# GPW (GAPW) electronic structure calculations¶

Matt Watkins

School of Mathematics and Physics, University of Lincoln, UK

http://www.cp2k.org

# DFT¶

Why DFT?

• Explicit inclusion of electronic structure
• predictable accuracy (at least when benchmarked for related systems)
• many observables available (spectroscopy)
• cheaper (better scaling) than many wavefunction based approaches
• ab initio molecular dynamics / Monte Carlo

## Hohenberg-Kohn Theorems¶

Theorem 1

• the potential and hence the total energy are a unique functional of the electronic density, $n(\mathbf{r})$
• $n(\mathbf{r}) \leftrightarrow V_{ext}(\mathbf{r},\mathbf{R})$

Theorem 2

• the total energy is variational
• $E[n] \geq E[n_{GS}]$

## Kohn-Sham: non-interacting electrons¶

We map the problem of interacting electrons onto an equivalent one with non-interacting electrons with an altered external potential (xc).

We expand the electron density, $n$, as a sum of one electron orbital contributions

the energy has several components:

$E_{tot}[n] = E_{kin}[n] + E_{ext}[n] + E_{H}[n] + E_{XC}[n]$

• $E_{kin}$ - kinetic energy of electrons
• $E_{ext}$ - energy due to external potential (pseudo potentials, external bias or electric fields ... )
• $E_{H}$ - Hartree energy, classical electrostatic interactions
• $E_{xc}$ - non-classical Coulomb energy: electron correlation

Electronic density

$$n(\mathbf{r}) = \sum_i f_i \mid \psi_i (\mathbf{r}) \mid ^2$$

where $f_i$ is the occupation number of the $i$th orbital

Kinetic energy of non-interacting electrons

$$T_s[n] = \sum_i f_i \big{<} \psi_i (\mathbf{r}) \mid -\frac{1}{2} \nabla^2 \mid \psi_i (\mathbf{r}) \big{>}$$

Electronic interaction with the external potential

$$E_{ext} [n] = \int_r n(\mathbf{r}) V_{ext} (\mathbf{r}) \text{d}\mathbf{r}, V_{ext} = \sum_I -\frac{Z_I}{\mid \mathbf{r} - \mathbf{R}_I \mid}$$

with the exact solution as a Slater determinant of the lowest $N$ orbitals

$$\Psi = \frac{1}{\sqrt{N!}} \text{det} [\psi_1 \psi_2 \psi_3 \cdots \psi_N]$$

## KS energy functional¶

We correct the non-interacting electron model by adding in an (in principle unknown) XC potential that accounts for all quantum mechanical many-body interactions (electron-electron repulsion)

Classical election-electron repulsion

$$J[n] = \frac{1}{2} \int_r \int_{r'} \frac{n(\mathbf{r})n(\mathbf{r'})}{\mid \mathbf{r} - \mathbf{r'}} = \frac{1}{2} \int_r n(\mathbf{r}) V_H(\mathbf{r}) \text{d} \mathbf{r}$$

Kohn-Sham functional

$$E_{KS}[n] = T_s[n] + E_{ext}[n] + J[n] + E_{XC}[n]$$$$E_{XC}[n] = E_{kin}[n] - T_s[n] + E_{ee}[n] - J[n]$$

The exact functional form for the electron-electron repulsion is not known, but various levels of approximation are available (Jacob's Ladder). The existence of this functional is guarenteed by the 1st Hohenberg-Kohn Theroem.

This maps mathematically onto the familiar Hartree-Fock model of electronic structure.

Using the 2nd Hohenberg-Kohn theorem, we find the electron density that minimises the energy of the system. But, like in Hartree-Fock theory, we have to ensure that the electron orbitals are orthonormal to prevent the system imploding.

Orthogonality constraint

$$\Omega_{KS} [\psi_i] = E_{KS}[n] - \sum_{ij} \epsilon_{ij} \int \psi_i^*(\mathbf{r})\psi_j(\mathbf{r})\text{d} \mathbf{r}$$

Variational search in the space of the orbitals

By finding the minimum (extremum) of the $\Omega$ functional $\partial \frac{\Omega_{KS} [\psi_i]}{\partial \psi_i}$ we get the KS equations

$$H_{KS} \psi_i = \big{[} -\frac{1}{2} \nabla^2 + V_{KS} \big{]} \psi_i = \sum_{ij} \epsilon_{ij} \psi_j$$

where $V_{KS}$ is given by

$$V_{KS}(\mathbf{r}) = V_{ext}(\mathbf{r}) + V_{H}(\mathbf{r}) + V_{XC}(\mathbf{r})$$

and, again, all the magic is in $V_{XC}$. The Lagrangian multipliers, $\epsilon_{ij}$ ensure that the orbitals stay orthonormal.

