Kinetic Energy Functionals

Kinetic Energy Functional Template Class

class functionals.KineticFunctional

Template kinetic functional class

This is a class that represents a kinetic functional which can be useful for functionals that require an initialized kernel to be saved for repeated use via interpolation or otherwise. This class is also useful for functionals with tunable parameters (e.g. machine-learned functionals) as it allows for Pytorch-based optimization of those parameters.

__init__(init_args=None)

Parameters:

init_args (tuple or None) – Functional parameters for initialization

initialize()

Sets the optimizer (default is Rprop) and make all functional parameters have requires_grad = False.

set_device(device=device(type='cpu'))

Moves all functional parameter tensors to the specified device. By default, the device is set to a GPU if is avaliable, otherwise the device is set as a CPU.

Parameters:

device (torch.device) – Device

param_grad(requires_grad=True)

Sets whether the functional parameters’ requires_grad is True or False.

Parameters:

requires_grad (boo) – Whether the functional parameters’ requires_grad is True or False

save(PATH)

Saves the functional parameters in a file.

Parameters:

PATH (string) – Path to the file where the parameters are saved

classmethod load(PATH)

Loads the functional parameters from a file.

Parameters:

PATH (string) – Path to the file where the parameters to be loaded are

grid_error(target, prediction, norm=False)

Utility function to compute errors on a grid.

Parameters:

• target (torch.Tensor) – The expected result

• prediction (torch.Tensor) – Prediction from the functional

• norm (boo) – Whether to normalize the error by the range of the target

Returns:

Mean error

Return type:

torch.Tensor

scalar_error(target, prediction)

Utility function to compute scalar errors.

Parameters:

• target (torch.Tensor) – The expected result

• prediction (torch.Tensor) – Prediction from the functional

Returns:

Relative error

Return type:

torch.Tensor

update_params(loss)

Updates the functional parameters based on a loss function to be minimized.

Parameters:

loss (torch.Tensor) – Quantity to be minimized

Exact Kinetic Energy Functionals

Thomas-Fermi Functional

functionals.ThomasFermi(box_vecs, den)

Thomas-Fermi functional

The Thomas-Fermi functional is exact for the free electron gas and can be considered the local density approximation (LDA) for non-interacting kinetic energy functionals. It is given by

\[T_\text{TF}[n] = \int d^3\mathbf{r} ~\frac{3}{10} (3\pi)^{2/3} n^{5/3}(\mathbf{r})\]

Parameters:

• box_vecs (torch.Tensor) – Lattice vectors

• den (torch.Tensor) – Electron density

Returns:

Thomas-Fermi kinetic energy

Return type:

torch.Tensor

von Weizsaecker Functional

functionals.Weizsaecker(box_vecs, den)

von Weizsaecker functional

The von Weizsaecker functional is exact for single-orbital systems It is given by

\[T_\text{vW}[n] = \int d^3\mathbf{r} ~\frac{1}{8} \frac{|\nabla n(\mathbf{r})|^2}{n(\mathbf{r})}\]

Parameters:

• box_vecs (torch.Tensor) – Lattice vectors

• den (torch.Tensor) – Electron density

Returns:

von Weizsaecker kinetic energy

Return type:

torch.Tensor

Semi-local Kinetic Energy Functionals

Semi-local functionals generally take the form

\[T_\text{S}[n] = T_\text{vW}[n] + \int d^3\mathbf{r} ~ F_\theta[n](\mathbf{r}) \tau_\text{TF}(\mathbf{r})\]

where \(T_\text{vW}[n]\) is the von Weizsaecker kinetic energy functional, \(\tau_\text{TF}(\mathbf{r})\) is the Thomas-Fermi kinetic energy density and \(F_\theta[n](\mathbf{r})\) is the Pauli enhancement factor to be approximated.

