Heitler-London function

One inconvenience of the expression of FvB-fuii (Eq. 3.4) is its relative complexity compared to the simple Heitler—London function (Eq. 3.2). 

The essence of this method, when illustrated with H2, is to write the two orbitals for the covalent Heitler-London function as... 

Here the first two determinants are the determinantal form of the Heitler-London function (eq 1), and represent a purely covalent interaction between the atoms. The remaining determinants represent zwitterionic structures, H-H+ and H+H, and contribute 50% to the wave function. The same constitution holds for any interatomic distance. This weight of the ionic structures is clearly too much at equilibrium distance, and becomes absurd at infinite separation where the ionic component is expected to drop to zero. Qualitatively, this can be corrected by including a second configuration where both electrons occupy the antibonding orbital, Gu, i.e. the doubly excited configuration. The more elaborate wave function T ci is shown in eq. 4, where C and C2 are coefficients of the two MO configurations ... 

It should be remarked that, while the Heitler-London function (1.93) for H2 is a two-determinant wave function, the Heitler-London function (1.94) for He2 is a single determinant wave function, so that in this case HL and MO approaches coincide. 

In the last two cases, the various localized pairs may be written in the form of either MO or Heitler-London functions. The interaction... 

Consider some improvements on the Heitler-London function (13.101). One obvious step is the introduction of an orbital exponent C in the Is function. This was done by Wang in 1928. The optimum value of C is 1.166 at Rg, and and R are improved to 3.78 eV and 0.744 A. Recall that Dickinson in 1933 improved the Finkelstein-Horowitz HJ trial function by mixing in some 2p character into the atomic orbitals (hybridization). In 1931 Rosen used this idea to improve the Heitler-London-Wang function. He took the trial function... 

Before going on to IA2, let us express the Heitler-London valence-bond functions for H2 as Slater determinants. The ground-state Heitler-London function (13.101) can be written as... 

Consider the valence-bond ground state wave function of HF. We expect a single bond to be formed by the pairing of the hydrogen Is electron and the unpaired fluorine 2pa electron. The Heitler-London function corresponding to this pairing is [Eq. (13.119)]... 

A simple example is the H2 ground state. The Heitler-London function is... 

This leads lu a very bad description of the H2 molecule at long iiiicinuclcai disianecs with the Haitree-Fock method. Indeed, for long internuclear distances, the Heitler-London function should dominate, because it corresponds to the (correct) dissociation limit (two ground-state hydrogen atoms). The trouble is that with fixed coefficients, the Hartree-Fock function overestimates the role of the ionic structure for long interatomic distances. Fig. 10.5 shows that the Heitler-London function describes the electron correlation (Coulomb hole), whereas the Haitree-Fock function does not. 

Fig. 10.5. Illustration of electron correlation in the hydrogen molecule. Hie nuclear positions are fO. 0.01 and (4.0.01 in a.u. Slater oifcitals of Is type have an orbital exponent equal to 1. (a) Visualizatirai of the xv cross-section of the wave function of electron 2, assuming that electron I resides on the nucleus (either the first or (he second onel, has spin coordinate Of = j, whereas electron 2 has spin coordinate

The DA structure corresponds to double occupation of n and n. From Eq. (10.28) we know that in the VB language the corresponding Slater determinant contains three structures one of the Heitler-London type and the two ionic structures. In our case of a polarized bond, the Heitler-London function would continue to treat the C and Y nuclei on the equal footing, the polarity of the CY bond would be correctly restored by different weights of the ionic structures, b > a. In conclusion this will he the CY double bond but polarized. 

The first two structures are famous Kekule structures, the next three are Dewar structures, the sixth is an example of the possible mixed covalent-ionic structures. From these graphs, we may deduce which atomic orbitals (out of the 2p, orbital of carbon atoms, z is perpendicular to the plane of the benzene ring) takes part in the covalent bond (of the tt type). As far as the mathematical form of the carbon atomic orbitals), the first for electrons 1,2, the second for electrons 3,4, and the third for 5,6. Within the functions d>/, the ionic structures can... 

Valence bond (VB) method (p. 520) covalent structure (p. 521) resonance theory (p. 520) Heitler-London function (p. 521) ionic structure (p. 521)... 

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