Anharmonic vibrational transition moment

In the present paper we calculate frequencies and intensities of OH-stretching and SOH-bending vibrational transitions as well as their combinations and overtones— the dominant vibrational transitions from 1000 cm to 20000 cm . The vibrational calculation is based on the harmonically coupled anharmonic oscillator (HCAO) local mode model [42-44] combined with ab initio calculated dipole moment functions [50]. This local mode method has been successful in the calculation of OH- and CH-stretching overtone spectra [19,51,52], The local mode parameters, frequency and anharmonicity, are obtained either from the observed experimental transitions or calculated ab initio [53-56]. 

The study of the rotation-vibration spectra of polyatomic molecules in the gas phase can provide extensive information about the molecular structure, the force field and vibration-rotation interaction parameters. Such IR-spectra are sources of rotational information, in particular for molecules with no permanent dipole moment, since for these cases a pure rotational spectrum does not exist. Vibrational frequencies from gas phase spectra are desirable, because the molecular force field is not affected by intermolecular interactions. Besides, valuable support for the assignment of vibrational transitions can be obtained from the rotational fine structure of the vibrational bands. Even spectra recorded with medium resolution can contain a wealth of information hot bands , for instance, provide insight into the anharmonicity of vibrational potentials. Spectral contributions of isotopic molecules, certainly dependent on their abundance, may also be resolved. 

The first of these two terms is zero, since the wavefunctions i . and j are defined as orthogonal hence vibrations are only infrared active when Q 7 0. If the harmonic model is assumed, the transition moment is only nonzero for transitions where An = 1 although this restriction is lifted for the anharmonic oscillator, transitions where An = 2, 3, etc., are still much weaker than An = 1. 

Another effect of the anharmonic terms is to change the transition probabilities of vibrational transitions. If the electric moment were a linear function of the displacements from equilibrium and if the vibrational wave functions were accurately given by harmonic oscillator functions, no overtones or combinations should appear in infrared spectra. The fact that such bands do occur shows that one or the other of these conditio)is is not met in fact, it is probable that neither condition is lived up to in actual molecules. It is evident from the convergerice of overtone levels that the harmonic oscillator approximation is not exact, while considerations of intensities indicate that in addition the electric moment is not a strictly linear function of the displacements. For a further discussion of the effect of these factors on the intensities, the reader may refer to the work of Crawford and collaborators. ... 

From a perturbative standpoint, anharmonicity in the stretching potential mixes harmonic wavefunctions. If the anharmonicity were to produce a mixing of the zero-order n = l and n = 2 states, the nonzero transition integral between the harmonic n = 0 and n = l states would be distributed among the perturbed first and second excited states. This would make both the n = 0 to n = 1 and n = 0 to n = 2 transitions allowed in the anhar-monically perturbed states. Anharmonicity in the potential and nonlinearity in the dipole moment function make it possible to observe vibrational transitions other than An = 1. However, such other transitions are usually weaker or less intense because the harmonic nature of the potential and the linearity of the dipole moment function have the major role. 

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. 

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. 

Avk=(vk+jic)= Avj=(vj+jj)=0, j k. A photon-induced transition via an external field occurs only for a given fundamental frequency, for w = between adjacent vibrational states, Vk and (vk l). Non-vanishing terms, to higher order in Eq. (28), mechanical anharmonicity, and non-linear terms in the dipole-moment function, Eq. (29), lead to breakdown in these strict selection rules, and frequencies other than the set, a n l>u2> u... . . to, become observable in the dipole-allowed spectrum. 

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. 

In the visible region of the spectrum water vapour is transparent and all further absorptions of interest occur in the infrared or at even longer wavelengths. These are associated with transitions between vibrational levels of the molecule, the fundamental modes for which are shown in fig. 1.4, and have a fine structure dependent upon the rotational levels involved. Since each of the three normal modes has a direct effect upon the dipole moment of the molecule, they aU lead to absorption bands. Because the interatomic potentials have appreciable anharmonic components from terms of cubic or higher order in the displacements, the relation between... 

Combination and difference bands Besides overtones, anharmonicity also leads to the appearance of combination bands and difference bands in the IR spectrum of a polyatomic molecule. In the harmonic case, only one vibration may be excited at a time (the transition dipole moment integral vanishes when the excited state is given by a product of more than one Hermite polynomial corresponding to different excited vibrations). This restriction is relaxed in the anharmonie case and one photon can simultaneously excite two different fundamentals. A weak band appears at a frequency approximately equal to the sum of the fundamentals involved. (Only approximately because the final state is a new one resulting from the anharmonie perturbation to the potential energy mixing the two excited state vibrational wave functions.)... 

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