The Crude Born-Oppenheimer Basis Set

In this and the following sections, we will discuss ways of selecting the basis function by separating the nuclear and electronic motions in a manner different from that in the previous section. In the present approach, the electronic Hairultonian is 

In the crude adiabatic (CA) approximation [1,32-40], the electronic wavefimctions p (r q) defined at a specific nuclear configuration qg satisfy the following Schrodin-ger equation  

As before, Xbv(q) are initially treated as expansion coefficients, which must be determined. Inserting Equation 1.19 in Equation 1.4 results in the usual infinite set of coupled equations for the Xbv(q)- 

The functions %av (q) are therefore determined by the set of coupled equations (1.20). The potential functions ( p A[/ pp ) are usually represented as power series expansions in the normal coordinates around qo. where qo is usually chosen at the minimum of the ground state. 

The diagonal matrix elements ( p Al/ p ) are the effective potential energy surface that governs nuclear motion. From Equations 1.10 and 1.23, it is evident that the vibrational wavefunction x differs from the adiabatic wavefunction x As long as the basis set p (r qo) is complete in the electronic space, the CA basis is perfectly adequate (independent of the choice of qg). The two matrix representations 1.8 and (1.20) are merely two different representations of the same operator. 

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