Bond formation in sp-valent dimers

The cylindrical symmetry about the inter-nuclear axis leads to the solutions of the molecular Schrodinger equation, eqn (3.3), having either a or character. Taking the z axis along the axis of the molecule, the a eigenfunctions will comprise linear combinations of the , , and atomic orbitals so that we can write the molecular orbital as 

By analogy with our previous treatment of the s-valent dimer, substituting eqn (3.54) into the Schrodinger equation (3.3) leads to the LCAO secular equation (neglecting the overlap integral, S) 

Since the potential is unchanged with respect to inversion about the centre of the molecule (i.e. l B(r) = ( ) with the origin at the molecular centre), the solutions will be either even (gerade) or odd (iungerade). The even solutions correspond to (cA = = -cBi), the odd solutions to 

The resultant two quadratic equations can be solved directly. However, 

The behaviour of these eigenvalues can best be understood by considering the two limiting cases (i) AEgp/ h - oo and (ii) AEsp/ h -+ 0. For the case where the sp splitting is large we have from eqs (3.59) and (3.60) that 

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