Multiconfigurational short-range density functional theory for nuclear magnetic resonance shielding constants with gauge-including atomic orbitals.

In this paper, we present the theory and implementation of nuclear magnetic resonance shielding constants with gauge-including atomic orbitals for the hybrid multiconfigurational short-range density functional theory model. As a special case, this implementation also includes Hartree-Fock srDFT (HF-srDFT). Choosing a complete-active space (CAS) wave function as the multiconfigurational parameterization of the wave function, we investigate how well CAS-srDFT reproduces experimental trends of nuclear shielding constants compared to DFT and complete active space self-consistent field (CASSCF). Calculations on the nucleobases adenine and thymine show that CAS-srDFT performs on average the best of the tested methods, much better than CASSCF but only marginally better than HF-srDFT. The performance, compared to regular DFT, is similar when functionals containing exact exchange are used. We generally find that the inclusion of exact exchange is important for an accurate description of the shielding constants. In cases where no exact exchange is included, we observe that the HF- and CAS-srDFT often outperform regular DFT. For calculations on transition metal nuclei in organometallic compounds with significant static correlation, the CAS-srDFT method again outperforms CASSCF compared to experimental shielding constants, and the change from HF-srDFT is substantial. In conclusion, the static correlation posed by the metal complexes seems to be captured by CAS-srDFT, which is promising since this type of correlation is not well described by regular DFT.

DFT exchange: sharing perspectives on the workhorse of quantum chemistry and materials science.

In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchange views on DFT in the form of 302 individual contributions, formulated as responses to a preset list of 26 questions. Supported by a bibliography of 777 entries, the paper represents a broad snapshot of DFT, anno 2022.

Exact two-component Hamiltonians for relativistic quantum chemistry: Two-electron picture-change corrections made simple.

Based on self-consistent field (SCF) atomic mean-field (amf) quantities, we present two simple yet computationally efficient and numerically accurate matrix-algebraic approaches to correct both scalar-relativistic and spin-orbit two-electron picture-change effects (PCEs) arising within an exact two-component (X2C) Hamiltonian framework. Both approaches, dubbed amfX2C and e(xtended)amfX2C, allow us to uniquely tailor PCE corrections to mean-field models, viz. Hartree-Fock or Kohn-Sham DFT, in the latter case also avoiding the need for a point-wise calculation of exchange-correlation PCE corrections. We assess the numerical performance of these PCE correction models on spinor energies of group 18 (closed-shell) and group 16 (open-shell) diatomic molecules, achieving a consistent ≈10-5 Hartree accuracy compared to reference four-component data. Additional tests include SCF calculations of molecular properties such as absolute contact density and contact density shifts in copernicium fluoride compounds (CnFn, n = 2,4,6), as well as equation-of-motion coupled-cluster calculations of x-ray core-ionization energies of 5d- and 6d-containing molecules, where we observe an excellent agreement with reference data. To conclude, we are confident that our (e)amfX2C PCE correction models constitute a fundamental milestone toward a universal and reliable relativistic two-component quantum-chemical approach, maintaining the accuracy of the parent four-component one at a fraction of its computational cost.

Multi-configurational short-range density functional theory can describe spin-spin coupling constants of transition metal complexes.

The multi-configurational short-range (sr) density functional theory has been extended to the calculation of indirect spin-spin coupling constants (SSCCs) for nuclear magnetic resonance spectroscopy. The performance of the new method is compared to Kohn-Sham density functional theory and the ab initio complete active space self-consistent field for a selected set of molecules with good reference values. Two density functionals have been considered, the local density approximation srLDA and srPBE from the GGA class of functionals. All srDFT calculations are of Hartree-Fock-type HF-srDFT or complete active space-type CAS-srDFT. In all cases, the calculated SSCC values are of the same quality for srLDA and srPBE functionals, suggesting that one should use the computationally cost-effective srLDA functionals in applications. For all the calculated SSCCs in organic compounds, the best choice is HF-srDFT; the more expensive CAS-srDFT does not provide better values for these single-reference molecules. Fluorine is a challenge; in particular, the FF, FC, and FO couplings have much higher statistical errors than the rest. For SSCCs involving fluorine and a metal atom CAS-srDFT with singlet, generalized Tamm-Dancoff approximation is needed to get good SSCC values although the reference ground state is not a multi-reference case. For VF6 -1, all other considered models fail blatantly.

Triplet excitation energies from multiconfigurational short-range density-functional theory response calculations.

