Performance of range-separated long-range SOPPA short-range density functional theory method for vertical excitation energies.

In this paper, benchmark results are presented on the calculation of vertical electronic excitation energies using a long-range second-order polarization propagator approximation (SOPPA) description with a short-range density functional theory description based on the Perdew-Burke-Ernzerhof (PBE) functional. The excitation energies are investigated for 132 singlet states and 71 triplet states across 28 medium-sized organic molecules. The results show that overall SOPPA-srPBE always performs better than PBE and that SOPPA-srPBE performs better than SOPPA for singlet states, but slightly worse than SOPPA for triplet states when CC3 results are the reference values.

Multiconfigurational SCF and Short-Range DFT Combined with Polarizable Density Embedding: Comparison of Linear-Response and State-Specific Solvatochromic Shifts of Acrolein and Para-nitrophenolate in Water.

The polarizable density embedding model is combined with the multiconfigurational self-consistent field and MC-srDFT electronic structure methods to calculate solvatochromic shifts of the n-π* absorption of acrolein and the π-π* absorption of the para-nitrophenolate anion in aqueous solution. Differences between linear-response (LR) and state-specific (SS) solvent shifts are analyzed by assessing the contributions of different terms in the solvent potential. This comparison shows that the differences are not only due to the intrinsically different response of LR and SS excitation energies to the polarizability of the environment but also due to a different response to the static part of the environment potential. These observations show that even in nonpolarizable environments, LR and SS calculations based on SCF (orbital optimization) methods do not necessarily agree on the spectral shift. The difference can be as large as, or even dominate, the difference due to dynamical polarization.

An efficient implementation of time-dependent linear-response theory for strongly orthogonal geminal wave function models.

We present an implementation of time-dependent linear-response equations for strongly orthogonal geminal wave function models: the time-dependent generalized valence bond (TD-GVB) perfect-pairing theory and the antisymmetrized product of strongly orthogonal geminals. The geminal wave functions are optimized using a restricted-step second-order algorithm suitable for handling many geminals, and the linear-response equations are solved in an efficient way using a direct iterative approach. The wave function optimization algorithm features an original scheme to create initial orbitals for the geminal functions in a black-box fashion. The implementation is employed to examine the accuracy of the geminal linear response for singlet excitation energies of small and medium-sized molecules. In systems dominated by dynamic correlation, geminal models constitute only a minor improvement with respect to time-dependent Hartree-Fock. Compared to the linear-response complete active space self-consistent field, TD-GVB either misses or gives large errors for states dominated by double excitations.

Implementation of Relativistic Coupled Cluster Theory for Massively Parallel GPU-Accelerated Computing Architectures.

In this paper, we report reimplementation of the core algorithms of relativistic coupled cluster theory aimed at modern heterogeneous high-performance computational infrastructures. The code is designed for parallel execution on many compute nodes with optional GPU coprocessing, accomplished via the new ExaTENSOR back end. The resulting ExaCorr module is primarily intended for calculations of molecules with one or more heavy elements, as relativistic effects on the electronic structure are included from the outset. In the current work, we thereby focus on exact two-component methods and demonstrate the accuracy and performance of the software. The module can be used as a stand-alone program requiring a set of molecular orbital coefficients as the starting point, but it is also interfaced to the DIRAC program that can be used to generate these. We therefore also briefly discuss an improvement of the parallel computing aspects of the relativistic self-consistent field algorithm of the DIRAC program.

The DIRAC code for relativistic molecular calculations.

DIRAC is a freely distributed general-purpose program system for one-, two-, and four-component relativistic molecular calculations at the level of Hartree-Fock, Kohn-Sham (including range-separated theory), multiconfigurational self-consistent-field, multireference configuration interaction, electron propagator, and various flavors of coupled cluster theory. At the self-consistent-field level, a highly original scheme, based on quaternion algebra, is implemented for the treatment of both spatial and time reversal symmetry. DIRAC features a very general module for the calculation of molecular properties that to a large extent may be defined by the user and further analyzed through a powerful visualization module. It allows for the inclusion of environmental effects through three different classes of increasingly sophisticated embedding approaches: the implicit solvation polarizable continuum model, the explicit polarizable embedding model, and the frozen density embedding model.

