In this chapter we examine some results for four model systems consisting of rings of H atoms. These calculations show how the number of atoms in a complex reaction may influence rates of reaction, particularly through the activation energy. The systems are as follows.

• Four H atoms in a rectangular geometry of D2h symmetry. The rectangle is characterized by two distances, RA and RB. We map out a region of the ground state energy for this four-electron system as a function of the two distances.

• Six H atoms in a hexagonal geometry of D3h symmetry. This is not a regular hexagon, in general, but, like the system of four H atoms, is characterized by two distances we also label RA and RB. These two distances alternate around the ring. We also calculate the map of the ground state energy for this six-electron system.

• Eight H atoms in an octagonal geometry of D4h symmetry and the specific shape characterized by the RA and RB variables as above. For this larger system we only determine the saddle point with respect to the same sort of variables.

• Ten H atoms in a decagonal geometry of D5h symmetry and the RA and RB variables. Again, we determine only the saddle point.

Since the geometries of these systems are in most regions not regular polygons, we will symbolize them as (H2)n, emphasizing the number of H2 molecules rather than the total number of atoms.

As was seen in the last chapter, the effect of permutations on portions of the wave function is important in enforcing their correct character. The permutations of n entities form a group in the mathematical sense that is said to be one of the symmetric groups. In particular, when we have all of the permutations of n entities the group is symbolized Sn. In this chapter we give, using the theory of the symmetric groups, a generalization of the special treatment of three electrons discussed above.

There are several more or less equivalent methods for dealing with the twin problems of constructing antisymmetric functions that are also eigenfunctions of the spin. Where orbitals are orthogonal the graphical unitary group approach (GUGA), based upon the symmetric group and unitary group representations, is popular today. With VB functions, which perforce have nonorthogonal orbitals, a significant problem centers around devising algorithms for calculating matrix elements of the Hamiltonian that are efficient enough to be useful. In the past symmetric group methods have been criticized as being overcomplicated. Nevertheless, the present author knows of no other techniques for obtaining what appears to be the optimal algorithm for these calculations.

This chapter is the most complicated and formal in the book. Looking back to Chapter 4 we can obtain an idea of what is needed in general.

Benzene is the archetypal aromatic hydrocarbon and its study has been central to the understanding of aromaticity and resonance from the early times. In addition, it has the physical property of having its π electrons reasonably independent from those in σ bonds, leading early quantum mechanics workers to treat the π electrons alone. Since benzene is a ring and the rules for forming Rumer diagrams have one draw noncrossing lines between orbital symbols written in a circle, the Rumer diagrams correspond to the classical Kekulé and Dewar bond schemes that chemists had postulated far earlier than the VB treatments occurred. This parallel has intrigued people since its first observation and led to many discussions concerning its significance. It has also led to considerable work in more qualitative “graphical methods” for which the reader is directed to the literature. (See, inter alia, Randić[59].)

We will examine benzene with different bases and also discuss some of the ideas that consideration of this molecule has led to, such as resonance and resonance energy.

We show again the traditional five covalent Rumer diagrams for six electrons and six orbitals in a singlet coupling and emphasize that the similarity between the ring of orbitals and the shape of the molecule considerably simplifies the understanding of the symmetry for benzene.

The existence of many ionic structures in MCVB wave functions has often been criticized by some workers as being unphysical. This has been the case particularly when a covalent bond between like atoms is being represented. Nevertheless, we have seen in Chapter 2 that ionic structures contribute to electron delocalization in the H2 molecule and would be expected to do likewise in all cases. Later in this chapter we will see that they can also be interpreted as contributions from ionic states of the constituent atoms. When the bond is between unlike atoms, it is to be expected that ionic structures in the wave function will also contribute to various electric moments, the dipole moment being the simplest. The amounts of these ionic structures in the wave functions will be determined by a sort of “balancing act” in the variation principle between the “diagonal” effects of the ionic state energies and the “off-diagonal” effect of the delocalization.

In this chapter we will also focus on the dipole moment of molecules. With these, some of the most interesting phenomena are the molecules for which the electric moment is in the “wrong” direction insofar as the atomic electronegativities are concerned. CO is probably the most famous of these cases, but other molecules have even more striking disagreements. One of the larger is the simple diatomic BF. We will take up the question of the dipole moments of molecules like BF in Chapter 12.

The separation of spin and space variables

One of the pedagogically unfortunate aspects of quantum mechanics is the complexity that arises in the interaction of electron spin with the Pauli exclusion principle as soon as there are more than two electrons. In general, since the ESE does not even contain any spin operators, the total spin operator must commute with it, and, thus, the total spin of a system of any size is conserved at this level of approximation. The corresponding solution to the ESE must reflect this. In addition, the total electronic wave function must also be antisymmetric in the interchange of any pair of space-spin coordinates, and the interaction of these two requirements has a subtle influence on the energies that has no counterpart in classical systems.

