater orbitals. A minimal basis set for a molecular SCF calculation consists of a single basis function for each mner-shell AO and each valence-shell AO of each atom. An extended basis set is a set that is larger than a minimal set. Minimal-basis-set S(2F calculations are easier than extended-basis-set calculations, but the latter are considerably more acciurate. 

Let us generalize the above results to obtain an expression involving spatial integrals for the Hartree-Fock energy of an iV-electron system containing an even number of electrons. The analogue of the minimal basis H2 Hartree-Fock wave function,... 

So fiir in this chapter we have discussed the Hartree-Fock equations from a formal point of view in terms of a general set of spin orbitals xj. We are now in a position to consider the actual calculation of Hartree-Fock wave functions, and we must be more specific about the form of the spin orbitals. In the last chapter we briefly discussed two types of spin orbitals restricted spin orbitals, which are constrained to have the same spatial function for a (spin up) and jS (spin down) spin functions and unrestricted spin orbitals, which have different spatial functions for a and P spins. Later in this chapter we will discuss the unrestricted Hartree-Fock formalism and unrestricted calculations. In this section we are concerned with procedures for calculating restricted Hartree-Fock wave functions and, specifically, we consider here... 

Hartree-Fock wave function, and, in fact, the most general derivation of the Hartree-Fock equations is possible through the Brillouin theorem which can be proved directly from the variation principle (Mayer 1971,1973,1974). We shall not prove here the complete equivalence of the Hartree-Fock equations and Eq. (11.1), it will be shown only that the Brillouin theorem is fulfilled for the Hartree-Fock wave function. The proof will make use of second quantization which helps us to evaluate the matrix element easily. To this goal, Eq. (11.1) should be rewritten in the second quantized notation. The ground state is simply represented the Fermi vacuum ... 

Spin contamination and noncollinearity in general complex Hartree-Fock wave functions... 

Abstract An expression for the square of the spin operator expectation valne, S, is obtained for a general complex Hartree-Fock wave fnnction and decomposed into four contributions the main one whose expression is formally identical to the restricted (open-shell) Hartree-Fock expression. A spin contamination one formally analogous to that found for spin nnrestricted Hartree-Fock wave functions. A noncollinearity contribntion related to the fact that the wave fnnction is not an eigenfunction of the spin- S operator. A perpendicularity contribution related to the fact that the spin density is not constrained to be zero in the xy-plane. All these contributions are evaluated and compared for the H2O+ system. The optimization of the collinearity axis is also considered. 

Most of the theory in the present chapter is concerned with perturbational corrections to states that are dominated by a single electronic configuration, usually represented by a Hartree-Fock wave function. However, in Section 14.7, we consider multiconfigurational generalizations of Mpller-Plesset perturbation theory, in particular the second-order perturbation theory developed within the framework of CASSCF theory. 

In Fock space, the exact solution to the SchrOdinger equation is an eigenfunction of the total and projected spins. Often, we would like the approximate solutions to exhibit the same spin symmetries. However, according to the discussion in Section 4.4, it does not follow that the Hartree-Fock wave function - which is not an eigenfunction of the exact Hamiltonian - possesses the same symmetries as the exact solution. In general, therefore, these and other symmetries of the exact state must be imposed on the Hartree-Fock solution. 

As the next standard model of quantum chemisUy, we consider here the generalization of the Hartree-Fock wave function to systems dominated by more than one electronic configuration the multiconfigurational self-consistent field (MCSCF) wave function. This flexible model may be useful for describing the electronic structure of bonded molecular systems, in particular for excited states. Perhaps more important, however, is its ability to describe bond breakings and molecular dissociation processes. 

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