Coordinates, atomic inertial

An improvement on the rg structure is the substitution structure, or structure. This is obtained using the so-called Kraitchman equations, which give the coordinates of an atom, which has been isotopically substituted, in relation to the principal inertial axes of the molecule before substitution. The substitution structure is also approximate but is nearer to the equilibrium structure than is the zero-point structure. 

In the original version of the r0-method, ground state inertial moments calculated for all isotopomers in terms of internal coordinates are least-squares fitted to the experimental moments 1°. The internal coordinates represent a reference system which is identical for all isotopomers and the resulting restructure is obtained as the final set of internal coordinates determined by the criterion of optimum fit. All atomic positions must either be included in the list of those to be determined, or estimated values must be supplied and then kept fixed in the fit. The result depends on these assumed values. Schwendeman has suggested a useful r0-derived variant [6], the p-Kr method , where the isotopic differences between the calculated inertial moments of the isotopomers and the parent species are fitted to the respective experimental differences in the attempt to compensate (the isotopomer-independent) part of the rovib contribution. The same result is achieved explicitly by the r/e-method, a r0-derived variant which is presented later in this chapter, where the calculated inertial moments plus three isotopomer-independent rovib contributions eg are fitted to the experimental ground state moments I°g. ... 

The original rs-method is based on equations presented by Kraitchman [4], He had noticed that the numerically dominant part of the inertial moment tensor for an isotopomer in which (exactly) one atom has been substituted, can be replaced by the known inertial moments of the parent molecule, and that the remainder of the tensor then depends only on the Cartesian coordinates of the substituted atom. Equating the roots of the secular equation for this hybrid tensor to the experimental inertial moments of the isotopomer, Kraitchman obtained his famous equations which, quite in contrast to any r0-type method, always aim at the determination of the Cartesian coordinates of just this one substituted atom. The rest of the molecule is irrelevant, it does not influence the result and need not be known. Costain has checked, for several small molecules, the invariance of -determined bond lengths when employing different pairs of parent molecule and isotopomer [7]. He found it superior to the comparable r0-data and recommended to prefer -structures for... 

The basic equations of the -method will be presented later within the framework of the more general r -fit problem. A rigid mass point model, which is strictly true only for the equilibrium configuration, is assumed. The application of Kraitch-man s equations (see below) to localize an atomic position requires (1) the principal planar moments (or equivalent inertial parameters) of the parent or reference molecule with known total mass, and (2) the principal planar moments of the isotopomer in which this one atom has been isotopically substituted (with known mass difference). The equations give the squared Cartesian coordinates of the substituted atom in the PAS of the parent. After extracting the root, the correct relative sign of a coordinate usually follows from inspection or from other considerations. The number, identity, and positions of nonsubstituted atoms do not enter the problem at all. To determine a complete molecular structure, each (non-equivalent) atomic position must have been substituted separately at least once, the MRR spectra of the respective isotopomers must all have been evaluated, and as many separate applications of Kraitchman s equations must be carried out. 

One of the first computer programs written to obtain the Cartesian atomic coordinates referred to the PAS of the parent by means of a least-squares fit to the inertial or planar moments of a number of isotopomers (also multiply substituted) appears to have been the program STRFIT coded by Schwendeman [6]. It is a versatile r0-type program incorporating many useful features, it is not a rs-fit program in the sense in which this term is used in this paper. 

Recently, van den Heuvel et al. [378 a] studied the bioconcentration of [ H]avermectin Bj in an 28-d uptake flow-through test with bluegill sunfish Lepomis macrochirus). Avermectin Bj (see Fig. 16), the major component of abamectin, possesses a molecular mass of 872. The molecular dimensions are 17.0 X 18.7 X 18.4 A and were determined by Nachbar (cited in [378]) by finding the smallest parallelepiped whose faces were centered on the inertial axes of the molecule and would enclose the van der Waals surface of the molecule. A van der Waals radius of 1.2 A for hydrogen was used and the atomic coordin-... 

Equation (14) demonstrates that it is very difficult to locate atoms that are close to a principal inertial plane. However, if only one a, b, or c coordinate is close to an inertial plane, the appropriate first moment relation may be used to compute that coordinate. For example,... 

Two special procedures employed in structure calculations will be mentioned here. One is the double substitution method introduced by Pierce26 and used for the determination of the coordinates of atoms near a principal inertial plane, which is the situation for which the Kraitchman equations become unreliable. In Pierce s method rotational constants must be available for four isotopic species, two of which are... 