Often, the Pauli enhancement factor is made a function of the reduced gradient,

\[s = \frac{|\nabla f|}{2(3\pi)^{1/3} n^{4/3}}\]

and the reduced Laplacian,

\[q = \frac{\nabla^2 f}{4(3\pi)^{2/3} n^{5/3}}\]

vWGTF Functionals

functionals.vWGTF1(box_vecs, den)

vWGTF1 functional

The vWGTF1 functional [Phys. Rev. B 91, 045124] has a Pauli enhancement factor given by

\[F_\theta^\text{vWGTF1}(d) = 0.9892 \cdot d^{-1.2994}\]

where \(d = n/n_0\) for average density \(n_0\).

Parameters:

• box_vecs (torch.Tensor) – Lattice vectors

• den (torch.Tensor) – Electron density

Returns:

vWGTF1 kinetic energy

Return type:

torch.Tensor

functionals.vWGTF2(box_vecs, den)

vWGTF2 functional

The vWGTF2 functional [Phys. Rev. B 91, 045124] has a Pauli enhancement factor given by

\[F_\theta^\text{vWGTF2}(d) = \sqrt{\frac{1}{\text{ELF}} - 1}\]

with the electron localization function (ELF) parameterized as

\[\text{ELF} = \frac{1}{2} \left(1 + \tanh(5.7001 \cdot d^{0.2563} - 5.7001) \right)\]

where \(d = n/n_0\) for average density \(n_0\).

Parameters:

• box_vecs (torch.Tensor) – Lattice vectors

• den (torch.Tensor) – Electron density

Returns:

vWGTF2 kinetic energy

Return type:

torch.Tensor

Luo-Karasiev-Trickey (LKT) Functional

functionals.LuoKarasievTrickey(box_vecs, den)

Luo-Karasiev-Trickey (LKT) functional

The Luo-Karasiev-Trickey (LKT) GGA kinetic functional [Phys. Rev. B 98, 041111(R)] has a Pauli enhancement factor given by

\[F_\theta^\text{LKT}(s) = \frac{1}{\cosh(1.3 s)}\]

Parameters:

• box_vecs (torch.Tensor) – Lattice vectors

• den (torch.Tensor) – Electron density

Returns:

LKT kinetic energy

Return type:

torch.Tensor

Pauli-Gaussian Functionals

class functionals.PauliGaussian

Pauli-Gaussian functional

The Pauli-Gaussian class of GGA kinetic energy functionals [J. Phys. Chem. Lett. 2018, 9, 15, 4385–4390, J. Chem. Theory Comput. 2019, 15, 5, 3044–3055] have Pauli enhancement factors with the form

\[F_\theta^\text{PG}(s,q) = e^{-\mu s^2} + \beta q^2 - \lambda q s^2 + \sigma s^4\]

__init__(init_args=None)

Parameters:

init_args (tuple) – \((\mu,~\beta,~\lambda,~\sigma)\) where each parameter is a float. These are key parameters of the Pauli-Gaussian functionals. The default parameters are that of the PGSL0.25 functional’s, \((\mu,~\beta,~\lambda,~\sigma) = (40/27,~0.25,~0,~0)\).

set_PG1()

Set \((\mu,~\beta,~\lambda,~\sigma) = (1,~0,~0,~0)\)

set_PGS()

Set \((\mu,~\beta,~\lambda,~\sigma) = (40/27,~0,~0,~0)\)

set_PGSL025()

Set \((\mu,~\beta,~\lambda,~\sigma) = (40/27,~0.25,~0,~0)\)

set_PGSLr()

Set \((\mu,~\beta,~\lambda,~\sigma) = (40/27,~0.25,~0.4,~0.2)\)

forward(box_vecs, den)

Parameters:

• box_vecs (torch.Tensor) – Lattice vectors

• den (torch.Tensor) – Electron density

Returns:

