Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Chemical Theory and Computation

Linked-Cluster Formulation of Electron–Hole Interaction Kernel in Real-Space Representation without Using Unoccupied States

Michael G. Bayne, Jeremy A. Scher, Benjamin H. Ellis, Arindam Chakraborty

Published May 21, 2018• 3 citations
Publisher

Abstract

Electron-hole or quasiparticle representation plays a central role in describing electronic excitations in many-electron systems. For charge-neutral excitation, the electron-hole interaction kernel is the quantity of interest for calculating important excitation properties such as optical gap, optical spectra, electron-hole recombination, and electron-hole binding energies. The electron-hole interaction kernel can be formally derived from the density-density correlation function using both Green's function and time-dependent density functional theory (TDDFT) formalism. The accurate determination of the electron-hole interaction kernel remains a significant challenge for precise calculations of optical properties in the GW+BSE formalism. From the TDDFT perspective, the electron-hole interaction kernel has been viewed as a path to systematic development of frequency-dependent exchange-correlation functionals. Traditional approaches, such as many-body perturbation theory formalism, use unoccupied states (which are defined with respect to Fermi vacuum) to construct the electron-hole interaction kernel. However, the inclusion of unoccupied states has long been recognized as the leading computational bottleneck that limits the application of this approach for larger finite systems. In this work, an alternative derivation that avoids using unoccupied states to construct the electron-hole interaction kernel is presented. The central idea of this approach is to use explicitly correlated geminal functions for treating electron-electron correlation for both ground and excited state wave functions. Using this ansatz, it is derived using both diagrammatic and algebraic techniques that the electron-hole interaction kernel can be expressed only in terms of linked closed-loop diagrams. It is proved that the cancellation of unlinked diagrams is a consequence of linked-cluster theorem in real-space representation. The electron-hole interaction kernel derived in this work was used to calculate excitation energies in many-electron systems, and results were found to be in good agreement with the EOM-CCSD and GW+BSE methods. The numerical results highlight the effectiveness of the developed method for overcoming the computational barrier of accurately determining the electron-hole interaction kernel to applications of large finite systems such as quantum dots and nanorods.

DOI
10.1021/acs.jctc.8b00123

Related papers

Many-body Theory for Excited-states Process

Meichun Huang

The most of the experimental measurable quantities in many-body electron system, such as the optical absorption, luminescent spectra and exciton effect, are related to the excitation states description. The density functional theory with local density approximation (DFT-LDA) as a ground state theory, i.e. the solutions based on the Kohn-Sham equation is a powerful tool for studying the ground state properties in many-particle systems. However, the first principles computation of excited states is more complexity than ground-state calculations. A key problem is that the exchange-correlation interaction in excited states is differ from the ground-states. In recent years, nevertheless, several electronic-excitation theories have been developed. The most important theoretical and computational method include the many-body perturbation theory which is based on the quasi-particle concept and the Green function equation, the time-dependent DFT and the Bethe-Salpeter equation for describing the electron-hole interaction. Among them the central ingredient is an electron's self-energy operator Σ that describes the exchange correlation effect beyond Hartree approximation. The implementations of the above theories are unavoidable to introduce some of approximations. A good approximation for Σ is the GW approach by Hedin. It is shown from computation simulations for many real condensed matter systems that the GW approach is a successful method for electronic-excitation problems. Another effective method for excited-states is so-called screen-exchange local-density approximation (sX-LDA). It is a combination of LDA and Hartree-Fock (HF)theory, in which the intrinsic local screen-exchange interaction has been replaced by a non-local one and a generalized Kohn-Sham equation (GKS) is given. Based on a PWscf package, the sX-LDA-PWscf method has been tested and indicates that an available approach for treating band-gap problem in semiconductors. In this paper, the many-body theories for the excitation process, its development and signification are reviewed. The relation and differences consist in different theoretical methods are discussed. On the based of above discussions, the applications of many-body theory to the band-band transition (band-gap underestimation problem) and exciton effect in semiconductors are presented.

