The Hartree-Fock-Roothaan Equations for 2n Electrons

As with the pi-electron model we want to treat the orbital coefficients as variation variables and use something like the Clementi-Raimondi-Slater atomic orbitals for the basis functions, or at least something like them which are easy to integrate. We want to minimize the energy by varying the values of the but we also want to maintain the orthonormality of the linear combination of basis functions as orthonormal one-electron orbitals. They are formed from linear combinations of basis 

Thus we use the idea of Lagrangian multipliers to remove from the minimization procedure that part which might change the orthonormal nature of the one-electron orbitals i ,. We therefore need the 

Although the process is for an arbitrary c, ., in effect it removes the summation over p and summations associated with the index for electron i. As with the Hiickel pi-electron derivation, this is the minimization requirement for just one (Cp,) coefficient for the orbital of just one electron (i) and just one equation in the rows of equations that form the system that leads to the Cayley— Hamilton situation. From the pi-electron case we can see that this can be put into a matrix equation where we will need to use diagonalization. 

There remains one very mysterious step. Note the subscripts on y which implies that each orbital vj ,- (which we do not know yet) is dependent somehow on all the other ) , to maintain orthogonality to them as well as normalization. Recall that the big wave function (1, 2, 3.) is a determinant which is a single number at each (x, y, z) point in space and we know some sort of unitary transformation [T] [X,j][T ] = [X ]d,ag could be applied to the orbitals vj (if we knew them ) which would not change the value of the overall wave function, but only make the y interactions diagonal. If we assume this has been done and solve the equation subject to that constraint we will make it tmel Thus we end up with an equation for aU the electrons in a given basis set as a Cayley-Hamilton problem to find the coefficients c ,- and from them find the energy once the coefficients are known. A diagonal X eliminates the Xy for the overlap term and leads to 

Before we go further, note that the secular equation to solve for the Cjj, coefficients involves the C j,j coefficients the answer depends on the answer We wdl need to make a good guess at the orbital coefficients and then iterate the calculation in some way to find convergence  

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