Atomic Cartesian displacement coordinates

The Eckart Sayvetz conditions imply that, if during the vibration a small translation of the center of masses is invoked, the origin of the Cartesian reference system is displaced so that no linear momentum is produced. The second Sayvetz condition, expressed in the last diree equations of (2.8), imposes the constraint that, during vibrational displacements, no angular momentum is produced. Eq. (2.8) implies that the reference Cartesian system translates and rotates with the molecule in such a way that the displacement coordinates Ax, Ay and Az reflect pure vibrational distortions. It is evident that through Eq. (2.8) certain mass-dependency is imposed on the atomic Cartesian displacement coordinates. 

The 3N atomic Cartesian displacement coordinates describe not only vibrational motion but die translation and rotation of die molecule in space as well. Therefore, Qt in expression (4.5) include also the six rototranslational normal coordinates. Thus, (dp/dQth divided into two parts. The derivatives of p with respect to normal... 

An alternative roach for analysis of vibrational intensities has been put forward by Migrants and Averbukh [116,129,130]. An extensive review on die mediod has been published by Ruppredit [37], Hereafter, we shall follow with few exceptions the notation used by Riqiprecht which is closer to die notation used so far. Instead, on die basis of atomic Cartesian displacement coordinates, the dp/dQi quantities are transformed into the coordinate space of bond displacement vectors. The change of dipole moment is defined as... 

Eq. (9.91) reveals the possibility of expressing molecular polarizability derivatives with respect to atomic Cartesian displacement coordinates in teims of electiYK>ptica] parameters. The validity of this relation will be checked in the case of SO2 molecule. 

The effect of the symmetry operations on the Cartesian displacement coordinates of the two hydrogen atoms in die water molecule. The sharp ( ) indicates the inversion of a coordinate axis, resulting in a change in handedness of the Cartesian coordinate system. 

Take an N-atomic molecule with the nuclei each at their equilibrium internuclear position. Establish a Cartesian x, y, z coordinate system for each of the nuclei such that, for Xj with i = 1,.., 3N, xi is the Cartesian x displacement of nucleus 1, x2 is the Cartesian y displacement coordinate for nucleus 1, X3 is the Cartesian z displacement coordinate for nucleus 1,..., x3N is the Cartesian z displacement for nucleus N. Use of one or another quantum chemistry program yields a set of force constants I ij in Cartesian displacement coordinates... 

Symmetry Types of the Normal Modes. For this nonlinear four-atomic molecule there are 3(4) -6 = 6 genuine internal vibrations. Using a set of three Cartesian displacement coordinates on each atom, we obtain the following representation of the group C3l, ... 

One may also wish to impose an additional requirement on the connection, namely that it is translationally and rotationally invariant. This may seem to be a trivial requirement. However, a connection is conveniently defined in terms of atomic Cartesian displacements rather than in terms of a set of nonredundant internal coordinates. This implies that each molecular geometry may be described in an infinite number of translationally and rotationally equivalent ways. The corresponding connections may be different and therefore not translationally and rotationally invariant. In other words, the orbital basis is not necessarily uniquely determined by the internal coordinates when the connections are defined in terms of Cartesian coordinates. Conversely, a rotationally invariant connection picks up the same basis set regardless of how the rotation is carried out and so the basis is uniquely defined by the internal coordinates. [For a discussion of translationally and rotationally invariant connections, see Carlacci and Mclver (1986).]... 

Now, check the rules with a larger basis set, the Cartesian displacement coordinates of the atoms of HNNH (see Figure 4-8). Operation E leaves all the 12 vectors unchanged, so its character will be 12. C2 brings each atom into a different position so their vectors will also be shifted. This means that all vectors will have zero contribution to the character. The same applies to the inversion operation. Finally, as already worked out before, the horizontal reflection leaves all the x and y vectors unchanged and brings the four z vectors into their negative selves. The result is... 

Cuony and Hug70) and Hug38) have discussed sum rules for vibrational ROA in the limited context of molecules that are chiral due to isotopic substitution. By expressing the normal coordinates in (8aafj/0Qp)o etc. in terms of atomic cartesian displacement vectors, they were able to show that the optical activity in the fundamentals sums to zero if the parent achiral group is other than and Cj. Their proof requires the assumption that the isotopic substitution does not affect the equilibrium electronic distribution, in which case the corresponding Rayleigh optical activity is zero. 

Computational and interpretational aspects of vibrational spectroscopy are greatly simplified by the introduction of normal coordinates . First, mass-weighted Cartesian displacement coordinates for an N-atom molecule q, qi,. ..,qm are defined according to... 

