Wavefunctions variation theory

A completely difierent approach to scattering involves writing down an expression that can be used to obtain S directly from the wavefunction, and which is stationary with respect to small errors in die waveftmction. In this case one can obtain the scattering matrix element by variational theory. A recent review of this topic has been given by Miller [32]. There are many different expressions that give S as a ftmctional of the wavefunction and, therefore, there are many different variational theories. This section describes the Kohn variational theory, which has proven particularly useftil in many applications in chemical reaction dynamics. To keep the derivation as simple as possible, we restrict our consideration to potentials of die type plotted in figure A3.11.1(c) where the waveftmcfton vanishes in the limit of v -oo, and where the Smatrix is a scalar property so we can drop the matrix notation. 

Another approach for developing approximations to CC and CS reactive scattering calculations is to use distorted wave theory. In this approach, one considers that reaction is only a small perturbation on the nonreactive collision dynamics. As a result, the reactive scattering matrix can be approximated by the matrix element of a perturbative Hamiltonian operator using reagent and product nonreactive wavefunctions. Variations on this idea can be developed by using different approximations to the nonreactive wavefunctions. At the top of the hierachy of these methods is the coupled channel distorted wave (CCDW) method, followed by coupled states distorted wave (CSDW). 

The second major approximation theory used in quantum mechanics is called variation theory. Variation theory is based on the fact that any test wavefunction for a system has an average energy that is equal to or greater than the true ground-state... 

One way of stating the basic idea behind variation theory is the following For a system having a Hamiltonian operator H, true wavefunctions I true> nd some lowest-energy eigenvalue E, the variation theorem states that for any normalized trial wavefunction [Pg.409]

This minimum energy is the best energy that this trial wavefunction can provide. When there are multiple variables in the trial wavefunction, then the absolute minimum with respect to all variables simultaneously is the lowest energy that such a trial wavefunction produces. Although variation theory does provide more complicated expressions for the energies of excited states, the above relatively simple expressions apply only to the ground state of a system. 

FIGURE 12.8 Trial wavefunctions for a variation-theory treatment of the ground state of the particle-in-a-box. The solid line is the trial parabolic wave-function, and the dotted line is the true wavefunction. 

The trial wavefunctions can have any number of variable parameters, limited mostly by the efficiency in determining the energy minimum. Variation theory is best illustrated by example. We will start by using a trial wavefunction without parameters to show that equation 12.23 is satisfied. For the particle-in-a-box of length a, assume that instead of a sine function, the ground-state wavefunction is instead an upside-down parabola at the center of the box, a/2 ... 

This form of variation theory is also best illustrated by example. Although the same idea can be applied to a trial wavefunction having any number of terms, a simple example involves the use of a two-term linear combination for the trial wavefunction ... 

Researchers in quantum mechanical calculations should understand the limitations and strengths of each method. Typically, the method used is the one that provides the information a particular researcher wants about a real system. If a well-defined Hamiltonian and wavefunction are desired, then perturbation theory provides that. If the absolute energy is important, variation theory provides a way to get better and better results. The calculational cost is also a factor. Those with access to supercomputers can work with a lot of equations in a relatively short time. Those without may find themselves limited to a small number of corrections to ideal wavefunctions. 

As might be expected, no known analytic wavefimctions are eigenfunctions of the Hamiltonian operator in equation 12.35. Some simplifications are needed in order to determine approximate solutions using perturbation or variation theory. One of the complications of this system is that there are now two nuclei, and a proper wavefunction should take into account not only the behavior of the electron but also the behavior of the nuclei. It should be clear that if the relative positions of the nuclei change (for example, during a vibration in which the nuclei are moving alternately closer and farther apart) then the electronic motion will also change to compensate. Any true wavefunction for electrons needs to consider nuclear behavior as well. 

Just as in linear variation theory, the coefficients can be determined using a secular determinant. But unlike the earlier examples using secular determinants, in this case some of the integrals are not identically zero or 1 due to orthonormality. In cases where there is an integral in terms of h(i) h(2) or vice versa, we cannot assume that the integral is identically zero. This is because the wavefunctions are centered on different atoms. The orthonormality conditions to this point are only strictly... 

The Rayleigh-Ritz variational theory is the basis for so-called variational methods in which an estimate of the energy of a system is calculated for an approximate trial wavefunction usually assembled from combinations of atomic orbitals. Expectation values of the energy may be calculated accurately for many trial wavefunctions and are upper bounds to the true energy. If the parameters of the trial wavefunctions are varied systematically, the lowest upper bound to the energy for a particular form of trial wavefunction may be determined (thus the term variational ). The trial functions must satisfy certain restrictions such... 

Problem Use the functional form of the ground state wavefunction of the harmonic oscillator as a trial wavefunction and apply variation theory to find an approximate wavefunction and energy for the quartic oscillator, an oscillator with V x) = kx. ... 

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. 

A special and powerful use of variation theory is with linear variational parameters. That means that the trial wavefunction is taken to be a linear combination of fimctions in some chosen set. The adjustable parameters are the expansion coefficients of each of these functions. This is, of course, a specialization of the way in which variation theory can be used, but it is powerful because the resulting equations take the form of matrix expressions. Solving the Schrodinger equation becomes a problem in linear algebra, and such problems are ideally suited to computer solution. 

Variation theory states that the energy expectation value e is greater than or equal to the tme ground-state energy, Eo, of the system. The equality occurs only when the trial wavefunction is the tme ground-state wavefunction of the system. 

Variation theory can be proven as follows. Take the trial wavefunction, H/triai, as a linear combination of the tme eigenfunctions of the Hamiltonian,... 

Variation theory states that the energy calculated from any trial wavefunction will never be less than the true ground-state energy of the system. This means that the smaller the value of e, the closer it is to the tme ground-state energy of the system and the more xp sai represents the true ground-state wavefunction. The trial wavefunction is set up with one or more adjustable parameters, pi, making the function flexible to minimize the value of e. An n number of adjustable parameters will set up an n number of differential equations ... 

Using the following trial wavefunction, determine the ground-state energy of a hydrogen atom using variational theory. 

To properly describe electronic rearrangement and its dependence on both nuclear positions and velocities, it is necessary to develop a time-dependent theory of the electronic dynamics in molecular systems. A very useful approximation in this regard is the time-dependent Hartree-Fock approximation (34). Its combination with the eikonal treatment has been called the Eik/TDHF approximation, and has been implemented for ion-atom collisions.(21, 35-37) Approximations can be systematically developed from time-dependent variational principles.(38-41) These can be stated for wavefunctions and lead to differential equations for time-dependent parameters present in trial wavefunctions. 

HF (HF) theory is based on the idea that one takes an antisymmetrized product wavefunction and uses the variational principle to obtain the best possible approximation to the A -particle wavefunction that cannot be represented by such a single determinant. Thus, one inserts the single determinant into the Rayleigh-Ritz functional and performs a constraint variation of the orbitals. The results of the variational process are the famous HF equations that are satisfied by each of the orbitals ... 

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