HamiltonianDerivatives¶

Usage Examples¶

Calculate the Hamiltonian derivatives for a system repeated five times in the B direction and three times in the C direction.

hamiltonian_derivatives = HamiltonianDerivatives(
    configuration,
    filename='HamiltonianDerivatives.hdf5',
    object_id='hamiltonian_derivatives',
    repetitions=(1,5,3),
    )
hamiltonian_derivatives.update()

When using repetitions=Automatic, the cell is repeated such that all atoms within a pre-defined, element-pair dependent interaction range are included.

hamiltonian_derivatives = HamiltonianDerivatives(
    configuration,
    filename='HamiltonianDerivatives.hdf5',
    object_id='hamiltonian_derivatives',
    repetitions=Automatic,
    )
hamiltonian_derivatives.update()

The default number of repetitions i.e. repetitions=Automatic can be found before a calculation using the function checkNumberOfRepetitions().

(nA, nB, nC)  = checkNumberOfRepetitions(configuration)

Notes¶

The Hamiltonian derivatives are calculated using the central finite difference method in a repeated cell constituting a super cell. That is, the Hamiltonian derivatives are calculated for each atom in the central cell by displacing the atom in the supercell and determining the Hamiltonian for two displacements from its original position along each of the Cartesian directions. In DFT, the derivatives of the Hamiltonian \(\hat{H}\) can be expressed as the derivatives of the effective potential \(V_{\text{eff}}\) since the kinetic term does not contribute. Thus the Hamiltonian derivatives for the \(i\) and \(j\) basis functions can be approximated as

where \(R_{I, \alpha}\) is the \(\alpha\) cartesian coordinate for atom \(I\) in the central unit cell, and \(\delta\) is the atomic displacement. For Slater-Koster calculators, the Hamiltonian derivatives are described by the on-site and off-site parameters and their distance dependence, hence

Aborted HamiltonianDerivatives calculations can be resumed by re-running the same script or reading the study object from file and calling update() on it. The study object will automatically detect which displacement calculations have already been carried out and only run the calculations that are not yet completed.

Notes for DFT¶

Fig. 131 (a) a unit cell for a 1D system confined in the \(x\) and \(y\) -directions. The k-point sampling is correspondingly \((1, 1, N_{\text{C}})\) , where \(N_{\text{C}}\) is the sampling in the \(z\) -direction. (b) a super cell created by five repetitions in the \(z\) -direction of the unit cell in (a). In the calculation of the Hamiltonian derivatives the atoms in the center unit cell of the super cell (the two atom indicated with arrows) are displaced by \(\pm\delta\) in each of the Cartesian directions. (c) the change in the effective potential \(\text{d} V_{\text{eff}}\) in the super cell as an atom is displaced \(\delta\) . The number of repetitions must ensure that the change in the effective potential goes to zero at the boundaries of the super cell in the non-confined directions for every atomic displacement.

When calculating the Hamiltonian derivatives both the number of sampled k-points for the super cell and repetitions in the confined the directions must be 1. In the following the number of sampled k-points for the super cell and repetitions in the non-confined directions will be adressed. To simplify things a system with only one non-confined direction will be used, see Fig. 131 (a), but the relations for the one non-confined direction applies to all non-confined directions. The bulk configuration in Fig. 131 (a) is converged in the total energy with respect to the number of k-points of \((1, 1, N_{\text{C}})\). The number of repetitions in the super cell in the non-confined direction, see Fig. 131 (b), is chosen large enough such that the change in the effective potential \(\text{d} V_{\text{eff}}\) goes to zero at the boundaries of the super cell in the non-confined directions for every atomic displacement, confer Fig. 131 (c). For this system five repetitions of the unit cell in the confined direction is enough for the change in the effective potential to go to zero at the boundaries. The recommended k-point sampling for the non-confined direction is the number of k-points in the non-confined direction for the unit cell divided by the number of repetitions in the non-confined direction. The k-point sampling then becomes \((1, 1, \frac{N_{\text{C}}}{\text{repetitions in C}})\) and in this particular case \((1, 1, N_{\text{C}}/5)\) .

The Hamiltonian derivatives calculations generally requires a low tolerance in the IterationControlParameters settings, e.g. a tolerance of 1e-6. Finally, it should be noted that the HuckelCalculator currently does not support calculation of the Hamiltonian derivatives.