## KS equations¶

$$H_{KS} \psi_i = \big{[} -\frac{1}{2} \nabla^2 + V_{KS} \big{]} \psi_i = \sum_{ij} \epsilon_{ij} \psi_j$$

We can transform the generalised eigenvalue problem given by the KS equations above, to get rid of the nasty sum on the right hand side. In this case $\epsilon_{ij}$ is diagonal, and we get an eigenvalue problem:

$$\big{[} -\frac{1}{2} \nabla^2 + V_{KS} \big{]} \psi_i = \epsilon_i \psi_i$$
• KS equations look like the Schr$\ddot{o}$dinger equations
• coupled and highly non-linear ($V_{KS}$ depends on the orbitals!)
• self consistent solution (Self Consistent Field - the $V_{KS}$ is consistent with the orbitals)
• KS scheme is, in principle, exact (if we knew the exact $E_{XC}$)

## Self-consistency¶

generate an initial guess, from a superposition of atomic densities (typical PW code) or atomic block diagonal density matrix (CP2K)

• generate a starting density $\Rightarrow n^{init}$
• generate the KS potential $\Rightarrow V_{KS}^{init}$
• solve the KS equations $\Rightarrow \epsilon, \psi$

then repeat until convergence

• calculate the new density $\Rightarrow n^{1}$
• calculate the new KS potential $\Rightarrow V_{KS}^{1}$
• new orbitals and energies $\Rightarrow \epsilon^1, \psi^1$

calculate properties from final density and orbitals

Stopping criteria in CP2K are either that the largest derivative of the in energy wrt a MO coefficient ($\frac{\delta E(C)}{\delta C}$) is smaller than EPS_SCF (OT), or the largest change in the density matrix ($P$) is smaller than EPS_SCF (traditional diagonalization).

## Basis Set Representation(s)¶

to actually do calculations we need to expand the KS equations into a basis. In CP2K we use Gaussian functions, as the primary basis set:

the KS orbitals are then represented as

$$\psi_i (\mathbf{r}) = \sum_{\alpha} C_{\alpha i} \phi_{\alpha} (\mathbf{r})$$

the density as

$$n (\mathbf{r}) = \sum_i \sum_{\alpha \beta} f_i C_{\alpha i} C_{\beta i} \phi_{\alpha} (\mathbf{r})\phi_{\beta} (\mathbf{r}) = P_{\alpha \beta} \phi_{\alpha} (\mathbf{r})\phi_{\beta} (\mathbf{r})$$

where the density matrix $\mathbf{P}$ is defined.

There is a complication - the basis functions are not orthogonal. We have an overlap matrix, $\mathbf{S}$,

$$S_{\alpha \beta} = \int_r \phi_{\alpha} (\mathbf{r})\phi_{\beta} (\mathbf{r}) \text{d}\mathbf{r}$$

KS total energy

$$E[\{\psi_i\};\{\mathbf{R}_I\}] = T[\{\psi_i\}] + E_{ext}[n] + E_H [n] + E_{XC}[n] + E_{II}$$

for local of semi-local functionals or

$$E[\{\psi_i\};\{\mathbf{R}_I\}] = T[\{\psi_i\}] + E_{ext}[n] + E_H [n] + E_{XC}[\{\psi_i\}] + E_{II}$$

for hybrid functionals.

Where $E[\{\psi_i\};\{\mathbf{R}_I\}]$ indicates that the energy depends on the (occupied) KS orbitals (which then give a density) and parametrically on all the $I$ nuclear coordinates, $\mathbf{R}_I$.

Matrix formulation of the KS equations

using the variational principle we get

$$\mathbf{K}(C)\mathbf{C} = \mathbf{SC}\epsilon$$ $$\mathbf{K}(C) = \mathbf{T} + \mathbf{V}_{ext} + \mathbf{V}_H(C) + \mathbf{V}_{XC}(C)$$

## Algorithmic challenges¶

• construct the Kohn-Sham matrix
• Hartree potential
• XC potential
• HF/exact exchange
• core/pseudopotential terms
• fast and efficient minimization of the energy functional
• efficient calculation of the density matrix and construction of the Molecular Orbitals ($\mathbf{C}$)

Construction of the KS matrix should be (and is in CP2K) linear scaling (O(N)) in basis set size. Using dual basis set (GPW) techniques - coming up.

Minimization / diagonalization is harder - linear scaling routines are available but are only more efficient for large systems and require considerable computational resources.

