Molecular internal coordinates

FIGURE 12. Molecular internal coordinates (bond lengths and angles) to be combined in symmetry coordinates (see Table 2). 

The central construct of the expert system DOCENT which we are developing is its capacity to represent a macromolecule by "generalized cylinders", and to permit manipulation of the cylinders directly, instead of adjusting the molecular internal coordinates or space-fixed axes. The inverse problem, to recover reasonable values of the underdetermined atomic coordinates from the disposition of the generalized cylinders, is posed. 

FogarasI G, Zhou X, Taylor P W and Pulay P 1992 The calculation of ab initio molecular geometries efficient optimization by natural Internal coordinates and empirical correction by offset forces J. Am. 

The full dynamical treatment of electrons and nuclei together in a laboratory system of coordinates is computationally intensive and difficult. However, the availability of multiprocessor computers and detailed attention to the development of efficient software, such as ENDyne, which can be maintained and debugged continually when new features are added, make END a viable alternative among methods for the study of molecular processes. Eurthemiore, when the application of END is compared to the total effort of accurate determination of relevant potential energy surfaces and nonadiabatic coupling terms, faithful analytical fitting and interpolation of the common pointwise representation of surfaces and coupling terms, and the solution of the coupled dynamical equations in a suitable internal coordinates, the computational effort of END is competitive. 

Z-matriccs arc commonly used as input to quantum mechanical ab initio and serai-empirical) calculations as they properly describe the spatial arrangement of the atoms of a molecule. Note that there is no explicit information on the connectivity present in the Z-matrix, as there is, c.g., in a connection table, but quantum mechanics derives the bonding and non-bonding intramolecular interactions from the molecular electronic wavefunction, starting from atomic wavefiinctions and a crude 3D structure. In contrast to that, most of the molecular mechanics packages require the initial molecular geometry as 3D Cartesian coordinates plus the connection table, as they have to assign appropriate force constants and potentials to each atom and each bond in order to relax and optimi-/e the molecular structure. Furthermore, Cartesian coordinates are preferable to internal coordinates if the spatial situations of ensembles of different molecules have to be compared. Of course, both representations are interconvertible. 

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... 

I J, J C Cole, J P M Lommerse, R S Rowland, R Taylor and M L Verdonk 1997. Isostar A Libraij )f Information about Nonbonded Interactions. Journal of Computer-Aided Molecular Design 11 525-531. g G, W C Guida and W C Still 1989. An Internal Coordinate Monte Carlo Method for Searching lonformational Space. Journal of the American Chemical Scociety 111 4379-4386. leld C and A J Collins 1980. Introduction to Multivariate Analysis. London, Chapman Hall, ig C-W, R M Cooke, A E I Proudfoot and T N C Wells 1995. The Three-dimensional Structure of 1 ANTES. Biochemistry 34 9307-9314. 

To an observer at the center of mass, the overall motion of the system (translation) is irrelevant. The only important motions are those motions relative to the center of mass. Distances from the center of mass to each particle are internal coordinates of the system, usually denoted r and r2 to emphasize that they are internal coordinates of a molecular system. 

IV. INTERNAL COORDINATE MOLECULAR DYNAMICS A. Main Problems and Historical Perspective... 

These difficulties have led to a revival of work on internal coordinate approaches, and to date several such techniques have been reported based on methods of rigid-body dynamics [8,19,34-37] and the Lagrange-Hamilton formalism [38-42]. These methods often have little in common in their analytical formulations, but they all may be reasonably referred to as internal coordinate molecular dynamics (ICMD) to underline their main distinction from conventional MD They all consider molecular motion in the space of generalized internal coordinates rather than in the usual Cartesian coordinate space. Their main goal is to compute long-duration macromolecular trajectories with acceptable accuracy but at a lower cost than Cartesian coordinate MD with bond length constraints. This task mrned out to be more complicated than it seemed initially. 

Internal coordinate molecular modeling is an efficient instrument with specific advantages that make it an indispensable complement to other existing approaches. It is best suited for simulation and analysis of large-scale structural transformations in biomacro-mole-cules, and at present ICMD is generally considered the most powerful tool in conforma-... 

If tbe molecular geometry is optimized by the program, only a rough estimate of the parameters is necessary, hi term of internal coordinates, this is fairly easy. Some typical bond lengths (A) and angles are given below. 

Lipton, M Still, W. C. The multiple minimum problem in molecular modeling. Tree searching internal coordinate conformational space. J. Comp. Chem. 1988, 9, 343-355. 

The final step in the molecular-mechanics calculation of molecular conformation involves the minimization of the energy Approximations are involved whose importance is not always clear. Usually, all first derivatives with respect to the various internal coordinates are set equal to zero - although these coordinates are often not independent (see Section 10.6). Furthermore, the final conformation obtained depends on the assumed initial structure. Therefore, (he method must be applied with care and a certain amount of chemical intuition. In spite of these uncertainties the molecular mechanics method has been employed with considerable success, particularly in the conformational analysis of branched alkanes. For molecules containing hetero-atoms, it can be applied, but with somewhat less confidence. 

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