Double harmonic approximation

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. 

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). 

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 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. 

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 ... 

Vibrational first hyperpolarizabilities. The double harmonic approximation... 

The errors encountered with a given level of theory in calculation of vibrational intensities are considerably more profound than those in the frequencies. Meaningless results are common unless the basis set is of double-C quality and contains polarization functions. The double-harmonic approximation is another source of error. Nonetheless, the effects of H-bonding upon the intensity of each mode can be profitably studied within this framework. Analysis of these intensities sheds light upon the electronic density perturbations caused by formation of the bond. 

In the double harmonic approximation (mechanical and electrical), the Fermi Golden Rule reduces to (4) [18] ... 

Kirtman and Luis" have presented a simple extension of their Finite Field-Nuclear Relaxation procedure for calculating the vibrational nonlinear optical contributions to the Degenerate Four-Wave Mixing or Intensity-Dependent Refractive Index process and have illustrated it by application to the water molecule at both HF and MP2 levels of approximation. This approach, which is based on the idea of substracting the unwanted electric dipole terms, enables to go beyond the simple double harmonic approximation, used so far for DFWM." ... 

At room temperature, only the vibrational ground state is in general populated. Truncating the expansion of the geometry dependence of the dipole moment in Eq. 2. 28 at the linear term (often referred to as the double-harmonic approximation [62] since we are also using a harmonic approximation for the vibrational wavefunction), there will be only one surviving term in the SOS expression for the vibrational wavefunction, and we can write the vibrational polarizability as... 

The strategy for evaluating these vibrational matrix elements is now clear, and invoking the double-harmonic approximation, we obtain the resonance contribution to the CARS intensity as... 

As mentioned in the introduction, most of the QM calculations of vibrational frequencies are performed within the double-harmonic approximation, that is, the truncation of the expansion of the potential energy as a function of the nuclear coordinates to the quadratic term (mechanical harmonic approximation) and the consideration of the hnear term only in the expansion of the dipole moment as a function of the nuclear coordinates (electric harmonic approximation). In such a framework, the QM calculation of vibrational frequencies can be reduced to the evaluation of the components of the Hessian matrix followed by diagonalization of the corresponding mass-weighted matrix [1]. Let us start by writing out the energy of a... 

Following Placzek s approach [e.g., 5-8], for the nonresonant Raman scattering of a randomly oriented ensemble within the double-harmonic approximation, and by assuming = ctijn, it is possible to obtain an expression of (k) as a function of derivatives of the firequency-dependent electronic polarizability with respect to normal modes Q. ... 

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