Vibrational Hyperpolarizabilities Research Articles

Vibrational polarizability and hyperpolarizability of p-nitroaniline

Within the double harmonic oscillator approximation scheme based on an RHF/6–31G electronic structure, the vibrational contributions to the polarizability, and first and second hyperpolarizability tensors of p-nitroaniline have been evaluated for different optical processes by adopting the infinite frequency or enhanced approximation. The static vibrational longitudinal first hyperpolarizability reaches 1352 au whereas the corresponding electronic counterpart is 1264 au. The [ α 2] 0,0 and [ μβ] 0,0 contributions to the vibrational longitudinal second hyperpolarizability are, in the static limit, equal to 44750 and 61112 au, respectively, in comparison with the 70874 au electronic value. These vibrational contributions remain important when considering the different dynamic optical processes with the exception of second- and third-harmonic generation. The main contributions to both the first and second vibrational hyperpolarizabilities are due to vibrational normal modes which describe the oscillation between aromatic-like and quinoid-like structures.

Molecular first hyperpolarizability of push-pull polyenes: Relationship between electronic and vibrational contribution by a two-state model.

In this work we present a theoretical justification, based on a two-state model, of the recently observed close resemblance between the molecular first-order hyperpolarizability (${\mathrm{\ensuremath{\beta}}}^{\mathit{e}}$) obtained with traditional experimental or theoretical methods and the vibrational (or relaxation) contribution (${\mathrm{\ensuremath{\beta}}}^{\mathit{r}}$) for several classes of polyconjugated molecules. The vibrational hyperpolarizabilities have been evaluated according to a semiclassical model previously presented by the authors in which molecular polarizabilities are expressed in terms of vibrational (infrared and Raman) intensities. Here we prove that in the case of polyconjugated molecules, the analytic expressions of ${\mathrm{\ensuremath{\beta}}}^{\mathit{e}}$ and ${\mathrm{\ensuremath{\beta}}}^{\mathit{r}}$ are functions of the same physical parameters. This implies that ${\mathrm{\ensuremath{\beta}}}^{\mathit{e}}$ and ${\mathrm{\ensuremath{\beta}}}^{\mathit{r}}$ provide a measure of the same physical property. \textcopyright{} 1996 The American Physical Society.

Role of collective modes in vibrational polarizabilities and hyperpolarizabilities of polyacetylene and other quasilinear polymers

It is shown that the low frequency collective modes of quasilinear polymers can contribute to the vibrational polarizability and hyperpolarizability despite the fact that their infrared or Raman intensity vanishes in the infinite chain limit. The most significant of these modes are characterized and the dynamic vibrational nonlinear optical (NLO) properties are obtained, at the doubly harmonic Hartree–Fock level of approximation, from static field calculations on polyacetylene as well as polyyne, polysilane, and polyethylene. Comparison is made to a pair of unit cell modes, one Raman-active and the other infrared-active, that are expected to generate a large vibrational hyperpolarizability. The combined effect of the collective and unit cell vibrations is potentially of considerable importance for the enhancement of NLO properties. Further studies are suggested to exploit this potential.

Vibrational hyperpolarizabilities and the Kerr effect in CH4, CF4, and SF6

The hyperpolarizabilities γ of CH4, CF4, and SF6 were measured by the dc Kerr effect at wavelengths from 457.9 to 1092 nm. Vibrational hyperpolarizabilities γv were obtained by combining these measurements with electric-field-induced second harmonic generation (ESHG) measurements. The vibrational contribution to the hyperpolarizability ranges from 6% to 35% of the total. At high optical frequency the difference between γv for Kerr and γv for ESHG is approximately constant, and has values 18, 31, and 51×10−63 C4 m4 J−3 for CH4, CF4, and SF6, respectively. The experimental results are in good quantitative agreement with the results of recent ab initio calculations of the frequency dependence of γv for CH4, except for a small but non-negligible discrepancy at high frequency.

Analysis of the vibrational, static and dynamic, second hyperpolarizability of five small molecules

Various approximate methods for the calculation of vibrational second hyperpolarizabilities are assessed on the basis of results for CH4, NH3, H2O, HF, and CO2. Three specific variations of the perturbation-theoretic approach are analyzed. These are defined by the types of terms retained in expansions which involve electrical and mechanical anharmonicities. The lowest-order approximation being the double-harmonic one and the next higher one the relaxation method. For the dynamic hyperpolarizabilities, the replacement of the optical frequencies (ω) by ω→∞ (the infinite-frequency approximation) is also explored. On the evidence at hand it is concluded that the relaxation/infinite-frequency method is a satisfactory approximation for the nonlinear optical vibrational hyperpolarizabilities.

Theoretical investigation of the Kerr effect for CH4

The vibrational contributions to the Kerr effect and to electric-field-induced second-harmonic generation (ESHG) are calculated for methane for a number of optical frequencies. The latter results, together with the experimental ESHG values of the total mean second hyperpolarizability, allow for the determination of the ω2L-dispersion curve for the mean electronic hyperpolarizability. Since this curve is identical, to fourth order, for both processes, we are able to combine it with the calculated Kerr vibrational hyperpolarizabilities and predict the total Kerr hyperpolarizabilities for CH4 for several laser frequencies.

A simple method for determining approximate static and dynamic vibrational hyperpolarizabilities

A simple method is presented for calculating approximate static and dynamic vibrational hyperpolarizabilities. It involves determining electrical properties in the presence of a static field with and without geometry optimization. This method is readily applicable to all tensor components of a general polyatomic molecule.