Linear response theory for the multiconfigurational short-range density functional theory (MC-srDFT) model is extended to triplet response with a singlet reference wave function. The triplet linear response equations for MC-srDFT are derived for a general hybrid srGGA functional and implemented in the Dalton program. Triplet excitation energies are benchmarked against the CC3 model of coupled cluster theory and the complete-active-space second-order perturbation theory using three different short-range functionals (srLDA, srPBE, and srPBE0), both with full linear response and employing the generalized Tamm-Dancoff approximation (gTDA). We find that using gTDA is required for obtaining reliable triplet excitations; for the CAS-srPBE model, the mean absolute deviation decreases from 0.40 eV to 0.26 eV, and for the CAS-srLDA model, it decreases from 0.29 eV to 0.21 eV. As expected, the CAS-srDFT model is found to be superior to the HF-srDFT model when analyzing the calculated triplet excitations for molecules in the benchmark set where increased static correlation is expected.

Generalized Valence Bond Perfect-Pairing Made Versatile Through Electron-Pairs Embedding.

We present an electron-pairs-based method employing a generalized valence bond perfect-pairing (GVB-PP) ansatz that provides a uniformly accurate description of systems where various types of electron correlation play a role and the GVB-PP wave function is a suitable reference. In the proposed EERPA-GVB approach, a GVB-PP energy is amended by adding correlation among electron pairs. The latter is achieved by embedding single pairs or couples of pairs in the environment of the other electron fragments and separately accounting for intra- and interfragment correlation effects. For this purpose, we employ truncated extended random phase approximation equations. Application of EERPA-GVB to systems governed by both short-range (energy barriers) and long-range (molecular interactions) correlation effects proves the good accuracy of the method. Moreover, EERPA-GVB is shown to cure a notorious problem of uncorrelated electron-pair models, namely, spatial symmetry breaking in aromatic molecules, using the example of benzene. We have also successfully applied EERPA-GVB to a challenging problem of a phase transition of the boron chain system, where the correlation changes its character along the reaction path. The accuracy and versatility of EERPA-GVB are accompanied by its attractively low computational cost. By truncation of the extended RPA equations and consideration of only at most two-fragment correlation contributions, the cost of computing the EERPA correlation energy is reduced to scale only quadratically with the number of pairs of electrons.

Multiconfigurational short-range density-functional theory for open-shell systems.

Many chemical systems cannot be described by quantum chemistry methods based on a single-reference wave function. Accurate predictions of energetic and spectroscopic properties require a delicate balance between describing the most important configurations (static correlation) and obtaining dynamical correlation efficiently. The former is most naturally done through a multiconfigurational (MC) wave function, whereas the latter can be done by, e.g., perturbation theory. We have employed a different strategy, namely, a hybrid between multiconfigurational wave functions and density-functional theory (DFT) based on range separation. The method is denoted by MC short-range DFT (MC-srDFT) and is more efficient than perturbative approaches as it capitalizes on the efficient treatment of the (short-range) dynamical correlation by DFT approximations. In turn, the method also improves DFT with standard approximations through the ability of multiconfigurational wave functions to recover large parts of the static correlation. Until now, our implementation was restricted to closed-shell systems, and to lift this restriction, we present here the generalization of MC-srDFT to open-shell cases. The additional terms required to treat open-shell systems are derived and implemented in the DALTON program. This new method for open-shell systems is illustrated on dioxygen and [Fe(H2O)6]3+.

A quantum-mechanical perspective on linear response theory within polarizable embedding.

We present a derivation of linear response theory within polarizable embedding starting from a rigorous quantum-mechanical treatment of a composite system. To this aim, two different subsystem decompositions (symmetric and nonsymmetric) of the linear response function are introduced and the pole structures as well as residues of the individual terms are discussed. In addition to providing a thorough justification for the descriptions used in polarizable embedding models, this theoretical analysis clarifies which form of the response function to use and highlights complications in separating out subsystem contributions to molecular properties. The basic features of the presented expressions and various approximate forms are illustrated by their application to a composite model system.

Relativistic Polarizable Embedding.

