Inverse Matrix Calculation

Question

I am working on the charge transport calculation, based on the non-equilibrium green function method, for the Hamiltonian with the non-orthogonal basis. Suppose $H$ and $S$ are the Kohn-Sham Hamiltonian and overlap matrice respectively. I want to calculate the transport conductance for the following equation. $$I=\frac{e_{0}}{\hbar}\int\frac{dE}{2\pi}Tr[{\Sigma}_{\textrm{left}}G^{R}{\Sigma}_{right}G^{A}][n_L(E)-n_R(E)]$$ ${\Sigma}_{left}$ and ${\Sigma}_{right}$ are the self energy for left and right leads; $G^{R}$ and $G^{A}$ are retarded and advanced green function for the scattering zone and they are written as the following equation. $$G^{R}=\left[(i+\varepsilon_{Fermi}) \times S - H - ({\Sigma}_{left}+{\Sigma}_{right}) \right]^{-1}$$.

I cannot compute the trace through multipling ${\Sigma}_{left}G^{R}{\Sigma}_{right}G^{A}$ with the eigen vector because I only have the eigen vector for the whole system; not for the scattering zone.

This is why I need to compute the inverse matrix. I am trying to use ZGETRF and ZGETRI subroutines in the MKL library to compute it; however, I find that ZGETRF function, the LU fraction subroutine, gives matrix with zero value on the diagonal part and this makes the following inverse matrix calculation fails with ZGETRI function.

When doing the coding for the non-equilibrium green function (NEGF) method, how does one compute such matrix for the non-orthogonal Hamiltonian system? Is there any subroutine to compute the inverse matrix of the complex matrix? Would anyone please give me some hints?

Thank you in advance.

Answer 1

This sounds like you are doing something wrong.

Even in a non-orthogonal basis the diagonal $\mathbf S$ matrix has the identity matrix in the diagonal.

There could be a number of reasons why this occurs:

1. Your basis set orbitals are too close together resulting in a singular matrix (non-invertible)
2. your imaginary part $\eta$ is too small, or even non-existing. The whole reason for the imaginary part is that one makes the matrix invertible since the energy value will surely not hit an eigenvalue of the system.

Others have noted that calculating the eigenspectrum is the way to go, but as the OP says, this is an open system and calculating the eigenspectrum in this, infinite, open system is too costly compared to the inversion. Hence one will always calculate the inverse like this. Additionally the imaginary part would/should cause the matrix to be non-singular.

It is hard to know what is causing this since we don't know the full details of your code, calling convention etc.