Pauli-Gaussian kinetic energy

Return type:

torch.Tensor

Yukawa GGA Functionals

class functionals.YukawaGGA

Yukawa GGA functional

The Yukawa GGA class of kinetic energy functionals [Phys. Rev. B 103, 155127, Computation 2022, 10(2), 30] have Pauli enhancement factors \(F_\theta(y, s^2, q)\) that depend on the Yukawa potential term,

\[y_{\alpha \beta}(\mathbf{r}) = \frac{3\pi\alpha^2}{4 k_F(\mathbf{r}) n^{\beta-1}(\mathbf{r})} \int d^3\mathbf{r}' \frac{n^\beta(\mathbf{r}') e^{-\alpha k_F(\mathbf{r}) |\mathbf{r}-\mathbf{r}'|}}{|\mathbf{r}-\mathbf{r}'|}\]

where \(k_F(\mathbf{r}) = [3\pi^2 n(\mathbf{r})]^{1/3}\), as an ingredient besides the reduced gradient and laplacian.

__init__(init_args=None)

Parameters:

init_args (tuple) – \((\alpha,~\beta,~F_\theta,~\kappa)\) where (alpha,~beta) are key parameters of the Yukawa GGA functional, \(F_\theta\) is the Pauli enhancement factor that takes as arguments the Yukawa descriptor, reduced gradient and reduced laplacian, i.e. \(F_\theta(y, s^2, q)\), and \(\kappa\) is a parameter for the spline-based field dependent convolution. Note that a geometric progression based spline is used so \(\kappa > 1\) The default parameters are that of the Yuk1 functional’s, \((\mu,~\beta,~f,~\kappa) = (1,~1,~y,~1.2)\).

set_yuk1()

Set \((\alpha,~\beta) = (1,~1)\) and

\[F_\theta(y,~s^2,~q) = y\]

set_yuk2()

Set \((\alpha,~\beta) = (1.36297,~1)\) and

\[F_\theta(y,~s^2,~q) = y~ \left(1 + \frac{40}{27} (q-s^2) \right)\]

Warning

Unstable for density optimizations as it violates Pauli positivity

set_yuk3(a=4)

Set \((\alpha,~\beta) = (1.36297,~1)\) and

\[F_\theta(y,~s^2,~q) = y~ T_a \left(-\frac{40}{27} (q-s^2) \right)\]

where

\[T_a(x) = \frac{4}{a} \frac{e^{ax}}{e^{ax} + 1} + \frac{a-2}{a}.\]

set_yuk4(a=3.3)

Set \((\alpha,~\beta) = (1.36297,~1)\) and

\[F_\theta(y,~s^2,~q) = y~ T_a \left(-\frac{40}{27} s^2 \right) T_2\left(-\frac{40}{27} q \right)\]

where

\[T_a(x) = \frac{4}{a} \frac{e^{ax}}{e^{ax} + 1} + \frac{a-2}{a}.\]

set_yuk2beta(alpha, beta)

Set \((\alpha,~\beta)\) according to user input and

\[F_\theta(y,~s^2,~q) = 1 - G_0 + y(G_0 + G)\]

where

\[G_0 = \frac{\alpha^2 (\alpha^2 - 60)}{108 \beta (9\beta - 10)}\]

and

\[G = \left(\frac{40}{27\beta} - \frac{4}{\alpha^2} (\beta-1) G_0\right) (q - \beta s^2).\]

Warning

Unstable for density optimizations as it violates Pauli positivity

set_yuk3beta(alpha, beta, a=2)

Set \((\alpha,~\beta)\) according to user input and

\[F_\theta(y,~s^2,~q) = T_a\left( - G_0 + y(G_0 + G) \right)\]

where

\[G_0 = \frac{\alpha^2 (\alpha^2 - 60)}{108 \beta (9\beta - 10)},\]

\[G = \left(\frac{40}{27\beta} - \frac{4}{\alpha^2} (\beta-1) G_0\right) (q - \beta s^2)\]

and

\[T_a(x) = \frac{4}{a} \frac{e^{ax}}{e^{ax} + 1} + \frac{a-2}{a}.\]

forward(box_vecs, den)

Parameters:

• box_vecs (torch.Tensor) – Lattice vectors

• den (torch.Tensor) – Electron density

Returns:

Yukawa GGA kinetic energy

Return type:

torch.Tensor

Non-local Kinetic Energy Functionals

Wang-Teter (WT) Functional

functionals.WangTeter(box_vecs, den)

Wang-Teter (WT) functional

The Wang-Teter (WT) functional [Phys. Rev. B 45, 13196] is a Wang-Teter style non-local kinetic functional with a density-independent kernel and parameters \((\alpha,~\beta) = (5/6,~5/6)\).

