Trial variation function method

Because of the Pauli principle antisymmetry requirement, the ground-state wave function has nodal surfaces in 3n-dimensional space, and to ensure that the walkers converge to the ground-state wave function, one must know the locations of these nodes and must eliminate any walker that crosses a nodal surface in the simulation. In the fixed-node (FN) DQMC method, the nodes are fixed at the locations of the nodes in a known approximate wave function for the system, such as found firom a large basis-set Hartree-Fock calculation. This approximation introduces some error, but FN-DQMC calculations are variational. (In practice, the accuracy of FN-DQMC calculations is improved by a procedure called importance sampling. Here, instead of simulating the evolution of with t, one simulates the evolution off, where / = where is a known accurate trial variation function for the ground state.)... 

This is perhaps the easiest method to understand. It is based on the variational principle (Appendix B), analogous to the HF method. The trial wave function is written as a linear combination of determinants with the expansion coefficients determined by requiring that the energy should be a minimum (or at least stationary), a procedure known as Configuration Interaction (Cl). The MOs used for building the excited Slater determinants are taken from a Hartree-Fock calculation and held fixed. Subscripts S, D, T etc. indicate determinants which are singly, doubly, triply etc. excited relative to the... 

Variational approximation methods are identified by the form of the variational trial function, particularly by the number and types of Slater determinants. 

Several other calculations of the first few partial-wave phase shifts for positron-helium scattering have been carried out using a variety of approximation methods in all cases, however, rather simple uncorrelated helium wave functions have been used. Drachman (1966a, 1968) and McEachran et al. (1977) used the polarized-orbital method, whereas Ho and Fraser (1976) used a formulation based on the static approximation, with the addition of several short-range correlation terms, to determine the s-wave phase shifts only. The only other elaborate variational calculations of the s-wave phase shift were made by Houston and Drachman (1971), who employed the Harris method with a trial wave function similar to that used by Humberston (1973, 1974), see equation (3.77), and with the same helium model HI. Their results were slightly less positive than Humberston s HI values, and are therefore probably less... 

Among the classes of the trial wave functions, those employing the form of the linear combination of the functions taken from some predefined basis set lead to the most powerful technique known as the linear variational method. It is constructed as follows. First a set of M normalized functions dy, each satisfying the boundary conditions of the problem, is selected. The functions dy are called the basis functions of the problem. They must be chosen to be linearly independent. However we do not assume that the set of fdy is complete so that any T can be exactly represented as an expansion over it (in contrast with exact expansion eq. (1.36)) neither is it assumed that the functions of the basis set are orthogonal. A priori they do not have any relation to the Hamiltonian under study - only boundary conditions must be fulfilled. Then the trial wave function (D is taken as a linear combination of the basis functions dyy... 

The original Heitler-London calculation, being for two electrons, did not require any complicated spin and antisymmetrization considerations. It merely used the familiar rules that the spatial part of two-electron wave functions are symmetric in their coordinates for singlet states and antisymmetric for triplet states. Within a short time, however, Slater[10] had invented his determinantal method, and two approaches arose to deal with the twin problems of antisymmetrization and spin state generation. When one is constructing trial wave functions for variational calculations the question arises as to which of the two requirements is to be applied first, antisymmetrization or spin eigenfunction. 

In order to implement the variational method, one needs to introduce an orthonormal basis set to represent the trial wave functions in the inner part of the potential. 

Fig. 17 Cluster size dependence of the overlap integral between the ground state wave function GS) obtained by the Lanczos method and the trial wave function [6]. In the variational wave function the...

Gorecki and Byers Brown [14,15] proposed an approach based on the variational boundary perturbation theory. In this method the trial wave function for the confined system / is constructed as the product of the wave function for the free (unbounded) system /, times a non-singular cut off function /, to ensure fulfillment of the boundary condition /(ro, cp, 0) = 0. The cut-off function clearly vanishes at ro, /(ro) = 0... 

Simulation of confinement by penetrable boxes represents a more realistic physical model. A very simple approach was proposed by Marin and Cruz [18], where they used the Rayleigh-Ritz variational method via a trial wave function for the ground state, which consists of two piecewise functions, one for the inner region (r < ro), and the other for the outer region (r > ro). The trial wave function is defined as follows ... 

The variational method represents an application of the variational principle. The trial wave function 4) is taken in an analytical form (with the variables denoted by the vector jc and automatically satisfying Postulate V). In the key positions in the formula for , we introduce the parameters c = (cq, ci, C2,..., cp), which we may change smoothly. The parameters play the role of tuning, and their particular values listed in vector c result in a certain shape of (jc c). The integration in flie formula for e pertains to the variables jc therefore, the result depends uniquely on c. Our function e(c) has the form... 

To solve the corresponding Eq. (6.8), we have at our disposal the variational and the perturbation methods. The latter should have a reasonable starting point (i.e., an unperturbed system). This is not the case in the problem that we want to consider at the moment. Thus, only the variational method remains. If so, a class of the trial functions should be proposed. In this chapter, the trial wave function will have a very specific form, bearing significant importance for the theory. We mean here the so-called Slater determinant, which is composed of molecular orbitals. At a certain level of approximation, each molecular orbital is a parking place for two electrons. We wUl now learn on how to get the optimum molecular orbitals (using the Hartree-Fock method). Despite some quite complex formulas, which will appear below, the main idea behind them is extremely simple. It can be expressed in the following way. 

Configuration Interaction A variational method with the trial wave function in the form of a linear combination of the given set of the Slater determinants. 

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