Slater determinants, trial wavefunctions

Much of the development of the previous chapter pertains to the use of a single Slater determinant trial wavefunction. As presented, it relates to what has been called the unrestricted Hartree-Fock (UHF) theory in which each spin-orbital (ftj has its own orbital energy 8i and LCAO-MO coefficients Cv,i there may be different Cv,i for a spin-orbitals than for (3 spin-orbitals. Such a wavefunction suffers from the spin contamination difficulty detailed earlier. 

Given the trial wavefunction - the Slater determinant eq. (11.37) - we then use the variational principle to minimize the energy - the expectation value of the Hamiltonian H - with respect to the orbital coefficients cy (eq. (11.39)). This leads after a fair amount of algebra to the self-consistent Hartree-Fock equations ... 

Let us now indicate how local-scaling transformations can be used in order to carry out the constrained minimization of the kinetic energy functional [85-88]. The strategy that we have adopted is first to select a Slater determinant such as the one appearing in Eq. (73), as the trial wavefunction which generates the particular orbit d c Sn- For the case of atoms, the one-particle orbitals ( >g,i r) from which this Slater determinant is constructed are explicitly given by g,nim(r) = Rg,ni r)Yiimi(0,), where the subindex i has been replaced by the quantum numbers n, /, m. The radial functions are expanded as... 

Undoubtedly, the methods most widely used to solve the Schrodinger equation are those based on the approach originally proposed by Hartree [1] and Fock [2]. Hartree-Fock (HF) theory is the simplest of the ab initio or "first principles" quantum chemical theories, which are obtained directly from the Schrodinger equation without incorporating any empirical considerations. In the HF approximation, the n-electron wavefunction is built from a set of n independent one-electron spin orbitals which contain both spatial and spin components. The HF trial wavefunction is taken as a single Slater determinant of spin orbitals. 

M-electron wavefunction can be expanded as a linear combination of an infinite set of Slater determinants that span the Hilbert space of electrons. These can be any complete set of M-electron antisymmetric functions. One such choice is obtained from the Hartree-Fock method by substituting all excited states for each MO in the determinant. This, of course, requires an infinite number of determinants, derived from an infinite AO basis set, possibly including continuum functions. As in Hartree-Fock, there are no many-body terms explicitly included in Cl expansions either. This failure results in an extremely slow convergence of Cl expansions [9]. Nevertheless, Cl is widely used, and has sparked numerous related schemes that may be used, in principle, to construct trial wavefunctions. 

A typical VMC computation to estimate the energy or other expectation values for a given 4/x(R) might involve the calculation of the wavefunction value, gradient, and Laplacian at several millions points distributed in configuration space. Computationally this is the most expensive part. So a desirable feature of TVR), from the point of view of Monte Carlo, is its compactness. It would be highly impractical to use a trial wavefunction represented, for example, as a Cl expansion of thousands (or more) of Slater determinants. 

One of the main advantages of the Monte Carlo method of integration is that one can use any computable trial function, including those going beyond the traditional sum of one-body orbital products (i.e., linear combination of Slater determinants). Even the exponential ansatz of the coupled cluster (CC) method [27, 28], which includes an infinite number of terms, is not very efficient because its convergence in the basis set remains very slow. In this section we review recent progress in construction and optimization of the trial wavefunctions. 

The wavefunction for an SCF calculation is one or more antisymmetrized products of one-electron spin-orbitals. We have already seen (Chapter 5) that a convenient way to produce an antisymmetrized product is to use a Slater determinant. Therefore, we take the trial function to be made up of Slater determinants containing spin-orbitals (j>. If we are dealing with an atom, then the (f> s, are atomic spin-orbitals. For a molecule, they are molecular spin-orbitals. 

The typical trial wavefunction for QMC calculations on molecular systems consists of the product of a Slater determinant multiplied by a second function, which accounts to some extent for electron correlation with use of interelectron distances. The trial wavefunctions are most often taken from relatively simple analytic variational calculations, in most cases from calculations at the SCF level. Thus, for the 10-electron system methane," the trial function may be the product of the SCF function, which is a 10 x 10 determinant made up of two 5x5 determinants, and a Jastrow function for each pair of electrons. 

There is, however, one particularly simple and commonly usefvil case that is appropriate to consider here. That is the case where the trial wavefunction is simply a single Slater determinant of doubly occupied orbitals and it is a commonly useful form because many molecules have an even number of electrons and their electronic ground states are, in many cases, totally symmetric singlet states. The expected value of the Hamiltonian for such a trial function is... 

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