Recent Advances in Wave Function-Based Methods of Molecular-Property Calculations

Analytic first-order derivatives of partially contracted n-electron valence state second-order perturbation theory (PC-NEVPT2)

A balanced treatment of dynamic and static electron correlation is important in computational chemistry, and multireference perturbation theory (MRPT) is able to do this at a reasonable computational cost. In this paper, analytic first-order derivatives, specifically gradients and dipole moments, are developed for a particular MRPT method, state-specific partially contracted n-electron valence state second-order perturbation theory (PC-NEVPT2). Only one linear equation needs to be solved for the derivative calculation if the Z-vector method is employed, which facilitates the practical application of this approach. A comparison of the calculated results with experimental geometrical parameters of O₃ indicates excellent agreement although the calculated results for O₃ ^- are slightly outside the experimental error bars. The 0-0 transition energies of various methylpyrimidines and trans-polyacetylene are calculated by performing geometry optimizations and seminumerical second-order geometrical derivative calculations. In particular, the deviations of 0-0 transition energies of trans-polyacetylene from experimental values are consistently less than 0.1 eV with PC-NEVPT2, indicating the reliability of the method. These results demonstrate the importance of adding dynamic electron correlation on top of methods dominated by static electron correlation and of developing analytic derivatives for highly accurate methods.

RPA(D) and HRPA(D): Two new models for calculations of NMR indirect nuclear spin-spin coupling constants

In this article, the RPA(D) and HRPA(D) models for the calculation of linear response functions are presented. The performance of the new RPA(D) and HRPA(D) models is compared to the performance of the established RPA, HRPA, and SOPPA models in calculations of indirect nuclear spin-spin coupling constants using the CCSD model as a reference. The doubles correction offers a significant improvement on both the RPA and HRPA models; however, the improvement is more dramatic in the case of the RPA model. For all coupling types investigated in this study, the results obtained using the HRPA(D) model are comparable in accuracy to those given by the SOPPA model, while requiring between 30% and 90% of the calculation time needed for SOPPA. The RPA(D) model, while of slightly lower accuracy compared to the CCSD model than HRPA(D), offered calculation times of only approximately 25% of those required for SOPPA for all the investigated molecules. © 2018 Wiley Periodicals, Inc.

DFT and spatial confinement: a benchmark study on the structural and electrical properties of hydrogen bonded complexes

An extended set of 37 exchange correlation functionals, representing different DFT approximations, has been evaluated on a difficult playground represented by the dipole moment (μ_z), polarizability (α_zz), first hyperpolarizability (β_zzz), and the corresponding interaction-induced electrical properties (Δμ_z, Δα_zz, Δβ_zzz) of spatially confined hydrogen bonded (HB) dimers. A two-dimensional harmonic oscillator potential was used to exert the effect of spatial restriction. The performance of DFT methods in predicting hydrogen bond lengths in the studied molecular complexes upon confinement has also been examined. The data determined using a high-level CCSD(T) method serve as a reference. The conducted analyses allow us to conclude that methods rooted in DFT constitute a precise tool for the calculation of μ_z and α_zz as well as Δμ_z and Δα_zz, as most of the tested functionals provide results affected by rather small relative errors. On the other hand, an accurate description of the nonlinear optical response of the studied HB systems remains a great challenge for most of the analyzed DFT functionals, both in vacuum and in the presence of an analytical confining potential. Some of the tested DFT methods are found to be prone to catastrophic failure in the prediction of β_zzz as well as Δβ_zzz. The obtained results indicate that there is no great chasm in performance between functionals belonging to different DFT approximations or functionals including different amount of Hartree-Fock exchange when the values of dipole moment and first hyperpolarizability as well as the corresponding interaction-induced electrical properties are considered. However, a higher fraction of Hartree-Fock exchange improves the quality of predictions of α_zz and Δα_zz. Additionally, it has been shown that only three functionals from the examined set, namely B2PLYP, B3LYP and ωB97X-D, provide highly accurate structural parameters for the investigated systems. Of significant importance is the conclusion that the ωB97X-D functional, representing a modern and highly parametrized range-separated hybrid, demonstrates the most coherent behavior, showing rather small deviations from the reference data in the case of μ_z, α_zz, Δμ_z and Δα_zz as well as the structural parameters of the studied HB dimers. Moreover, our results indicate that the presence of spatial confinement has a rather small effect on the performance of DFT methods.