Dalton Project: A Python platform for molecular- and electronic-structure simulations of complex systems.

The Dalton Project provides a uniform platform access to the underlying full-fledged quantum chemistry codes Dalton and LSDalton as well as the PyFraME package for automatized fragmentation and parameterization of complex molecular environments. The platform is written in Python and defines a means for library communication and interaction. Intermediate data such as integrals are exposed to the platform and made accessible to the user in the form of NumPy arrays, and the resulting data are extracted, analyzed, and visualized. Complex computational protocols that may, for instance, arise due to a need for environment fragmentation and configuration-space sampling of biochemical systems are readily assisted by the platform. The platform is designed to host additional software libraries and will serve as a hub for future modular software development efforts in the distributed Dalton community.

The Second-Order-Polarization-Propagator-Approximation (SOPPA) in a four-component spinor basis.

A theoretical framework for understanding molecular structures is crucial for the development of new technologies such as catalysts or solar cells. Apart from electronic excitation energies, however, only spectroscopic properties of molecules consisting of lighter elements can be computationally described at a high level of theory today since heavy elements require a relativistic framework, and thus far, most methods have only been derived in a non-relativistic framework. Important new technologies such as those mentioned above require molecules that contain heavier elements, and hence, there is a great need for the development of relativistic computational methods at a higher level of accuracy. Here, the Second-Order-Polarization-Propagator-Approximation (SOPPA), which has proven to be very successful in the non-relativistic case, is adapted to a relativistic framework. The equations for SOPPA are presented in their most general form, i.e., in a non-canonical spin-orbital basis, which can be reduced to the canonical case, and the expressions needed for a relativistic four-component SOPPA are obtained. The equations are one-index transformed, giving more compact expressions that correspond to those already available for the four-component RPA. The equations are ready for implementation in a four-component quantum chemistry program, which will allow both linear response properties and excitation energies to be calculated relativistically at the SOPPA level.

Remarkable reversal of 13C-NMR assignment in d1, d2 compared to d8, d9 acetylacetonate complexes: analysis and explanation based on solid-state MAS NMR and computations.

13C solid-state MAS NMR spectra of a series of paramagnetic metal acetylacetonate complexes; [VO(acac)2] (d1, S = ½), [V(acac)3] (d2, S = 1), [Ni(acac)2(H2O)2] (d8, S = 1), and [Cu(acac)2] (d9, S = ½), were assigned using modern NMR shielding calculations. This provided a reliable assignment of the chemical shifts and a qualitative insight into the hyperfine couplings. Our results show a reversal of the isotropic 13C shifts, δiso(13C), for CH3 and CO between the d1 and d2versus the d8 and d9 acetylacetonate complexes. The CH3 shifts change from about -150 ppm (d1,2) to roughly 1000 ppm (d8,9), whereas the CO shifts decrease from 800 ppm to about 150 ppm for d1,2 and d8,9, respectively. This was rationalized by comparison of total spin-density plots and computed contact couplings to those corresponding to singly occupied molecular orbitals (SOMOs). This revealed the interplay between spin delocalization of the SOMOs and spin polarization of the lower-energy MOs, influenced by both the molecular symmetry and the d-electron configuration. A large positive chemical shift results from spin delocalization and spin polarization acting in the same direction, whereas their cancellation corresponds to a small shift. The SOMO(s) for the d8 and d9 complexes are σ-like, implying spin-delocalization on the CH3 and CO groups of the acac ligand, cancelled only for CO by spin polarization. In contrast, the SOMOs of the d1 and d2 systems are π-like and a large CO-shift results from spin polarization, which accounts for the reversed assignment of δiso(13C) for CH3 and CO.