Chinese Journal of Luminescence•Jan 1, 2005•0 citations

Excitonic effects in time-dependent density-functional theory: An analytically solvable model

Ralf Stubner, I. V. Tokatly, Oleg Pankratov

We investigate the description of excitonic effects within time-dependent density-functional theory (TDDFT). The exchange-correlation kernel ${f}_{xc}$ introduced in TDDFT allows a clear separation of quasiparticle and excitonic effects. Using a diagrammatic representation for ${f}_{xc}$, we express its excitonic part ${f\phantom{\rule{0.2em}{0ex}}}_{xc}^{\mathrm{Ex}}$ in terms of the effective vertex function $\ensuremath{\Lambda}$. The latter fulfills an integral equation that thereby establishes the exact correspondence between TDDFT and the standard many-body approach based on the Bethe-Salpeter equation (BSE). The diagrammatic structure of the kernel in the equation for $\ensuremath{\Lambda}$ suggests the possibility of strong cancellation effects. Should the cancellation take place, already the first-order approximation to ${f\phantom{\rule{0.2em}{0ex}}}_{xc}^{\mathrm{Ex}}$ is sufficient. A potential advantage of TDDFT over the many-body BSE method is thus dependent on the efficiency of the above-quoted cancellation. We explicitly verify this for an analytically solvable two-dimensional two-band model. The calculations confirm that the low-order ${f\phantom{\rule{0.2em}{0ex}}}_{xc}^{\mathrm{Ex}}$ perfectly describes the bound exciton as well as the excitonic effects in the continuous spectrum in a wide range of the electron-hole coupling strength.

arXiv:cond-mat/0406344•Physical Review B•Dec 23, 2004•43 citations

Diagrammatic techniques for time-dependent density-functional theory

Ralf Stubner

The density-functional theory (DFT) is one of the most important ab initio methods for calculating properties of many-body systems. The success of DFT is based on the Kohn-Sham (KS) method and the approximations for the exchange-correlation (xc) effects. These approximations are well understood. However, DFT is first and foremost a theory for the ground state. It is very difficult to obtain properties of excited states. The time-dependent density-functional theory (TDDFT) can overcome this restriction and in the linear-response regime can give access to the properties of excited states. However, the xc effects, which are contained in the xc kernel f_xc , have to be approximated. The approximations for f_xc are not well understood up to now. Green-functions-based conventional many-body theory (MBT) is an alternative approach to the many-body problem. Especially when combined with diagrammatic techniques, MBT allows to derive physically motivated approximations. This work investigates the use of diagrammatic techniques from MBT for TDDFT . The aim of the investigations is to contribute to the development, testing, and understanding of the approximations involved in the time-dependent case. It is shown that it is possible to expand the xc kernel in a diagrammatic series. Explicit diagrammatic rules for constructing f_xc in any order of the perturbation theory are given. The derived diagrammatic rules enable us to study the analytic properties of f_xc. It is shown that f_xc is finite at KS excitation energies in every order of the perturbation theory. If this were not the case, finite-order approximations to f_xc would not give sensible results for the kernel which plays the role of a mass operator connecting the KS density--density response function with the exact density--density response function. However, there is one important difference between f_xc and the self-energy, the latter playing the role of the mass operator connecting free and exact Green function. This becomes apparent when the perturbative expansion for f_xc is used to calculate the correction to the KS excitation energies. This correction is obtained up to the second-order with respect to the interaction and with respect to one- and two-particle irreducible elements. It is shown that if a perturbative approximation for f_xc is used, the correction to the KS excitation energies has to be calculated using a consistent approach based on the same order of perturbation theory for both quantities. One of the most promising applications of TDDFT in the linear response regime is the calculation of electronic excitation spectra of solids, where TDDFT could replace the computationally expensive solution of the BSE , which is normally used for describing the excitonic correlations. It is shown that the xc kernel can be separated into a quasiparticle and a excitonic part in an exact and unambiguous way. The diagrammatic rules derived for f_xc apply to these parts separately. This allows us to express the excitonic part in terms of the three-point function, for which an integral equation is derived. This integral equation is similar to the BSE and gives the exact translation of the BSE into the TDDFT language. There exists the possibility of cancellation effects in the kernel of the integral equation. If these effects are effective, a first-order approximation to the excitonic f_xc is sufficient. The cancellation effects in the equation for Lambda are explicitly studied for a two-band model semiconductor. Using a general short-range interaction between the quasiparticles (QPs) all relevant equations are solved analytically. It is shown that the cancellation is most effective directly at the band-gap energy. Consequently, the first-order and the exact excitonic f_xc are very similar close to the band-gap energy for all interaction strengths. For a weak interaction the first-order approximation to the excitonic f_xc describes the different excitonic correlations in the density©density response function very well. Qualitatively similar results are found when the QPs interact via the long-ranged Coulomb interaction.

