Heitler-London integral

If the overlap integral is neglected, the Heitler-London equation becomes... 

In the case of the hydrogen molecule-ion H2" ", we defined certain integrals Saa, Taa, Tab, Labra- The electronic part of the energy appropriate to the Heitler-London (singlet) ground-state wavefunction, after doing the integrations... 

Figure 5.9 Potential-energy curves of integral-order dicarbon bonds, calculated by the Heitler-London method for screened nuclei, superimposed on the point-charge covalency curve.

At smaller distances the exchange interaction comes into play. In the Heitler-London approximation there will be two types of exchange integrals without and with excitation transfer. These are... 

It has been pointed out that any relationship between the exchange integral and the Weiss field is only valid at 0 K, since the former considers magnetic coupling in a pair-wise manner and the latter results from a mean-field theory (Goodenough, 1966). Finally, it is also essential to understand that Eq. 8.43 is strictly valid only for localized moments (in the context of the Heitler-London model). One might wonder then whether the Weiss model is applicable to the ferromagnetic metals, in which the electrons are in delocalized Bloch states, for example, Fe, Co, and Ni. This will be taken up later. 

In these relations the operator B (Bn) describes the creation (annihilation) of a molecular excitation at lattice site n. We assume below that n 1,2,. ..,Ar, where N is the number of molecules in the chain and we consider one electronically excited molecular state. Then E-p is the on-site energy of a Frenkel exciton and Mnni is the hopping integral for molecular excitation transfer from molecule n to molecule n. In the summation in HF the terms with n = n are omitted. The Hamiltonian HF describes the Frenkel excitons in the Heitler-London approximation. 

By applying the Heitler-London approximation to transition metals, Bethe (1933) calculated exchange integrals for Fe, Co, Ni, Cr and Mn, as a function of interatomic distance and radii of 3d orbitals. 

The Heitler-London VB wave function for ground-state Ha is [Eq. (13.101)] ls (l)lsj(2) ls (2)lsj(l) multiphed by a normalization constant and a spin function. The GVB ground-state Ha wave function replaces this spatial function by/(l)g(2) + /(2)g(l), where the functions/and g are found by minimization of the variational integral. To find / and g, one expands each of them in terms of a basis set of AOs and finds the expansion coefficients by iteratively solving one-electron equations that resemble the equations of the SCF MO method. 

The generalization of these ideas to more than 2 electrons is summarized in Box 5. We shall now demonstrate that when localized MC-SCF orbitals are used we can partition the Cl hamiltonian. The projection onto the space of covalent configurations yields an effective hamiltonian that has the property that it reproduces a subset of the MC-SCF eigenvalues explicitly. The final step involves expressing this effective hamiltonian in terms of the Heitler-London Q and K integrals via a Heisenberg hamiltonian... 

Here, Q, J and S are integrals of combined Is orbitals. Q is the coulombic, J the exchange and S the overlap integral. The Heitler-London expression is often approximated neglecting the overlap S term, such that E 2 /. Here, 2 - - / is the H2 bonding orbital and E - y is the H2 antibonding orbital. Thus, the London expression for the ABC triatom adopts the form... 

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