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

The Journal of Chemical Physics

Linear-response time-dependent density-functional theory with pairing fields

Degao Peng, Helen van Aggelen, Yang Yang, Weitao Yang

Published Mar 14, 2014• 39 citations
Publisher

Abstract

Recent development in particle-particle random phase approximation (pp-RPA) broadens the perspective on ground state correlation energies [H. van Aggelen, Y. Yang, and W. Yang, Phys. Rev. A 88, 030501 (2013), Y. Yang, H. van Aggelen, S. N. Steinmann, D. Peng, and W. Yang, J. Chem. Phys. 139, 174110 (2013); D. Peng, S. N. Steinmann, H. van Aggelen, and W. Yang, J. Chem. Phys. 139, 104112 (2013)] and N ± 2 excitation energies [Y. Yang, H. van Aggelen, and W. Yang, J. Chem. Phys. 139, 224105 (2013)]. So far Hartree-Fock and approximated density-functional orbitals have been utilized to evaluate the pp-RPA equation. In this paper, to further explore the fundamentals and the potential use of pairing matrix dependent functionals, we present the linear-response time-dependent density-functional theory with pairing fields with both adiabatic and frequency-dependent kernels. This theory is related to the density-functional theory and time-dependent density-functional theory for superconductors, but is applied to normal non-superconducting systems for our purpose. Due to the lack of the proof of the one-to-one mapping between the pairing matrix and the pairing field for time-dependent systems, the linear-response theory is established based on the representability assumption of the pairing matrix. The linear response theory justifies the use of approximated density-functionals in the pp-RPA equation. This work sets the fundamentals for future density-functional development to enhance the description of ground state correlation energies and N ± 2 excitation energies.

DOI
10.1063/1.4867540

Related papers

Exchange-correlation energy from pairing matrix fluctuation and the particle-particle random-phase approximation

Helen van Aggelen, Yang Yang, Weitao Yang

We formulate an adiabatic connection for the exchange-correlation energy in terms of pairing matrix fluctuation. This connection opens new channels for density functional approximations based on pairing interactions. Even the simplest approximation to the pairing matrix fluctuation, the particle-particle random phase approximation (pp-RPA), has some highly desirable properties. It has no delocalization error with a nearly linear energy behavior for systems with fractional charges, describes van der Waals interactions similarly and thermodynamic properties significantly better than particle-hole RPA, and eliminates static correlation error for single-bond systems. Most significantly, the pp-RPA is the first known functional that has an explicit and closed-form dependence on the occupied and unoccupied orbitals and captures the energy derivative discontinuity in strongly correlated systems. These findings illustrate the potential of including pairing interactions within a density functional framework.

arXiv:1306.4957•Physical Review A•Sep 24, 2013•105 citations

Time-Dependent Quantum Many-Body Systems: Linear Response, Electronic Transport, and Reduced Density Matrices

Heiko Appel

Time-dependent density functional theory (TDDFT) provides a successful approach to calculate excitation energies of atomic and molecular systems. In part I of this work we present a double-pole approximation (DPA) to the response equations of TDDFT. The double-pole approximation provides an exact description of systems with two strongly coupled excitations which are isolated from the rest of the spectrum. In contrast to the traditional single- pole approximation of TDDFT the DPA also yields corrections to the Kohn-Sham oscillator strengths. Several critical pole separations can be identified, e.g. we find that the pole coupling can cause transitions to vanish entirely from the optical spectrum. We also demonstrate how to invert the double-pole solution which allows us to predict matrix elements of the exchange- correlation kernel fxc from experimental input. This can serve as benchmark for the construction of future approximations for the kernel fxc. Reduced density matrix functional theory (RDMFT) has emerged recently as promising candidate to treat strongly correlated electronic many-body systems beyond traditional density functional theory (DFT). The research within RDMFT was so far focussed on the static theory. In this work we attempt some first steps towards a time-dependent generalization of RDMFT. In part II we derive equations of motion for natural orbitals and occupation numbers. Using the equation of motion for the occupation numbers we show that an adiabatic extension of presently known ground-state functionals of static RDMFT always leads to occupation numbers which are constant in time. From the stationary conditions of the equations of motion for the N-body correlations (correlated parts of the N-body matrices) we derive a new class of ground-state functionals which can be used in static RDMFT. Applications are presented for a one-dimensional model system where the time-dependent many-body Schrödinger equation can be propagated numerically. We use optimal control theory to find optimized laser pulses for transitions in a model for atomic Helium. From the numerically exact correlated wavefunction we extract the exact time evolution of natural orbitals and occupation numbers for (i) laser-driven Helium and (ii) electron-ion scattering. Part III of this work considers time-dependent quantum transport within TDDFT. We present an algorithm for the calculation of extended eigenstates of single- particle Hamiltonians which is especially tailored to a finite-difference discretization of the Schrödinger equation. We consider the propagation of finite mesoscopic systems and demonstrate the limitations of such an approach. To overcome the shortcomings of a description of quantum transport in terms of finite systems we develop a time-propagation scheme for extended states which utilizes a mixed basis representation. Our discretization scheme allows to treat central device and lead regions on the same footing thus preventing artificial reflections at grid boundaries.

