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Gap edge eigenpairs from density matrix purification using moments of the Dirac distribution

Lionel Alexandre Truflandier

Abstract

In this work, we propose a simple method to resolve the eigenstates located at the band gap edges of an electronic eigenspectrum using only the quasi-purified one-particle density matrix as input. The theoretical framework relies on the decomposition of the occupation number variance into a particle and hole moment. These moments, when purified using power narrowing iterations, allow to isolate the higher occupied and lower unoccupied single state projectors, giving readily access to the corresponding eigenpairs. We demonstrate that when degeneracy is encountered, power narrowing remains able to deliver relevant mixed states. From a benchmark of selected molecules, we show that the method is robust and efficient since it requires no more that a dozen of matrix-matrix multiplications at worst. The possibility of reducing the computational cost using Lanczos subspace approach is discussed. The very low algorithmic complexity of power narrowing makes it very easy to implement in electronic structure codes or libraries already featuring Fermi operator expansion or density matrix purifications.

Gap edge eigenpairs from density matrix purification using moments of the Dirac distribution

Abstract

In this work, we propose a simple method to resolve the eigenstates located at the band gap edges of an electronic eigenspectrum using only the quasi-purified one-particle density matrix as input. The theoretical framework relies on the decomposition of the occupation number variance into a particle and hole moment. These moments, when purified using power narrowing iterations, allow to isolate the higher occupied and lower unoccupied single state projectors, giving readily access to the corresponding eigenpairs. We demonstrate that when degeneracy is encountered, power narrowing remains able to deliver relevant mixed states. From a benchmark of selected molecules, we show that the method is robust and efficient since it requires no more that a dozen of matrix-matrix multiplications at worst. The possibility of reducing the computational cost using Lanczos subspace approach is discussed. The very low algorithmic complexity of power narrowing makes it very easy to implement in electronic structure codes or libraries already featuring Fermi operator expansion or density matrix purifications.

Paper Structure

This paper contains 22 equations, 3 figures, 1 table.

Figures (3)

  • Figure 1: (a) Product of the Fermi-Dirac $\rho_{\beta}$-distribution with $f(\epsilon)$ as a function of the energy, for different temperatures in units of $k_\textrm{B}$. The step function is plotted with a solid purple line. The band gap is fixed to $0.4$ eV and centered on the chemical potential $\mu=0$. (b) Corresponding Dirac $\delta_{\beta}$-distributions. (c) and (d) Particle and hole filters, $\omega_{\beta}$ (red colormap) and $\bar{\omega}_{\beta}$ (blue color map), respectively. In these 4 plots the dashed lines represent the expected behavior without band gap. (e) and (f) Normalized $\omega_{\beta}$ and $\bar{\omega}_{\beta}$ as a function of temperature. Positions of the last occupied ($\epsilon_{N}$), first unoccupied ($\epsilon_{N+1}$) and $\mu$ energies are indicated on the $x$-axis.
  • Figure 2: (a) Estimates [Eq. (\ref{['eq:momeigs']})] and purified [Eq. (\ref{['eq:mompureigs']})] LU energy obtained by power narrowing, as a function of temperature for $\epsilon\in[-40;+40]$ and a grid spacing $\delta\epsilon=10^{-5}$ eV. Expected value is 0.5 eV. The number of iterations used in power narrowing for $k=3$ is given by the blue circles (right $y$-axis). (b) Convergence of the power narrowing, for different values of $k$ at $T=250$ K.
  • Figure 3: From left to right: molecular structure of quetiapine, HOMO obtained with power narrowing purification and diagonalisation, respectively.