Normal co-ordinates

Using assumed molecular models and force constants based on the force constants derived from the paraffin series, normal co-ordinate calculations for the simple alkylcarbonium ions were carried out. These calculations were made in order to predict the vibrational spectra. Comparison with the experimentally obtained infra-red spectra show that the main observed features can indeed be reasonably explained in terms of the modes calculated for the planar models of the ions and allowed an assignment of the fundamentals (Table 11). 

Chloro-, Bromo-, and lodo-complexes. The effect of pressure on the kinetics of the reaction between solid NaCl and ZrC or HfCl4 vapours has been investigated. The i.r. and Raman spectra of solid ZrCl4,2NOCl and HfCl4,2NOCl have shown that they consist of NO and octahedral hexachlorometallate ions. °° Normal-co-ordinate analyses have been reported for the [ZrX ] " and [HfXg] ions (X = Cl or Br). ° ... 

Raman and i.r. spectra of [MgOig] - (M = Nb or Ta) salts have been determined and normal-co-ordinate analyses suggest that the ratio of the force constants of the terminal, bridging, and central M—O bonds is close to 8 4 1. 

The AH and AS values for the dissociation W2Clio(g) — 2WCl5(g) have been estimated as 26.0 Jmol and 66.5 JmoP respectively. The results of a normal co-ordinate analysis of WCl suggest that the molecules do not have octahedral symmetry. ... 

I.r. and Raman spectral studies of the cage compound 0s404(C0)i2 have indicated a tetrahedral symmetry in solution. A partial normal-co-ordinate analysis gave approximate CO stretching force constants. An interesting... 

X -Ray powder pattern studies of the complexes [M(phen)2X2]" and [M(bipy)2YZ]"+ [M = Co, Rh, Ir, or Os X = Cl, H2O, or ox YZ = CI2 or (0H)(H20)] show that each metal in each series is isomorphous. A cis configuration is therefore assigned to all complexes. Luminescence quantum yields have been measured for a series of [IrCl2(N—N)2C1 complexes (N—N = phen or bipy, or diphenyl derivatives), permitting a quantitative estimate of the effect of ligand phenyl substituents.A normal-co-ordinate analysis (i.r. has been carried out for [Ir(NH3)5Cl]Cl2. ... 

Assign all bands and performed c a normal co-ordinate analysis. v(M—N) decreases in order Pt > Pd > Ni, suggesting same trend for strength of co-ordinate bonds. 

Performed a normal co-ordinate b analysis of vibrations of the PtNX2CX3 model. Calculated force constants. 

This equation can be solved several ways [40]. One method involves diagonalizing the Liouville matrix, iL + R+ K. The matrix, iL + R + K, is precisely the matrix that Binsch deals with [22, 35]. If U is the matrix with the eigenvectors as columns, and A is the diagonal matrix with the eigenvalues down the diagonal, then we can write (11) as (12). This is similar to other eigenvalue problems in quantum mechanics, such as the transformation to normal co-ordinates in vibrational spectroscopy. 

This book attempts to describe alternative approaches to ligand reactivity involving normal co-ordination complexes as opposed to organometallic compounds. In part, a justification for this view comes from a study of natural systems. With very few exceptions, organometallic compounds are not involved in biological systems it is equally true that numerous enzymes bind or require metal ions that are essential for their activity. If enzymes can utilise metal ions to perform complex and demanding organic chemical reactions in aqueous, aerobic conditions at ambient temperature and pressure, it would seem to be worthwhile to ask the question whether this is a better approach to catalysis. 

AsPh3, thiourea bipy. or phen) [Pd(MeCOCHCSMe)2] Ni. Zn. Cd substituents on L Normal-co-ordinate analysis 9... 

