Hamiltonian rigid-rotor

The rigid-rotor Hamiltonian for a diatomic molecule with the moment of inertia I = pf o is... 

The first terms of (8.109) and (8.112) together constitute the rigid rotor Hamiltonian, as follows ... 

In this equation Av, Bv, and Cv are the rotational constants and La, Lb, and Lc are the projections of the rotational angular momentum on the principal inertial axes of the molecule. The Hamiltonian in Eq. (1) is often referred to as a rigid-rotor Hamiltonian, even though significant vibrational effects appear in the rotational constants. To good approximation... 

The first high resolution spectroscopic measurements of FAD with fully resolved rotational-vibrational-tunneling transitions were reported by Madeja and Havenith in 2002 [39]. They measured the C-O vibrational band of (DC00H)2 which could be analyzed in terms of an asymmetric top rigid rotor Hamiltonian. The vibrational frequency of the C-O stretch in (DCOOH)2 was determined to be 1244.8461(2) cm i. This deviated considerably from values for the band center as given by Wachs et al. [55] (1231.85 cm i). Previous measurements include the Raman transition as reported by Bertie et al. (1230 2 cm ) [54] and the value obtained by Millikan and Pitzer (1239 cm ) [56]. 

The classical normal-mode/rigid-rotor Hamiltonian is composed of the vibrational energy vib of Eq. (2.4) and the classical rotational energy... 

The rigid rotor Hamiltonian for a molecule in vibrational level v as the form ... 

This sampling, with the normal-mode/rigid-rotor Hamiltonian, provides an exact microcanonical ensemble for this Hamiltonian, but an approximate microcanonical ensemble for the actual anharmonic and reactive Hamiltonian with vibrational-rotational coupling. 

Here eR is the rotational energy of a rigid rotor and r0 is the equilibrium ground state bond length of the diatomic molecule. The total Hamiltonian is thus... 

In fact, the frequency ofthe torsional oscillation mode V4 is found to be more than double that ofthe ground state. The frequency ofthe torsional oscillation mode was reevaluated by Mukheijee et al [56], using a very accurate representation of the one-dimensional vibrational Hamiltonian of the non-rigid rotor in terms of a Fourier series [76-78], and other spectroscopic parameters calculated for the first time taking care of anharmonicity. A new assignment of the experimental spectrum was given. The results are displayed in Table 8. For reference purpose the vibrational frequencies of the ionic states are also listed... 

The Hamiltonian for a non-linear rigid rotor is quite complicated [1] and the derivation of the expressions for the sum and density of states is cumbersome. We know, however, that the partition function is given by Eq. (A.20), and it is quite easy to find the expressions for the sum and density of states that are consistent with Eq. (A.20). 

In the previous section, we have shown that switching the picture from the nearly integrable Hamiltonian to the Hamiltonian with internal structures may make it possible to solve several controversial issues listed in Section IV. In this section we shall examine the validity of an alternative scenario by reconsidering the analyses done in MD simulations of liquid water. As mentioned in Section III, since a water molecule is modeled by a rigid rotor, and has both translational and rotational degrees of freedom. So, the equation of motion involves the Euler equation for the rigid body, coupled with ordinary Hamiltonian equations describing the translational motions. The precise Hamiltonian is therefore different from that of the Hamiltonian in Eq. (1), but they are common in that the systems have internal structures, and the separation of the time scale between subsystems appears if system parameters are appropriately set. 

The 2-3 splitting is of the order of a typical low-J rotational spacing and non-rigid-rotor spectra result. The rovibrational levels were therefore computed from the Hamiltonian of Eqs. (4.9a, b) with the help of a second-order perturbation correction used by Butcher and Costain66 for cyclopentene rather than by direct matrix... 

Huttner and Flygare4 have discussed in detail the rotational average of the Hamiltonian in Eq. (20) in both the nuclear coupled (discussed later) and uncoupled cases. In the absence of nuclear spin, the rotational energy for a molecule is computed by first obtaining the eigenfunctions Ofjlrr J from Eq. (7) at zero field. These rigid-rotor and... 

For a RRKM calculation without any approximations, the complete vibrational/rotational Hamiltonian for the unimolecular system is used to calculate the reactant density and transition state s sum of states. No approximations are made regarding the coupling between vibration and rotation. However, for many molecules the exact nature of the coupling between vibration and rotation is uncertain, particularly at high energies, and a model in which rotation and vibration are assumed separable is widely used to calculate the quantum RRKM k(E, J) [4,16]. To illustrate this model, first consider a linear polyatomic molecule which decomposes via a linear transition state. The rotational energy for the reactant is assumed to be that for a rigid rotor, i.e. 

With the above approach we can combine the use of curvilinear normal coordinates with the Eckart frame. When we do so, the harmonic oscillator, rigid rotor, and, to lowest order, the Coriolis and centrifugal coupling contributions to H have exactly the same form as those found for the more commonly used Watson Hamiltonian (58). 

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