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Eigenvalue equations approximation

In applying quantum mechanics to real chemical problems, one is usually faced with a Schrodinger differential equation for which, to date, no one has found an analytical solution. This is equally true for electronic and nuclear-motion problems. It has therefore proven essential to develop and efficiently implement mathematical methods which can provide approximate solutions to such eigenvalue equations. Two methods are widely used in this context- the variational method and perturbation theory. These tools, whose use permeates virtually all areas of theoretical chemistry, are briefly outlined here, and the details of perturbation theory are amplified in Appendix D. [Pg.57]

You will see shortly that an exact solution of the electronic Schrodinger equation is impossible, because of the electron-electron repulsion term g(ri, r2). What we have to do is investigate approximate solutions based on chemical intuition, and then refine these models, typically using the variation principle, until we attain the required accuracy. This means in particular that any approximate solution will not satisfy the electronic Schrodinger equation, and we will not be able to calculate the energy from an eigenvalue equation. First of all, let s see why the problem is so difficult. [Pg.87]

The matrix Bis numerically much less important than A and accordingly y << x. This suggests a simplification of the foregoing equations by setting B = 0. This Tamm-Dancoff approximation [11] leads to a symmetric matrix eigenvalue equation... [Pg.20]

Using the extended Hiickel approximation, we obtain the corresponding eigenvalues E (k) and coefficients C (k) from the eigenvalue equation " ... [Pg.602]

Work is currently in progress to determine other ways of enhancing the speed of eigenvalue determination. One method that shows promise is the diffusion equation approximation. The basis of this method is that it can be shown that under certain circumstances the integral operator on the right hand side of equation (2.30) can be replaced by a differential operator that is similar in form (and solution) to a diffusion equation. Such equations can sometimes yield analytic results and even when this is not the case they are much more amenable to numerical solution often with considerable savings in CPU time. [Pg.169]

As the dimension of the blocks of the Hessian matrix increases, it becomes more efficient to solve for the wavefunction corrections using iterative methods instead of direct methods. The most useful of these methods require a series of matrix-vector products. Since a square matrix-vector product may be computed in 2N arithmetic operations (where N is the matrix dimension), an iterative solution that requires only a few of these products is more efficient than a direct solution (which requires approximately floating-point operations). The most stable of these methods expand the solution vector in a subspace of trial vectors. During each iteration of this procedure, the dimension of this subspace is increased until some measure of the error indicates that sufficient accuracy has been achieved. Such iterative methods for both linear equations and matrix eigenvalue equations have been discussed in the literature . [Pg.185]

The Contracted Schrodinger Equation is studied here in a spin-orbital representation coupled with the S2 eigenvalue equation as an auxiliary condition. A set of new algorithms for approximating RDM s in terms of the lower order ones are reported here. These new features improve significantly the method. [Pg.33]

To progress further toward practical implementation, specific choices must be made for how one is going to approximate the neutral molecule wave function lO, N) and at what level one is going to truncate the expansion of the operator Q K) given in Eq. (5). It is also conventional to reduce Eq. (7) to a matrix eigenvalue equation by projecting this equation onto an appropriately chosen space of A + 1-electron functions. Let us first deal with the latter issue. [Pg.447]

The work Wnfr) is retained in the equation to ensure there is no self-interaction). In contrast to the Kohn-Sham equation, this differential equation can in practice be solved because the dependence of the Fermi hole p, (r, r ), and thus of the work W (r), on the orbitals is known. Furthermore, since the solution of this equation leads to the exact asymptotic structure of vj (r), and the fact that Coulomb correlation effects are generally small for finite systems, the highest occupied eigenvalue should approximate well the exact (nonrelativistic) removal energy. This conclusion too is borne out by results given in Sect. 5.2.2. [Pg.194]

Other kinds of approximants can also be used. For example, Olsen et al. [20] analyzed MP series convergence using a 2 X 2 matrix eigenvalue equation [38,39], which implicitly incorporates a square-root branch point. It is of course possible simply to explicitly construct an approximant as an arbitrary function with the singularity structure that E z) is expected to have. We suggest, for example, approximants of the form ... [Pg.200]

The random phase approximation (RPA) was first introduced into many-body theory by Pines and Bohm.This approximation was shown to be equivalent to the TDHF for the linear opticcd response of many-electron systems by Lindhard. ° (See, for example, Chapter 8.5 in ref 83. The electronic modes are identical to the transition densities of the RPA eigenvalue equation.) The textbook of D. J. Thouless contains a good overview of Hailree—Fock and TDHF theory. [Pg.4]

Here, the e, value is the orbital s energy eigenvalue. Equation (6.9) is remarkably similar to the original Schrodinger equation. Equation (6.2), but the wave functions have been replaced with the KS orbitals and the exchange and correlation terms have been isolated. Thus, we have replaced the iV-body coupled electronic wave function with a collection of uncorrelated wave fimctions while at the same time defining precisely what the uncertain many-body terms in need of approximation are. [Pg.164]


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