Ziegler–Biersack–Littmark (ZBL)¶

Description¶

The zbl_compute_force operator calculates the screened nuclear repulsion using the universal ZBL potential:

\[ E(r) = \frac{Z_1 Z_2 e^2}{4\pi \varepsilon_0} \, \frac{\Phi(r/a)}{r} \quad \text{for} \quad r<r_c, \]

where the screening function \(\Phi(x)\) is

\[ \Phi(x)= 0.1818\,e^{-3.2x} + 0.5099\,e^{-0.9423x} + 0.2802\,e^{-0.4029x} + 0.02817\,e^{-0.2016x}, \]

and the screening length \(a\) is

\[ a = 0.8854\,a_0\,\big(Z_1^{0.23} + Z_2^{0.23}\big)^{-1}, \]

with \(a_0\) the Bohr radius.

\[ E_{ij} = \frac{1}{4 \pi \epsilon_0} \frac{Z_i Z_j e^2}{r_{ij}} \phi \left(r_{ij}/a \right) + S\left(r_{ij}\right) \]

where \(e\) is the electron charge density, \(\\epsilon_0\) the vacuum electrical permittivity, \(Z_i\) and \(Z_j\) the nuclear charges of the two interacting atoms. The scaling factor \(a\) is defined as:

\[ a = \frac{0.46850}{Z_i^{0.23}+Z_j^{0.23}} \]

and the switch function reads

\[ \phi\left(x\right) = 0.18175e^{-3.19980x}+0.50986e^{-0.94229x}+0.28022e^{-0.40290x}+0.02817e^{-0.20162x} \]

Parameter	Units	Description
\(Z_1\)	—	Atomic number of species A
\(Z_2\)	—	Atomic number of species B
\(r_c\)	distance	Cutoff radius

YAML syntax¶

zbl_compute_force:
  rcut: VALUE UNITS
  parameters: { Z1: VALUE , Z2: VALUE }

VALUE = Physical value of the intended parameter.
UNITS = Units of the provided value that will be passed to the conversion helper for internal units conversion.

Usage examples¶

Systems with a single atomic specy

# Default variant (Al–Al)
zbl_compute_force:
  parameters: { Z1: 13 , Z2: 13 }
  rcut: 2.0 ang

# Symetric variant
zbl_compute_force_symetric:
  parameters: { Z1: 13 , Z2: 13 }
  rcut: 2.0 ang

Systems with multiple atomic species

zbl_multi_force:
  rcut: 2.0 ang
  common_parameters: { Z1: 0 , Z2: 0 }
  parameters:
    - { type_a: Al , type_b: Al , rcut: 2.0 ang , parameters: { Z1: 13 , Z2: 13 } }
    - { type_a: Al , type_b: Cu , rcut: 2.0 ang , parameters: { Z1: 13 , Z2: 29 } }

# Ziegler-Biersack-Littmark

The Ziegler-Biersack-Littmark pair potential computes the screened nuclear repulstion that describes high-enery collisions between atoms. Energy, forces and curvature are smoothly rapmed to zero between and inner and an outer cutoff thanks to a switch function. The energy contribution for to atoms \(i\) and \(j\) at a idstance \(r_{ij}\) from each other reads

.. math::

E_{ij} = \frac{1}{4 \pi \epsilon_0} \frac{Z_i Z_j e^2}{r_{ij}} \phi \left(r_{ij}/a \right) + S\left(r_{ij}\right)

where \(e\) is the electron charge density, \(\\epsilon_0\) the vacuum electrical permittivity, \(Z_i\) and \(Z_j\) the nuclear charges of the two interacting atoms. The scaling factor \(a\) is defined as:

.. math::

a = \frac{0.46850}{Z_i^{0.23}+Z_j^{0.23}}

and the switch function reads

.. math::

\phi\left(x\right) = 0.18175e^{-3.19980x}+0.50986e^{-0.94229x}+0.28022e^{-0.40290x}+0.02817e^{-0.20162x}

.. list-table:: Ziegler-Biersack-Littmark Parameters :widths: 40 40 :header-rows: 1 :align: center

- Denomination
- Units
- \(Z_i\)
- atomic charge
- \(Z_j\)
- atomic charge
- \(\r_1\)
- distance
- \(r_c\)
- distance

Below is a usage example for a system containing one single atomic specy

.. code-block:: yaml

# Basic force computation zbl_compute_force: &zbl_params parameters: { r1: 5. ang , rc: 5.68 ang } rcut: 5.68 ang

# Symetric interaction variant zbl_compute_force_symetric: *zbl_params

# General force calculation block called in the integration scheme compute_force: - zbl_compute_force

The atomic charges are not passed as parameters of the potential since they are already attached to particle species and defined in the YAML block species defined in the :ref:particles-species section. In the case of a system with multiple species, the force operator can be defined as follows

.. code-block:: yaml

# Basic force computation compute_force_pair_multimat: potentials: - { type_a: Si , type_b: Si , potential: zbl , rcut: 5.68 ang , parameters: { r1: 5 ang, rc: 5.68 ang } } - { type_a: Si , type_b: O , potential: zbl , rcut: 5.4 ang , parameters: { r1: 5.1 ang, rc: 5.4 ang } }