Pure Vibrational Contributions Research Articles

Coupled cluster investigation of the vibrational and electronic second and third harmonic scattering hyperpolarizabilities of the water molecule

The vibrational contributions to the average polarizability (α¯), to the second harmonic scattering (SHS) first hyperpolarizability (βSHS), and depolarization ratio (DRSHS), as well as to the third harmonic scattering (THS) second hyperpolarizability (γTHS) and depolarization ratio (DRTHS), have been evaluated for the water molecule using the Bishop and Kirtman perturbative theory approach, in combination with finite differentiation techniques to evaluate the higher-order derivatives. From a hierarchy of coupled cluster techniques and extended atomic basis sets, the CCSD/d-aug-cc-pVTZ level has been selected to assess the importance of the zero-point vibrational average (ZPVA) contributions and of the pure vibrational contributions with respect to their electronic counterparts. This is the first investigation demonstrating electronic and vibrational SHS, and THS responses can be computed for small molecules, with the perspective of performing comparisons with recent experimental data [Van Steerteghem et al., Anal. Chem. 89, 2964 (2017) and V. Rodriguez, J. Phys. Chem. C 121, 8510 (2017)]. Numerical results on the water molecule highlight that (i) the vibrational contributions to the dynamic α¯, βSHS, and γTHS are small but non negligible; (ii) they amount to 3%, 10%, and 4% at the typical 1064 nm wavelength, respectively; (iii) the mechanical anharmonicity term dominates the ZPVA contribution; (iv) the double harmonic terms dominate the pure vibrational contributions; (v) the stretching vibrations provide the largest contributions to the dynamic (hyper)polarizabilities; and (vi) these conclusions are strongly impacted in the static limit where the vibrational contributions are much larger, in particular the double harmonic pure vibrational terms, and even more in the case of the first hyperpolarizability.

Non-linear optical response of meso substituted dipyrromethene boron difluoride dyes: Synthesis, photophysical, DFT and Z scan study

2-(Naphthalen-2-yl)benzo[d]thiazole meso fused dipyrromethene boron difluoride dyes are synthesised and their linear and non-linear optical properties are studied. The third order nonlinear susceptibility (χ3) value obtained from the Z-scan experiment is 11.21 × 10−13 and 13.30 × 10−13 esu. The global hybrid (B3LYP and BHHLYP) and range-separated hybrid (CAM-B3LYP) functional with the basis set 6-311++G(d,p) performed better in computing the linear and non-linear optical properties. The computed β0 value (56.36 × 10−30 esu and 48.64 × 10−30 esu for Kr-OH-BDY and Kr-OMe-BDY respectively) of both the dyes are superior to that of urea (βo = 0.371 × 10−30 esu). Thermal and reorientational study display the dominating electronic contribution to the linear polarizability (α) whereas for first-order hyperpolarizability (β), pure vibrational contributions found to be more operative in both the dyes. The geometries of the dyes were optimized by using B3LYP/6-311++ G(d,p) level of theory.

Analytic Density Functional Theory Calculations of Pure Vibrational Hyperpolarizabilities: The First Dipole Hyperpolarizability of Retinal and Related Molecules

We present a general approach for the analytic calculation of pure vibrational contributions to the molecular (hyper)polarizabilities at the density functional level of theory. The analytic approach allows us to study large molecules, and we apply the new code to the study of the first dipole hyperpolarizabilities of retinal and related molecules. We investigate the importance of electron correlation as described by the B3LYP exchange-correlation functional on the pure vibrational and electronic hyperpolarizabilities and compare the computed hyperpolarizabilities with available experimental data. The effects of electron correlation on the pure vibrational corrections vary signficantly even between these structurally very similar molecules, making it difficult to estimate these effects without explicit calculations at the density functional theory level. As expected, the frequency-dependent first hyperpolarizability, which determines the experimentally observed second-harmonic generation, is dominated by the electronic term, whereas for the static hyperpolarizability, the vibrational contribution is equally important. As a consequence, frequency extrapolation of the measured optical hyperpolarizabilities can only provide an estimate for the electronic contribution to the static hyperpolarizability, not its total value. The relative values of the hyperpolarizabilities for different molecules, obtained from the calculations, are in reasonable agreement with experimental data.

