H. Ness^{1,2} , L.K. Dash^{1,2} and R.W. Godby^{1,2}

^{1}Department of Physics, University of York, Heslington, York YO10 5DD, United Kingdom

^{2}European Theoretical Spectroscopy Facility (ETSF)

Physical Review B 82 085426 (2010) (12 pages)

Using non-equilibrium Green's functions (NEGF), we calculate the current through an interacting region connected to non-interacting leads. The problem is reformulated in such a way that a Landauer-like term appears in the current as well as extra terms corresponding to non-equilibrium many-body effects. The interaction in the central region renormalizes not only the Green's functions but also the coupling at the contacts between the central region and the leads, allowing the total current to be further expressed as a generalized Landauer-like current formula. The general expression for the dynamical functional that renormalizes the contacts is provided. We analyze in detail under what circumstances Landauer-like approaches to the current, i.e. without contact renormalization, are valid for interacting electron-electron and/or electron-phonon systems. Numerical NEGF calculations are then performed for a model electron-phonon coupled system in order to validate our analytical approach. We show that the conductance for the off-resonant transport regime is adequately described by Landauer-like approach in the small-bias limit, while for the resonant regime the Landauer-like approach results depart from the exact results even at small finite bias. The validity of applying a Landauer-like approach to inelastic electron tunneling spectroscopy is also studied in detail.

PACS numbers: 71.38.-k, 73.40.Gk, 85.65.+h, 73.63.-b

Keywords: Transport

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H. Ness^{1,2}, L. K. Dash^{1,2}, M. Stankovski^{3,2} and R. W. Godby^{1,2}

^{1}Department of Physics, University of York, Heslington, York YO10 5DD, United Kingdom

^{2}European Theoretical Spectroscopy Facility (ETSF)

^{ 3}IMCN-NAPS, Université Catholique de Louvain, Place Croix du Sud 1 bte 3, B-1348 Louvain-la-Neuve, Belgium

Physical Review B 84 195114 (2011) [13 pages].

We provide the formal extension of Hedin's GW equations for single-particle Green's functions with electron-electron interaction onto the Keldysh time-loop contour. We show an application of our formalism to the plasmon model of a core electron within the plasmon-pole approximation. We study in detail the diagrammatic perturbation expansion of the core-electron/plasmon coupling on the spectral functions of the so-called S-model which provides an exact solution, concentrating especially on the effects of self-consistency and vertex corrections on the GW self-energy. For the S-model, self-consistency is essential for GW-like calculations to obtain the full spectral information. The second- order exchange diagram (i.e. a vertex correction) is crucial to obtain a better spectral description of the plasmon peak and side-band peaks in comparison to GW-like calculations. However, the vertex corrections are well reproduced within a non-self-consistent calculation. We also consider conventional equilibrium GW calculations for the pure jellium model. We find that with no second-order vertex correction, we cannot obtain the full set of plasmon side-band peaks. Finally, we address the issues of formal connection for the Dyson equations of the time-ordered Green's function and the Keldysh Green's functions at equilibrium in the cases of zero and finite temperature.

PACS numbers: 71.38.-k, 73.40.Gk, 85.65.+h, 73.63.-b

Keywords: GW Transport

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L.K. Dash^{1,2}, H. Ness^{1,2} and R.W. Godby^{1,2}

^{1}Department of Physics, University of York, Heslington, York YO10 5DD, United Kingdom

^{2}European Theoretical Spectroscopy Facility (ETSF)

Physical Review B 84 085433 (2011) (9 pages)

We study the effect of electron-vibron interactions on the inelastic transport properties of single-molecule nanojunctions. We use the non-equilibrium Green's functions technique and a model Hamiltonian to calculate the effects of second-order diagrams (double-exchange DX and dressed-phonon DPH diagrams) on the electron-vibration interaction and consider their effects across the full range of parameter space. The DX diagram, corresponding to a vertex correction, introduces an effective dynamical renormalization of the electron-vibron coupling in both the purely inelastic and the inelastic-resonant features of the IETS. The purely inelastic features correspond to an applied bias around the energy of a vibron, while the inelastic-resonant features correspond to peaks (resonance) in the conductance. The DPH diagram affects only the inelastic resonant features. We also discuss the circumstances in which the second-order diagrams may be approximated in the study of more complex model systems.

