The Hellmann–feynman Theorem and Chemical Binding

Abstract

The advantages of the electrostatic interpretation of chemical bincling are illustrated. It is show11 that the forces exerted by the electrons in a molecular orbital may be determined by general expressions. These are functions of only one parameter, the value of which is easily specified for a particular case. From the genera1 expressions it is possible to obtain quantitative measures of the bonding or antibonding power of a molecular orbital. It is shown that the usual interpretations of bonding and antibonding properties in molecular orbital theory arc misleading and that few orbitals are, in actual fact, arltibonding. If molecular orbitals are classified as net bonding or net antibonding, terms which are precisely defined and have a definite physical significance, a much closer correlation with previous concepts of bonding is obtained. The effects of mutual ortl~ogonalization within a set of orbitals are studied. I t is found that this procedure decreases the bonding ability of bonding orbitals, but increases that of antibonding orbitals. The dangers of disregarding such orthogonalization in a conventional calculation are clearly illustrated. I. INTRODUC'I'IONThe FIellmann-Feyn~~~an theorem permits one to relate the forces acting on a nucleus in a molecule to the classical electrostatic forces exerted on the nucleus by the other nuclei and by the particular distribution of electrons found in the nlolecule. This relationship may be expressed as where p(x, y, z,) is the electron density, ~y~, the electric field produced by the at11 ilucleus in the x direction a t the position of the it11 electron, and 2, is the nuclear charge of atom a. In a previous paper ( I ) we have shown that equation [ I ] is capable of providing reasonable results for the binding energies of simple molecules. At that time it was stated that the Hellmann-Feynman theorem provides a qualitative interpretation of chemical binding that is perhaps more readily understandable than is the usual discussion which is couched in terms of energies. The reason for this is that the electron density present in a molecule is a real three-dimensional quantity. In employing the I-Iellmann-Feynlnan theorem one reduces a problem in molecular structure to a problem in classical electrostatics, a problem of the interaction of a system of nuclei and some continuous charge distribution. In this paper we wish to illustrate the advantages of this approach. Rather than press the Hellmann-Feynman theorem by attempting to calculate molecular energies through the use of approxin~ately determined electron densities, we \visl~ instead to demonstrate the ability of the theorem to provide us with useful semiquantitative results regarding the bonding character of various electron distributions. T H E HEL1,MANN-FEYNMAN THEOREM A N D T H E MOLECULXIP OliRIrTi\l. APPROACHWhen the molecular orbitals used in the description of the electronic structure o l a ~nolecule are all orthonormal, the electron density may be expressed as For personal use only. the occupatioil n...

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“…The methods described in this paper offer no advantages in the evaluation of this third The basis of the present method is in the use of the Gegenbauer polynomials for the expailsion of the radial dependence of the electric field about another atomic center. In both the two-and three-center integrals, gaz and gau are expanded about center fi (see are the incomplete galnilla functions defined by Equation [3] terminates after the first few terlns in any particular case due t o the orthogonality and recurrence relationships of the Legeitdre polynomials. In Table I we have listed the resultiilg general formulae for all of the nonvanishing penetration integrals involving s, P, and d atomic orbitals of ally principal quantum number.…”

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“…The manner in which equation [3] reduces in a particular case will be illustrated for nFf n', I = I ' = 1 , and rn = nz' = 0. Consider first the two integrations over 0 in equation [3]. IiVhen 1 = I ' = 1 the first integral is zero u~lless k is an even integer and the second is zero unless k is odd.…”