Trial functions optimization

It may seem that minimization of the energy should be a valid approach to trial function optimization. As it turns out, minimization of the energy using a fixed sample is problematic because it is easy for a few very low energies to dominate the sample. In this case, the optimization is not reliable and a good trial function is not obtained. This is not a concern in variance minimization because the variance is bounded from below by zero. However, even for variance minimization, care... 

The representation of trial fiinctions as linear combinations of fixed basis fiinctions is perhaps the most connnon approach used in variational calculations optimization of the coefficients is often said to be an application of tire linear variational principle. Altliough some very accurate work on small atoms (notably helium and lithium) has been based on complicated trial functions with several nonlinear parameters, attempts to extend tliese calculations to larger atoms and molecules quickly runs into fonnidable difficulties (not the least of which is how to choose the fomi of the trial fiinction). Basis set expansions like that given by equation (A1.1.113) are much simpler to design, and the procedures required to obtain the coefficients that minimize are all easily carried out by computers. 

In all the variational methods, the choice of trial function is the basic problem. Here we are concerned with the choice of the trial function for the polarization orbitals in the calculation of polarizabilities or hyperpolarizabilities. Basis sets are usually energy optimized but recently we can find in literature a growing interest in the research of adequate polarization functions (27). 

The orthogonal collocation method has several important differences from other reduction procedures. Jn collocation, it is only necessary to evaluate the residual at the collocation points. The orthogonal collocation scheme developed by Villadsen and Stewart (1967) for boundary value problems has the further advantage that the collocation points are picked optimally and automatically so that the error decreases quickly as the number of terms increases. The trial functions are taken as a series of orthogonal polynomials which satisfy the boundary conditions and the roots of the polynomials are taken as the collocation points. A major simplification that arises with this method is that the solution can be derived in terms of its value at the collocation points instead of in terms of the coefficients in the trial functions and that at these points the solution is exact. 

As improved trial functions were developed, simple optimization methods for obtaining parameter values were no longer adequate. The most common application is minimization of local energy fluctuations about a reference energy chosen as, or close to, an estimate of the nonrelativistic BO energy. The motivation is that the local energy approaches the exact energy as the trial function approaches the exact function. 

There are several important implications of this result the TD functional theory and the Baym functional theory are versions of the many-body theory in that they invoke different trial functionals in obtaining the stationary solution of the same variational functional, namely, the action functional. The two theories explore the optimization of the action functional in terms of different functional variables the TD functional employs density, current, ion-coordinates, etc. while the -derivable theory employs the two-point functions such as the Green functions associated with electrons, ions, etc. The above demonstration shows that different optimization strategies may be employed to explore the stationar-ity of the action functional. 

Computer calculations of molecular electronic structure use the orbital approximation in exactly the same way. Approximate MOs are initially generated by starting with trial functions selected by symmetry and chemical intuition. The electronic wave function for the molecule is written in terms of trial functions, and then optimized through self-consistent field (SCF) calculations to produce the best values of the adjustable parameters in the trial functions. With these best values, the trial functions then become the optimized MOs and are ready for use in subsequent applications. Throughout this chapter, we provide glimpses of how the SCF calculations are carried out and how the optimized results are interpreted and applied. 

The methodologies described in the previous section can be invoked to handle the problem of optimization of linear and non-linear parameters in trial J irrespective of the details of the trial function, if the... 

The optimization of trial functions for many-body systems is time consuming, particularly for complex trial functions. The dimension of the parameter space increases rapidly with the complexity of the system and the optimization can become very cumbersome since it is, in general, a nonlinear optimization problem. Here we are not speaking of the computer time, but of the human time to decide which terms to add, to program them and their derivatives in the VMC code. This allows an element of human bias into VMC the VMC optimization is more likely to be stopped when the expected result is obtained. The basis set problem is still plaguing quantum chemistry even at the SCF level where one only has 1-body orbitals. VMC shares this difficulty with basis sets as the problems get more complex. 

