Linear expansions and eigenvalue equations

A particularly simple reahzation of the variation method arises if we make a linear ansatz for the wave function, expanding the approximate electronic state in an m-dimensional set of normalized antisymmetric A-electron functions (e.g. Slater determinants or CSFs)  

We here assume that the wave function is real. The energy function for this state depends on the numerical parameters C, and is given by 

The first derivatives are the elements of a vector called the electronic gradieru and the second derivatives form a matrix known as the electronic Hessian To obtain the elements of the gradient and the Hessian, it is convenient to rewrite (4.2.12) in the form 

Returning to the variational problem, we note that the conditions for the stationary points 

Assuming that the antisymmetric A-electron functions i) are orthonormal 

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