Symmetry related internal coordinates

If the molecules has some symmetry the solution of the secular determinant can be simplified by the use of symmetry coordinates. Symmetry coordinates are linear combinations of the internal coordinates. After a little experience one can frequently select symmetry coordinates intuitively by taking combinations of symmetry related internal coordinates. The symmetry coordinates picked out are not necessarily unique but must have definite properties. They must be normal, orthogonal, and they must transform properly as will be described below. A symmetry coordinate has the form... 

The symmetry and internal coordinates Si and Rj, respectively, are related by the matrix equations S = VR and R = U S. It is further assumed for this derivation that the CO-stretching vibrations are factored out of the secular equation governing all the vibrations of the molecule. In this case, the matrix U is orthogonal, so that... 

In the case of H2O, Bis a 3 x 9matrix, whereas U,r, and Fr are 3 x 3 matrices. To take advantage of symmetry properties, internal coordinates are further transformed to internal symmetry coordinates (R ) by the relation... 

Qualitative ways of analyzing a problem in molecular vibrations, that is, methods for determining the number of normal modes of each symmetry type which will arise in the molecule as a whole and in each set of equivalent internal coordinates, have been developed. There is also the quantitative problem of how the frequencies of these vibrations, which can be obtained by experiment, are related to the masses of the atoms, the bond angles and bond lengths, and most particularly the force constants of the individual bonds and interbond angles. In this section we shall show how to set up the equations which express these relationships, making maximum use of symmetry to simplify the task at every stage. 

The nature of these six vibrations may be further specified in terms of the contribution made to each of them by the various internal coordinates. We first note that Ag and Bu vibrations must involve only motions within the molecular plane, since the characters of the representations Ag and Bu with respect to ah are positive. The Au vibration will, however, involve out-of-plane deformation, since the character of Au with respect to oh is negative. Thus we may describe the normal mode of Au symmetry as the out-of-plane deformation. In order to treat the remaining five in-plane vibrations we need a set of five internal coordinates so chosen that changes in them may occur entirely within the molecular plane. A suitable set, related to the bonding in... 

Again this relation follows from the symmetries (ii) and (iii) it expresses that the electronic energy function assumes the same value for all isometric NCs. Equations (3.5) and (3.9) show that e°(Xk( )) is symmetric w.r.t. to the full isometric group. Whereas the symmetry of e°(Xk( )) w.r.t. ( ) merely expresses that e°(Xk( )) is a function of the internal coordinates only, its symmetry w.r.t. Jr( ) is a genuine symmetry. 

Fig. 23. Structures of the tetrasubstituted TEEs. Below the molecules the symmetry is given in international and Schonflies notation. In the upper right corner the different conjugation paths and the coordinate system are depicted as used in the derivation of the symmetry relations...

The symmetry coordinates show themselves to be particularly useful for the functional representation of the molecular potential. For example, the potential function of a X3-type molecule must be invariant with respect to the interchange of any internal coordinate ft, (/ = 1, 2, 3) hence it must be totally symmetric in relation to those coordinates. Thus, in terms of the coordinates Qi (/ =1,2, 3), such a function can only be written in terms of or totally symmetric combinations of Q2 and Q3. Such combinations may in fact be obtained by using the projection-operator technique.16"27 In fact, one can demonstrate16 27 that any totally symmetric function of three variables is representable in terms of the integrity basis,28... 

Other than in the preceding section, the matrix D must now relate the increments of symmetrically equivalent internal (or bond) coordinates with the increment of the respective base internal coordinate. Let us return to the example, Eq. 56, and let the matrix D, as depicted there, be a portion of the whole matrix D. The row and column headings of each 3x3 block of D for a particular atom now specify the increments of bond length Ar, bond angle Atp, and dihedral (or torsional) angle AO used for the attachment of this atom to the chain, instead of the increments of Cartesian coordinates of the atom. It is assumed that the description by internal coordinates has been chosen for the best possible preservation of molecular symmetry. The three symmetrically equivalent atoms a, a, and a" will then be attached to an interatomic bond along thejr-axis (or, by convention, to an equivalent... 

The eigenvectors of polypeptide chain modes, as in the case of NMA, can be described by PEDs in terms of symmetry coordinates, which in turn are related to internal coordinates. A list of the internal coordinates for (Ala) is given in Table IV, and the local symmetry coordinates are given in Table V (Moore and Krimm, 1976b). These serve as the general local symmetry coordinates for most polypeptide chain structures [for the particular set for (Gly) I, see Dwivedi and Krimm (1982a)]. 

