# 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 or dav. The default setting cg is recommended, which is band-by-band conjugate gradient diagonalization method. There is a large probability that the use of setting of dav , which is block Davidson diagonalization method, can be tried to improve performance.

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(https://github.com/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.