Rotational Part of

The first three terms of HROT have diagonal matrix elements exclusively. This diagonal part of HROT is the rotational energy of the JMflAST,) basis function. The eigenfunctions [defined by Eq. (2.3.40)] of the rotational eigenvalue equation, 

In general, electronic wavefunctions of molecules are not eigenfunctions of L2, hence L(L + 1) is not quantized, but the R-dependent quantity, 

Born-Oppenheimer approximation perturbation AA AE AS AH spin-orbitalsrf 

The final three terms of the rotational operator in Eq. (3.1.13), which couple the orbital, spin, and total angular momenta, are responsible for perturbations between different electronic states  

+(l/2pf 2)L S F gives rise to homogeneous (Aft = 0) spin-electronic perturbations between basis functions of the same ft and S, but different A and E. 

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Observe that, in principle, it is possible to introduce quaternions in the solution of the free rotational part of a Hamiltonian splitting, although there is no compelling reason to do so, since the rotation matrix is usually a more natural coordinatization in which to describe interbody force laws. 

For a non-linear polyatomie moleeule, again with the eentrifugal eouplings to the vibrations evaluated at the equilibrium geometry, the following terms form the rotational part of the nuelear-motion kinetie energy ... 

The rotational part of the I>f i i> integral is not of the expeetation value form beeause the initial rotational funetion ( )ir is not the same as the final ([). This integral has the form ... 

The rotating part of a centrifugal pump in contact with the water, converts centrifugal force into pressure. 

In general a rotating part of a machine in the mbber industry the term refers particularly to the contoured rolls of an internal mixer and to the mushroom-shaped rotor of the Mooney Viscometer. 

Derivative, for a viscoelastic liquid or solid in homogeneous deformation, of the rotational part of the deformation-gradient tensor at reference time, t. 

Barbette. A mound of earth or a specially protected platform on which guns are mounted to fire over a parapet a cylinder of armor on a warship that gives protection to the rotating part of the turret below the gunhouse a fixed superstructure on an armored vehicle, usually with gun mount o limited traverse... 

A detailed analysis of the rotational degrees of freedom and of bending motion has been carried out by Freed and Band (2) and by Morse and Freed (52-54). The former authors considered the photodissociation of a linear triatomic molecule. In that work the rotational part of the initial wavefunction is written in the form... 

If D has cut points, then the order of det W (D) can, possibly, be further reduced by successively rotating parts of D around these points such that, in the resulting figure D, ... 

We recall also that the y l i,, (< >) are proportional to the eigenfunctions of a symmetric top rotational Hamiltonian (section 5.3.4). Realising that a diatomic molecule behaves as a symmetric top (albeit a rather special one), we write the rotational part of the wave function as... 

The first two terms in the purely rotational part of (8.361) are wholly diagonal in our basis set and may be replaced by their respective eigenvalues. The remaining scalar products are expanded in the molecule-fixed coordinate system, q, and in the sum over q we separate the e/ = 0 terms from those with q = 1 (denoted by a superscript prime). We also take note of the anomalous commutation rules for the components of J. Equation (8.361) becomes... 

As a general guide, CH2 groups close to a stereogenic centre are more likely to be revealed as diastereotopic than those further away. Those in part of a structure with a fixed conformation are more likely to be revealed as diastereotopic than those in a flexible, freely rotating part of the molecule. 

Figures 2-13-2-15 illustrate rotational symmetries in flowers, rotating parts of machinery, and hubcaps. Seldom does exclusively rotational symmetry have functional importance in flowers. In contrast, the motion of rotating parts in machinery is reinforced by having only rotational symmetry and no symmetry planes. There...

Figure 2-14. Rotating parts of machinery (photographs by the authors).

In principle, the sums over ai and other terms can be most easily evaluated by applying the gradient formula in spherical tensor form (Edmonds, 1957) ... 

The harmonic approximation consists of expanding the potential up to second order in the atomic or molecular displacements around some local minimum and then diagonalizing the quadratic Hamiltonian. In the case of molecular crystals the rotational part of the kinetic energy, expressed in Euler angles, must be approximated, too. The angular momentum operators that occur in Eq. (26) are given by... 

In this connection it is particulary interesting to study the general form of the pure rotational part of the G-matrix, usually referred to as the /r-tensor. We find... 

The present stage is suitable for the introduction of the Coriolis coupling constants, f%k 63, 64). This may seem curious, but it is in accordance with the fact that these constants are appropriate only when rectilinear coordinates are involved. This will be further discussed below in relation to the vibration-rotation part of the G-matrix. Here it is convenient to give a general definition,... 

Once again returning to a general set of vibrational coordinates we shall now study the pure rotational part of the kinetic energy, given by... 

Thus the remaining derivation can proceed exactly as for nonlinear molecules (Sect. 3.5), and the vibration-rotation part of the kinetic energy finally becomes... 

The matrices Mjg and phases can be deduced from the relations between the Fourier coefficients and atomic basis functions [Equation (54)]. The matrices Mp correspond, in the case of commensurate magnetic structures, to the rotational parts of the magnetic Shubnikov group acting on magnetic moments. 

We calculate the pure rotational part of our spectrum taking into account all the light scattering mechanisms described in Eq. (56). Then the resulting spectrum takes the form of the convolution of the rotational and translational parts [see Eqs. (31) and (32)]. We deal numerically with the rotational and translational spectra and their convolution [15,16,36] by methods described in the previous section for optically isotropic molecules. 

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