Polarizability vibrational averaged

Abstract Although the electronic structure and the electrical properties of molecules in first approximation are independent of isotope substitution, small differences do exist. These are usually due to the isotopic differences which occur on vibrational averaging. Vibrational amplitude effects are important when considering isotope effects on dipole moments, polarizability, NMR chemical shifts, molar volumes, and fine structure in electron spin resonance, all properties which must be averaged over vibrational motion. 

PEC, are in general in better agreement with the CAS results than the pure SOPPA results. This applies to the equilibrium geometry results as well as to the vibrational averaged results. This shows that SOPPA(CCSD) performs better in the calculation of polarizabilities for LiH than SOPPA as might have been expected [36,41]. 

We have vibrationally averaged the CAS /daug-cc-pVQZ dipole and quadmpole polarizability tensor radial functions (equation (14)) with two different sets of vibrational wavefunctions j(i )). One was obtained by solving the one-dimensional Schrodinger equation for nuclear motion (equation (16)) with the CAS /daug-cc-pVQZ PEC and the other with an experimental RKR curve [70]. Both potentials provide identical vibrational... 

In Table 7 we compare the ZPVCs for the dipole and quadrupole polarizabilities of HF. In the same way as for LiH, we have calculated the vibrational averages for each method with two different wavefunctions - one obtained from the PEC of the same or related method as used in the calculation of the property curve and the other obtained from the loo CAS PEC. Compared with the equivalent results for LiH we observe significant differences between the calculations on the two molecules. Eirst of all the vibrational corrections are smaller than in LiH but roughly in the same ratio as the polarizabilities. The influence of the PEC is larger than in LiH. 

We have compared our MCSCE results for the vibrational ground state with CCSD, SOPPA, and SOPPA(CCSD) calculations. In particular we have investigated the importance of the PEC on the ZPVCs and find that there are significant differences between LiH and HF. In LiH the CCSD results for the ZPVC are very close to the MCSCF results independent on whether the CCSD or MCSCF PEC was employed. Similarly, the differences between SOPPA(CCSD) calculations with either the CCSD or the MCSCF energy surface are very small. In HF, on the other hand somewhat larger differences are found if the CCSD polarizabilities are averaged over the CCSD PEC and the difference between... 

SOPPA(CCSD) calculations with the CCSD or MCSCF PEC are also larger. In general the differences in the ZPVC are larger between the different PEC than between the different linear response methods. The SOPPA(CCSD) results for the equilibrium geometry as well as the vibrationally averaged polarizabilities are in both molecules in better agreement with the MCSCF results than the pure SOPPA values. 

H2 quadrupole moment, <72(re) at the fixed equilibrium position, and thus the long-range coefficient of the quadrupole-induced dipole component, Eq. 4.3, is about 5% too small relative to the proper vibrational average, <12 = (v = 0 < 2(r) f = 0) [216, 217, 209], A 5% difference of the dipole moment amounts to a 10% difference of the associated spectral intensities. Furthermore, the effects of electron correlation on this long-range coefficient can be estimated. Correlation increases the He polarizability by 5% but decreases the H2 quadrupole moment by 8% [275], a net change of-3% of the leading induction term B R). 

Recent work improved earlier results and considered the effects of electron correlation and vibrational averaging [278], Especially the effects of intra-atomic correlation, which were seen to be significant for rare-gas pairs, have been studied for H2-He pairs and compared with interatomic electron correlation the contributions due to intra- and interatomic correlation are of opposite sign. Localized SCF orbitals were used again to reduce the basis set superposition error. Special care was taken to assure that the supermolecular wavefunctions separate correctly for R —> oo into a product of correlated H2 wavefunctions, and a correlated as well as polarized He wavefunction. At the Cl level, all atomic and molecular properties (polarizability, quadrupole moment) were found to be in agreement with the accurate values to within 1%. Various extensions of the basis set have resulted in variations of the induced dipole moment of less than 1% [279], Table 4.5 shows the computed dipole components, px, pz, as functions of separation, R, orientation (0°, 90°, 45° relative to the internuclear axis), and three vibrational spacings r, in 10-6 a.u. of dipole strength [279]. 

