Molecule-fixed coordinate system

In a molecule-fixed coordinate system, the multipolar Hamiltonian of ref. 

This is referred as BO ansatz. This ansatz is taken as a variational trial function. Terms beyond the leading order in m/M are neglected m is the electronic and M is nuclear mass, respectively). The problem with expansion (4) is that functions /(r, R) contain except bound states also continuum function since it includes the centre of mass (COM) motion. Variation principle does not apply to continuum states. To avoid this problem we can separate COM motion. The remaining Hamiltonian for the relative motion of nuclei and electrons has then bound state solution. But there is a problem, because this separation mixes electronic with nuclear coordinates and also there is a question how to define molecule-fixed coordinate system. This is in detail discussed by Sutcliffe [5]. In the recent paper by Kutzelnigg [8] this problem is also discussed and it is shown how to derive adiabatic corrections using, as he called it, the Bom-Handy ansatz. There are few important steps to arrive at formula for a diabatic corrections. Firstly, one separates off COM motion. Secondly, (very important step) one does not specify the relative coordinates (which are to some extent arbitrary). In this way one arrives at relative Hamiltonian Hrd [8] with trial wavefunction If we make BO ansatz... 

We will use the 7D CRS Hamiltonian which has been determined and analyzed in Ref. [10] (DFT/B3LYP, 6-31+G(d,p)). In short, the large-amplitude motion of the H/D atom is restricted to the (x,y) plane of the molecule (cf. Fig. 1). The origin of the molecule-fixed coordinate system is at the center of mass, with the axes pointing along the principal axes of inertia for the enol configuration. The H/D motion couples strongly to 5 in-plane skeleton modes, Q = (Q4, Q, Qu, Q26, Q3o)> which are described in harmonic approximation... 

Now consider fa. We set up the space-fixed and molecule-fixed coordinate systems with a common origin on the internuclear axis, midway between the nuclei, as in Fig. 4.11. (Previously in this chapter, we put the origin at the center of mass, but the difference is of no consequence.) The electronic wave function depends on the electronic spatial and spin coordinates and parametrically on R. The parity operator does not affect spin coordinates, and we shall only be considering transformations of spatial coordinates in this section. 

Fig. 8. C2vF(C2vT)2SRM (a) definition of the internal coordinates -n

The transformation between the space- and molecule-fixed coordinate systems is thus expressed by... 

The operators in (2.36) are easily re-expressed in the molecule-fixed coordinate system since the V" operator merely becomes the V, operator in the new coordinate system and, as mentioned earlier, Fei,nuci becomes independent of the Euler angles. We must also consider the transformation of the partial differential operators 3/3[Pg.52]

Relationship between operators in space-fixed and molecule-fixed coordinate systems... 

It is tempting to construct spherical tensors from J acting within the molecule-fixed coordinate system. From Table 5.2 the components would be expected to have the form ... 

Figure 6.24. The effectofthe space-fixed inversion operator E on the molecule-fixed coordinate system (x, y, z). The molecule-fixed coordinate system is always taken to be right-handed. After the inversion of the electronic and nuclear coordinates in laboratory-fixed space, the (x, y, z) coordinate system is fixed back onto the molecule so that the z axis points from nucleus 1 to nucleus 2 and the y axis is arbitrarily chosen to point in the same direction as before the inversion. As a result, the new values of the Euler angles (ip 6, x ) are related to the original values , 9, x)by

The molecular physics underlying the nuclear spin-rotation interaction has been discussed by Flygare [107], In the general case of a polyatomic molecule the spin-rotation interaction is represented by a second-rank tensor in a molecule fixed coordinate system x, y, z, the diagonal component in the x direction may be written as the sum of a nuclear part (k labelling the nucleus under consideration) and an electronic part ... 

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... 

Since we shall be interested in the electric field gradient with respect to a molecule-fixed coordinate system, we need to transform (8.492) from space-fixed to molecule-fixed axes the relationships between the two are illustrated in figure 8.53. Denoting molecule-fixed axes with primes, and space-fixed axes without primes, the spherical harmonic addition theorem gives the result ... 

The analysis is not yet complete because we have to consider the left-hand term in equation (8.495) which relates the space-fixed and molecule-fixed coordinate systems. Although we have already selected the q = 0 component, we will retain q as a variable in order to facilitate later discussion. First we note that and can be expressed in terms of the Euler angles 0 and x as follows ... 

In our discussion of the FIR laser magnetic resonance spectrum of CH in its a 4 " state we encountered the reduced matrix element of P(.S. . S. . S ). The result was presented in equation (9.155), which we now derive. First we note that, by the Wigner Eckart theorem, the following result applies in the molecule-fixed coordinate system with... 

If we expand the scalar product in the molecule-fixed coordinate system, the diagonal elements (q = 0) are readily seen to be... 

For an isolated system, treatment of the intramolecular Jahn-Teller effect is relatively simple. As the system is isolated, we may ignore molecular rotation and consider a molecule-fixed coordinate system. Within this frame of reference, the electronic and vibrational states can be formulated in terms of the irreducible representations (irreps) of the reference configuration. Overall, the system Hamiltonian is generally written in the form... 

The conformers give spectra as if they were separate molecules. Molecular motions that are fast on the NMR time scale but slow relative to molecular tumbling are considered to be between conformers. The observed NMR spectra correspond to a spin Hamiltonian that is a weighted mean of those of the individual conformers. In general, each such con-former will be characterized by its own structure, molecule-fixed coordinate system, and motional constants. 

If one assumes that benzene is a planar molecule with a sixfold axis of symmetry, which we take to be the z axis of the molecule-fixed coordinate system, then only the motional constant c3z is nonzero. By making use of Eq. (8) the dipolar interaction for any pair of nuclei may be written as ... 

As the second step in our derivation we will introduce a molecule fixed coordinate system and we will rewrite the Lagrangian using the corresponding generalized coordinates. 

One normally proceeds by eliminating the translational motion and chooses the origin of a molecule-fixed coordinate system. A suitable choice is the center of mass... 

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