Anharmonicity vibrational wavefunctions

They obtained relative frequency shift between the C-F stretching fundamentals of the S- and i7-enantiomer of about - -3 x 10 in the harmonic and —8 x 10 in the anharmonic approximation. The authors also computed the relative shifts in rotational constants, which were of the same order of magnitude. Laerdahl, Schwerdtfeger and Quiney [65] obtained within the DHF approach -f-5 x 10 in the harmonic approximation and —6 x 10 for an anharmonic vibrational wavefunction and Viglione, Zanasi, Lazzeretti and Libague [138] estimated within their RPA scheme a harmonic relative frequency shift of +4 x 10 or -1-5 x 10 depending... 

Notice that as one moves to higher vf values, the energy spacing between the states (Evf -Evf-i) decreases this, of course, reflects the anharmonicity in the excited state vibrational potential. For the above example, the transition to the vf = 2 state has the largest Franck-Condon factor. This means that the overlap of the initial state s vibrational wavefunction Xvi is largest for the final state s %vf function with vf = 2. 

Henry s group is also involved in theoretical studies to determine sources of local mode overtone intensity. These investigators have developed a very successful approach that uses their harmonically coupled anharmonic oscillator local mode model to obtain the vibrational wavefunctions, and ab initio calculations to obtain the dipole moment functions. The researchers have applied these calculations to relatively large molecules with different types of X-H oscillator. Recently they have compared intensities from their simple model to intensities from sophisticated variational calculations for the small molecules H20 and H2CO. For example, for H2CO they generated a dipole moment function in terms of all six vibrational degrees of freedom.244 This comparison has allowed them to determine the quality of basis set needed to calculate dipole moment... 

Figure 8. Vibrational wavefunctions for the harmonic (a) and the anharmonic wells (b). In the harmonic case the wavefunction is indicated by a dashed line and its square by a solid line. In (b) only the square of the wavefunction is depicted. The energy (Y-axis) and the distance (X-axis) are in arbitrary units. The amplitude of the wavefunctions superposed on the potential energy surface is also arbitrary...

Pij(r) for a given pair of atoms could in principle be calculated from a knowledge of the vibrational wavefunctions of the molecule and the population of each excited vibrational state, i.e. the Boltzmann factors for the excited vibrational states. The vibrational wavefunctions should (again in principle) be derived from a potential energy expression which includes anharmonic terms. 

Fig. 6.5 Wavefunctirais and transition dipole magnitudes for an anharmonic vibrational mode. (A) Relative amplitudes of wavefunctions 0-3 of an oscillator with the Morse potential illustrated in Fig. 2.1 (curves 0,7,2 and i, respectively). Wavefimction 13 is shown in (B), and 14 in (C). The abscissa is the relative departure of the vibrational coordinate (r) fixnn its equilibrium value (ro). The curves are normalized to the same integrated probabilities (squares of the wavefimction amplitudes) in the range 0 < (r — r )/r < 11.5, and are scaled relative to the peak of wavefimction 0. This normalization considers only part of wavefimctitm 14, which is at the dissociation energy and continues indefinitely off scale to the right. (D) The relative magnitudes of the transition dipoles ((Xm Xo)) for excitati

The vibrational wavefunctions as a function of a normal coordinate Q are illustrated in Fig. 3.13 (see also Fig. 1.12). These wavefunctions have symmetry properties related to those of the normal coordinate. For example, if a symmetry operation leaves the normal coordinate unchanged, the wave-function will remain unchanged (regardless of anharmonicity). If a nondegenerate antisymmetrical normal coordinate Q is reversed in sign by a symmetry operation Q — Q) then, as seen in Fig. 3.13, the wavefunction is left unchanged if the quantum number is even (0o 06> 2 2) or is... 

Since molecular vibrations in general are slightly anharmonic, both the infrared and Raman spectrum may contain weak overtone and combination bands. A combination energy level is one which involves two or more normal coordinates with different frequencies that have vibrational quantum numbers greater than zero. For example, a combination band which appears at the sum of the wavenumbers of two different fundamentals involves a transition from the ground vibrational level (belonging to the totally symmetric species) to an excited combination level where two different normal coordinates each have a quantum number of one and all the others have a quantum number zero. To obtain the spectral activity of the combination band transition it is necessary to determine the symmetry species of the excited wave-function. In quantum mechanics the total vibrational wavefunction is equal... 

Here in the strong coupling region, it is apparent that the magnitude of the displacement in the CT state can significantly affect the observed value of S. We note that in the adiabatic limit, the vibrational wavefunctions are anharmonic. However, for small vibrational quantum numbers, the wavefunctions can be approximated by linearly displaced oscillators with renormalized temperature dependent frequencies and displacements. A large S in the CT state can now appear to substantially increase the observed S value in the P band. 

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. 

In our discussion the usual Born-Oppenheimer (BO) approximation will be employed. This means that we assume a standard partition of the effective Hamiltonian into an electronic and a nuclear part, as well as the factorization of the solute wavefunction into an electronic and a nuclear component. As will be clear soon, the corresponding electronic problem is the main source of specificities of QM continuum models, due to the nonlinearity of the effective electronic Hamiltonian of the solute. The QM nuclear problem, whose solution gives information on solvent effects on the nuclear structure (geometry) and properties, has less specific aspects, with respect the case of the isolated molecules. In fact, once the proper potential energy surfaces are obtained from the solution of the electronic problem, such a problem can be solved using the standard methods and approximations (mechanical harmonicity, and anharmonicity of various order) used for isolated molecules. The QM nuclear problem is mainly connected with the vibrational properties of the nuclei and the corresponding spectroscopic observables, and it will be considered in more detail in the contributions in the book dedicated to the vibrational spectroscopies (IR/Raman). This contribution will be focused on the QM electronic problem. 

When anharmonic terms of the vibrational potential are introduced in the calculation, the probability of reaching each level m directly upon N-H stretch decay (points in Fig. 7) becomes non-negligible. Above 300 meV the molecule can translate classically into other sites. The classical threshold is attained at m = 30 state of the anharmonic frustrated translation mode. The change in wavefunction above the threshold leads to an extra kink in the decay rate function. The probability of populating states above the 300 meV diffusion barrier is in the order of 10 5, compatible with yield values found in experiments [43]. 

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. 

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