Molecular Modeling of Crystal Structures molecules surfaces crystals
Dec 19, 2015
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