Bond formation in s-valent dimers

Let us consider what happens as two s-valent atoms A and are brought together from infinity to form the AB diatomic molecule as illustrated schematically in Fig. 3.1. The more deeply bound energy level EA could represent, for example, the hydrogenic Is orbital (EA = —13.6 eV), whereas the less deeply bound energy level EB could represent lithium s 2s orbital (EB = — 5.5 eV. cf Fig. 2.16). Each free atomic orbital satisfies its own effective one-electron Schrodinger equation (cf eqn (2.59)), namely 

We wish to solve the one electron Schrodinger equation for the AB dimer, 

The molecular Schrodinger equation can be solved exacdy for the case of Hj when VAB is simply the sum of two hydrogen ion potentials. In general, however, an exact solution is not possible. Following the well-worn tracks of MO theory we look instead for an approximate solution that is given by some linear combination of atomic orbitals (LCAO). Considering the AB dimer illustrated in Fig. 3.1 we write 

Premultiplying by (or ) and integrating over all space we find the LCAO secular equation (taking = as is real for s orbitals) 

The Hamiltonian matrix elements can be evaluated directly by assuming VAB = VA + Vb as in eqn (3.4). The diagonal element HAA may then be written 

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