Vibrational spectra and static vibrational contribution to first hyperpolarizability of naphthopyrans—A combined experimental and DFT study

Two pairs of isomeric photochromic naphthopyrans were investigated by FT-IR and FT-Raman spectroscopy and by density functional theory (DFT) methods. The vibrational frequencies and contribution to first hyperpolarizabilities of the four naphthopyrans were determined at the B3LYP/6-31G (d,p) level. Comparison of the experimental and theoretical vibrational frequencies of the naphthopyrans was made. The assignments of the vibrational spectra have been carried out with the help of normal co-ordinate analysis (NCA). These results show that the ratio between pure vibrational contribution to the electronic contribution of first hyperpolarizability (βv/βe) increases when the 4-methoxyphenyl groups in the symmetrical isomers is replaced by 4-morpholinophenyl and phenyl group resulting in an asymmetrically substituted isomer pair. At the static limit, the low vibrational frequency modes have dominating contributions to the vibrational dipole polarizability and first hyperpolarizability.

A Lanczos-chain driven approach for calculating damped vibrational configuration interaction response functions

We present an approach based on the Lanczos method for calculating the vibrational configuration interaction response functions necessary for evaluating the pure vibrational contributions to the polarizabilities and first hyperpolarizabilities of molecules. The method iteratively builds a tridiagonal representation of the central response matrix, which is subsequently used for solving the response equations. From the same chain, the response functions can be evaluated approximately for any frequency as well as using any complex damping factor. Applications to formaldehyde, cyclopropene, and uracil illustrate the concepts presented.

David Bishop's approach to vibrational dynamic contributions to molecular properties: Application to Jones and magnetoelectric birefringences in diatomic molecules

AbstractThe expressions derived by D. M. Bishop for the pure vibrational contributions to the dynamic electric dipole polarizabilities and hyperpolarizabilities of polyatomic molecules (see Bishop, D. M. Rev Mod Phys, 1990, 62, 343) are generalized to arbitrary linear, quadratic and cubic response functions, and then applied to determine the effect of nuclear motion on two diatomic molecules (N2 and CO) on Jones (and Magnetoelectric) linear birefringences. The influence of nuclear motion on the anisotropy of the refractive index appears to be quite contained: the observable differs from the equilibrium electronic estimate by ≈4% for N2, and by ≈8–9% in CO. As a case study, also the effect of vibrations on the temperature independent contribution to Buckingham birefringence of H2 and D2 is analyzed and discussed. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

Static electric polarizabilities and first hyperpolarizabilities of molecular ions RgH + (Rg = He, Ne, Ar, Kr, Xe): ab initio study

Extensive ab initio calculations of static electric properties of molecular ions of general formula RgH + (Rg = He, Ne, Ar, Kr, Xe) involving the finite field method and coupled cluster CCSD(T) approach have been done. The relativistic effects were taken into account by Douglas–Kroll–Hess approximation. The numerical stability and reliability of calculated values have been tested using the systematic sequence of Dunning’s cc-pVXZ-DK and ANO-RCC-VQZP basis sets. The influence of ZPE and pure vibrational contribution has been discussed. The component α zz has increasing trend in RgH + while the relativistic effect on α zz leads to a small increase of this molecular parameter.

Atomic orbital-based cubic response theory for one-, two-, and four-component relativistic self-consistent field models

We present the first analytic calculations of the second hyperpolarizability in a relativistic framework. The calculations are made possible by our recent developments of a response theory built on a quasienergy formalism, in which the basis set may be both time and perturbation dependent. The approach is formulated for an arbitrary self-consistent field state in the atomic orbital basis. The implementation consists of a stand-alone code that only requires the unperturbed density in the atomic orbital basis as input, as well as a linear response solver by which we can determine the perturbed density matrices to different orders, at each new order solving equations that have the same structure as the linear response equation. Using these features of our formalism, we extend in this paper our approach to the relativistic domain, utilizing both two- and four-component relativistic wave functions. We apply the formalism to the calculation of the electronic and pure vibrational contributions to the second hyperpolarizability tensor for the hydrogen halides. Our results demonstrate that relativistic effects can be substantial for frequency-dependent second hyperpolarizabilities. Due to changes in the pole structure when going to the relativistic domain, the relativistic corrections to the hyperpolarizabilities are not transferable between different optical processes, except for very low frequencies.

