Frequency-dependent polarizabilities and hyperpolarizabilities

The experimental measures of these molecular electric properties involve oscillating fields. Thus, the frequency-dependence effects should be considered when comparing the experimental results . Currently, there are fewer calculations of the frequency-dependent polarizabilities and hyperpolarizabilities than those of the static properties. Recent advances have enabled one to study the frequency dispersion effects of polyatomic molecules by ab initio methods In particular, the frequency-dependent polarizability a and hyperpolarizability y of short polyenes have been computed by using the time-dependent coupled perturbed Hartree-Fock method. The results obtained show that the dispersion of a increases with the increase in the optical frequency. At a given frequency, a and its relative dispersion increase with the chain length. Also, like a, the hyperpolarizability y values increase with the chain length. While the electronic static polarizability is smaller than the dynamic one, the vibrational contribution is smaller at optical frequencies. ... 

The expansion coefficients are then obtained via least-squares fitting. This procedure can be (and has been) implemented directly to the ab initio calculation of the frequency-dependent polarizabilities and hyperpolarizabilities. 

P. K. Korambath and H. A. Kurtz, in Nonlinear Optical Materials Theory and Modeling, S. P. Kama and A. T. Yeates, Eds., American Chemical Society, Washington, DC, 1996, pp. 133-144. Frequency-Dependent Polarizabilities and Hyperpolarizabilities of Polyenes. 

Runge and Gross were among those to provide a solid theoretical framework for TDDFT. They showed that DFT can be extended to problems where a time-dependent external perturbation is present, such as the oscillating electric field of a laser. The theory was originally applied in physics to simple systems, such as atoms or idealized metal surfaces. Initially the focus was on frequency-dependent polarizabilities and hyperpolarizabilities. TDDFT also allows the treatment of problems that fall outside the scope of standard perturbation theory, such as high harmonic generation, but this field is still in its infancy and will not be further discussed here. 

Aiga, F. and Itoh, R. (1996). Calculation of frequency-dependent polarizabilities and hyperpolarizabilities by the second-order Mpller-Plesset perturbation theory. Chem. Phys. Lett, 251, 372-380. 

Sasagane, K., Aiga, F., 8c Itoh, R. (1993)). Higher-order response theory based on the quasienergy derivatives The derivation of the frequency-dependent polarizabilities and hyperpolarizabilities. Journal of Chemical Physics, 99, 3738. 

Static polarizabilities and hyperpolarizabilities were mentioned above. Frequency-dependent polarizabilities and hyperpolarizabilities can be calculated using two different approaches. First, the time-dependent Hartree-Fock (TDHF) approach, which is not correlated. This formalism enables a wide range of optical properties to be computed analytically and for any order. Correlated approaches to frequency-dependent polarizabilities and hyperpolarizabilities are available in the EOM-CCSD formalism. Obviously, this is significantly more expensive than TDHF, and it is sometimes useful to compute static properties with correlated methods and estimate the dynamic dispersion contribution from TDHF. 

This article considers ways of calculating nonresonant frequency-dependent polarizabilities and hyperpolarizabilities of molecules in the gas phase. This means that there is no consideration of solvent or surface effects. Nor is there any consideration of vibrational effects, which are covered in the article by Bishop. This article will concentrate on recent work. It has become clear that. Just as for static properties, it is necessary to use a high level of theory to get sensible results. In particular it is necessary to include electron correlation, and to use large, carefully chosen basis sets. Therefore this article considers ab initio methods rather than semi-empirical ones. Recent advances in density functional methods, however, are included, as they show promise for the future. 

Real-time TDDFT has been used to evaluate the frequency-dependent polarizability and first hyperpolarizability of YL156 (Fig. 1, 5), a push-pull 7i-conjugated system. It helps in assessing the reliability of the two-state approximation (TSA), which turns out to underestimate the effects of the solvent. 

In principle, density functional theory calculations should be able to give answers that are more reliable than Hartree-Fock but at similar cost. Static a and can be calculated by finite field methods or by coupled perturbed Kohn-Sham theory (CPKS) and give answers that are broadly comparable with MP2. In 1986 Sennatore and Subbaswamy did some calculations of the dynamic polarizability and second hyperpolarizability of rare gas atoms, but there have been no calculations of frequency dependent polarizabilities or hyperpolarizabilities of molecules until very recently. 

