Anharmonicity of the potential well

As with the Schrodinger equation itself, the most useful estimate of the form of the potential well in real molecules was an inspired hypothesis, rather than a function derived rigorously from first principles. The Morse Curve is a very realistic approximation to the potential energy well in real molecular bonds. 

The expression for potential energy within the well becomes  

When the function V(r) is used in the vibrational Hamiltonian instead of the simple harmonic V(x), the quantised vibrational energy levels are 

There are two effects of the anharmonicity of the quantised energy levels described above, which have significance 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. Secondly, the rigorous selection rule that An = 1 is relaxed, so that weak absorptions can occur with An = 2 (first overtone band), or 3 (second overtone band), etc. 

For the transition from n = 0 to n = 1 (the fundamental vibration) one sees the absorption at 

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While the idea of LF explained the H2 data quite well [28], we were surprised by the magnitude of the oscillations in our I2 data [16], as, unlike H2,12 is not vibrationally cold at room temperature - the conditions for our experiment. Generally, thermal motion is detrimental to observing coherent motion. Thus, we took a long time scale run to get a more accurate measurement of the frequency of the vibrations, shown in Fig. 1.5. These data also exhibit a vibrational revival, from which the anharmonicity of the potential well can be determined. Indeed, the vibrational frequency accurately matched that of the ground state. 

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. 

To obtain the anharmonic terms in the potential, on the other hand, the choice of coordinates is important 130,131). The reason is that the anharmonic terms can only be obtained from a perturbation expansion on the harmonic results, and the convergence of this expansion differs considerably from one set of coordinates to another. In addition it is usually necessary to assume that some of the anharmonic interaction terms are zero and this is true only for certain classes of internal coordinates. For example, one can define an angle bend in HjO either by a rectilinear displacement of the hydrogen atoms or by a curvilinear displacement. At the harmonic level there is no difference between the two, but one can see that a rectilinear displacement introduces some stretching of the OH bonds whereas the curvilinear displacement does not. The curvilinear coordinate follows more closely the bottom of the potential well (Fig. 12) than the linear displacement and this manifests itself in rather small cubic stretch-bend interaction constants whereas these constants are larger for rectilinear coordinates. A final and important point about the choice of curvilinear coordinates is that they are geometrically defined (i.e. independent of nuclear masses) so that the resulting force constants do not depend on isotopic species. At the anharmonic level this is not true for rectilinear coordinates as it has been shown that the imposition of the Eckart conditions, that the internal coordinates shall introduce no overall translation or rotation of the body, forces them to have a small isotopic dependence 132). 

The anharmonicities of the potential contribute by the terms involving the constants x, g, y,. .. as well as the energy shifts AEx = 0(h2),. .. and the frequency shifts Aw, = 0(h2),. These anharmonic constants can be calculated by the Van Vleck contact transformations [20] as well as by a semi-classical method based on an h expansion around the equilibrium point [14], which confirms that the Dunham expansion (2.8) is a series in powers of h. Systematic methods have been developed to carry out the Van Vleck contact transformations, as in the algebraic quantization technique by Ezra and Fried [21]. It should be noted that the constants x and g can also be obtained from the classical-mechanical Birkhoff normal forms [22], The energy shifts AEx,... 

This form of the effective potential incorporates some degree of quantum effects as well as the anharmonicity of the potential. In the case of the free-particle reference system, the centroid-constrained propagator is... 

Clemenger [42] has studied the effect of ellipsoidal deformations in alkali clusters with N less than 100, using a modified three-dimensional harmonic oscillator model. The model considers different oscillator frequencies along the z axis (taken as symmetry axis) and perpendicular to the z axis. The model Hamiltonian used by Clemenger also contains an anharmonic term. Its purpose is to flatten the bottom of the potential well and to make it to resemble a rounded square-well potential. 

A comparison of the Morse potential (blue) and the harmonic oscillator potential (green), showing the effects of anharmonicity of the potential energy curve, where is the depth of the well. [ Mark M Sa moza/CCC-BY-SA 3.0/G FDL /Wikimedia Commons reproduced from http //en.wikipedia.org/wiki /Morse potential (accessed December 27, 2013).]... 

The sensitivity as given by Equation 16.60 for the variation of a is in addition reduced by the anharmonicity of the potential. For the highest vibrational levels of the electronic ground state as well as for all levels of the upper (weakly bound) electronic state, the separation between the nuclei is large, 7 > 12 au (see Figure 16.2). Thus, both electronic wavefunctions are close to either symmetric (for E+) or antisymmetric combination (for E+) of the atomic 6 functions,... 

The Morse oscillator allows the first anharmonic correction to be related directly to the curvature and depth of the potential well ... 

Whereas the quasi-chemical theory has been eminently successful in describing the broad outlines, and even some of the details, of the order-disorder phenomenon in metallic solid solutions, several of its assumptions have been shown to be invalid. The manner of its failure, as well as the failure of the average-potential model to describe metallic solutions, indicates that metal atom interactions change radically in going from the pure state to the solution state. It is clear that little further progress may be expected in the formulation of statistical models for metallic solutions until the electronic interactions between solute and solvent species are better understood. In the area of solvent-solute interactions, the elastic model is unfruitful. Better understanding also is needed of the vibrational characteristics of metallic solutions, with respect to the changes in harmonic force constants and those in the anharmonicity of the vibrations. 

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