A comparative analysis of two methods for the calculation of electric-field-induced perturbations to molecular vibration

Two common methods of accounting for electric-field-induced perturbations to molecular vibration are analyzed and compared. The first method is based on a perturbation-theoretic treatment and the second on a finite-field treatment. The relationship between the two, which is not immediately apparent, is made by developing an algebraic formalism for the latter. Some of the higher-order terms in this development are documented here for the first time. As well as considering vibrational dipole polarizabilities and hyperpolarizabilities, we also make mention of the vibrational Stark effect.

Calculation of vibrational dynamic hyperpolarizabilities for H2O, CO2, and NH3

Dynamic vibrational first and second hyperpolarizabilities, applicable to a number of nonlinear optical processes, have been calculated for H2O, CO2, and NH3 at a typical laser frequency. As a percentage of their electronic counterparts, they range from being very small to being on the order of 20–30 %. The static values have also been calculated and are much larger. The calculations were based on perturbation theory and required knowledge of (a) first and second derivatives of the electrical properties (dipole moment, polarizability, and first hyperpolarizability functions) with respect to the normal coordinates, and (b) the harmonic and first anharmonic force constants. These were found at both the SCF and MP2 levels of approximation. In most cases electron correlation has a marked effect on the vibrational hyperpolarizabilities. Anharmonicity contributions are relatively small except for the static case and for the dc-Kerr effect.

Static polarizabilities and hyperpolarizabilities, and multipole moments for Cl 2 and Br 2. Electron correlation and molecular vibration effects

Ab initio calculations of the quadrupole and hexadecapole moments, and the static dipole polarizabilities and hyperpolarizabilities are reported for Cl 2 and Br 2. These calculations allow for both electron correlation and molecular vibration. The electronic contributions to the various properties are calculated by finite-field, fourth-order, many-body perturbation theory at three bond lengths, and then averaged over the zero point vibrations. The pure vibrational hyperpolarizabilities are calculated using sum-over-vibrational-states expressions; the requisite matrix elements are calculated with numerical vibrational wavefunctions and the electronic dipole polarizabilities as a function of bond length. The vibrational hyperpolarizabilities are found to be sizable fractions of the electronic hyperpolarizabilities, especially for the longitudinal component. Electron correlation effects are shown to be important for both the electronic and the vibrational hyperpolarizabilities. The calculations are in good agreement with available experimental values.

Compact formulas for vibrational dynamic dipole polarizabilities and hyperpolarizabilities

Compact, readily-programmable formulas are given for the vibrational dynamic dipole polarizability (αv) and hyperpolarizabilities (β v and γv) for polyatomic molecules. They are based on perturbation theory and should replace those presented by us previously.

Linear and nonlinear polarizabilities of t r a n s-polysilane from a b i n i t i o oligomer calculations

Linear and nonlinear static electronic polarizabilities of the σ-conjugated polymer trans-polysilane (t-PSi) are obtained at the Hartree–Fock level by extrapolation of ab initio calculations on the oligomers H3Si–(SiH2)N–SiH3 (N odd). For the longitudinal components, in particular, a 6–31 G basis is shown to yield accurate values if the chains are long enough. No computational advantage is found in using effective core potentials. Chain lengths through N=13 are sufficient to give limiting infinite chain polarizabilities with relatively small uncertainty. The calculated longitudinal linear polarizability per SiH unit is about the same as the corresponding property in the prototype π-conjugated polymers polyacetylene (PA) and polydiacetylene (PDA), but the longitudinal second hyperpolarizability is significantly less in the σ-conjugated polymer. In t-PSi the conjugation length is about half that in either PA or PDA. Frequency-dependent vibrational distortion contributions are estimated in the doubly harmonic approximation and turn out to be important in trisilane and pentasilane even at optical frequencies. An analysis is developed which sheds light on the expected behavior for increasing chain lengths. Reasonable agreement with experiment is obtained for both the electronic and vibrational longitudinal second hyperpolarizabilities.

A perturbation method for calculating vibrational dynamic dipole polarizabilities and hyperpolarizabilities

Perturbation formulas are derived for calculating the vibrational dynamic polarizability (αv) and hyperpolarizabilities (βv and γv ) of polyatomic molecules. These formulas, based on an initial harmonic oscillator approximation, include corrections for mechanical anharmonicity (cubic) terms in the vibrational potential and electrical anharmonicity (quadratic) terms in the dependence of the electrical field polarization potential on nuclear coordinates. Results are presented for FH and CO2. In the former case, comparison is made to ‘‘exact’’ numerical values. Fully documented computer programs for the perturbation treatment are available.

Evaluation of vibrational hyperpolarizabilities

A practical working formula is developed for calculating the dynamic (or static) vibrational hyperpolarizability, γ v αβγδ, of an arbitrary polyatomic molecule. First-order contributions due to electrical and mechanical anharmonicity are included. Contrary to previous work, we find that the so-called “μ 4” terms vanish to the order considered.

Vibrational hyperpolarizability of SF6

Expressions are derived for the vibrational second hyperpolarizability γv of SF6, in the static limit and for the nonlinear-optical processes of the dc Kerr effect, electric-field-induced second harmonic generation (ESHG), and third harmonic generation (THG). These expressions give γv in terms of vibrational transition dipole, polarizability, and first hyperpolarizability matrix elements (μ,α,β). The matrix elements needed in the numerical evaluation of γv are obtained from the literature and from the Raman scattering measurements reported here. In the static limit it is found that γv≈20 γe, where γe is the electronic contribution to γ. At optical frequencies γv is still comparable to γe in magnitude, and it shows significant frequency dependence.