Molecular Modeling of 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 neglect of some parts of the ab-initio calculation. MOPAC (mopac6, 7, 93, 2000) Empirical methods: observed/fitted values for interactions between atoms. Sybyl, Cerius 2, 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 equations giving U as function of atomic coordinates G = U + pV - TS H = U + pV E MM = U

Molecular Mechanics system from atoms + bonds stretching bending torsion CCH H HH H H E MM = E stretch + E bend + E torsion + E vdw + E coul +... bondednon-bonded

MM: interactions via bonds bond stretch r E s = 1/2 k s (r-r 0 ) 2 … + C 3 (r-r 0 ) 3 + C 4 (r-r 0 ) 4 r E - True.. modeled via (r-r 0 ) 2 r0r0

Force field parameters: bond lengths (Dreiding) CC C C Bond type r 0 (Å) k s (kcal/mol.A 2 ) C(sp 3 )--C(sp 3 ) C(sp 3 )--C(sp 2 ) C(sp 2 )--C(sp 2 ) C(sp 3 )--H E s = 1/2 k s (r-r 0 ) 2 k S =700:  E=3 kcal ~  r=0.09Å E

MM: interactions via bonds bending E b = 1/2 k b (  -  0 ) 2   E 00

Force field parameters: bond angles (Dreiding) CO H Angle type  0 (°) k b (kcal/mol.rad 2 ) X--C(sp 3 )--X X--O(sp 3 )--X  E=3 kcal ~  =14° E

Force field parameters: torsion angles (Dreiding) E tor = V 1 [1 - cos (  -  01 ) ] V 2 [1 - cos 2(  -  02 )] V 3 [1 - cos 3(  -  03 )] E CC V3V3

Force field parameters: torsion angles (Dreiding) torsion type n V (kcal/mol)  0 (°) X--C(sp 3 )--C(sp 3 )--X X--C(sp 2 )--C(sp 2 )--X CC C C

Non-bonded interactions: Van der Waals E=D 0 [(r 0 /r) 12 -2(r 0 /r) 6 ] (Lennard-Jones) E=D 0 {exp[a(r 0 /r)]-b(r 0 /r) 6 } (Buckingham; “exp-6”) repulsive: ~r -10 attractive: ~r -6

Non-bonded interactions: Coulomb (electrostatic) - - atomic partial charges: E ij =(q i xq j )/(  r ij ) atomic/molecular multipoles: E=  i x  j /Dr

additional energy terms in force fields * out-of-plane energy term * Hydrogen bond energy term

MM energy calculation E MM = E stretch + E bend + E torsion + E vdw + E coul +... bondednon-bonded bonded non-bonded 1…2 1…3 1…4 1…4: scaled 1…5 1…6/7/

Some available force fields FF software focus Gromos Gromos bio Charmm Charmm; Quanta bio Amber Amber bio Tripos Sybyl general Dreiding Cerius general Compass Cerius general CVFF Cerius general Glass2.01 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-initio calculations. Examples: Dreiding, Compass. + outcome will be ‘reasonable’, predictable; extension to new systems 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 a particular 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 charges Why? 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 sp 2 oxygens are more negative than others.

Atomic charges What 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 -- Gasteiger Dreiding -- ESP Compass -- Compass

Atomic charges: charges fitted to the ElectroStatic Potential (ESP) mechanism: Coulomb interactions result from the electrostatic potential around a molecule. H O H H+H+

Atomic charges: charges fitted to the ElectroStatic Potential (ESP) molecule QM wave function electron density sample true ESP mathematical fit atomic charges that reproduce the true ESP sample point for each sample point:  atoms q/  r= ESP QM * 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 q H in CH 3 equal). * Fitting to multiple conformations. Known-to-be-good fitting schemes: ChelpG RESP