•Jan 1, 2005•0 citations

Self-consistent density-functional approach to the correlated ground states and an unrestricted many-body perturbation theory

Behnam Farid

We introduce a scheme through which the ground-state electronic density n and total energy Ev [n] of inhomogeneous interacting systems can be determined in a self-consistent (SC) way. It is based on a SC perturbation expansion for the single-particle Green function G. Our employed expansion is argued to be valid under all conditions provided that the ground-state charge density of the interacting system is pure-state non-interacting v representable. The formalism discussed in this paper has been applied to a quasi-one-dimensional periodic system. It has even been possible to carry out SC calculations with the dynamically screened exchange diagram (the so-called GW diagram, where W is the dynamically screened electron-electron interaction function) for the self-energy operator Σ. A coherent mode of interpretation of a number of our results entails that the overwhelming success of the GW framework in relatively accurately reproducing the experimental energy gaps of a number of semiconductors and insulators may be a direct consequence of non-self-consistency effects; whereas a non-self-consistent calculation in our tests increases the value of the local-density approximation (LDA) gap in the range of 40–65% of the ‘experimental’ gaps, self-consistency effects reduce this range to 14–17%. The same line of argument leads us to conclude that, in our model, 26–35% of the LDA error in the gap energies is due to the absence of a well known discontinuity in the LDA exchange-correlation potential, and the remaining part is due to the Kohn-Sham eigenvalues not being identifiable with one-particle excitation energies. A natural conclusion that may be drawn from these observations is that a correct description of excitation energies (e.g. of the gap energy) cannot be achieved without including, at least approximately, the vertex part of Σ, which is entirely absent in the GW scheme.

Philosophical Magazine B•Aug 1, 1997•18 citations

Development of multicomponent coupled-cluster method for investigation of multiexcitonic interactions

Benjamin H. Ellis, Somil Aggarwal, Arindam Chakraborty

Multicomponent systems are defined as chemical systems that require a quantum mechanical description of two or more different types of particles. Non-Born-Oppenheimer electron-nuclear interactions in molecules, electron-hole interactions in electronically excited nanoparticles, and electron-positron interactions are examples of physical systems that require a multicomponent quantum mechanical formalism. The central challenge in the theoretical treatment of multicomponent systems is capturing the many-body correlation effects that exist not only between particles of identical types (electron-electron) but also between particles of different types (electron-nuclear and electron-hole). In this work, the development and implementation of multicomponent coupled-cluster (mcCC) theory for treating particle-particle correlation in multicomponent systems is presented. This method provides a balanced treatment of many-particle correlation in a general multicomponent system while maintaining a size-consistent and size-extensive formalism. The coupled-cluster ansatz presented here is the extension of the electronic structure CCSD formulation for multicomponent systems and is defined as $\vert Ψ_\mathrm{mcCC} \rangle = e^{T_1^\mathrm{I}+T_2^\mathrm{I}+T_1^\mathrm{II}+T_2^\mathrm{II}+T_{11}^\mathrm{I,II}+T_{12}^\mathrm{I,II}+T_{21}^\mathrm{I,II}+T_{22}^\mathrm{I,II}}\vert 0^\mathrm{I}0^\mathrm{II}\rangle$. The applicability of the mcCC method was demonstrated by computing biexciton binding energies for multiexcitonic systems and benchmarking the results against full configuration interaction calculations. The results demonstrated that connected cluster operators that generate simultaneous excitation in type I and type II space are critical for capturing electron-hole correlation in multiexcitonic systems.

arXiv:1510.05762•Oct 20, 2015•0 citations

Ab initiocalculation of the exchange-correlation kernel in extended systems

Gianni Adragna, Rodolfo Del Sole, Andrea Marini

We develop a method to calculate the polarizability of a many-electron system within Green's function theory in a similar way as within time-dependent density functional theory and apply it to Si and GaAs. The method joins the computational simplicity of the latter with the accuracy of the former approach. The resulting exchange-correlation many-body kernel ${f}_{\mathrm{xc}}^{\mathrm{FQP}}$ and optical spectra are in good agreement with those obtained by the solution of the Bethe-Salpeter equation.