Parameters:

• box_vecs (torch.Tensor) – Lattice vectors

• den (torch.Tensor) – Electron density

Returns:

WT kinetic energy

Return type:

torch.Tensor

Perrot Functional

functionals.Perrot(box_vecs, den)

Perrot functional

The Perrot functional [J. Phys.: Condens. Matter 6 431] is a Wang-Teter style non-local kinetic functional with a density-independent kernel and parameters \((\alpha,~\beta) = (1,~1)\).

Parameters:

• box_vecs (torch.Tensor) – Lattice vectors

• den (torch.Tensor) – Electron density

Returns:

Perrot kinetic energy

Return type:

torch.Tensor

Smargiassi-Madden (SM) Functional

functionals.SmargiassiMadden(box_vecs, den)

Smargiassi-Madden (SM) functional

The Smargiassi-Madden (SM) functional [Phys. Rev. B 49, 5220] is a non-local kinetic Wang-Teter style functional with a density-independent kernel and parameters \((\alpha,~\beta) = (1/2,~1/2)\).

Parameters:

• box_vecs (torch.Tensor) – Lattice vectors

• den (torch.Tensor) – Electron density

Returns:

SM kinetic energy

Return type:

torch.Tensor

Wang-Govind-Carter 98 (WGC98) Functional

functionals.WangGovindCarter98(box_vecs, den)

Wang-Govind-Carter 98 (WGC98) functional

The Wang-Govind-Carter 98 (WGC98) functional [Phys. Rev. B 58, 13465] is a non-local kinetic Wang-Teter style functional with a density-independent kernel and parameters \((\alpha,~\beta) = ((5+\sqrt{5})/6,~(5-\sqrt{5})/6)\).

Parameters:

• box_vecs (torch.Tensor) – Lattice vectors

• den (torch.Tensor) – Electron density

Returns:

WGC98 kinetic energy

Return type:

torch.Tensor

Wang-Teter Style Functional

class functionals.WangTeterStyleFunctional

Wang-Teter style functional

This class represents a general Wang-Teter style non-local kinetic energy functional with user-chosen \((\alpha,~\beta)\) parameters. Conventional choices of these parameters include

• \((\alpha,~\beta)=(5/6,5/6)\) in the Wang-Teter functional [Phys. Rev. B 45, 13196]

• \((\alpha,~\beta)=(1,1)\) in the Perrot functional [J. Phys.: Condens. Matter 6 431]

• \((\alpha,~\beta)=(1/2,1/2)\) in the Smargiassi-Madden functional [Phys. Rev. B 49, 5220]

• \((\alpha,~\beta) = ((5+\sqrt{5})/6,~(5-\sqrt{5})/6)\) in the Wang-Govind-Carter 98 functional [Phys. Rev. B 58, 13465]

This class also allows for the use of a Pauli-positivity stabilization function [J. Phys. Chem. A 2021, 125, 7, 1650–1660].

__init__(init_args=None)

Parameters:

init_args (tuple) – \((\alpha,~\beta,~f)\) where \((\alpha,~\beta)\) are floats and \(f\) is a function. \((\alpha,~\beta)\) are key parameters of the Wang-Teter style functionals while \(f\) is the Pauli-positivity stabilization function, which must obey \(f(0) = 1\). The default parameters are \((\alpha,~\beta,~f) = (5/6,~5/6,~f(x) = 1 +x)\).

forward(box_vecs, den)

Parameters:

• box_vecs (torch.Tensor) – Lattice vectors

• den (torch.Tensor) – Electron density

Returns:

WT-style functional kinetic energy

Return type:

torch.Tensor

Wang-Govind-Carter 99 (WGC99) Functional

class functionals.WangGovindCarter99

Wang-Govind-Carter 99 (WGC99) functional

The Wang-Govind-Carter 99 (WGC99) functional [Phys. Rev. B 60, 16350] is non-local kinetic functional that extends the Wang-Govind-Carter 98 functional to have a density-dependent kernel. In practice however, a Taylor-expansion is used to avoid the unattractive computational scaling resulting from the density-dependent kernel.

