Block tridiagonal matrix inversion and fast transmission calculations

A method for the inversion of block tridiagonal matrices encountered in electronic structure calculations is developed, with the goal of efficiently determining the matrices involved in the Fisher–Lee relation for the calculation of electron transmission coefficients. The new method leads to faster transmission calculations compared to traditional methods, as well as freedom in choosing alternate Green’s function matrix blocks for transmission calculations. The new method also lends itself to calculation of the tridiagonal part of the Green’s function matrix. The effect of inaccuracies in the electrode self-energies on the transmission coefficient is analyzed and reveals that the new algorithm is potentially more stable towards such inaccuracies.