Slave mode expansion for obtaining ab initio interatomic potentials

Here we propose an approach for performing a Taylor series expansion of the first-principles computed energy of a crystal as a function of the nuclear displacements. We enlarge the dimensionality of the existing displacement space and form new variables (i.e., slave modes) which transform like irreducible representations of the point group and satisfy homogeneity of free space. Standard group theoretical techniques can then be applied to deduce the nonzero expansion coefficients a priori. At a given order, the translation group can be used to contract the products and eliminate terms which are not linearly independent, resulting in a final set of slave mode products. While the expansion coefficients can be computed in a variety of ways, we demonstrate that finite difference is effective up to fourth order. We demonstrate the power of the method in the strongly anharmonic system PbTe. All anharmonic terms within an octahedron are computed up to fourth order. A proper unitary transformation demonstrates that the vast majority of the anharmonicity can be attributed to just two terms, indicating that a minimal model of phonon interactions is achievable. The ability to straightforwardly generate polynomial potentials will allow precise simulations at length and time scales which were previously unrealizable.

Figure 1

Top panel: Normal modes for the dimer cluster in the square lattice. The red X's designate modes that are removed from the expansion. Middle panel: Normal modes for the square cluster in the square lattice. Bottom panel: A schematic illustrating the summation of overlapping slave modes for the case of the dimer (left) and the square cluster (right). The slave clusters for a central unit cell are illustrated in red while translationally shifted.

Figure 2

A section of the rocksalt structure. The primitive unit cell is given in green. The two slave clusters associated with the primitive unit cell are denoted by atoms connected with bold lines.

Figure 3

Octahedral modes transforming as the irreducible representations of the point group. The three $T_{1u}$ modes which shift the octahedron have been removed. Reading from left to right and top to bottom, the modes are $A_{1g},E_g,T_{1g},T_{2g},2T_{1u},\text{and}{T}_{2u}$. Our choice of coordinate system and numbering convention is given in the bottom right. Displacement vectors within a given mode have relative magnitudes of 1, 2, or 4, which can be identified by inspection.

Figure 4

Fourth order derivatives computed using central step finite difference as a function of $\Delta$ for a conventional supercell choice of $2\times 2\times 3$ (i.e., 96 atoms).

Figure 5

Fourth order derivatives computed using central step finite difference as a function of conventional supercell size in the $y$ direction (top panel) and the $z$ direction (bottom panel) for $\Delta =0.07\text{Å}$.

Figure 6

A plot of the third and fourth order slave mode product coefficients $\Phi$. The values are ordered in decreasing magnitude for the Pb-centered coefficients, and the same absolute ordering is used for the Te-centered coefficients. Only a fraction of the fourth order terms are shown for clarity.

Figure 7

Top panel: Energy as a function of triaxial engineering strain. Bottom panel: True stress as a function of triaxial engineering strain.

Figure 8

$L$-point phonon frequencies as a function of triaxial engineering strain.

Figure 9

$\Gamma$-point optical phonon frequencies as a function of different engineering strain states: triaxial (top panel), uniaxial (middle panel), and shear ${\gamma }_{xy}$ (bottom panel). In the top panel, the green curve uses the minimal slave mode expansion which has only two expansion coefficients. In the bottom panel, the orange curve uses only the third order slave mode terms.

Figure 10

Energy as a function of displacing a single Pb atom in a 216-atom supercell along the $\langle -3,1,1\rangle$ direction. The green curve uses the minimal slave mode expansion which has only two expansion coefficients.

Figure 11

A plot of the transformed third and fourth order slave mode product coefficients ${\Phi }^{\prime }$. The values are ordered in decreasing magnitude. Only a fraction of the fourth order coefficients are shown.