21Chapter 3 Calculation MethodsHyperChem uses two types of methods in calculations: molecular mechanics and quantum mechanics. The quantum mechanics methods implemented in HyperChem include semi-empirical, ab initio, and density functional quantum mechanics methods. The molecular mechanics and semi-empirical quantum mechanics methods have several advantages over ab initio and density func-tional methods. Most importantly, these methods are fast. While this may not be important for small molecules, it is certainly important for biomolecules. Another advantage is that for specific and well-parameterized molecular systems, these methods can cal-culate values that are closer to experiment than lower level ab initio and density functional techniques.The accuracy of a molecular mechanics or semi-empirical quan-tum mechanics method depends on the database used to parame-terize the method. This is true for the type of molecules and the physical and chemical data in the database. Frequently, these methods give the best results for a limited class of molecules or phenomena. A disadvantage of these methods is that you must have parameters available before running a calculation. Develop-ing parameters is time-consuming.The ab initio or density functional methods may overcome this problem. However they are slower than any molecular mechanics and semi-empirical methods.Molecular MechanicsMolecular 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.22 Chapter 3Molecular MechanicsThis component is often approximated as a harmonic oscillator and can be calculated using Hooke’s law.(7)The bonding between two atoms is analogous to a spring connect-ing two masses. Using this analogy, equation 7 gives the potential energy of the system of masses, Vspring, and the force constant of the spring, Kr. The equilibrium and displaced distances of the at-oms in a bond are r0 and r. Both Kr and r0 are constants for a spe-cific pair of atoms connected by a certain spring. Kr and r0 are force field parameters.The potential energy of a molecular system in a force field is the sum of individual components of the potential, such as bond, angle, and van der Waals potentials (equation 8). The energies of the individual bonding components (bonds, angles, and dihe-drals) are functions of the deviation of a molecule from a hypo-thetical compound that has bonded interactions at minimum val-ues.(8)The absolute energy of a molecule in molecular mechanics has no intrinsic physical meaning; ETotal values are useful only for com-parisons between molecules. Energies from single point calcula-tions are related to the enthalpies of the molecules. However, they are not enthalpies because thermal motion and temperature-dependent contributions are absent from the energy terms (equa-tion 8).Unlike quantum mechanics, molecular mechanics does not treat electrons explicitly. Molecular mechanics calculations cannot de-scribe bond formation, bond breaking, or systems in which elec-tronic delocalization or molecular orbital interactions play a major role in determining geometry or properties.This discussion focuses on the individual components of a typical molecular mechanics force field. It illustrates the mathematical functions used, why those functions are chosen, and the circum-stances under which the functions become poor approximations. Part 2 of this book, Theory and Methods, includes details on the im-plementation of the MM+, AMBER, BIO+, and OPLS force fields in HyperChem.Vspring12---Krrr0–()2=ETotalterm1term2…termn+++=Calculation Methods 23Molecular MechanicsBonds and AnglesHyperChem uses harmonic functions to calculate potentials for bonds and bond angles (equation 9). (9)Example:For the AMBER force field, a carbonyl C–O bond has an equilibrium bond length of 1.229 Å and a force constant of 570 kcal/mol Å2. The potential for an aliphatic C–C bond has a mini-mum at 1.526 Å. The slope of the latter potential is less steep; a C–C bond has a force constant of 310 kcal/mol Å2.VstretchKrbond∑rr0–()2=VbendKθθθ0–()2angle∑=Kr = 310 kcal/mol Å2; r0 = 1.526Åenergy (kcal/mol)bond length (Å)Kr = 570 kcal/mol Å2; r0 = 1.229Å24 Chapter 3Molecular MechanicsA Morse function best approximates a bond potential. One of the obvious differences between a Morse and harmonic potential is that only the Morse potential can describe a dissociating bond.The Morse function rises more steeply than the harmonic poten-tial at short bonding distances. This difference can be important especially during molecular dynamics simulations, where thermal energy takes a molecule away from a potential minimum.In light of the differences between a Morse and a harmonic poten-tial, why do force fields use the harmonic potential? First, the har-monic potential is faster to compute and easier to parameterize than the Morse function. The two functions are similar at the po-tential minimum, so they provide similar values for equilibrium structures. As computer resources expand and as simulations of thermal motion (See “Molecular Dynamics”, page 71) become more popular, the Morse function may be used more often.harmonicMorsebond length (Å)energy (kcal/mol)Calculation Methods 25Molecular MechanicsTorsionsIn molecular mechanics, the dihedral potential function is often implemented as a truncated Fourier series. This periodic function (equation 10) is appropriate for the torsional potential. (10)In this representative dihedral potential, Vn is the dihedral force constant, n is the periodicity of the Fourier term, φ0 is the phase angle, and φ is the dihedral angle. Example:This example of an HN–C(O) amide torsion uses the AMBER force field. The Fourier component with a periodicity of one (n = 1) also has a phase shift of 0 degrees. This component shows a maximum at a dihedral angle of 0 degrees and minima at both –180 and 180 degrees. The potential uses another Fourier component with a periodicity of two (n = 2).The relative sizes of the potential barriers indicate that the V2 force constant is larger than the V1 constant. The phase shift is 180 degrees for the Fourier component with a two-fold barrier.