The spin functions

When there are only two electrons the analysis is much simplified. Even quite elementary textbooks discuss two-electron systems. The simplicity is a consequence of the general nature of what is called the spin-degeneracy problem, which we describe in Chapters 4 and 5. For now we write the total solution for the ESE θ(1, 2), where the labels 1 and 2 refer to the coordinates (space and spin) of the two electrons. Since the ESE has no reference at all to spin, θ(1, 2) may be factored into separate spatial and spin functions.

In previous chapters we have repeatedly emphasized that the principal difficulty in calculating the dissociation energy of a bond is the correct treatment of the change in electron correlation as the bond distance changes. This observation also applies to reactions where bonds are both formed and broken. In many important cases, however, the particular atom–atom distances that change significantly during a reaction are relatively few in number, and a method for accurately treating the correlation in only those “bonds” would have a clear advantage in efficiency. The MCVB method provides a method for targeting certain bonds to treat the correlation in them as well as possible. We call this procedure targeted correlation, and in this chapter we give examples using it. The SCVB method could also be used in this context.

In our previous work we have used SCF solutions of the atoms as the ingredients of the n-electron VB basis functions. With targeted correlation we go one step up and use SCF solutions of molecular fragments as the ingredients. As the name implies, this must be tailored to the specific example and must be done with a careful eye to the basic chemistry and physics of the situation at hand.

Methylene, ethylene, and cyclopropane

In this section we consider some molecules that can be viewed as consisting of methylene radicals in some combination. Earlier publications[39, 66] have covered some of the aspects of the subjects covered here.

The consideration of isoelectronic sequences can provide considerable physical understanding of structural details. We here give details of the calculation of a series of isoelectronic diatomic molecules from the second row of the periodic table, N2, CO, BF, and BeNe. By studying this sequence we see how the competition between nuclear charges affects bonding. All of these are closed-shell singlet systems, and, at least in the cases of the first two, conventional bonding arguments say there is a triple bond between the two atoms. We expect, at most, only a Van der Waals type of bond between Be and Ne, of course. Our calculations should predict this.

The three polar molecules in the series are interesting because they all have anomalous directions to their dipole moments, i.e., the direction is different from that predicted by an elementary application of the idea of electronegativity, accepting the fact that there may be ambiguity in the definition of electronegativity for Ne. We will see how VB ideas interpret these anomalous dipole moments.

We do the calculations with a 6-31G* basis in the same way as was done in Chapter 11 and for three arrangements of STO3G bases. This will allow us both to judge the stability of the qualitative predictions to the basis and to assess the ability of the calculations to obtain quantitative answers.

We have already treated N2 in Chapter 11, but will look at it here from a somewhat different point of view.

In Chapter 11 we discussed the properties of the atoms in the second row of the periodic table and how these might influence molecules formed from them. We focus on carbon in this chapter and examine how the bonding changes through the series CH, CH2, CH3, and CH4. The first three of these are known only spectroscopically, in matrix isolation, or as reaction intermediates, but many of their properties have been determined. The reader will recall that carbon exhibits relatively low-energy excited valence configurations. For carbon the excitation energy is around 4 eV, and among the atoms discussed in Chapter 11, only boron has a lower excitation energy. If this excited configuration is to have an important role in the bonding, the energy to produce the excitation must be paid back by the energy of formation of the bond or bonds. We shall see that VB theory predicts this happens between CH and CH2. After our discussion of these single carbon compounds, we will consider ethane, CH3CH3, as an example for dealing with larger hydrocarbons.

CH, CH2, CH3, and CH4

STO3G basis

We first give calculations of these four molecules with an STO3G basis. The total energies and first bond dissociation energies are collected in Table 13.1. We see that, even with the minimal basis, the bond energies are within 0.4eV of the experimental values except for CH3, which has considerable uncertainty. The calculated values tend to be smaller, as expected for a minimal VB treatment.

Since, for any but the smallest of systems, a full VB calculation is out of the question, it is essential to devise a useful and systematic procedure for the arrangement of the bases and for the selection of a manageable subset of structures based upon these orbitals. These two problems are interrelated and cannot be discussed in complete isolation from one another, but we will consider the basis question first. In our two-electron calculations we have already addressed some of the issues, but here we look at the problems more systematically.

The AO bases

The calculations described in this section of the book have, for the most part, been carried out using three of the basis sets developed by the Pople school.

STO3G A minimal basis. This contains exactly the number of orbitals that might be occupied in each atomic shell.

6-31G A valence double-ζ basis. This basis has been constructed for atoms up through Ar.

6-31G* A valence double-ζ basis with polarization functions added. Polarization functions are functions of one larger l-value than normally occurs in an atomic shell in the ground state.

Any departures from these will be spelled out at the place they are used.

Our general procedure is to represent the atoms in a molecule using the Hartree–Fock orbitals of the individual atoms occurring in the molecule. (We will also consider the interaction of molecular fragments where the Hartree–Fock orbitals of the fragments are used.)