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We investigate the optimization of flexible tailored real-space Jastrow factors for use in the transcorrelated (TC) method in combination with highly accurate quantum chemistry methods such as initiator full configuration interaction quantum Monte Carlo (FCIQMC). Jastrow factors obtained by minimizing the variance of the TC reference energy are found to yield better, more consistent results than those obtained by minimizing the variational energy. We compute all-electron atomization energies for the challenging first-row molecules C2 , CN, N2 , and O2 and find that the TC method yields chemically accurate results using only the cc-pVTZ basis set, roughly matching the accuracy of non-TC calculations with the much larger cc-pV5Z basis set. We also investigate an approximation in which pure three-body excitations are neglected from the TC-FCIQMC dynamics, saving storage and computational cost, and show that it affects relative energies negligibly. Our results demonstrate that the combination of tailored real-space Jastrow factors with the multi-configurational TC-FCIQMC method provides a route to obtaining chemical accuracy using modest basis sets, obviating the need for basis-set extrapolation and composite techniques., Comment: Submitted to J Chem Phys

In this work we investigate the performance of a recently proposed transcorrelated (TC) approach based on a single-parameter correlation factor [JCP, 154, 8, 2021] for systems involving more than two electrons. The benefit of such an approach relies on its simplicity as efficient numerical-analytical schemes can be set up to compute the two- and three-body integrals occuring in the effective TC Hamiltonian. To obtain accurate ground state energies within a given basis set, the present TC scheme is coupled to the recently proposed TC-full configuration interaction quantum Monte Carlo method [JCP, 151, 6, 2019]. We report ground state total energies on the Li-Ne series, together with their first cations, computed in increasing large basis sets and compare to more elaborate correlation factors involving electron-electron-nucleus coordinates. Numerical results on the Li-Ne ionization potentials show that the use of the single-parameter correlation factor brings on average only a slightly lower accuracy (1.2 mH) in a triple-zeta quality basis set with respect to a more sophisticated correlation factor. However, already using a quadruple-zeta quality basis set yields results within chemical accuracy to complete basis set limit results when using this novel single-parameter correlation factor.

Transcorrelated coupled cluster and distinguishable cluster methods are presented. The Hamiltonian is similarity transformed with a Jastrow factor in the first quantisation, which results in up to three-body integrals. The coupled cluster with singles and doubles equations on this transformed Hamiltonian are formulated and implemented. It is demonstrated that the resulting methods have a superior basis set convergence and accuracy to the corresponding conventional and explicitly correlated methods. Additionally, approximations for three-body integrals are suggested and tested.

By expressing the electronic wavefunction in an explicitly-correlated (Jastrow-factorised) form, a similarity-transformed effective Hamiltonian can be derived. The effective Hamiltonian is non-Hermitian and contains three-body interactions. The resulting ground-state eigenvalue problem can be solved projectively using a stochastic configuration-interaction formalism. Our approach permits use of highly flexible Jastrow functions, which we show to be effective in achieving extremely high accuracy, even with small basis sets. Results are presented for the total energies and ionisation potentials of the first-row atoms, achieving accuracy within a mH of the basis-set limit, using modest basis sets and computational effort., Comment: 6 pages, 1 figure

Exact pieces of information on the adiabatic connection integrand $W_{\lambda}[\rho]$, which allows to evaluate the exchange-correlation energy of Kohn-Sham density functional theory, can be extracted from the leading terms in the strong coupling limit ($\lambda\to\infty$, where $\lambda$ is the strength of the electron-electron interaction). In this work, we first compare the theoretical prediction for the two leading terms in the strong coupling limit with data obtained via numerical implementation of the exact Levy functional in the simple case of two electrons confined in one dimension, confirming the asymptotic exactness of these two terms. We then carry out a first study on the incorporation of the fermionic statistics at large coupling $\lambda$, both numerical and theoretical, confirming that spin effects enter at orders $\sim e^{-\sqrt{\lambda}}$.

