Analytical differentiation Schrodinger equation

To date, there is no known analytic solution to the second-order differential Schrodinger equation for the helium atom. This does not mean that there is no solution, or that wavefunctions do not exist. It simply means that we know of no mathematical function that satisfies the differential equation. In fact, for atoms and molecules that have more than one electron, the lack of separability leads directly to the fact that there are no known analytical solutions to any atom larger than hydrogen. Again, this does not mean that the wavefunctions do not exist. It simply means that we must use other methods to understand the behavior of the electrons in such systems. (It has been proven mathematically that there is no analytic solution to the so-called three-body problem, as the He atom can be described. Therefore, we must approach multielectron systems differently.)... 

The difference equation or numerical integration method for vibrational wavefunctions usually referred to as the Numerov-Cooley method [111] has been extended by Dykstra and Malik [116] to an open-ended method for the analytical differentiation of the vibrational Schrodinger equation of a diatomic. This is particularly important for high-order derivatives (i.e., hyperpolarizabilities) where numerical difficulties may limit the use of finite-field treatments. As in Numerov-Cooley, this is a procedure that invokes the Born-Oppenheimer approximation. The accuracy of the results are limited only by the quality of the electronic wavefunction s description of the stretching potential and of the electrical property functions and by the adequacy of the Born-Oppenheimer approximation. 

This differential equation does have an analytic solution. The method we use here is one general technique for solving differential equations We define the wave-function as a power series. What we will ultimately find is that in order to solve the Schrodinger equation, the power series must have a special form. 

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. 

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. 

The reason why the random flight model has proved so popular theoretically stems from its simplicity, which offers hope for the development of analytic solutions. The problem can usually be cast in the form of a diffusionlike or a Schrodinger-wave-equation-like differential equation, the solutions of which are reasonably well explored. A tendency has developed in recent times to apply extremely sophisticated mathematical procedures to what are really very primitive models for polymer chains (see, e.g. Levine et al., 1978). Whether the ends merit the means in such instances cannot yet be assessed objectively. A strategy that might be more productive in terms of the development of a practical theory for steric stabilization is to aim for a simpler mathematical description of more complex models of polymer chains. It should also be borne in mind in developing ab initio theories that a simple model that may well suffice in polymer solution thermodynamics may be quite inadequate for the simulation of the conformational properties of polymers. Polymer solution thermodynamics seem to be relatively insensitive to molecular architecture per se whereas the conformation of a polymer chain is extremely sensitive to it. 

Both the Newton/Einstein and Schrodinger/Dirac dynamical equations are differential equations involving the derivative of either the position vector or wave function with respect to time. For two-particle systems with simple interaction potentials V, these can be solved analytically, giving r(t) or F(r,i) in terms of mathematical functions. For systems with more than two particles, the differential equation must be solved by numerical techniques involving a sequence of small finite time steps. 

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