# GPW method in CP2K¶

after the general discussion of DFT above, what does CP2K do to solve the KS equations efficiently?

What should you know / look out for, to get good use of the code?

http://www.cp2k.org

## Gaussian's and Plane Wave method (GPW)¶

• localized Gaussians, centred at atomic positions
• plane waves

The idea of GPW is to use the plane-waves as an auxiliary basis, primarily to construct the Hartree potential.

This leads to linear scaling KS construction for Gaussian Type Orbitals (GTO)

## GPW ingredients¶

As the name suggests, this method uses two different types of functions

• Guassian basis sets (many matrix elements can be done analytically)
we go a bit further than implied above - to be more accurate, we contract several Gaussians to form approximate atomic orbitals $$\phi_{\alpha} (\mathbf{r}) = \sum_m d_{m\alpha}g_m(\mathbf{r})$$ where a primitive cartesian Gaussian centred at the origin is given by $$g_m(\mathbf{r}) = x^{m_x}y^{m_y}z^{m_z} e^{-\alpha_m r^2}$$ and $m_x + m_y + m_z = l$, the angular momentum quantum number of the functions.

• Pseudo potentials

• Plane waves auxiliary basis for Coulomb integrals
• Regular grids and FFT for the density
• Sparse matrices (KS and P)
• Efficient screening

### Gaussian Basis set¶

• localised atom-centred GTO basis
$$\phi_{\alpha} (\mathbf{r}) = \sum_m d_{m\alpha}g_m(\mathbf{r})$$
• expansion of the density using the density matrix
$$n (\mathbf{r}) = \sum_{\alpha \beta} P_{\alpha \beta} \phi_{\alpha}^* (\mathbf{r})\phi_{\beta} (\mathbf{r})$$
• operator matrices are sparse. By neglecting overlap elements smaller than some number indirectly specified by EPS_DEFAULT, the Gaussian functions are localized with finite support. This means that S, H and P are sparse.
$$S_{\alpha \beta} = \int_r \phi_{\alpha} (\mathbf{r})\phi_{\beta} (\mathbf{r}) \text{d}\mathbf{r}$$ $$H_{\alpha \beta} = \int_r \phi_{\alpha} (\mathbf{r}) V(\mathbf{r}) \phi_{\beta} (\mathbf{r}) \text{d}\mathbf{r}$$

### Basis set libraries¶

There are two main types of basis sets supplied with CP2K

• GTH_BASIS_SETS: atomically optimized sets. These were the first shipped with the code and vary systematically in quality from DZ to QZ for lighter elements. Can be very good for molecular systems, but can be very bad for condensed matter systems.

• BASIS_MOLOPT: molcularly optimized basis sets. These cover most elements of the periodic table, but only with fairly good quality DZVP-MOLOPT-SR-GTH. Should be a good starting point for most condensed matter calculations.

The basis set files provide the contraction coefficients ($d_{m \alpha}$) and exponents ($\alpha_m$) of the Gaussian functions. $$\phi_{\alpha} (\mathbf{r}) = \sum_m d_{m\alpha}g_m(\mathbf{r})$$$$g_m(\mathbf{r}) = x^{m_x}y^{m_y}z^{m_z} e^{-\alpha_m r^2}$$

Generally the first column is the exponents ($\alpha_m$ above) and the later columns give the $d_{m \alpha}$, each column being a set $\alpha$. Details can be found in the header of the BASIS_MOLOPT file

A MOLOPT basis for nitrogen is

N  DZVP-MOLOPT-SR-GTH DZVP-MOLOPT-SR-GTH-q5
1
2 0 2 5 2 2 1
7.341988051825  0.113789156500  0.077765588400 -0.053744330400 -0.007627243700  0.033688455200
2.542637110957  0.097294516500  0.108655219900 -0.165752516200  0.015163333100  0.109813343200
0.888574967229 -0.445077422600 -0.374125427100 -0.317365165600 -0.129388247500  0.856542971300
0.333802200435 -0.584142233900  0.024021712400 -0.312039675200  0.554905847400  0.509681657500
0.112012109029 -0.139562383500  0.979415132500 -0.117936008100  1.001020469600  0.047030652200


Here there are gaussians with five different exponents ( in Bohr$^{-2}$). From these 5 sets of functions are built, two $s$ functions (2nd and 3rd columns), 2 sets of $p$ functions (4th and fifth columns) and one set of $d$ functions (last column).

Note that the contraction coefficients are not varied during calculation. For the nitrogen basis above we have $2 + 2 \times 3 + 1 \times 5 = 13$ variables to optimize for each nitrogen atom in the system.

