Solving the Hamiltonian

Explicit Diagonalization

Method of explicit solving KS-equation can be chosen by variable “ks_solver” in INPUT file.

When “basis_type = pw”, ks_solver can be cg, bpcg, dav, dav_subspace, or ppcg. The default setting cg is recommended, which is a band-by-band conjugate-gradient diagonalization method. The dav and dav_subspace settings use Davidson-style subspace diagonalization and can be tried to improve performance.

The ppcg setting uses the projection preconditioned conjugate-gradient method, a restarted block method with bounded workspace. It is optimized and validated for CPU plane-wave calculations; non-CPU devices use a transitional host/device bridge. PPCG is unavailable for esolver_type = sdft, including mixed stochastic-deterministic calculations.

For PPCG, pw_diag_ndim controls the block size. pw_diag_rr_step sets the interval between full Rayleigh–Ritz diagonalizations; its default of 1 performs one on every iteration. Initialization, startup correction, tentative convergence, final validation and failed overlap orthogonalization can also trigger a full Rayleigh–Ritz step. Larger intervals reduce projected eigensolves but may require more iterations, so energies, forces and stresses should be checked for the intended calculation. Refreshing the cached Hamiltonian and overlap images is independent of this interval: periodic and numerical safeguards recompute them, and convergence is confirmed with freshly applied operators.

When “basis_type = lcao”, ks_solver can be genelpa or scalapack_gvx. The default setting genelpa is recommended, which is based on ELPA (EIGENVALUE SOLVERS FOR PETAFLOP APPLICATIONS) (https://elpa.mpcdf.mpg.de/) and the kernel is auto choosed by GENELPA(pplab/GenELPA), usually faster than the setting of “scalapack_gvx”, which is based on ScaLAPACK(Scalable Linear Algebra PACKage)

Stochasic DFT

We support stochastic DFT calculation (SDFT) or mixed stochastic-deterministic DFT (MDFT) with plane-wave basis [Phys. Rev. B 106, 125132 (2022)]. Different from traditional KSDFT with the explicit diagonalization method, SDFT and MDFT calculate physical quantities with trace of the corresponding operators. The advantages of SDFT and MDFT compared to the traditional KSDFT are the ability to simulate larger sizes and higher temperatures. In our package, SDFT and MDFT can be used by setting the esolver_type parameter to sdft for SCF calculations or MD calculations. To start with, you can refer to two examples and an explanation of the input variables.

When we have a hamiltonian, the electronic density can be calculated with:

\(\rho(\mathbf{r})={\rm Tr}[f(\hat{H})\ket{\mathbf{r}}\bra{\mathbf{r}}]\),

where the Fermi-Dirac function \(f(\hat{H})=\frac{1}{1+\exp(\frac{\hat{H}-\mu}{kT})}\) and it can be calculated with the Chebyshev expansion. Here we only support the “fd” or “fermi-dirac” smearing_method, the parameter smearing_sigma is equal the temperature \(T\) (in Ry) and nche_sto represents the order of the expansion.

For physical quantities represented by operator \(\hat{O}\), SDFT calculates its trace with:

\({\rm Tr}[\hat{O}]=\sum_{i=1}^{N_\chi}{\bra{\chi_i}\hat{O}\ket{\chi_i}}\),

while MDFT calculates the trace as:

\({\rm Tr}[\hat{O}]=\sum_{n=1}^{N_\phi}{\bra{\phi_n}\hat{O}\ket{\phi_n}}+\sum_{i=1}^{N_\chi}{\bra{\tilde \chi_i}\hat{O}\ket{\tilde \chi_i}}\),

where \(\{\ket{\tilde\chi_i}\}\) are obtaiend by projecting stochastic orbitals onto the subspace orthogonal to KS orbitals \(\{\phi_n\}\):

\(\ket{\tilde\chi_i}=\ket{\chi_i}-\sum_{n=1}^{N_\phi}\braket{\phi_n|\chi_i}\ket{\phi_n}\).

Here the number of KS orbitals \(N_\phi\) is controlled by the parameter nbands while the number of stochastic orbitals \(N_\chi\) is controlled by nbands_sto.

Besides, although SDFT does not diagonalize the hamiltonian, it can also caluclate DOS and electronic conductivities with parameters out_dos and cal_cond separately.