•Jan 1, 2007•10 citations

Response Function Technique for Calculating the Random-Phase Approximation Correlation Energy

Y. R. Shimizu, Paola Donati, R.A. Broglia

We develop a scheme to exactly evaluate the correlation energy in the random-phase approximation, based on linear response theory [Y. R. Shimizu, J. D. Garrett, R. A. Broglia, M. Gallardo, and E. Vigezzi, Rev. Mod. Phys. 61, 131 (1989)]. It is demonstrated that our formula is equivalent to a contour integral representation recently proposed [F. Donau, D. Almehed, and R. G. Nazmitdinov, Phys. Rev. Lett. 83, 280 (1999)] being numerically more efficient for realistic calculations. Examples are presented for pairing correlations in rapidly rotating nuclei.

arXiv:nucl-th/9911039•Physical Review Letters•Sep 11, 2000•16 citations

The formal framework of linear-response TDDFT

Carsten A. Ullrich

This chapter discusses the linear-response framework of time-dependent density-functional theory (TDDFT). It begins with a thorough and self-contained exposition of general linear-response theory for many-body systems. The time-dependent as well as the frequency-dependent response are discussed. Several exact representations and properties of the density-density response function are reviewed, such as the Lehmann representation and fluctuation-dissipation theorem, and the Kramers-Kronig relations. It is shown how several important spectroscopic observables are obtained from linear-response theory. Then, the TDDFT formulation of linear-response theory is derived. The key quantity in this formalism is the so-called exchange-correlation kernel. The remainder of the chapter demonstrates how this linear-response formalism can be used to calculate excitation energies in principle exactly with TDDFT. The Casida equation is derived, as well as several approximate schemes. A comparison is made with time-dependent Hartree-Fock theory.

Oxford University Press eBooks•Dec 22, 2011•0 citations

Range-separated density-functional theory with the random-phase approximation: Detailed formalism and illustrative applications

Julien Toulouse, Wuming Zhu, János G. Ángyán +1 more

Using Green's-function many-body theory, we present the details of a formally exact adiabatic-connection fluctuation-dissipation density-functional theory based on range separation, which was sketched in Toulouse et al. [Phys. Rev. Lett. 102, 096404 (2009)]. Range-separated density-functional theory approaches combining short-range density-functional approximations with long-range random-phase approximations (RPAs) are then obtained as well-identified approximations on the long-range Green's-function self-energy. Range-separated RPA-type schemes with or without long-range Hartree-Fock exchange response kernel are assessed on rare-gas and alkaline-earth-metal dimers and compared to range-separated second-order perturbation theory and range-separated coupled-cluster theory.

arXiv:1006.2061•Physical Review A•Sep 9, 2010•130 citations

The adiabatic approximation in time-dependent density matrix functional theory: Response properties from dynamics of phase-including natural orbitals