It is of interest to note that the identical result (2.38) is obtained for the librations by a treatment using cartesian displacement coordinates for the atoms of a diatomic molecule (Suzuki, 1970). Although we do not face the problems of separability of coordinates in the kinetic energy in this case, the assiunption of harmonicity of the potential energy presumably causes the same inaccuracies. In other words, it is immaterial which coordinates are used but in terms of Eulerian angle displacement coordinates the harmonic approximation is not confined to the potential energy but also appears in the kinetic energy. 

As pointed out in Section IIB, it is possible to approach the lattice dynamics problem of a molecular crystal by choosing the cartesian displacement coordinates of the atoms as dynamical variables (Pawley, 1967). In this case, all vibrational degrees of freedom of the system are included, i.e., translational and librational lattice modes (external modes) as well as intramolecular vibrations perturbed by the solid (internal modes). It is then obviously necessary to include all intermolecular and intramolecular interactions in the potential function O. For the intramolecular part a force field derived from a molecular normal coordinate analysis is used. The force constants in such a case are calculated from the measured vibrational frequencies, The intermolecular part of O is usually expressed as a sum of terms, each representing the interaction between a pair of atoms on different molecules, as discussed in Section IIA. 

The potential energy is written in terms of internal coordinates or here explicitly atom-atom distances, and it takes the form of a sum of terms as in (2.2). All these distances (including those within the molecule and those between different molecules) form the components of a column vector R. The harmonic force constant matrix in terms of the corresponding displacements r is As already noted, however, it is advantageous to use cartesian displacement coordinates since the kinetic energy matrix G is then diagonal. It is therefore necessary to express also the potential 4> in terms of the cartesian displacements (vector x) and to write the force constant matrix accordingly ... 

To use this scheme we require then a linear transformation B between the internal displacement coordinates r (usually atom-atom distance changes in terms of which the potential is expanded to second order) and the cartesian displacement coordinates of the atoms x. However, the relation between these coordinates is not linear and therefore an approximation must be made at this point. Shimanouchi, Tsuboi, and Miyazawa (1961) expand the change in distance between atoms a and b, Tab, from its equilibrium value i ab to first order in the cartesian displacement of the atoms as given in (2.44) ... 

Changes in interatomic distances or in the angles between chemical bonds, or both, can be used to provide a set of 3A — 6 (or 37V — 5 for linear molecules) internal coordinates (Sec. 2-6), i.e., coordinates which are unaffected by translations or rotations of the molecule as a whole. These are particularly important because they provide the most physically significant set for use in describing the potential emwgy of the molecule. The kinetic energy, on the other hand, is more easily set up in terms of cartesian displacement coordinates of the atoms (Sec. 2-6). A relation between the two types is therefore needed. 

Here the kinetic energy expressed in terms of the momenta, Pt, conjugate to the coordinates Sa, contains no cross terms because the transformation from cartesian to external symmetry coordinates is orthogonal. Moreover, since each external symmetry coordinate is a linear combination of the cartesian displacement coordinates of a single equivalent set of atoms, the coefficient Gaa = ma- will be the reciprocal mass of an atom of the set from which Sa is constructed. Since the product of the characteristic values... 

In the second-order methods we have described, the choice of coordinate system was not made explicit. Prom a quantum-chemical perspective, analytical derivatives are most conveniently computed in Cartesian (or symmetry-adapted Cartesian) coordinates. Indeed, second-order methods are not particularly sensitive to the choice of coordinate system and second-order implementations based on Cartesian coordinates usually perform quite well. As we discussed above, however, if the Hessian is to be estimated empirically, a representation in which the Hessian is diagonal, or close to diagonal, is highly desirable. This is certainly not true for Cartesian coordinates some set of internal coordinates that better resemble normal coordinates would be required. Two related choices are popular. The first choice is the internal coordinates suggested by Wilson, Decius and Cross [25], which comprise bond stretches, bond angle bends, motion of a bond relative to a plane defined by several atoms, and torsional (dihedral) motion of two planes, each defined by a triplet of atoms. Commonly, the molecular geometry is specified in Cartesian coordinates, and a linear transformation between Cartesian displacement coordinates and internal displacement coordinates is either supplied by the user or generated automatically. Less often, the (curvilinear) transformation from Cartesian coordinates to internals may be computed. The second choice is Z-matrix coordinates, popularized by a number of semiempirical... 

If we want to draw a diagram of the atomic displacements during the vibration we must evaluate the cartesian coordinate displacements per unitary change in the normal coordinates. The transformation from cartesian displacement coordinates to normal coordinates is given by... 

The kinetic energy T of a molecule can be expressed in cartesian displacement coordinates where i is a column matrix of the time derivatives thereof and M is a diagonal matrix with atomic masses as diagonal elements (zeros elsewhere). 

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