Internal coordinate choice

The matrix M is an inverse generalised inertia tensor while G is is an inverse metric matrix. The operator A< is associated with the Coriolis coupling and so no coordinate system can be found in which it will vanish. The operator t, is dependent on internal coordinate choice and it is possible to choose a coordinate system in which this term vanishes. 

Figure 1 The course of energy minimization of a DNA duplex with different choices of coordinates. The rate of convergence is monitored by the decrease of the RMSD from the final local minimum structure, which was very similar in all three cases, with the number of gradient calls. The RMSD was normalized by its initial value. CC, IC, and SG stand for Cartesian coordinates, 3N internal coordinates, and standard geometry, respectively.

Another way of removing the six translational and rotational degrees of freedom is to use a set of internal coordinates. For a simple acyclic system these may be chosen as 3N — I distances, 3N — 2 angles and 3N -3 torsional angles, as illustrated in the construction of Z-matrices in Appendix E. In internal coordinates the six translational and rotational modes are automatically removed (since only 3N — 6 coordinates are defined), and the NR step can be formed straightforwardly. For cyclic systems a choice of 3A — 6 internal variables which span the whole optimization space may be somewhat more problematic, especially if symmetry is present. 

Vibrational Spectra Many of the papers quoted below deal with the determination of vibrational spectra. The method of choice is B3-LYP density functional theory. In most cases, MP2 vibrational spectra are less accurate. In order to allow for a comparison between computed frequencies within the harmonic approximation and anharmonic experimental fundamentals, calculated frequencies should be scaled by an empirical factor. This procedure accounts for systematic errors and improves the results considerably. The easiest procedure is to scale all frequencies by the same factor, e.g., 0.963 for B3-LYP/6-31G computed frequencies [95JPC3093]. A more sophisticated but still pragmatic approach is the SQM method [83JA7073], in which the underlying force constants (in internal coordinates) are scaled by different scaling factors. 

The state mixing term, the first in the r.h.s., usually dominates, at least in the presence of avoided crossings. Its determination reduces to a simple problem of interpolation of the Hu matrix elements, according to eq.(16). The second term corresponds, for large R, to the electron translation factor (see for instance [38]). This term depends on the choice of the reference frame that is, for baricentric frames, it depends on the isotopic masses. It contains the Gn matrix, which may be determined by numerical differentiation of the quasi-diabatic wavefunctions [16] this calculation is more demanding, especially in the case of many internal coordinates. It is therefore interesting to adopt the approximation ... 

From the above analysis, as shown in the last column of Table 2, it should be obvious that two linear combinations of the internal coordinates Ar can be formed whose symmetry corresponds to the IRs k and B2. They are Arj + Ar2(A]) and Ar — Ar2(B2), as can be verified by inspection of the character table. Tbe choice of the correct linear combinations is in this case simple. However, more generally, it can be made by application of the projection-operator method described in Section 8.11. 

The PES is only a function of the shape of the molecule, as described by the internal coordinates (there are ZN — 6 of them for a non-collinear molecule of N atoms). There are many possible choices of internal coordinates, including atom-atom distances (bond lengths), bond angles, dihedral or out-of-plane angles, or some combination of these. However, the PES can be expressed solely in terms of the atom-atom distances (a result that follows from the group theory of functions like the PES which are invariant to rotation).48,49 For N atoms there are NCi = N(N — l)/2 such distances, which are easily calculated from the Cartesian coordinates. Rather than use these atom-atom distances, Rn, we actually use the reciprocal distances, Zn ... 

For a quantitative description of molecular geometries (i.e. the fixing of the relative positions of the atomic nuclei) one usually has the choice between two possibilities Cartesian or internal coordinates. Within a force field, the potential energy depends on the internal coordinates in a relatively simple manner, whereas the relationship with the Cartesian nuclear coordinates is more complicated. However, in the calculations described here, Cartesian coordinates are always used, since they offer a number of computational advantages which will be commented on later (Sections 2.3. and 3.). In the following we only wish to say a few words about torsion angles, since it is these parameters that are most important for conformational analysis, a topic often forming the core of force field calculations. 

