Eigenvalues rotational motions

We next solve the secular equation F — I = 0 to obtain the eigenvalues and eigenvectors o the matrix F. This step is usually performed using matrix diagonalisation, as outlined ii Section 1.10.3. If the Hessian is defined in terms of Cartesian coordinates then six of thes( eigenvalues will be zero as they correspond to translational and rotational motion of th( entire system. The frequency of each normal mode is then calculated from the eigenvalue using the relationship ... 

Thus, the /, in Eq. (2.46) are one of the 3Arows of L while the l i in Eq. (2.35) are one of the 3N columns. For a nonlinear molecule there are six zero eigenvalues in A, which correspond to translation and external rotation motions. The remaining 3N — 6 nonzero eigenvalues equal where the v, s are the normal mode vibrational frequencies. 

The molecule also has angular momentum, which you would expect it to have because it is rotating. The quantum number / is used to define the total angular momentum of the molecule rotating in three dimensions. The total angular momentum of a molecule is given by the same eigenvalue equation from three-dimensional rotational motion ... 

Suppose that W(r,Q) describes the radial (r) and angular (0) motion of a diatomic molecule constrained to move on a planar surface. If an experiment were performed to measure the component of the rotational angular momentum of the diatomic molecule perpendicular to the surface (Lz= -ih d/dQ), only values equal to mh (m=0,1,-1,2,-2,3,-3,...) could be observed, because these are the eigenvalues of ... 

It is possible (see, for example, J. Nichols, H. E. Taylor, P. Schmidt, and J. Simons, J. Chem. Phys. 92, 340 (1990) and references therein) to remove from H the zero eigenvalues that correspond to rotation and translation and to thereby produce a Hessian matrix whose eigenvalues correspond only to internal motions of the system. After doing so, the number of negative eigenvalues of H can be used to characterize the nature of the... 

Figure 1.4 The very anharmonic energy levels of the C6H6 - Ar stretch motion (cf. Figure 0.3) (adapted from Neusser, Sussman, Smith, Riedle, and Weber, 1992). The computed values of the rotational constant Bv [the coefficient of 1(1+1) in the expression for the energy eigenvalues] are given in the figure, as are the vibrational spacings.

With these approximations, the rotational and vibrational motion is completely separated and the eigenvalue problem... 

Eqn (9-3.1) can then be separated into three eigenvalue equations for the three types of motion. The three eigenvalues will be Wu (translational energy , W 1 (rotational energy) and Wyih (vibrational energy)... 

Whereas the group jr and its representations are relevant and sufficient for problems which are completely defined by relative nuclear configurations (RNCs) of a SRM, primitive period isometric transformations have to be considered as nontrivial symmetry operations in all those applications where the orientation of the NC w.r.t. the frame and laboratory coordinate system is relevant, e.g. the rotation-internal motion energy eigenvalue problem of a SRM. Inclusion of such primitive period operations leads to the internal isometric group ( ) represented faithfully by... 

If the rotational quantum number J is zero, the molecule possesses no angular momentum arising from the motion of the nuclei nuclear motion is purely vibrational. The vibrational energy levels depend on the shape of the potential function 1/(7 ), most often of the well-known diatomic form. Near the minimum the potential function approximates to a parabola. The eigenvalues and functions are thus approximately those appropriate for a harmonic oscillator,... 

Six of these normal coordinates (five for a linear molecule) have a frequency eigenvalue identically equal to zero. These motions are translations and rotations of the molecule. Although the approach through Cartesian displacement coordinates is theoretically elegant, it is generally more practical to express the vibrational motions in terms of internal coordinates, such as bond stretches and distortions of bond angles. The method is discussed in detail in Chapter 4 of Wilson, Decius and Cross [57]. Since the distortions of the molecule can be described in terms of 3A — 6 of these internal coordinates there are no redundant dimensions to be removed when the analysis is complete. 

After calculating the ground and excited mean field states of a- and y-nitrogen, we have included the correlation between the molecular motions, as well as the translational-rotational coupling, by determining the eigenvalues of the RPA matrix M(q) ]Eq. (129)]. The expansion of the potential in the translational displacements (u/ ) of the molecules [see Eq. 

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