Emmanuel Fromager

Multi-configuration time-dependent density-functional theory based on range separation.

Multi-configuration range-separated density-functional theory is extended to the time-dependent regime. An exact variational formulation is derived. The approximation, which consists in combining a long-range Multi-Configuration-Self-Consistent Field (MCSCF) treatment with an adiabatic short-range density-functional (DFT) description, is then considered. The resulting time-dependent multi-configuration short-range DFT (TD-MC-srDFT) model is applied to the calculation of singlet excitation energies in H2, Be, and ferrocene, considering both short-range local density (srLDA) and generalized gradient (srGGA) approximations. As expected, when modeling long-range interactions with the MCSCF model instead of the adiabatic Buijse-Baerends density-matrix functional as recently proposed by Pernal [J. Chem. Phys. 136, 184105 (2012)], the description of both the 1(1)D doubly-excited state in Be and the 1(1)Σu(+) state in the stretched H2 molecule are improved, although the latter is still significantly underestimated. Exploratory TD-MC-srDFT/GGA calculations for ferrocene yield in general excitation energies at least as good as TD-DFT using the Coulomb attenuated method based on the three-parameter Becke-Lee-Yang-Parr functional (TD-DFT/CAM-B3LYP), and superior to wave-function (TD-MCSCF, symmetry adapted cluster-configuration interaction) and TD-DFT results based on LDA, GGA, and hybrid functionals.

Rigorous formulation of two-parameter double-hybrid density-functionals.

A two-parameter extension of the density-scaled double hybrid approach of Sharkas et al. [J. Chem. Phys. 134, 064113 (2011)] is presented. It is based on the explicit treatment of a fraction of multideterminantal exact exchange. The connection with conventional double hybrids is made when neglecting density scaling in the correlation functional as well as second-order corrections to the density. In this context, the fraction a(c) of second-order Møller-Plesset (MP2) correlation energy is not necessarily equal to the square of the fraction a(x) of Hartree-Fock exchange. More specifically, it is shown that a(c)≤a(x)(2), a condition that conventional semi-empirical double hybrids actually fulfill. In addition, a new procedure for calculating the orbitals, which has a better justification than the one routinely used, is proposed. Referred to as λ(1) variant, the corresponding double hybrid approximation has been tested on a small set consisting of H(2), N(2), Be(2), Mg(2), and Ar(2). Three conventional double hybrids (B2-PLYP, B2GP-PLYP, and PBE0-DH) have been considered. Potential curves obtained with λ(1)- and regular double hybrids can, in some cases, differ significantly. In particular, for the weakly bound dimers, the λ(1) variants bind systematically more than the regular ones, which is an improvement in many but not all cases. Including density scaling in the correlation functionals may of course change the results significantly. Moreover, optimized effective potentials based on a partially-interacting system could also be used to generate proper orbitals. Work is currently in progress in those directions.

Analysis of double-hybrid density functionals along the adiabatic connection

We present a graphical analysis of the adiabatic connections underlying double-hybrid density-functional methods that employ second-order perturbation theory. Approximate adiabatic connection formulae relevant to the construction of these functionals are derived and compared directly with those calculated using accurate ab initio methods. The discontinuous nature of the approximate adiabatic integrands is emphasised, the discontinuities occurring at interaction strengths which mark the transitions between regions that are: (i) described predominantly by second-order perturbation theory; (ii) described by a mixture of density-functional and second-order perturbation theory contributions; and (iii) described purely by density-functional theory. Numerical examples are presented for a selection of small molecular systems and van der Waals dimers. The impacts of commonly used approximations in each of the three sections of the adiabatic connection are discussed along with possible routes for the development of improved double-hybrid methodologies.

Assessment of charge-transfer excitations with time-dependent, range-separated density functional theory based on long-range MP2 and multiconfigurational self- consistent field wave functions

Charge transfer excitations can be described within Time-Dependent Density Functional Theory (TD-DFT), not only by means of the Coulomb Attenuated Method (CAM) but also with a combination of wave function theory and TD-DFT based on range separation. The latter approach enables a rigorous formulation of multi-determinantal TD-DFT schemes where excitation classes, which are absent in conventional TD-DFT spectra (like for example double excitations), can be addressed. This paper investigates the combination of both the long-range Multi-Configuration Self-Consistent Field (MCSCF) and Second Order Polarization Propagator Approximation (SOPPA) ansätze with a short-range DFT (srDFT) description. We find that the combinations of SOPPA or MCSCF with TD-DFT yield better results than could be expected from the pure wave function schemes. For the Time-Dependent MCSCF short-range DFT ansatz (TD-MC-srDFT) excitation energies calculated over a larger benchmark set of molecules with predominantly single reference character yield good agreement with their reference values, and are in general comparable to the CAM-B3LYP functional. The SOPPA-srDFT scheme is tested for a subset of molecules used for benchmarking TD-MC-srDFT and performs slightly better against the reference data for this small subset. Beyond the proof-of-principle calculations comprising the first part of this contribution, we additionally studied the low-lying singlet excited states (S1 and S2) of the retinal chromophore. The chromophore displays multireference character in the ground state and both excited states exhibit considerable double excitation character, which in turn cannot be described within standard TD-DFT, due to the adiabatic approximation. However, a TD-MC-srDFT approach can account for the multireference character, and excitation energies are obtained with accuracy comparable to CASPT2, although using a much smaller active space.

