Molecular Modelingof Crystal Structures

molecules

surfaces

crystals

1. Potential energy functions

QM ab initio: distribution of electrons over the system.Gaussian94, Gamess, ...

Semi-empirical methods: pre-calculated values or neglectof some parts of the ab-initio calculation.MOPAC (mopac6, 7, 93, 2000)

Empirical methods: observed/fitted values for interactionsbetween atoms.Sybyl, Cerius2, Gromos, ...

Potential energy functions

Differences:* Speed (as a function of system size)* Accuracy* Intended use (heat of fusion; conformational energies; transition states; vibrations/spectra; …)* Transferability / applicability* Availability / user interface

Potential energy functions

Focus: Molecular Mechanics (MM)

“Ball and Spring” model of molecules, based on simple equationsgiving U as function of atomic coordinates

G = U + pV - TSH = U + pV

EMM = U

Molecular Mechanicssystem from atoms + bonds

• stretching• bending• torsion

C CH

H

H H

H

H

EMM = Estretch + Ebend + Etorsion + Evdw + Ecoul + ...

bonded non-bonded

MM: interactions via bonds

bond stretchr Es = 1/2 ks(r-r0)2

… + C3(r-r0)3 + C4(r-r0)4

r

E

- True.. modeled via (r-r0)2

r0

Force field parameters: bond lengths (Dreiding)

C C

C

C

Bond type r0 (Å) ks (kcal/mol.A2)

C(sp3)--C(sp3) 1.53 700C(sp3)--C(sp2) 1.43 700C(sp2)--C(sp2) 1.33 1400C(sp3)--H 1.09 700

Es = 1/2 ks(r-r0)2

kS=700: E=3 kcal ~ r=0.09ÅE

MM: interactions via bonds

bending Eb = 1/2 kb(-0)2

E

0

Force field parameters: bond angles (Dreiding)

C O

H

Angle type 0 (°) kb(kcal/mol.rad2)

X--C(sp3)--X 109.471 100X--O(sp3)--X 104.510 100

E=3 kcal ~ =14°E

Force field parameters: torsion angles (Dreiding)

Etor = V1[1 - cos (-01) ] V2[1 - cos 2(-02)] V3[1 - cos 3(-03)]

E

0 60 120 180

C C

V3

Force field parameters: torsion angles (Dreiding)

torsion type n V (kcal/mol) 0 (°)

X--C(sp3)--C(sp3)--X 3 1.0 180X--C(sp2)--C(sp2)--X 2 22.5 0

C C

C

C

Non-bonded interactions: Van der Waals

E=D0[(r0/r)12-2(r0/r)6]

(Lennard-Jones)

E=D0{exp[a(r0/r)]-b(r0/r)6}

(Buckingham; “exp-6”)

repulsive: ~r-10

attractive: ~r-6

Non-bonded interactions: Coulomb (electrostatic)

--

atomic partial charges:

Eij=(qixqj)/(rij)

atomic/molecular multipoles:

E=ixj/Dr3

+

+

+

+

additional energy terms in force fields

* out-of-plane energy term

* Hydrogen bond energy term

MM energy calculation

EMM = Estretch + Ebend + Etorsion + Evdw + Ecoul + ...

bonded non-bonded

1

3

6

52

bonded non-bonded1…21…31…4 1…4: scaled 1…5 1…6/7/8

4

78

Some available force fieldsFF software focusGromos Gromos bioCharmm Charmm; Quanta bioAmber Amber bioTripos Sybyl general Dreiding Cerius generalCompass Cerius generalCVFF Cerius generalGlass2.01 Cerius ionic

Force field parameters:where do they come from?

1. Mimic physical properties of individual elements or atom types,producing a “physical” force field.

Properties can be taken from experimental data, or ab-initiocalculations.

Examples: Dreiding, Compass.

+ outcome will be ‘reasonable’, predictable; extension to newsystems relatively straightforward.- performance not very good.

Force field parameters:where do they come from?

2. Optimize all parameters with respect to a set of test data,producing a “consistent” force field.

Test set can be chosen to represent the system under investigation.

Examples: CFF, CVFF.

+ outcome often good for a particular type of systems, or aparticular property (e.g. IR spectrum).- extension to new systems can be difficult; no direct link to‘physical reality’

Force field parameters:where do they come from?

3. Apply common sense and look at what the neighbors do.

Examples: Gromos.

+ does not waste time on FF parameterization;resonable results.- ?

Atomic chargesWhy? To include the effect of the charge distribution over the system.

How?Assign a small charge to each atom.

Caveat: interaction with other force field parameters (e.g. VdW).

Some sp2 oxygens aremore negative than others.

Atomic chargesWhat is the atomic charge?

* Based on atomic electronegativity, optimized for a given FF.example: Gasteiger charges.

•Based on atomic electronegativity and the resulting electrical field.example: Charge Equilibrium charges (QEq).

* Based on the electronic distribution calculated by QM.example: Mulliken charges.

* Based on the electrostatic potential near the molecule,calculated by a non-empirical method (or determined experimentally).examples: Chelp, ChelpG, RESP.

Atomic charges

Properties and features of different charge schemes:

* Depends on molecular conformation?* Easy (=quick) to calculate?* Performance in combination with force field?

Known-to-be-good combinations:Tripos -- GasteigerDreiding -- ESPCompass -- Compass

Atomic charges:charges fitted to the ElectroStatic Potential (ESP)

mechanism:Coulomb interactions result from the electrostatic potentialaround a molecule.

HO

H

+ +

+

+

+

++

+

+

+--

- -

----

-- H+

Atomic charges:charges fitted to the ElectroStatic Potential (ESP)

molecule

QM

wave functionelectron density

sample true ESP

mathematical fit

atomic charges thatreproduce the true ESP

HO

H

sample point

for each sample point:atomsq/r= ESPQM

* atomic q as variables

Atomic charges:charges fitted to the ElectroStatic Potential (ESP)

Properties and features of different fitting schemes:

* Number of sample points.* Position of sample points.* Additional restraints (e.g. all qH in CH3 equal).* Fitting to multiple conformations.

Known-to-be-good fitting schemes:ChelpGRESP