Symmetry coordinates parameters

Atom multiplicity (Wyckoff pos.) site symmetry coordinates parameters... 

Atom Position Point symmetry Coordinates Parameters... 

For f systems, parameterization in the CFT scheme using the expansions and coefficients of equations (11) or (12) follows on directly from the application to dn systems. The number of parameters rises, and even in cubic symmetry two parameters are necessary to describe the splittings of the/orbitals. These parameters involve respectively 7f and Tf terms. Using the same coordinate system and defining quantities equivalent to Dq for af-orbitals... 

Before we apply the formalism developed in Section 3 to the vibration—inversion-rotation spectra of ammonia, we shall discuss in this section certain group theoretical problems concerning the classification of the states of ammonia, the construction of the symmetry coordinates, the symmetry properties of the molecular parameters, and the GF matrix problem for the ammonia molecule. 

The vibrations may be described by different sets of basis coordinates. To start with, there are the changes of the 3n Cartesian coordinates X of the molecule. Chemists favor descriptions of the motions and the force constants in terms of bond lengths and bond angles. These are known as internal coordinates R. The equivalent internal coordinates of a molecule which possesses a certain symmetry, may change either in-phase or out-of-phase. The simultaneously occurring relative changes of the bond parameters of equivalent bonds are described by symmetry coordinates S. Normal coordinates Q describe motions as linear combinations of any set of basis coordinates. Different coordinate. systems can be transformed into each other by matrix multiplication. For further details, see Sec. 5.2. 

In the space group I4mm the origin of coordinates along the Z-axis is not fixed by symmetry. Therefore, the z-coordinate of one atom in the unit cell must be excluded from the least squares at all times to avoid severe correlation problems. We choose the z-coordinate of Ce atom (z = 0.000) as the fixed coordinate parameter. 

The various stabilities of these complexes were later rationalized by analysis of differences in the Ln 4f"6s 4P 5d 6s promotion energies, Ln—bond enthalpies and Ln ionic radii [43]. Compounds 21a-l were characterized by various spectroscopic and magnetometric methods but solid state structures were only obtained for 21d and 21g, revealing eclipsed arene rings and staggered tert-butyl groups, and approximate symmetry. Metrical parameters confirmed that the two coordinated q -arenes are neutral donors, with planar, fully delocalized arene rings [e.g., 21d C -C 1.400(6)-1.426(5) A Gd-C 2.630(4) A... 

A crystal structure has been determined for the Fe(III) (benzohydro-xamate)3 complex by Lindner and Gottlicher (41) and shows the 3-fold propeller coordination of the ferric ion. This may be the case for the solid but one wonders if this coordination persists in solution or frozen solution since the spin Hamiltonian parameters from Mossbauer experiments seem to indicate rhombic symmetry. These parameters are listed in Table 4. If in fact there is a 3-fold symmetry in the molecular arrangement, then one must question the relationship between the symmetry of the spin Hamiltonian and the symmetry of the local crystal or ligand field (52). 

Hitchcock has described a bis(NiOEP) derivative where the two porphyrin rings are joined by a me50,meio -ethylene bridge. The notable point is a lack of interaction between the two rings. The two rings are required to be parallel to each other by a center of symmetry located at the center of the ethylene bridge. Metal coordination parameters are normal. 

If an optimization is required to retain the symmetry of a structure, then symmetry-equivalent parameters must remain equal. Thus only displacements belonging to the totally symmetric irreducible representation are permitted other displacements are forbidden since they would change the symmetry. If it can be determined (e.g. from the Stanton-Mciver symmetry rules) that the transition vector does not belong to the totally symmetric representation, then the position of the transition structure along the reaction path is fixed by symmetry, and only the coordinates perpendicular to the reaction path need to be optimized. Since a transition structure must be a minimum with respect to all displacements other than along the reaction path, optimization of the transition structures is reduced to a simple minimization. 

The existence of aryloxide oxygen to metal r-bonding should be manifested in a variety of ways. The extent of jr-donation would depend on a large number of interrelated factors (formal metal oxidation state, molecular symmetry, coordination number, nature of ancillary ligands, etc.), which ultimately control the electron deficiency of the metal centre and the availability of suitable empty r-acceptor orbitals. How some of these factors influence various parameters is discussed below. 

By die transformation (3.3) a substantial step is made in die transition from experimental intensities into quantities characterizing molecular structure. At die first place, a natural separation between dipole derivatives associated with bond stretchings and angle deformations is achieved. In some cases the dp/dSj derivatives can be associated with vibrations localized within certain atomic groupings. Such distortions may be described by local group symmetry coordinates. Snyder [27] first applied dipole moment derivatives with respect to group symmetry coordinates as basic parameters in infrared intensity analysis on a series of crystalline n-alkanes. The procedure described in his work will be discussed later in this section. 

