Finite field treatment of vibrational polarizabilities and hyperpolarizabilities: On the role of the Eckart conditions, their implementation, and their use in characterizing key vibrations

HeH+ under Spatial Confinement

In the present study, the influence of spatial confinement on the bond length as well as dipole moment, polarizability and (hyper)polarizabilities of HeH+ ion was analyzed. The effect of spatial confinement was modelled by cylindrically symmetric harmonic oscillator potential, that can be used to mimic high pressure conditions. Based on the conducted research it was found that the spatial confinement significantly affects the investigated properties. Increasing the confinement strength leads to a substantial decrease of their values. This work may be of particular interest for astrochemistry as HeH+ is believed to be the first compound to form in the Universe.

Molecular Electrides: An In Silico Perspective

Electrides are defined as the ionic compounds where the electron(s) serves as an anion. These electron(s) is (are) not bound to any atoms, bonds, or molecules rather than they are localized into the space, crystal voids, or interlayer between two molecular slabs. There are three major categories of electrides, known as organic electriades, inorganic electrides, and molecular electrides. The computational techniques have proven as a great tool to provide emphasis on the electride materials. In this review, we have focused on the computational methodologies and criteria that help to characterize molecular electrides. A detailed account of the computational methods and basis sets applicable for molecular electrides have been discussed along with their limitation(s) in this field. The main criterion for the identification of the electrides has also been discussed thoroughly with proper examples. The molecular electrides presented here have been justified with all the required criteria that support and proved their electride characteristics. We have also presented a few systems which have similar properties but are not considered as molecular electrides. Moreover, the applicability of the electrides in catalytic processes has also been presented.

First-order hyperpolarizabilities of propellanes: elucidating structure-property relationships

Following recent experimental work demonstrating strong nonlinear optical properties, namely second harmonic generation of light, in crystals composed of 16,20-dinitro-(3,4,8,9)-dibenzo-2,7-dioxa-5,10-diaza[4.4.4]propellane molecules [A. Miniewicz, S. Bartkiewicz, E. Wojaczyńska, T. Galica, R. Zaleśny and R. Jakubas, J. Mater. Chem. C, 2019, 7, 1255-1262] in this paper we aim to investigate "structure-property" relationships for a series of 16 propellanes presenting a wide palette of substituents with varying electron-accepting/donating capabilities. To that end, we use electronic- and vibrational-structure theories and a recently developed generalized few-state model combined with a range-separated CAM-B3LYP functional to analyze electronic and vibrational contributions to the first hyperpolarizability for the whole series of molecules. The variations in computed properties are large among the studied set of substituents and can reach an order of magnitude. It has been demonstrated that the maximum values of frequency-independent first hyperpolarizability are expected for strong electron-accepting NO₂ substituents, but only at the preferred position with respect to the electronegative oxygen atom in the 1,4-oxazine moiety. This holds for electronic as well as vibrational counterparts.

Partitioning of interaction-induced nonlinear optical properties of molecular complexes. II. Halogen-bonded systems

Following our study on hydrogen-bonded (HB) complexes [Phys. Chem. Chem. Phys., 2018, 20, 19841], the physical nature of interaction-induced (non)linear optical properties of another important class of molecular complexes, namely halogen-bonded (XB) systems, was analyzed in this study. The excess electronic and nuclear relaxation (hyper)polarizabilities of nine representative XB complexes covering a wide range of halogen-bond strengths were computed. The partitioning of the excess properties into individual interaction-energy components (electrostatic, exchange, induction, dispersion) was performed by using the variational-perturbational energy decomposition scheme at the MP2/aug-cc-pVTZ level of theory and further supported by calculations with the SCS-MP2 method. In the case of the electronic interaction-induced properties, the physical composition of Δαel and Δγel was found to be very similar for the two types of bonding, despite the different nature of the binding. For Δβel, the XB complexes exhibit a more systematic interplay of interaction-energy contributions compared to the HB systems studied in the previous work. Our analysis revealed that the patterns of interaction-energy contributions to the interaction-induced nuclear-relaxation contributions to the linear polarizability and the first hyperpolarizability are very similar. For both properties the exchange repulsion term is canceled out by the electrostatic and delocalization terms. The physical composition of these contributions is analogous to those observed for the HB complexes.

