Molecular moment of inertia

Molecular moments of inertia are about 10 g/cm thus 7 values for benzene, N2, and NH3 are 18, 1.4, and 0.28, respectively, in those units. For the case of benzene gas, a = 6 and n = 3, and 5rot is about 21 cal K mol at 25°C. On adsorption, all of this entropy would be lost if the benzene were unable to rotate, and part of it if, say, rotation about only one axis were possible (as might be the situation if the benzene was subject only to the constraint of lying flat... 

The rotational states are characterized by a quantum number J = 0, 1, 2,. .. are degenerate with degeneracy (2J + 1) and have energy t r = ) where 1 is the molecular moment of inertia. Thus... 

Each atom of a molecule that rotates about an axis through its centre of mass, describes a circular orbit. The total rotational energy must therefore be a function of the molecular moment of inertia about the rotation axis and the angular momentum. The energy calculation for a complex molecule is of the same type as the calculation for a single particle moving at constant (zero) potential on a ring. 

Rotational relaxation must depend to some extent on the molecular moment of inertia. 

The first-order part describes the nuclear and the second-order the electronic contribution to the molecular moment of inertia. The A-doubling terms, on the other hand, have no counterpart in the zeroth-order Hamiltonian. For a 2 n state, the operator form is... 

The microwave experiment studies rotational structure at a given vibrational level. The spectra are analyzed in terms of rotational models of various symmetries. The vibration of a diatomic molecule is, for instance, approximated by a Morse potential and the rotational frequencies are related to a molecular moment of inertia. For a rigid classical diatomic molecule the moment of inertia I = nr2 and the equilibrium bond length may be calculated from the known reduced mass and the measured moment, assuming zero centrifugal distortion. 

If adequate spectral resolution is available, infrared spectra of molecules in the gas phase can be highly structured. The varying molecular complexity and symmetry makes it necessary to employ different models for spectrum interpretation. These models may be characterized by considering the molecular moments of inertia. Generally, the moment of inertia la of a rigid body with n point masses rotating about the axis ri is defined as... 

I, y, k denotes a circular permutation / Ij, are the molecular moments of inertia and F is an external potential depending on molecular orientation and /(I) white stochastic forces defined by... 

Coordinates relative to center-of-mass, principal molecular moment-of-inertia frame. 

Here A(u,) is the Laplacian and Ja,Jb, and Jc are the components of the angular momentum operator with respect to the principal a, b, and c axes of the molecule / , h, and Ic are the principal values of the molecular moment of inertia. For simplicity we have assumed that the crystal is composed of just one type of molecules. Otherwise, the molecular mass M and the moments of inertia have to carry the sublattice label i, appearing in P = n, /. ... 

While the tg structure represents the most well-defined molecular geometry, it is not, unfortunately, one that exists in nature. Real molecules exist in the quantum states of the 3N-6 (or 5) vibrational states with quantum numbers (vj, V2.-..V3N-6 (or 5)). Vj = 0, 1, 2,. Even in the lowest (ground) (0,0...0) vibrational state, the N atoms of the molecule undergo their zero point vibrational motions, oscillating about the equilibrium positions defined by the B-O potential energy surface. It is necessary then to speak of some type of average or effective structures, and to account for the vibrational motions, which vary with vibrational state and isotopic composition. In spectroscopy, a molecule s structural information is carried most straightforwardly by its molecular moments of inertia (or their inverses, the rotational constants), which are determined hy analysis of the pure rotational spectrum or fire resolved rotational structure of vibration-rotation bonds. Thus, the spectroscopic determination of molecular structure boils down to how one uses the rotational constants of a molecule... 

In all cases, by virtue of Eqs. (2), it is clear that file molecular moments of inertia of a particular set of atomic isotopes is entirely determined by the atomic coordinates, i.e., the structure. It is also obvious that the moments are isotopomer-dependent, since they depend upon atomic mass. 

I80. In practice, effects are much smaller than this, largely because bonds are not completely broken at the transition state. With these small effects, the effect of change of mass on molecular moments of inertia cannot be neglected. Such ponderal kinetic isotope effects have a lower dependence on temperature than zero point effects, for which rin( L/ H) = constant (equation 1.14) holds generally, T being the absolute temperature. 

Because of their relation to the molecular moments of inertia, spectroscopic rotational constants have been one of the two most important sources of information for the determination of bond distances and bond angles in free molecules. The purpose of this paper is to describe several computational methods by which bond distances and bond angles may be derived from rotational constants. The sources and magnitudes of the uncertainties in molecular parameters that result from the methods will also be discussed. 

In Eq. (I.l) Ja, Jt>, and Je stand for the a, b, and c-components of the overall angular momentum measured in units of h. They are referred to the principal axis system of the molecular moment of inertia tensor. Because of the mutual compensation of positive and negative contributions, the absolute values of the g-tensor elements are usually smaller than 1 (typically on the order of 0.01 to 0.1 as may be checked in Table AI of the Appendix). In many cases the off-diagonal elements of the g-tensor in Eq. (I.l) will be zero because of molecular symmetry. Formaldehyde or 1,2-difluorobenzene may serve as examples. 

The energies of molecular rotational levels, and hence energy differences, depend on the masses that are rotating the heavy nuclei and their geometrical arrangement in space, i.e. the molecular moments of inertia Ix, ly, and (or Ig, lb, and Ic, depending on the coordinate system). For a diatomic molecule of nuclear mass mi and m2, and internuclear distance r, the moment of inertia I is ... 

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