Symmetry-adapted coordinate

More generally, it is possible to combine sets of Cartesian displacement coordinates qk into so-called symmetry adapted coordinates Qrj, where the index F labels the irreducible representation and j labels the particular combination of that symmetry. These symmetry adapted coordinates can be formed by applying the point group projection operators to the individual Cartesian displacement coordinates. 

Distances and angles. Structures may be presented in an internal coordinate system (symmetry-adapted coordinates used in spectroscopy or Z-matrices - i.e., inter-... 

In some highly symmetrical molecules, a symmetry-adapted coordinate as a linear combination of internal coordinates can be identical with a normal coordinate. This is true for the following totally symmetric vibrations ... 

Force constant calculations are facilitated by applying symmetry concepts. Group theory is used to find the appropriate linear combination of internal coordinates to symmetry-adapted coordinates (symmetry coordinates). Based on these coordinates, the G matrix and the F matrix are factorized, which makes it possible to carry out separate calculations for each irreducible representation (c.f. Secs. 2.133 and 5.2). The main problem in calculating force constants is the choice of the potential function. Up until now, it has not been possible to apply a potential function in which the number of force constants corresponds to the number of frequencies. The number of remaining constants is only identical with the number of internal coordinates (simple valence force field SVFF) if the interaction force constants are neglected. If this force field is applied to symmetric molecules, there are often more frequencies than force constants. However, the values are not the same in different irreducible representations, a fact which demonstrates the deficiencies of this force field (Becher, 1968). 

Internal coordinates Distances and angles. Structures can be presented in an internal coordinate system (symmetry adapted coordinates used in spectroscopy or Z-matrices, that is, interatomic distances, three center angles, and four center angles) instead of a global coordinate system (coordinate triples, e.g., Cartesian, crystal, cylindrical, or spherical coordinates). 

We can now determine the symmetry-adapted coordinates by applying the projection operators, but the results can be written down almost immediately by again using the criterion of overlap with central symmetry functions. The A g, T u, and Eg -I- l2g SALCs reflect the nodal patterns of central p, and d functions, respectively. The T g mode corresponds to the rotation and evidently consists of tangential displacements of ligands in the equator perpendicular to the rotation axis. Finally, the T2u is a buckling mode, which has the symmetry of central f orbitals, viz. 

A more elegant analysis of the dynamics of coupled pairs is based on the crystal structure in which the face-to-face methyl groups are indistinguishable and form centrosymmetric pairs. Consequently, the dynamics are better represented with symmetry-adapted coordinates corresponding to in-phase 0 = (0i - - 02)/2 and antiphase 0a = (01 — 02)/2 rotation. In contrast, again, to the harmonic case, the choice of the symmetry-adapted coordinates is much more constrained. It is imposed by the periodicity of the rotors. The coordinates must retain the threefold and sixfold periodicity for in-phase and anti-phase rotation. The Hamiltonian can be thus rewritten as ... 

An important consequence of the symmetry-adapted coordinates is that the rotational constant is divided by a factor 2 compared to the single rotor. The maximum frequency for free rotation is now in the range 325-350 xeV for a pair of CH3 and 0.162-0.175 xeV for a pair of CD3. Therefore, transitions observed around 270 ijreV may correspond to pairs rotating almost freely, in accordance with measurements of the spin-lattice relaxation [83]. 

Another frequently used approach is to preprocess the set of input coordinates by a transformation to so-called symmetry functions or symmetry-adapted coordinates. " An illustrative example is the study of the three-body interaction energies in the H20-A1 -H20 complex. If the interatomic distances would be used directly as input for the NN, the result would depend on the order of the distances in the NN input. Including the permutation symmetry of equivalent atoms explicitly has several advantages The training data set is kept as small as possible and reference calculations need to be done only for symmetry-unique structures. Further, if the symmetry functions are chosen properly, the symmetry is included exactly. Finally, the number of NN weight parameters can be reduced, because the complexity of the PES is reduced. 

To construct symmetry-adapted coordinates we have to symmetrize all types of intermolecular distances r in the system. For simplicity, the H2O... 

To illustrate, again consider the H2O molecule in the coordinate system described above. The 3N = 9 mass weighted Cartesian displacement coordinates (Xl, Yl, Zl, Xq, Yq, Zq, Xr, Yr, Zr) can be symmetry adapted by applying the following four projection operators ... 

Symmetry-adapted combinations 6 Symmetry coordinates 13 Synthons 767, 788... 

In the symmetry-adapted formulation, the 43- term no longer occurs because the d-orbital density contains a vertical mirror plane even if such a plane is absent in the point group. This is illustrated as follows. Point groups without vertical mirror planes differ from those with vertical mirror planes by the occurrence of both dlm+ and d(m functions, with m being restricted to n, the order of the rotation axis. But the coordinate system can be rotated around the main symmetry axis such that P4 becomes zero. As proof, we write the (p dependence as... 

We have now developed the analogy between the decomposition of a vector into components along coordinate axes, and the decomposition of a set of objects into symmetry-adapted combinations. 

Just as group theory enables one to find symmetry-adapted orbitals, which simplify the solution of the MO secular equation, group theory enables one to find symmetry-adapted displacement coordinates, which simplify the solution of the vibrational secular equation. We first show that the matrices describing the transformation properties of any set of degenerate normal coordinates form an irreducible representation of the molecular point group. The proof is based on the potential-energy expression for vibration, (6.23) and (6.33) ... 

If the TV-particle basis were a complete set of JV-electron functions, the use of the variational approach would introduce no error, because the true wave function could be expanded exactly in such a basis. However, such a basis would be of infinite dimension, creating practical difficulties. In practice, therefore, we must work with incomplete IV-particle basis sets. This is one of our major practical approximations. In addition, we have not addressed the question of how to construct the W-particle basis. There are no doubt many physically motivated possibilities, including functions that explicitly involve the interelectronic coordinates. However, any useful choice of function must allow for practical evaluation of the JV-electron integrals of Eq. 1.7 (and Eq. 1.8 if the functions are nonorthogonal). This rules out many of the physically motivated choices that are known, as well as many other possibilities. Almost universally, the iV-particle basis functions are taken as linear combinations of products of one-electron functions — orbitals. Such linear combinations are usually antisymmetrized to account for the permutational symmetry of the wave function, and may be spin- and symmetry-adapted, as discussed elsewhere ... 

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