Trial wavefunctions optimization

Trial wavefunction Optimized parameter value(s) IE(calc) (eV) ... 

Alexander et al. [55] used VMC to compute cross sections for the elastic and inelastic scattering of fast electrons and X rays by H2. Novel trial wavefunction optimization schemes has been proposed by Huang and Cao... 

Umrigar C J, Wilson K G and Wilkins J W 1988 Optimized trial wavefunctions for quantum Monte Carlo calculations Phys. Rev. Lett. 60 1719-22... 

This condition is termed the variational principle. Thus, the trial wavefunction can be optimized using standard techniques43 until the system energy is minimized. At this point, the final solution can be regarded as Mf. for all practical purposes. It is clear, however, that the wavefunction that is obtained following this iterative procedure will depend on the assumptions employed in the optimization procedure. 

The most straightforward way to introduce the concept of optimal molecular orbitals is to consider a trial wavefunction of the form which was introduced earlier in Chapter 9.II. The expectation value of the Hamiltonian for a wavefunction of the multiconfigurational fomn... 

Optimization of linear and non-linear parameters in a trial wavefunction by the method of simulated annealing... 

Thus, the wavefunction giving the lowest eigenvalue E will be the best. Having defined a trial wavefunction (P with adjustable parameters, we want to optimize it by determining those values of the parameters that give the lowest expectation value for the energy. If we use a trial function that is a linear combination (LC) of an orthonormal 1 basis set, e.g. a set of orthonormal AOs , (LCAO) (Equation 1.15),... 

Variational intra-orbit optimization of trial wavefunctions... 

The initial results for helium listed in Table I were obtained using just one value of a for each type of transformation. This value was obtained by carrying out the above-described optimization procedure for the nine-term trial wavefunction. For transformation 1 an optimum value of a = 1.05 was obtained, whereas that found for 2 was a = 1.35. 

The excellent agreement obtained for hydrogen and helium between sum-over-points and variational eigenvalues using relatively small numbers of Diophantine points for trial wavefunctions for which the scale factor was optimized. 

Suppose we knew the exact wavefunction in that case the local energy would be a constant over all configuration space, and the error in its estimate would be rigorously zero. Similarly, if we could obtain very good wavefunctions the fluctuations of the local energy would be very small. (This is an amusing property of VMC the more accurate the trial wave-function, the easier the simulations. However, if accuracy of the wavefunction entails evaluation of massive expressions, such an approach makes life harder and is counterproductive.) The error thus provides a measure of the quality of the trial wavefunction. This leads naturally to the idea of optimizing the wavefunction to minimize the error of the estimate. 

The review is organized as follows. In the next section we introduce the three main methods VMC, DMC, and PIMC. In the following section we describe the forms and optimization of trial wavefunctions. Then we discuss the treatment of atomic cores. Next, we outline selected applications to atoms, molecules, clusters, and a few results for extended systems. We conclude with prospects for future progress. 

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. 

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