Total energy optimisation

(39) |

This minimisation must be performed under the constraints of constant electron number,

where the overlap matrix, is given by

In order to avoid explicitly imposing the idempotency constraint (41), we use the method suggested by Li, Nunes and Vanderbilt [21] and independently by Daw [22], and generalised to the case of non-orthogonal functions by Nunes and Vanderbilt [23]. Our implementation follows the simplified version of Millam and Scuseria [24]. We define the following function of the density kernel :

(43) |

The contravariant, tensor-corrected gradient [15,16] that is used in the steepest descent or conjugate gradient iterative minimisation is given by

(45) |

The value of is set at each step such that . This ensures that the total electron number remains unchanged, thus we simply require that our initial guess for the density kernel gives the correct number of electrons.