Bond displacement vectors

An alternative roach for analysis of vibrational intensities has been put forward by Migrants and Averbukh [116,129,130]. An extensive review on die mediod has been published by Ruppredit [37], Hereafter, we shall follow with few exceptions the notation used by Riqiprecht which is closer to die notation used so far. Instead, on die basis of atomic Cartesian displacement coordinates, the dp/dQi quantities are transformed into the coordinate space of bond displacement vectors. The change of dipole moment is defined as... 

As the starting geometries for iterative calculation, we take all the possible structures in which bond lengths are distorted so that the set of displacement vectors may form a basis of an irreducible representation of the full symmetry group of a molecule. For example, with pentalene (I), there are 3, 2, 2 and 2 distinct bond distortions belonging respectively to a, b2 and representations of point group D21,. 

Kuhn has shown how a real polymer chain may be approximated by an equivalent freely jointed chain. Instead of taking the individual bonds as statistical elements, one may for this purpose choose sequences of m bonds each. In Fig. 79, arbitrarily chosen statistical elements consisting of five bonds are indicated, the displacement vectors for these elements being shown by the dashed lines. The direction assumed by a statistical element will be nearly independent of the direction of the preceding element, provided the number m of bonds per... 

Where the quantities d, d% and are the projections of the Drude displacement vector d on orthogonal axis defined on a local intramolecular reference frame. The intramolecular reference frame may be defined, for example, by the C=0 vector and the N—C=0 plane of an amide bond [65],... 

There are two totally symmetric ( ,) normal modes and one b2 normal mode. (The convention is to use lowercase letters for the symmetry species of the normal modes.) The symmetry species of the normal modes have been found without solving the vibrational secular equation. Moreover, since there is only one b2 normal mode, the form of this vibration must be determined from symmetry considerations together with the requirement that the vibration have no translational or rotational energy associated with it. Thus (Fig 6.1), any bent XYX molecule has a b2 normal mode with the X atoms vibrating along the X—Y bonds and the Y atom vibrating in the plane of the molecule and perpendicular to the symmetry axis. On the other hand, there are two ax symmetry coordinates and the two ax normal vibrations are linear combinations of the ax symmetry coordinates, where the coefficients are dependent on the nuclear masses and the force constants. Thus the angles between the displacement vectors of the X atoms and the X—Y bonds for the ax modes of a bent XYX molecule vary from molecule to molecule. 

Consider the symmetry behavior of the normal modes of H20 and COz. For water, the symmetry operations are E, C2 b), ov(ab), and av(bc), where the a and b principal axes lie in the molecular plane the bond angle in water is relatively obtuse, so that the C2 axis is the axis of intermediate moment of inertia. Each of these operations converts the displacement vectors of p, in Fig. 6.1 to an indistinguishable configuration the same is true for v2. Normal modes that are symmetric with respect to all the molecular symmetry operations are called totally symmetric. For H20, the modes and v2 are totally symmetric, but v3 is not. Figure 6.3 shows that v3 is antisymmetric with respect to C2(b). We can tabulate the behavior of the H20 normal modes with respect to the molecular symmetry operations ... 

Stacking faults are characterised by a fault plane and a fault displacement vector. On one side of the fault plane, the atoms that are located fer from the fault are displaced by a vector R in relation to the positions they would occupy in the absence of the fault. Strain fields emanating from any reconstructive bonding that is present near the fault plane will lead to additional displacements for atoms near the fault plane. Thus, the specification of R determines the positions of the atoms that are sufficiently distant so that the strain field generated by the fault is below some specified tolerance. For a planar fault, R may be determined experimentally by analysis of the diffraction contrast obtained with different diffraction vectors g. The positions of atoms near the fault may be determined theoretically by total energy minimisation calculations. Knowledge of these positions is essential to determine the electronic structure of the fault. 

All of the operations of the point group leave the displacement vectors unchanged during the symmetric stretching of Pd-Cl bonds in the trans isomer. Therefore, all of the operations have a character of 1. This corresponds to the first row in the Dyn character table, which is the Ag symmetry type. 

In this expression, the dependences on the nuclear positions and the displacements of the Drude particle are indicated by r and d, respectively, f/bond (r) and f/Lj(r) are, respectively, intramolecular bonding and non-bonding LJ energetic contributions. l/eiec(r d) is the sum over all Coulombic interactions between atomic core charges located at r[j and the Drude charges —and located at rjj and r ) respectively. The displacement vector for the Drude particle... 

If the point of reference (point 1) coincides with the center of mass (point G), the displacement vector between point 1 and point G is null. The transformer relating the velocity due to rotation of point G disappears resulting in the bond graph in... 

Dimensions of Ideal Chains Now we obtain Rp and R for ideal chains whose conformations are given as trajectories of random walkers. They include a random walk on a lattice, a freely jointed chain, a bead-spring model, and any other model that satisfies the requirement of Markoffian property (Eq. 1.19). The bond vector r, - r, i of the ith bond is then the displacement vector Ar, of the ith step. We assume Eq. 1.19 only. Then the end-to-end distance is Nb. To calculate / g, we note that a part of the ideal chain is also ideal. The formula of the mean square end-to-end distance we obtained for a random walk applies to the mean square distance between the ith and/th monomers on the chain just by replacing N with i - j. ... 

A much more detailed discussion of the choice of basis for a quantitative description of molecular vibrations is given in the text by Bright Wilson et al. referenced in this chapter s Further Reading section. This covers the use of mass-weighted coordinates and systems of internal coordinates based on bond vectors, bond angles and dihedral angles. Here, we are interested in the application of symmetry to vibrational spectroscopy to understand selection rules, and usually the much simpler basis of a few carefully chosen atom or bond displacements will suffice. 

Owing to the different bonding environment around an APB, atoms near the APB are expected to deviate from ideal lattice positions, as a result of which both the direction and the magnitude of the APB displacement vector p differ from that for an ideal APB. In general, the deviation vector 2p should vary as a function of the position relative to the APB. In most cases, however, a simplification is made in which the APB still. separates two rigidly displaced crystals but with a displacement vector in the form... 

Not only functions but also displacement vectors can be analyzed for symmetry properties. Displacements can be associated with movement of atoms (vibration, rotation, and translation of molecules), and these may transform as symmetry representations. Consider a vector that corresponds to simultaneously stretching both water O-H bonds by like amounts. A picture of this displacement is that of two vectors of the same magnitude on the two hydrogen centers. 

If we apply the symmetry operators of water s point group to this picture, we find that it looks the same for all operations. This combination of two atomic displacement vectors transforms as the or totally symmetric representation. On the other hand, when a composite displacement consists of contracting one O-H bond by as much as the other is being stretched, then a different representation is obtained. 

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