Atomic Cartesian displacement

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).]... 

This problem has already been solved by the early infrared spectroscopists who calculated the transformation between the atomic Cartesian displacement coordi-... 

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. 

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. 

AXq are atomic Cartesian displacements equilibrium atomic charges and... 

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. 

Wfi cri you perform a single poin t sem i-cmpirical or ah initio ealeu-laliori, you obtain th c en ei gy and tli e first dci ivalives of the eu ei gy with respect to Cartesian displacement of the atoms. Since the wave function for the molecule is computed in the process, there are a n urn ber of oth er molecti lar properties th at could be available to you. Molecularproperties arc basically an average over th e wave fun ction of certain operatorsdescribin g the property. For exam pie, the electron ic dipole operator is basical ly ju st the operator for the position of an electron and the electron ic con tribution to the dipole iTi otTi en t is... 

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... 

An approximate form of the NMO model, the atomic polar tensor (APT) model, has also proved effective (42). Rather than considering local contributions, this model considers contributions from the derivative of the total molecular dipole moment with respect to Cartesian displacement of a given nucleus, (3p/9R )o. The latter ate the elements of the atomic polar tensor for atom n. 

Stable adsorption complexes are characterized by local minima on the potential energy hypersurface. The reaction pathway between two stable minima is determined by computation of a transition state structure, a saddle point on the potential energy hypersurface, characterized by a single imaginary vibrational mode. The Cartesian displacements of atoms that participate in this vibration characterize movements of these atoms along the reaction coordinate between sorption complexes. 

We shall now illustrate this, using CO5" as an example. A number of further examples will be found in Section 10.7. The first step must be to determine the symmetry group to which the molecule belongs, as described in Chapter 3, especially Section 3.14. We find that CO5" belongs to the DVt group. Figure 10.3 shows the CO3 ion with the sets of Cartesian displacement vectors attached to.each atom. There are of course 3n - 12 in all, and the representation will therefore be of dimension 12. 

The details of the F-G matrix procedure are best explained by working through a simple example, such as the water molecule. This belongs to the point group C2l.. The nine Cartesian displacement vectors, three on each atom, give rise to the representation... 

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, ... 

The determination of the eigenvalues wy(q)2 may be simplified by an orthogonal transformation to symmetry coordinates, which are linear combinations of the Cartesian displacements of the atoms which represent the actual displacements of the atoms in the unit cell. Simultaneously, the eigenvectors undergo the same orthogonal transformation (see Section 9.4, especially eqs. (9.4.4) and (9.4.6)). In matrix notation,... 

The advantage of the effective Hamiltonian (7), beside the reduced space of nuclear coordinates, is that it involves effective force constants (elements of the matrix K) which rapidly decrease with the distance between centres. This allows for efficient models involving only few force constants. Furthermore, the force constants can be calculated. For this purpose the local nuclear coordinates, at the centre n, are expanded in Cartesian displacements, uKa n ), of surrounding atoms ... 

In VFF the molecular vibrations are considered in terms of internal coordinates qs (s = 1..3N — 6, where N is the number of atoms), which describe the deformation of the molecule with respect to its equilibrium geometry. The advantage of using internal coordinates instead of Cartesian displacements is that the translational and rotational motions of the molecule are excluded explicitly from the very beginning of the vibrational analysis. The set of internal coordinates q = qs is related to the set of Cartesian atomic displacements x = Wi by means of the Wilson s B-matrix [1] q = Bx. In the harmonic approximation the B-matrix depends only on the equilibrium geometry of the molecule. 

---