PACS numbers: 71.38.-k, 73.40.-c, 85.65.+h, 73.63.-b

Keywords: Transport

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L.K. Dash^{1,2}, H. Ness^{1,2}, M.J. Verstraete^{3,2} and R.W. Godby^{1,2}

^{1}Department of Physics, University of York, Heslington, York YO10 5DD, United Kingdom

^{ 2}European Theoretical Spectroscopy Facility (ETSF)

^{ 3}Institut de Physique, Université de Liège, B-4000 Sart Tilman, Belgium

Journal of Chemical Physics 136 064708 (2012) [11 pages]

We analyze how functionality could be obtained within single-molecule devices by using a combination of non-equilibrium Green's functions and ab-initio calculations to study the inelastic transport properties of single-molecule junctions. First we apply a full non-equilibrium Green's function technique to a model system with electron-vibration coupling. We show that the features in the inelastic electron tunneling spectra (IETS) of the molecular junctions are virtually independent of the nature of the molecule-lead contacts. Since the contacts are not easily reproducible from one device to another, this is a very useful property. The IETS signal is much more robust versus modifications at the contacts and hence can be used to build functional nanodevices. Second, we consider a realistic model of a organic conjugated molecule. We use ab-initio calculations to study how the vibronic properties of the molecule can be controlled by an external electric field which acts as a gate voltage. The control, through the gate voltage, of the vibron frequencies and (more importantly) of the electron-vibron coupling enables the construction of functionality: non-linear amplification and/or switching is obtained from the IETS signal within a single-molecule device.

PACS numbers: 85.65.+h, 81.07.Nb, 73.40.Gk, 71.15.-m, 02.30.-f

Keywords: Transport

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J.D. Ramsden^{1,2} and R.W. Godby^{1,2}

^{1}Department of Physics, University of York, Heslington, York YO10 5DD, United Kingdom

^{ 2}European Theoretical Spectroscopy Facility (ETSF)

Physical Review Letters 109 036402 (2012) (4 pages). (Editors' Suggestion, APS Viewpoint)

We calculate the exact Kohn-Sham potential that describes, within time-dependent density-functional theory, the propagation of an electron quasiparticle wavepacket of non-zero crystal momentum added to a ground-state model semiconductor. The potential is observed to have a highly nonlocal functional dependence on the charge density, in both space and time, giving rise to features entirely lacking in local or adiabatic approximations. The dependence of the non-equilibrium part of the Kohn-Sham electric field on the local current and charge density is identified as a key element of the correct Kohn-Sham functional.

PACS numbers: 71.15.Mb, 73.63.-b, 73.23.-b, 85.35.Be

Keywords: DFT Transport

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J.D. Ramsden^{1,2} and R.W. Godby^{1,2}

^{1}Department of Physics, University of York, Heslington, York YO10 5DD, United Kingdom

^{ 2}European Theoretical Spectroscopy Facility (ETSF)

Physical Review B ** 88** 195115 (2013) [6 pages].

We calculate the exact Kohn-Sham (KS) scalar and vector potentials that reproduce, within current-density functional theory, the steady-state density and current density corresponding to an electron quasiparticle added to the ground state of a model quantum wire. Our results show that, even in the absence of an external magnetic field, a KS description of a steady-state system in general requires a non-zero exchange-correlation magnetic field that is purely mechanical in origin. The KS paramagnetic current density is not, in general, that of the interacting system in any gauge.

PACS numbers: 71.15.Mb, 73.63.-b, 73.23.-b, 85.35.Be

Keywords: DFT Transport

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M.J.P. Hodgson^{1,2}, J.D. Ramsden^{1,2}, J.B.J. Chapman^{1,2}, P. Lillystone^{1,2}, and R.W. Godby^{1,2}

^{1}Department of Physics, University of York, Heslington, York YO10 5DD, United Kingdom

^{2}European Theoretical Spectroscopy Facility (ETSF)

Physical Review B (Rapid Communications) ** 88** 241102(R) (2013) [4 pages]

By propagating the many-body Schrödinger equation, we determine the exact time-dependent Kohn-Sham potential for a system of strongly correlated electrons which undergo field-induced tunneling. Numerous features are entirely absent from the approximations commonly used in time-dependent density-functional theory. The self-interaction correction is strong and time dependent, owing to electron localization, and prominent dynamic spatial potential steps arise from minima in the charge density, as modified by the Coulomb interaction experienced by the partially tunneled electron.

