Internal coordinate system

Rotation of the internal coordinate system e for an angle about... 

Translation of the center of mass and rotation of the internal coordinate system as in Step2... 

It is always possible to convert internal to Cartesian coordinates and vice versa. However, one coordinate system is usually preferred for a given application. Internal coordinates can usefully describe the relationship between the atoms in a single molecule, but Cartesian coordinates may be more appropriate when describing a collection of discrete molecules. Internal coordinates are commonly used as input to quantum mechanics programs, whereas calculations using molecular mechanics are usually done in Cartesian coordinates. The total number of coordinates that must be specified in the internal coordinate system is six fewer... 

In our non-BO calculations performed so far, we have considered atomic systems with only -electrons and molecular systems with only a-electrons. The atomic non-BO calculations are much less complicated than the molecular calculations. After separation of the center-of-mass motion from the Hamiltonian and placing the atom nucleus in the center of the coordinate system, the internal Hamiltonian describes the motion of light pseudoelectrons in the central field on a positive charge (the charge of the nucleus) located in the origin of the internal coordinate system. Thus the basis functions in this case have to be able to accurately describe only the electronic correlation effect and the spherically symmetric distribution of the electrons around the central positive charge. 

Another problem comes in examining the polarizability. In the physical picture, the spherically symmetric molecule, just like an atom, has isotropic polarizability. In the chemical picture, for a diatomic molecule we have two unique polarizabilities (1) and in the internal coordinate system or (2) dzz = 5 (o xc + (isotropic polarizability) and Aa = — [polar-... 

The number of beads in the model macromolecule is n, and is the Stokes law friction coefficient of each bead. The are to be evaluated for each macromolecule in its own internal coordinate system, with origin at the molecular center of gravity and axes (k = 1,2,3) lying along the principal axes of the macromolecule. The coordinates of the ith bead in this frame of reference are (x ]),-, (x2)i, and (x3)f. The averaging indicated by < > is performed over all macromolecules in the system. Thus, < i + 2 + 3) is simply S2 for the macromolecules. The viscosity is therefore identical, for all free-draining models with the same molecular frictional coefficient n and the same radius of gyration, to the expression from the Rouse theory ... 

Coordinates of molecules may be represented in a global or in an internal coordinate system. In a global coordinate system each atom is defined with a triplet of numbers. These might be the three distances x,, y,-, z, in a crystal coordinate system defined by the three vectors a, b, c and the three angles a, / , y or by a, b, c, a, P, y with dimensions of 1,1,1,90°, 90°, 90° in a cartesian, i. e. an orthonormalized coordinate system. Other common global coordinate systems are cylindrical coordinates (Fig. 3.1) with the coordinate triples r, 6, z and spherical coordinates (Fig. 3.2) with the triples p, 9, . 

Internal coordinate systems include normal coordinates which are symmetry adapted and used in spectroscopy, and coordinate systems based on interatomic distances ( bond lengths ), three-center angles ( valence angles ) and four-center angles ( torsion angles ). In the latter case a Z-matrix of the form shown in Table 3.1 defines the structure of a molecule. The input and output files of nearly all molecular mechanics programs are in cartesian 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-... 

We shall not describe the internal coordinate system nor the symmetry coordinates, the symmetry force constants and the kinetic energy matrix, as they are detailed in McCullough s paper (/). Wilson s book, "Molecular vibrations , gives further explanations of the methods used (2). 

There is no imique coordinate system in which to perform the vibrational analysis. We have chosen the Cartesian system but in the early literature it was common to use complicated internal coordinate systems to facilitate the computations and exploit molecular symmetry. However, easy access to robust and well proven programs has removed much of the necessity of such complexities. Moreover, the modem... 

As a first example the states of any pure rotor series should all have maximum probability for r, = r2 and 0l2 = n, and the probability densities for members of a given rotor series should have very similar spatial distributions in their internal coordinate system (called the intrinsic coordinate system in the context of nuclear physics). The total wavefunctions of different rotor states in any series should differ primarily only in the parts that describe the rotation of the figure axis in space these parts do not affect the distributions in their internal coordinate systems. At a higher level of approximation, the distributions for the states of a given series may be expected to differ a little because of centrifugal distortion such differences, of course, are apparent in the internal coordinate system. 

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). 

The initial step in locating a stationary point is defining the internal coordinate system. An effort should be made to choose a coordinate system where the parameters are not interdependent (coupled). As shown below, there is a coupling (i.e., the bond length will affect the optimum bond angle) between the CH distance, rj, and the HCC angle, aj. The coupling can be... 

In Sect. 2 we discussed the practical advantages of valence coordinates for modelling anharmonicity. As discussed there, when the valence coordinates are non-redundant the valence-coordinate force field can be calculated directly from a global PES in any internal coordinate system. 

One of the simplest approximations employed in reducing the number of independent force constants is the assumption of central forces. It is assumed that the forces holding the atoms in their ciiuilibrium positions act only along the lines joining pairs of atoms and that every pair of atoms is connected by such a force. This type of force function would result if the molecule were held together by purely ionic interactions. Also, this type of force yields only diagonal terms in the force constant matrix when the internal coordinate system is the complete set of interatomic distances (the central force coordinates). 

The algorithmic steps for the constrained aBB approach can be generalized to any force field model or routine for solving constrained optimization problems. Here, the otBB approach is interfaced with PACK [74] and NPSOL [28]. PACK is used to transform to and from Cartesian and internal coordinate systems, as well as to obtain function and gradient contributions for the ECEPP/3 force field and the distance constraint equations. NPSOL is a local nonlinear optimization solver that is used to locally solve the constrained upper and lower bounding problems in each subdomain. 

---