(CP2K tends to use general contractions for efficiency, allowing maximum use of recursion relations to generate matrix elements for high angular momentum functions).

Gaussian basis sets are used to analytically calculate the kinetic energy and non-local pseudo-potential terms in the KS matrix, which are the first two terms in $$\mathbf{K}(C) = \mathbf{T} + \mathbf{V}_{ext} + \mathbf{V}_H(C) + \mathbf{V}_{XC}(C)$$

What about the last two terms?

## Multigrids¶

We collocate the Gaussian representation of the density onto the plane-wave basis set.

When we want to put (collocate) a Gaussian type function onto the realspace grid, we can gain efficiency by using multiple grids with differing cutoff / spacing.

More diffuse Gaussians can be collocated onto coarser grids specified in input as:


&MGRID
CUTOFF 400
REL_CUTOFF 60
NGRIDS 5
&END MGRID


## Electrostatics¶

• long-range term: Non-local Hartree Potential
Poisson equation solved using the auxiliary plane-wave basis $$E_H[n_{tot}] = \frac{1}{2} \int_r \int_{r'} \frac{n(\mathbf{r})n(\mathbf{r'})}{\mid \mathbf{r} - \mathbf{r'}}$$ where $n_{tot}$ includes the nuclear charge as well as the electronic.
(The nuclear charge density is (of course) represented as a Gaussian distribution with parameter $R_I^c$ chosen to cancel a similar term from the local part of the pseudopotential)

• FFT (scaling as $N\text{log}N$) gives $$\tilde{n}(\mathbf{r}) = \frac{1}{\Omega} \sum_G \tilde{n}(\mathbf{G})e^{i\mathbf{G \cdot r}}$$ In the G space representation the Poisson equation is diagonal and the Hartree energy is easily evaluated $$E_H[n_{tot}] = 2 \pi \Omega \sum_G \frac{\tilde{n}(\mathbf{G}) ^* \tilde{n}(\mathbf{G}) }{\mathbf{G}^2}$$

• The electrostatic energy converges rapidly with the grid spacing / planewave cutoff

• ## Real space integration¶

Finite cutoff and simulation box define a realspace grid

• density collocation

$$n(\mathbf{r}) = \sum_{\alpha \beta} P_{\alpha \beta} \phi_{\alpha}^* (\mathbf{r})\phi_{\beta} (\mathbf{r}) \rightarrow \sum_{\alpha \beta} P_{\alpha \beta} \bar{\phi}_{\alpha \beta} (\mathbf{r}) = n(\mathbf{R})$$ where $n(\mathbf{R})$ is the density at grid points in the cell
• numerical approximation of the gradient $n(\mathbf{R}) \rightarrow \nabla n(\mathbf{R})$

• $\epsilon_{xc}$ and derivatives evaluated on the grid $$v_{XC}[n](\mathbf{r}) \rightarrow V_{XC}(\mathbf{R}) = \frac{\partial \epsilon_{xc}}{\partial n} (\mathbf{R})$$

• real space integration $$H_{HXC}^{\mu \nu} = < \mu \mid V_{HXC} (\mathbf{r}) \mid \nu > \rightarrow \sum_R V_{HXC} (\mathbf{R}) \bar{\phi}_{\alpha \beta} (\mathbf{R})$$

## Energy ripples¶

Low density regions can cause unphysical behaviour of $XC$ terms (such as $\frac{\mid \nabla n \mid ^2}{n^{\alpha}}$)

GTH pseudos have small density at the core - graph of density and $v_{XC}$ through a water molecule. These spikes can cause ripples in the energy as atoms move relative to the grid.

There are smoothing routines &XC_GRID / XC_DERIV, that can help get consistent forces for AIMD. But probably best to stick with the defaults if you are performing a static calculation. Whatever you do don't change settings between simulations you want to compare.

You can minimize these effects with a higher cutoff, or using the GAPW methodology.

These ripples (egg-box effect) can be very problematic when trying to calculate vibrational frequencies.

## Multigrids¶

you can see in the output


-------------------------------------------------------------------------------
----                             MULTIGRID INFO                            ----
-------------------------------------------------------------------------------
count for grid        1:          24832          cutoff [a.u.]          200.00
count for grid        2:          11934          cutoff [a.u.]           66.67
count for grid        3:           6479          cutoff [a.u.]           22.22
count for grid        4:           1976          cutoff [a.u.]            7.41
count for grid        5:            215          cutoff [a.u.]            2.47
total gridlevel count  :          45436

For this system (formaldeyhde with an aug-TZV2P-GTH basis) that 45436 density matrix elements ($\bar{\phi}_{\alpha \beta}$) were mapped onto the grids. To be efficient, all grids should be used. To fully converge calculations both ''CUTOFF'' and ''REL_CUTOFF'' need to be increased together otherwise the highest grid may not be used.

