Electrical anharmonicities

Equations (6.5) and (6.12) contain terms in x to the second and higher powers. If the expressions for the dipole moment /i and the polarizability a were linear in x, then /i and ot would be said to vary harmonically with x. The effect of higher terms is known as anharmonicity and, because this particular kind of anharmonicity is concerned with electrical properties of a molecule, it is referred to as electrical anharmonicity. One effect of it is to cause the vibrational selection mle Au = 1 in infrared and Raman spectroscopy to be modified to Au = 1, 2, 3,. However, since electrical anharmonicity is usually small, the effect is to make only a very small contribution to the intensities of Av = 2, 3,. .. transitions, which are known as vibrational overtones. 

A molecule may show both electrical and mechanical anharmonicity, but the latter is generally much more important and it is usual to define a harmonic oscillator as one which is harmonic in the mechanical sense. It is possible, therefore, that a harmonic oscillator may show electrical anharmonicity. 

One effect of mechanical anharmonicity is to modify the Au = t infrared and Raman selection rule to Au = 1, 2, 3,. .., but the overtone transitions with Au = 2, 3,... are usually weak compared with those with Au = t. Since electrical anharmonicity also has this effect both types of anharmonicity may contribute to overtone intensities. 

However, unlike electrical anharmonicity, mechanical anharmonicity modifies the vibrational term values and wave functions. The harmonic oscillator term values of Equation (6.3) are modified to a power series in (u + ) ... 

Owing to the effects of mechanical anharmonicity - to which we shall refer in future simply as anharmonicity since we encounter electrical anharmonicity much less frequently -the vibrational wave functions are also modified compared wifh fhose of a harmonic oscillator. Figure 6.6 shows some wave functions and probabilify densify functions (iA A ) for an anharmonic oscillator. The asymmefry in and (iA A ) 5 compared wifh fhe harmonic oscillator wave functions in Figure f.i3, increases fheir magnitude on the shallow side of the potential curve compared with the steep side. 

At last, because of the whole strong anharmonicity involved in the H-bond system, the possibility of electrical anharmonicity cannot be a priori ignored [20],... 

T. Di Paolo, C. Bourderon, and C. Sandorfy, Model calculations on the influence of mechanical and electrical anharmonicity on infrared intensities Relation to hydrogen bonding. Can. 

The first term in Eq. (1.41) is the dipole moment, while the second is the electric anharmonicity. The expansion (1.40) diverges for large r. More appropriate forms of expansions are... 

This variation (electrical anharmonicity) can be taken into account, within the algebraic approach, by expanding the operator as... 

A further improvement can be obtained by introducing even more complex electrical anharmonicities, for example, by replacing the operator D in Eq. (2.140) with... 

An additional point that should be considered is that in the harmonic oscillator approximation, the selection mle for transitions between vibrational states is Ay = 1, where v is the vibrational quantum number and Ay > 1, that is, overtone transitions, which involve a larger vibrational quantum number change, are forbidden in this approximation. However, in real molecules, this rule is slightly relaxed due to the effect of anharmonicity of the oscillator wavefunction (mechanical anharmonicity) and/or the nonlinearity of the dipole moment function (electrical anharmonicity) [55], affording excitation of vibrational states with Ay > 1. However, the absorption cross sections for overtone transitions are considerably smaller than for Ay = 1 transitions and sharply decrease with increasing change in Av. 

Overtones and combination tones are forbidden in the harmonic oscillator approximation. Mechanical anharmonicity is one factor that might give them intensity through violating the Av = I selection rule. There is another possible cause for this, however, which might operate even when the oscillator is perfectly harmonic. It is electrical anharmonicity. 

Thus electrical anharmonicity is the non-linear part of the variation of the dipole moment with normal coordinates. There is no reason to expect it to be negligibly small. It can give intensity to overtones and combination tones. In the general case both mechanical and electrical anharmonicities contribute to the intensity. As will be mentioned later they cannot be disregarded when spectra of H-bond complexes are dealt with. 

It is good to remember too that we are dealing with combination tones. For example (0, 0)- (l, 1) or (vj + v3) is a binary combination (summation) band. Should (0,0)—+(1,1) be more intense than (0, 0)- (l, 0) that means that the combination is more intense than the fundamental. This is quite possible but it requires a large X13 coupling constant or/and a large amount of electrical anharmonicity. Finally, whether or not (0,0)->(l, 0) is the strongest band among the subbands due to combinations of Vj and v3 with various quantum numbers depends on several... 

Finally there are effects of electrical anharmonicity incorporating higher derivatives of the dipole moment to consider. The intensities of transitions between successive levels of the vibrator are not necessarily given accurately by harmonic or anharmonic matrix elements. For example, a ratio of (IM1212/2 /x0i 2) = (1 + (A/m)) is obtained for a Morse oscillator. 

If the potential is anharmonic, the higher terms of the potential energy according to Eq. 2.1-16 are not negligible, the mechanical anharmonicity gives rise to overtones, vibrations with the double, triple, or multiple frequency of the fundamentals. Overtones are also produced by the higher terms of the dipole moment according to Eq. 2.3-1, the electrical anharmonicity. Since both equations may contain mixed terms, combinations of two or more normal vibrations, i.e., sums or differences, are produced. These appear in the spectra, but usually only with small intensity. Also Raman spectra show overtones and combinations due to mechanical and electrical anharmonicities (see below). 

Raman spectra, as infrared spectra show overtones and combination bands. These are due to mechanical anharmonicities (Eq, 2.1-16) as well as to electrical anharmonicities (higher terms in Eq. 2.4-3). 

HREELS generally does not possess enough sensitivity to detect higher-order effects due to mechanical and/or electrical anharmonicity. 

De Almeida, W. B., Craw, J. S., and Hinchliffe, A., Ab initio vibrational spectra ofHCN -HF and HF-HCN hydrogen-bonded dimers Mechanical and electrical anharmonicities, J. Mol. Struct. (Theochem) 200, 19-31 (1989). 

Quadratic terms in die property expansions are considered to be first-order in electrical anharmonicity, cubic terms are taken to be second-order, etc. Similarly, cubic terms in the vibrational potential are considered to be first-order in mechanical anharmonicity, quartic terms are second-order, and so forth. The notation (n, m) is used hereafter for the order of electrical (n) and mechanical (m) anharmonicity whereas the total order (n -I- m) is denoted by I, II,. Although our definition of orders is reasonable other choices are possible. Two key questions are (1) How important are anharmonicity contributions to vibrational NLO properties and (2) What is the convergence behavior of the double perturbation series in electrical and mechanical anharmonicity Both questions will be addressed later. Here we note that compact expressions, complete through order II in electrical plus mechanical anharmonicity, have been presented [19]. The formulas of order I contain either cubic force constants or second derivatives of the electrical properties with respect to the normal coordinates. Depending upon the level of calculation at least one order of numerical differentiation is ordinarily required to determine these anharmonicity parameters. For electrical properties, the additional normal coordinate derivative may be replaced by an electric field derivative using relations such as d p./dQidQj = —d E/dldAj.ACd, = —dk,/rjF where F is the field and k j is... 

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