Relativistic quantum chemical calculations show that the uranium molecule U2 has a quadruple bond.

Understanding the bonding, reactivity and electronic structure of actinides is lagging behind that of the rest of the periodic table. This can be partly explained by the challenges that one faces in experimental studies of such radioactive compounds and also by the need to properly account for relativistic effects in theoretical studies. A further challenge is the very complicated electronic structures encountered in actinide chemistry, as vividly illustrated by the naked diuranium molecule U2. Here we report a computational study of this emblematic molecule using state-of-the-art relativistic quantum chemical methods. Notably, the variational inclusion of spin-orbit interactions leads not only to a different electronic ground state, but also to a lower bond multiplicity compared with those in previous studies.

Electron correlation within the relativistic no-pair approximation.

This paper addresses the definition of correlation energy within 4-component relativistic atomic and molecular calculations. In the nonrelativistic domain the correlation energy is defined as the difference between the exact eigenvalue of the electronic Hamiltonian and the Hartree-Fock energy. In practice, what is reported is the basis set correlation energy, where the "exact" value is provided by a full Configuration Interaction (CI) calculation with some specified one-particle basis. The extension of this definition to the relativistic domain is not straightforward since the corresponding electronic Hamiltonian, the Dirac-Coulomb Hamiltonian, has no bound solutions. Present-day relativistic calculations are carried out within the no-pair approximation, where the Dirac-Coulomb Hamiltonian is embedded by projectors eliminating the troublesome negative-energy solutions. Hartree-Fock calculations are carried out with the implicit use of such projectors and only positive-energy orbitals are retained at the correlated level, meaning that the Hartree-Fock projectors are frozen at the correlated level. We argue that the projection operators should be optimized also at the correlated level and that this is possible by full Multiconfigurational Self-Consistent Field (MCSCF) calculations, that is, MCSCF calculations using a no-pair full CI expansion, but including orbital relaxation from the negative-energy orbitals. We show by variational perturbation theory that the MCSCF correlation energy is a pure MP2-like correlation expression, whereas the corresponding CI correlation energy contains an additional relaxation term. We explore numerically our theoretical analysis by carrying out variational and perturbative calculations on the two-electron rare gas atoms with specially tailored basis sets. In particular, we show that the correlation energy obtained by the suggested MCSCF procedure is smaller than the no-pair full CI correlation energy, in accordance with the underlying minmax principle and our theoretical analysis. We also show that the relativistic correlation energy, obtained from no-pair full MCSCF calculations, scales at worst as X(-2) with respect to the cardinal number X of our correlation-consistent basis sets optimized for the two-electron atoms. This is better than the X(-1) scaling suggested by previous studies, but worse than the X(-3) scaling observed in the nonrelativistic domain. The well-known 1/Z- expansion in nonrelativistic atomic theory follows from coordinate scaling. We point out that coordinate scaling for consistency should be accompanied by velocity scaling. In the nonrelativistic domain this comes about automatically, whereas in the relativistic domain an explicit scaling of the speed of light is required. This in turn explains why the relativistic correlation energy to the lowest order is not independent of nuclear charge, in contrast to nonrelativistic theory.

Investigation of Multiconfigurational Short-Range Density Functional Theory for Electronic Excitations in Organic Molecules.

Computational methods that can accurately and effectively predict all types of electronic excitations for any molecular system are missing in the toolbox of the computational chemist. Although various Kohn-Sham density-functional methods (KS-DFT) fulfill this aim in some cases, they become inadequate when the molecule has near-degeneracies and/or low-lying double-excited states. To address these issues we have recently proposed multiconfiguration short-range density-functional theory-MC-srDFT-as a new tool in the toolbox. While initial applications for systems with multireference character and double excitations have been promising, it is nevertheless important that the accuracy of MC-srDFT is at least comparable to the best KS-DFT methods also for organic molecules that are typically of single-reference character. In this paper we therefore systematically investigate the performance of MC-srDFT for a selected benchmark set of electronic excitations of organic molecules, covering the most common types of organic chromophores. This investigation confirms the expectation that the MC-srDFT method is accurate for a broad range of excitations and comparable to accurate wave function methods such as CASPT2, NEVPT2, and the coupled cluster based CC2 and CC3.

