Molecular axis system

In Equation 7.33 we have written out both the g-value and the zero-field coefficient of the basic S2 interaction term in the form of diagonal 3x3 matrices in which all off-diagonal elements are equal to zero. The diagonal elements were indexed with subscripts x, y, z, corresponding to the Cartesian axes of the molecular axes system. But how do we define a molecular axis system in a (bio)coordination complex that lacks symmetry The answer is that if we would have made a wrong choice, then the matrices would not be diagonal with zeros elsewhere. In other words, if the spin Hamiltonian would have been written out for a different axes system, then, for example, the g-matrix would not have three, but rather six, independent elements ... 

The vector d, conveniently expresses the directionality of hybrid h with respect to the chosen molecular axis system. Note that each d, is a unit vector (dt dt = 1), and the scalar product of dt and dj is related to the angle (twy) between these vectors in the usual way (see Fig. 3.10),... 

Fig.l Chemical structure of the gramicidin S analogue GS-3/3 with 4F-Phg (Fluorine atom is marked red) substituted for Leu3/Leu3 and N-labeled Vall/Vall (Nitrogen atom is marked blue). The peptide is displayed from one hydrophilic surface (A) and from one side (B) to define the molecular axis system used in the structure analysis... 

The isotropic chemical shift is the average value of the diagonal elements of the chemical shift tensor. Advances in solid state NMR spectroscopy allow one to determine the orientation dependence, or anisotropy, of the chemical shift interaction. It is now possible to determine the principal elements of a chemical shift powder pattern conveniently, and the orientation of the principal axes with more effort. Hence, instead of settling for just the average value of the chemical shift powder pattern, one can now aim for values of the three principal elements and the corresponding orientations in a molecular axis system. 

An advantage of this redefinition is that 6 vanishes for a spherical charge distribution, and it is often advantageous to work in terms of principal axes, i.e. an axis system related to the molecular axis system by an orthogonal transformation such that the off-diagonal elements of the tensor vanish. 

In conclusion we summarise the total Hamiltonian (excluding nuclear spin effects), written in a molecule-fixed rotating coordinate system with origin at the nuclear centre of mass, for a diatomic molecule with electron spin quantised in the molecular axis system. We number the terms sequentially, and then describe their physical significance. The Hamiltonian is as follows ... 

Let us consider the scalar product of a rotation operator J and an internal angular momentum P which is quantised in the molecule-fixed axis system. Although it might appear sensible to evaluate J P in a molecule-fixed axis system where both angular momenta operate, we shall instead expand the tensor product in a space-fixed axis system and then refer the components of P to the molecular axis system using a rotation matrix 3) ( >) ... 

In case (a) coupling two main possibilities arise. The first, which is expected to arise very rarely, if at all, implies that the magnetic interaction of the nuclear spin magnetic moment with the electronic orbital and spin moments is sufficiently strong to force the nuclear spin to be quantised in the molecular axis system. The basis kets may be expressed in the form rj, A S, S, A, 2 2, Iz, 2 2, N, J) this scheme is known... 

We have taken the direction of the electric field to define the space-fixed p = 0 direction in the expansion of the scalar product. The electric dipole moment of the molecule, fj,e, is, however, quantised in the molecule-fixed axis system (q) we therefore rotate the space-fixed component of fie into the molecular axis system using a rotation matrix, so that the perturbation (6.318) becomes... 

The principal components (defined in the molecular axis system) of the shielding and susceptibility tensors can be determined from solid state studies which have sufficient accuracy and resolution. In the case of HF in the gas phase, with J = 1, the spin-rotation and dipolar constants were determined accurately from the earlier electric resonance studies, so that de Leeuw and Dymanus [89] were able to use their Zeeman studies to measure the anisotropy of the screening and susceptibility tensors, with the following results ... 

As well known, the NRG of acetone is the Gae. The first step should be to determine the representations of the Gae group under whidi the electric moment components transform. For that purpose, let us consider acetone in a molecular axis system, in which the z axis coincides with the Ga axis of the molecule, the y axis lies perpendicular to z axis in the CCC plane, and the x axis stands perpendicular to the CCC plane. 

Figure 2 Examples of global and local axis systems, (a) Molecular axis system for a homonuclear diatomic. Wth this system, all central multipoles with k 0 otl odd will be zero, and no S functions with k 0 oxl odd can appear in a molecule-molecule expansion of U(R, Q). The atomic multipoles Q q (all / 0 allowed) on the two atoms will be related by Qio = (-l) Qio- ( ) Local atomic axis system for a homonuclear diatomic molecule. With this definition QJq = Qio- (c) Molecular axis system for water. The nonzero atomic multipole moments for the O atom would be QoO> QlO) QzOj Qz2c = (Q22 + Qz-2)1 > QsOJ Q32c = (Qs2 + on the hydrogen atoms Qjo = Qoo. Qio = Qio> Qiic = (-Q11 + Qi-i) = -Qiic...

The macroscopic property observed in sum-frequency experiments, Xs . is a sum of the molecular hyperpolarizabilities, over all vibrational modes and all of the molecules at the interface, which takes into account the orientation of each molecule. Orientational information is obtained from the experimental spectra through consideration of the relationship between the observed Cartesian components of the macroscopic second-order susceptibility Xuk, the corresponding spectroscopically active components of the molecular hyperpolarizability, This is accomplished through an Euler angle rotation of the molecular axis system into the laboratory axis system as defined through the use of the rotational matrix iiuK imn- The general expression for the transformation from a molecular-fixed axis system to a laboratory-fixed system is... 

The magnetic field strength and the direction cosines between the field and the molecular axis system enter linearly in the g term ( linear Zeeman effect contribution ). Since the direction cosines are proportional to M, the quantum number for the component of the angular momentum in the direction of the exterior field (— /field dependent splitting of the rotational level into a pattern of 2/ +1 sublevels which are symmetrically arranged around the zero-field position. 

The use of standard NMR spectroscopy without any selective averaging techniques has generally had little importance in the field of catalysis. An exception is high-field V NMR, which yields characteristic lineshapes in the solid state that are easily interpreted in terms of the chemical shift anisotropy ] 11 ]. Generally, we can distinguish the three situations illustrated in Fig. 1 The spectrum in Fig. Ic is observed for compounds with asymmetric coordination environments. It shows three distinct features, which can be identified with the three cartesian chemical shift components 8, 8yy, 8 in the molecular axis system. Figure lb corresponds to the case of cylindrical symmetry, where 8 = 8yy 8, and hence two distinct lineshape components appear. Finally, for chemical environments with spherical symmetry, the chemical shift is the same in all three directions. The solid-state NMR spectrum then contains only a single symmetric peak (Fig. la). 

The chemical shift anisotropy is usually described in a principal axis system, which is usually not the molecular axis system. In the principal axis system, the chemical shift tensor is diagonal. The elements of this tensor contribute to the NMR spectrum via these two equations ... 

Figure 8.16 shows the geometrical relationship for the molecular axis system (x, y, z) and the optical axis ( ) parallel to the orientation axis. 

Figure 1. Molecular structure of pentacene (C22H14). molecular axis system and labeling of the inequivalent carbon positions.

---