Can Density Functional Theory Be Trusted for High-Order Electric Properties? The Case of Hydrogen-Bonded Complexes

This work reports on an extensive assessment of the performance of a wide palette of density functional approximations in predicting the (high-order) electric properties of hydrogen-bonded complexes. To this end, we compute the electronic and vibrational contributions to the electric polarizability and the first and second hyperpolarizabilities, using the CCSD(T)/aug-cc-pVTZ level of theory as reference. For all the studied properties, the average absolute errors below 20% can only be obtained using the CAM-B3LYP functional, while LC-BLYP and MN15 are shown to be only slightly less accurate (average absolute errors not exceeding 30%). Among Minnesota density functionals, i.e., M06, M06-2X, and MN15, we only recommend the latter one, which quite accurately predicts the electronic and vibrational (hyper)polarizabilities. We also analyze the optimal tuning of the range-separation parameter μ for the LC-BLYP functional, finding that this approach does not bring any systematic improvement in the predictions of electronic and vibrational (hyper)polarizabilities and the accuracy of computed properties is largely system-dependent. Finally, we report huge errors in predicting the vibrational second hyperpolarizability by ωB97X, M06, and M06-2X functionals. Based on the explicit evaluation of anharmonic terms contributing to the second hyperpolarizability, this failure is traced down to a poor determination of third- and fourth-order energy derivatives with respect to normal modes. These results reveal serious flaws of some density functional approximations and suggest caution in selecting the appropriate functional to calculate not only electronic and vibrational (hyper)polarizabilities but also other molecular properties that contain vibrational anharmonic contributions.

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.

Partitioning of interaction-induced nonlinear optical properties of molecular complexes. I. Hydrogen-bonded systems

Interaction-induced electronic and vibrational (hyper)polarizabilities were decomposed into different interaction types (electrostatic, exchange, induction and dispersion).

Vibrational nonlinear optical properties of spatially confined weakly bound complexes

This study focuses on the theoretical description of the influence of spatial confinement on the electronic and vibrational contributions to (hyper)polarizabilities of two dimeric hydrogen bonded systems, namely HCNHCN and HCNHNC. A two-dimensional analytical potential is employed to render the confining environment (e.g. carbon nanotube). Based on the results of the state-of-the-art calculations, performed at the CCSD(T)/aug-cc-pVTZ level of theory, we established that: (i) the influence of spatial confinement increases with increasing order of the electrical properties, (ii) the effect of spatial confinement is much larger in the case of the electronic than vibrational contribution (this holds for each order of the electrical properties) and (iii) the decrease in the static nuclear relaxation first hyperpolarizability upon the increase of confinement strength is mainly due to changes in the harmonic term, however, in the case of nuclear relaxation second hyperpolarizability the anharmonic terms contribute more to the drop of this property.

About the nature of halogen bond interaction under the spatial confinement

Nowadays, much attention is put toward the description of noncovalent complexes exposed to the high pressure or embedded in confining environments. Such conditions may strongly modify the physical and chemical properties of molecular systems. This study focuses on the theoretical description of the confinement induced changes in geometry and energetic parameters of the halogen bonded FCl⋯CNF complex. A model analytical potential is applied to render the effect of orbital compression. In order to analyze the nature of halogen bond interaction, in the presence of spatial confinement, the supermolecular approach together with the symmetry-adapted perturbation theory is used. Furthermore, a thorough analysis of topological parameters, characterizing the halogen bond upon orbital compression, is performed within the quantum theory of atoms in molecules. The calculations are carried out using the ωB97x and CCSD(T) methods in connection with the aug-cc-pVTZ basis set. Among others, the obtained results indicate that the spatial confinement not only modifies the nature of halogen bond interaction but also induces the appearance of a completely new form of the studied FCl⋯CNF system.