Figure 3 Left Calculated structure of [d(G4T4G4) ]2 dh internal NH4 ions based on NMR-derived distance restraints For clarity, the dT residues of the loops are drawn with thinner lines than the dG residues. Right Topology diagram of [d(G4T4G4)]2 indicating the two distinct coordination sites, the symmetry-related outer sites and the unique inner site that lies on the axis of rotational symmetry...

The symmetry coordinate iR (y4,) in Eq. 12.21 represents a redundant coordinate (see Eq. 8.1). In such a case, a coordinate transformation reduce.s the order of the matrix by one, since all the G matrix elements related to this coordinate become zero. Conversely, this result provides a general method of finding redundant coordinates. Suppose that the elements of the G matrix are calculated in terms of internal coordinates such as those in Table 1-10. If a suitable combination of internal coordinates is made so that Gy =0 (where j refers to ail the equivalent internal coordinates), such a combination is a redundant coordinate. By using the U matrices in Eqs. 12.20 and 12.21, the problem of solving a tenth-order secular equation for a tetrahedral XY4 molecule is reduced to that of solving two first-order (y4, and E) and one quadratic (F2) equation. 

This number applies only to unsymmetrical (Ci) molecules, how ever, since the existence of symmetry implies further relations between constants and hence a still smaller number of independent parameters. As an illustration, consider the water molecule (62,.). A convenient set of internal coordinates is formed from ri and r, the stretches of the two OH bonds, and a, the distortion of the valence angle. The potential energy can be w ritten as... 

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... 

Translational and unit cell symmetries greatly simplify the vibrational spectra of infinite polymers. In the case of /-PA, whose Cyi unit cell is shown in Fig. 6.3, we have just six in-plane, optically active q = 0 modes. The four g Raman-active modes are even under inversion at bond centers, whereas the two IR-active hu modes are odd. Only C—C stretches couple to it electrons in Hiickel theory, and they remain the strongly coupled modes of PPP models. Since C—C stretches do not appear in the subspace, the IR modes are not coupled efficiently to tt electrons. Formally, the symmetry of the polymer is exploited by defining a set of symmetry coordinates S related to the internal coordinates by the... 

Thus, a rule of thumb says that valid normal coordinates must be either entirely symmetric or entirely antisymmetric over the most common symmetry operations if m and n label any two internal coordinates related by symmetry, then either Ukm = cikn or akm = —akn- This rule of thumb holds also in the construction of the coefficients of atomic orbitals in molecular orbitals (see Section 3.5). The bending mode of water straddles the mirror plane and is already symmetry-adapted if the displacements of the two hydrogen atoms are equal and in opposite directions (if the motions were in the same direction, that would be a molecular rotation, not a vibration). 

Considerable simplifications in describing molecular vibrations are attained by introducing symmetry coordinates. These are related to the ordinary internal coordinates by the transformation, in matrix notation [4]... 

The geometric parameters of H2O are roH = 0.9572 A, ZHOH = 104.5° [99]. The reference Cartesian system and internal coordinates are defined in Fig. 3.6. The symmetry coordimtes have their usual form. The non-zero elements of P obtained are given in Table 3.7. From these the application of Eq. (3.24) produces the following equations relating /dSj derivatives with eop s for the two symmetry classes [100]... 

To describe the contents of a unit cell, it is sufficient to specify the coordinates of only one atom in each equivalent set of atoms, since the other atomic positions in the set are readily deduced from space group symmetry. The collection of symmetry-independent atoms in the unit cell is called the asymmetric unit of the crystal structure. In the International Tables, a portion of the unit cell (and hence its contents) is designated as the asymmetric unit. For instance, in space group P2 /c, a quarter of the unit cell within the boundaries 0asymmetric unit. Note that the asymmetric unit may be chosen in different ways in practice, it is preferable to choose independent atoms that are connected to form a complete molecule or a molecular fragment. It is also advisable, whenever possible, to take atoms whose fractional coordinates are positive and lie within or close to the octant 0 < x < 1/2,0 < y < 1/2, and 0 < z < 1 /2. Note also that if a molecule constitutes the asymmetric unit, its component atoms may be related by non-crystallographic symmetry. In other words, the symmetry of the site at which the molecule is located may be a subgroup of the idealized molecular point group. 

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