Excitation Spectra of Nucleobases with Multiconfigurational Density Functional Theory.

Range-separated hybrid methods between wave function theory and density functional theory (DFT) can provide high-accuracy results, while correcting some of the inherent flaws of both the underlying wave function theory and DFT. We here assess the accuracy for excitation energies of the nucleobases thymine, uracil, cytosine, and adenine, using a hybrid between complete active space self-consistent field (CASSCF) and DFT methods. The method is based on range separation, thereby avoiding all double-counting of electron correlation and is denoted long-range CASSCF short-range DFT (CAS-srDFT). Using a linear response extension of CAS-srDFT, we compare the first 7-8 excited states of the nucleobases with perturbative multireference approaches as well as coupled cluster based methods. Our results show that the CAS-srDFT method can provide accurate excitation energies in good correspondence with the computationally more expensive methods.

The Dalton quantum chemistry program system.

Dalton is a powerful general-purpose program system for the study of molecular electronic structure at the Hartree-Fock, Kohn-Sham, multiconfigurational self-consistent-field, Møller-Plesset, configuration-interaction, and coupled-cluster levels of theory. Apart from the total energy, a wide variety of molecular properties may be calculated using these electronic-structure models. Molecular gradients and Hessians are available for geometry optimizations, molecular dynamics, and vibrational studies, whereas magnetic resonance and optical activity can be studied in a gauge-origin-invariant manner. Frequency-dependent molecular properties can be calculated using linear, quadratic, and cubic response theory. A large number of singlet and triplet perturbation operators are available for the study of one-, two-, and three-photon processes. Environmental effects may be included using various dielectric-medium and quantum-mechanics/molecular-mechanics models. Large molecules may be studied using linear-scaling and massively parallel algorithms. Dalton is distributed at no cost from http://www.daltonprogram.org for a number of UNIX platforms.

Correlated four-component EPR g-tensors for doublet molecules.

The first correlated ab initio four-component calculations of electron paramagnetic resonance (EPR) g-tensors for doublet radicals are reported. We have implemented a first-order degenerate perturbation theory approach based on the four-component Dirac-Coulomb Hamiltonian and fully relativistic configuration interaction wave functions in the DIRAC program package. We find that the correlation effects on the g-tensors can be sufficiently well described with manageable basis sets of triple-zeta quality and manageable configuration spaces. The new fully relativistic EPR module in DIRAC should be useful for benchmarking density functional theory approaches, however, with future optimization of the code we believe it will also be useful for applications.

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.

A multiconfigurational hybrid density-functional theory.

We propose a multiconfigurational hybrid density-functional theory which rigorously combines a multiconfiguration self-consistent-field calculation with a density-functional approximation based on a linear decomposition of the electron-electron interaction. This gives a straightforward extension of the usual hybrid approximations by essentially adding a fraction λ of exact static correlation in addition to the fraction λ of exact exchange. Test calculations on the cycloaddition reactions of ozone with ethylene or acetylene and the dissociation of diatomic molecules with the Perdew-Burke-Ernzerhof and Becke-Lee-Yang-Parr density functionals show that a good value of λ is 0.25, as in the usual hybrid approximations. The results suggest that the proposed multiconfigurational hybrid approximations can improve over usual density-functional calculations for situations with strong static correlation effects.

Molecular-Level Insight into the Spectral Tuning Mechanism of the DsRed Chromophore.