Theoretical Assessment of the Second-Order Nonlinear Optical Responses of Lindqvist-Type Organoimido Polyoxometalates

The second-order nonlinear optical properties of Lindqvist-type organoimido polyoxometalates bearing donor and acceptor substituents are evaluated by employing density functional theory and time-dependent density functional theory using the ωB97X-D range-separated hybrid exchange-correlation functional to describe accurately the field-induced effects. The hyper-Rayleigh scattering responses, β_HRS (-2ω; ω, ω), and the depolarization ratio are the targeted quantities. They are analyzed by resorting to the two-state model, which reduces the full summation-over-state expression to a single diagonal term and relates the response to a few spectroscopic quantities. The validity of this model is demonstrated by its ability to reproduce the β_HRS variations as a function of the nature of the ligand, owing to the dominant 1D character of these organic-inorganic hybrids. The calculated values are in good agreement with the recent experimental work of Al-Yasari et al. (Inorg. Chem. 2017, 56, 10181-10194), which demonstrates that the hexamolybdate moiety plays the role of an electron acceptor group. On the contrary, they contradict previous studies, which attributed an electron donor character to the polyoxometalate moiety. Calculations highlight that (i) combining the hexamolybdate unit with an organic ligand bearing a strong donor substituent leads to an enhanced first hyperpolarizability, associated with a dominant low-energy excited state, characterized by a large excitation-induced electron transfer from the donating ligand to the hexamolybdate, therefore coupling the polyoxometalate (POM) and its substituted ligand; (ii) in the case of weaker donor substituents, the hexamolybdate still behaves as an electron acceptor, but the first hyperpolarizability is smaller and the coupling has a reduced spatial extension; and, on the contrary, (iii) in the presence of an acceptor substituent, there is a competition between the hexamolybdate and this group so that the first hyperpolarizability becomes very small. The whole set of results demonstrates that polyoxometalate moieties are good candidates to achieve large second-order nonlinear optical (NLO) responses while keeping a rather large transparency window and also that there remains space to improve their integration into NLO efficient organic-inorganic hybrids.

General framework for calculating spin-orbit couplings using spinless one-particle density matrices: Theory and application to the equation-of-motion coupled-cluster wave functions

Standard implementations of nonrelativistic excited-state calculations compute only one component of spin multiplets (i.e., M_s = 0 triplets); however, matrix elements for all components are necessary for deriving spin-dependent experimental observables. Wigner-Eckart's theorem allows one to circumvent explicit calculations of all multiplet components. We generate all other spin-orbit matrix elements by applying Wigner-Eckart's theorem to a reduced one-particle transition density matrix computed for a single multiplet component. In addition to computational efficiency, this approach also resolves the phase issue arising within Born-Oppenheimer's separation of nuclear and electronic degrees of freedom. A general formalism and its application to the calculation of spin-orbit couplings using equation-of-motion coupled-cluster wave functions are presented. The two-electron contributions are included via the mean-field spin-orbit treatment. Intrinsic issues of constructing spin-orbit mean-field operators for open-shell references are discussed, and a resolution is proposed. The method is benchmarked by using several radicals and diradicals. The merits of the approach are illustrated by a calculation of the barrier for spin inversion in a high-spin tris(pyrrolylmethyl)amine Fe(II) complex.

Data-Driven Acceleration of the Coupled-Cluster Singles and Doubles Iterative Solver

Solving the coupled-cluster (CC) equations is a cost-prohibitive process that exhibits poor scaling with system size. These equations are solved by determining the set of amplitudes (t) that minimize the system energy with respect to the coupled-cluster equations at the selected level of truncation. Here, a novel approach to predict the converged coupled-cluster singles and doubles (CCSD) amplitudes, thus the coupled-cluster wave function, is explored by using machine learning and electronic structure properties inherent to the MP2 level. Features are collected from quantum chemical data, such as orbital energies, one-electron Hamiltonian, Coulomb, and exchange terms. The data-driven CCSD (DDCCSD) is not an alchemical method because the actual iterative coupled-cluster equations are solved. However, accurate energetics can also be obtained by bypassing solving the CC equations entirely. Our preliminary data show that it is possible to achieve remarkable speedups in solving the CCSD equations, especially when the correct physics are encoded and used for training of machine learning models.

