Elements of the Secular- Matrix

To begin a more general approach to molecular orbital theory, we shall describe a variational solution of the prototypical problem found in most elementary physical chemistry textbooks the ground-state energy of a particle in a box (McQuanie, 1983) The particle in a one-dimensional box has an exact solution 

Let the dimension of the box be 1 in any units and consider only the ground state, for which n = 1. Now 

There are two functions, so we shall obtain two eigenvalues. The ground-state energy will be the lower of the two. The full secular matrix is 

If we do not make any simplifying assumptions, we must calculate the matrix 

In this problem, the integral over all spaee dx is in only one dimension, x. The limits of integration are the dimensions of the box, 0 and 1 in whatever unit was ehosen. 

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If we divide each element of the secular matrix by p and perform the substitution... 

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