Analytic Calculations of Vibrational Hyperpolarizabilities in the Atomic Orbital Basis

We present an analytic scheme for the calculation of pure vibrational contributions to linear and nonlinear optical properties such as the polarizability and the first and second hyperpolarizabilities. The formalism is fully expressed in terms of a perturbation- and time-dependent atomic orbital basis, using the elements of the density matrix in the atomic orbital basis as the basic variables. We calculate perturbed densities up to third order with respect to the electric field in accordance with the n + 1 rule, and the approach is therefore applicable for the calculation of pure vibrational contributions involving all vibrational coordinates in large molecular complexes. In the case of static electric fields, we therefore only need to calculate 19 response equations, independent of the size of the molecule. If we can determine the molecular energy and force field, the calculation of pure vibrational contributions to the nonlinear optical properties of the molecule is therefore a rather straightforward task. We illustrate the implementation by calculating pure vibrational contributions to the first and second hyperpolarizabilities of molecules containing up to 66 atoms using basis sets of good quality.

Vibrational effects on the dynamic electric properties of hydrogen peroxide

In this work we present a method based on the perturbation theoretic approach of Bishop and co-workers [J. Chem. Phys. 95, 2646 (1991); 97, 5255 (1992); 108, 10013 (1998)] to calculate the effect of torsional motion on the polarizability and hyperpolarizabilities of hydrogen peroxide. The frequency dependence has been evaluated using the time-dependent Hartree-Fock method. The results obtained show that the zero-point vibrational averaging contributions are small compared to the corresponding electronic contributions. In the static limit the pure vibrational contributions are very large, specially for beta and gamma. These contributions are significant for the hyperpolarizabilities even in the visible region, except for the second harmonic generation and third harmonic generation processes.

Linear response functions for a vibrational configuration interaction state

Linear response functions are implemented for a vibrational configuration interaction state allowing accurate analytical calculations of pure vibrational contributions to dynamical polarizabilities. Sample calculations are presented for the pure vibrational contributions to the polarizabilities of water and formaldehyde. We discuss the convergence of the results with respect to various details of the vibrational wave function description as well as the potential and property surfaces. We also analyze the frequency dependence of the linear response function and the effect of accounting phenomenologically for the finite lifetime of the excited vibrational states. Finally, we compare the analytical response approach to a sum-over-states approach.

Theoretical Investigation of the (Hyper)polarizabilities of Pyrrole Homologues C4H4XH (X = N, P, As, Sb, Bi). A Coupled-Cluster and Density Functional Theory Study

Geometries, inversion barriers, static and dynamic electronic and vibrational dipole polarizability (alpha), and first (beta) and second (gamma) hyperpolarizability of the pyrrole homologues C(4)H(4)XH (X = N, P, As, Sb, Bi) have been calculated by Hartree-Fock, Møller-Plesset second-order perturbation theory, coupled-cluster theory accounting for singles, doubles, and noniterative triple excitations methods, as well as density functional theory using B3LYP and PBE1PBE functionals and Sadlej's Pol and 6-311G basis sets. Relativistic effects on the heavier homologues stibole and bismole have been taken into account within effective core potential approximation. The results show that the electronic (hyper)polarizabilities monotonically increase with the atomic number of the heteroatom, consistent with the decrease in the molecular hardness. Ring planarization reduces the carbon-carbon bond length alternation of the cis-butadienic unit, enhancing the electronic polarizability values (alpha(e)) by 4-12% and the (hyper)polarizability values (and gamma(e)) by 30-90%. Pure vibrational and zero-point vibrational average contributions to the (hyper)polarizabilities have been determined within the clamped nucleus approach. In the static limit, the pure vibrational hyperpolarizabilities have a major contribution. Anharmonic corrections dominate the pure vibrational hyperpolarizabilities of pyrrole, while they are less important for the heavier homologues. Static and dynamic electronic response properties of the pyrrole homologues are comparable to or larger than the corresponding properties of the furan and cyclopentadiene homologue series.

