Further examples of penalty functionals

Before studying the efficiency of the minimisation procedure when applied to the total functional described above in section 6.2.1, we mention two further examples of penalty functionals which are suitable for this approach.

The first is applicable for positive semi-definite trial density-matrices. This requirement can be satisfied in practice by writing the density-kernel $K$ in terms of an auxiliary matrix $T$ as $K = T T^{\dag }$ (see section 4.4.2). Since the eigenvalues of such a density-matrix must be non-negative, variation of the energy functional alone is sufficient to drive the occupation numbers of unoccupied bands to zero, and the penalty functional need only impose the occupation numbers of the occupied bands to lie close to unity. An appropriate penalty functional is then

$$P[{\rho}] = \int {\mathrm d}{\bf r} \left[ {\rho} \left( 1 -... ...ght] ({\bf r},{\bf r}) = \sum_i f_i \left( 1 - f_i \right)^2 ,$$ (6.26)

and the corresponding energy correction is

$$E_0 \approx E[{\bar \rho}] - \alpha \sum_i^{\rm all} (1 - 3 ... ... \alpha \sum_i^{\rm unocc} (1 - 3{\bar f}_i)(1 - {\bar f}_i) .$$ (6.27)

Numerical investigation has shown that the occupation numbers of the unoccupied bands do indeed become very small but positive when this scheme is used.

The second penalty functional is applicable only when no unoccupied bands are included in the calculation. In this case, all of the occupation numbers should equal unity and so an appropriate penalty functional is

$$P[{\rho}] = \int {\mathrm d}{\bf r} \left( 1 - {\rho} \right)^2 ({\bf r},{\bf r}) = \sum_i \left( 1 - f_i \right)^2 .$$ (6.28)

The corresponding correction to the total energy in this case is

$$E_0 \approx E[{\bar \rho}] + 2 \alpha \sum_i^{\rm all} (1 - {\bar f}_i)^2 .$$ (6.29)

Both of these penalty functionals have been tested, and the results are very similar to those presented for the original functional in the previous section. These penalty functionals are plotted in figure 6.9.

Figure 6.9: Two further examples of analytic penalty functionals.