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Subsections

# 7.1 Total energy and Hamiltonian

A quantity which is frequently required is the overlap matrix defined by

 (7.3)

The overlap matrix elements between the spherical-wave basis functions can be calculated analytically (section 5.4), and are denoted where
 (7.4)

so that the overlap matrix elements are given by
 (7.5)

recalling that the support functions may be assumed real in the case of -point Brillouin zone sampling.

## 7.1.1 Kinetic energy

The kinetic energy of the non-interacting Kohn-Sham system is given by

 (7.6)

in which
 (7.7)

are the matrix elements of the kinetic energy operator in the representation of the support functions. Since all of the matrix elements between the spherical-wave basis functions can be calculated analytically,
 (7.8)

where denotes matrix elements of the kinetic energy operator between spherical-wave basis functions.

## 7.1.2 Hartree energy and potential

The Hartree and exchange-correlation terms are calculated by determining the electronic density on a real-space grid , and Fast Fourier Transforms (FFTs) are used to transform between real- and reciprocal-space7.1to obtain . The Hartree energy is then given by

 (7.9)

where is the volume of the supercell and the (infinite) term is omitted because the system is charge neutral overall. This term is therefore cancelled by similar terms in the ion-ion and electron-ion interaction energies. The Hartree potential in real-space is given by
 (7.10)

but is calculated in reciprocal-space as
 (7.11)

and then transformed back into real-space by a FFT.

## 7.1.3 Exchange-correlation energy and potential

Having calculated the electron density on the grid points, the exchange-correlation energy is obtained by summing over those grid points

 (7.12)

in the local density approximation. is the volume of the supercell divided by the number of grid points. The exchange-correlation potential is similarly calculated at each grid point as
 (7.13)

In practice, the values of and are tabulated for various values of the electronic density and then interpolated during the calculation.

## 7.1.4 Local pseudopotential

Like the Hartree potential, the local pseudopotential is also calculated in reciprocal-space as

 (7.14)

where the summation is over ionic species , is the local pseudopotential for an isolated ion of species in reciprocal-space and is the structure factor for species defined by
 (7.15)

where the sum is over all ions of species with positions . We note that in general the calculation of the structure factor is an operation, but since it only has to be calculated once for each atomic configuration, it is not a limiting factor of the overall calculation at this stage. Within the quantum chemistry community, work on generalised multipole expansions and new algorithms [156,157,158,159,160,161,162] has led to the development of methods to calculate Coulomb interaction matrix elements which scale linearly with system-size. The local pseudopotential energy can be calculated in reciprocal-space as
 (7.16)

where is the pseudopotential core energy, and the number of ions of species . The matrix elements are defined by
 (7.17)

The Hartree potential and local pseudopotential can be summed and then transformed back together into real-space and added to the exchange-correlation potential to obtain the local part of the Kohn-Sham potential in real-space.

We note that the FFT is not strictly an operation but an operation (where is some small number which depends upon the prime factors of the number of grid points), but in practice (section 9.2) this scaling is not observed.

## 7.1.5 Non-local pseudopotential

The non-local pseudopotential energy is given by

 (7.18)

The matrix elements of the non-local pseudopotential in the representation of the support functions are calculated by summing over all ions whose cores overlap the support regions of and , and using the method described in section 5.6.2 to calculate the spherical-wave basis function matrix elements analytically. The result is therefore
 (7.19)

which is of exactly the same form as the kinetic energy, so that in practice the basis function matrix elements for the kinetic energy and non-local pseudopotential are summed and the two contributions to the energy combined.

Next: 7.2 Energy gradients Up: 7. Computational implementation Previous: 7. Computational implementation   Contents
Peter Haynes