PACS numbers: 71.15.Mb, 73.23.Hk, 73.63.Nm

Keywords: DFT Transport iDEA code

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L. Mancini^{1,2}, M.J.P. Hodgson^{1}, J.D. Ramsden^{1} and R.W. Godby^{1}

^{1}Department of Physics, University of York, Heslington, York YO10 5DD, United Kingdom

^{2}University of Bologna, Department of Physics and Astronomy, viale Berti Pichat 6/2 40127, Bologna, Italy

Physical Review B **89** 195114 (2014) [6 pages]

We obtain the exact Kohn-Sham potentials *V*_{KS} of time-dependent density-functional theory for 1D Hubbard chains, driven by a d.c. external field, using the time-dependent electron density and current density obtained from exact many-body time-evolution. The exact *V*_{xc} is compared to the adiabatically-exact *V*_{xc}^{ad} and the "instantaneous ground state" *V*_{xc}^{igs}. The latter is shown to work effectively in some cases when the former fails. Approximations for the exchange-correlation potential *V*_{xc} and its gradient, based on the local density and on the local current density, are also considered and both physical quantities are observed to be far outside the reach of any possible local approximation. Insight into the respective roles of ground-state and excited-state correlation in the time-dependent system, as reflected in the potentials, is provided by the pair correlation function.

PACS numbers: 31.15.E-, 71.15.Mb, 71.10.Fd

Keywords: DFT Transport

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Rex Godby

Department of Physics, University of York, Heslington, York YO10 5DD, United Kingdom

Physics **7** 70 (2014) [3 pages]

A new model shows that quantum fluctuations make a significant contribution to a metal's low temperature resistance. A Viewpoint on "Dynamical many-body corrections to the residual resistivity of metals", V. U. Nazarov, G. Vignale, and Y.-C. Chang, Physical Review B **89**, 241108 (2014).

Keywords: DFT

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M.J.P. Hodgson, J.D. Ramsden, T.R. Durrant and R.W. Godby

Department of Physics, University of York, Heslington, York YO10 5DD, United Kingdom

Physical Review B (Rapid Communications) **90** 241107(R) (2014)

We introduce a new functional for simulating ground-state and time-dependent electronic systems within density-functional theory. The functional combines an expression for the exact Kohn-Sham (KS) potential in the limit of complete electron localization with a measure of the actual localization. We find accurate self-consistent charge densities, even for systems where the exact exchange-correlation potential exhibits non-local dependence on the density, such as potential steps. We compare our results to the exact KS potential for each system. The self-interaction correction is accurately described, avoiding the need for orbital-dependent potentials.

PACS numbers: 71.15.Mb, 31.15.A-, 73.63.Nm, 73.63.-b

Keywords: DFT Transport iDEA code

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M.J.P. Hodgson, J.D. Ramsden and R.W. Godby

Department of Physics, University of York, Heslington, York YO10 5DD, United Kingdom

Physical Review B **93** 155146 (2016) [11 pages]. (Editors' Suggestion)

Knowledge of exact properties of the exchange-correlation (xc) functional is important for improving the approximations made within density functional theory. Features such as steps in the exact xc potential are known to be necessary for yielding accurate densities, yet little is understood regarding their shape, magnitude and location. We use systems of a few electrons, where the exact electron density is known, to demonstrate general properties of steps. We find that steps occur at points in the electron density where there is a change in the 'local effective ionization energy' of the electrons. We provide practical arguments, based on the electron density, for determining the position, shape and height of steps for ground-state systems, and extend the concepts to time-dependent systems. These arguments are intended to inform the development of approximate functionals, such as the mixed localization potential (MLP), which already demonstrate their capability to produce steps in the Kohn-Sham potential.

PACS numbers: 31.15.E-, 71.15.Mb, 31.15.A, 73.63.-b

Keywords: DFT Transport iDEA code

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M.T. Entwistle^{1,2}, M.J.P. Hodgson^{1}, J. Wetherell^{1}, B. Longstaff^{1}, J.D. Ramsden^{1} and R.W. Godby^{1}

^{1}Department of Physics, University of York, Heslington, York YO10 5DD, United Kingdom

^{2}Fitzwilliam College, University of Cambridge, Cambridge CB3 0DG, United Kingdom

Physical Review B **94** 205134 (2016) (11 pages).

