Force Fields: Evaluation and development

• Structures

• Energies

• Other properties

• Validation

Evaluating Force Fields

Structures:

Energies:

Vibrations:

Charges:

Gas phase data

Crystal dataQ

M structures

Experimental energy types

Comparing com

puted and experimental energies

Shape of the potential energy surfaceIR data or Q

M data?

Not observable. W

hat to do?

How

reliable are the calculated properties? How

to compare?

Molecular M

echanics Structures

Structures calculated in vacuo (“gas phase”)

Equilibrium positions of nuclei (energy m

inima)

Gas phase structures

All structures >0 K

fi H

igher vibrational states occupied

Bond anharmonicity fi

Elongates observed bonds

Not nuclear positions!

• Electron diffraction• M

icrowave

reBoltzm

ann average,gives rg , rz , …

Crystal structures

X-ray gives electron densities, not nuclear positions

— very different for hydrogen

Neutron diffraction gives nuclear positions

Other error sources:

• Crystal packing• Libration, disorder

Comparison m

ethods:• O

verlay• Bond &

angle list• Crystal structure energy

CH

Center of electron density

Nucleus

Crystal packing

41°

crystalgas

Tight packing is favored.

Crystal structures are more “planar”

than ideal in vacuo structures

Only soft m

odes are distorted —

Bonds and most angles are good

— Torsional angles can deviate

Small changes accum

ulate

Small angle error

Large distance error

HH

HH

Libration & disorder

Atom

ic positions from crystallography are average positions.

Thermal w

agging motions (librations) or disorder m

ay giveunphysical structures.

Libration of a bond

Average atom

position

Apparent

short bond

Phenyl librationA

pparent phenyl

PdPd

120°D

isorder

PdA

verage structureLarge angle, short bonds

Overlaying structures

Root mean square error:

Summ

ation is done over all cartesian coordinatesRotate and translate structure until rm

s minim

ized (automatic)

Final rms value gives agreem

ent between tw

o structures

• Sensitive to soft mode errors

• Sensitive to accumulation of sm

all errors

Not recom

mended as “goodness” indicator. If used, select

overlay atoms carefully!

†

rms

=xobs -

xcalc(

) 2Â

Comparing structures

Compare lists of bonds and angles

Calculate rms for bonds and angles separately

• Insensitive to torsional errors• N

o error accumulation

Inspect largest errors, find a rationale

• Inaccurate force field or experimental error?

• Soft modes (som

e angles) and crystal packing

Crystal structure energies

The energy of a crystal structure, compared to the energy of a

force field minim

um structure, can be used as a “goodness”

measure.

Small bond length deviations give large energy penalties.

Structures should be relaxedby constrained optim

ization.Flat-bottom

potential gives nopenalty for sm

all deviations

Neutral structures should be w

ithin 10-20 kJ/mol of m

inimum

E

Comparing com

puted structure

QM

structures are also energy minim

a, can be directly compared

(Zero K, in vacuo, no vibrational contributions)

re

Well-developed force fields

are usually at least as accurateas standard Q

M m

ethods. Use

validation against QM

resultsw

ith caution.

Molecular M

echanics Energies

Empirical force fields calculate distortion energies.

The energy from each term

differs from “chem

ical” potentialenergy by a constant, unknow

n term.

Comparison of force field energies are only valid w

hen all suchunknow

n terms cancel (i.e., com

paring conformers)

∆Econf can be com

pared to experimental conform

ational energies

†

E=

ks l-

l0(

) 2

bondsÂ

+kb

q-

q0

() 2

anglesÂ

+vn cosnw

torsionsÂ

+q

i qj

er+

Ar 12-

Br 6Ê Ë Á

ˆ ¯ ˜ r≥3 bonds

Â

Energies

Experimental energies are usually free energies obtained from

equilibrium constants.

∆G = RT lnK

∆G = ∆H

– T∆S

Free energies differ from calculated potential energies

Gtot = E

0 + ZPE + Hvib – T S

vib + Grot/trans – T S

conf + Gsolv

Each contribution can be calculated before comparison.