We discuss now the choice of the spin orbitals. The spin-orbitals are conceptually more important than the pseudopotential because they provide the nodal structure of the trial function. With the fixed node approximation in RQMC, the projected ground state has the same nodal surfaces of the trial function, while the other details of the trial function are automatically optimized for increasing projection time. It is thus important that the nodes provided by given spin-orbitals be accurate. Moreover, the optimization of nodal parameters (see below) is, in general, more difficult and unstable than for the pseudopotential parameters [6]. 

The unknown functions, Cij r),riij r) in (57) and (58) need to be parameterized in some way. In a first attempt we have chosen gaussians with variance and amplitude as new variational parameters [16]. This form was shown to be suitable for homogeneous electron gas [13]. Approximate analytical forms for ij r) and r]ij r), as well as for the two-body pseudopotential, have been obtained later in the framework of the Bohm-Pines collective coordinates approach [14]. This form is particularly suitable for the CEIMC because there are no parameters to be optimized. This trial function is faster than the pair product trial function with the LDA orbitals, has no problems when protons move around and its nodal structure has the same quality as the corresponding one for the LDA Slater determinant [14]. We have extensively used this form of the trial wave function for CEIMC calculations of metallic atomic hydrogen. 

For metallic hydrogen we have described a parameter-free trial function which does not need optimization. However, if we use the pair proton action both for molecular or LDA orbitals, we are left with free parameters in the Jastrow factor and with the width of the gaussians for molecular orbitals. Optimization of the parameters in a trial function is crucial for the success of VMC. Bad upper bounds do not give much physical information. Good trial functions will be needed in the Projector Monte Carlo method. First, we must decide on what to optimize and then how to perform the optimization. There are several possibilities for the quantity to optimize and depending on the physical system, one or other of the criteria may be best. 

The overlap with the exact wave function f If we maximize the overlap, we find the trial function closest to the exact wave function in the least squares sense. This is the preferred quantity to optimize if you want to calculate correlation functions, not just ground state energies since, then, the VMC correlation functions will be closest to the true correlation functions. Optimization of the overlap will involve a Projector Monte Carlo calculation to determine the change of the overlap with respect to the trial function so it is more complicated and rarely used. 

Calculations by this method have shown remarkable Insensitivity to the nonlinear parameters of the complex part of the trial function. For the 2s autolonlzlng state, for which Equation 14 Is the trial function, Chung and Davis (33) obtained Ej. 57.8483 eV and r 0.12468 eV which compare well with the experimental (34) values of Ej. 57.8210.04 eV and T 0.13810.15 eV. The saddle point complex-rotation method Is strictly speaking another variant of CCI, but It demonstrates the premium In accuracy and efficiency to be gained from a well chosen trial function. In this case one In which the Feshbach Q-space (resonance) part of the trial wave-function Is optimized. 

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),... 

A concrete example of the variational principle is provided by the Hartree-Fock approximation. This method asserts that the electrons can be treated independently, and that the -electron wavefimction of the atom or molecule can be written as a Slater determinant made up of orbitals. These orbitals are defined to be those which minimize the expectation value of the energy. Since the general mathematical form of these orbitals is not known (especially in molecules), then the resulting problem is highly nonlinear and formidably difficult to solve. However, as mentioned in subsection (A 1.1.3.2). a common approach is to assume that the orbitals can be written as linear combinations of one-electron basis functions. If the basis functions are fixed, then the optimization problem reduces to that of finding the best set of coefficients for each orbital. This tremendous simplification provided a revolutionary advance for the application of the Hartree-Fock method to molecules, and was originally proposed by Roothaan in 1951. A similar form of the trial function occurs when it is assumed that the exact (as opposed to Hartree-Fock) wavefimction can be written as a linear combination of Slater determinants (see equation (A 1.1.104 ) ). In the conceptually simpler latter case, the objective is to minimize an expression of the form... 

The basic Monte Carlo methods reviewed in this chapter have been used in many different contexts and under many different names for many decades, but we emphasize the solution of eigenvalue problems by means of Monte Carlo methods and present the methods from a unified point of view. A vital ingredient in the methods discussed here is the use of optimized trial functions. Section IV deals with this topic briefly, but in general we suppose that optimized trial functions are given. We refer the reader to Ref. 3 for more details on their construction. 

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