As already discussed, an important step towards reducing the experimental absorption intensities into molecular intensity parameters is die evaluation of the matrix Ps [Eq (3.3)]. Dipole moment derivatives with respect to symmetry coordinates nuty, however, contain contributions fiom molecular rotation for certain vibrations. These contributicms can be eliminated by using the following relation... 

Structural parameters, symmetry coordinates and rotational correction terms to dipole moment derivatives for H2CO ... 

Vibrations belonging to Bi and B2 have to be treated for contributions from compensatory molecular rotation. The geometric parameters and the definition of symmetry coordinates are given in Table 3.4. Cartesian reference system and definition of internal coordinates are shown in Fig. 3.3(A). 

Structural parameters, definition of symmetry coordinates and rotational correction terms for 1,1,1-trifluoroethane... 

If dipole moment derivatives with respect to symmetry coordinates are used as initial parameters, udiich in most cases is qrpropriate, the following matrix equation is obtained... 

The experimental gas-phase infiared intensities used are diose determined by Kondo et al. [101], The Lg matrix is obtained with the force field of Duncan et al. [24], The definitions of internal coordinates and of the Cartesian reference system are given in Fig. 3.7. Geometric parameters and symmetry coordinates are shown in Table 3.8. The Lg matrix for CH3CI is as follows ... 

The solution of the inverse intensity problem, which in this formulation implies die evaluation of bond polar parameters from experimental dipole moment derivatives with respect to normal coordinates, can only be performed fm a molecule possessing higfrer symmetry. The situation in this respect is the same as in the alternative theoretical formulations. The requirement is that the direction of the vibrational transition dipole is fixed by symmetry. In odier words, there should be only one non>zero element in each colunm of the Pq matrix [Eq. (3.1)]. Again, all calculations are considerably simplified if the dp/dQi derivatives are first transformed into dipole moment derivatives with respect to internal symmetry coordinates. The determination of the elements of P(, can then be realized using die following general expression, in matrix notation... 

Gussoni and Abbate [151] have derived a general formula relating atomic polar tensors and electro-optical parameters. Eq. (3.27) is an expression of die dipole moment derivatives with respect to symmetry coordinates in terms of electro-optical parameters. 

The heavy-isotope approach to evaluate rotational contributions to polarizability derivatives [288] will be illustrated with calculations on a series of molecules consisting of acetonitrile (C3V synunetiy), dichloromethane (C2v symmetry) and acetone (C2v symmetry). Structural parameters and polarizability tensors employed in die calculations are surtunarized in Table 9.1. Since the axes of the Cartesian reference systems (Fig. 9.1) are chosen to coincide with the respective inertial axes, the static polarizability tensors acquire sirtqile diagonal form. The symmetry coordinates corresponding to vibrations which may crmtain contributions from compensatory molecular rotation for the three molecules are given in Tables 9.2, 9.3 and 9.4, respectively. The following heavy isotopes are employed ... 

In this section the general equations (9.33) and (9.34) of VOTR are applied in interpreting Raman intensities of SO2. As was pointed out in Section 8.11, the Raman intensity experiment for bent XY2 molecules is not favorable in deteimining a complete set of molecular polarizability derivatives with respect to normal coordinates. That is why, in order to overcome the indeterminacy problem, a set of polarizability derivatives, with respect to symmetry coordinates for SO2 evaluated by means of ab initio MO calculations, is used [301]. The do/dSj derivatives forming the ag matrix are computed by applying the numerical differentiation approach. Other entries needed in solving the problem are taken from the same source [301]. Structural parameters for the sulfur dioxide molecule are given in Table 9.5. The Cartesian reference system and definition of internal coordinates and unit bond vectors are shown in Fig. 9.2. The a tensor employed in the calculations is as follows (in units of or rad" ) [301],... 

Cartesian reference fiame, geometrical data, definition of internal and symmetry coordinates and a matrix for sulfiir dioxide were already given in Section 9.I1.B. The application of valence-optical theory of Raman intensities results in the set of electro-optical parameters given by Eq. (9.43). If these quantities are substituted inside the brace of Eqs. (9.33) and (9.34), the elements of ([a]) array are obtained... 

A thorough study on the interaction between xylene pairs will be performed, and thus it is essential to give a good definition of their relative position with respect to each other. The relative position and orientation of a xylene pair is characterized by attaching a Cartesian coordinate frame to each xylene (Fig. 2a) [30] denoted as (Oi, JCi,yi, Zi) and (O2, X2, >2, Z2) for the two xylenes, Xi and X2, respectively. The jc-axis always coincides with the twofold rotational symmetry axis of each xylene. Note that in the three-dimensional structure, the orientation of the methyl groups may prevent the existence of this C2 symmetry, but the symmetry axis is always present in the projection on paper (Fig. 2a). The sense of Z2 is chosen such that the angle between Zi and Z2 is less than 90°. In principle, the relative orientation of two molecules is completely characterized by six parameters. For this work, we only consider the four most important internal coordinate parameters to describe... 

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