Nuclear inertia tensor

Starting from the Coulomb Hamiltonian, to arrive at a Hamiltonian which had an angular part of this form, a neutral and natural choice would seem to be to choose the Eulerian angles to define an orthogonal matrix C that diagonalises the nuclear inertia tensor. This yields moments of inertia and puts the Hamiltonian in principal axis form, just the form appropriate to describe a rigid rotator in classical mechanics. The principal axis approach to the molecule was attempted in classical mechanics by Eckart [Eckart, 1934] shortly before the approach [Eckart, 1935] that we have referred to in 2. It was tried too, almost simultaneously, by Hirschfelder... 

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]. 

We will prove this explicitly as an example for how S3mimetries of the nuclear frame may be used to show that certain off-diagonal elements of the moment of inertia tensor, the jgf-tensor, and -tensor must be zero. 

For ethylene oxide this Hamiltonian is considerably simplified due to the C2t -symmetry of the nuclear frame. Using arguments similar to those which have been used to show that there are no nonzero off-diagonal elements in the molec-tdar moment of inertia tensor, it may be shown that the - and /-tensors must be... 

Because of the assumption of a rigid nuclear frame, it is most convenient to identify the molecular system with the principal axis system of the nuclear moment of inertia tensor (compare Fig. IV. 1). 

Fig. IV. 1. Coordinate systems used in the derivation of the classical Hamiltonian. The difference between the nuclear center of mass (n.c.m.) and the molecular center of mass (m.c.m.) which is typically on the order of 10 to 10- A is vastly exagerated for illustration. C r, By, (bz) are the basis vectors of the space fixed coordinate system. Ba, et, (Bc) are the rotating basis vectors of the principle moment of inertia tensor of the rigid nuclear frame...

H2D is the coplanar Hamiltonian and Ttumble represents the instantaneous tumbling of the body fixed nuclear plane. The components of the inertia tensor are given explicitly in Ref. 14. The x axis is oriented perpendicular to the nuclear plane. 

For linear molecules and symmetric top molecules with a nuclear quadrupole located on the symmetiy axis, the principal axes of the field gradient tensor and the inertia tensor coincide. Since two components of the field gradient tensor are equal because of symmetry, only one independent component of the quadrupole coupling tensor remains to be determined in an analysis of the quadrupole hfs ... 

In the previous section we have seen that the rotational g tensor is related to the paramagnetic contribution to the magnetizability, Eq. (6.29). Here, we will explore a relation between the rotational g tensor and the electric dipole moment. We will see that the latter is related to the difference between the rotational g tensor of two isotopologues of the same molecule, i.e. two molecules that differ only in the isotopes of one or more nuclei. We consider therefore a component of the rotational g tensor, d japy of one isotopologue with moment of inertia tensor I and centre of nuclear masses, R qm, which is shifted by a vector D = R q — Rcm from the centre of nuclear masses of the second isotopologue with moment of inertia tensor I... 

In O Eq. 11.151 we have introduced the moment of inertia tensor I and we have used the definition of as given in OEq. 11.149, which is applicable both when using London orbitals and when using conventional basis sets. Both the nuclear positions and magnetic dipole operators are defined with respect to the center of mass of the molecule, as this is the point about which the molecule rotates. 

Table 16-1. Energies, electric dipolar moments, net atomic populations, vibrational polarizabilities and mean vibrational molecular polarization, magnetizability and contributions thereto, isotropic g tensor and nuclear and electronic paramagnetic and diamagnetic contributions thereto, principal moments of inertia and rotational parameters calculated for H2 C N2 in seven structural isomers... 

Nr. Molecule v Nu- cleus Nuclear quadrupole coup in system of principal axes of inertia x, [MHz] ling tensor components in system of principal coupling tensor or bond axes x,p, Xxy [MHz] position of the axes 0 Ref. V aria, remarks See T ab., Fig. in Chap. 3, Nr. subvol. b... 

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