Nonvariational methods

The Nonvariational Method (Method I) An initial possibility is to keep the same orbitals that optimize the adiabatic state for the diabatic state something that seems simple and appealing. In practice, this would be done as follows ... 

CPMET is obviously a nonvariational method. However, the advantage of having an upper bound to the energy is probably not so important when the method is accurate enough to give the correlation energy with an accuracy of a few percent. 

Recently Lindroth (30) has introduced the concept of complex coordinates into nonvariational methods. The idea of Lindroth is to produce a basis set of one-particle complex rotated functions as solutions of the one-particle complex-rotated problem, and then to apply this basis within a bound-state method for a many-particle system. In this way the complex rotation has been combined with the many-body perturbation theory (30) and with the coupled cluster method (31). Apart from the fact that the resulting energy is complex and its real and imaginary part can be interpreted as the position and width of a resonance under consideration, the use of complex coordinates has the advantage that singularities caused by the degeneracy on the real axis between the resonance and the adjacent continuum are not present in the complex energy plane. 

A nonvariational method does not behave according to the variational principle, such that a lowered energy is not necessarily connected with a better wave function, and vice versa. 

Care must be taken in using the expressions above for obtaining nonlinear optical properties, because the values obtained may not be the same as those obtained from Eq. [4]. The results will be equivalent only if the Hellmann-Feyn-man theorem is satisfied. For the case of the exact wavefunction or any fully variational approximation, the Hellmann-Feynman theorem equates derivatives of the energy to expectation values of derivatives of the Hamiltonian for a given parameter. If we consider the parameter to be the external electric field, F, then this gives dE/dP = dH/d ) = (p,). For nonvariational methods, such as perturbation theory or coupled cluster methods, additional terms must be considered. 

ABSTRACT. We explore the factors responsible for the rapid convergence of the Schwinger and Newton variational principles in scattering theory. We find that, contrary to conventional wisdom, these variational methods yield high accuracy not because the error associated with the computed quantity is second oide in the error in the wavefunction, but because variational methods find wavefiinctions that are far more accurate in relevent regions of the potential, compared to nonvariational methods. 

Variational methods are at present used extensively in the study of inelastic and reactive scattering involving atoms and diatomic molecules[l-5]. Three of the most commonly used variational methods are due to Kohn (the KVP)[6], Schwinger (the SVP)[7] and Newton (the NVP)[8]. In the applications of these methods, the wavefunction is typically expanded in a set of basis functions, parametrized by the expansion coefficients. These linear variational parameters are then determined so as to render the variational functional stationary. Unlike the variational methods in bound state calculations, the variational principles of scattering theory do not provide an upper or lower bound to the quantity of interest, except in certain special cases.[9] Neverthless, variational methods are useful because, the minimum basis size with which an acceptable level of accuracy can be achieved using a variational method is often much smaller than those required if nonvariational methods are used. The reason for this is generally explained by showing (as... 

A large number of nonvariational methods for scattering calculations are known. Of these, we examine the method of moments for the amplitude density (MMAD), as an example of a nonvariational method. This method is attractive because of its simplicity. It also represents a large class of nonvariational methods that can be derived from it, or formulated as extensions of it.[lO] Moreover, the studies of Staszweska and Truhlar indicate that the method of moments for the amplitude density (the MMAD) is capable of generating a more accurate wavefunction than the analogous method for the wavefunction.[lO]... 

The great populariQr of variational methods in scattering theory has always been attributed to the fact that they achieve a higher level of accuracy for a given basis size N. Their superiority over nonvariational methods have been, as al dy mention explained by the fact that e variational functionals for quantities such as K, S or T matrix elements are... 

Moreover, the explanation above does not prohibit the same analysis being extended to variationally correct, but nonstadonary methods, in which a wavefunction fi om a nonvariational method is inserted into a variational functional. This renders die functional nonstadonaiy, but the error in the quantity computed by the functional is still, in principle, a smallo quantity. Howevo, from Table I, we see that in practice, this is not always the... 

This suggests, it seems to us, that the traditional explanation for the superiority of variational methods must be incorrect. From the previous two subsections, we see that the main difference between the two variational methods, the SVP and the NVP, and the nonvariational method MMAD, is that for a given basis size, the variational methods find a far more accurate wavefunction than the nonvariational method. The variational methods also differ fixim the nonvariational ones in that the former are characterized by a stationary principle. Let us now examine the consequences of this aspect of variational methods, taking the SVP functional as an example. 

We have examined the nature of the error in the wavefunctions for potential scattering in three one dimensional potentials, computed by one nonvariational method, die MMAD, and two commonly used variational methods, the SVP and the NVP. We have found that the nature of the error is dramatically different in each case. All three methods generate wavefunctions that are more accurate in the internal region than in the asymptotic region. However, the MMAD wavefunctions have errors in the internal region that tend to disappear rather slowly, whereas the variational methods banish the error to the asymptotic regions extremely rapicUy, as the basis size is increased. 

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