How many atoms lie near principal inertial planes, and how have the small coordinates been treated ... 

Molecular structures may be described and compared in terms of external or internal coordinates. The question of which is to be preferred depends on the type of problem that is to be solved. For example, one problem that is much easier to solve in a Cartesian system is that of finding the principal inertial axes of a molecule indeed, if only internal coordinates are given then, in general, the first step is to convert them to Cartesian ones and then proceed as described in Section 1.2.4. Similarly, the optimal superposition of two or more similar molecules or molecular fragments, i.e. with the condition of least-squared sums of distances between all pairs of corresponding atoms, is best done in a Cartesian system. On the other hand, systematic trends in a collection of molecular structures and correlations among their structural parameters are more readily detectable in internal coordinates. 

There are, however, considerable problems involved in the generation and interpretation of numerical results derived from inertial axis coordinate sets. Firstly, we must use 3 A descriptors, more than is strictly necessary, to define each A/-atom fragment. Secondly, and most important, it is difficult (if not impossible) to interpret differences in the external coordinates of superimposed fragments in basic chemical terms. Essentially, then, it is the arbitrariness and lack of chemical significance of most axial frames that preclude the use of external coordinates for systematic studies external coordinates are only of use when the axial frame is directly related to the structure of the fragment. 

In the scheme the coordinates of an atom located far from a principal inertial plane can be determined accurately, whereas those of an atom located close to an inertial plane are poorly defined, irrespective of the atomic mass, hi the latter case the relative signs of the coordinates are difficult to determine, because Kraitchman s equations give only the absolute values. For small coordinates, doubly-substituted... 

Imagine a well isolated space ship observed in an inertial coordinate system. Its energy is preserved, its centre of mass moves along a straight line with constant velocity the total, or centre-of-mass, momentum vector is preserved), it rotates about an axis with an angular velocity total angular momentum preserved ). The same is true for a molecule or atom, but the conservation laws have to be formulated in the language of quantum mechanics. 

Table 4 Atom coordinates in the principal inertial axis system and substitution (r ) structure for the diketo tautomer of uracil. Distances are given in A and angles in degrees... 

Once the nuclear coordinates are expressed in the inertial reference frame, one can locate the extreme points occupied by a nucleus in the molecule along each of the three inertial axes, thus defining three limiting molecular dimensions (Table 1.1 and Fig. 1.1). The radii of the peripheral atoms (see below) may be added to these dimensions. The axis corresponding to the highest moment is perpendicular to the plane of maximum spread of nuclear positions - like, for example, the molecular... 

The first step in the study of a rotational spectrum is to evaluate the moments of inertia, or rotational constants, from which the rigid rotor spectrum (discussed in the following section) can be predicted. The rotational problem is treated mathematically in terms of a molecule-fixed axis system with its origin at the center of mass of the molecule and its axes oriented along the principal inertial axes. With respect to these axes the moments of inertia are constant, and the intertia matrix is diagonal. The principal axes of inertia are designated by a, b, and c. The corresponding moments of inertia are denoted by la, h, and Ic, where, by convention, the inertial axes are labeled so that la = h = Join terms of the coordinates of the atoms in the principal axis system, the principal moments of inertia are defined by... 

The coordinate system is now drawn such that the origin is at the center of mass of the molecule. In order to describe the rotational motion of a nonlinear molecule, three angular coordinates are needed resulting in three moments of inertia. The position of each atom is now expressed in a coordinate system whereby the center of mass of the molecule is at the origin and each atom is along the axes labeled by convention as a, b, and c. This coordinate system is called the principal inertial axis system. The three moments of inertia that result from the principal inertial axis system are called the principal moments of inertia. 

Fig. 5 shows the picture of MO s along the reaction coordinate with the inertial axis fixed in the same direction. The bending motion is dominated by the fast rotation of CO as we have seen in equation (1). The most critical region for the non-adiatetic transition is between 80 - 120 . An adiabatic process retains an electron in the 13a orbital that is the p-oibital parallel to the chemical bond, while the non-adiabatic process due to electron deactivation from 13a to 12a corresponds to rotation of the p-orbital. In other words, the p-lobe of 13a on the S atom is pa in character while diat of 12a is pit. The non-adiabatic transition of OCS firom the 2A to 1 A state is essentially an unlocking of an electron oribital fit>m the bond axis that is similar in nature to a pa-im transition observed in atomic collision dynamics. 

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