Single Point Energy Calculations

A single point energy (SPE) calculation calculates the wavefunction
and charge density, and hence the energy, of a particular (arbitrary)
arrangement of nuclei. The total energy functional of a system of electrons
and nuclei can be written as

The minimization of the total energy functional can be done in a number of ways. One method proceeds by direct diagonalization of the matrix equation (2.30). Starting from an initial trial density is calculated and inserted into (2.30). A new density is then calculated by inverting the matrix equation (2.30). If this new density gives an energy that is consistent with the old density (if the change in energy between iterations is smaller than a given tolerance) it is then inserted into the total energy functional and the energy calculated. Otherwise this new density is used to calculate a new . This is repeated until the density and potential are consistent with each other, within a given tolerance.

The matrix diagonalization method has the disadvantage that the
computational cost of matrix diagonalization scales as the number of plane
waves cubed. An alternative method is to minimize the energy functional
directly [46]. The energy functional is a functional of the
density, which is determined by the expansion coefficients
. The ground state density is found from the set
of
that minimize the energy functional.
Standard functional optimization methods [47] can then be
used to find the minima of the total energy functional. The simplest of these
techniques is the method of * Steepest Descents* (SD)
[47]. The SD method produces a series of points

In this case are sets of plane wave expansion coefficients and is the energy functional. The scalar in (2.33) is the distance along the direction from that a minima is located. The SD method proceeds by moving in the steepest downhill direction from a point until a minima along that direction is located at point . The steepest downhill direction from is then determined ( ) and the minima located along that direction is found. This is repeated until the change in the function is lower than a preset tolerance. The speed at which the SD method will find a minima is limited as at each step only the information at that point is taken into account. It is easy to think of examples where this will lead to slow convergence. A better method is the

where is the Hessian given by

The CG method produces a series of points as in (2.33) but is given by

where . Using the quadratic form (2.35) the step needed to get to the minima in the direction is given by [47]

This however can be found from a one-dimensional minimization along , i.e. without explicitly calculating the Hessian. As the Hessian is an matrix, for DFT calculations with plane waves the CG method has a large advantage over methods that explicitly use the Hessian. The CG method will find a minimum of an dimensional function in iterations. When using the CG and SD methods to minimize the total energy functional , the requirement that the wavefunctions are to be orthonormal places an additional constraint on the minimzation. For a DFT calculation with 10 plane waves this is still a monumental task. The number of iterations required can be substantially reduced by preconditioning the function [46].