JAX Exchange Correlation Library

This library contains direct translations of exchange correlation functionals in libxc to jax. The core calculations in libxc are implemented in maple. This gives us the opportunity to translate them directly into python with the help of CodeGeneration.

Usage

Installation

pip install jax-xc

Invoking the Functionals

jax_xc's API is functional: it receives $\rho$ a function of Callable type, and returns the $\varepsilon_{xc}$ as a function of Callable type.

$$E_{xc} = \int \rho(r) \varepsilon_{xc}(r) dr$$

LDA and GGA

Unlike libxc which takes pre-computed densities and their derivative at certain coordinates. In jax_xc, the API is designed to directly take a density function.

import jax
import jax.numpy as jnp
import jax_xc


def rho(r):
  """Electron number density. We take gaussian as an example.

  A function that takes a real coordinate, and returns a scalar
  indicating the number density of electron at coordinate r.

  Args:
  r: a 3D coordinate.
  Returns:
  rho: If it is unpolarized, it is a scalar.
      If it is polarized, it is a array of shape (2,).
  """
  return jnp.prod(jax.scipy.stats.norm.pdf(r, loc=0, scale=1))

# create a density functional
gga_xc_pbe = jax_xc.gga_x_pbe(polarized=False)

# a grid point in 3D
r = jnp.array([0.1, 0.2, 0.3])

# pass rho and r to the functional to compute epsilon_xc (energy density) at r.
# corresponding to the 'zk' in libxc
epsilon_xc_r = gga_xc_pbe(rho, r)
print(epsilon_xc_r)

mGGA

Unlike LDA and GGA that only depends on the density function, mGGA functionals also depend on the molecular orbitals.

import jax
import jax.numpy as jnp
import jax_xc


def mo(r):
  """Molecular orbital. We take gaussian as an example.

  A function that takes a real coordinate, and returns the value of
  molecular orbital at this coordinate.

  Args:
    r: a 3D coordinate.
  Returns:
    mo: If it is unpolarized, it is a array of shape (N,).
        If it is polarized, it is a array of shape (N, 2).
  """
  # Assume we have 3 molecular orbitals
  return jnp.array([
      jnp.prod(jax.scipy.stats.norm.pdf(r, loc=0, scale=1)),
      jnp.prod(jax.scipy.stats.norm.pdf(r, loc=0.5, scale=1)),
      jnp.prod(jax.scipy.stats.norm.pdf(r, loc=-0.5, scale=1))
  ])


rho = lambda r: jnp.sum(mo(r)**2, axis=0)
mgga_xc_cc06 = jax_xc.mgga_xc_cc06(polarized=False)

# a grid point in 3D
r = jnp.array([0.1, 0.2, 0.3])

# evaluate the exchange correlation energy per particle at this point
# corresponding to the 'zk' in libxc
print(mgga_xc_cc06(rho, r, mo))

Hybrid Functionals

Hybrid functionals expose the same API, with extra attributes for the users to access parameters needed outside of libxc/jax_xc (e.g. the fraction of exact exchange).

import jax
import jax.numpy as jnp
import jax_xc


def rho(r):
  """Electron number density. We take gaussian as an example.

  A function that takes a real coordinate, and returns a scalar
  indicating the number density of electron at coordinate r.

  Args:
    r: a 3D coordinate.
  Returns:
    rho: If it is unpolarized, it is a scalar.
        If it is polarized, it is a array of shape (2,).
  """
  return jnp.prod(jax.scipy.stats.norm.pdf(r, loc=0, scale=1))


hyb_gga_xc_pbeb0 = jax_xc.hyb_gga_xc_pbeb0(polarized=False)

# a grid point in 3D
r = jnp.array([0.1, 0.2, 0.3])

# evaluate the exchange correlation energy per particle at this point
# corresponding to the 'zk' in libxc
print(hyb_gga_xc_pbeb0(rho, r))

# access to extra attributes
cam_alpha = hyb_gga_xc_pbep0.cam_alpha  # fraction of full Hartree-Fock exchange

The complete list of extra attributes can be found below:

cam_alpha: float
cam_beta: float
cam_omega: float
nlc_b: float
nlc_C: float

The meaning for each attribute is the same as libxc:

• cam_alpha: fraction of full Hartree-Fock exchange, used both for usual hybrids as well as range-separated ones
• cam_beta: fraction of short-range only(!) exchange in range-separated hybrids
• cam_omega: range separation constant
• nlc_b: non-local correlation, b parameter
• nlc_C: non-local correlation, C parameter

Experimental

We support automatic functional derivative!

import jax
import jax_xc
import autofd.operators as o
from autofd import function
import jax.numpy as jnp
from jaxtyping import Array, Float32

@function
def rho(r: Float32[Array, "3"]) -> Float32[Array, ""]:
  """Electron number density. We take gaussian as an example.

  A function that takes a real coordinate, and returns a scalar
  indicating the number density of electron at coordinate r.

  Args:
  r: a 3D coordinate.
  Returns:
  rho: If it is unpolarized, it is a scalar.
      If it is polarized, it is a array of shape (2,).
  """
  return jnp.prod(jax.scipy.stats.norm.pdf(r, loc=0, scale=1))

# create a density functional
gga_x_pbe = jax_xc.experimental.gga_x_pbe
epsilon_xc = gga_x_pbe(rho)

# a grid point in 3D
r = jnp.array([0.1, 0.2, 0.3])

# pass rho and r to the functional to compute epsilon_xc (energy density) at r.
# corresponding to the 'zk' in libxc
print(f"The function signature of epsilon_xc is {epsilon_xc}")

energy_density = epsilon_xc(r)
print(f"epsilon_xc(r) = {energy_density}")

vxc = jax.grad(lambda rho: o.integrate(rho * gga_x_pbe(rho)))(rho)
print(f"The function signature of vxc is {vxc}")
print(vxc(r))

Support Functionals

Please refer to the functionals section in jax_xc's documentation for the complete list of supported functionals.

