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2014 | OriginalPaper | Buchkapitel

Piecewise Linearity and Spectroscopic Properties from Koopmans-Compliant Functionals

verfasst von : Ismaila Dabo, Andrea Ferretti, Nicola Marzari

Erschienen in: First Principles Approaches to Spectroscopic Properties of Complex Materials

Verlag: Springer Berlin Heidelberg

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Abstract

Density-functional theory is an extremely powerful and widely used tool for quantum simulations. It reformulates the electronic-structure problem into a functional minimization with respect to the charge density of interacting electrons in an external potential. While exact in principle, it is approximate in practice, and even in its exact form it is meant to reproduce correctly only the total energy and its derivatives, such as forces, phonons, or dielectric properties. Quasiparticle levels are outside the scope of the theory, with the exception of the highest occupied state, since this is given by the derivative of the energy with respect to the number of electrons. A fundamental property of the exact energy functional is that of piecewise linearity at fractional occupations in between integer fillings, but common approximations do not follow such piecewise behavior, leading to a discrepancy between total and partial electron removal energies. Since the former are typically well described, and the latter provide, via Janak’s theorem, orbital energies, this discrepancy leads to a poor comparison between predicted and measured spectroscopic properties. We illustrate here the powerful consequences that arise from imposing the constraint of piecewise linearity to the total energy functional, leading to the emergence of orbital-density-dependent functionals that (1) closely satisfy a generalized Koopmans condition and (2) are able to describe with great accuracy spectroscopic properties.

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Vorheriges Kapitel Gas-Phase Valence-Electron Photoemission Spectroscopy Using Density Functional Theory

Nächstes Kapitel Optical Response of Extended Systems Using Time-Dependent Density Functional Theory

Note that besides functional approximations, the KS-DFT empty states need to be corrected for the derivative discontinuity of the potential upon infinitesimal electron addition. Such derivative discontinuity is usually neglected by approximate functionals which also tend to downshift further the orbital energies of empty states.

Otherwise, an infinitesimal transfer of charge δf > 0 from the highest occupied orbital to a state of energy ε _i < ε _ℋ would decrease the total energy by an amount (ε _i – ε _ℋ)δf < 0.

Reference [49] also highlights the limitations of conventional DFT approximations in capturing static correlation in spin-degenerate systems (the H₂ dissociation problem). Self-interaction errors arising from fractional occupations are nevertheless distinct from static correlation errors arising from fractional spins. In this work, only the self-interaction problem is addressed.

Koopmans’ theorem has been originally proven for the HF method considering frozen orbitals [109]. Here we refer to this case as the restricted Koopmans theorem. The generalized version of the theorem has been introduced later [110] in order to include orbital relaxation. We note in passing that the generalized Koopmans theorem is a property of the exact many-body Green’s function G. The performance of GW approximations in this regard has been recently discussed by Bruneval [48]. In fact, when adopting the Lehmann representation, the poles of G, playing the role of (Dyson) orbital energies, are exactly given by total energy differences corresponding to many-body states with different number of particles (with one electron added or removed).

One could rely on other definitions to measure the lack of Koopmans compliance. In particular, (30) has recently been exploited in [64] within the frozen orbital approximation. The comparative assessment of these closely related definitions is beyond the scope of this introductory review and will be discussed in detail elsewhere.

It is very instructive to note that the linear-response DFT + U method of Cococcioni and de Gironcoli [57] is obtained from a similar expansion to evaluate the U parameters for the N _I preselected orbitals χ _Ii of the Ith atom. In fact, in its simplest form, the nonlinearity correction reads

$$ {E}_U\left[{f}_1,{f}_2,\dots, {\varphi}_1,{\varphi}_2,\dots \right]={\displaystyle \sum_{I=1}^{N_{\mathrm{atom}}}{\displaystyle \sum_{i=1}^{N_I}{\scriptscriptstyle \frac{U_{Ii}}{2}}{n}_{Ii}\left(1-{n}_{Ii}\right)}} $$

with

$$ {U}_{Ii}={\displaystyle \int {d}^3\mathbf{r}{d}^3{\mathbf{r}}^{\mathbf{\prime}}{d}^3{\mathbf{r}}^{\mathbf{{\prime\prime}}}\left|{\chi}_{Ii}\right|{}^2\left(\mathbf{r}\right){\tilde{\varepsilon}}^{-1}\left(\mathbf{r},{\mathbf{r}}^{\mathbf{\prime}}\right){f}_{\mathrm{Hxc}}\left({\mathbf{r}}^{\mathbf{\prime}},{\mathbf{r}}^{\mathbf{{\prime\prime}}}\right)\left|{\chi}_{Ii}\right|{}^2\left({\mathbf{r}}^{\mathbf{{\prime\prime}}}\right)}\kern1em \mathrm{and}\kern1em {n}_{Ii}={\displaystyle \sum_{j=1}^{+\infty }{f}_j\left|\left\langle {\chi}_{Ii}\Big|{\varphi}_j\right\rangle \right|{}^2.} $$

The spirit of the Koopmans-compliant correction is identical with the advantage of not requiring preselected atomic orbitals.

We note that in Figs. 2 and 4 that we have used the Koopmans-compliant functional defined in (40), where the α screening coefficient has been included. We have adopted the same value for α in both figures. If no α were used [(35)], the K panel in Fig. 2 would show a flat curve, while that of Fig. 4 would display a negative slope as the HF method.

For instance, one could compute the average dielectric screening coefficient related to the orbital ψ _i through

$$ {\alpha}_i=\frac{{\displaystyle \int {d}^3\mathbf{r}{d}^3{\mathbf{r}}^{\mathbf{\prime}}{d}^3{\mathbf{r}}^{\mathbf{{\prime\prime}}}\left|{\psi}_i\right|{}^2\left(\mathbf{r}\right){\tilde{\varepsilon}}^{-1}\left(\mathbf{r},{\mathbf{r}}^{\mathbf{\prime}}\right){f}_{\mathrm{Hxc}}\left({\mathbf{r}}^{\mathbf{\prime}},{\mathbf{r}}^{\mathbf{{\prime\prime}}}\right)\left|{\psi}_i\right|{}^2\left({\mathbf{r}}^{\mathbf{{\prime\prime}}}\right)}}{{\displaystyle \int {d}^3\mathbf{r}{d}^3{\mathbf{r}}^{\mathbf{\prime}}\left|{\psi}_i\right|{}^2\left(\mathbf{r}\right){f}_{\mathrm{Hxc}}\left(\mathbf{r},{\mathbf{r}}^{\mathbf{\prime}}\right)\left|{\psi}_i\right|{}^2\left({\mathbf{r}}^{\mathbf{\prime}}\right)}}+\cdots, $$

where it is understood that each quantity that appears in the integrals must be calculated self-consistently.

The same approach is adopted when computing virtual orbital levels and band gaps within, e.g., hybrid DFT and DFT+U approximations.

A detailed sensitivity analysis of this approximation is presented in [62].

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Titel: Piecewise Linearity and Spectroscopic Properties from Koopmans-Compliant Functionals
verfasst von: Ismaila Dabo
Andrea Ferretti
Nicola Marzari
Verlag: Springer Berlin Heidelberg
Buch: First Principles Approaches to Spectroscopic Properties of Complex Materials
Print ISBN: 978-3-642-55067-6

Electronic ISBN: 978-3-642-55068-3

Copyright-Jahr: 2014
DOI: https://doi.org/10.1007/128_2013_504

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