Cartesian axes

For a distribution of charges (one not restricted to lie along one of the Cartesian axes), th dipole moment is given by ... 

The dielectric constant is concerned with the electrical properties of a material. The dielectric constant for a solid is a 3 x 3 matrix with different components according to the Cartesian axes. These elements are given by ... 

The electric dipole moment operator jx has components along the cartesian axes ... 

In the sixth column of the main body of the character table is indicated the symmetry species of translations (7) of the molecule along and rotations (R) about the cartesian axes. In Figure 4.14 vectors attached to the nuclei of H2O represent these motions which are assigned to symmetry species by their behaviour under the operations C2 and n (xz). Figure 4.14(a) shows that... 

The dipole moment vector /i must be totally symmetric, and therefore symmetric to all operations of the point group to which the molecule belongs otherwise the direction of the dipole moment could be reversed by carrying out a symmetry operation, and this clearly cannot happen. The vector /i has components fiy and along the cartesian axes of the molecule. In the examples of NH3 and NF3, shown in Figures 4.18(b) and 4.18(e), respectively, if the C3 axis is the z-axis, 7 0 but = 0. Similarly in H2O and cis-... 

The polarizability ellipsoid rotates with the molecule at a frequency say, and the radiation sees the polarizability changing at twice the frequency of rotation since, as can be seen from Figure 5.14, the ellipsoid appears the same for a rotation by n radians about any of the cartesian axes. The variation of a with rotation is given by... 

CFlBrClF (bromochlorofluoromethane) dipole moment, 99ff enantiomers, 79 symmetry elements, 79ff CF12F2 (difluoromethane) cartesian axes, 89 symmetry elements, 77, 83 CF13F (methyl fluoride) dipole moment, 116 symmetry elements, 74, 83 C Fl3F (methyl fluoride) vibration-rotation band, 178... 

The starting points for many conventions in spectroscopy are the paper by R. S. Mulliken in the Journal of Chemical Physics (23, 1997, 1955) and the books of G. Herzberg. Apart from straightforward recommendations of symbols for physical quantities, which are generally adhered to, there are rather more contentious recommendations. These include the labelling of cartesian axes in discussions of molecular symmetry and the numbering of vibrations in a polyatomic molecule, which are often, but not always, used. In such cases it is important that any author make it clear what convention is being used. 

Depending on the orientation of the Cartesian axes, the matrix of quadrupole moments... 

Consider a matrix A expressed in a coordinate system [x, X2,X3,..., xn - The coordinate axes are the X/ vectors, these may be simple Cartesian axes, or one-variable functions, or many-variable functions. The matrix A is typically defined by an operator working on the coordinates. Some examples are ... 

Remember the spatial arrangement of the p or- atom has partially filled valence orbitals. Elec-bitals Each one protrudes along one of the tron sharing can occur, placing electrons close three cartesian axes (as shown in Figure 15-9). to two nuclei simultaneously. Hence a stable If the electrons have the orbital occupancy of bond can occur. This is shown in representations 20), then two electrons occupy the p orbital (22) and (23). 

Describe the orientation of the lobes of the px, px, and p -orbitals with respect to the reference Cartesian axes. 

Cartesian axes. You may wish to refer to the animation w of the atomic orbitals found on the Web site for this text. 

The three p orbitals are directed along the three cartesian axes and so, in an octahedral crystal field, suffer equal repulsion from point charges sited on those axes. The energies of the three p orbitals, therefore, remain degenerate. Similarly, a free-ion P term remains unsplit in octahedral or tetrahedral crystal fields and is labelled Tig or Ti respectively. 

The molecule contains atoms with mass m at positions r and a and J in the above equations correspond to Cartesian axes (x,y,z). 

To sum up, the basic idea of the Doppler-selected TOF technique is to cast the differential cross-section S ajdv3 in a Cartesian coordinate, and to combine three dispersion techniques with each independently applied along one of the three Cartesian axes. As both the Doppler-shift (vz) and ion velocity (vy) measurements are essentially in the center-of-mass frame, and the (i j-componcnl, associated with the center-of-mass velocity vector can be made small and be largely compensated for by a slight shift in the location of the slit, the measured quantity in the Doppler-selected TOF approach represents directly the center-of-mass differential cross-section in terms of per velocity volume element in a Cartesian coordinate, d3a/dvxdvydvz. As such, the transformation of the raw data to the desired doubly differential cross-section becomes exceedingly simple and direct, Eq. (11). 

In the case of (8.49), the situation is once again simplified by prior knowledge of the pulling direction es can be taken without loss of generality to lie along one of the natural Cartesian axes of the system, so that the p and p systems are equivalent, and no axis rotations need be performed. 

The p-states have the symmetry of dumbbells (Figure 3.1). The axes of the dumbells can have three orientations px, py, pz each directed along one of the Cartesian axes. Consider pz, for example. Let M = normalization coefficient then the form is ... 

Figure 6.4 shows the magnetic field vector B in the molecular Cartesian axes system xyz whose orientation is defined by the polar angles 0 (between B and z) and < ) (between the projection of B on the x-y plane and x). The third dimension of the... 

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 expansion of Fourier components of the dipole interaction tensor in the vicinity of the minimum point at the boundary of the first Brillouin zone, with the Cartesian axes Ox and Oy respectively chosen along bi and b2 (see Fig. 2.9b), has the form... 

Cartesian axes. Based on previously reported results,47"49,52,78,79 the lateral... 

Then in the initial Hamiltonian function (Al.l), all the indices, a and P, of the projections onto the Cartesian axes are the same and thus can be omitted, while the... 

Figure 43 a shows the angular dependence of the four Co-ENDOR transitions for rotations of the crystal around three cartesian axes. For rotation I, the angular dependence shows a noteworthy peculiarity near

[Pg.86]

---