Potential energy surface, Force field & Molecular Mechanics

3N (or 3N-6 or 3N-5) Dimension PES for N-atom system

x

E’ = ki (li l0,i) + ki’ (θi θ0,i) + etc…bonds angles

E’’ = ki + ki’ + etc…bonds angles

For geometry optimization, evaluate

E, E’ (& E’’) at the input structure X

(x1,y1,z1,…,xi,yi,zi,…,xN,yN,zN) or {li,θi,i}.

How do we obtain the potential energy E?

• QM: Solve Schrödinger equation• MM: Evaluate analytic functions (FF)

li

θi

rij

i

• 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

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.

Bond Stretching Energy• Based on Hooke's law (Simple harmonic functions). • Energy associated with vibration about the equilibrium bond length. • "kb" parameter controls the stiffness of the bond spring (force constant).

• "ro" parameter defines its equilibrium length.

• "kb" & "ro" 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.

Taylor expansion of energy about equilibrium position ro

212 12 12( ) ( )oU r k r r harmoni

c

122

12( ) 1 rU r D e

Bond Stretching Energy

22

12 12 12 12 12 122( ) ( ) ( ) ( )

o o

o o o

r r r r

dU d UU r U r r r r r

dr dr

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

...)()()()( 44

33

22 ABABABABABABAB

str RkRkRkRE

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

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, CH4 can form liquid phases)

• Stronger for larger, more polarizable molecules.e.g. CCl4 > CH4 ; Kr > Ar > He

• a.k.a. “London” or “dispersion” forces-+-+ - +- +-+ -+-+ -+ - +- +- +

• Overlap of electron clouds

• Lennard-Jones Exp-6 (Buckingham)

Non-bonded Energy: vdW Repulsion

12 6

A CU

r r 6

Br CU Ae

r

10

8

6

4

2

0

2.01.81.61.41.21.0

LJ Exp-6

Non-bonded EnergyElectrostatic (Coulombic) Interaction

Energy

1.5

1.0

0.5

0.0

-0.5

-1.0

4321

Lennard-Jones Coulomb

1.5

1.0

0.5

0.0

-0.5

-1.0

4321

Lennard-Jones Coulomb

• Coulomb’s law• Variables: Interatomic distance (rij)

• Parameters: atomic charges (qi) 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

Beyond two-body potential: Angle Bending Energy

• Also based on Hooke's law (Harmonic function). • Energy associated with vibration about the equilibrium bond angle.• "kθ" parameter controls the stiffness of the angle spring (force constant).

• "θo" 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.).

Taylor expansion of energy about equilibrium position

improvements by including higher-order terms

Out-of-plane bending

22

2( ) ( ) ( ) ( )

o o

o o odU d UU U

d d

2( ) ( )oU k

2( ) ( )oU k

minimum

defined as 0

higher term neglected

harmonic

Angle Bending Energy

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.). • CH3-CH3, for example, ought to repeat its energy every 120. • The cis conformation of a dihedral angle is assumed to be the zero by convention.

• 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 (CH3CH3) vs. ethylene (CH2CH2)• Usually at most n = 1, 2, and/or 3 terms are included

Torsion Energy (for A-B-C-D bond)

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.

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

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, CH4 can form liquid phases)

• Stronger for larger, more polarizable molecules.e.g. CCl4 > CH4 ; Kr > Ar > He

• a.k.a. “London” or “dispersion” forces-+-+ - +- +-+ -+-+ -+ - +- +- +

• Overlap of electron clouds

• Lennard-Jones Exp-6 (Buckingham)

Non-bonded Energy: vdW Repulsion

12 6

A CU

r r 6

Br CU Ae

r

10

8

6

4

2

0

2.01.81.61.41.21.0

LJ Exp-6

Non-bonded EnergyElectrostatic (Coulombic) Interaction

Energy

1.5

1.0

0.5

0.0

-0.5

-1.0

4321

Lennard-Jones Coulomb

1.5

1.0

0.5

0.0

-0.5

-1.0

4321

Lennard-Jones Coulomb

• Coulomb’s law• Variables: Interatomic distance (rij)

• Parameters: atomic charges (qi) 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

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

Only interactions consideredOnly interactions considered

5000-atom system

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)

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

Estimating Non-bonded (esp. Electrostatic) Energy for Periodic Systems: Ewald

Summation

For details, read Leach (pp.324-343) and reading materials (Kofke)

(Goddard)

(Kollman)

(Goddard)

Some Commonly Used Force Fields

in Materials Studio

Atom Types

Example of Atom Types (MM2)

Force Field Parametrization

X-ray, neutron, electron diffraction, NMR (structure),Calorimetry (enegy), IR spectroscopy, elastic properties