Bond vectors, linear combination

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. 

The four linear combinations with Ai and T2 symmetries of the four vectors zi,. .., Z4 forming a bonds with orbitals on atom A may be easily obtained by the technique of projection operators. Therefore, only the results are given in Table 7.1.8, where all linear combinations of ligand orbitals will be listed. It is noted that the four combinations of the z vectors are identical to the combinations of hydrogen Is orbitals obtained for methane. 

When one bond is stretched, no moments are produced in other bonds. In other words, the total moment resulting from the simultaneous displacement of several bonds is assumed to be the vector sum of the moments produced by each individual bond. Since the molecular symmetry coordinates are linear combinations of internal bond coordinates R 135),... 

Having found the independent linear combinations of bond vectors [see Eqs. (2.1.17), (2.1.27), and (2.1.28)], through the central-limit theorem it is easy to construct the Gaussian joint distribution and the associated quadratic potential. Adopting for simplicity the periodic-chain transform, we have... 

As stated earlier, we can derive this same result by considering linear combinations of local bond quantities transforming as a vector, under point group operations [i.e., in the same way as the pair of coordinates (jc, y) associated with the degenerate species (for in-plane vibrations)]. Thus we are left to determine two arbitrary parameters y, fB. The first one is typically fixed by a normalization procedure, while /3 is obtained by explicitly accounting for the observed slope of the infrared intensities. 

Basis functions In the context of solutions of the electronic Schrodinger s equation for hydrogen-bonded system, this term refers to Gaussian functions of the form exp[-ar ], where r is the position vector, multiplied by powers of the coordinates x, y, and z. The basis functions are usually located at the nuclear positions and near the midpoint of the hydrogen bond (midbond functions). Linear combinations of such basis functions form molecular orbitals. 

From the vector properties of orbitals, the bond orbitals oah> and of Figure 7.6 may be decmnposed into atomic orbital contributions as shown in 7 Jl. So we can easily see that the MOs of bent AHj are approxiinated by the bond orbitals as shown in Figure 7.7. Consider for instance the linear combinations of the two bonding orbitals oah The positive combination of the two leads to the Os MO, and the s character of A is retained. The negative combination leads to the Ox, and removes the s character of A. Therefore, the two linear combinations of the degenerate bonding orbitals oah become different in energy. Similarly, linear combinations of the two a orbitals or the two orbitals lift the de-... 

These are linear combinations of the type suggested by Equation (6.23). The symmetric mode (Equation (6.25)) has been arbitrarily chosen as mode 1, so we have c = Cu = 1, and the asymmetric mode as mode 2 has Cji = I.C22 = -1. In this way the individual modes can be thought of as sets of coeflhcients for the basis functions. Because the basis is simplified to two bond vectors, this analysis does not give the O atom motion that is shown in Figure 5.3 however, since the symmetry analysis for these two modes does not require this part of the motion, we will ignore it for now. 

To find the linear combinations of the basis vectors for Aj and the first part of the E representation we can apply the projection method arbitrarily taking b (the N—Hi vector) as the generating vector. The results are summarized in Table 6.13, where the three mirror planes in the 3ay class are considered separately using A, B and C superscripts to refer to planes containing the N—Hj, N—Hj and N—Hj bonds respectively. 

The C=0 bond on the C2 axis, with basis vector bs, is separate from bi and 2 because it is not exchanged with them by any of the symmetry operations, b lies on the symmetry axis, so it can only have an Ai representation. This means it can only be involved in linear combinations with the Ai function found for the bi and 2 set, i.e. 

Figure 1.10. Illustration of covalent bonding in diamond. Top panel representation of the sp linear combinations of s and p atomic orbitals appropriate for the diamond structure, as defined in Eq. (1.3), using the same convention as in Fig. 1.8. Bottom panel on the left side, the arrangement of atoms in the three-dimensional diamond lattice an atom A is at the center of a regular tetrahedron (dashed lines) formed by equivalent B, B, B", B " atoms the three arrows are the vectors that connect equivalent atoms. On the right side, the energy level diagram for the s, p atomic states, their sp linear combinations the bonding (V f) and antibonding ( ) states. The up-down arrows indicate occupation by electrons in the two possible spin states. For a perspective view of the diamond lattice, see Fig. 1.5.

Allegra has shown that the probability distribution for more general linear combinations of the skeletal bond vectors of an unperturbed chain is also Gaussianly distributed. In particular, if we decompose the conformation into the Fourier modes, 7 (p), defined by... 

From the vector properties of orbitals, the bond orbitals (Toh. nd (T oh of Figure 7.8 may be decomposed into atomic orbital contributions as shown on the left-hand side of 7.29 for the two n bond orbitals. One can force the bond orbitals to have the full symmetry properties associated with the molecule by taking linear combinations of each degenerate set. Consider for instance the linear combinations of the two nonbonding orbitals no- The positive combination of the two, 7.29, leads to a molecular orbital we shall call n r- By decomposing each bond... 

These are expressed in terms of scalar products between the unit axis system vectors on sites 1 and 2 (on different molecules) and the unit vector 6. from site 1 to 2. The S functions that can have nonzero coefficients in the intermolecular potential depend on the symmetry of the site. This table includes the first few terms that would appear in the expansion of the atom-atom potential for linear molecules. The second set illustrate the types of additional functions that can occur for sites with other than symmetry. These additional terms happen to be those required to describe the anisotropy of the repulsion between the N atom in pyridine (with Zj in the direction of the conventional lone pair on the nitrogen and yj perpendicular to the ring) and the H atom in methanol (with Z2 along the O—H bond and X2 in the COH plane, with C in the direction of positive X2). The important S functions reflect the different symmetries of the two molecules.Note that coefficients of S functions with values of k of opposite sign are always related so that purely real combinations of S functions appear in the intermolecular potential. 

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