HL wave function

Compared to the overlap of the undistorted atomic orbitals used in the HL wave function, which is just 5ab. it is seen that the overlap is increased (c is positive), i.e. the orbitals distort so that they overlap better in order to make a bond. Although the distortion is fairly small (a few %) this effectively eliminates the need for including ionic VB terms. When c is variationally optimized, the MO-CI, VB-HL and VB-CF wave functions (eqs. (7.4), (7.7) and (7.8)) are all completely equivalent. The MO approach incorporates the flexibility in terms of an excited determinant, the VB-FIL in terms of ionic structures, and the VB-CF in terms of distorted atomic orbitals. 

Generally, normalization factors for determinants are larger than unity, with the exception of those VB determinants that do not have more than one spin-orbital of each spin variety, for example, as is the case of the determinants that compose the HL wave function. For these latter determinants the normalizing factor is unity, that is, N = 1. 

As we already argued, the origin of the barrier is G. Since R in Fig. 6.3 is just the VB image of the product HL wave function in the geometry of the reactants, this excited state displays a covalent-bond coupling between the infinitely separated fragments X and A, and an uncoupled fragment Y in the vicinity... 

The state R in Equation 6.4 strictly keeps the HL wave function of the product P, and is hence a quasi/spectroscopic state that has a finite overlap with R. If one orthogonalizes the pair of states R and R, by, for example, a Graham—Schmidt procedure (see Exercise 6.3), the excited state becomes a pure spectroscopic state in which the A—Y is in a triplet state and is coupled to X to yield a doublet state. In such an event, one could simply use, instead of Equation 6.5, the spectroscopic gap Gs in Equation 6.6 that is simply the singlet—triplet energy gap of the A—Y bond ... 

Exercise 6.1 Consider two radicals, R and X (not Na and Cl of course) combining to form a bond. Since the bond is polar covalent, its wave function will be dominated by the HL structure. Letting r and x represent the singly occupied orbitals of the two radicals, the unnormalized HL wave function is... 

This simple wave function, so called the Heitler-London (HL) wave function, was able to account for about 66% of the bonding energy of H2, and performed a little better than the rival MO method that appeared almost at the same time. 

The early VB point of view was based solely on the purely covalent HL wave function. In this wave function the electrons are never allowed to approach each other and therefore their electron repulsion is minimized and their Coulomb correlation is at maximum. Thus, while the Hartree-Fock model has no electron correlation, giving equal weight to covalent and ionic structures, the early VB models overestimated the correlation. The true situation is about half-way in-between. In the same way as the Hartree-Fock wave function is improved by Cl, the purely covalent VB function can be improved by admixture of ionic structures as in eq 5, in which the coefficients X and p would be directly optimized in the VB framework. Both improved models thus lead to wave functions that are strictly equivalent and physically correct, even though their initial expressions appear entirely different. This... 

Compared to the overlap of the undistorted atomic orbitals used in the HL wave function, which is just 5ab> it is seen that the overlap is increased (c is positive), i.e. the orbitals distort so that they overlap better in order to make a bond. Although the distortion is fairly small (a few %) this effectively eliminates the need for including ... 

This section aims to illustrate the origin of the quantum mechanical exchange-overlap densities and their different behaviour in the case of the chemical bond in ground state H2 and the Pauli repulsion in He2. We choose as starting point for the 1Xg+ ground state of the systems the normalized Heitler-London (HL) wave functions (Magnasco, 2008) ... 

Also shown on Figure 4 is the potential curve given by the original HL wave function (equation 1) and that given by MO theory. It can be seen that the SCF wave function does not describe dissociation correctly. 

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