Matrix Formulation of the Linear Variation Method

In Chapter 7 we developed a method for performing linear variational calculations. The method requires solving a determinantal equation for its roots, and then solving a set of simultaneous homogeneous equations for coefficients. This procedure is not the most efficient for programmed solution by computer. In this chapter we describe the matrix formulation for the linear variation procedure. Not only is this the basis for many quantum-chemical computer programs, but it also provides a convenient framework for formulating the various quantum-chemical methods we shall encounter in future chapters. 

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Chapter 9 Matrix Formulation of the Linear Variation Method... 

That is, the method of linear variations is identical to the matrix formulation of the Schrodinger equation. Another way of viewing this result is that only solutions to eq. (12) are energetically stable with respect to variations in the linear expansion coefficients. 

Direct minimization techniques. The variational principle indicates that we want to minimize the energy as a function of the MO coefficients or the corresponding density matrix elements, as given by eq. (3.54). In this formulation, the problem is no different from other types of non-linear optimizations, and the same types of technique, such as steepest descent, conjugated gradient or Newton-Raphson methods can be used (see Chapter 12 for details). 

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