## Timings¶

At the end of the run you'll see timings - these can be very useful for understanding performance. Here we can see the timings for the formaldehyde example. It is in a large box, and the operations on grids dominate the cost on a single processor.


-------------------------------------------------------------------------------
-                                                                             -
-                                T I M I N G                                  -
-                                                                             -
-------------------------------------------------------------------------------
SUBROUTINE                       CALLS  ASD         SELF TIME        TOTAL TIME
MAXIMUM       AVERAGE  MAXIMUM  AVERAGE  MAXIMUM
CP2K                                 1  1.0    0.013    0.013   24.048   24.048
qs_energies                          1  2.0    0.000    0.000   23.417   23.417
scf_env_do_scf                       1  3.0    0.000    0.000   22.962   22.962
scf_env_do_scf_inner_loop           14  4.0    0.001    0.001   22.962   22.962
qs_ks_update_qs_env                 14  5.0    0.000    0.000   19.555   19.555
rebuild_ks_matrix                   14  6.0    0.000    0.000   19.554   19.554
qs_ks_build_kohn_sham_matrix        14  7.0    0.002    0.002   19.554   19.554
qs_vxc_create                       14  8.0    0.000    0.000    8.977    8.977
xc_vxc_pw_create                    14  9.0    0.273    0.273    8.977    8.977
xc_rho_set_and_dset_create          14 10.0    0.086    0.086    8.147    8.147
pw_transfer                        203  9.2    0.009    0.009    7.539    7.539
fft_wrap_pw1pw2                    203 10.2    0.001    0.001    7.530    7.530
pw_poisson_solve                    14  8.0    0.458    0.458    7.457    7.457
xc_functional_eval                  14 11.0    0.000    0.000    7.395    7.395
pbe_lda_eval                        14 12.0    7.394    7.394    7.394    7.394
fft_wrap_pw1pw2_200                 87 10.8    0.400    0.400    7.147    7.147
fft3d_s                            204 12.1    4.799    4.799    4.805    4.805
qs_rho_update_rho                   15  5.0    0.000    0.000    3.631    3.631
calculate_rho_elec                  15  6.0    0.711    0.711    3.631    3.631
ps_wavelet_solve                    14  9.0    3.418    3.418    3.496    3.496
density_rs2pw                       15  7.0    0.001    0.001    2.882    2.882
sum_up_and_integrate                14  8.0    0.040    0.040    1.917    1.917
integrate_v_rspace                  14  9.0    0.378    0.378    1.877    1.877
potential_pw2rs                     14 10.0    0.005    0.005    1.498    1.498
pw_gather_s                        104 11.5    1.274    1.274    1.274    1.274
pw_scatter_s                        99 12.8    1.028    1.028    1.028    1.028
pw_nn_compose_r                     84 10.5    0.968    0.968    0.968    0.968
pw_poisson_rebuild                  14  9.0    0.000    0.000    0.897    0.897
ps_wavelet_create                    1 10.0    0.000    0.000    0.897    0.897
RS_z_slice_distribution              1 11.0    0.897    0.897    0.897    0.897
qs_init_subsys                       1  2.0    0.001    0.001    0.567    0.567
qs_env_setup                         1  3.0    0.000    0.000    0.563    0.563
qs_env_rebuild_pw_env                2  3.5    0.000    0.000    0.563    0.563
pw_env_rebuild                       1  5.0    0.000    0.000    0.563    0.563
pw_grid_setup                        5  6.0    0.000    0.000    0.516    0.516
pw_grid_setup_internal               5  7.0    0.012    0.012    0.516    0.516
-------------------------------------------------------------------------------


## Warnings¶

Also check if you get an output like


The number of warnings for this run is : 0

if there are warnings, you should try and understand why!

## GAPW¶

• The Gaussian and Augmented Plane Wave method works similarly to the GPW method. But Gaussians with larger exponents are not mapped onto the grids, instead onto an extra set of atom centered radial grids (there is more to it that this).
• This allows core electrons to be represented and all-electron calculations to be run
• It also allows a smaller plane-wave cutoff to be used.
• Activate GAPW with the METHOD GAPW keywords in the &QS section.
• To run all-electron use POTENTIAL ALL in the &KIND SECTION and an all-electron basis.