Molecular force field, analytical energy

With the rapid evolution of the ab initio MO technique involving complete geometry optimization, quantum chemists are now seeking analytical expressions of molecular force fields for their expensive results (215). It should be possible, in principle, to construct a complete artificial force field that faithfully describes the Hartee-Fock limit energy surface, but still reflects important properties of the real surface sufficiently to allow useful applications (149). 

The complexity of polymeric systems make tire development of an analytical model to predict tlieir stmctural and dynamical properties difficult. Therefore, numerical computer simulations of polymers are widely used to bridge tire gap between tire tlieoretical concepts and the experimental results. Computer simulations can also help tire prediction of material properties and provide detailed insights into tire behaviour of polymer systems. A simulation is based on two elements a more or less detailed model of tire polymer and a related force field which allows tire calculation of tire energy and tire motion of tire system using molecular mechanisms, molecular dynamics, or Monte Carlo teclmiques 1631. 

Molecular mechanical force fields use the equations of classical mechanics to describe the potential energy surfaces and physical properties of molecules. A molecule is described as a collection of atoms that interact with each other by simple analytical functions. This description is called a force field. One component of a force field is the energy arising from compression and stretching a bond. 

Concerning quantum chemical computations, we have used the MOLE-COLE program [18a], for HF and MP2 type computations. The Molecular Dynamics simulations with analytical force fields have been performed with the DINAMICA program [18b], The MOLECOLE-DFT program [18c] has been used for both the DFT energy minimization and for the DFT-Molecular Dynamics. 

Nowadays a wide variety of quantum-chemical programs are disposable, which permit to calculate with high accuracy the equilibrium geometry of the molecules and their energy of formation. Theoretical methods have been developed for analytical calculation of the first and second derivatives of energy [8,9], so that the force-constant matrix FHT and the harmonic frequencies can be extracted from the quantum-mechanical calculations. Since as a rule the molecular orbitals (MO) obtained by the quantum-mechanical methods are spread around the entire molecule, the corresponding quantum-mechanical force fields incorporate the important effects of the off-diagonal interactions. 

In molecular-mechanics calculations, the atoms are considered to move in a force field defined by an energy function based on classical (rather than quantum) mechanics. Thus, the energy of a given molecular conformation is not calculated in an iterative SCF procedure, as in quantum-chemical approaches, but rather uses an analytical formula based on effective potentials. 

If we could analytically evaluate the action of the operator in Eq. [144] for any potential energy function or force field, there would be no need for molecular dynamics simulation. Since that is generally not possible, we need to devise a numerical scheme that will solve Eq. [144] to a desired accuracy, while preserving the symmetry of the equations of motion (e.g., time-reversal symmetry). 

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