Klaas J. H. Giesbertz, O. V. Gritsenko, Evert Jan Baerends

The adiabatic approximation is problematic in time-dependent density matrix functional theory. With pure density matrix functionals (invariant under phase change of the natural orbitals) it leads to lack of response in the occupation numbers, hence wrong frequency dependent responses, in particular α(ω→0)≠α(0) (the static polarizability). We propose to relinquish the requirement that the functional must be a pure one-body reduced density matrix (1RDM) functional, and to introduce additional variables which can be interpreted as phases of the one-particle states of the independent particle reference system formed with the natural orbitals, thus obtaining so-called phase-including natural orbital (PINO) functionals. We also stress the importance of the correct choice of the complex conjugation in the two-electron integrals in the commonly used functionals (they should not be of exchange type). We demonstrate with the Löwdin-Shull energy expression for two-electron systems, which is an example of a PINO functional, that for two-electron systems exact responses (polarizabilities, excitation energies) are obtained, while writing this energy expression in the usual way as a 1RDM functional yields erroneous responses.

The Journal of Chemical Physics•Nov 4, 2010•32 citations
Recent development in particle-particle random phase approximation (pp-RPA) broadens the perspective on ground state correlation energies [H. van Aggelen, Y. Yang, and W. Yang, Phys. Rev. A 88, 030501 (2013), Y. Yang, H. van Aggelen, S. N. Steinmann, D. Peng, and W. Yang, J. Chem. Phys. 139, 174110 (2013); D. Peng, S. N. Steinmann, H. van Aggelen, and W. Yang, J. Chem. Phys. 139, 104112 (2013)] and N ± 2 excitation energies [Y. Yang, H. van Aggelen, and W. Yang, J. Chem. Phys. 139, 224105 (2013)]. So far Hartree-Fock and approximated density-functional orbitals have been utilized to evaluate the pp-RPA equation. In this paper, to further explore the fundamentals and the potential use of pairing matrix dependent functionals, we present the linear-response time-dependent density-functional theory with pairing fields with both adiabatic and frequency-dependent kernels. This theory is related to the density-functional theory and time-dependent density-functional theory for superconductors, but is applied to normal non-superconducting systems for our purpose. Due to the lack of the proof of the one-to-one mapping between the pairing matrix and the pairing field for time-dependent systems, the linear-response theory is established based on the representability assumption of the pairing matrix. The linear response theory justifies the use of approximated density-functionals in the pp-RPA equation. This work sets the fundamentals for future density-functional development to enhance the description of ground state correlation energies and N ± 2 excitation energies.
We formulate an adiabatic connection for the exchange-correlation energy in terms of pairing matrix fluctuation. This connection opens new channels for density functional approximations based on pairing interactions. Even the simplest approximation to the pairing matrix fluctuation, the particle-particle random phase approximation (pp-RPA), has some highly desirable properties. It has no delocalization error with a nearly linear energy behavior for systems with fractional charges, describes van der Waals interactions similarly and thermodynamic properties significantly better than particle-hole RPA, and eliminates static correlation error for single-bond systems. Most significantly, the pp-RPA is the first known functional that has an explicit and closed-form dependence on the occupied and unoccupied orbitals and captures the energy derivative discontinuity in strongly correlated systems. These findings illustrate the potential of including pairing interactions within a density functional framework.
Time-dependent density functional theory (TDDFT) provides a successful approach to calculate excitation energies of atomic and molecular systems. In part I of this work we present a double-pole approximation (DPA) to the response equations of TDDFT. The double-pole approximation provides an exact description of systems with two strongly coupled excitations which are isolated from the rest of the spectrum. In contrast to the traditional single- pole approximation of TDDFT the DPA also yields corrections to the Kohn-Sham oscillator strengths. Several critical pole separations can be identified, e.g. we find that the pole coupling can cause transitions to vanish entirely from the optical spectrum. We also demonstrate how to invert the double-pole solution which allows us to predict matrix elements of the exchange- correlation kernel fxc from experimental input. This can serve as benchmark for the construction of future approximations for the kernel fxc. Reduced density matrix functional theory (RDMFT) has emerged recently as promising candidate to treat strongly correlated electronic many-body systems beyond traditional density functional theory (DFT). The research within RDMFT was so far focussed on the static theory. In this work we attempt some first steps towards a time-dependent generalization of RDMFT. In part II we derive