Double Harmonic

Figure 5-13 A Double Harmonic Oscillator. Displacements jci and X2, shown positive, may also be negative or zero.

The change in the dipole moment with respect to a geometry displacement along a normal coordinate is approximately proportional to the intensity of an IR absorption. In the so-called double harmonic approximation (terminating the expansion at first order in the electric field and geometry), the intensity is (except for some constants)... 

In the double harmonic approximation, only fundamental bands can have an intensity different from zero. Including higher-order terms in the expansion allows calculation... 

To be precise, this expression employs the so-called double-harmonic approximation, where cubic and higher force constants as well as second and higher dipole derivatives are ignored. This approximation is common to all current implementations of calculating IR and Raman intensities. For details see Amos, 1987. 

Fig. 4. Schematic representation of a 3-atom Double Harmonic Oscillator (DHO) with r, r2 interatomic distances and k, k2 force constants, and of TR-A, TR-B, TR-C temperature ranges corresponding to the DHO model.

The quantities of interest in vibrational spectra are frequencies and intensities. Within the double harmonic approximation, vibrational frequencies and normal modes for solvated molecules are related, within the continuum approach, to free energy second derivatives with respect to nuclear coordinates calculated at the equilibrium nuclear configuration. The QM analogues for vibrational intensities , depend on the spectroscopy under study, but in any case derivative methods are needed. 

In the limit of static fields, the nuclear relaxation contribution (from now on just vibrational ) to the polarizabilities can be computed in the double harmonic approximation, i.e. assuming that the expansions of both the potential energy and the electronic properties with respect to the normal coordinates can be limited to the quadratic and the linear terms, respectively (i.e. assuming both mechanical and electric harmonicity). 

As shown in ref. [20], the double harmonic procedure can be reformulated within the PCM so as to obtain the analogues of the classical expressions in terms of summations of derivatives of dipoles and polarizabilities with respect to normal coordinates but with all the properties computed in the presence of the solvent (i.e. exploiting effective properties), namely we obtain ... 

The derivatives of the star-quantities in Equations (2.195) and (2.196) can be obtained including the contributions due to the external charge qex in the expansion of G with respect to the field to be used in the derivation of the PCM double-harmonic scheme, exactly as we have done in the previous section to evaluate the orientational averaging namely ... 

Most ab initio analyses of vibrational spectra invoke a double-harmonic assumption wherein the potential energy surface in the vicinity of the minimum is fit to a function that involves only quadratic dependence of the energy with respect to the nuclear motions. The intensities of the normal vibrational modes are extracted from the derivatives of the dipole moment, taken as linear with respect to nuclear coordinates. Within this approximation, the intensities of the fundamentals are proportional to the square of the dipole moment derivatives with respect to normal coordinates ". ... 

Suitieri97 has also evaluated the anharmonic contributions to the nuclear relaxation y for some push-pull polyenes using analytical methods in a valence bond charge transfer model. Saal and Ouamerali98 have investigated the vibrational ft of N-fluorophemyl-2,5-dimethypyrrole in the double harmonic... 

The zeroth-order approximation in the BK perturbation treatment of pure vibrational NLO is the double harmonic model. As far as electrical properties are concerned this approximation includes just the terms in the instantaneous property expression that are linear in the normal coordinates (there is no vibrational contribution from the constant term). To these are added the quadratic terms in the pure vibrational (or mechanical) potential which constitute the usual harmonic approximation. Then, in zeroth-order roughly half of the square brackets vanish leaving ... 

Indeed, in the Bom-Oppenheimer approximation, the effects can be interpreted at a first-order level of approximation referred to as the double harmonic approximation. 

In the BO approximation, the perturbation theory implies the evaluar tion of the first terms of the expansion of the free energy function and of the properties with respect to the nuclear coordinates and to the external field. At level of double harmonicity (electric and mechanical) only linear... 

In this section we report a second extract of the study we have published on the Journal of the American Chemical Society about solvent effects on electronic and vibrational components of linear and nonlinear optical properties of Donor-Acceptor polyenes. In a previous section we have presented the analysis on geometries, here we report the results obtained for the electronic and vibrational (in the double harmonic approximation) static polarizability and hyperpolarizability for the two series of noncentrosym-metric polyenes NH2(CH=CH) R (n=l,2), with R=CHO (series I) and with R=N02 (series II) both in vacuo and in water. 

Numerical Solution for SchrOdlnger s Equation for Double Harmonic Oscillator... 

Figure 17 QuickBASIC user interface for numerical solutions to Schrodinger s equation for the double harmonic oscillator.

We consider, here, the case of infrared absorption spectroscopy. Raman intensities have been treated in a parallel way [6]. In the double harmonic approximation (where the forces and the changes in the total molecular dipole moment are linear with atomic splacements) the integrated intensity of the i-th normal mode is related to molecular properties by the relation ... 

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