A stochastic minimization method for a real-space wavefunction, $\Psi({\bf r}_{1},{\bf r}_{2}\ldots{\bf r}_{n})$, constrained to a chosen density, $\rho({\bf r})$, is developed. It enables the explicit calculation of the Levy constrained search $F[\rho]=\min_{\Psi\rightarrow\rho}\langle\Psi|\hat{T}+\hat{V}_{ee}|\Psi\rangle$ (Proc. Natl. Acad. Sci. 76 6062 (1979)), that gives the exact functional of density functional theory. This general method is illustrated in the evaluation of $F[\rho]$ for two-electron densities in one dimension with a soft-Coulomb interaction. Additionally, procedures are given to determine the first and second functional derivatives, $\frac{\delta F}{\delta\rho({\bf r})}$ and $\frac{\delta^{2}F}{\delta\rho({\bf r})\delta\rho({\bf r}')}$. For a chosen external potential, $v({\bf r})$, the functional and its derivatives are used in minimizations only over densities to give the exact energy, $E_{v}$ without needing to solve the Schr\"odinger equation., Comment: 5 pages, 4 figures

One of the great challenges of electronic structure theory is the quest for the exact functional of density functional theory. Its existence is proven, but it is a complicated multivariable functional that is almost impossible to conceptualize. In this paper, the asymmetric two-site Hubbard model is studied, which has a two-dimensional universe of density matrices. The exact functional becomes a simple function of two variables whose three dimensional energy landscape can be visualized and explored. A walk on this unique landscape, tilted to an angle defined by the one-electron Hamiltonian, gives a valley whose minimum is the exact total energy. This is contrasted with the landscape of some approximate functionals, explaining their failure for electron transfer in the strongly correlated limit. We show concrete examples of pure-state density matrices that are not $v$-representable due to the underlying non-convex nature of the energy landscape. For the first time, the exact functional is calculated for all numbers of electrons, including fractional, allowing the derivative discontinuity to be visualized and understood. The fundamental gap for all possible systems is obtained solely from the derivatives of the exact functional., Comment: 5 pages with 6 pages supplementary information also three movies are available as gifs in the arXiv source file or by hyperlinks at the end of the article

The stretching of closed-shell molecules is a qualitative problem for restricted Hartree-Fock that is usually circumvented by the use of unrestricted Hartree-Fock (UHF). UHF is well known to break the spin symmetry at the Coulson-Fischer point, leading to a discontinuous derivative in the potential energy surface and incorrect spin density. However, this is generally not considered as a major drawback. In this work, we present a set of two electron molecules which magnify the problem of symmetry breaking and lead to drastically incorrect potential energy surfaces with UHF. These molecules also fail with unrestricted density-functional calculations where a functional such as B3LYP gives both symmetry breaking and an unphysically low energy due to the delocalization error. The implications for density functional theory are also discussed.

The derivative discontinuity is a key concept in electronic structure theory in general and density functional theory in particular. The electronic energy of a quantum system exhibits derivative discontinuities with respect to different degrees of freedom that are a consequence of the integer nature of electrons. The classical understanding refers to the derivative discontinuity of the total energy as a function of the total number of electrons ($N$), but it can also manifest at constant $N$. Examples are shown in models including several Hydrogen systems with varying numbers of electrons or nuclear charge ($Z$), as well as the 1-dimensional Hubbard model (1DHM). Two sides of the problem are investigated: first, the failure of currently used approximate exchange-correlation functionals in DFT and, second, the importance of the derivative discontinuity in the exact electronic structure of molecules, as revealed by full configuration interaction (FCI). Currently, all approximate functionals miss the derivative discontinuity, leading to basic errors that can be seen in many ways: from the complete failure to give the total energy of H$_2$ and H$_2^+$, to the missing gap in Mott insulators such as stretched H$_2$ and the thermodynamic limit of the 1DHM, or a qualitatively incorrect density in the HZ molecule with two electrons and incorrect electron transfer processes. Description of the exact particle behavior of electrons is emphasized, which is key to many important physical processes in real systems, especially those involving electron transfer, and offers a challenge for the development of new exchange-correlation functionals.

Discontinuous changes in the electronic structure upon infinitesimal changes to the Hamiltonian are demonstrated. Remarkably, these are revealed in one and two electron molecular systems if the realm of the nuclear charge is extended to be fractional. Dramatic changes in the electron density from full configuration interaction are observed in real space illustrating key intricacies of electronic structure including the transfer, hopping and removal of electrons. Physically, this is due to the particle nature of electrons and manifests itself theoretically as a diverging linear density response function or an energy derivative discontinuity that occurs at constant number of electrons. This is essential to correctly describe real physical processes, from chemical reactions to electron transport and metal-insulator transitions. The dramatic errors of DFT are seen in real space as this physics is missing from currently used approximations and poses a great challenge for the development of new electronic structure methods., Comment: 5 pages, 4 figures

