1. Potential energy surface, Force field & Molecular Mechanics 3N (or 3N-6 or 3N-5) Dimension PES for N-atom system x E’ =  k i (l i  l 0,i ) +  k i ’ (θ i  θ 0,i ) + etc… bondsangles E’’ =  k i +  k i ’ + etc… bondsangles For geometry optimization, evaluate E, E’ (& E’’) at the input structure X (x 1,y 1,z 1,…,x i,y i,z i,…,x N,y N,z N ) or {l i,θ i,  i }. How do we obtain the potential energy E? QM: Solve Schrödinger equation MM: Evaluate analytic functions (FF) lili θiθi r ij ii

3. Bypass the time-consuming QM procedure (ignore electrons) Write E(R) as a parametric function of nuclear coordinates R Derive the function E(R) from an intuitive ball-and-spring model Fit the parameters to experiment and/or higher-level computational data Force Fields and Molecular Mechanics

4. Two-body case first. Example of pairwise interaction potentials 1. Based on Hooke's law (Simple harmonic functions). Energy associated with vibration about the equilibrium bond length.

5. Bond Stretching Energy Based on Hooke's law (Simple harmonic functions). Energy associated with vibration about the equilibrium bond length. "k b " parameter controls the stiffness of the bond spring (force constant). "r o " parameter defines its equilibrium length. "k b " & "r o " parameters are assigned to each pair of bonded atoms based on their types (e.g. C-C, C-H, O-C, etc.). The model breaks down as a bond is stretched toward the point of dissociation. But bonds are usually so stiff that it works for moderate T.

6. Taylor expansion of energy about equilibrium position r o harmonic Bond Stretching Energy minimum defined as 0 higher term neglected unrealistic when the bond is stretched. fails in strained geometries. Polynomial with higher terms The limiting behavior is not correct for 3rd, 5th order… Morse potential

7. Two-body case first. Example of pairwise interaction potentials. Pair-wise sum of the energies of all possible interacting non-bonded atoms i and j

8. Non-bonded Energy van der Waals (vdW) Attraction Correlation of electron fluctuations. Only attraction which is present between nonpolar molecules (the reason that Ar, He, CH 4 can form liquid phases) Stronger for larger, more polarizable molecules. e.g. CCl 4 > CH 4 ; Kr > Ar > He a.k.a. “London” or “dispersion” forces Overlap of electron clouds Lennard-Jones Exp-6 (Buckingham) Non-bonded Energy: vdW Repulsion

9. Non-bonded Energy Electrostatic (Coulombic) Interaction Energy Coulomb’s law Variables: Interatomic distance (r ij ) Parameters: atomic charges (q i ) calculated using QM Dielectric constant (  0 ) for the attenuation of electrostatic interaction by the environment (80 for aq solution; 1 for vacuum) Very long-range

10. Beyond two-body potential: Angle Bending Energy Also based on Hooke's law (Harmonic function). Energy associated with vibration about the equilibrium bond angle. "k θ " parameter controls the stiffness of the angle spring (force constant). "θ o " parameter defines its equilibrium angle. Parameters are assigned to each bonded triplet of atoms based on their types (e.g. C-C-C, C-O-C, C-C-H, etc.).

11. Taylor expansion of energy about equilibrium position improvements by including higher-order terms Out-of-plane bending  minimum defined as 0 higher term neglected harmonic Angle Bending Energy

12. Beyond two-body potential: Torsion (A-B-C-D bond) Modeled by a simple periodic function "A" parameter controls the amplitude of the curve. "n" parameter controls its periodicity and reflects the symmetry in torsion angle. "  " parameter shifts the entire curve along the rotation angle axis (  ). The parameters are determined from curve fitting. Parameters are assigned to each bonded quartet of atoms based on their types (e.g. C-C-C-C, C-O-C-N, H-C-C-H, etc.). CH 3 -CH 3, for example, ought to repeat its energy every 120 . The cis conformation of a dihedral angle is assumed to be the zero by convention.

13. Periodic Weak (~100 times less stiff than bond stretching motions) (Taylor expansion in  not appropriate) Fourier series Terms are included to capture appropriate minima/maxima Depends on substituent atoms e.g., ethane has three mimum-energy conformations (n = 3) Depends on type of bond e.g. ethane (CH 3  CH 3 ) vs. ethylene (CH 2  CH 2 ) Usually at most n = 1, 2, and/or 3 terms are included Torsion Energy (for A-B-C-D bond)

14. Bonded interaction (valence terms) Nonbonded interaction (non-bonding term) van der Waals (dispersion) Coulombic interaction (electrostatic) + cross terms (coupling) Sum of individual components The mathematical form of the energy terms varies from force-field to force-field.

16. Two-body case first. Example of pairwise interaction potentials. Pair-wise sum of the energies of all possible interacting non-bonded atoms i and j

17. Non-bonded Energy van der Waals (vdW) Attraction Correlation of electron fluctuations. Only attraction which is present between nonpolar molecules (the reason that Ar, He, CH 4 can form liquid phases) Stronger for larger, more polarizable molecules. e.g. CCl 4 > CH 4 ; Kr > Ar > He a.k.a. “London” or “dispersion” forces Overlap of electron clouds Lennard-Jones Exp-6 (Buckingham) Non-bonded Energy: vdW Repulsion

18. Non-bonded Energy Electrostatic (Coulombic) Interaction Energy Coulomb’s law Variables: Interatomic distance (r ij ) Parameters: atomic charges (q i ) calculated using QM Dielectric constant (  0 ) for the attenuation of electrostatic interaction by the environment (80 for aq solution; 1 for vacuum) Very long-range

19. x x x x Direct method (simplest) Interactions are calculated to a cutoff distance. Interactions beyond this distance are ignored. Leads to discontinuities in energy and derivatives. As a pair distance moves in and out of the cutoff range between calculation steps, the energy jumps. (since the non-bond energy for that pair is included in one step and excluded from the next.) Cutoff for Long-Range Non-bonded Interactions 5000-atom system

20. Minimizing discontinuity. Spline, a possible choice Switching function S(r) = 1 for small r = 1  0 smoothly at intermediate r = 0 for large r Should be continuously differentiable (so that forces can be calculated). Smoothly turns off non-bond interactions over a range of distances. Switching range is important. Upper limit = the cut-off distance. Too large lower limit (small spline width)  Unrealistic forces may result.  Too small lower limit  The feature of the equilibrium region may be lost. Effective potential = actual potential  smoothing function S(r)

21. Number of non-bond interactions for a 5000-atom system as a function of cutoff distance vdW energy of a hexapeptide crystal as a function of cutoff distance, which does not converge until 20 Å Cutoff for Long-Range Non-bonded Interactions

22. Estimating Non-bonded (esp. Electrostatic) Energy for Periodic Systems: Ewald Summation For details, read Leach (pp.324-343) and reading materials (Kofke)

23. (Goddard) (Kollman) (Goddard) Some Commonly Used Force Fields

24. in Materials Studio

25. Atom Types

26. Example of Atom Types (MM2)

27. Force Field Parametrization X-ray, neutron, electron diffraction, NMR (structure), Calorimetry (enegy), IR spectroscopy, elastic properties