We have now described a scheme for finding the electronic ground state for a system of atoms within the Born-Oppenheimer approximation. It would also be desirable to know the forces on the ionic cores, the combination of nucleus and core electrons, which for these purposes will be considered as classical point particles. There are two possible ways in which this information could be used. Firstly, the forces could be used to solve the classical equations of motion for the ions, allowing dynamical simulations to be performed. Alternatively, the ground state ionic positions may be found, a procedure known as ionic minimisation. This requires the ionic configuration to be found at which the ionic forces are zero. The latter application is of most importance in the context of this thesis as the true ionic geometry is rarely known a priori for the systems studied.
The ionic forces may be calculated in a straightforward manner at a relatively low cost. The force on an ion in a given direction is given by the derivative of the energy with respect to that degree of freedom,
The total energy may be written,
However, is an Hermitian operator and, when the electrons are in their ground state, is an eigenfunction of this operator,
Thus, Equation becomes
But is simply the normalisation of the wavefunction, a constant, and hence its derivative will be zero. The result of this, known as the Hellmann-Feynman theorem [35, 36], is
which may be easily evaluated.
The solution is not as straightforward when an atom-centred basis set is used. In this case, care must be taken to consider the effects of the change in basis set when the cores move. This requires the calculation of a Pulay correction  which is computationally expensive. The absence of a Pulay correction represents a significant advantage for plane-wave expansion methods in most practical applications.
Once the ionic forces have been obtained, there is a wide range of possible methods to find the minimum energy ionic configuration at which the forces are zero. Both the joint electron-ion relaxation method  and the BFGS scheme  have been used. A comparison between these techniques and a simple steepest descent algorithm may be found in .