Most chemistry, including chemistry where relativistic effects are important, occurs in an environment, and in many cases, this environment has a significant effect on the chemistry. In nonrelativistic quantum chemistry, a lot of progress has been achieved with respect to including environments such as a solvent or protein in the calculations, and now is the time to extend the possibilities for also doing this in relativistic quantum chemistry. The polarizable embedding (PE) model efficiently incorporates electrostatic effects of the environment by describing it as a collection of localized electric multipoles and polarizabilities obtained through quantum chemical calculations. In this article, we present the theory and implementation of four- and exact two-component Hamiltonians within a PE framework. We denote the methods the PE-4c-DFT and PE-X2C-DFT models. The models include a linear response formalism to calculate time-dependent (TD) properties: PE-TD-4c-DFT and PE-TD-X2C-DFT. With this first implementation, we calculate the PE-TD-4c-PBE0 excitation energies of the TcO4- and ReO4- ions in an explicit water solvent. This initial investigation focuses on the relative size of relativistic and solvent contributions to the excitation energies. The solvent effect is divided into an indirect solvent effect due to the structural perturbation of the XO4- ion and a direct electrostatic effect. The relativistic effects as well as both types of solvent effects are found to contribute to a shift in the excitation energies, but they do so to different extents depending on the ion and the electronic transition in question.

Local electric fields and molecular properties in heterogeneous environments through polarizable embedding.

In spectroscopies, the local field experienced by a molecule embedded in an environment will be different from the externally applied electromagnetic field, and this difference may significantly alter the response and transition properties of the molecule. The polarizable embedding (PE) model has previously been developed to model the local field contribution stemming from the direct molecule-environment coupling of the electromagnetic response properties of molecules in solution as well as in heterogeneous environments, such as proteins. Here we present an extension of this approach to address the additional effective external field effect, i.e., the manifestations of the environment polarization induced by the external field, which allows for the calculation of properties defined in terms of the external field. Within a response framework, we report calculations of the one- and two-photon absorption (1PA and 2PA, respectively) properties of PRODAN-methanol clusters as well as the fluorescent protein DsRed. Our results demonstrate the necessity of accounting for both the dynamical reaction field and effective external field contributions to the local field in order to reproduce full quantum chemical reference calculations. For the lowest π→π* transition in DsRed, inclusion of effective external field effects gives rise to a 1.9- and 3.5-fold reduction in the 1PA and 2PA cross-sections, respectively. The effective external field is, however, strongly influenced by the heterogeneity of the protein matrix, and the resulting effect can lead to either screening or enhancement depending on the nature of the transition under consideration.

Beyond the electric-dipole approximation: A formulation and implementation of molecular response theory for the description of absorption of electromagnetic field radiation.

We present a formulation of molecular response theory for the description of a quantum mechanical molecular system in the presence of a weak, monochromatic, linearly polarized electromagnetic field without introducing truncated multipolar expansions. The presentation focuses on a description of linear absorption by adopting the energy-loss approach in combination with the complex polarization propagator formulation of response theory. Going beyond the electric-dipole approximation is essential whenever studying electric-dipole-forbidden transitions, and in general, non-dipolar effects become increasingly important when addressing spectroscopies involving higher-energy photons. These two aspects are examined by our study of the near K-edge X-ray absorption fine structure of the alkaline earth metals (Mg, Ca, Sr, Ba, and Ra) as well as the trans-polyenes. In following the series of alkaline earth metals, the sizes of non-dipolar effects are probed with respect to increasing photon energies and a detailed assessment of results is made in terms of studying the pertinent transition electron densities and in particular their spatial extension in comparison with the photon wavelength. Along the series of trans-polyenes, the sizes of non-dipolar effects are probed for X-ray spectroscopies on organic molecules with respect to the spatial extension of the chromophore.

Density matrix renormalization group with efficient dynamical electron correlation through range separation.

We present a new hybrid multiconfigurational method based on the concept of range-separation that combines the density matrix renormalization group approach with density functional theory. This new method is designed for the simultaneous description of dynamical and static electron-correlation effects in multiconfigurational electronic structure problems.

Polarizable embedding with a multiconfiguration short-range density functional theory linear response method.

We present here the coupling of a polarizable embedding (PE) model to the recently developed multiconfiguration short-range density functional theory method (MC-srDFT), which can treat multiconfigurational systems with a simultaneous account for dynamical and static correlation effects. PE-MC-srDFT is designed to combine efficient treatment of complicated electronic structures with inclusion of effects from the surrounding environment. The environmental effects encompass classical electrostatic interactions as well as polarization of both the quantum region and the environment. Using response theory, molecular properties such as excitation energies and oscillator strengths can be obtained. The PE-MC-srDFT method and the additional terms required for linear response have been implemented in a development version of Dalton. To benchmark the PE-MC-srDFT approach against the literature data, we have investigated the low-lying electronic excitations of acetone and uracil, both immersed in water solution. The PE-MC-srDFT results are consistent and accurate, both in terms of the calculated solvent shift and, unlike regular PE-MCSCF, also with respect to the individual absolute excitation energies. To demonstrate the capabilities of PE-MC-srDFT, we also investigated the retinylidene Schiff base chromophore embedded in the channelrhodopsin protein. While using a much more compact reference wave function in terms of active space, our PE-MC-srDFT approach yields excitation energies comparable in quality to CASSCF/CASPT2 benchmarks.

