Normal coordinate transformation matrix

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... 

The evaluation of the first and second derivatives of molecular energy with respect to an tqtpropriate set of coordinates defining die positions of nuclei is reqtiired for calculations of frequencies and intensities in vibrational spectra. The second derivatives of energy, the force constants, are used in determining the frequencies and tiie normal coordinates. The normal coordinate transformation matrix is applied together with theoretical estimates of the dipole moment derivatives in evaluating vibrational absorption intensities. 

The transfrumation of vibrational intensities in Raman spectra into molecular parameto s involves sevoal calculation stages. An essential initial step is the reduction of intensity data to polarizability derivatives with respect to symmetiy vibrational coordinates. As pointed out in previous ciutyters, the inverse electro-optical problem of vibrational intensities can be performed with success only for molecules possessing sufficient symmetry. The transformation between da/dQ and do/dSj derivatives is carried out widi die aid of the normal coordinate transformation matrix Lg according to the expression ... 

The identification of the IRs according to which the normal coordinates transform can greatly reduce the computational labor associated with implementing the FG matrix method. It is frequently easy to set up symmetry coordinates, as linear combinations of internal coordinates, which transform according to IRs of the point group G. For CIF3, one choice of symmetry coordinates would be... 

The normal coordinate transformation requires diagonahzation of the matrix A, appearing in Eq. (4.46), in such a way that the transformed matrix will have only the diagonal components. The method of obtaining such a matrix is well documented in many standard textbooks. An excellent exposition of this method can be found in the textbook by Hildebrand (1952). 

Qi are normal comdinates, Rj internal coordinates, and Ljj elements of the normal coordirurte transformation matrix. From expression (3.22) we can see diat electro-optical parameters are die following quantities magnitudes of die bond moments, and... 

In the last equation M is die number of internal coordinates R], and is an elemoit of die normal comdinate transformation matrix L. The off-diagonal term can be rearranged in die following manner ... 

It should be noted that the force constant matrix can be calculated at any geometry, but the transformation to nonnal coordinates is only valid at a stationary point, i.e. where the first derivative is zero. At a non-stationary geometry, a set of 3A—7 generalized frequencies may be defined by removing the gradient direction from the force constant matrix (for example by projection techniques, eq. (13.17)) before transformation to normal coordinates. 

Hence, as is often stated, the determination of the normal coordinates is equivalent to the successful search for a matrix L that diagonalizes the product GF via a similarity transformation. This system of linear, simultaneous homogeneous equations can be written in the form... 

A complementary approach to the parabolic barrier problem is obtained by considering the Hamiltonian equivalent representation of the GLE. If the potential is parabolic, then the Hamiltonian may be diagonalized" using a normal mode transformation. One rewrites the Hamiltonian using mass weighted coordinates q Vmd. An orthogonal transformation matrix... 

The Qx, QtN are called normal coordinates and what we have done is to transform the coordinates y, to another sot Qt such that eqn (9-2.10) is true. We can form the matrix C by using the coefficients for each X value as columns ... 

Because of the statement made in the beginning of this section as to the Gaussian nature of the subchains, the matrix for transformation into the normal coordinates is the same for the x- and the -directions. This means that matrix (-4ir) is transformed in the same way as matrix aik), i.e. by a congruence transformation [see eq. (3.12)]. One obtains instead of eq. (3.33) ... 

In this equation is the internal friction factor of thej-th normal mode and Qjj1 is the inverse transformation matrix of Zimm. In other words, Cerf assumed that one can ascribe a separate internal friction factor to every normal mode. This assumption is critisized by Budtov and Gotlib (183) as, in this way, the elements of the internal friction matrix in the laboratory coordinate system x, y, z, viz. 

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. 

The coefficients of the B matrix have been developed by Wilson and Califano (Wilson, 1955 Califano, 1976). The corresponding formulas are included in all available normal coordinate packages. In the system of internal coordinates the force constants are defined as the second derivatives of the potential energy with respect to two coordinates. In order to obtain the Fx matrix, the Fr matrix with internal coordinates has to be transformed according to... 

An efficient method to visualize the properties of the normal coordinates is to calculate activity measurements, AM. They show which components of the dipole moment vector and the polarizabilty tensor are modulated by the vibration, and the relative sign of the infrared and Raman optical activity (Schrader et al., 1984 Schrader, 1988). The necessary transformation of the eigenvectors (Eq. 5.2-13) needs only seconds of computer time. The AMs are useful to assign vibrations to symmetry species and to check the input of the frequency calculation the symmetry of the Cartesian coordinates of the atoms as well as of the force constant matrix. This program is part of the SPSIM program package (Fischer et al., 1989). 

In equation 27 is the distance vector of the equilibrium internuclear distance between the atoms k and i. S and S are the vector components associated with the matrix that transforms normal coordinate displacements to internal coordinate displacements (along and perpendicular to bond directions). The sum is over the atoms and transformation vectors associated with the yth vibrational normal coordinate. 

Matsen and Franklin make use of an analogy between the formulation of monomolecular systems and normal mode analysis of the vibrations of polyatomic molecules. The too strict use of this analogy leads them to make an assumption that is wrong for monomolecular systems in general. Normal mode analysis of vibration spectra treats symmetric matrices except in very special cases, the rate constant matrix for the reactions of the various species A, is asymmetric when the amounts are expressed in the A system of coordinates. The heart of their formulation is expressed by Eqs. (2), (3), and (4) of their paper, which will be designated (MF2), (MF3), and (MF4) when expressed in our notation. Matsen and Franklin begin by assuming that a transformation matrix X exists such that... 

If for a given molecule we consider sets of curvilinear and rectilinear coordinates that are interrelated in the sense discussed in Sect. 3.1.1, we will obtain identical leading terms in corresponding expansions, irrespective of the class of coordinates. Since the harmonic part of the potential function is also independent of whether we use curvi- or rectilinear coordinates6, we can use the same linear transformation to normal coordinates as well. In both cases the transformation matrix, L, is determined by the set of equations,... 

From the L-matrix and its inverse it is possible to calculate s- and /-vectors corresponding to normal coordinates, Qlt Q2, Qin-6- All previously derived formulae apply equally well to these transformed vectors. But, when normal coordinates are considered, the zeroth order vibrational G-matrix becomes a unit matrix and the vectors present some special properties,... 

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