__init__(init_args=None)

Parameters:

init_args (tuple) – \((\alpha,~\beta,~\gamma,~\kappa)\) where each parameter is a float. \(\alpha,~\beta,~\gamma\) are key parameters of the WGC99 functional, while \(\kappa\) is an enhancement factor for the reference uniform density such that \(n^* = κ n_0\) The default parameters are \((\alpha,~\beta,~\gamma,~\kappa) = ((5+\sqrt{5})/6,~(5-\sqrt{5})/6,~2.7,~1)\)

generate_kernel(num_terms=100)

Generates the WGC99 kernel based on its analytical form given in Phys. Rev. B 78, 045105.

Parameters:

num_terms (int) – Number of terms at which the infinite summation is truncated

forward(box_vecs, den)

Parameters:

• box_vecs (torch.Tensor) – Lattice vectors

• den (torch.Tensor) – Electron density

Returns:

WGC99 kinetic energy

Return type:

torch.Tensor

Foley-Madden (FM) Functional

class functionals.FoleyMadden

Foley-Madden (FM) functional

The Foley-Madden (FM) functional [Phys. Rev. B 53, 10589] is a non-local kinetic functional that includes a kernel meant to enforce the correct quadratic response in the homogeneous electron gas limit, in addition to the Wang-Teter style non-local kernel integral term meant to enforce the correct linear response.

__init__(init_args=None)

Parameters:

init_args (tuple) – \((\alpha,~\beta,~f)\) where \((\alpha,~\beta)\) are floats and \(f\) is a function. \((\alpha,~\beta)\) are key parameters of the Foley-Madden functional while \(f\) is the Pauli-positivity stabilization function, which must obey \(f(0) = f'(0) = 1\). The default parameters are \((\alpha,~\beta,~f) = (5/6,~1,~f(x) = 1 +x)\).

forward(box_vecs, den)

Parameters:

• box_vecs (torch.Tensor) – Lattice vectors

• den (torch.Tensor) – Electron density

Returns:

FM kinetic energy

Return type:

torch.Tensor

KGAP Functional

functionals.KGAP(box_vecs, den, E_gap, f=<function <lambda>>)

KGAP functional

The KGAP functional [Phys. Rev. B 97, 205137] is a Wang-Teter style non-local kinetic functional with a density-independent kernel that allows the functional to satisfy the homogeneous limit linear response of a gapped-jellium, instead of the usual Lindhard response of the free electron gas.

Parameters:

• box_vecs (torch.Tensor) – Lattice vectors

• den (torch.Tensor) – Electron density

• E_gap (float) – Band gap of the system of interest (in eV)

Returns:

KGAP kinetic energy

Return type:

torch.Tensor

Huang-Carter (HC) Functional

class functionals.HuangCarter

Huang-Carter (HC) functional

The Huang-Carter (HC) functional [Phys. Rev. B 81, 045206] is a non-local kinetic functional that has a single-point density dependent kernel.

__init__(init_args)

Parameters:

init_args (tuple) – \((\lambda,~\beta,~\kappa)\) where each parameter is a float. \(\lambda,~\beta\) are key parameters of the HC functional, while \(\kappa\) is a parameter for the spline-based field dependent convolution. Recommended values for the parameters are \((\lambda,~\beta) = (0.01177,~0.7143)\). Note that a geometric progression based spline is used so \(\kappa > 1\). It is recommended to start with \(\kappa = 1.2\) and reduce \(\kappa\) until the energy is converged.

generate_kernel(eta_max=50, N_eta=10000)

Generates the Huang-Carter kernel, \(\omega(\eta)\), by solving an initial value problem with Xitorch. The associated ordinary differential eqaution (ODE) is obtained by imposing the Lindhard response of a homogeneous electron gas on the Huang-Carter functional.

