Density Functional Theory Implementation in CP2K

combines the advantages of two representations of density in terms of Gaussians and plane waves using the Gaussian and plane wave method (GPW)95 _{or the Gaussian Augmented Plane} wave method (GAPW)96 _{where augmented plane waves are used.}

2.11.7.1 Gaussian and Plane Wave Method

The starting point for the GAPW method is the hybrid Gaussian and Plane Wave (GPW) method.95 _{The GPW method uses an atom-centred Gaussian type basis to describe the wave} function and plane wave basis to describe the density. Using a dual representation allows for an efficient treatment of the electrostatic interactions, and results in a scheme which has linear scaling cost for the computation of the total energy with respect to the number of atoms in a system.94

The representation of the electron density 𝜌(𝐫) can be written in terms of Gaussian functions: 𝜌(𝐫) = ∑ 𝑃𝑖𝑗𝜑𝑖(𝐫)

𝑖𝑗

42 where 𝑃𝑖𝑗 is the density matrix element in the atomic basis, and 𝜑𝑖(𝐫) are contracted Gaussian

basis functions. Core electrons are represented using pseudopotentials such as GTH pseudopotentials described in Section 2.6.2.

The representation of the electron density in plane waves is given by: 𝜌̃(𝐫) = 1

Ω∑ 𝜌̃(𝐆)𝑒

(𝑖𝐆 .𝐫) 𝐆

𝐄𝐪. 𝟐. 𝟒𝟔

where Ω is the volume of the unit cell, and 𝐆 are the reciprocal lattice vectors. The expansion coefficients 𝜌̃(𝑮) are such that 𝜌(𝐫) is equal to 𝜌̃(𝐫) on a regular grid in a unit cell. Conversion between the two representations is achieved by expressing Gaussians numerically on a real space grid and the use of Fast Fourier Transforms (FFTs).

The Kohn-Sham DFT energy expression as employed within the GPW framework is defined in 𝐄𝐪. 𝟐. 𝟑𝟐. All integrals and derivatives are computed using Gaussian basis functions, except for the integrals of the Coulomb term 𝐽[𝜌] which is evaluated using plane waves according to 𝐄𝐪. 𝟐. 𝟒𝟔. ΕXC[𝜌] is approximated by functionals such as those described in Section 2.11.4.

The GPW method combines the benefits of Gaussians and plane waves as basis sets. The use of FFTs to convert between real and reciprocal space makes the algebraic manipulations simple in the plane wave basis set. However, a combination of high energy plane waves is needed to reproduce the wavefunction close to the nuclei to compute the Coulomb term. While this is alleviated by the use of pseudopotentials it can lead to unreasonably large basis sets with some elements. In regions of empty space, a plane wave basis set needs to represent empty space with the same accuracy as the atom filled region which can be very computationally demanding. To remedy these deficiencies while preserving the advantages the Gaussian Augmented Plane wave (GAPW) method can be used.95,96

2.11.7.2 Gaussian Augmented Plane Wave Method

The GAPW method96 _{as used in this thesis extends the GPW method using augmented plane} waves instead of pure plane waves to describe the electron density. The GPW method assumes that the plane wave cutoff is high enough that the plane wave basis is sufficiently large to describe the electron density correctly. However, even with the use of pseudopotentials, the cutoff needed for a converged calculation can lead to an unfavourably high number of FFT grid points. The use of augmented plane waves in the GAPW method reduces the energy cutoff needed, and as a result reduces the number of FFT grid points needed for the plane wave representation of the density.

43 The electron density of a molecular system has different characteristics depending on the region in space. 𝑈𝐴 is the spherical atomic region around nucleus 𝐴 where the density is

strongly varying. In the interstitial region, I, away from the nucleus the density varies slowly. It can be assumed the total density is the sum of three contributions:

𝜌(𝐫) = 𝜌̃(𝐫) − 𝜌̃1_{(𝐫) + 𝜌}1_{(𝐫) 𝐄𝐪. 𝟐. 𝟒𝟕}

where 𝜌̃(𝐫) is smoothed and distributed over all space, and 𝜌1_{(𝐫) and 𝜌̃}1_{(𝐫) are the sums of}

atom-centred contributions, 𝜌𝐴1(𝐫) and 𝜌̃𝐴1(𝐫), which are hard and soft respectively.

𝜌1(𝐫) = ∑ 𝜌𝐴1(𝐫) 𝐴 𝐄𝐪. 𝟐. 𝟒𝟖 𝜌̃1_{(𝐫) = ∑ 𝜌̃} 𝐴 1_(𝐫) 𝐴 𝐄𝐪. 𝟐. 𝟒𝟗

It is assumed the difference between 𝜌𝐴1 and 𝜌̃𝐴1 is zero outside 𝑈𝐴. For 𝑈𝐴 it is assumed that

the atomic regions around different atoms do not overlap. Inside 𝑈𝐴, the soft density, 𝜌̃ is

equal to its atom-centred contribution 𝜌̃1_:

𝜌̃(𝐫) = 𝜌̃1(𝐫) for 𝐫 ∈ 𝑈𝐴 𝐄𝐪. 𝟐. 𝟓𝟎

Outside the atomic regions in the interstitial region, 𝐼, it is assumed that the soft density is equal to the total density:

𝜌̃(𝐫) = 𝜌(𝐫) for 𝐫 ∈ 𝐼 𝐄𝐪. 𝟐. 𝟓𝟏

These requirements lead to the four assumptions made to set up the GAPW representation of the electron density:

𝜌(𝐫) − 𝜌̃(𝐫) = 0 for 𝒓 ∈ 𝐼 𝐄𝐪. 𝟐. 𝟓𝟐

𝜌𝐴1(𝐫) − 𝜌̃𝐴1(𝐫) = 0 for 𝒓 ∈ 𝐼 𝐄𝐪. 𝟐. 𝟓𝟑

𝜌̃(𝐫) − 𝜌̃𝐴1(𝐫) = 0 for 𝒓 ∈ 𝑈𝐴 𝐄𝐪. 𝟐. 𝟓𝟒

𝜌(𝐫) − 𝜌_𝐴1_{(𝐫) = 0 for 𝒓 ∈ 𝑈}

𝐴 𝐄𝐪. 𝟐. 𝟓𝟓

Partitioning the density in this way means that the Coulomb and exchange correlation terms can be calculated independently for 𝜌̃, 𝜌1 _{and 𝜌̃}1_{. The Coulomb potential is separated into}

smooth non-local contributions expanded in planewaves and local contributions described with Gaussians and treated analytically.

44 The exchange correlation functional for the GAPW method can be written as:

ΕXC[𝜌] = ΕXC[𝜌̃] − ∑ ΕXC[𝜌̃𝐴1] + ∑ ΕXC[𝜌𝐴1] 𝐴

𝐴

𝐄𝐪. 𝟐. 𝟓𝟔

The first term can be evaluated according using plane waves as in the GPW method, and the last two terms are evaluated using atom centred expanded Gaussians.

The GAPW method maintains the advantages of the plane basis sets, including periodic boundary conditions naturally but reducing the planewave cutoff needed to converge the system which reduces the computational cost of calculations.