Moment of inertia tensor

For molecules that are non-linear and whose rotational wavefunctions are given in terms of the spherical or symmetric top functions D l,m,K, the dipole moment Pave can have components along any or all three of the molecule s internal coordinates (e.g., the three molecule-fixed coordinates that describe the orientation of the principal axes of the moment of inertia tensor). For a spherical top molecule, Pavel vanishes, so El transitions do not occur. 

Azurmendi, H. F., Bush, C. A. Tracking alignment from the moment of inertia tensor (TRAMITE) of biomolecules in neutral dilute liquid crystal solutions. J. Am. Chem. Soc. 2002, 124, 2426-2427. 

A better estimate of the shape of the polymer molecules, since they are highly anisotropic, is a representation of each molecule in terms of an equivalent spheroid with the same moment of inertia [45,46]. This is achieved by diagonalizing the moment of inertia tensor to obtain the eigenvectors a, b, and c and the principal moments 7, I/,/, and Icc. The moment of inertia tensor of molecule j is given by... 

Here I denotes the moment of inertia tensor defined with nuclear masses i cM is the position vector of the center of nuclear mass. The electrons, with position vectors have to he treated quantum mechanically which implies that their contribution is obtained as expectation value of the corresponding electronic operator over the ground state wavefunction... 

In practical applications of the expressions for the rotational g tensor, equations (2), (4), (5), or (6), the nuclear masses in the nmment of inertia tensor I are generally approximated with atomic masses and is approximated with the center of atomic masses. This introduces a correction term to the moment of inertia tensor which is actually closely related to the nuclear contribution to the rotational g tensor [3,11,38]. Going to second order of perturbation theory for the electronic contributions one would obtain a further correction term to the moment of inertia tensor which is similar to the electronic contribution to the rotational g tensor [3,4]. 

All together one would obtain an effeetive moment of inertia tensor which includes the rotational g tensor again. This correction is normally ignored for polyatomic molecules, but allows to estimate the rotational g factor of diatomic molecules from field-free rotation-vibration spectra [5,10,11]. 

For less symmetric molecules one has to resort to computer programs [164] to solve the Woessner equations. The orientation of the rotational diffusion tensor is usually defined by assuming that its principal axes coincide with those of the moment of inertia tensor. This assumption is probably a good approximation for molecules of low polarity containing no heavy atoms, since under these conditions the moment of inertia tensor roughly represents the shape of the molecule. 

This study is the first where semiquantitative use of relaxation data was made for conformational questions. A similar computer program was written and applied to the Tl data of several small peptides and cyclic amino acids (Somorjai and Deslauriers, 1976). The results, however, are questionable since in all these calculations it is generally assumed that the principal axis of the rotation diffusion tensor coincides with the principal axis of the moment of inertia tensor. Only very restricted types of molecules can be expected to obey this assumption. There should be no large dipole moments nor large or polar substituents present. Furthermore, the molecule should have a rather rigid backbone, and only relaxation times of backbone carbon atoms can be used in this type of calculation. 

The calculated Euler angles (a = 50°, /3 = 60°, and y = 40°), which determine the relative orientation between the principal-axis system of the rotational diffusion tensor and that of the moment of inertia tensor, indicate a significant shift between the two tensors. This result is expected because of the fact that molecule 31 contains a number of polar groups and hydrogen-bonding centers, leading to strong intermolecular interactions. 

Somewhat similar conclusions apply to the rotational magnetic moment g tensor for a diatomic molecule. The component of the moment of inertia tensor along the intemuclear axis is zero, and the two perpendicular components are, of course, equal. Consequently the rotational magnetic moment Zeeman interaction can be represented by the simple term... 

All the correlation functions above are normalized, therefore equations (4 and 5) are identical to correlation functions over linear momentum p = mv and angular momentum J — lu, respectively. Note that, in this context I is the moment of inertia tensor The correlation function in equation (6) is calculated over the spherical harmonics. If m = 0, this reduces to time correlation function over Legendre polynomials ... 

The gauge-invariant metric tensor of internal space and classical equations of internal motion in terms of the PAHC are given explicitly as follows, from which the general properties of a metric force associated with this coordinate system will be deduced. By applying Eq. (20) to Eq. (6), we obtain for the moment of inertia tensor referred to the body frame as... 

We define the order of the singular values as a > a2 > 31. The planar and collinear configurations give a3 0 and a2 a3 = 0, respectively. Furthermore, we let the sign of a3 specify the permutational isomers of the cluster [14]. That is, if (det Ws) = psl (ps2 x ps3) > 0, which is the case for isomer (A) in Fig. 12, fl3 >0. Otherwise, a3 < 0. Eigenvectors ea(a = 1,2,3) coincide with the principal axes of instantaneous moment of inertia tensor of the four-body system. We thereby refer to the principal-axis frame as a body frame. On the other hand, the triplet of axes (u1,u2,u3) or an SO(3) matrix U constitutes an internal frame. Rotation of the internal frame in a three-dimensional space, which is the democratic rotation in the four-body system, is parameterized by three... 

Next, we construct the Lagrangian for the four-body system of vanishing total angular momentum. The moment of inertia tensor M in Eq. (6) is diagonal as... 

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