Terms vibrational constants

Here, the last term is constant for all materials with the same number of Cu atoms within the approximation that entropic contributions from atomic vibrations are equivalent in each material. 

The main difficulty in solving (2.153) lies in the evaluation of the potential energy term (2.154). Even in the case ofH2, calculation of V from the electronic waveftmctions for different values of R is no easy matter. Usually, therefore, the vibrational wave equation is solved by inserting a restricted form of the potential experimental data on the rovibrational levels are then expressed in terms of constants introduced semiempirically, as we shall show. 

In order to calculate Go, one needs to know the potential parameters k, a, and b. These should be obtainable from spectroscopically observed quantities. Thus, k can be obtained from observed Teller has shown that the rotational-vibrational constant ae (in cm. ) can be expressed in terms of the cubic potential constant a and other parameters in the following manner [see Reference 6]. 

We now introduce the term values of the vibrational levels in terms of the vibrational constants co I and oo via... 

SCs correspond to SC II and SC I, respectively. In early experimental work of Linton et al. (1983a) the configurational assignment of the 0=0 ground state to the SC II was only tentative and based primarily on the similarity of the vibrational constant to that of EuO. Only two electronic states at term energies of 910 cm and 1010 cm were observed in low-resolution work. It was not possible to assign Q values for them (possible values were... 

Values for the vibrational constants are based on varying numbers of anharmonic terms. 

TABLE 9.1 Ground state molecular rotational and vibrational constants for selected diatomic molecules. Values for the vibrational constants are based on varying numbers of anharmonic terms. In this table Dq is the v = 0 rotational distortion constant. Missing values indicate the constant was not measured in the corresponding experiment... 

To define the rotation-vibration constants in terms of more fundamental parameters or to understand the origin of various nonrigidity effects in the rotational spectrum, the general rotation-vibration Hamiltonian must be employed. This Hamiltonian contains pure rotation and vibration terms as well as interaction terms between rotation and vibration. Perturbation treatments to various orders are required to characterize the different rotation-vibration effects. Space does not permit further discussion of this however, we mention that such a perturbation treatment shows that the a constants depend on the cubic potential energy constants of the molecule. 

Figure 6-8. The contribution of the first anharmonicity, centrifugal distortion, and rotation-vibration coupling for H Cl vibration/rotation energy levels relative to the energy values computed from the rigid rotor harmonic oscillator approximation (RRHO). The numbers in parenthesis correspond to the contribution of each correction term. The constants were obtained from Table 6-2.

Since depends on nuclear coordinates, because of the term, so do and but, in the Bom-Oppenheimer approximation proposed in 1927, it is assumed that vibrating nuclei move so slowly compared with electrons that J/ and involve the nuclear coordinates as parameters only. The result for a diatomic molecule is that a curve (such as that in Figure 1.13, p. 24) of potential energy against intemuclear distance r (or the displacement from equilibrium) can be drawn for a particular electronic state in which and are constant. 

The rotational constants B and D are both slightly vibrationally dependent so that the term values of Equation (5.19) should be written... 

The transition is fully classical and it proceeds over the barrier which is lower than the static one, Vo = ntoColQl- Below but above the second cross-over temperature T 2 = hcoi/2k, the tunneling transition along Q is modulated by the classical low-frequency q vibration. The apparent activation energy is smaller than V. The rate constant levels off to its low-temperature limit k only at 7 < Tc2, when tunneling starts out from the ground state of the initial parabolic term. The effective barrier in this case is neither V nor Vo,... 

P Q-) =p Q-,Q-,p), which in the harmonic approximation is described by (3.16), PhiQ-iQ-,P) exp(— CO Q1 tanh co ). Having reached the point Q, the particle is assumed to suddenly tunnel along the fast coordinate Q+ with probability A id(Q-), which is described in terms of the usual one-dimensional instanton. The rate constant comes from averaging the onedimensional tunneling rate over positions of the slow vibration mode,... 

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