Computer Simulation of the Linear and Nonlinear Optical Susceptibilities of p-Nitroaniline in Cyclohexane, 1,4-Dioxane, and Tetrahydrofuran in Quadrupolar Approximation. I. Molecular Polarizabilities and Hyperpolarizabilities

This is the first part of a study of the local field effects on (non)linear optical susceptibilities of solutions of para-nitroaniline (pNA) in three different solvents, cyclohexane (CH), 1,4-dioxane (DI), and tetrahydrofuran (THF), using a discrete molecular representation of the condensed phase. To account for dipolar and quadrupolar effects, the latter of which are especially important for DI solution, all the electric properties necessary to compute the local fields and local field gradients in quadrupolar approximation as well as the dipolar hyperpolarizabilities for the four molecules are computed, including frequency dispersion and vibrational contributions to the dipolar properties. The convergence of the perturbation treatment for the pure vibrational (PV) contributions is examined by comparison of the values obtained at the lowest order with those of partially computed second order in mechanical and electrical anharmonicity. For pNA, for which previous computations of the hyperpolarizabilities have generally found poor agreement with experimental results, a thorough investigation of the effects of solvent-induced geometry changes, dynamic and static correlation, frequency dispersion, and classical thermal averaging over the torsional modes of the substituent groups and the inversion mode of the amino group on the dipolar properties is carried out. Computations using self-consistent continuum reaction field models show that the amino group is substantially less pyramidalized in polar solvents than in the gas phase. With all the effects taken into account, reasonable agreement with the experimental electric-field induced second harmonic generation (EFISH) result on pNA vapor of Kaatz, Donley, and Shelton is obtained.

Vibrational contributions to nonlinear optical properties of methanol, ethanol and propanol

Static and dynamic vibrational contributions to the linear polarizability and the first and second nonlinear hyperpolarizabilities of methanol, ethanol and propanol have been calculated. Both the pure vibrational contribution and the zero-point vibrational averaging contribution have been determined by adopting the perturbation treatment of Bishop and Kirtman at the Hartree–Fock level. The pure vibrational contribution is quite important at the static limit, while in the optical region it varies for different nonlinear optical processes. The zero-point vibrational averaging contribution is a quite large correction to the electronic nonlinear optical (NLO) properties, especially for the second hyperpolarizability at finite fundamental frequencies. Vibrational analysis shows that the swinging modes at around 300 cm −1 and the C–H stretching modes at around 3160 cm −1 often give large contributions.

Vibrational properties and first hyperpolarizability of cyclopentadiene homologues C 4H 4XH 2 (X=C, Si, Ge, Sn): ab initio HF and DFT investigation

IR and Raman spectra and static electronic and vibrational first hyperpolarisabilities, β, of the cyclopentadiene homologues C 4H 4XH 2 (X=C, Si, Ge, Sn) have been calculated by conventional ab initio and DFT-B3LYP methods using Sadlej POL and correlation consistent Dunning basis sets to investigate on the role of the heteroatom on the response property. The pure vibrational contribution to β has been evaluated within the double harmonic oscillator approximation. A clear heavy atom effect was found in the vibrational frequencies and intensities localised on the XH 2 and C–X–C fragments, which decrease and increase, respectively, as the size of the heteroatom increases. Both β e and β v show an increase along the series, the vibrational contribution to β being modest in cyclopentadiene, but substantial in metalloles, especially for silole, where it is about two times greater than the electronic counterpart. The β v/ β e ratio is strongly dependent on the basis set, decreasing as the quality of the basis set increases. Both β v and total β e+ β v values of the cyclopentadiene homologues are higher than the corresponding ones in the furan series, except for stannole and tellurophene which have close total β e+ β v values.

Trends in the electronic and vibrational contributions to the dipole moment, polarizabilities, and first and second hyperpolarizabilities of the hydrides of Li, Na and K

The dipole moments μ, polarizabilities α, and first and second hyperpolarizabilities, β and γ of LiH, NaH and KH (MeH) have been computed using Hartree-Fock, MP2 and CCSD(T) theories. The static electronic and vibrational contributions to these properties are presented. The vibrational properties have been analysed into contributions due to zero-point vibrational averaging and pure vibrational terms. An alternative dissection of the vibrational properties into nuclear relaxation and curvature terms has also been considered. KH has been selected as a model system to study how the number of electrons, which are correlated (2, 10 and 20) affect both electronic and vibrational properties. The 10 electron approximation gives results that are practically the same as those computed by taking into account all 20 electrons of KH. The double-harmonic approximation has been shown to give satisfactory results for the pure vibrational contributions to the polarizability and the first hyperpolarizability, while this approximation is useful for demonstrating, qualitatively, the significance of the pure vibrational contributions to the second hyperpolarizability. In many cases the vibrational contributions are rather small percentages of the corresponding electronic contribution. However, several exceptions to the above observation have been noted. In all the cases considered the vibrational properties should be computed if reasonably accurate property values are sought. Electron correlation is important for both electronic and vibrational contributions to the electrical properties of the hydrides considered. The results are in satisfactory agreement with most of the best theoretical and experimental data concerning bond lengths, vibrational frequencies and electrical properties.