The exact conditions for density functionals and density matrix functionals in terms of fractional charges and fractional spins are known, and their violation in commonly used functionals has been shown to be the root of many major failures in practical applications. However, approximate functionals are not normally expressed in terms of the fractional variables. Here we develop a general framework for extending approximate density functionals and many-electron theory to fractional-charge and fractional-spin systems. Our development allows for the fractional extension of any approximate theory that is a functional of $G^{0}$, the one-electron Green's function of the non-interacting reference system. The extension to fractional charge and fractional spin systems is based on the ensemble average of the basic variable, $G^{0}$. We demonstrate the fractional extension for the following theories: (1) any explicit functional of the one-electron density, such as the LDA and GGA; (2) any explicit functional of the one-electron density matrix of the non-interacting reference system, such as the exact exchange functional and hybrid functionals; (3) many-body perturbation theory; and (4) random-phase approximations. A general rule for such an extension has also been derived through scaling the orbitals and should be useful for functionals where the link to the Green's function is not obvious. The development thus enables the examination of approximate theories against known exact conditions on the fractional variables and the analysis of their failures in chemical and physical applications in terms of violations of exact conditions of the energy functionals. The present work should facilitate the calculation of chemical potentials and fundamental band gaps with approximate functionals and many-electron theories through the energy derivatives with respect to the fractional charge., Comment: 33 pages + 11 page appendix

The random phase approximation (RPA) to the correlation energy is extended to fractional occupations and its performance examined for exact conditions on fractional charges and fractional spins. RPA satisfies the constancy condition for fractional spins that leads to correct bond dissociation and no static correlation error for H$_2$ but massively fails for fractional charges, with an enormous delocalization error even for a one-electron system such as H$_2^+$. Other methods such as range-separated RPA can reduce this delocalization error but only at the cost of increasing the static correlation error. None of the RPA methods seem to have the discontinuous nature required to satisfy both exact conditions and the full unified condition, emphasizing the need to go further than just smooth functionals of the orbitals., Comment: 4 pages, 7 figures

Standard approximations for the exchange-correlation functional have been found to give big errors for the linearity condition of fractional charges, leading to delocalization error, and the constancy condition of fractional spins, leading to static correlation error. These two conditions are now unified for states with both fractional charge and fractional spin: the exact energy functional is a plane, linear along the fractional charge coordinate and constant along the fractional spin coordinate with a line of discontinuity at the integer. This sheds light on the nature of the derivative discontinuity and calls for explicitly discontinuous functionals of the density or orbitals that go beyond currently used smooth approximations. This is key for the application of DFT to strongly correlated systems., Comment: 5 pages 2 figures

Electronic states with fractional spins arise in systems with large static correlation (strongly correlated systems). Such fractional-spin states are shown to be ensembles of degenerate ground states with normal spins. It is proven here that the energy of the exact functional for fractional-spin states is a constant, equal to the energy of the comprising degenerate pure spin states. Dramatic deviations from this exact constancy condition exist with all approximate functionals, leading to large static correlation errors for strongly correlated systems, such as chemical bond dissociation and band structure of Mott insulators. This is demonstrated with numerical calculations for several molecular systems. Approximating the constancy behavior for fractional spins should be a major aim in functional constructions and should open the frontier for DFT to describe strongly correlated systems. The key results are also shown to apply in reduced density-matrix functional theory., Comment: 6 pages, 4 figures

The band-gap problem and other systematic failures of approximate functionals are explained from an analysis of total energy for fractional charges. The deviation from the correct intrinsic linear behavior in finite systems leads to delocalization and localization errors in large or bulk systems. Functionals whose energy is convex for fractional charges such as LDA display an incorrect apparent linearity in the bulk limit, due to the delocalization error. Concave functionals also have an incorrect apparent linearity in the bulk calculation, due to the localization error and imposed symmetry. This resolves an important paradox and opens the possibility to obtain accurate band-gaps from DFT., Comment: 4 pages 4 figures

The calculation of the band-gap by density-functional theory (DFT) methods is examined by considering the behavior of the energy as a function of number of electrons. It is found that the incorrect band-gap prediction with most approximate functionals originates mainly from errors in describing systems with fractional charges. Formulas for the energy derivatives with respect to number of electrons are derived which clarify the role of optimized effective potentials in prediction of the band-gap. Calculations with a recent functional that has much improved behavior for fractional charges give a good prediction of the energy gap and also $\epsilon_{{\rm homo}}\simeq-I$ for finite systems. Our results indicate it is possible, within DFT, to have a functional whose eigenvalues or derivatives accurately predict the band-gap.