Molecular quantum mechanical gradients within the polarizable embedding approach--application to the internal vibrational Stark shift of acetophenone.

We present an implementation of analytical quantum mechanical molecular gradients within the polarizable embedding (PE) model to allow for efficient geometry optimizations and vibrational analysis of molecules embedded in large, geometrically frozen environments. We consider a variational ansatz for the quantum region, covering (multiconfigurational) self-consistent-field and Kohn-Sham density functional theory. As the first application of the implementation, we consider the internal vibrational Stark effect of the C=O group of acetophenone in different solvents and derive its vibrational linear Stark tuning rate using harmonic frequencies calculated from analytical gradients and computed local electric fields. Comparisons to PE calculations employing an enlarged quantum region as well as to a non-polarizable embedding scheme show that the inclusion of mutual polarization between acetophenone and water is essential in order to capture the structural modifications and the associated frequency shifts observed in water. For more apolar solvents, a proper description of dispersion and exchange-repulsion becomes increasingly important, and the quality of the optimized structures relies to a larger extent on the quality of the Lennard-Jones parameters.

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.

The multi-configuration self-consistent field method within a polarizable embedded framework.

We present a detailed derivation of Multi-Configuration Self-Consistent Field (MCSCF) optimization and linear response equations within the polarizable embedding scheme: PE-MCSCF. The MCSCF model enables a proper description of multiconfigurational effects in reaction paths, spin systems, excited states, and other properties which cannot be described adequately with current implementations of polarizable embedding in density functional or coupled cluster theories. In the PE-MCSCF scheme the environment surrounding the central quantum mechanical system is represented by distributed multipole moments and anisotropic dipole-dipole polarizabilities. The PE-MCSCF model has been implemented in DALTON. As a preliminary application, the low lying valence states of acetone and uracil in water has been calculated using Complete Active Space Self-Consistent Field (CASSCF) wave functions. The dynamics of the water environment have been simulated using a series of snapshots generated from classical Molecular Dynamics. The calculated shifts from gas-phase to water display between good and excellent correlation with experiment and previous calculations. As an illustration of another area of potential applications we present calculations of electronic transitions in the transition metal complex, [Fe(NO)(CN)5](2-) in a micro-solvated environment. This system is highly multiconfigurational and the influence of solvation is significant.

Relativistic effects in the intermolecular interaction-induced nuclear magnetic resonance parameters of xenon dimer.

Relativistic effects on the (129)Xe nuclear magnetic resonance shielding and (131)Xe nuclear quadrupole coupling (NQC) tensors are examined in the weakly bound Xe(2) system at different levels of theory including the relativistic four-component Dirac-Hartree-Fock (DHF) method. The intermolecular interaction-induced binary chemical shift delta, the anisotropy of the shielding tensor Deltasigma, and the NQC constant along the internuclear axis chi( parallel) are calculated as a function of the internuclear distance. DHF shielding calculations are carried out using gauge-including atomic orbitals. For comparison, the full leading-order one-electron Breit-Pauli perturbation theory (BPPT) is applied using a common gauge origin. Electron correlation effects are studied at the nonrelativistic (NR) coupled-cluster singles and doubles with perturbational triples [CCSD(T)] level of theory. The fully relativistic second-order Moller-Plesset many-body perturbation (DMP2) theory is used to examine the cross coupling between correlation and relativity on NQC. The same is investigated for delta and Deltasigma by BPPT with a density functional theory model. A semiquantitative agreement between the BPPT and DHF binary property curves is obtained for delta and Deltasigma in Xe(2). For these properties, the currently most complete theoretical description is obtained by a piecewise approximation where the uncorrelated relativistic DHF results obtained close to the basis-set limit are corrected, on the one hand, for NR correlation effects and, on the other hand, for the BPPT-based cross coupling of relativity and correlation. For chi( parallel), the fully relativistic DMP2 results obtain a correction for NR correlation effects beyond MP2. The computed temperature dependence of the second virial coefficient of the (129)Xe nuclear shielding is compared to experiment in Xe gas. Our best results, obtained with the piecewise approximation for the binary chemical shift combined with the previously published state of the art theoretical potential energy curve for Xe(2), are in excellent agreement with the experiment for the first time.