Approximate Wavefunctions

Approximate Wavefunctions.—We now turn to the problem of constructing an acceptable wavefunction from an arbitrary spatial function (n, r%. .i v) which we might select according to some model. This is achieved by forming the following functions  

That the functions f sM-k are in fact of the form of equation (7) can be easily demonstrated by substituting equation (19) into equation (18) and making use of equation (17). One obtains the following alternative form for the approximate wavefunction [equation (18)]  

Except for a trivial normalization factor, these are just the usual group 

The most general wavefunction that can be formed from a given approximate spatial function d is a linear combination of the functions WgM.t  

Substituting expression (18) into these equations, noting that H and si commute and that, p/2=j /, and using equations (17) and (13), one arrives at the expressions 

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Notice that the Heilman-Feymnan theorem only applies to exact wavefunc-jB 8, not to variational approximations. All the enthusiasm of the 1960s and jWOs evaporated when it was realized that approximate wavefunctions them-Mves also depend on nuclear coordinates, since the basis functions are usually... 

General expressions for the force constants and dipole derivatives of molecules are derived, and the problems arising from their practical application are reviewed. Great emphasis is placed on the use of the Hartree-Fock function as an approximate wavefunction, and a number... 

Figure 10.1. Exaggerated reaction coordinate diagram showing approximate wavefunctions for the zero-point and first vibrational levels of the reactant.

Semi-empirical quantum-mechanical methods combine fundamental theoretical treatments of electronic behavior with parameters obtained from experiment to obtain approximate wavefunctions for molecules composed of hundreds of atoms20-22. Originally developed in response to the need to evaluate the electronic properties of organic molecules, especially those possessing unusual structures and/or chemical reactivity in organic chemistry,... 

The energy of an approximate wavefunction P is given by the expectation value of the Hamiltonian H ... 

An important characteristic of ab initio computational methodology is the ability to approach the exact description - that is, the focal point [11] - of the molecular electronic structure in a systematic manner. In the standard approach, approximate wavefunctions are constructed as linear combinations of antisymmetrized products (determinants) of one-electron functions, the molecular orbitals (MOs). The quality of the description then depends on the basis of atomic orbitals (AOs) in terms of which the MOs are expanded (the one-electron space), and on how linear combinations of determinants of these MOs are formed (the n-electron space). Within the one- and n-electron spaces, hierarchies exist of increasing flexibility and accuracy. To understand the requirements for accurate calculations of thermochemical data, we shall in this section consider the one- and n-electron hierarchies in some detail [12]. 

For approximate wavefunctions, however, the various formulations give rise to different theoretical predictions. This has been demonstrated in detail, for example, by Hush and Williams (31) for large aromatic systems. Thus, when we wish to obtain exact values of J, we must be very careful in deciding which formalism to use. A final point here is that the one-electron model does not take into account configuration interaction. Calculations for relatively simple systems would be useful here. 

In variational treatments of many-particle systems in the context of conventional quantum mechanics, these symmetry conditions are explicitly introduced, either in a direct constructive fashion or by resorting to projection operators. In the usual versions of density functional theory, however, little attention has b n payed to this problem. In our opinion, the basic question has to do with how to incorporate these symmetry conditions - which must be fulfilled by either an exact or approximate wavefunction - into the energy density functional. 

Figure 3. The shapes of the potential energy curves of the OH radical from the 2-RDM methods with DQG and DQGT2 conditions as well as the approximate wavefunction methods UMP2 and UCCSD are compared with the shape of the FCl curve. The potential energy curves of the approximate methods are shifted by a constant to make them agree with the FCl curve at equilibrium or 1.00 A. The 2-RDM method with the DQGT2 conditions yields a potential curve that within the graph is indistinguishable in its contour from the FCl curve.

As Eqs. (18) and (2) demonstrate, the exact wavefunction is no longer explicitly needed nor is an approximate wavefunction used. Instead, the generating geminals determine the wavefunction as well as the reduced density matrices and thereby the energy of an AGP wavefunction. 

All real-valued wavefunctions and the overwhelming majority of other approximate wavefunctions also produce [156] momentum densities that satisfy... 

Eq. (5.34). However, it is possible to construct approximate wavefunctions that lead to electron momentum densities that do not have inversion symmetry. Within the Born-Oppenheimer approximation, the total electronic system must be at rest the at-rest condition... 

The simplest approximate wavefunction for an open-shell molecule is the spin-unrestricted Hartree-Fock function... 

There are essentially two different quantum mechanical approaches to approximately solve the Schrodinger equation. One approach is perturbation theory, which will be described in a different set of lectures, and the other is the variational method. The configuration interaction equations are derived using the variational method. Here, one starts out by writing the energy as a functional F of the approximate wavefunction ip>... 

When the variational method is applied to the functional (3.1) the convergence of the energy E is particularly efficient and much faster than the convergence of other properties which can be derived from the same wavefunction. This can be seen by the following set of operations. Say that the approximate wavefunction ip has a small error A which can be chosen orthogonal to the exact wavefunction. The energy functional (3.1) can then be written,... 

When considering the effects of an approximate wavefunction, W, it is useful to remember that 9 is usually variationally determined. Effectively the quantity... 

The MO concept is directly related to an approximate wavefunction consisting of a Slater determinant of occupied one-particle wavefunctions, or molecular orbitals. The Hartree-Fock orbitals are by definition the ones that minimize the expectation value of the Hamiltonian for this Slater determinant. They are usually considered to be the best orbitals, although it should not be forgotten that they are only optimal in the sense of energy minimization. 

The HF equations are approximate mainly because they treat electron-electron repulsion approximately (other approximations are mentioned in the answer suggested for Chapter 5, Harder Question 1). This repulsion is approximated as resulting from interaction between two charge clouds rather than correctly, as the force between each pair of point-charge electrons. The equations become more exact as one increases the number of determinants representing the wavefunctions (as well as the size of the basis set), but this takes us into post-Hartree-Fock equations. Solutions to the HF equations are exact because the mathematics of the solution method is rigorous successive iterations (the SCF method) approach an exact solution (within the limits of the finite basis set) to the equations, i.e. an exact value of the (approximate ) wavefunction l m.. 

The three Equations (4.108)—(4.110) are only true for exact wavefunctions and they do indeed provide crude and problematic methods for calculating molecular properties. The advantage of these equations is that they indicate what one is able to obtain from this method but for actual calculations of molecular properties using approximative wavefunctions, it is important to use modern versions of response theory where the summation over states is eliminated [1,10-14,88-90],... 

Table 3.2.1 summarizes the results of various approximate wavefunctions for the hydrogen molecule. This list is by no means complete, but it does show that, as the level of sophistication of the trial function increases, the calculated dissociation energy and bond distance approach closer to the experimental values. In 1968, W. Kolos and L. Wolniewicz used a 100-term function to obtain results essentially identical to the experimental data. So the variational treatment of the hydrogen molecule is now a closed topic. 

It will be recalled that the approach of molecular orbital (MO) theory starts, on the other hand, from an independent-particle model (IPM) in which both electrons occupy the same bonding MO , 1 = Xa + Xb, similar to the one used [4] for the hydrogen molecule ion, Hj. The bonding MO is in fact the approximate wavefunction for a single electron in the field of the two nuclei and allocating two electrons to this same MO, with opposite spins, yields the 2-electron wavefunction... 

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