The NDF is very similar to the PDFs introduced in the previous section to describe turbulent reacting flows. However, the reader should not confuse them and must keep in mind that they are introduced for very different reasons. The NDF is in fact an extension of the finite-dimensional composition vector laminar flow where the PDFs are not needed, the NDF still introduces an extra dimension (1) to the problem description. The choice of the state variables in the CFD model used to solve the PBE will depend on how the internal coordinate is discretized. Roughly speaking (see Ramkrishna (2000) for a more complete discussion), there are two approaches that can be employed ... 

The preferred choice of internal coordinates is discipline dependent. Nevertheless, the conservation of solid mass will imply constraints on particular moments of the PSD. In general, given the relationship between the various choices of coordinates, it is possible (although not always practical) to rewrite the PBE in terms of any choice of internal coordinate. 

When it comes to polyatomic molecules, there are two problems that complicate the issue, as already discussed in Note 1 of Chapter 3. One is the separation of the overall rotation of the molecule (Jellinek and Li, 1989). The other is that, depending on the choice of internal coordinates, certain coupling terms can be assigned to be kinetic or potential terms. A simple and familiar case is a linear triatomic, when one uses bond coordinates versus Jacobi coordinates. The case for Fermi coupling for a bending motion is discussed in Sibert, Hynes, and Reinhardt (1983). 

In practice, MC simulations are primarily applied to collections of molecules (e.g., molecular liquids and solutions). The perturbing step involves the choice of a single molecule, which is randomly translated and rotated in a Cartesian reference frame. If the molecule is flexible, its internal geometry is also randomly perturbed, typically in internal coordinates. The ranges on these various perturbations are adjusted such that 20-50% of attempted moves are accepted. Several million individual points are accumulated, as described in more detail in Section 3.6.4. 

A stable nuclear configuration on a potential energy surface is associated with a point for which there is zero slope in any direction and for which there is no direction in which the curvature is negative or zero. Such points are uniquely defined in any system of internal coordinates but we shall see that some other characteristic features of a surface are dependent on the choice of coordinate. 

Within the harmonic approximation the choice of a system of internal coordinates is irrelevant provided they are independent and that a complete potential function is considered ). For example, the vibrations of HjO can be analysed in terms of valence coordinates (r, >2, or interatomic coordinates (r, r, 3) and any difference in the accuracy to which observed energy levels are fitted (considering all the isotopic species H2O, HDO and D2O) will be due to the neglect of anharmonic terms. If one makes the approximation of a diagonal force field so that one is comparing the two potentials... 

To obtain the anharmonic terms in the potential, on the other hand, the choice of coordinates is important 130,131). The reason is that the anharmonic terms can only be obtained from a perturbation expansion on the harmonic results, and the convergence of this expansion differs considerably from one set of coordinates to another. In addition it is usually necessary to assume that some of the anharmonic interaction terms are zero and this is true only for certain classes of internal coordinates. For example, one can define an angle bend in HjO either by a rectilinear displacement of the hydrogen atoms or by a curvilinear displacement. At the harmonic level there is no difference between the two, but one can see that a rectilinear displacement introduces some stretching of the OH bonds whereas the curvilinear displacement does not. The curvilinear coordinate follows more closely the bottom of the potential well (Fig. 12) than the linear displacement and this manifests itself in rather small cubic stretch-bend interaction constants whereas these constants are larger for rectilinear coordinates. A final and important point about the choice of curvilinear coordinates is that they are geometrically defined (i.e. independent of nuclear masses) so that the resulting force constants do not depend on isotopic species. At the anharmonic level this is not true for rectilinear coordinates as it has been shown that the imposition of the Eckart conditions, that the internal coordinates shall introduce no overall translation or rotation of the body, forces them to have a small isotopic dependence 132). 

Assuming that most of the atomic displacements in the reactant molecule are small, the obvious choices for constraints on the internal coordinates not directly participating in the reaction are fixed values of the bond lengths and bond angles (say, / in number). The technical details concerning the construction of mathematical representations of these constraints and their incorporation into the appropriate projection operators can be found elsewhere [24], The result is, formally,... 

It should be pointed out that the occurrence of primitive period transformations is closely connected to the choice of the internal coordinates or equivalently to the choice of the frame system. Expectation values of all observable quantities whether dependent only on the RNC or dependent on the NC of a SRM must be independent of the choice of internal coordinates (frame system). If introduction of a certain internal coordinate (frame system) gives rise to a primitive period transformation and as a consequence to an extension of J to J ", then still observable quantities should be classifyable according to the symmetry group jr. 

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