Physical review. B, Condensed matter•Oct 29, 2003•88 citations
Electron-hole or quasiparticle representation plays a central role in describing electronic excitations in many-electron systems. For charge-neutral excitation, the electron-hole interaction kernel is the quantity of interest for calculating important excitation properties such as optical gap, optical spectra, electron-hole recombination, and electron-hole binding energies. The electron-hole interaction kernel can be formally derived from the density-density correlation function using both Green's function and time-dependent density functional theory (TDDFT) formalism. The accurate determination of the electron-hole interaction kernel remains a significant challenge for precise calculations of optical properties in the GW+BSE formalism. From the TDDFT perspective, the electron-hole interaction kernel has been viewed as a path to systematic development of frequency-dependent exchange-correlation functionals. Traditional approaches, such as many-body perturbation theory formalism, use unoccupied states (which are defined with respect to Fermi vacuum) to construct the electron-hole interaction kernel. However, the inclusion of unoccupied states has long been recognized as the leading computational bottleneck that limits the application of this approach for larger finite systems. In this work, an alternative derivation that avoids using unoccupied states to construct the electron-hole interaction kernel is presented. The central idea of this approach is to use explicitly correlated geminal functions for treating electron-electron correlation for both ground and excited state wave functions. Using this ansatz, it is derived using both diagrammatic and algebraic techniques that the electron-hole interaction kernel can be expressed only in terms of linked closed-loop diagrams. It is proved that the cancellation of unlinked diagrams is a consequence of linked-cluster theorem in real-space representation. The electron-hole interaction kernel derived in this work was used to calculate excitation energies in many-electron systems, and results were found to be in good agreement with the EOM-CCSD and GW+BSE methods. The numerical results highlight the effectiveness of the developed method for overcoming the computational barrier of accurately determining the electron-hole interaction kernel to applications of large finite systems such as quantum dots and nanorods.
The most of the experimental measurable quantities in many-body electron system, such as the optical absorption, luminescent spectra and exciton effect, are related to the excitation states description. The density functional theory with local density approximation (DFT-LDA) as a ground state theory, i.e. the solutions based on the Kohn-Sham equation is a powerful tool for studying the ground state properties in many-particle systems. However, the first principles computation of excited states is more complexity than ground-state calculations. A key problem is that the exchange-correlation interaction in excited states is differ from the ground-states. In recent years, nevertheless, several electronic-excitation theories have been developed. The most important theoretical and computational method include the many-body perturbation theory which is based on the quasi-particle concept and the Green function equation, the time-dependent DFT and the Bethe-Salpeter equation for describing the electron-hole interaction. Among them the central ingredient is an electron's self-energy operator Σ that describes the exchange correlation effect beyond Hartree approximation. The implementations of the above theories are unavoidable to introduce some of approximations. A good approximation for Σ is the GW approach by Hedin. It is shown from computation simulations for many real condensed matter systems that the GW approach is a successful method for electronic-excitation problems. Another effective method for excited-states is so-called screen-exchange local-density approximation (sX-LDA). It is a combination of LDA and Hartree-Fock (HF)theory, in which the intrinsic local screen-exchange interaction has been replaced by a non-local one and a generalized Kohn-Sham equation (GKS) is given. Based on a PWscf package, the sX-LDA-PWscf method has been tested and indicates that an available approach for treating band-gap problem in semiconductors. In this paper, the many-body theories for the excitation process, its development and signification are reviewed. The relation and differences consist in different theoretical methods are discussed. On the based of above discussions, the applications of many-body theory to the band-band transition (band-gap underestimation problem) and exciton effect in semiconductors are presented.