There are two ways in which molecular vibrations affect non-linear optical properties. The first, which is well understood, is zero-point-vibrational averaging of the calculated electronic properties. This need not delay us long. The second comes about from the effect that the electromagnetic radiation has on the vibrational motions themselves and this leads to the vibrational polarizabilities and hyperpolarizabilities which are the exact counterparts of the electronic ones which stem from the effect that the radiation has on the electronic motions. This phenomenon is now receiving long overdue attention and will be the main subject of this section. A more extensive review is available elsewhere [2]. 

In the following the polarizability and the first and second hyperpolarizabilities for urea calculated at the SCF level in vacuo and in water are reported. Both static and frequency dependent nonlinear properties have been calculated, with the Coupled Perturbed Hartree-Fock (CPHF) and Time Dependent-CPHF procedures that have been described above. The solvent model is the Polarizable Continuum Model (PCM) whereas vibrational averaging of the optical properties along the C-0 stretching coordinate has been obtained by the DiNa package both in vacuo and in solution. 

Table 1.8 Vibrational averaged static polarizability (10 esu) and first hyperpolarizability (10 esu) in vacuo and in water at 298K.

The vibrational averaging has a little effect on the calculated polarizability (around -2%), and a greater influence on the longitudinal component of the hyperpolarizability. It is worth noting that in aqueous solution the vibrational correction to the Pz) value is markedly greater than in gas phase. Thus it is advisable to consider both the solvent and the vibrational effects, and their possible coupling, to get reliable values of molecular nonlinear properties. 

Santiago et a/. ° have calculated at the Time-Dependent Hartree-Fock level the vibrational contributions to the dynamic (hyper)polarizabilities of H2O2 and have demonstrated that, though smaller than their electronic counterparts, the zero-point vibrational average contributions increase faster with the frequency. 

The Stark effect of a rotational transition of a specific vibronic state will only yield the permanent electric dipole moment and the anisotropy of the electric polarizability (an — aj.). In principle, both molecular parameters are functions of the intemuclear distance. The measured Stark effect yields, therefore, the vibrational average of the molecular parameter which in trun is represented as a power series expansion in (t) + 1/2) and J J + 1) ... 

This then gives again two contributions a vibrationally averaged electronic polarizability... 

In this section, we will describe in more detail how the vibrational averaging of the pure electronic polarizability and the calculation of the vibrational polarizability is carried out. We will hereby distinguish between diatomic and polyatomic molecules. 

Whereas all the theoretical papers mentioned here discuss the vibrational contribution to the (hyper)polarizabilities, only Whitehouse and Buckingham [4] made an attempt to calculate these properties for the type of system in which we are interested. They used a much simplified pofenfial, in conjunction with a classical analysis— estimated to be valid above 20 K—to obtain a (temperature-dependent) expression for the vibrationally averaged dipole moment. Then, from the field-dependence of this expression, formulas for the vibrational linear polarizability and second hyperpolarizability were extracted and the former quantity was evaluated for the [Li C6o] ... 

We have described our most recent efforts to calculate vibrational line shapes for liquid water and its isotopic variants under ambient conditions, as well as to calculate ultrafast observables capable of shedding light on spectral diffusion dynamics, and we have endeavored to interpret line shapes and spectral diffusion in terms of hydrogen bonding in the liquid. Our approach uses conventional classical effective two-body simulation potentials, coupled with more sophisticated quantum chemistry-based techniques for obtaining transition frequencies, transition dipoles and polarizabilities, and intramolecular and intermolecular couplings. In addition, we have used the recently developed time-averaging approximation to calculate Raman and IR line shapes for H20 (which involves... 

The methods described above are all based on the Born-Oppenheimer approximation. Therefore, they can be used to calculate polarizabilities of diatomic molecules for a given internuclear distance R. However, if one is interested in values of the polarizability tensors, and C", for a particular vibrational state /i )), one has to average the polarizability radial functions a(R) and C(R) with the vibrational wavefunction i.e., one has to... 

For the vibrational ground state with quantum number u = 0 the averaged polarizabilities are often expressed as e sum of the polarizability at an equilibrium geometry, i e, and a zero-point-vibrational correction (ZPVC)... 

The polarizability tensor, a, introduced in section 4.1.2, is a measure of the facility of the electron distribution to distortion by an imposed electric field. The structure of the electron distribution will generally be anisotropic, giving rise to intrinsic birefringence. This optical anisotropy reflects the average electron distribution whereas vibrational and rotational modes of the molecules making up a sample will cause the polarizability to fluctuate in time. These modes are discrete, and considering a particular vibrational frequency, vk, the oscillating polarizability can be modeled as... 

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