Exact Excited-State Functionals of the Asymmetric Hubbard Dimer

Giarrusso, Sara; Loos, Pierre-François

doi:10.1021/acs.jpclett.3c02052

Cited by 9 publications

(3 citation statements)

References 95 publications

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“…The Hubbard dimer is a simple but nontrivial two‐site lattice model that can be used, for example, for describing diatomic molecules 71 . As it can be solved exactly, 72 it is often used as a toy system for testing new ideas in connection with the many‐body problem 31,42,53,54,71–79 . The basic idea of the model is to simplify the (second‐quantized) ab initio Hamiltonian as follows,

Ĥ \to \overset{scriptH}{true} = \overset{𝒯}{true} + \overset{𝒰}{true} + {\overset{𝒱}{true}}_{e x t},

where the analogue for the kinetic energy operator

\overset{𝒯}{true}

(the so‐called hopping operator), the on‐site electron repulsion operator

\overset{𝒰}{true}

, and the local (external) potential operator

{\overset{𝒱}{true}}_{e x t}

read

\begin{matrix} alignleft \\ rightalign-odd \hat{𝒯} & align-even = - t \sum_{σ = ↑, ↓} (ĉ_{0 σ}^{†} ĉ_{1 σ} + ĉ_{1 σ}^{†} ĉ_{0 σ}), \end{matrix}

\begin{matrix} alignleft \\ rightalign-odd \hat{𝒰} & align-even = U \sum_{i = 0}^{1} {\hat{n}}_{i ↑} {\hat{n}}_{i ↓}, \end{matrix}

\begin{matrix} alignleft \\ rightalign-odd {\hat{𝒱}}_{ext} & align-even \end{matrix}

…”

Section: Exact Implementation For the Two‐electron Hubbard Dimermentioning

confidence: 99%

Extended N‐centered ensemble density functional theory of double electronic excitations

Cernatic,

Fromager

2024

J Comput Chem

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A recent work (arXiv:2401.04685) has merged N‐centered ensembles of neutral and charged electronic ground states with ensembles of neutral ground and excited states, thus providing a general and in‐principle exact (so‐called extended N‐centered) ensemble density functional theory of neutral and charged electronic excitations. This formalism made it possible to revisit the concept of density‐functional derivative discontinuity, in the particular case of single excitations from the highest occupied Kohn–Sham (KS) molecular orbital, without invoking the usual “asymptotic behavior of the density” argument. In this work, we address a broader class of excitations, with a particular focus on double excitations. An exact implementation of the theory is presented for the two‐electron Hubbard dimer model. A thorough comparison of the true physical ground‐ and excited‐state electronic structures with that of the fictitious ensemble density‐functional KS system is also presented. Depending on the choice of the density‐functional ensemble as well as the asymmetry of the dimer and the correlation strength, an inversion of states can be observed. In some other cases, the strong mixture of KS states within the true physical system makes the assignment “single excitation” or “double excitation” irrelevant.

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Ĥ \to \overset{scriptH}{true} = \overset{𝒯}{true} + \overset{𝒰}{true} + {\overset{𝒱}{true}}_{e x t},

where the analogue for the kinetic energy operator

\overset{𝒯}{true}

(the so‐called hopping operator), the on‐site electron repulsion operator

\overset{𝒰}{true}

, and the local (external) potential operator

{\overset{𝒱}{true}}_{e x t}

read

\begin{matrix} alignleft \\ rightalign-odd \hat{𝒯} & align-even = - t \sum_{σ = ↑, ↓} (ĉ_{0 σ}^{†} ĉ_{1 σ} + ĉ_{1 σ}^{†} ĉ_{0 σ}), \end{matrix}

\begin{matrix} alignleft \\ rightalign-odd \hat{𝒰} & align-even = U \sum_{i = 0}^{1} {\hat{n}}_{i ↑} {\hat{n}}_{i ↓}, \end{matrix}

\begin{matrix} alignleft \\ rightalign-odd {\hat{𝒱}}_{ext} & align-even \end{matrix}

…”

Section: Exact Implementation For the Two‐electron Hubbard Dimermentioning

confidence: 99%

Extended N‐centered ensemble density functional theory of double electronic excitations