Experimental energies

1/T

RlnK

∆Sslope = –∆Hx

x xx

xx

∆G = ∆H

– T∆S = RTlnK fi

RlnK = ∆S – ∆H

/T

Comm

on assumption: ∆H

≈ ∆E (∆ZPE ≈ 0, ∆∆S ≈ 0, ∆∆G

solv ≈ 0)

Example energy evaluation

CH3

HH

∆E‡=7.5 kJ/m

ol

H

H HH

HH

∆E‡=12.0 kJ/m

ol

H

HCH

3

CH3

HH

CH3

HH

∆H=4.1

kJ/mol

t-Bu

t-Bu

t-But-Bu

t-Bu

t-Bu

∆G=4.4

kJ/mol

OH

O H∆H

=2.9kJ/m

olR

R

F

HH

H

HF

F

HH

F

HH

∆H=3.3

kJ/mol

-6 - 23kJ/m

ol

R'R'

Force field comparison

K. G

undertofte, T. Liljefors, P.-O. N

orrby, I. Pettersson, J. Comput. Chem

. 1996, 17, 429-449

0 2 4 6 8 10 12 14

Amber*

CFF91

CFF99

CHARMm_23

CVFF

Dreiding2.21

MM2*

MM2(91)

MM3*

MM3(92)

MMFF

OPLS_AA

Sybyl5.21

UFF1.1

PM3

HF/6-31G*

B3LYP/6-31G*

Conjugated

Halocyclohexanes

Halides

Cyclohexanes

Nitrogen

Oxygen

Hydrocarbons

Rotation barriers

Mean A

bsolute Error (MA

E), kJ/mol

Molecular vibrations

Vibrations are very sensitive probes for the quality of the

potential energy surface (PES), must be good for M

D.

Experimental: IR (bond and angle vibrations)

Very hard to assign exactly (isotopic substitution)

Computational: H

armonic approxim

ation,diagonalization of the m

ass-weighted H

essian.Sm

all changes in curvature can give largeshifts in calculates frequencies

Recomm

endation: compare H

essian elements to Q

M calculation

†

1mi m

j

∂2E

∂xi ∂xj

Electrostatics

Force fields use atomic charges or bond dipoles

– Neither are observables

Good quality needed for non-bonded interactions, solvation, ...

Obtain from

molecular dipoles or Q

M charges

Validate by force field perform

ance for solvation, complexation

constants, etc. (usually with M

D &

explicit solvent)

d+d+

d--

Force Field example

W

hat method to trust? Validate!

NH

N

H

H3 CO

OC H

H

R

H

OCH

3

R = ethyl, vinyl

Coalescence in NM

R, what dynam

ic process isresponsible? ∆H

* = 60 kJ/mol for R = vinyl,

70#kJ/mol for R = ethyl.

MM

? Unusual push-pull system

, parameters?

Solvation needed.M

MFF and A

mber* quite different!

A few

B3LYP-calculations validated A

mber*. N

-inversion fast, side group rotation hasa large barrier. In this case, ethyl is m

uch larger than vinyl!

Force Fields: Tailoring and Application

• Quantitative evaluation of force fields

• Parameter estim

ation

• Parameter refinem

ent

• Examples

Force Field BasicsA D

iagonal, Harm

onic Force Field

l

q

w

r

E=

ksl-

l0(

) 2

bondsÂ

+

kbq

-q

0(

) 2

anglesÂ

+

vn cosnwtorsions

Â+

qi q

j

er+

Ar 12-

Br 6Ê Ë Á

ˆ ¯ ˜ r≥3 bonds

Â

Extending MM

Developing param

eters

Es = ks (l–l0 ) 2

Eb = kb (q–q

0 ) 2

observable

Et = vcos(nw

+f) E

nb = qi q

j /er + Ar -12 – Br -6

l

q

w

r

parameter

parameter

1) Nonbonded param

eters2) Reproduce structures3) Energies, vibrations, …4) Properties

Estimating param

eters

1) From related param

eters Sam

e type of parameter, sim

ilar atom types

2) From reference data

Ideal bond lengths and angles from unstrained structures,

force constants from com

paring strained and unstrained structures, charges from

QM

calculations, ...

3) Rule based Bond lengths from

sum of covalent radii, angles and torsions from

gross hybridization type, charges from electronegativity.

Parameters &

observables

Es = ks (l–l0 ) 2

Bond parameters

Refined estimate

Assum

e that the strain on a bond is independent of small param

eter changes

kest (lcalc –l0,est ) = kreal (lobs –l0,real )¤

lobs = (kest /kreal )(lcalc –l0,est ) + l0,real

Optim

ize all reference structures with

estimated param

eters. Plotting theobserved length lobs vs. the calculateddistortion (lcalc –l0,est ) yields an im

provedl0 as intercept. The slope can give abetter force constant.

lcalc –l0,est

lobs

new l0

Torsional parameters

Fitting the rotation profile

Calculate a rigid QM

profile, discard structures with very high energy. Calculate the

MM

energy using estimated param

eters for each rigid structure. Assum

e that thedifference only arises from

torsional parameter error plus a constant.