The local density approximation (LDA) constructed through quantum Monte Carlo calculations of the homogeneous electron gas (HEG) is the most common approximation to the exchange-correlation functional in density functional theory. We introduce an alternative set of LDAs constructed from slab-like systems of one, two and three electrons that resemble the HEG within a *finite* region, and illustrate the concept in one dimension. Comparing with the exact densities and Kohn-Sham potentials for various test systems, we find that the LDAs give a good account of the self-interaction correction, but are less reliable when correlation is stronger or currents flow.

PACS numbers: 71.15.Mb, 71.10.Ca, 31.15.E-, 31.15.ac

Keywords: DFT iDEA code

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T.R. Durrant^{1}, M.J.P. Hodgson^{2}, J.D. Ramsden and R.W. Godby

Department of Physics, University of York, Heslington, York YO10 5DD, United Kingdom, and European Theoretical Spectroscopy Facility

^{1}
Present address: Department of Physics and Astronomy and London Centre for Nanotechnology, University College London, Gower Street, London, WC1E 6BT, United Kingdom

^{2} Present address: Max-Planck-Institut für Mikrostrukturphysik, Weinberg 2, D-06120 Halle, Germany

Journal of Physics: Condensed Matter **30** 065901 (2018) [6 pages]

Electron localisation is the tendency of an electron in a many-body system to exclude other electrons from its vicinity. Using a new natural measure of localisation based on the exact many-electron wavefunction, we find that localisation can vary considerably between different ground-state systems, and can also be strongly disrupted, as a function of time, when a system is driven by an applied electric field. We use our new measure to assess the well-known electron localisation function (ELF), both in its approximate single-particle form (often applied within density-functional theory) and its full many-particle form. The full ELF always gives an excellent description of localisation, but the approximate ELF fails in time-dependent situations, even when the exact Kohn-Sham orbitals are employed.

PACS numbers: 72.15.Rn, 71.15.Mb, 31.15.A-

Keywords: DFT Transport iDEA code

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J. Wetherell^{1}, M.J.P. Hodgson^{2} and R.W. Godby^{1}

^{1} Department of Physics, University of York, Heslington, York YO10 5DD, United Kingdom

^{2} Max-Planck-Institut für Mikrostrukturphysik, Weinberg 2, D-06120 Halle, Germany

Physical Review B (Rapid Communications) **97** 121102(R) (2018) [5 pages].

The self-screening error in electronic structure theory is the part of the self-interaction error that would remain within the *GW* approximation if the exact dynamically screened Coulomb interaction, *W*, were used, causing each electron to artificially screen its own presence. This introduces error into the electron density and ionization potential. We propose a simple, computationally efficient correction to *GW* calculations in the form of a local density functional, obtained using a series of finite training systems; in tests, this eliminates the self-screening errors in the electron density and ionization potential.

Keywords: GW iDEA code

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A.H. Skelt, R.W. Godby and I. D'Amico

Department of Physics, University of York, Heslington, York YO10 5DD, United Kingdom

Physical Review A **98** 012104 (2018) [6 pages].

Understanding out-of-equilibrium quantum dynamics is a critical outstanding problem, with key questions regarding characterizing adiabaticity for applications in quantum technologies. We show how the metric-space approach to quantum mechanics naturally characterizes regimes of quantum dynamics, and provides an appealingly visual tool for assessing their degree of adiabaticity. Further, the dynamic trajectories of quantum systems in metric space suggest a lack of "ergodicity", thus providing a better understanding of the fundamental one-to-one mapping between densities and wavefunctions.

Keywords: iDEA code

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A.H. Skelt, R.W. Godby and I. D'Amico

Department of Physics, University of York, Heslington, York YO10 5DD, United Kingdom

Brazilian Journal of Physics **48** 467–471 (2018) *[special issue for the workshop "Density Functional Theory meets Quantum Information Theory", São Paulo, October 2017]*

Metrics have been used to investigate the relationship between wavefunction distances and density distances for families of specific systems. We extend this research to look at random potentials for time-dependent single electron systems, and for ground-state two electron systems. We find that Fourier series are a good basis for generating random potentials. These random potentials also yield quasi-linear relationships between the distances of ground-state densities and wavefunctions, providing a framework in which Density Functional Theory can be explored.