On the physical origins of interaction-induced vibrational (hyper)polarizabilities

This paper presents the results of a pioneering exploration of the physical origins of vibrational contributions to the interaction-induced electric properties of molecular complexes. In order to analyze the excess nuclear relaxation (hyper)polarizabilities, a new scheme was proposed which relies on the computationally efficient Bishop-Hasan-Kirtman method for determining the nuclear relaxation contributions to electric properties. The extension presented herein is general and can be used with any interaction-energy partitioning method. As an example, in this study we employed the variational-perturbational interaction-energy decomposition scheme (at the MP2/aug-cc-pVQZ level) and the extended transition state method by employing three exchange-correlation functionals (BLYP, LC-BLYP, and LC-BLYP-dDsC) to study the excess properties of the HCN dimer. It was observed that the first-order electrostatic contribution to the excess nuclear relaxation polarizability cancels with the negative exchange repulsion term out to a large extent, resulting in a positive value of Δα(nr) due to the contributions from the delocalization and the dispersion terms. In the case of the excess nuclear relaxation first hyperpolarizability, the pattern of interaction contributions is very similar to that for Δα(nr), both in terms of their sign as well as relative magnitude. Finally, our results show that the LC-BLYP and LC-BLYP-dDsC functionals, which yield smaller values of the orbital relaxation term than BLYP, are more successful in predicting excess properties.

Comparison of Property-Oriented Basis Sets for the Computation of Electronic and Nuclear Relaxation Hyperpolarizabilities

In the present work, we perform an assessment of several property-oriented atomic basis sets in computing (hyper)polarizabilities with a focus on the vibrational contributions. Our analysis encompasses the Pol and LPol-ds basis sets of Sadlej and co-workers, the def2-SVPD and def2-TZVPD basis sets of Rappoport and Furche, and the ORP basis set of Baranowska-Łączkowska and Łączkowski. Additionally, we use the d-aug-cc-pVQZ and aug-cc-pVTZ basis sets of Dunning and co-workers to determine the reference estimates of the investigated electric properties for small- and medium-sized molecules, respectively. We combine these basis sets with ab initio post-Hartree-Fock quantum-chemistry approaches (including the coupled cluster method) to calculate electronic and nuclear relaxation (hyper)polarizabilities of carbon dioxide, formaldehyde, cis-diazene, and a medium-sized Schiff base. The primary finding of our study is that, among all studied property-oriented basis sets, only the def2-TZVPD and ORP basis sets yield nuclear relaxation (hyper)polarizabilities of small molecules with average absolute errors less than 5.5%. A similar accuracy for the nuclear relaxation (hyper)polarizabilites of the studied systems can also be reached using the aug-cc-pVDZ basis set (5.3%), although for more accurate calculations of vibrational contributions, i.e., average absolute errors less than 1%, the aug-cc-pVTZ basis set is recommended. It was also demonstrated that anharmonic contributions to first and second hyperpolarizabilities of a medium-sized Schiff base are particularly difficult to accurately predict at the correlated level using property-oriented basis sets. For instance, the value of the nuclear relaxation first hyperpolarizability computed at the MP2/def2-TZVPD level of theory is roughly 3 times larger than that determined using the aug-cc-pVTZ basis set. We link the failure of the def2-TZVPD basis set with the difficulties in predicting the first-order field-induced coordinates. On the other hand, the aug-cc-pVDZ and ORP basis sets overestimate the property in question only by roughly 30%. In this study, we also propose a low-cost composite treatment of anharmonicity that relies on the combination of two basis sets, i.e., a large-sized basis set is employed to determine lowest-order derivatives with respect to the field-induced coordinates, and a medium-sized basis set is used to compute the higher-order derivatives. The results of calculations performed at the MP2 level of theory demonstrate that this approximate scheme is very successful at predicting nuclear relaxation hyperpolarizabilities.

On the particular importance of vibrational contributions to the static electrical properties of model linear molecules under spatial confinement

The influence of the spatial confinement on the electronic and vibrational contributions to longitudinal electric-dipole properties of model linear molecules including HCN, HCCH and CO₂ is analyzed.

