Some Basic Theory - Energetics Energies can be calculated at several levels of theory Molecular Mechanics l Ab Initio l Most theoretically rigorous Chem 8711/7711 l Energies calculated from electronic structure l Requires no experimental parameters l Semi-Empirical l Simplifying assumptions made l Experimental parameters compensate l Molecular Mechanics l Electrons essentially ignored l Many experimental parameters required Reasonable Simplification Additional Approximations l Born-Oppenheimer Approximation: l Potential energy can be represented by terms l Movements of electrons are so rapid relative to describing deformation from “standard” movements of nuclei that they adapt essentially values instantly to the nuclear positions - thus the l Deformations from “standard” values can be motions of electrons and nuclei can be separated represented harmonically l Molecular (classical) mechanics relies on this approximation to the extreme of allowing l Potential functions and parameters are electrons to be IMPLIED by their associated transferable across molecules nuclei Transferability Exercise l Forcefields are generally parameterized to l Build a simple molecule in MOE or Spartan give accurate relative energies for isomeric (cyclohexane, an amino acid, a structures monosaccharide, etc) l Compute and record its energy using every l Energy values will not be comparable from available forcefield one forcefield to another l Forcefields can be changed in MOE using the Window->Potential Control dialog box l Forcefield control in Spartan is controlled during calculation setup l What is the range of energies you noted? 1 Molecular Mechanics - Goals Molecular Mechanics l To reproduce molecular geometries and l Energy broken down into terms RELATIVE energies l Bond stretching l Bond lengths: + 0.005 Å l Angle bending l Bond angles: + 1° l Torsion angles: + 5° l Torsional potential l DHf: + 0.7 kcal/mol l Non-bonded interactions l Van derWaals, electrostatic, dipolar interactions Forcefields Question l The combination of mathematical formulae and parameters used to represent the energy of a chemical system 1. Sketch the energy as a function of distance (r) between two bonded atoms (start @ r=0 Å) l Different forcefields are optimized for different problems: 2. If this interaction is expressed harmonically [V = l MMFF94: optimized for small organic compounds - wide k ( r - r )2] what will the curve look like? structural variety s 0 l Sybyl: general purpose – reasonable (but not excellent) 3. Where will the greatest error be? parameters for wide variety of atom environments 4. What chemical process(es) will therefore not be l AMBER94: optimized for proteins - often missing parameters for other organics modeled accurately? l PEFSAC95: optimized for carbohydrates l UFF: universal forcefield, contains parameters even for metals Bond Stretching Experimental Bond Distances l Approximated with a harmonic l r : equilibrium bond distance - bottom of energy well potential e l rav: average distance (slightly longer than re) l V = k ( r - r )2 What value of r is s 0 0 l ra : thermal average, from electron diffraction radial distribution appropriate? harmonic potential function l Two parameters per pair of atom l rg : derived from ra (~0.002 Å longer) averaged over all molecular types Morse potential vibrations l ra: distance between mean atom positions at a given T huge errors at relatively large l ra°: ra extrapolated to 0 K Energy interatomic distances l ro: directly obtained from microwave } l rs: directly obtained from microwave l r : microwave result with vibrational correction (should agree with r ° Interatomic distance z a 2 Experimental Bond Distances Angle Bending l Treated Harmonically l Electron Diffraction 2 l Thermal average of occupied states E = k (q - q0) } l Gives ra, rg, ra, ra° l Microwave Displacement from equilibrium bond angle l Values for the state examined l Gives ro, rs, rz l Requires two parameters for each l Molecular Mechanics l Usually parameterized to give room-temperature vibrationally- combination of three atom types averaged structures: ra l Comparable to x-ray or electron diffraction (usually) l NOT identical to Ab Initio (which gives re)! Question Torsional Potential Typical of sp3-sp3 bond s=1 n=3 1. Sketch the energy as a function of rotation around the central bond of butane 2. Draw structures of minimum energy structures and transition structures Typical of sp2-sp2 bond s=-1 n=2 0 60 120 180 240 300 360 Ew = Vn (1 + s cos nw) (Two--fold term added to compensate for non-equivalent minima) Three parameters needed for each combination of atom types Question Van der Waals l Usually expressed as a Lennard-Jones l What value of n do you expect for an sp2-sp3 potential (6-12 shown): bond? 12 6 éæ s ö æ s ö ù s r0 VVDW = 4eêç ÷ - ç ÷ ú ëêè r ø è r ø ûú l Sketch the energy as a function of distance repulsive attractive for non-bonded atoms Energy -1/6 r0 = 2 s é r 12 r 6 ù Interatomic distance æ 0 ö æ 0 ö VVDW = e êç ÷ - 2ç ÷ ú ëêè r ø è r ø ûú 3 Ionic Interactions Dipolar Interactions l Directional l Generally approximated using partial point l Often computed using Jeans’ formula q1q2 charges V = mi m j l Point charges come from Dr a i V = 3 (cos x - 3cosai cosa j ) mi Dr l Forcefield mj ij x aj l Quantum mechanics Effective dielectric constant q1 r Angles dipoles form with q2 Angle formed by dipole tails vector between midpoints Reading l Required l Chapter 1 in Leach (will discuss questions only) l Chapter 3 in Leach l Historical Interest l Molecular Mechanics, Burkert and Allinger, ACS Monograph, 1982 4.