We investigate the description of excitonic effects within time-dependent density-functional theory (TDDFT). The exchange-correlation kernel fxc{f}_{xc}fxc​ introduced in TDDFT allows a clear separation of quasiparticle and excitonic effects. Using a diagrammatic representation for fxc{f}_{xc}fxc​, we express its excitonic part fxcEx{f\phantom{\rule{0.2em}{0ex}}}_{xc}^{\mathrm{Ex}}fxcEx​ in terms of the effective vertex function $\ensuremath{\Lambda}$. The latter fulfills an integral equation that thereby establishes the exact correspondence between TDDFT and the standard many-body approach based on the Bethe-Salpeter equation (BSE). The diagrammatic structure of the kernel in the equation for $\ensuremath{\Lambda}$ suggests the possibility of strong cancellation effects. Should the cancellation take place, already the first-order approximation to fxcEx{f\phantom{\rule{0.2em}{0ex}}}_{xc}^{\mathrm{Ex}}fxcEx​ is sufficient. A potential advantage of TDDFT over the many-body BSE method is thus dependent on the efficiency of the above-quoted cancellation. We explicitly verify this for an analytically solvable two-dimensional two-band model. The calculations confirm that the low-order fxcEx{f\phantom{\rule{0.2em}{0ex}}}_{xc}^{\mathrm{Ex}}fxcEx​ perfectly describes the bound exciton as well as the excitonic effects in the continuous spectrum in a wide range of the electron-hole coupling strength.
The density-functional theory (DFT) is one of the most important ab initio methods for calculating properties of many-body systems. The success of DFT is based on the Kohn-Sham (KS) method and the approximations for the exchange-correlation (xc) effects. These approximations are well understood. However, DFT is first and foremost a theory for the ground state. It is very difficult to obtain properties of excited states. The time-dependent density-functional theory (TDDFT) can overcome this restriction and in the linear-response regime can give access to the properties of excited states. However, the xc effects, which are contained in the xc kernel f_xc , have to be approximated. The approximations for f_xc are not well understood up to now. Green-functions-based conventional many-body theory (MBT) is an alternative approach to the many-body problem. Especially when combined with diagrammatic techniques, MBT allows to derive physically motivated approximations. This work investigates the use of diagrammatic techniques from MBT for TDDFT . The aim of the investigations is to contribute to the development, testing, and understanding of the approximations involved in the time-dependent case. It is shown that it is possible to expand the xc kernel in a diagrammatic series. Explicit diagrammatic rules for constructing f_xc in any order of the perturbation theory are given. The derived diagrammatic rules enable us to study the analytic properties of f_xc. It is shown that f_xc is finite at KS excitation energies in every order of the perturbation theory. If this were not the case, finite-order approximations to f_xc would not give sensible results for the kernel which plays the role of a mass operator connecting the KS density--density response function with the exact density--density response function. However, there is one important difference between f_xc and the self-energy, the latter playing the role of the mass operator connecting free and exact Green function. This becomes apparent when the perturbative expansion for f_xc is used to calculate the correction to the KS excitation energies. This correction is obtained up to the second-order with respect to the interaction and with respect to one- and two-particle irreducible elements. It is shown that if a perturbative approximation for f_xc is used, the correction to the KS excitation energies has to be calculated using a consistent approach based on the same order of perturbation theory for both quantities. One of the most promising applications of TDDFT in the linear response regime is the calculation of electronic excitation spectra of solids, where TDDFT could replace the computationally expensive solution of the BSE , which is normally used for describing the excitonic correlations. It is shown that the xc kernel can be separated into a quasiparticle and a excitonic part in an exact and unambiguous way. The diagrammatic rules derived for f_xc apply to these parts separately. This allows us to express the excitonic part in terms of the three-point function, for which an integral equation is derived. This integral equation is similar to the BSE and gives the exact translation of the BSE into the TDDFT language. There exists the possibility of cancellation effects in the kernel of the integral equation. If these effects are effective, a first-order approximation to the excitonic f_xc is sufficient. The cancellation effects in the equation for Lambda are explicitly studied for a two-band model semiconductor. Using a general short-range interaction between the quasiparticles (QPs) all relevant equations are solved analytically. It is shown that the cancellation is most effective directly at the band-gap energy. Consequently, the first-order and the exact excitonic f_xc are very similar close to the band-gap energy for all interaction strengths. For a weak interaction the first-order approximation to the excitonic f_xc describes the different excitonic correlations in the density©density response function very well. Qualitatively similar results are found when the QPs interact via the long-ranged Coulomb interaction.