On the vibrational linear and nonlinear optical properties of compounds involving noble gas atoms: HXeOXeH, HXeOXeF, and FXeOXeF

The vibrational (hyper)polarizabilities of some selected Xe derivatives are studied in the context of Bishop-Kirtman perturbation theory (BKPT) and numerical finite field methodology. It was found that for this set of rare gas compounds, the static vibrational properties are quite large, in comparison to the corresponding electronic ones, especially those of the second hyperpolarizability. This also holds for the dc-Pockels β(-ω;ω,0), Kerr γ(-ω;ω,0,0) and electric field second harmonic generation γ (-2ω;ω,ω,0) effects, although the computed nuclear relaxation (nr) vibrational contributions are smaller in magnitude than the static ones. HXeOXeH was used to study the effects of electron correlation, basis set, and geometry. Geometry effects were found to lead to noticeable changes of the vibrational and electronic second hyperpolarizability. A limited study of the effect of Xe insertion to the nr vibrational properties is also reported. Assessment of the results revealed that Xe insertion has a remarkable effect on the nr (hyper)polarizabilities. In terms of the BKPT, this is associated with a remarkable increase of the electrical and mechanical anharmonicity terms. The latter is consistent with the anharmonic character of several vibrational modes reported for rare gas compounds.

Performance of density functional theory in computing nonresonant vibrational (hyper)polarizabilities

A set of exchange-correlation functionals, including BLYP, PBE0, B3LYP, BHandHLYP, CAM-B3LYP, LC-BLYP, and HSE, has been used to determine static and dynamic nonresonant (nuclear relaxation) vibrational (hyper)polarizabilities for a series of all-trans polymethineimine (PMI) oligomers containing up to eight monomer units. These functionals are assessed against reference values obtained using the Møller-Plesset second-order perturbation theory (MP2) and CCSD methods. For the smallest oligomer, CCSD(T) calculations confirm the choice of MP2 and CCSD as appropriate for assessing the density functionals. By and large, CAM-B3LYP is the most successful, because it is best for the nuclear relaxation contribution to the static linear polarizability, intensity-dependent refractive index second hyperpolarizability, static second hyperpolarizability, and is close to the best for the electro-optical Pockels effect first hyperpolarizability. However, none of the functionals perform satisfactorily for all the vibrational (hyper)polarizabilities studied. In fact, in the case of electric field-induced second harmonic generation all of them, as well as the Hartree-Fock approximation, yield the wrong sign. We have also found that the Pople 6-31+G(d) basis set is unreliable for computing nuclear relaxation (hyper)polarizabilities of PMI oligomers due to the spurious prediction of a nonplanar equilibrium geometry.

Variational calculation of static and dynamic vibrational nonlinear optical properties

The vibrational configuration interaction method used to obtain static vibrational (hyper)polarizabilities is extended to dynamic nonlinear optical properties in the infinite optical frequency approximation. Illustrative calculations are carried out on H(2)O and NH(3). The former molecule is weakly anharmonic while the latter contains a strongly anharmonic umbrella mode. The effect on vibrational (hyper)polarizabilities due to various truncations of the potential energy and property surfaces involved in the calculation are examined.

Refractive index of liquid water in different solvent models

We present a combined molecular dynamics/quantum chemical perturbation method for calculating the refractive index of liquid water at different temperatures. We compare results of this method with the refractive index obtained from other solvent models. The best agreement with the experimental refractive index of liquid water and its temperature dependence is obtained using correlated gas-phase polarizabilities in the classical Lorentz-Lorenz expression. Also, the iterative self-consistent reaction field approach in the semicontinuum implementation matches the experimental refractive index reasonably well.