Initial convergence of the perturbation series expansion for vibrational nonlinear optical properties

Ab initio Hartree–Fock and MP2 calculations of the longitudinal (hyper)polarizability—including the static electronic, static zero-point vibrational average (ZPVA), and pure vibrational (static and dynamic) contributions—have been carried out on a set of seven typical medium size conjugated nonlinear optical (NLO) molecules. The ZPVA is obtained through first-order in mechanical plus electrical anharmonicity. Based on physical “nuclear relaxation” considerations the individual (square bracket) terms that contribute to the pure vibrational (hyper)polarizability are then taken into account through third-, fourth-, or fifth-order depending upon the type of term. In order to carry out the correlated treatment, field-induced coordinates and a special finite field technique are utilized. Correlation leads to very substantial differences in the absolute and relative values of the various contributions. In comparison to the electronic term the ZPVA correction is usually small but in one case is over two-thirds as large. On the other hand, both static and dynamic pure vibrational contributions are commonly of a magnitude that is comparable to, or are larger than, the electronic term. The higher-order pure vibration terms are often large. For dynamic processes they can be almost as important as the lowest-order terms; for static (hyper)polarizabilities they can be more important. Thus, for typical NLO molecules, the initial convergence behavior of the perturbation series in mechanical and electrical anharmonicity requires further investigation.

Vibrational corrections to electric properties of weakly bound systems

The pure electronic and vibrational contributions to electric dipole moments, dipole polarizabilities, and first hyperpolarizabilities have been evaluated for the HF and H2O dimers. The zero-point vibrational average corrections to dipole moments and dipole polarizabilities turn out to be relatively small. However, the corresponding contributions to the first hyperpolarizability are found to be of the same magnitude as the pure electronic values. The so-called pure vibrational corrections to the dipole polarizability and first hyperpolarizability of hydrogen bonded dimers are exceptionally large and indicate that the perturbation theory method used for their evaluation fails to account properly for the high mechanical and electric anharmonicities present in these systems. The analysis of different harmonic and anharmonic contributions to the pure vibrational correction to the first hyperpolarizability shows explicitly the importance of the low frequency intermolecular modes.

Vibrational Corrections to Transition Properties

The vibrational contributions to second-order transition properties are considered in the framework of the Born—Oppenheimer approximation. It is shown that the usual formula for vibrational second-order transition matrix elements is incomplete and needs to be supplemented by a term of purely vibrational origin. This pure vibrational contribution is calculated for vibrational transition polarizabilities in LiH and BeF and found to be quite significant. Its inclusion in theoretically calculated data for the Raman intensities appears to be necessary.

Theoretical determination of the vibrational and electronic (hyper)polarizabilities of C4H4X (X=O, S, Se, Te) heterocycles

Electronic, pure vibrational and dispersion contributions to the linear polarizability, α, and first hyperpolarizability, β, of the group 16 chalcogenides C4H4X (X=O, S, Se, Te) have been studied by ab initio calculations using the 6-31G*, the Sadlej POL, correlation consistent Dunning and aug-CEP-31G basis sets. Electron correlation was introduced through the MP2 theory. Vibrational (hyper)polarizabilities, evaluated within the double harmonic oscillator approximation, were analyzed in terms of single normal mode contributions. They were found to be a fraction of the electronic counterpart. βv does not show a clear dependence on the heavy atom, contrary to βe, which steadily increases along the series. The βv/βe ratio is very dependent on the basis set, decreasing and also changing sign in the presence of diffuse functions. Frequency-dependent electronic (hyper)polarizabilities were evaluated in the 0.02–0.16 a.u. frequency range. In the low-frequency range they can be fitted into a power series expansion in ω. In accord with the theory, the second-order coefficients of the β expansion are the same for both the SHG and EOPE non-linear optical processes, while in the α expansion they can be related to the lowest electronic excitation energy.