Trial wavefunctions energy optimization

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. 

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),... 

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 lower value for E is the lowest energy attainable with the trial basis set. The corresponding coefficients give the character of the most bonding orbital. It should be noted that a secular determinant must be solved whenever a linear combination of wavefunctions is optimized. 

The problem of node locations—the sign problem in quanmm Monte Carlo —remains one of the major obstacles to obtaining exact solutions for systems of more than a few electrons. In analytic variational calculations and in VQMC, the locations of the nodal smfaces of a trial wavefunction may be and usually are optimized along with the rest of the wavefunction in the attempt to reach a minimum in the expectation value of the energy. In DQMC and GF-QMC, the node locations are not so easily varied. 

The first step is to write the Hamiltonian for the problem. Then an educated guess is made at a reasonable wavefunction called formally the trial wavefunction, v /triai. The trial wavefunction will have one or more adjustable parameters, pi, that will be used for optimization. An energy expectation value in terms of the adjustable parameters, s, is obtained by using the same form as in Equation 2-23. 

Since pL = 1.1331 results in a lower value for the energy of the ground-state this value is adopted. This optimizes the trial wavefunction to ... 

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