Implementation of EDM using the continued fraction representation of Fermi function|
- Date: 2016/10/17 22:03
- Name: Ari Ojanperä
I have attempted to implement the equilibrium energy density matrix to the ATK program by QuantumWise using the contour integration method based on the continued fraction representation of the Fermi function as described in  Phys. Rev. B, 75 (035123), 2007 (http://dx.doi.org/10.1103/PhysRevB.75.035123) and  Phys. Rev. B, 81 (035116), 2010 (http://dx.doi.org/10.1103/PhysRevB.81.035116). I've followed the equations presented in Appendix B of Ref. . However, I've been unable to obtain the correct EDM, as compared to other contour integration methods. The system I've tested with is a simple carbon wire at the gamma point, and the Green's function includes self-energies in addition to the Hamiltonian and overlap matrices.
The density matrix implementation in ATK based on the same contour integration method (using Eqs. in Ref ) has been well tested and works as expected. However, for the same poles and residues as for the density matrix, Eqs. (B4) and (B1) in Ref.  lead to a diverging EDM as a function of number of poles. Has anyone else encountered problems in the implementation of the EDM? Mathematically, looking at the sum term on the right-hand side of Eq. (B4), the Green's function must decay rapidly to zero at large \alpha_p in order for the sum to converge as the number of poles increases. Could there be some rare physical systems for which the sum diverges, or should the poles and residues perhaps be calculated differently than for the density matrix?
Thank you for the assistance!