Second harmonic generation second hyperpolarizability of water calculated using the combined coupled cluster dielectric continuum or different molecular mechanics methods

In this article we report the first calculations of second harmonic generation second hyperpolarizability of liquid water using coupled cluster/molecular mechanics (CC/MM) methods or coupled cluster/dielectric continuum (CC/DC) methods. The latter approach treats the solvent as an isotropic homogeneous fluid while the former accounts for the discrete nature of the solvent molecules. The CC/MM approach may include or exclude polarization effects explicitly. Alternatively, polarization effects may be included using perturbation theory. The CC descriptions implemented are the coupled cluster second-order approximate singles and doubles (CC2) and coupled cluster singles and doubles models. The second harmonic generation second hyperpolarizabilities are, depending on the model, obtained using either an analytical implementation of the cubic response function or using an analytical implementation of the quadratic response function combined with the finite field technique. The CC/MM results for the second harmonic generation second hyperpolarizability compare excellently with experimental data while a significant overestimation is found when using the CC/DC model. Particular, the cavity radius in the CC/DC calculations have an enormous effects on this fourth-order property.

Variational calculation of vibrational linear and nonlinear optical properties

A variational approach for reliably calculating vibrational linear and nonlinear optical properties of molecules with large electrical and/or mechanical anharmonicity is introduced. This approach utilizes a self-consistent solution of the vibrational Schrodinger equation for the complete field-dependent potential-energy surface and, then, adds higher-level vibrational correlation corrections as desired. An initial application is made to static properties for three molecules of widely varying anharmonicity using the lowest-level vibrational correlation treatment (i.e., vibrational Møller-Plesset perturbation theory). Our results indicate when the conventional Bishop-Kirtman perturbation method can be expected to break down and when high-level vibrational correlation methods are likely to be required. Future improvements and extensions are discussed.

Ab initiodipole polarizability surfaces of water molecule: Static and dynamic at 514.5nm

Coupled cluster calculations with a carefully designed basis set have been performed to obtain both static, alpha, and dynamic at 514.5 nm, alpha(514.5 nm), dipole polarizability surfaces of water. We employed a medium size basis set (13s10p6d3f9s6p2d1f)[9s7p6d3f6s5p2d1f] consisting of 157 contracted Gaussian-type functions that yields values near the Hartree-Fock limit for alpha [G. Maroulis, J. Chem. Phys. 94, 1182 (1991)]. The alpha and alpha(514.5 nm) surfaces were able to reproduce all the experimentally available information about the dipole polarizability of water, especially the Raman spectra of gaseous H(2)O, D(2)O, and HDO. Vibrational averages for the dipole polarizability of water molecule are also reported.

Basis set and electron correlation effects on initial convergence for vibrational nonlinear optical properties of conjugated organic molecules

Using three typical pi-conjugated molecules (1,3,5-hexatriene, 1-formyl-6-hydroxyhexa-1,3,5- triene, and 1,1-diamino-6,6-dinitrohexa-1,3,5-triene) we investigate the level of ab initio theory necessary to produce reliable values for linear and nonlinear optical properties, with emphasis on the vibrational contributions that are known to be important or potentially important. These calculations are made feasible by employing field-induced coordinates in combination with a finite field procedure. For many, but not all, purposes the MP2/6-31+G(d) level is adequate. Based on our results the convergence of the usual perturbation treatment for vibrational anharmonicity was examined. Although this treatment is initially convergent in most circumstances, a problematic situation has been identified.

Intensity-carrying modes important for vibrational polarizabilities and hyperpolarizabilities of molecules: derivation from the algebraic properties of formulas and applications

The intensity-carrying mode (ICM) theory is developed for analyzing the vibrational motions that mainly contribute to vibrational polarizabilities and hyperpolarizabilities, which are important for describing intermolecular electrostatic interactions and nonlinear optical properties of molecules. The ICMs are derived from dipole derivatives, polarizability derivatives, and first hyperpolarizability derivatives by using algebraic properties of intensity formulas. The way to obtain explicit forms of ICMs, including the optimization method of the basis of the ICM vector space, is discussed in detail. One- and two-dimensional models are constructed on the basis of the ICMs. The theory is applied to three molecules (a push-pull type polyene, a streptocyanine dye cation, and a symmetric neutral polyene) taken as typical examples. It is shown that the ICM theory provides a reasonable picture on the vibrational polarization properties of these molecules. On the basis of this result, the validity of the valence-bond charge transfer (VB-CT) model, which is a one-dimensional model and is widely used to describe the electronic and vibrational properties of dye molecules, is also discussed.