Generalised adiabatic connection in ensemble density-functional theory for excited states: example of the H2 molecule

A generalised adiabatic connection for ensembles (GACE) is presented. In contrast to the traditional adiabatic connection formulation, both ensemble weights and interaction strength can vary along a GACE path while the ensemble density is held fixed. The theory is presented for non-degenerate two-state ensembles but it can in principle be extended to any ensemble of fractionally occupied excited states. Within such a formalism an exact expression for the ensemble exchange–correlation density-functional energy, in terms of the conventional ground-state exchange–correlation energy, is obtained by integration over the ensemble weight. Stringent constraints on the functional are thus obtained when expanding the ensemble exchange–correlation energy through second order in the ensemble weight. For illustration purposes, the analytical derivation of the GACE is presented for the H2 model system in a minimal basis, leading thus to a simple density-functional approximation to the ensemble exchange–correlation energy. Encouraging results were obtained with this approximation for the description in a large basis of the first 1Σ+g excitation in H2 upon bond stretching. Finally, a range-dependent GACE has been derived, providing thus a pathway to the development of a rigorous state-average multi-determinant density-functional theory.

On the exact formulation of multi-configuration density-functional theory: electron density versus orbitals occupation

The exact formulation of multi-configuration density-functional theory is discussed in this work. As an alternative to range-separated methods, where electron correlation effects are split in the coordinate space, the combination of configuration interaction methods with orbital occupation functionals is explored at the formal level through the separation of correlation effects in the orbital space. When applied to model Hamiltonians, this approach leads to an exact site-occupation embedding theory (SOET). An adiabatic connection expression is derived for the complementary bath functional and a comparison with density matrix embedding theory is made. Illustrative results are given for the simple two-site Hubbard model. SOET is then applied to a quantum chemical Hamiltonian, thus leading to an exact complete active space site-occupation functional theory (CASSOFT) where active electrons are correlated explicitly within the CAS and the remaining contributions to the correlation energy are described with an orbital occupation functional. The computational implementation of SOET and CASSOFT as well as the development of approximate functionals are left for future work.

Alternative separation of exchange and correlation energies in multi-configuration range-separated density-functional theory

The alternative separation of exchange and correlation energies proposed by Toulouse et al. [Theor. Chem. Acc. 114, 305 (2005)] is explored in the context of multi-configuration range-separated density-functional theory. The new decomposition of the short-range exchange-correlation energy relies on the auxiliary long-range interacting wavefunction rather than the Kohn-Sham (KS) determinant. The advantage, relative to the traditional KS decomposition, is that the wavefunction part of the energy is now computed with the regular (fully interacting) Hamiltonian. One potential drawback is that, because of double counting, the wavefunction used to compute the energy cannot be obtained by minimizing the energy expression with respect to the wavefunction parameters. The problem is overcome by using short-range optimized effective potentials (OEPs). The resulting combination of OEP techniques with wavefunction theory has been investigated in this work, at the Hartree-Fock (HF) and multi-configuration self-consistent-field (MCSCF) levels. In the HF case, an analytical expression for the energy gradient has been derived and implemented. Calculations have been performed within the short-range local density approximation on H2, N2, Li2, and H2O. Significant improvements in binding energies are obtained with the new decomposition of the short-range energy. The importance of optimizing the short-range OEP at the MCSCF level when static correlation becomes significant has also been demonstrated for H2, using a finite-difference gradient. The implementation of the analytical gradient for MCSCF wavefunctions is currently in progress.

Double hybrid density-functional theory using the coulomb-attenuating method

A double hybrid approximation using the Coulomb-attenuating method (CAM-DH) is derived within range-separated density-functional perturbation theory, in the spirit of a recent work by Cornaton {\it et al.} [Phys. Rev. A 88, 022516 (2013)]. The energy expression recovered through second order is linear in the parameters

An analysis of core effects on shape-consistent pseudopotentials

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Linear interpolation method in ensemble Kohn-Sham and range-separated density-functional approximations for excited states

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