Cernatic,

Fromager

2024

J Comput Chem

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“…In a conceptually simple approach, the orbitals are variationally optimized to find solutions to the KS equations higher in energy than the ground state. These mean-field solutions corresponding to single Slater determinants with nonaufbau occupation represent the excited states. ,− Thus, unlike in TD-DFT, both ground and excited states are treated variationally, resulting in a more balanced description of the different electronic states. This approach is sometimes called delta self-consistent field (ΔSCF), − referring to the calculation of the excitation energy as the energy difference between the individually optimized ground and excited state solutions.…”

Section: Introductionmentioning

confidence: 99%

Orbital-Optimized Versus Time-Dependent Density Functional Calculations of Intramolecular Charge Transfer Excited States

Selenius,

Sigurdarson,

Schmerwitz

et al. 2024

J. Chem. Theory Comput.

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The performance of time-independent, orbital-optimized calculations of excited states is assessed with respect to charge transfer excitations in organic molecules in comparison to the linear-response time-dependent density functional theory (TD-DFT) approach. A direct optimization method to converge on saddle points of the electronic energy surface is used to carry out calculations with the local density approximation (LDA) and the generalized gradient approximation (GGA) functionals PBE and BLYP for a set of 27 excitations in 15 molecules. The time-independent approach is fully variational and provides a relaxed excited state electron density from which the extent of charge transfer is quantified. The TD-DFT calculations are generally found to provide larger charge transfer distances compared to the orbital-optimized calculations, even when including orbital relaxation effects with the Z-vector method. While the error on the excitation energy relative to theoretical best estimates is found to increase with the extent of charge transfer up to ca. −2 eV for TD-DFT, no correlation is observed for the orbital-optimized approach. The orbital-optimized calculations with the LDA and the GGA functionals provide a mean absolute error of ∼0.7 eV, outperforming TD-DFT with both local and global hybrid functionals for excitations with a long-range charge transfer character. Orbital-optimized calculations with the global hybrid functional B3LYP and the range-separated hybrid functional CAM-B3LYP on a selection of states with short-and long-range charge transfer indicate that inclusion of exact exchange has a small effect on the charge transfer distance, while it significantly improves the excitation energy, with the best-performing functional CAM-B3LYP providing an absolute error typically around 0.15 eV.

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“…More recently, orbital-optimized DFT, a time-independent framework in which the state of interest is selected by converging the KS self-consistent procedure to saddle points, has demonstrated relative success in computing certain kinds of excitations including those that are typically challenging for TDDFT [77][78][79][80]. Theoretically, functional approximations specific for the state of interest should be used [81][82][83], however such functionals have not been developed yet (to the best of our knowledge) and therefore most applications of DFT to excited states are performed using ground state functionals. On these bases, the development of state-specific functionals appears to be an area of DFT where progress would be particularly useful.…”

Section: Introductionmentioning

confidence: 99%

Electronic vector potential from the exact factorization of a complex wavefunction

Giarrusso,

Gori-Giorgi,

Agostini

2024

Preprint

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We generalize the definitions of local scalar potentials named vkin and vN−1, which are relevant to properly describe phenomena such as molecular dissociation with density-functional theory, to the case in which the electronic wavefunction corresponds to a complex current-carrying state. In such a case, an extra term in the form of a vector potential appears which cannot be gauged away. Both scalar and vector potentials are introduced via the exact factorization formalism which allows to expand the given Schrödinger equation in two coupled equations, one for the marginal and one for the conditional amplitude. The electronic vector potential is directly related to the paramagnetic current density carried by the total wavefunction and to the diamagnetic current density in the equation for the marginal amplitude. An explicit example of this vector potential in a triplet state of two non-interacting electrons is showcased together with its associated circulation, giving rise to a non-vanishing geometric phase. Some connections with the exact factorization for the full molecular wavefunction beyond the Born-Oppenheimer approximation are also discussed.

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Exact Excited-State Functionals of the Asymmetric Hubbard Dimer

Cited by 9 publications

References 95 publications

Extended N‐centered ensemble density functional theory of double electronic excitations

Extended N‐centered ensemble density functional theory of double electronic excitations

Orbital-Optimized Versus Time-Dependent Density Functional Calculations of Intramolecular Charge Transfer Excited States

Electronic vector potential from the exact factorization of a complex wavefunction

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