Heitler-London procedure

The valence electron of a promoted atom readily interacts with other activated species in its vicinity to form chemical bonds. The mechanism is the same for all atoms, since the valence state always consists of a monopositive core, loosely associated with a valence electron, free to form new liaisons. Should the resulting bond be of the electron-pair covalent type, its properties, such as bond length and dissociation energy can be calculated directly by standard Heitler-London procedures, using valence-state wave functions (section 5.3.4). 

Shortly after quantum mechanics evolved Heitler and London[l] applied the then new ideas to the problem of molecule formation and chemical valence. Their treatment of the H2 molecule was qualitatively very successful, and this led to numerous studies by various workers applying the same ideas to other substances. Many of these involved refinements of the original Heitler-London procedure, and within three or four years, a group of ideas and procedures had become reasonably well codified in what was called the valence bond (VB) method for molecular structure. 

This paper attracted Pauling s attention, who recognized that the Heitler-London procedure could not only be extended to larger molecules but also serve as a bridge between Lewis classical idea of chemical bond and the new quantum theory. In a series of papers ", where he introduced the concepts of hybridization, resonance, electronegativity, etc., and also in a famous book that influenced many generations of chemists, Pauling developed his ideas which formed the basis of the classical Valence-Bond theory. 

On equating the atomic radius to a characteristic atomic radius, r, a single curve of d vs D describes homonuclear covalent interaction, irrespective of bond order. Practical use of the formulae requires definition of a complex set of characteristic radii, which could be derived empirically [1] and was used subsequently to calculate molecular shape descriptors [2] and as the basis of a generalized Heitler-London procedure, valid for all pairwise covalent interactions [3,4], In all of these applications, interaction is correctly described by the dimensionless curves of Fig. 1. 

When the ionization spheres of two neighbouring atoms interpenetrate, their valence electrons become delocalized over a common volume, from where they interact equally with both atomic cores. The covalent interaction in the hydrogen molecule was modelled on the same assumption in the pioneering Heitler-London simulation, with the use of free-atom wave functions. By the use of valence-state functions this H-L procedure can be extended to model the covalent bond between any pair of atoms. The calculated values of interatomic distance and dissociation energy agree with experimentally measured values. 

The procedure of taking a linear combination of atomic orbitals, which we have considered with respect to the H2 molecule, is very fruitful when applied to other covalent bonds. Consider, for example, the hydrogen fluoride molecule, HF, formed from a hydrogen atom with one electron in the Is state and a fluorine atom with an electron configuration of ls 2s 2p. Fluorine has an unpaired 2p electron, and we can form a wave function of the Heitler-London type by making use of the atomic orbitals for this 2p electron and for the Is electron in the hydrogen atom ... 

The reason why the Heitler-London method gives such a bad Hellmann-Feynman force-constant is thus that R is "attached to the nuclei so that 9 / R is not zero the same applies to the Weinbaum function 94), to the Wang function (95), and to the Coulson function 96) for Hg. To yield better force results, the variable parameters ) must be "detached from the nuclei and their values determined at each internuclear configuration. A wave function in which the parameters are determined by the variational procedure is called a floating function by Hurley 93, 97, 30, 31, 32) (this is eqtuvalent to Hall s stable wave functions 88)). This procedure can be extended to the scale factors, as discussed by McLean 81) and Lowdin 83). The vibrational frequency of H2 determined by Ross and Phillipson using the differentiation of the virial theorem (which assumed that all the variable parameters are variationally... 

Several important modifications have been made of the semiempirical procedure of Eyring and Polanyi for calculating the potential-energy surface. Sato [4] has used the Heitler-London formulation to obtain the following... 

In the companion article in this volume [1] we have presented [16] the methodology for the transformation of MC-SCF wavefunctions to the space of VB wavefunction through the use of effective hamiltonians. The important point is that the simple VB wavefunction obtained in this procedure reproduces the energetics of the MC-SCF computations exactly. Thus we can understand the topology of the potential surfaces in terms of the simple parameters of Heitler-London type VB theory (For a thorough discussion of VB theory the reader is referred to the standard textbooks [25,26]). We now demonstrate that the potential surface topology for forbidden and allowed cycloaddition reactions can be understood using a simple model of 4 electrons in 4 orbitals that is summarized in BOX 4 of Part I. 

The recursive projection procedure described above provides an alternative and potentially more efficient method for obtaining eigenstates than does the development of Eqs. (8) to (10). In the two-electron case, this procedure serves to separate the symmetric and antisymmetric subspaces spanned in the absence of unphysical representations, and can accelerate the convergence relative to that of Table II through incorporation of explicitly symmetric or antisymmetric test functions. In Figure 3 are shown and potential energy curves in H2 obtained from the recursive development and the basis states of Table I employing Heitler-London test functions in each case. These functions serve as appropriate chemical reference states... 

Figure 3. Potential energy (a.u.) curves for the and states of H2 as functions of atomic separation E(ao). Solid lines refer to Heitler-London (HL) (4) and previously determined accurate values (KW) (SO), whereas the dashed lines give the present results obtained from the recursion procedure indicated in the text employing the [s], [sp], and [spd basis states of Table I and Heitler-London test functions in each case.

Various approximate methods are employed. The general nature of the procedure is illustrated quite well by the original method of Heitler and London, which is based upon a so-called perturbation ... 

Using truncating method (as in Heitler and London method - to be amended in the Section 1.4) and then applying the perturbation approximations (viz. adiabatic coupling), leads to uncertain procedures. 

F. Himd proposed estimating the electronic terms of the molecules in this fashion. W. Heitler and F. London instead point out that in this procedure not all calculated eigenvalues will represent stationary states of the molecule, but many will lie in the continuous spectrum. 

---