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Implementation of EDM using the continued fraction representation of Fermi function
Date: 2016/10/17 22:03
Name: Ari Ojanper&#228;   <>


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 [1] Phys. Rev. B, 75 (035123), 2007 ( and [2] Phys. Rev. B, 81 (035116), 2010 ( I've followed the equations presented in Appendix B of Ref. [2]. 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 [1]) 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. [2] 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!

Kind regards,
Ari Ojanper&#228;
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Re: Implementation of EDM using the continued fraction representation of Fermi function ( No.1 )
Date: 2016/10/18 00:23
Name: T. Ozaki


The calculation of EDM is more sensitive to numerical round-off error than that for the DM.
After the publication of Ref. [2], I also noticed a similar problem, and proposed a revised
scheme as shown by Eq. (15) in [3] PRB 82, 075131 (2010). By looking at Eq. (10) of [3] carefully,
one may notice that the second and third terms are in the same order as a function of number
of poles, but the sign is opposite. Thereby the second term (zero-th order moment) has to be very
accurately calculated. Eq.(13) of [3] allows us to accurately calculate the zero-th and first
order moments. As shown in Table III of [3], it turns out that the scheme is accurate and stable.



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