DFT computational schemes for 15 N NMR chemical shifts of the condensed nitrogen-containing heterocycles

A systematic density functional theory (DFT) study of the accuracy factors (functionals, basis sets, and solvent effects) for the computation of ¹⁵ N NMR chemical shifts has been performed in the series of condensed nitrogen-containing heterocycles. The behavior of the most representative functionals was examined based on the benchmark calculations of ¹⁵ N NMR chemical shifts in the reference set of compounds. It was found that the best agreement with experiment was achieved with OLYP functional in combination with aug-pcS-3(N)//pc-2 locally dense basis set scheme providing mean absolute error of 5.2 ppm in the range of about 300 ppm. Taking into account solvent effects was performed within a general Tomasi's polarizable continuum model scheme. It was also found that computationally demanding supermolecular solvation model computations essentially improved some "difficult" cases, as was illustrated with phenanthroline dissolved in methanol. Based on the performed calculations, some 200 unknown ¹⁵ N NMR chemical shifts were predicted with a high level of confidence for about 50 real-life condensed nitrogen-containing heterocycles, which could serve as a practical guide in structural elucidation of this class of compounds.

Accurate Band Gap Predictions of Semiconductors in the Framework of the Similarity Transformed Equation of Motion Coupled Cluster Theory

In this work, we present a detailed comparison between wave-function-based and particle/hole techniques for the prediction of band gap energies of semiconductors. We focus on the comparison of the back-transformed Pair Natural Orbital Similarity Transformed Equation of Motion Coupled-Cluster (bt-PNO-STEOM-CCSD) method with Time Dependent Density Functional Theory (TD-DFT) and Delta Self Consistent Field/DFT (Δ-SCF/DFT) that are employed to calculate the band gap energies in a test set of organic and inorganic semiconductors. Throughout, we have used cluster models for the calculations that were calibrated by comparing the results of the cluster calculations to periodic DFT calculations with the same functional. These calibrations were run with cluster models of increasing size until the results agreed closely with the periodic calculation. It is demonstrated that bt-PNO-STEOM-CC yields accurate results that are in better than 0.2 eV agreement with the experiment. This holds for both organic and inorganic semiconductors. The efficiency of the employed computational protocols is thoroughly discussed. Overall, we believe that this study is an important contribution that can aid future developments and applications of excited state coupled cluster methods in the field of solid-state chemistry and heterogeneous catalysis.

In this work, it is shown that a combination of the embedded cluster approach with wave-function-based ab initio methods in the framework of the Similarity Transformed Equation of Motion Coupled Cluster (bt-PNO STEOM-CC) provides an accurate protocol for band gap energy predictions in classes of organic and inorganic semiconductors.

Computational protocols for calculating 13C NMR chemical shifts

The most recent results dealing with the computation of ¹³C NMR chemical shifts in chemistry (small molecules, saturated, unsaturated and aromatic compounds, heterocycles, functional derivatives, coordination complexes, carbocations, and natural products) are reviewed, paying special attention to theoretical background and accuracy, the latter involving solvent effects, vibrational corrections, and relativistic effects.

CASSCF linear response calculations for large open-shell molecules

The complete active space self-consistent-field (CASSCF) linear response method for the simulation of ultraviolet-visible (UV/Vis) absorption and electronic circular dichroism (ECD) spectra of large open-shell molecules is presented. By using a one-index transformed Hamiltonian, the computation of the most time-consuming intermediates can be pursued in an integral-direct fashion, which allows us to employ the efficient resolution-of-the-identity and overlap-fitted chain-of-spheres approximation. For the iterative diagonalization, pairs of Hermitian and anti-Hermitian trial vectors are used which facilitate, on the one hand, an efficient solution of the pair-structured generalized eigenvalue problem in the reduced space, and on the other hand, make the full multiconfigurational random phase approximation as efficient as the corresponding Tamm-Dancoff approximation. Electronic transitions are analyzed and characterized in the particle-hole picture by natural transition orbitals that are introduced for CASSCF linear response theory. For a small organic radical, we can show that the accuracy of simulated UV/Vis absorption spectra with the CASSCF linear response approach is significantly improved compared to the popular state-averaged CASSCF method. To demonstrate the efficiency of the implementation, the 50 lowest roots of a large Ni triazole complex with 231 atoms are computed for the simulated UV/Vis and ECD spectra.

New and Efficient Equation-of-Motion Coupled-Cluster Framework for Core-Excited and Core-Ionized States

We present a fully analytical implementation of the core-valence separation (CVS) scheme for the equation-of-motion (EOM) coupled-cluster singles and doubles (CCSD) method for calculations of core-level states. Inspired by the CVS idea as originally formulated by Cederbaum, Domcke, and Schirmer, pure valence excitations are excluded from the EOM target space and the frozen-core approximation is imposed on the reference-state amplitudes and multipliers. This yields an efficient, robust, practical, and numerically balanced EOM-CCSD framework for calculations of excitation and ionization energies as well as state and transition properties (e.g., spectral intensities, natural transition, and Dyson orbitals) from both the ground and excited states. The errors in absolute excitation/ionization energies relative to the experimental reference data are on the order of 0.2-3.0 eV, depending on the K-edge considered and on the basis set used, and the shifts are systematic for each edge. Compared to a previously proposed CVS scheme where CVS was applied as a posteriori projection only during the solution of the EOM eigenvalue equations, the new scheme is computationally cheaper. It also achieves better cancellation of errors, yielding similar spectral profiles but with absolute core excitation and ionization energies that are systematically closer to the corresponding experimental data. Among the presented results are calculations of transient-state X-ray absorption spectra, relevant for interpretation of UV-pump/X-ray probe experiments.

