Ensuring that the HF energy is a minimum and the correct minimum

The standard iterative procedure produces a solution where the variation of the HF energy is stationary with respect to all orbital variations, i.e. the first derivatives of the energy with respect to the MO coefficients are zero. In order to ensure that this corresponds to an energy minimum, the second derivatives should also be calculated.24 This is a matrix the size of the number of occupied MOs multiplied by the number of virtual MOs (identical to that arising in quadratic convergent SCF methods (Section 3.8.1)), and the eigenvalues of this matrix should all be positive in order to be an energy minimum. Of course only the lowest eigenvalue is required to probe whether the solution is a minimum. A negative eigenvalue means that it is possible to get to a lower energy state by "exciting" an electron from an occupied to an unoccupied orbital, i.e. the solution is unstable. In practice, the stability is rarely checked - it is assumed that the iterative procedure has converged to a minimum. It should be noted that a positive definite second-order matrix only ensures that the solution is a local minimum; there may be other minima with lower energies.

The problem of convergence to saddle points in the wave function parameter space and the existence of multiple minima is rarely a problem for systems composed of elements from the first two rows in the periodic table. For systems having more than one metal atom with several partially filled d-orbitals, however, care must be taken to ensure that the iterative procedure converges to the desired solution. Consider for example the Fe2S2 system in Figure 3.6, where the d-electrons of two Fe atoms are coupled through the sulfur bridge atoms.

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