Bond displacement coordinates

Figure Al.2.6. Anliannonic stretch nonnal modes of a synnnetric triatoniic. The plot is similar to figure Al.2.5. except the nonnal modes are now anliannonic and can be curvilinear hi the bond displacement coordinates r, r. The antisyimnetric stretch is curved, but the synnnetric stretch is linear because of synnnetry.

The relevant equations for the derivative Numerov-Cooley (DNC) method closely follow Cooley s [111] presentation. Let R be the radial coordinate or bond displacement coordinate, P R) a radial eigenfunction, and U R) the potential function. The one-dimensional Schrodinger equation is then... 

If one treats the anharmonicities and to second and first order respectively in a harmonic oscillator basis. Here q is the C-H bond displacement coordinate, and the expansion coefficients in the potential are related to the overtone spectrum parameters by... 

Ls is a normal coordinate transformation matrix of rank 3N-3 which includes the three rotations. It defines the relation between the bond displacement coordinates Xg and an... 

I is a linear array containing the set of bond displacement coordinates Ai, A9j( and A i. Using the standard notation we can also write... 

Experimental intensity data, reference Cartesian system, internal coordinates, force fields and L matrices for methyl chloride are the same as given in section m.C. Bond displacement coordinates are defined in Fig. 4.6. Rotational corrections to the dipole moment derivatives with respect to symmetry coordinates are evaluated using the heavy isotope method [34]. The rotational correction terms are given in Table 4.8. To illustrate the calculations in more detail the entire V matrix of methyl chloride is presented in Table 4.9. To remove die rotational terms fi om the sets of linear equations for symmetiy... 

Bond displacement coordinates are defined by relations (4.96) ai (4.97). By e q)ressing the btMid coordinates as differences between the respective Cartesian displacements of the two atoms forming a txnid die translational motion is eliminated. Thus, there are no redundancies associated widi translational modem between die elements of bond charge tensors [129]. The elements of D matrix [Eq. (4.98)] are expressed in terms of a space-fixed Cartesian reference system. The elements of D matrix may contain considerable contribudons associated with the equilibrium dipole moment value. The resulting implicit redundancies are expressed by six reladtms [Eq. (4.118)]. 

In zero-order approximation of the bond polarizability model the following three types of electro-optical parameters representing derivatives of the bond polarizability with respect to bond displacement coordinates are defined [296,297]... 

In this equation ax(v) is the atomic polarizability tensor free from any rotational contribution. Its elements are, however, still interrelated through the dependency condition (9.84). The problem can be solved if a set of bond displacement coordinates [Eqs. (4.96) and (4.97)] instead of atomic displacement coordinates is used. A rotation-free bond polarizability tensor is defined as... 

The two most useful sets are the bond displacements themselves, and the symmetry coordinates. The use of the latter leads naturally to a scheme in which the Hamiltonian for bent molecules is no longer diagonal in the total 0(4) quantum numbers (ti, x2), and thus one loses the simple form of the secular equation (Figure 4.11). The secular equation must be now diagonalized in the full space with dimensions that become rapidly larger. This scheme, developed by Leviatan and Kirson (1988), can be implemented only if the vibron numbers N are relatively small, N < 10. 

The fact that coordination of the diolefin occurs by displacing the arene also favors the hypothesis that the diolefin coordinates in the cis conformation by the two double bonds. Displacement reactions of this type are well known in organometallic chemistry (See e.g. (2)) and, in the cases already clarified, it has been shown that diolefin coordinates in the cis conformation by the two double bonds. This is not surprising if we consider that the diolefin, coordinated in this way to a transition metal, has a quasi-aromatic character owing to the complete delocalization of the 7r-electrons. 

Initially, the 1 1 adduct between 1 and LiC6Hs is formed, and this reacts with more LiC6H5 to give 8. The CDT in 8, of which only one double bond is coordinated to nickel, is then displaced by N2 in the presence of LiC H5. Each N2 molecule bonds two nickel atoms, forming [(LiQHjlaNi Nj units. A dimer of two such units is observed in the X-ray structure of the end product (14). 

Franck-Condon factor with an exponential factor with the form of the WKB approximation for timneling through intersecting parabolas. This is intrinsically a multidimensional problem, and a generalized bond-length displacement coordinate was introduced to make it tractable. In the process of reducing the coordinates, a measure of the relative munber of identical oscillators (same p and to) was included in the exponential factor, rj, and the timneling rate was expressed as (50)... 

Six of these normal coordinates (five for a linear molecule) have a frequency eigenvalue identically equal to zero. These motions are translations and rotations of the molecule. Although the approach through Cartesian displacement coordinates is theoretically elegant, it is generally more practical to express the vibrational motions in terms of internal coordinates, such as bond stretches and distortions of bond angles. The method is discussed in detail in Chapter 4 of Wilson, Decius and Cross [57]. Since the distortions of the molecule can be described in terms of 3A — 6 of these internal coordinates there are no redundant dimensions to be removed when the analysis is complete. 

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