On the geometric dependence of the molecular dipole polarizability in water: A benchmark study of higher-order electron correlation, basis set incompleteness error, core electron effects, and zero-point vibrational contributions

In this work, we investigate how geometric changes influence the static dipole polarizability (α) of a water molecule by explicitly computing the corresponding dipole polarizability surface (DPS) across 3125 total (1625 symmetry-unique) geometries using linear response coupled cluster theory including single, double, and triple excitations (LR-CCSDT) and the doubly augmented triple-ζ basis set (d-aug-cc-pVTZ). Analytical formulae based on power series expansions of this ab initio surface are generated using linear least-squares analysis and provide highly accurate estimates of this quantity as a function of molecular geometry (i.e., bond and angle variations) in a computationally tractable manner. An additional database, which consists of 25 representative molecular geometries and incorporates a more thorough treatment of both basis sets and core electron effects, is provided as a current benchmark for this quantity and the corresponding leading-order C ₆ dispersion coefficient. This database has been utilized to assess the importance of these effects as well as the relative accuracy that can be obtained using several quantum chemical methods and a library of density functional approximations. In addition to high-level electron correlation methods (like CCSD) and our analytical least-squares formulae, we find that the SCAN0, PBE0, MN15, and B97-2 hybrid functionals yield the most accurate descriptions of the molecular polarizability tensor in H₂O. Using first-order perturbation theory, we compute the zero-point vibrational correction to α at the CCSDT/d-aug-cc-pVTZ level and find that this correction contributes approximately 3% to the isotropic (α _iso) and nearly 50% to the anisotropic (α _aniso) polarizability values. In doing so, we find that α _iso = 9.8307 bohr³, which is in excellent agreement with the experimental value of 9.83 ± 0.02 bohr³ provided by Russell and Spackman. The DPS reported herein provides a benchmark-quality quantum mechanical estimate of this fundamental quantity of interest and should find extensive use in the development (and assessment) of next-generation force fields and machine-learning based approaches for modeling water in complex condensed-phase environments.

Assessment of DFT for endohedral complexes' dipole moment: PNO-LCCSD-F12 as a reference method

We present a systematic evaluation of the performance of a wide range of exchange-correlation functionals and related dispersion correction schemes for the computation of dipole moments of endohedral complexes, formed through the encapsulation of an AB molecule (AB = LiF, HCl) inside carbon nanotubes (CNTs) of different diameter. The consistency and accuracy of (i) generalized gradient approximation, (ii) meta GGA, (iii) global hybrid, and (iv) range-separated hybrid density functionals are assessed. In total, 37 density functionals are tested. The results obtained using the highly accurate pair natural orbitals based explicitly correlated local coupled cluster singles doubles (PNO-LCCSD-F12) method of Werner and co-workers [Schwilk et al., J. Chem. Theory Comput., 2017, 13, 3650; Ma et al., J. Chem. Theory Comput., 2017, 13, 4871] with the aug-cc-pVTZ basis set serve as a reference. The static electric dipole moment is computed via the finite field response or, when possible, as the expectation value of the dipole operator. Among others, it is shown that functionals belonging to the class of range-separated hybrids, provide results closest to the coupled cluster reference data. In particular, the ωB97X as well as the M11 functional may be considered as a promising choice for computing electric properties of noncovalent endohedral complexes. On the other hand, the worst performance was found for the functionals which do not include the Hartree-Fock exchange. The analysis of both the coupled cluster and the DFT results indicates a strong coupling of dispersion and polarization that may also explain why lower level DFT methods, as well as Hartree-Fock and MP2, cannot yield dipole moments beyond a qualitative agreement with the higher order reference data. Interestingly, the much smaller and less systematically constructed basis sets of Pople of moderate size provide results of accuracy at least comparable with the extended Dunning's aug-cc-pVTZ basis set.

