Angular Dependent Potential (ADP)

GPUMD supports the Angular Dependent Potential (ADP), which is an extension of the embedded atom method (EAM) that includes angular forces through dipole and quadrupole distortions of the local atomic environment.

The ADP was developed to provide a more accurate description of directional bonding and angular forces in metallic systems, particularly for materials where traditional EAM potentials fail to capture the full complexity of atomic interactions. The ADP formalism is especially useful for modeling complex crystal structures, defects, and phase transformations in metals and alloys.

Potential form

General form

The ADP is described in detail by Mishin et al. [Mishin2005] and has been successfully applied to various metallic systems including the Cu-Ta system [Pun2015], U-Mo alloys [Starikov2018], etc. The total energy of atom \(i\) is given by:

\[E_i = F_\alpha\left(\sum_{j\neq i} \rho_\beta(r_{ij})\right) + \frac{1}{2} \sum_{j\neq i} \phi_{\alpha\beta}(r_{ij}) + \frac{1}{2} \sum_s (\mu_{is})^2 + \frac{1}{2} \sum_{s,t} (\lambda_{ist})^2 - \frac{1}{6} \nu_i^2\]

where:

  • \(F_\alpha\) is the embedding energy as a function of the total electron density at atom \(i\)

  • \(\rho_\beta(r_{ij})\) is the electron density contribution from atom \(j\) at distance \(r_{ij}\)

  • \(\phi_{\alpha\beta}(r_{ij})\) is the pair potential interaction between atoms of types \(\alpha\) and \(\beta\)

  • \(\alpha\) and \(\beta\) are the element types of atoms \(i\) and \(j\)

  • \(s\) and \(t\) are indices running over Cartesian coordinates (\(x, y, z\))

  • \(\mu_{is}\) is the dipole distortion tensor (3 components)

  • \(\lambda_{ist}\) is the quadrupole distortion tensor (6 independent components)

  • \(\nu_i\) is the trace of the quadrupole tensor

Angular terms

The dipole distortion tensor represents the first moment of the local environment:

\[\mu_{is} = \sum_{j\neq i} u_{\alpha\beta}(r_{ij}) r_{ij}^s\]

where \(u_{\alpha\beta}(r)\) is a tabulated function and \(r_{ij}^s\) is the \(s\)-component of the vector from atom \(i\) to atom \(j\).

The quadrupole distortion tensor represents the second moment of the local environment:

\[\lambda_{ist} = \sum_{j\neq i} w_{\alpha\beta}(r_{ij}) r_{ij}^s r_{ij}^t\]

where \(w_{\alpha\beta}(r)\) is another tabulated function. The trace of the quadrupole tensor is:

\[\nu_i = \lambda_{ixx} + \lambda_{iyy} + \lambda_{izz}\]

The angular terms \(\mu\) and \(\lambda\) introduce directional dependence into the potential energy, allowing the ADP to capture angular forces that are absent in the traditional EAM formalism. These terms are essential for accurately modeling materials with complex bonding environments.

File format

General structure

The ADP potential file follows the extended DYNAMO setfl format, which is compatible with LAMMPS and other molecular dynamics codes. The file structure consists of:

Header section (lines 1-5):

  • Lines 1-3: Comment lines (can contain any text, typically author and date information)

  • Line 4: Nelements Element1 Element2 … ElementN

    • Nelements: Number of elements in the potential

    • Element1, Element2, etc.: Element symbols (e.g., Cu, Ta, Mo)

  • Line 5: Nrho drho Nr dr cutoff

    • Nrho: Number of points in the embedding function \(F(\rho)\) tabulation

    • drho: Spacing between tabulated \(\rho\) values

    • Nr: Number of points in the pair potential and density function tabulations

    • dr: Spacing between tabulated \(r\) values

    • cutoff: Cutoff distance for all functions (in Angstroms)

Per-element sections (repeated Nelements times):

Each element section contains:

  • Line 1: atomic_number mass lattice_constant lattice_type

    • atomic_number: Atomic number of the element

    • mass: Atomic mass (in amu)

    • lattice_constant: Equilibrium lattice constant (in Angstroms)

    • lattice_type: Crystal structure (e.g., fcc, bcc, hcp)

  • Next Nrho values: Embedding function \(F(\rho)\)

    • Tabulated values of \(F\) at \(\rho = 0, \Delta\rho, 2\Delta\rho, ..., (N_\rho-1)\Delta\rho\)

    • Units: eV

  • Next Nr values: Electron density function \(\rho(r)\)

    • Tabulated values at \(r = 0, \Delta r, 2\Delta r, ..., (N_r-1)\Delta r\)

    • Units: electron density

Pair potential section:

For all element pairs \((i, j)\) with \(i \geq j\) (upper triangular, since \(\phi_{ij} = \phi_{ji}\)):

  • Nr values: Pair potential \(\phi_{ij}(r)\)

    • Tabulated as \(r \times \phi(r)\) (scaled by distance)

    • Units: eV·Angstrom

    • Order: (1,1), (2,1), (2,2), (3,1), (3,2), (3,3), etc.

Dipole function section:

For all element pairs \((i, j)\) with \(i \geq j\):

  • Nr values: Dipole function \(u_{ij}(r)\)

    • Tabulated as \(u(r)\) (NOT scaled by distance)

    • Units: electron density·Angstrom

    • Same ordering as pair potentials

Quadrupole function section:

For all element pairs \((i, j)\) with \(i \geq j\):

  • Nr values: Quadrupole function \(w_{ij}(r)\)

    • Tabulated as \(w(r)\) (NOT scaled by distance)

    • Units: electron density·Angstrom²

    • Same ordering as pair potentials

Table format

GPUMD supports the standard ADP format as defined in LAMMPS.

Note

The user needs to modify the first line of the potential file as follows:

adp <num_types> <list of elements>

The last two parts can be directly copied from the fourth line of the original potential file. For example:

adp 2 Cu Ta

Since the first three lines are comments in LAMMPS, this modification does not affect usage in LAMMPS.

Usage

To use an ADP potential in GPUMD, specify it in the run.in input file:

potential <potential_file>

where <potential_file> is the path to the ADP potential file.

Examples

Single-element system (pure Ta):

potential Ta.adp

Multi-element system (Cu-Ta alloy):

potential CuTa_LJ15_2014.adp.txt

Multi-element system (U-Mo alloy):

potential U_Mo.adp

Note

Element types are automatically matched between model.xyz and the ADP potential file based on element symbols. The order and number of atom types in model.xyz can be different from those in the potential file.

References

Pun2015

    1. Pun, K. A. Darling, L. J. Kecskes, and Y. Mishin, “Angular-dependent interatomic potential for the Cu–Ta system and its application to structural stability of nano-crystalline alloys,” Acta Mater. 100, 377 (2015).

Starikov2018

    1. Starikov, L. N. Kolotova, A. Y. Kuksin, D. E. Smirnova, and V. I. Tseplyaev, “Atomistic simulation of cubic and tetragonal phases of U-Mo alloy: Structure and thermodynamic properties,” J. Nucl. Mater. 499, 451 (2018).