We present a detailed study of the protein environmental effects on the one- and two-photon absorption (1PA and 2PA, respectively) properties of the S0-S1 transition in the DsRed protein using the polarizable embedding density functional theory formalism. We find that steric factors and chromophore-protein interactions act in concert to enhance the 2PA activity inside the protein while adversely blue-shifting the 1PA maximum. A two-state model reveals that the 2PA intensity gain is primarily governed by the increased change in the permanent dipole moment between the ground and the excited states acquired inside the protein. Our results indicate that this mainly is attributable to counter-directional contributions stemming from Lys163 and the conserved Arg95 with the former additionally identified as a key residue in the color tuning mechanism. The results provide new insight into the tuning mechanism of DsRed and suggest a possible strategy for simultaneous improvement of its 1PA and 2PA properties.

Analysis of self-consistency effects in range-separated density-functional theory with Møller-Plesset perturbation theory.

Range-separated density-functional theory combines wave function theory for the long-range part of the two-electron interaction with density-functional theory for the short-range part. When describing the long-range interaction with non-variational methods, such as perturbation or coupled-cluster theories, self-consistency effects are introduced in the density functional part, which for an exact solution requires iterations. They are generally assumed to be small but no detailed study has been performed so far. Here, the authors analyze self-consistency when using Møller-Plesset-type (MP) perturbation theory for the long range interaction. The lowest-order self-consistency corrections to the wave function and the energy, that enter the perturbation expansions at the second and fourth order, respectively, are both expressed in terms of the one-electron reduced density matrix. The computational implementation of the latter is based on a Neumann series which, interestingly, even though the effect is small, usually diverges. A convergence technique, which perhaps can be applied in other uses of Neumann series in perturbation theory, is proposed. The numerical results thus obtained show that, in weakly bound systems, self-consistency can be neglected since the long-range correlation does not affect the density significantly. Although MP is not adequate for multireference systems, it can still be used as a reliable analysis tool. Though the density change is not negligible anymore in such cases, self-consistency effects are found to be much smaller than long-range correlation effects (less than 10% for the systems considered). For that reason, a sensible approximation might be to update the short-range energy functional term while freezing its functional derivative, namely, the short-range local potential, in the wave function optimization. The accuracy of such an approximation still needs to be assessed.

Large-scale parallel configuration interaction. II. Two- and four-component double-group general active space implementation with application to BiH.

We present a parallel implementation of a large-scale relativistic double-group configuration interaction (CI) program. It is applicable with a large variety of two- and four-component Hamiltonians. The parallel algorithm is based on a distributed data model in combination with a static load balancing scheme. The excellent scalability of our parallelization scheme is demonstrated in large-scale four-component multireference CI (MRCI) benchmark tests on two of the most common computer architectures, and we also discuss hardware-dependent aspects with respect to possible speedup limitations. With the new code we have been able to calculate accurate spectroscopic properties for the ground state and the first excited state of the BiH molecule using extensive basis sets. We focused, in particular, on an accurate description of the splitting of these two states which is caused by spin-orbit coupling. Our largest parallel MRCI calculation thereby comprised an expansion length of 2.7x10(9) Slater determinants.

Gauge origin independent calculations of nuclear magnetic shieldings in relativistic four-component theory.

The use of perturbation-dependent London atomic orbitals, also called gauge including atomic orbitals, has proven efficient for calculations of NMR shielding constants and other magnetic properties in the nonrelativistic framework. In this paper, the theory of London atomic orbitals for NMR shieldings is extended to the four-component relativistic framework and our implementation is described. The relevance of London atomic orbitals in four-component calculations as well as computational aspects are illustrated with test calculations on hydrogen iodide. We find that the use of London atomic orbitals is an efficient method for reliable calculations of NMR shielding constants with standard basis sets, also for four-component calculations with spin-orbit coupling effects included in the wave function optimization. Furthermore, we find that it is important that the small component basis functions fulfill the magnetic balance for accurate description of the diamagnetic shielding and that the role of London atomic orbitals in the relativistic domain is to provide atomic magnetic balance even in the molecular case, thus greatly improving basis set convergence. The Sternheim approximation, which calculates the diamagnetic contribution as an expectation value, leads to significant errors and is not recommended.