Complex trial functions

The numerical value of S is listed in Table 9.1. The simple variation function (9.88) gives an upper bound to the energy with a 1.9% error in comparison with the exact value. Thus, the variation theorem leads to a more accurate result than the perturbation treatment. Moreover, a more complex trial function with more parameters should be expected to give an even more accurate estimate. 

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. 

In some cases it is necessary to consider complex trial functions, for instance in the presence of a magnetic field [29] or in the twist average method to be discussed later [30]. In these cases we have to deal with a trial function of the form... 

With increased experience, it became clear that improved results and better precision required more complex trial functions. The Be atom illustrates this point a single determinant... 

The method of weighted residuals comprises several basic techniques, all of which have proved to be quite powerful and have been shown by Finlayson (1972, 1980) to be accurate numerical techniques frequently superior to finite difference schemes for the solution of complex differential equation systems. In the method of weighted residuals, the unknown exact solutions are expanded in a series of specified trial functions that are chosen to satisfy the boundary conditions, with unknown coefficients that are chosen to give the best solution to the differential equations ... 

Since the resulting system after radial collocation is still too complex for direct mathematical solution, the next step in the solution process is discretization of the two-dimensional system by orthogonal collocation in the axial direction. Although elimination of the spatial derivatives by axial collocation increases the number of equations,8 they become ordinary differential equations and are easily solved using traditional techniques. Since the position and number of points are the only factors affecting the solution obtained by collocation, any set of linearly independent polynomials may be used as trial functions. The Lagrangian polynomials of degree N based on the collocation points... 

For the solution of sophisticated mathematical models of adsorption cycles including complex multicomponent equilibrium and rate expressions, two numerical methods are popular. These are finite difference methods and orthogonal collocation. The former vary in the manner in which distance variables are discretized, ranging from simple backward difference stage models (akin to the plate theory of chromatography) to more involved schemes exhibiting little numerical dispersion. Collocation methods are often thought to be faster computationally, but oscillations in the polynomial trial function can be a problem. The choice of best method is often the preference of the user. 

In one view the choice of trial functions is one of the features which distinguishes the spectral methods (SMs) from the spectral element Methods (SEMs). The finite element methods (FEMs) can thus be regarded as SEMs with linear expansion- and weight functions. The trial functions for spectral methods are infinitely differentiable global functions. In the case of spectral element methods, the domain is divided into small elements, and the trail function is specified in each element. The trial and test functions are thus local in character, and well suited for handling complex geometries. 

Complex Configuration Interaction (CCI). A logical extension of the CSCF Idea Is complex Cl with a CSCF reference configuration. The trial function Is now... 

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. 

The method expressed In Equations 18-23 has seen only a few numerical applications, but It represents the ultimate application of the generalized complex variational principle in that only real valued eigenvalue calculations are required. Doubtless the use of this idea coupled with more cleverly chosen trial functions, such as those of Junker (19,20) and Chung and Davis (33) described above will also be possible. 

By requiring Schrodinger s equation and its solutions to remain in complex 3-space, we have done something interesting. The trial functions are actually the... 

The difficulty with VMC is exactly identical in spirit to all the problems of traditional methods the basis-set problem. Although the wavefunction is vastly improved in VMC, it is difficult to know when the wavefunction form is sufficiently flexible, and therefore it is always necessary to show that the basis-set limit of a given class of trial function has been reached. Moreover, the accuracy of energy in no way implies accuracy of other properties. One can assume that many of the variational errors cancel out in going from one system to another, but it is not very hard to find counterexamples. With the current class of wavefunctions it seems that we are far from getting chemical accuracies from VMC when applied to systems more complex than the electron gas or a single atom. In addition, in VMC one can waste a lot of time trying new forms rather than have the computer do the work. This problem is solved in a different way in the next two methods we discuss. 

Instead of having imaginary time evolution as in DMC, one keeps the entire path in memory and moves it around. PIMC uses a sophisticated Metropolis Monte Carlo method to move the paths. One trades off the complexity of the trial function for more complex ways to move the paths [23]. One gains in this trade-off because the former changes the answer while the latter changes only the computational cost. 

Methods involving ri -correlated trial functions have also been developed for three-electron bound systems (117). One of them, the superposition of correlated configuration method of Woznicki (118), has been recently combined with the complex rotation method and succesfully applied to He autoionizing resonances (22,119-121). 

The use of complex coordinates either in the Hamiltonian or in the trial functions leads to complex symmetric matrices, which are often large due to the necessity of employing large basis. Some effective algorithms cind computer codes have been recently developed (130-132) for diagonalizing such matrices. 

Starting from expression (2.7), it is impossible to find trial functions satisfying a priori the essential boundary conditions, since in our problem those conditions are functions of the electric field. Therefore, the only way is to search trial functions satisfying the field equation and possibly - if the geometry is not too complex - the natural boundary conditions. 

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