Anharmonicity energy levels

Figure 1.4 The very anharmonic energy levels of the C6H6 - Ar stretch motion (cf. Figure 0.3) (adapted from Neusser, Sussman, Smith, Riedle, and Weber, 1992). The computed values of the rotational constant Bv [the coefficient of 1(1+1) in the expression for the energy eigenvalues] are given in the figure, as are the vibrational spacings.

To calculate n E-E, the non-torsional transitional modes have been treated as vibrations as well as rotations [26]. The former approach is invalid when the transitional mode s barrier for rotation is low, while the latter is inappropriate when the transitional mode is a vibration. Harmonic frequencies for the transitional modes may be obtained from a semi-empirical model [23] or by performing an appropriate normal mode analysis as a function of the reaction path for the reaction s potential energy surface [26]. Semiclassical quantization may be used to determine anharmonic energy levels for the transitional modes [27]. 

We need anharmonic energy level calculations with all six vibrational degrees of fteedom. Although we have a qualitatively correct semiclassical picture of the tunneling splitting and its dependence on monomer stretch and rotational excitations, the details are far fiom settled. We need calculations of the effect of 06 excitation on the tunneling splitting. 

Anharmonic energy levels and partition functions for surface 5SP... 

These harmonic-oscillator solutions predict evenly spaced energy levels (i.e., no anharmonicity) that persist for all v. It is, of course, known that molecular vibrations display anharmonicity (i.e., the energy levels move closer together as one moves to higher v) and that quantized vibrational motion ceases once the bond dissociation energy is reached. 

The reason that does not change with isotopic substitution is that it refers to the bond length at the minimum of the potential energy curve (see Figure 1.13), and this curve, whether it refers to the harmonic oscillator approximation (Section 1.3.6) or an anharmonic oscillator (to be discussed in Section 6.1.3.2), does not change with isotopic substitution. Flowever, the vibrational energy levels within the potential energy curve, and therefore tq, are affected by isotopic substitution this is illustrated by the mass-dependence of the vibration frequency demonstrated by Equation (1.68). 

Figure 6.4 Potential energy curve and energy levels for a diatomic molecule behaving as an anharmonic oscillator compared with those for a harmonic oscillator (dashed curve)...

In a diatomic molecule one of the main effects of mechanical anharmonicity, the only type that concerns us in detail, is to cause the vibrational energy levels to close up smoothly with increasing v, as shown in Figure 6.4. The separation of the levels becomes zero at the limit of dissociation. 

The concept of discrete or quantized energy levels can be superimposed on this diagram by representing them as a series of horizontal lines, the spacing of which becomes closer with increasing energy due to the anharmonic nature of the vibration. In quantum mechanical terms, these levels are labelled V =... 

There are two effects of the anharmonicity of the quantized energy levels described above, which have signiflcance for NIRS. First, the gap between adjacent energy levels is no longer constant, as it was in the simple harmonic case. The energy levels converge as n increases. Second, the rigorous selection rule that An = +1 is relaxed, so that weak absorptions can occur with n = 2 (flrst overtone band), or +3 (second overtone band), etc. 

Figure 3.2 shows the experimental potential well for the H2 molecule, compared with the harmonic fit and the Morse potential. Horizontal lines represent quantized energy levels. Note that, as vibrational quantum number n increases, the energy gap between neighboring levels diminishes and the equilibrium distance increases, due to the anharmonicity of the potential well. The latter fact is responsible for the thermal expansion of the substance. 

In experimental data such evenly spaced energy level patterns are seldom seen most commonly, one finds spacings En+i - En that decrease as the quantum number n increases. In such cases, one says that the progression of vibrational levels displays anharmonicity. 

Two of the most severe limitations of the harmonic oscillator model, the lack of anharmonicity (i.e., non-uniform energy level spacings) and lack of bond dissociation, result from the quadratic nature of its potential. By introducing model potentials that allow for proper bond dissociation (i.e., that do not increase without bound as x=>°°), the major shortcomings of the harmonic oscillator picture can be overcome. The so-called Morse potential (see the figure below)... 

The first term is the harmonic expression. The next is termed the first anharmonicity it (usually) produces a negative contribution to E(vj) that varies as (vj + 1/2)2. The spacings between successive Vj —> Vj + 1 energy levels is then given by ... 

To obtain the allowed energy levels, Ev, for a real diatomic molecule, known as an anharmonic oscillator, one substitutes the potential energy function describing the curve in Fig. 3.2c into the Schrodinger equation the allowed energy levels are... 

When exposed to electromagnetic radiation of the appropriate energy, typically in the infrared, a molecule can interact with the radiation and absorb it, exciting the molecule into the next higher vibrational energy level. For the ideal harmonic oscillator, the selection rules are Av = +1 that is, the vibrational energy can only change by one quantum at a time. However, for anharmonic oscillators, weaker overtone transitions due to Av = +2, + 3, etc. may also be observed because of their nonideal behavior. For polyatomic molecules with more than one fundamental vibration, e.g., as seen in Fig. 3.1a for the water molecule, both overtones and... 

In the LM model, molecular vibrations are treated as motions of individual anharmonic bonds [38] (usually Morse oscillators). They therefore include anharmonicity, but not coupling between bonds, thus requiring inclusion of interbond coupling for obtaining a better description. For the case of t identical Morse oscillators, the energy levels related to the LM Hamiltonian are given by... 

Energy levels in the anharmonic oscillator are not equal, although they become slightly closer as energy increases. This phenomenon can be seen in the following equation , ... 

Within the harmonic approximation the choice of a system of internal coordinates is irrelevant provided they are independent and that a complete potential function is considered ). For example, the vibrations of HjO can be analysed in terms of valence coordinates (r, >2, or interatomic coordinates (r, r, 3) and any difference in the accuracy to which observed energy levels are fitted (considering all the isotopic species H2O, HDO and D2O) will be due to the neglect of anharmonic terms. If one makes the approximation of a diagonal force field so that one is comparing the two potentials... 

A strong anharmonie interaction between the vibrations approximately described as rXH and vX.ll Y. There is independent evidence for a parametric relationship between the X Y and X—H interim clear distances from diffraction studies. The resulting effect on the vibrational spectrum increases with the anharmonicity and amplitude of both types of vibration, and seems to be most completely described by a type of energy level scheme proposed by Stepanov. A slight extension of this theory proposed here enables it to explain the persistence of broad vX l absorption regions at low temperatures. 

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