Time-Dependent Formulation of Resonance Raman Optical Activity Spectroscopy

In this work, we extend the theoretical framework recently developed for the simulation of resonance Raman (RR) spectra of medium-to-large sized systems to its chiral counterpart, namely, resonance Raman optical activity (RROA). The theory is based on a time-dependent (TD) formulation, with the transition tensors obtained as half-Fourier transforms of the appropriate cross-correlation functions. The implementation has been kept as general as possible, supporting adiabatic and vertical models for the PES representation, both in Cartesian and internal coordinates, with the possible inclusion of Herzberg-Teller (HT) effects. Thanks to the integration of this TD-RROA procedure within a general-purpose quantum-chemistry program, both solvation and leading anharmonicity effects can be included in an effective way. The implementation is validated on one of the smallest chiral molecule (methyloxirane). Practical applications are illustrated with three medium-size organic molecules (naproxen-OCD₃, quinidine and 2-Br-hexahelicene), whose simulated spectra are compared to the corresponding experimental data.

Polarization propagator theory and the entanglement between MO excitations

Entanglement is at the core of quantum physics and so, one may conjecture that it should have some influence on atomic and molecular response properties. The usual way of treating entanglement is by applying information theory via the von Newman entropy. Given that the principal propagator is the operator that contains the physical information that arises due to the transmission of the effects of two external perturbations through the electronic framework of a quantum system, it should have in it the information necessary to quantify the likely entanglement among molecular orbital excitations. In this article we first propose a proper density matrix and from it, the way to quantify entangled excitations by using information theory. The NMR J-couplings are among the best candidates to learn about the potentialities of this formalism. We applied this new tool to analyze the famous Karplus rule and found a relationship between the dihedral angular dependence and the entanglement. We also found that the entangled excitations are related to electron correlation. The new formalism can be applied to all other response properties.

Theoretical calculations of carbon-hydrogen spin-spin coupling constants

Structural applications of theoretical calculations of carbon-hydrogen spin-spin coupling constants are reviewed covering papers published mainly during the last 10-15 years with a special emphasis on the most notable studies of hybridization, substitution and stereoelectronic effects together with the investigation of hydrogen bonding and intermolecular interactions. The wide scope of different applications of calculated carbon-hydrogen couplings in the structural elucidation of particular classes of organic and bioorganic molecules is reviewed, concentrating mainly on saturated, unsaturated, aromatic and heteroaromatic compounds and their functional derivatives, as well as on natural compounds and carbohydrates. The review is dedicated to Professor Emeritus Michael Barfield in view of his invaluable pioneering contribution to this field.

GIAO-DFT calculation of 15 N NMR chemical shifts of Schiff bases: Accuracy factors and protonation effects

¹⁵ N NMR chemical shifts in the representative series of Schiff bases together with their protonated forms have been calculated at the density functional theory level in comparison with available experiment. A number of functionals and basis sets have been tested in terms of a better agreement with experiment. Complimentary to gas phase results, 2 solvation models, namely, a classical Tomasi's polarizable continuum model (PCM) and that in combination with an explicit inclusion of one molecule of solvent into calculation space to form supermolecule 1:1 (SM + PCM), were examined. Best results are achieved with PCM and SM + PCM models resulting in mean absolute errors of calculated ¹⁵ N NMR chemical shifts in the whole series of neutral and protonated Schiff bases of accordingly 5.2 and 5.8 ppm as compared with 15.2 ppm in gas phase for the range of about 200 ppm. Noticeable protonation effects (exceeding 100 ppm) in protonated Schiff bases are rationalized in terms of a general natural bond orbital approach.

Substitution effects in the 15 N NMR chemical shifts of heterocyclic azines evaluated at the GIAO-DFT level

A systematic study of the accuracy factors for the computation of ¹⁵ N NMR chemical shifts in comparison with available experiment in the series of 72 diverse heterocyclic azines substituted with a classical series of substituents (CH₃ , F, Cl, Br, NH₂ , OCH₃ , SCH₃ , COCH₃ , CONH₂ , COOH, and CN) providing marked electronic σ- and π-electronic effects and strongly affecting ¹⁵ N NMR chemical shifts is performed. The best computational scheme for heterocyclic azines at the DFT level was found to be KT3/pcS-3//pc-2 (IEF-PCM). A vast amount of unknown ¹⁵ N NMR chemical shifts was predicted using the best computational protocol for substituted heterocyclic azines, especially for trizine, tetrazine, and pentazine where experimental ¹⁵ N NMR chemical shifts are almost totally unknown throughout the series. It was found that substitution effects in the classical series of substituents providing typical σ- and π-electronic effects followed the expected trends, as derived from the correlations of experimental and calculated ¹⁵ N NMR chemical shifts with Swain-Lupton's F and R constants.