Matrix Analysis of Molecular Structures: Applying Engineering Knowledge to Molecular Dynamics
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The structure of a molecule is determined by its constituent atoms and the bonds that interconnect them. The interatomic bonds respond proactively to the eventual modification of the relative positions of the atoms, in a try to recover its stable equilibrium configuration. Since these interactions are governed by Quantum Mechanics, their exact modeling is extremely complicated. This paper presents a general formulation of the molecular behavior that is valid when the deformations (i.e.: the modifications of the relative positions of its atoms) are relatively small and the tensional response of the bonds (i.e.: the reactions that try to return the molecular structure to its stable equilibrium configuration) can be linearized. This formulation is a generalization of the Matrix Analysis of Structures in which the elements consist of groups of bonded atoms, and in which the concepts of generalized displacement, strain and stress, nodal force vector, stiffness and mass matrices, local and global numbering, assembly and renumbering emerge naturally. These refinements allows to deal with much larger problems with a much lower computational cost than the modeling by means of quadratic potentials that is typically employed in Molecular Dynamics. The proposed formulation allows the analysis of any three-dimensional molecule with the bonds usually considered in the field, i.e., stretch, angular and dihedral type bonds, including their degenerate configurations [1-3]. Possible applications include the determination of the natural frequencies of vibration of the capsid or spicules of viral particles for their possible destruction by resonance and the prediction of the macroscopic tensile-deformational properties of new materials.