Internal coordinate definition

Figure 5. Representation of the geometry of a cation in site E, illustrating the internal coordinate definitions used in the frequency and intensity calculations.

Another problem that can occur during the course of a minimization in redundant internal coordinates is the internal forces being reported as infinity or undefined [126]. As discussed in Section 10.2.3, the energy derivatives are typically computed in Cartesian coordinates and later converted to internals using Eq. (10). If the redundant internal coordinate definitions become ill defined and/or include linear dependences, then the conversion of forces and Hessians to internal coordinates can become problematic. The easy fix to this problem begins by inspecting the latest structure in the optimization using visualization software to ensure that the structure is reasonable. If all is well, start... 

The coordinate system choice is also important in TS optimization. As with minimization, redundant internal coordinates have been shown to be the best choice for TS optimization [52]. Table 10.5 compares the number of optimization steps required for convergence using the three-structure STQN method with Z-matrix and redundant internal coordinates. Clearly, redundant internals work best. In Section 10.3.5.2, we advised that users check the redundant internal coordinate definitions to ensure all of... 

Figure 2 Illustration of the definition of internal coordinates. Main chain atoms are shown as filled circles.

By definition, the vector of conjugate momenta corresponding to the vector of generalized internal coordinates q is... 

We then dehne an internal coordinate <() such that <() = 0 2ti denotes a a path that has described one complete loop around the Cl in the nuclear branching space. Other than this, we need specify no further details about < ). We do not even need to specify whether the complete set of nuclear coordinates give a direct product representation of the space. It is sufficient that closed loop has wound around the CL Using this definition of ((), we can express the effect of the GP on the... 

Equations (56) and (57) give six constrains and define the BF-system uniquely. The internal coordinates qk(k = 1,2, , 21) are introduced so that the functions satisfy these equations at any qk- In the present calculations, 6 Cartesian coordinates (xi9,X29,xi8,Xn,X2i,X3i) from the triangle Og — H9 — Oi and 15 Cartesian coordinates of 5 atoms C2,C4,Ce,H3,Hy are taken. These 21 coordinates are denoted as qk- Their explicit numeration is immaterial. Equations (56) and (57) enable us to express the rest of the Cartesian coordinates (x39,X28,X38,r5) in terms of qk. With this definition, x, ( i, ,..., 21) are just linear functions of qk, which is convenient for constructing the metric tensor. Note also that the symmetry of the potential is easily established in terms of these internal coordinates. This naturally reduces the numerical effort to one-half. Constmction of the Hamiltonian for zero total angular momentum J = 0) is now straightforward. First, let us consider the metric. 

As V is a unitary matrix, Y = VTX is just an equivalent set of Cartesian coordinates, and = UTZ is just an equivalent set of internal coordinates, simply linear combinations of the Zn. The i, , N-6, change independently, in proportion to changes in linear combinations of the Cartesian coordinates. So, locally, we have defined 3N — 6 independent internal coordinates. Every different configuration of the molecule, X, will have a different B matrix, and hence a different definition of local internal coordinates, defined automatically. 

Like many other chemical concepts the concept of strain is only semi-quantitative and lacks precise definition. Molecules are considered strained if they contain internal coordinates (interatomic distances (bond lengths, distances between non-bonded atoms), bond angles, torsion angles) which deviate from values regarded as normal and strain-free . For instance, the normal bond angle at the tetra-coordinated carbon atom is close to the tetrahedral value of 109.47°. In the course of force field calculations these normal values are defined more satisfactorily, though in a somewhat different way, as force field parameters. 

These equations differ from the previous definitions of the X and Y matrix elements since the derivatives of the internal coordinates with respect to vibrational coordinate are considered. 

These are invariant to translation and rotation of a stereomodel. Most experimental chemists are familiar with internal coordinates from crystal structure analyses and the employment of molecular modelling computer programs. With regard to fundamental concepts of stereochemistry, internal coordinates are of prime importance as they allow clear definitions of terms to be formulated. ... 

At the transition state the reaction coordinate is defined to be in the direction of the one normal coordinate which has a negative eigenvalue. Fukui (22) suggested that the reaction coordinate be defined as the line of steepest descents from the transition state to reactant or product regions on the potential surface. However, this definition fails to acknowledge the non-orthogonality of the internal coordinates and it would appear more satisfactory to define the reaction coordinate with respect to normal coordinates Q for all points on its path so that the line of steepest descent will be that for which (9r/9(2r) is a maximum (correspondingly all other derivatives (9V/ds r will be zero). 

From the definition (2.2) it is seen that the distances dkk- are functions of the internal coordinates. The set of transformations... 

It should be pointed out that the operators PF which by definition act only on internal coordinates, are not symmetries of H° ... 

Fig. 8. C2vF(C2vT)2SRM (a) definition of the internal coordinates -n

From what has been shown in the preceding sections (cf. Eqs. 61 and 73, 83), it is possible to present the molecular structure resulting from both the r -fit method and any of the r()-derived methods in a convenient and easily comparable form, as a structural description in both Cartesian and internal coordinates, and with consistent errors and correlations (for small and larger molecules). A detailed comparison would require a sufficiently large SDS to determine a complete molecular structure, but the requirements are still the least restrictive of all methods presented. The input data must include the covariance matrix of the rotational constants or moments. This matrix may have to be adequately modeled to avoid grossly different weighting of isotopomers which is usually not warranted. The definition of the input data set... 

In spectroscopic analysis the first term, Vo, is usually made zero by definition. This is done by assuming that each internal coordinate, let it be a given bond distance or bond angle, is strainless at its equilibriiun value. For example, the standard sp3-sp3 carbon-carbon bond length zq is 1.53 A. The harmonic stretching potential then is represented by ... 

Gallina et al. [20] introduced the hyperspherical symmetrical parametrization in a particle-physics context, as did Zickendraht later [21, 22], At the same time, F.T. Smith [23] gave the definitions of internal coordinates following Fock s work already mentioned [16], Clapp [24, 25] and others and established, for the symmetrical and asymmetrical parametrization, the basic properties and the notation we follow. Since then, applications have been extensive, especially for bound states. For example, the symmetrical coordinates have often been used in atomic [26], nuclear [27] and molecular [28-31] physics. This paper accounts for modem applications, with particular reference to the field of reaction dynamics, in view of the prominent role played by these coordinates for dealing with rearrangement problems. 

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