PAOFLOW.hamiltonian.do_d2Hd2k#

Attributes#

comm

rank

size

Functions#

do_d2Hd2k_ij(Hksp, Rfft, alat, npool, v_kp, bnd, degen)

Compute the six unique second-order derivatives of the k-space Hamiltonian.

Module Contents#

PAOFLOW.hamiltonian.do_d2Hd2k.comm[source]#
PAOFLOW.hamiltonian.do_d2Hd2k.rank[source]#
PAOFLOW.hamiltonian.do_d2Hd2k.size[source]#
PAOFLOW.hamiltonian.do_d2Hd2k.do_d2Hd2k_ij(Hksp, Rfft, alat, npool, v_kp, bnd, degen)[source]#

Compute the six unique second-order derivatives of the k-space Hamiltonian.

Parameters:

  • Hksp (np.ndarray, shape (snawf, nk1, nk2, nk3, nspin)) – Distributed real-space Hamiltonian in the FFT representation, where each element already contains the factor \(i \cdot a_{\text{lat}}\) (i.e. HR * 1j * alat).

  • Rfft (np.ndarray, shape (nk1, nk2, nk3, 3)) – Real-space grid vectors used as multiplication factors in FFT-based gradient computation.

  • alat (float) – Lattice constant in Bohr radii, used to scale the real-space Hamiltonian prior to the FFT.

  • npool (int) – Number of MPI pools used for the gather_scatter() redistribution.

  • v_kp (np.ndarray, shape (nkpnts, nawf, bnd, nspin)) – Bloch eigenvectors (columns are eigenstates) at each k-point, distributed over pools.

  • bnd (int) – Number of bands for which the curvature is computed.

  • degen (list) – Nested list of degenerate subspace indices, one entry per spin channel and k-point, as produced by the eigenvalue solver.

Returns:

  • M_ij (np.ndarray, shape (6, nkpnts, bnd, nspin)) – Diagonal matrix elements \(\langle n | d^2H / dk_i dk_j | n \rangle\) in the Bloch eigenstate basis for the six unique ij pairs xx, yy, zz, xy, xz, yz.

  • dvec_list (list) – Nested list of shape [6][nspin][nkpnts] containing the modified eigenvector arrays after degenerate-subspace diagonalisation (used by do_band_curvature() to compute the perturbative correction). Each element is either an empty array or an np.ndarray of shape (nawf, nawf).

Notes

For each of the six unique Cartesian index pairs \((i, j)\), the second derivative is obtained via the FFT convolution

\[d^2 H(\mathbf{k}) / dk_i dk_j = \mathcal{F}\left[ R_i R_j \cdot H(\mathbf{R}) \cdot i \cdot a_{\text{lat}} \right]\]

where \(R_i\) is the i-th Cartesian component of the real-space lattice vector grid Rfft. The result is then projected onto the Bloch eigenstate basis using perturb_split() to correctly handle degenerate bands.