Semi-empirical force field potential

The sum of all non-bonded (Lennard-Jones and electrostatic) and bonded (bond length, bond angle, dihedral angle) interactions is the potential energy of the system, also called a force field potential. Compactly, we write for the total potential energy of the system the following expression  

A typical classical mechanical potential energy for a system of M linear molecules, with Nm atoms each, for a total of N atoms in the system (N = M x Nm), is then written as 

This potential is semi-empirical because some terms and parameters are determined using quantum mechanical principles, while 

Calculation of the non-bonded empirical potential energy requires sums in Eq. 14.2 that may in principle extend to infinite distances. For the calculation to be tractable, we can truncate the potential, by picking a distance cutoff, Tc, such that 

The cutoff distance is typically chosen to be rc 2.5a. At this distance the dispersion force between any two atoms is indeed close to zero. 

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Molecular Energetics. Molecular energies can be computed in a variety of ways including empirical fixed valence potentials, full force field potentials, and semi-empirical molecular orbital techniques (CNDO-2, INDO, MINDO-3, MNDO, PCILO). 

For polyatomic molecules the situation is more complex, and detailed knowledge of the potential hypersurface is available only for few systems. Semi-empirical methods are therefore introduced. As in the construction of the LEPS surface one should in the semi-empirical method, use expressions which behave correctly in the asymptotic limit. Thus for the molecule ABC the potential should approach Vab for bc oo Vbc for Rab oo, and Vac for Rab oo. The LEPS function discussed above has this property. However, it is advantageous to have a recipe which uses the extensive information on force-field potentials which is available from spectroscopic measurements. For a molecule as CO2, the... 

Example Jensen and Gorden calculated the potential energy surface of glycine using ab initio and semi-empirical methods.This study is of special interest to developers of molecular mechanics force fields. They frequently check their molecular mechanics methods by comparing their results with ab initio and semi-empir-ical calculations for small amino acids. 

A classical description of M can for example be a standard force field with (partial) atomic charges, while a quantum description involves calculation of the electronic wave function. The latter may be either a semi-empirical model, such as AMI or PM3, or any of the ab initio methods, i.e. HF, MCSCF, CISD, MP2 etc. Although the electrostatic potential can be derived directly from the electronic wave function, it is usually fitted to a set of atomic charges or multipoles, as discussed in Section 9.2, which then are used in the actual solvent model. 

The success of any molecular simulation method relies on the potential energy function for the system of interest, also known as force fields [27]. In case of proteins, several (semi)empirical atomistic force fields have been developed over the years, of which ENCAD [28,29], AMBER [30], CHARMM [31], GRO-MOS [32], and OPLSAA [33] are the most well known. In principle, the force field should include the electronic structure, but for most except the smallest systems the calculation of the electronic structure is prohibitively expensive, even when using approximations such as density functional theory. Instead, most potential energy functions are (semi)empirical classical approximations of the Born-Oppenheimer energy surface. 

The use of Wigner type correlation correction to Hartree-Fock energies [78] and/or the inclusion of dispersion forces [79] and/or the use of Cl energies [80] to define different potentials in Monte Carlo simulations of liquid water, underscores the problem on the reliability of ab initio potentials for force fields. Note that at the time the force fields were obtained only semi-empirically, but I was championing the ab initio banner. 

Lii J-H and N L Allinger 1989. Molecular Mechanics. The MM3 Force Field for Hydrocarbons 2 Vibrational Frequencies and Thermod5mamics Journal of the American Chemical Society 111-8566-8582 London F 1930 Zur Theori und Systematik der Molekularkrafte Zeiischrift fur Physik 63 245-279 Luckhurst G R, R A Stephens and R W Phippen 1990 Computer Simulation Studies of Anisotropic Systems XIX Mesophases Formed by the Gay-Beme Model Mesogen Liquid Crystals 8-451-464 Luque F J, F lias and M Orozco 1990 Comparabve Study of the Molecular Electrostatic Potential Obtained from Different Wavefunctions - Reliability of the Semi-Empirical MNDO Wavefunction Journal of Computational Chemistry 11-416-430. 

In the past, force fields were parameterized based only on experimental data nowadays, most modem force fields include substantial quantum chemical information. According to the nature of the data used for parameterization, force fields can be classified as ab initio, semi-empirical, and empirical. Simple potentials, e.g., for argon, which require few parameters, can be fitted exclusively to macroscopic experimental data however, more complex force fields have numerous parameters and thus depend heavily on ab initio data. This chapter gives an introduction to the present state-of-the-art in this field. Attention is given to the way modeling and simulation on the scale of molecular force fields interact with other scales, which is mainly by parameter inheritance. Parameters are determined both bottom-up from quantum chemistry and top-down from experimental data. 

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