A new, more accurate electron diffraction study of gaseous mercuric chloride has been reported.159 The interatomic distances (Hg—Cl = 2.25 A, Cl—Cl = 4.48 A) are shorter than previously reported values by 0.02 to 0.09 A. A complete normal-co-ordinate analysis of bis(methylthio)mercury has also been reported.160 The Raman spectra of gaseous mercuric chloride, bromide, and iodide have been reported.161 The bond polarizability derivatives calculated from the data increase in the order Cl < Br < I, suggesting an increased degree of covalence in the mercury-halogen bond with increasing size of the halogen atom. 

The equilibrium and non-equilibrium characteristics of the macromolecular coil are calculated conveniently in terms of new co-ordinates, so-called normal co-ordinates, defined by... 

It can readily be seen that the determinant of the matrix given by (1.8) is zero, so that one of the eigenvalues, say Ao, is always zero. The normal co-ordinate corresponding to the zeroth eigenvalue... 

It is convenient to describe the behaviour of a macromolecule in a co-ordinate frame with the origin at the centre of the mass of the system. Thus p° = 0 and there are only N normal co-ordinates, numbered from 1 to N. 

In the case of an orthogonal transformation, the relationship between the normal co-ordinate corresponding to the zeroth eigenvalue and the position of the centre of mass of the chain is... 

The probability distribution function allows us readily to calculate equilibrium moments of the normal co-ordinates... 

In the normal co-ordinates (1.13), in the case of the orthogonal transformation, the mean square radius of gyration of the macromolecule (1.21) is expressed in equilibrium moments... 

When normal co-ordinates, defined by equations (1.13), are employed, it is possible to make use of the arbitrariness of the transform matrix to define matrix Q in such a way that matrix B in the right-hand side of equation (2.16) assumes a diagonal form after transformation. The problem of the simultaneous adjustment of the symmetrical matrices A and B to a diagonal form does have a solution. Since matrix A is defined non-negatively and B is defined positively, it is possible to find a transformation such that B is transformed into a unit matrix (with accuracy to constant multiplier), and A into a diagonal matrix. Therefore, one can write simultaneously the equations... 

In terms of the normal co-ordinates introduced by equation (1.13), the matrix of the internal friction can be written as follows... 

Now one can return to the equation (2.1) for the dynamics of the macromolecule in the flow of a viscous liquid. The dissipative forces acting on the particles of the chain have generally non-linear forms, but the assumptions, when these force can be written in linear approximation, were discussed in the previous sections, so that we are able to write, in terms of the normal co-ordinates introduced previously and by taking into account all the considerations described above, the dynamic equation... 

In this section we refer to the stochastic equation (2.29) to calculate the mode moments, that is, the averaged values of the products of the normal co-ordinates and their velocities. It is convenient in this section to omit the label of mode and to rewrite the dynamic equation for the relaxation mode in the form of two linear equations... 

The symmetrical numerical matrix Ga r represents the influence of motion of the particle 7 on the motion of the particle a. The only general requirement one ought to put on matrix G 7 in the last relation is the following in normal co-ordinates it has a zero eigenvalue. The simplest forms, satisfying the requirement (3.5), can be written as... 

One has to refer to dynamic equations (3.11) of the macromolecule to find independent modes of motion. The matrices A and G in these equations are defined by equations (1.8) and (3.10), and, in the normal co-ordinates (1.13), simultaneously have diagonal forms... 

The expression for the velocity of the normal co-ordinate follows from equation (4.3). Differentiating (4.3) with respect to time, and integrating by parts, we find, by using the above shown properties of the integrands, that... 

Iteration of (4.3) and (4.5) can be used to expand the normal co-ordinates and their velocities into a power series of small velocity gradients of the medium. We can write down the zero-order approximation... 

The expansion of normal co-ordinates and their velocities (4.7) allows us to calculate various moments of co-ordinates and velocities, which are needed to determine physical quantities, first of all of second-order moments. For simplicity, we shall omit the label of the normal co-ordinates at calculation of the moments. 

We use relation (4.4) to write an expression for the moment of the normal co-ordinate... 

---