We introduce a scheme through which the ground-state electronic density n and total energy Ev [n] of inhomogeneous interacting systems can be determined in a self-consistent (SC) way. It is based on a SC perturbation expansion for the single-particle Green function G. Our employed expansion is argued to be valid under all conditions provided that the ground-state charge density of the interacting system is pure-state non-interacting v representable. The formalism discussed in this paper has been applied to a quasi-one-dimensional periodic system. It has even been possible to carry out SC calculations with the dynamically screened exchange diagram (the so-called GW diagram, where W is the dynamically screened electron-electron interaction function) for the self-energy operator Σ. A coherent mode of interpretation of a number of our results entails that the overwhelming success of the GW framework in relatively accurately reproducing the experimental energy gaps of a number of semiconductors and insulators may be a direct consequence of non-self-consistency effects; whereas a non-self-consistent calculation in our tests increases the value of the local-density approximation (LDA) gap in the range of 40–65% of the ‘experimental’ gaps, self-consistency effects reduce this range to 14–17%. The same line of argument leads us to conclude that, in our model, 26–35% of the LDA error in the gap energies is due to the absence of a well known discontinuity in the LDA exchange-correlation potential, and the remaining part is due to the Kohn-Sham eigenvalues not being identifiable with one-particle excitation energies. A natural conclusion that may be drawn from these observations is that a correct description of excitation energies (e.g. of the gap energy) cannot be achieved without including, at least approximately, the vertex part of Σ, which is entirely absent in the GW scheme.
Multicomponent systems are defined as chemical systems that require a quantum mechanical description of two or more different types of particles. Non-Born-Oppenheimer electron-nuclear interactions in molecules, electron-hole interactions in electronically excited nanoparticles, and electron-positron interactions are examples of physical systems that require a multicomponent quantum mechanical formalism. The central challenge in the theoretical treatment of multicomponent systems is capturing the many-body correlation effects that exist not only between particles of identical types (electron-electron) but also between particles of different types (electron-nuclear and electron-hole). In this work, the development and implementation of multicomponent coupled-cluster (mcCC) theory for treating particle-particle correlation in multicomponent systems is presented. This method provides a balanced treatment of many-particle correlation in a general multicomponent system while maintaining a size-consistent and size-extensive formalism. The coupled-cluster ansatz presented here is the extension of the electronic structure CCSD formulation for multicomponent systems and is defined as ∣ΨmcCC⟩=eT1I+T2I+T1II+T2II+T11I,II+T12I,II+T21I,II+T22I,II∣0I0II⟩\vert Ψ_\mathrm{mcCC} \rangle = e^{T_1^\mathrm{I}+T_2^\mathrm{I}+T_1^\mathrm{II}+T_2^\mathrm{II}+T_{11}^\mathrm{I,II}+T_{12}^\mathrm{I,II}+T_{21}^\mathrm{I,II}+T_{22}^\mathrm{I,II}}\vert 0^\mathrm{I}0^\mathrm{II}\rangle∣ΨmcCC​⟩=eT1I​+T2I​+T1II​+T2II​+T11I,II​+T12I,II​+T21I,II​+T22I,II​∣0I0II⟩. The applicability of the mcCC method was demonstrated by computing biexciton binding energies for multiexcitonic systems and benchmarking the results against full configuration interaction calculations. The results demonstrated that connected cluster operators that generate simultaneous excitation in type I and type II space are critical for capturing electron-hole correlation in multiexcitonic systems.
We develop a method to calculate the polarizability of a many-electron system within Green's function theory in a similar way as within time-dependent density functional theory and apply it to Si and GaAs. The method joins the computational simplicity of the latter with the accuracy of the former approach. The resulting exchange-correlation many-body kernel fxcFQP{f}_{\mathrm{xc}}^{\mathrm{FQP}}fxcFQP​ and optical spectra are in good agreement with those obtained by the solution of the Bethe-Salpeter equation.