Gaussian can also predict some other properties dependent on the second and h er derivatives of the energy, such as the polarizabilities and hyperpolarizabilities. These depend on the second derivative with respect to an electric field, and are included automatically in every Hartree-Fock frequency calculation. 

Frequency-dependent polarizability a and second hyperpolarizability y corresponding to various third-order nonlinear optical processes have been... 

Our present focus is on correlated electronic structure methods for describing molecular systems interacting with a structured environment where the electronic wavefunction for the molecule is given by a multiconfigurational self-consistent field wavefunction. Using the MCSCF structured environment response method it is possible to determine molecular properties such as (i) frequency-dependent polarizabilities, (ii) excitation and deexcitation energies, (iii) transition moments, (iv) two-photon matrix elements, (v) frequency-dependent first hyperpolarizability tensors, (vi) frequency-dependent polarizabilities of excited states, (vii) frequency-dependent second hyperpolarizabilities (y), (viii) three-photon absorptions, and (ix) two-photon absorption between excited states. 

The extension of density functional theory (DFT) to the dynamical description of atomic and molecular systems offers an efficient theoretical and computational tool for chemistry and molecular spectroscopy, namely, time-dependent DFT (TDDFT) [7-11]. This tool allows us to simulate the time evolution of electronic systems, so that changes in molecular structure and bonding over time due to applied time-dependent fields can be investigated. Its response variant TDDF(R)T is used to calculate frequency-dependent molecular response properties, such as polarizabilities and hyperpolarizabilities [12-17]. Furthermore, TDDFRT overcomes the well-known difficulties in applying DFT to excited states [18], in the sense that the most important characteristics of excited states, the excitation energies and oscillator strengths, are calculated with TDDFRT [17, 19-26]. 

It is obviously useful to compare calculated and experimental values for response properties like (frequency-dependent) polarizabilities a, and hyperpolarizabilities j3... 

Calculations of the vibrational contributions to the static polarizability and hyperpolarizability have also been attempted. As far as the EFISH experiment is concerned, which depends on the square of an optical frequency field, it is assumed that there will be no direct contribution to (—2static contribution is comparable with the static electronic contribution to /1(0 0,0). An indirect vibrational effect through the linear polarizability of the solvent molecules is more important. Calculations of the vibrational effects in pNA cannot be carried out reliably even for the static case since the second term in the perturbation theory is much larger then the first and there is no evidence of convergence. 

X specifies the experimental angle between the external electric field and the light polarization at frequency p, and h is Planck s constant. The scalars and S, and the vectors R and R are functions of the transition moment polarizability and hyperpolarizability tensors, m is a unit vector oriented along the transition dipole moment and F is the internal electric field at the molecule, which depends on the externally applied field such that... 

Molecular polarizabilities and hyperpolarizabilities are now routinely calculated in many computational packages and reported in publications that are not primarily concerned with these properties. Very often the calculated values are not likely to be of quantitative accuracy when compared with experimental data. One difficulty is that, except in the case of very small molecules, gas phase data is unobtainable and some allowance has to be made for the effect of the molecular environment in a condensed phase. Another is that the accurate determination of the nonlinear response functions requires that electron correlation should be treated accurately and this is not easy to achieve for the molecules that are of greatest interest. Very often the higher-level calculation is confined to zero frequency and the results scaled by using a less complete theory for the frequency dependence. Typically, ab initio studies use coupled-cluster methods for the static values scaled to frequencies where the effects are observable with time-dependent Hartree-Fock theory. Density functional methods require the introduction of specialized functions before they can cope with the hyperpolarizabilities and higher order magnetic effects. 

Let us return to the problem of solving the response of the quantum mechanical system to an external electric field. The zeroth-order wave function of the quantum mechanical system is obtained by use of any of the standard approximate methods in quantum chemistry and the coupling to the field is described by the electtic dipole operator. There exist a number of ways to determine the response functions, some of which differ in formulation only, whereas others will be inherently different. We will give a short review of the characteristics of tire most common formulations used for the calculation of molecular polarizabilities and hyperpolarizabilities. The survey begins with the assumption that the external perturbing fields arc non-oscillatory, in which case we may determine molecular properties at zero frequencies, and then continues with the general situation of time-dependent fields and dynamic properties. 

