matscipy.calculators package¶
Submodules¶
matscipy.calculators.eam.calculator module¶
Embedded-atom method potential.
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class
matscipy.calculators.eam.calculator.EAM(fn=None, atomic_numbers=None, F=None, f=None, rep=None, cutoff=None, kind='eam/alloy')¶ Bases:
ase.calculators.calculator.Calculator-
calculate(atoms, properties, system_changes)¶ Do the calculation.
- properties: list of str
List of what needs to be calculated. Can be any combination of ‘energy’, ‘forces’, ‘stress’, ‘dipole’, ‘charges’, ‘magmom’ and ‘magmoms’.
- system_changes: list of str
List of what has changed since last calculation. Can be any combination of these six: ‘positions’, ‘numbers’, ‘cell’, ‘pbc’, ‘initial_charges’ and ‘initial_magmoms’.
Subclasses need to implement this, but can ignore properties and system_changes if they want. Calculated properties should be inserted into results dictionary like shown in this dummy example:
self.results = {'energy': 0.0, 'forces': np.zeros((len(atoms), 3)), 'stress': np.zeros(6), 'dipole': np.zeros(3), 'charges': np.zeros(len(atoms)), 'magmom': 0.0, 'magmoms': np.zeros(len(atoms))}
The subclass implementation should first call this implementation to set the atoms attribute.
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calculate_hessian_matrix(atoms, divide_by_masses=False)¶ Compute the Hessian matrix
The Hessian matrix is the matrix of second derivatives of the potential energy \(\mathcal{V}_\mathrm{int}\) with respect to coordinates, i.e.
\[\frac{\partial^2 \mathcal{V}_\mathrm{int}} {\partial r_{\nu{}i}\partial r_{\mu{}j}},\]where the indices \(\mu\) and \(\nu\) refer to atoms and the indices \(i\) and \(j\) refer to the components of the position vector \(r_\nu\) along the three spatial directions.
The Hessian matrix has contributions from the pair potential and the embedding energy,
\[\frac{\partial^2 \mathcal{V}_\mathrm{int}}{\partial r_{\nu{}i}\partial r_{\mu{}j}} = \frac{\partial^2 \mathcal{V}_\mathrm{pair}}{ \partial r_{\nu i} \partial r_{\mu j}} + \frac{\partial^2 \mathcal{V}_\mathrm{embed}}{\partial r_{\nu i} \partial r_{\mu j}}.\]The contribution from the pair potential is
\[\begin{split}\frac{\partial^2 \mathcal{V}_\mathrm{pair}}{ \partial r_{\nu i} \partial r_{\mu j}} &= -\phi_{\nu\mu}'' \left( \frac{r_{\nu\mu i}}{r_{\nu\mu}} \frac{r_{\nu\mu j}}{r_{\nu\mu}} \right) -\frac{\phi_{\nu\mu}'}{r_{\nu\mu}}\left( \delta_{ij}- \frac{r_{\nu\mu i}}{r_{\nu\mu}} \frac{r_{\nu\mu j}}{r_{\nu\mu}} \right) \\ &+\delta_{\nu\mu}\sum_{\gamma\neq\nu}^{N} \phi_{\nu\gamma}'' \left( \frac{r_{\nu\gamma i}}{r_{\nu\gamma}} \frac{r_{\nu\gamma j}}{r_{\nu\gamma}} \right) +\delta_{\nu\mu}\sum_{\gamma\neq\nu}^{N}\frac{\phi_{\nu\gamma}'}{r_{\nu\gamma}}\left( \delta_{ij}- \frac{r_{\nu\gamma i}}{r_{\nu\gamma}} \frac{r_{\nu\gamma j}}{r_{\nu\gamma}} \right).\end{split}\]The contribution of the embedding energy to the Hessian matrix is a sum of eight terms,