Numerical Correctness

We test all the functionals that are auto-generated from maple files against the reference values in libxc. The test is performed by comparing the output of libxc and jax_xc and make sure they are within a certain tolerance, namely atol=2e-10 and rtol=2e-10.

Performance Benchmark

We report the performance benchmark of jax_xc against libxc on a 64-core machine with Intel(R) Xeon(R) Silver 4216 CPU @ 2.10GHz.

We sample the points to evaluate the functionals by varying the number of points from 1 to $10^7$. The benchmark is performed by evaluating the runtime of the functional. Note that the runtime of jax_xc is measured by excluding the time of just-in-time compilation.

We visualize the mean value (averaged for both polarized and unpolarized) of the runtime of jax_xc and libxc in the following figure. The y-axis is log-scale.

jax_xc's runtime is constantly below libxc's for all batch sizes. The speed up is ranging from 3x to 10x, and it is more significant for larger batch sizes.

We hypothesize that the reason for the speed up is that Jax's JIT compiler is able to optimize the functionals (e.g. vectorization, parallel execution, instruction fusion, constant folding for floating points, etc.) better than libxc.

We visualize the distribution of the runtime ratio of jax_xc and libxc in the following figure. The ratio is closer to 0.1 for large batch sizes (~ 10x speed up). The ratio is constantly below 1.0.

Note that, we exclude one datapoint mgga_x_2d_prhg07 from the runtime ratio visualization because it is an outlier due to Jax's lack of support oflamberw function and we use tensorflow_probability.substrates.jax.math.lambertw.

Caveates

The following functionals from libxc are not available in jax_xc because some functions are not available in jax.

gga_x_fd_lb94          # Becke-Roussel not having a closed-form expression
gga_x_fd_revlb94       # Becke-Roussel not having a closed-form expression
gga_x_gg99             # Becke-Roussel not having a closed-form expression
gga_x_kgg99            # Becke-Roussel not having a closed-form expression
hyb_gga_xc_case21      # Becke-Roussel not having a closed-form expression
hyb_mgga_xc_b94_hyb    # Becke-Roussel not having a closed-form expression
hyb_mgga_xc_br3p86     # Becke-Roussel not having a closed-form expression
lda_x_1d_exponential   # Requires explicit 1D integration
lda_x_1d_soft          # Requires explicit 1D integration
mgga_c_b94             # Becke-Roussel not having a closed-form expression
mgga_x_b00             # Becke-Roussel not having a closed-form expression
mgga_x_bj06            # Becke-Roussel not having a closed-form expression
mgga_x_br89            # Becke-Roussel not having a closed-form expression
mgga_x_br89_1          # Becke-Roussel not having a closed-form expression
mgga_x_mbr             # Becke-Roussel not having a closed-form expression
mgga_x_mbrxc_bg        # Becke-Roussel not having a closed-form expression
mgga_x_mbrxh_bg        # Becke-Roussel not having a closed-form expression
mgga_x_mggac           # Becke-Roussel not having a closed-form expression
mgga_x_rpp09           # Becke-Roussel not having a closed-form expression
mgga_x_tb09            # Becke-Roussel not having a closed-form expression
gga_x_wpbeh            # jit too long for E1_scaled
gga_c_ft97             # jit too long for E1_scaled
lda_xc_tih             # vxc functional
gga_c_pbe_jrgx         # vxc functional
gga_x_lb               # vxc functional

Building from Source Code

Modify the .env.example to fill in your envrionment variables, then rename it to .env. Then run source .env to load them into your shell.

• OUTPUT_USER_ROOT: The path to the bazel cache. This is where the bazel cache will be stored. This is useful if you are building on a shared filesystem.
• MAPLE_PATH: The path to the maple binary.
• TMP_INSTALL_PATH: The path to a temporary directory where the wheel will be installed. This is useful if you are building on a shared filesystem.

Make sure you have bazel and maple installed. Your python envrionment has installed the dependencies in requirements.txt.

How to build python wheel.

bazel --output_user_root=$OUTPUT_USER_ROOT build --action_env=PATH=$PATH:$MAPLE_PATH @jax_xc_repo//:jax_xc_wheel

Once the build finished, the python wheel could be found under bazel-bin/external/jax_xc_repo. For example, the name for version 0.0.7 is jax_xc-0.0.7-cp310-cp310-manylinux_2_17_x86_64.whl.

Install the python wheel. If needed, specify the install path by

pip install {{wheel_name}} --target $TMP_INSTALL_PATH

Running Test

The test could be run without the command above that builds wheel from source, though it might take longer time to build all the components needed for the test. To run all the test:

bazel --output_user_root=$OUTPUT_USER_ROOT test --action_env=PATH=$PATH:$MAPLE_PATH //tests/...

To run a specific test, for example test_impl:

bazel --output_user_root=$OUTPUT_USER_ROOT test --action_env=PATH=$PATH:$MAPLE_PATH //tests:test_impl

The test output could be found in bazel-testlogs/tests/test_impl/test.log for the tests:test_impl and similar to the others. If you prefer output in command line, add --test_output=all to the above command.

License

Aligned with libxc, jax_xc is licensed under the Mozilla Public License 2.0. See LICENSE for the full license text.