Orbitals and the Secular Equation

Using the cf) basis functions are assumed to be fixed functions and only the 

In order to evaluate the energy as a function of the unknown coefficients Cj we set up the expectation process for the energy, but actually we will find N different energies, e which are the one-electron orbital energies. In this simple model the total energy will be the simple sum of the orbital energies but in more sophisticated treatments this is not exactly tme. For this model 

Then Yli Ey ciCj H jd r = ( ) 14 T t T and next we take the derivative with respect 

Suppose we seek the derivative of an expression of weighted elements of a Hermitian matrix, where Ay = Aji (all real) with respect to one coefficient, say C2, where Ay could be Hy or Sy. 

This is a linear system of equations in n variables where the unknown variables are the c, coefficients. Formally, this calls into effect the Cayley-Hamilton theorem [6] because the right-hand sides of the equations are all zero. The Cayley-Hamilton theorem [6] states that a square matrix. A, satisfies its characteristic equation and if we have a characteristic polynomial of the eigenvalues of the matrix 

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