Keywords: iDEA code

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A.R. Elmaslmane^{1}, J. Wetherell^{1,2}, M.J.P. Hodgson^{3,2}, K.P. McKenna^{1} and R.W. Godby^{1,2}

^{1} Department of Physics, University of York, Heslington, York YO10 5DD, United Kingdom

^{2} European Theoretical Spectroscopy Facility

^{3} Max-Planck-Institut für Mikrostrukturphysik, Weinberg 2, D-06120 Halle, Germany

Physical Review Materials (Rapid Communications) **2** 040801(R) (2018).

We evaluate the accuracy of electron densities and quasiparticle energy gaps given by hybrid functionals by directly comparing these to the exact quantities obtained from solving the many-electron Schrödinger equation. We determine the admixture of Hartree-Fock exchange to approximate exchange-correlation in our hybrid functional via one of several physically justified constraints, including the generalized Koopmans' theorem. We find that hybrid functionals yield strikingly accurate electron densities and gaps in both exchange-dominated and correlated systems. We also discuss the role of the screened Fock operator in the success of hybrid functionals.

Keywords: DFT GW iDEA code

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M.T. Entwistle^{1,2}, M. Casula^{3} and R.W. Godby^{1,2}

^{1} Department of Physics, University of York, Heslington, York YO10 5DD, United Kingdom

^{2} European Theoretical Spectroscopy Facility

^{3} Institut de Minéralogie, de Physique des Matériaux et de Cosmochimie (IMPMC), Sorbonne Université, CNRS UMR 7590, IRD UMR 206, MNHN, 4 Place Jussieu, 75252 Paris, France

Physical Review B **97** 235143 (2018) [8 pages].

The most commonly used approximation to the exchange-correlation functional in density functional theory is the local density approximation (LDA), typically derived from the properties of the homogeneous electron gas (HEG). We previously introduced a set of alternative LDAs constructed from one-dimensional systems of one, two and three electrons that resemble the HEG within a finite region. We now construct a HEG-based LDA appropriate for spinless electrons in one dimension and find that it is remarkably similar to the finite LDAs. As expected, all LDAs are inadequate in low-density systems where correlation is strong. However, exploring the small but significant differences between the functionals, we find that the finite LDAs give better densities and energies in high-density exchange-dominated systems, arising partly from a better description of the self-interaction correction.

Keywords: DFT iDEA code

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J. Wetherell^{1,2}, M.J.P. Hodgson^{3,2}, L. Talirz^{4} and R.W. Godby^{1,2}

Physical Review B **99** 045129 (2019) [5 pages].

For properties of interacting electron systems, Kohn-Sham (KS) theory is often favored over many-body perturbation theory (MBPT) owing to its low computational cost. However, the exact KS potential can be challenging to approximate, for example in the presence of localized subsystems where the exact potential is known to exhibit pathological features such as spatial steps. By modeling two electrons, each localized in a distinct potential well, we illustrate that the step feature has no counterpart in MBPTs (including Hartree-Fock and *GW*) or hybrid methods involving Fock exchange because the spatial non-locality of the self-energy renders such pathological behavior unnecessary. We present a quantitative illustration of the orbital-dependent nature of the non-local potential, and a numerical demonstration of Kohn's concept of the nearsightedness for self-energies, when two distant subsystems are combined, in contrast to the KS potential. These properties emphasize the value of self-energy-based approximations in developing future approaches within KS-like theories.

Keywords: DFT GW iDEA code

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M.T. Entwistle and R.W. Godby

Department of Physics, University of York, Heslington, York YO10 5DD, United Kingdom, and European Theoretical Spectroscopy Facility

Submitted

For two prototype systems, we calculate the exact exchange-correlation kernels *f*_{xc}(x,x′,ω) of time-dependent density functional theory. *f*_{xc}, the key quantity for optical absorption spectra of electronic systems, is normally subject to uncontrolled approximation. We find that, up to the first excitation energy, the exact *f*_{xc} has weak frequency-dependence and a simple, though non-local, spatial form. For higher excitations, the spatial behavior and frequency-dependence become more complex. The accuracy of the underlying exchange-correlation potential is of crucial importance.

PACS numbers: PACS

Keywords: DFT GW iDEA code

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End of abstracts