Bond eigenfunctions

When d eigenfunctions as well as s and p are available for bond-eigenfunction formation a number of bond configurations are possible. One d eigenfunction with s and p permits the construction of only four strong bonds, and these are directed towards the corners of a square. Such a configuration has been... 

Fig. 3.—Polar graph of V2 + / sin 6 in the xz plane, representing a tetrahedral eigenfunction, the best bond eigenfunction which can be formed from s and p eigenfunctions.

This eigenfunction is equivalent to and orthogonal to pi, and has its maximum value of 2 at 6 = 19°28

angle between the lines drawn from the center to two corners of a regular tetrahedron. The third and fourth best bond eigenfunctions... 

It should be borne in mind that the bond eigenfunctions actually are obtained from the expressions given in this paper by substituting for s the complete eigenfunction i nM(r,0,ip), etc. It is not necessary that the r. part of the eigenfunctions be identical the assumption made in the above treatment is that they do not affect the evaluation of the coefficients in the bond eigenfunctions. 

A double bond behaves differently, however. Let us introduce two substituents in the octants xyz and xyz of an atom, a carbon atom, say, using the bond eigenfunctions and fn,. 

S A discussion of rotation about a double bond on the basis of the quantum mechanics has been published by E. Hiickel, Z. Physik, 60, 423 (1930), which is, I feel, neither so straightforward nor so convincing as the above treatment, inasmuch as neither the phenomenon of concentration of the bond eigenfunctions nor that of change in quantization is taken into account. 

Assuming as before that the dependence on r of the s, p and d eigenfunctions under discussion is not greatly different, the best bond eigenfunctions can be determined by the application of the treatment already applied to 5 and p alone, with the following results. 

The best bond eigenfunction which can be obtained from s, p and d is... 

One of these is shown in Fig. 9. These all have their maxima in the xy plane, directed toward the comers of a square. The strength of these bond eigenfunctions, 2.694, is much greater than that of the four tetrahedral eigenfunctions formed from s and p alone (2.00). But if three d... 

Bond eigenfunctions for iron-group atoms are formed from the nine eigenfunctions 3dh, 4s and 4p3, as described in preceding sections. One bond eigenfunction is needed for each electron-pair bond. 

Examples of deductions regarding atomic arrangement, bond angles and other properties of molecules and complex ions from magnetic data, with the aid of calculations involving bond eigenfunctions, are given. 

Throughout this paper the word eigenfunction will be used to denote a singleelectron eigenfunction, such as one of the four tetrahedral bond eigenfunctions of a carbon atom. 

These questions can often be answered with the aid of a set of rules regarding the properties of electron-pair bonds derived from the quantum mechanics, and of a semi-quantitative method of treatment of bond eigenfunctions leading to information on the strengths and relative orientation of bonds2). The rules are the following. 

When tt o d eigenfunctions are available, as in trivalent cobalt, quadrivalent palladium and platinum, etc., six equivalent bond eigenfunctions of strength 2.923 and directed toward the comers of a regular octahedron can be formed. These form the bonds in a great many octahedral complexes. 

Such a "general form of wave function is easily written explicitly for each set of values of N, S, and MS- Any appropriate form of approximate wave functions, like determinantal functions composed of one-electron functions ( molecular spin orbitals ), the "bond eigenfunctions" used in the valence bond approach, and so on, is shown to fulfil this requirement. 

Fairly soon after the Heitler-London calculation, Slater, using his determi-nantal functions, gave a generalization to the n-electron VB problem[10]. This was a popular approach and several studies followed exploiting it. It was soon called the method of bond eigenfunctions. A little later Rumer[ll] showed how the use of these could be made more efficient by eliminating linear dependencies before matrix elements were calculated. 

Slater s bond eigenfunctions constitute one choice (out of an infinite number) of a particular sort of basis function to use in the evaluation of the Hamiltonian and overlap matrix elements. They have come to be called the Heitler-London-Slater-Pauling (HLSP) functions. Physically, they treat each chemical bond as a singlet-coupled pair of electrons. This is the natural extension of the original Heitler-London approach. In addition to Slater, Pauling[12] and Eyring and Kimbal[13] have contributed to the method. Our following description does not follow exactly the discussions of the early workers, but the final results are the same. 

The valence-bond treatment of polyatomic molecules is closely tied to chemical ideas of structure. One begins with the atoms that form the molecule and pairs up the unpaired electrons to form chemical bonds. There are usually several ways of pairing up (coupling) the electrons. Each pairing scheme gives a VB structure. A Heitler-London-type function (called a bond eigenfunction) is written for each structure i, and the molecular wave function is taken as a linear combination 2, c,4>, of the bond eigenfunctions. The variation principle is then applied to determine the coefficients c,. The VB wave function is said to be a resonance hybrid of the various structures. 

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