equations of motion for natural orbitals and occupation numbers. Using the equation of motion for the occupation numbers we show that an adiabatic extension of presently known ground-state functionals of static RDMFT always leads to occupation numbers which are constant in time. From the stationary conditions of the equations of motion for the N-body correlations (correlated parts of the N-body matrices) we derive a new class of ground-state functionals which can be used in static RDMFT. Applications are presented for a one-dimensional model system where the time-dependent many-body Schrödinger equation can be propagated numerically. We use optimal control theory to find optimized laser pulses for transitions in a model for atomic Helium. From the numerically exact correlated wavefunction we extract the exact time evolution of natural orbitals and occupation numbers for (i) laser-driven Helium and (ii) electron-ion scattering. Part III of this work considers time-dependent quantum transport within TDDFT. We present an algorithm for the calculation of extended eigenstates of single- particle Hamiltonians which is especially tailored to a finite-difference discretization of the Schrödinger equation. We consider the propagation of finite mesoscopic systems and demonstrate the limitations of such an approach. To overcome the shortcomings of a description of quantum transport in terms of finite systems we develop a time-propagation scheme for extended states which utilizes a mixed basis representation. Our discretization scheme allows to treat central device and lead regions on the same footing thus preventing artificial reflections at grid boundaries.
We develop a scheme to exactly evaluate the correlation energy in the random-phase approximation, based on linear response theory [Y. R. Shimizu, J. D. Garrett, R. A. Broglia, M. Gallardo, and E. Vigezzi, Rev. Mod. Phys. 61, 131 (1989)]. It is demonstrated that our formula is equivalent to a contour integral representation recently proposed [F. Donau, D. Almehed, and R. G. Nazmitdinov, Phys. Rev. Lett. 83, 280 (1999)] being numerically more efficient for realistic calculations. Examples are presented for pairing correlations in rapidly rotating nuclei.
This chapter discusses the linear-response framework of time-dependent density-functional theory (TDDFT). It begins with a thorough and self-contained exposition of general linear-response theory for many-body systems. The time-dependent as well as the frequency-dependent response are discussed. Several exact representations and properties of the density-density response function are reviewed, such as the Lehmann representation and fluctuation-dissipation theorem, and the Kramers-Kronig relations. It is shown how several important spectroscopic observables are obtained from linear-response theory. Then, the TDDFT formulation of linear-response theory is derived. The key quantity in this formalism is the so-called exchange-correlation kernel. The remainder of the chapter demonstrates how this linear-response formalism can be used to calculate excitation energies in principle exactly with TDDFT. The Casida equation is derived, as well as several approximate schemes. A comparison is made with time-dependent Hartree-Fock theory.
Using Green's-function many-body theory, we present the details of a formally exact adiabatic-connection fluctuation-dissipation density-functional theory based on range separation, which was sketched in Toulouse et al. [Phys. Rev. Lett. 102, 096404 (2009)]. Range-separated density-functional theory approaches combining short-range density-functional approximations with long-range random-phase approximations (RPAs) are then obtained as well-identified approximations on the long-range Green's-function self-energy. Range-separated RPA-type schemes with or without long-range Hartree-Fock exchange response kernel are assessed on rare-gas and alkaline-earth-metal dimers and compared to range-separated second-order perturbation theory and range-separated coupled-cluster theory.
The adiabatic approximation is problematic in time-dependent density matrix functional theory. With pure density matrix functionals (invariant under phase change of the natural orbitals) it leads to lack of response in the occupation numbers, hence wrong frequency dependent responses, in particular α(ω→0)≠α(0) (the static polarizability). We propose to relinquish the requirement that the functional must be a pure one-body reduced density matrix (1RDM) functional, and to introduce additional variables which can be interpreted as phases of the one-particle states of the independent particle reference system formed with the natural orbitals, thus obtaining so-called phase-including natural orbital (PINO) functionals. We also stress the importance of the correct choice of the complex conjugation in the two-electron integrals in the commonly used functionals (they should not be of exchange type). We demonstrate with the Löwdin-Shull energy expression for two-electron systems, which is an example of a PINO functional, that for two-electron systems exact responses (polarizabilities, excitation energies) are obtained, while writing this energy expression in the usual way as a 1RDM functional yields erroneous responses.