\[\begin{split}\frac{\mathcal{V}_\mathrm{embed}}{\partial r_{\mu j} \partial r_{\nu i}} &= T_1 + T_2 + T_3 + T_4 + T_5 + T_6 + T_7 + T_8 \\ T_1 &= \delta_{\nu\mu}U_\nu'' \sum_{\gamma\neq\nu}^{N}g_{\nu\gamma}'\frac{r_{\nu\gamma i}}{r_{\nu\gamma}} \sum_{\gamma\neq\nu}^{N}g_{\nu\gamma}'\frac{r_{\nu\gamma j}}{r_{\nu\gamma}} \\ T_2 &= -u_\nu''g_{\nu\mu}' \frac{r_{\nu\mu j}}{r_{\nu\mu}} \sum_{\gamma\neq\nu}^{N} G_{\nu\gamma}' \frac{r_{\nu\gamma i}}{r_{\nu\gamma}} \\ T_3 &= +u_\mu''g_{\mu\nu}' \frac{r_{\nu\mu i}}{r_{\nu\mu}} \sum_{\gamma\neq\mu}^{N} G_{\mu\gamma}' \frac{r_{\mu\gamma j}}{r_{\mu\gamma}} \\ T_4 &= -\left(u_\mu'g_{\mu\nu}'' + u_\nu'g_{\nu\mu}''\right) \left( \frac{r_{\nu\mu i}}{r_{\nu\mu}} \frac{r_{\nu\mu j}}{r_{\nu\mu}} \right)\\ T_5 &= \delta_{\nu\mu} \sum_{\gamma\neq\nu}^{N} \left(U_\gamma'g_{\gamma\nu}'' + U_\nu'g_{\nu\gamma}''\right) \left( \frac{r_{\nu\gamma i}}{r_{\nu\gamma}} \frac{r_{\nu\gamma j}}{r_{\nu\gamma}} \right) \\ T_6 &= -\left(U_\mu'g_{\mu\nu}' + U_\nu'g_{\nu\mu}'\right) \frac{1}{r_{\nu\mu}} \left( \delta_{ij}- \frac{r_{\nu\mu i}}{r_{\nu\mu}} \frac{r_{\nu\mu j}}{r_{\nu\mu}} \right) \\ T_7 &= \delta_{\nu\mu} \sum_{\gamma\neq\nu}^{N} \left(U_\gamma'g_{\gamma\nu}' + U_\nu'g_{\nu\gamma}'\right) \frac{1}{r_{\nu\gamma}} \left(\delta_{ij}- \frac{r_{\nu\gamma i}}{r_{\nu\gamma}} \frac{r_{\nu\gamma j}}{r_{\nu\gamma}} \right) \\ T_8 &= \sum_{\substack{\gamma\neq\nu \\ \gamma \neq \mu}}^{N} U_\gamma'' g_{\gamma\nu}'g_{\gamma\mu}' \frac{r_{\gamma\nu i}}{r_{\gamma\nu}} \frac{r_{\gamma\mu j}}{r_{\gamma\mu}}\end{split}\]- Parameters
atoms (ase.Atoms) –
divide_by_masses (bool) – Divide block \(\nu\mu\) by \(m_\num_\mu\) to obtain the dynamical matrix
- Returns
D – Block Sparse Row matrix with the nonzero blocks
- Return type
numpy.matrix
Notes
- Notation:
\(N\) Number of atoms
\(\mathcal{V}_\mathrm{int}\) Total potential energy
\(\mathcal{V}_\mathrm{pair}\) Pair potential
\(\mathcal{V}_\mathrm{embed}\) Embedding energy
\(r_{\nu{}i}\) Component \(i\) of the position vector of atom \(\nu\)
\(r_{\nu\mu{}i} = r_{\mu{}i}-r_{\nu{}i}\)
\(r_{\nu\mu{}}\) Norm of \(r_{\nu\mu{}i}\), i.e.\(\left(r_{\nu\mu{}1}^2+r_{\nu\mu{}2}^2+r_{\nu\mu{}3}^2\right)^{1/2}\)
\(\phi_{\nu\mu}(r_{\nu\mu{}})\) Pair potential energy of atoms \(\nu\) and \(\mu\)
\(\rho_nu\) Total electron density of atom \(\nu\)
\(U_\nu(\rho_nu)\) Embedding energy of atom \(\nu\)
\(g_{\delta}\left(r_{\gamma\delta}\right) \equiv g_{\gamma\delta}\) Contribution from atom \(\delta\) to \(\rho_\gamma\)
\(m_\nu\) mass of atom \(\nu\)
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property
cutoff¶
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default_parameters= {}¶
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energy_virial_and_forces(atomic_numbers_i, i_n, j_n, dr_nc, abs_dr_n)¶ Compute the potential energy, the virial and the forces.
- Parameters
atomic_numbers_i (array_like) – Atomic number for each atom in the system
j_n (i_n,) – Neighbor pairs
dr_nc (array_like) – Distance vectors between neighbors
abd_dr_n (array_like) – Length of distance vectors between neighbors
- Returns
epot (float) – Potential energy
virial_v (array) – Virial
forces_ic (array) – Forces acting on each atom
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implemented_properties= ['energy', 'stress', 'forces']¶
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name= 'EAM'¶
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