Perturbation theory SchrOdinger equation differentiation

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. 

In configuration space, such expansions are not amenable to regular perturbation theory methods, since the Schrodinger equation suffers an abrupt (non-smooth) transition between e = 0 and e = 0+. More precisely, the order of the differential equations abruptly changes from zero to two. 

The treatment of Schrodinger s perturbation theory based on the use of a series of inhomogeneous differential equations of iterative character is briefly surveyed. As an illustration, the method is used to derive the general expression for the expectation value of the Hamiltonian to any order which provides an upper bound for the ground-state energy. It is indicated how the well-known theory for inhomogeneous equations may be utilized also in this special case. 

For the most realistic treatment of chemical problems, the Schrodinger equation does not tend to be separable and it is not usually a differential equation easily solved by analytical means. Or, more to the point, the problems of interest are not simple and are not strictly the same as ideal model problems. Powerful techniques have been formulated for dealing with realistic problems. Variation theory and perturbation theory are two such techniques that have been widely employed in understanding many quantum chemical systems. 

As in the developments of variation theory and perturbation theory, we make use of the fact that any valid wavefunction for a system can be formed as a linear combination of the eigenfunctions of a model Hamiltonian, in this case Hq. The task of solving the time-dependent differential Schrodinger equation is then converted to the task of finding the proper linear combination. The linear combination is made from the stationary states of the TDSE involving just Hq (Equation 9.6). Thus, for the Schrodinger equation. 

These values can also be found anal5dically, without approximation, by solving equations obtained by differentiating the Schrodinger equation or by perturbation theory. 

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