The response to frequency-dependent external fields may be obtained from Hartree-Fock response theory, yielding dynamical polarizabilities and hyperpolarizabilities. The identification of excitation energies as the poles of the dynamical polarizability tensor may be invoked to calculate excitation energies as well as one-photon and two-photon transition moments from the time development of the ground state [40-42]. 

Maroulis117 has applied the finite field method to a study of HC1. In a systematic analysis with large basis sets, MBPT and CC techniques, the dipole, quadrupole, octupole and hexadecapole moments have been calculated at the experimental internuclear distance. The polarizability and several orders of hyperpolarizability have been calculated and the mean a and -values for the 18-electron systems HC1, HOOH, HOF, A, F2, H2S are compared. Fernandez et a/.118 have calculated the frequency dependent a, / and tensors for HC1 and HBr using the Multiple Configuration Self Consistent Field method (MCSCF), including the effect of molecular vibration. The results show good agreement with available experimental and theoretical data. 

Polarizabilities and Hyperpolarizabilities of Larger Molecules. - Ab Initio Calculations. At the most highly correlated level Christiansen et al.157 have used the CCS, CC2 and CCSD models to calculate the static polarizability of furan. Dispersion effects are included to make an estimate of the frequency-dependent polarizability. 

In the case of polarizability derivatives, however, the sparsity of results is not due to lack of interest, as this is a property that is just as important as the dipole moment derivative. Here the problem is that the calculations are more difficult, though not so much more difficult as to justify the comparatively small number of calculations in this area. There was a brief period of activity some five or six years ago in which various MC-SCF and Cl methods were tried on small molecules. ° Some earlier calculations are listed elsewhere.As with the quadrupole moment results, most of these could easily be improved upon with the aid of a large-scale multi-reference Cl calculation, which would be well within current capabilities. Some more recent polarizability derivative calculations, mostly SCF, may be found in Refs. 220 and 246-257. The most detailed of these is an M BPT calculation by Diercksen and Sadlej on CO. Another interesting group of calculations has considered the derivatives of the frequency-dependent polarizability. This shows some expected effects, for example that the frequency dependence in CI2 is noticeable, and some unexpected results, for example that the intensity of the V4 Raman-active mode of CH has a very marked frequency dependence. Dacre has provided some calculations on the polarizability of rare-gas dimers, which is of interest to the collision-induced Raman spectrum of such systems. Calculations of hyperpolarizabilities are confined to small systems. A recent example is for LiH. An example of the use of hyperpolarizability derivatives can be found where some fairly crude calculations were nevertheless useful in distinguishing two possible mechanisms in the collision-induced Raman spectrum of CO2. 

Finally, in the future, semiempirical methods will be applied to a wider and wider range of problems. At present, applications have been made to a wide range of ground-state properties, to vibrational frequencies, and consequently to thermodynamic quantities, such as entropies and heat capacities, to reaction mechanisms and transition states, to polarizabilities and hyperpolarizabilities, to time-dependent phenomena, and to quantum phenomena, such as tunneling, to molecules, ions, enzyme active sites, and to polymer properties including heats of polymerization and elastic moduli. With the passage of time the range of simulations will increase, and with it the ultimate limitations of the methods will become apparent. 

There are many approaches to compute the polarizabilities and hyperpolarizabilities and also different ways to classify them. One convenient division is between perturbation theory approaches, which express the (hyperjpolarizability using Summation-Over-States (SOS) expressions and those techniques, which are based on the evaluation of derivatives of the energy (or another property). SOS approaches consist in evaluating energies and transition dipoles that appear in the (hyper)polarizability expressions. For instance, in the case of the frequency-dependent electric-dipole electronic first hyperpolarizability, the SOS expression reads ... 

Many molecular properties may be calculated (static polarizability and hyperpolarizability frequency-dependent polarizability electric moments electric field and electric field gradient). 

An interesting application of the first case (i.e., an external oscillating field) is the study of the nonlinear properties of molecules in condensed matter. Once the approximate solutions of the corresponding time-dependent SchrOdinger equation are found, the frequency-dependent electric response functions (polarizability and hyperpolarizabilities tensors) of the molecular solute are easily calculated. 

Heterogeneous dielectric media models have included the developments of Jprgensen et al. [7-9] (reviewed here) and Corni and Tomasi [52,53], Generally, the number of methods for determining frequency-dependent molecular electronic properties, such as the polarizability or first- and second hyperpolarizability tensors of heterogeneously solvated molecules, is very limited. 

---