Parameters:

• eta_max (float) – Upper bound of \(\eta\) to solve the ODE from

• N_eta (int) – Number of data points in \([0, \eta_\text{max}]\)

forward(box_vecs, den)

Parameters:

• box_vecs (torch.Tensor) – Lattice vectors

• den (torch.Tensor) – Electron density

Returns:

HC kinetic energy

Return type:

torch.Tensor

Revised Huang-Carter (revHC) Functional

class functionals.RevisedHuangCarter

revised Huang-Carter (revHC) functional

The revised Huang-Carter (revHC) functional [Phys. Rev. B 104, 045118] is a non-local kinetic functional that has a single-point density dependent kernel. The revision of the Huang-Carter functional is to handle cases where the reduced gradient term is large due to small densities, such as for surface calculations.

__init__(init_args)

Parameters:

init_args (tuple) – \((a,~b,~\beta,~\kappa)\) where each parameter is a float. \(a,~b,~\beta\) are key parameters of the revHC functional, while \(\kappa\) is a parameter for the spline-based field dependent convolution. Recommended values for the parameters are \((a,~b,~\beta) = (0.45,~0.10,~2/3)\). Note that a geometric progression based spline is used so \(\kappa > 1\). It is recommended to start with \(\kappa = 1.15\) and reduce \(\kappa\) until the energy is converged.

generate_kernel(eta_max=50, N_eta=10000)

Generates the Huang-Carter kernel, \(\omega(\eta)\), by solving an initial value problem with Xitorch. The associated ordinary differential eqaution (ODE) is obtained by imposing the Lindhard response of a homogeneous electron gas on the Huang-Carter functional.

Parameters:

• eta_max (float) – Upper bound of \(\eta\) to solve the ODE from

• N_eta (int) – Number of data points in \([0, \eta_\text{max}]\)

forward(box_vecs, den)

Parameters:

• box_vecs (torch.Tensor) – Lattice vectors

• den (torch.Tensor) – Electron density

Returns:

revHC kinetic energy

Return type:

torch.Tensor

Mi-Genova-Pavanello (MGP) Functional

class functionals.MiGenovaPavanello

Mi-Genova-Pavanello (MGP) functional

The Mi-Genova-Pavanello (MGP) functional [J. Chem. Phys. 148, 184107] is a non-local kinetic functional based on line integrals.

__init__(init_args)

Parameters:

init_args (tuple) – \((a,~b)\) where each parameter is a float. They are key parameters of the Mi-Genova-Pavanello functional.

generate_kernel(eta_max=60, N_eta=2000, N_int=10000)

Generates the integral part of the MGP kernel in one-dimension for later interpolation. This process involves perform numerical integration.

Parameters:

• eta_max (float) – \(\eta_\text{max}\) is the upper bound for which the kernel \(K(\eta)\) is generated up to

• N_eta (int) – Number of data points in \([0, \eta_\text{max}]\)

• N_int (int) – Number of integration points for numerical integration of the kernel

forward(box_vecs, den)

Parameters:

• box_vecs (torch.Tensor) – Lattice vectors

• den (torch.Tensor) – Electron density

Returns:

MGP kinetic energy

Return type:

torch.Tensor

Xu-Wang-Ma (XWM) Functional

functionals.XuWangMa(box_vecs, den, kappa=0)

Xu-Wang-Ma (XWM) functional

The Xu-Wang-Ma (XWM) functional [Phys. Rev. B 100, 205132] is a non-local kinetic functional based on line integrals, with a density-dependent kernel. In practice however, a Taylor-expansion is used to avoid the unattractive computational scaling resulting from the density-dependent kernel.

Parameters:

• box_vecs (torch.Tensor) – Lattice vectors

• den (torch.Tensor) – Electron density

• kappa (float) – Adjustable parameter (default \(\kappa=0\))

Returns:

XWM kinetic energy

Return type:

torch.Tensor