∆E = EQ

M – EM

M = E0 +∆v cosnw

Plotting ∆E vs. cosnw yields the param

eterchange as the slope. This is easily extendedto three (or m

ore) independent parameters.

060

120180

240300

360

Ref

Est

∆E

Force field qualityThe penalty function

Reference data y : bond length l, angle q, torsion w

, energy E, dipole m, ...

Penalty function (error function)

c2 = ∑

wi 2(ycalc –yobs ) 2

Optim

al force field = minim

al c2

Force field qualityW

eight factors

c2 = ∑

wi 2(ycalc –yobs ) 2

Dim

ensionless

Conversion and weighting

An error of 1° is less im

portant than 1 Å !

Weight each data point by the inverse of the acceptable error

The rms error of the force field is acceptable w

hen c2 is low

erthan the num

ber of data points N.

Data type

bond lengthangleconform

ational energyQ

M charge

Weight factor

1001110

Acceptable error

0.01 Å1°

1 kJ/mol

0.1 au

Example:

Parameterization

P.-O. N

orrby T. Liljefors, J. Comput. Chem

. 1998, 19, 1146-1166 Minim

ize c2 by varying the param

eters

• Grid search

• Monte Carlo

• Simplex

• Genetic A

lgorithm

Utilize derivatives, ∂c

2 / ∂p

• Steepest descent• Conjugate gradient• N

ewton-Raphson

Penalty function:

c2 = ∑

w2(y-y

ref ) 2

Gather reference data

Create MM

model

Penalty function

Refine parameters

Simplex

12

3

4

56 7

89

p1

p2

bestworst

1) Initial Simplex

p1

p2

3) Expansion or contraction

reflectionexpansion

contractions

p1

p1

p2

4) Next cycleneww

orst

inversioncenter

p1

p2

bestworst

new

2) Reflection

p1

p2

bestworst

newm

ove inversioncenter tow

ards best point

Biasing

Least-squares optimization

The Newton-Raphson method

New

ton-Raphson, 1 dimension: ∆p = –

∂c2/∂p

∂2c

2/∂p2

Multidim

ensional: [∆pi ] = –[∂

2c2/∂p

i ∂pj ] –1[∂c

2/∂pi ]

Gauss-N

ewton: [∂y

k /∂pi ] T[∂y

k /∂pi ] [∂y

k /∂pi ] T [∆y

k ]

Approxim

ate:

Time-consum

ing, but good convergence

Example: (h

3-Allyl)Palladium Com

plexesStructures and energies, H

. Hagelin, B. Å

kermark, P.-O

. Norrby, O

rganometallics 1999, 18, 2884-2895

>200 parameters needed in M

M3*

Available data: X

-Ray structures, conformational energies

Added data: B3LY

P structures, charges.D

ata:V

alidation:

NN

!Pd!

Ph

GAFBAQ

NPPh2

!Pd!

O PhPh

LELKES

Ph2 PPPh2

!Pd!

LEGZO

M

NPh2 P !Pd!

O Ph

NANCIO, NANCO

U

NN

!Pd!

JERGES

ZIBVUB HNNH

OO Ph2 P

PPh2

!Pd!

QM

(B3LYP)

NN

!Pd!R

1R

1

R2

NN

!Pd!R

1R

1R

2

∆G

Energies

YH3

!Pd!H

3 XX, Y = N, P

Example: (h

3-Allyl)Palladium Com

plexesValidation

NPh2 P !Pd!

O Ph

NANCIO, NANCO

U

NANCIO + M

M3*, rm

s=0.049 Å

Overlay

Pd + attached atoms

(C–C–C, N, P)

NANCIO + NANCO

U, rms=0.079 Å

Example: (h

3-Allyl)Palladium Com

plexesCom

parison of methods

NPPh2

!Pd!

O PhPh

LELKES

Overlay

Pd + attached atoms

(C–C–C, N, P)

MM

3*, rms=0.041 Å

PM3(tm

), rms=0.055 Å

B3LYP, rms=0.130 Å

Tripos, rms=0.409 Å