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Molecular Modeling of Crystal Structures

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

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Molecular Modeling of Crystal Structures

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  1. Molecular Modelingof Crystal Structures molecules surfaces crystals

  2. 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, Cerius2, Gromos, ...

  3. 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

  4. 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 EMM = U

  5. EMM = Estretch + Ebend + Etorsion + Evdw + Ecoul + ... bonded non-bonded Molecular Mechanicssystem from atoms + bonds H H • stretching • bending • torsion H C C H H H

  6. MM: interactions via bonds r Es = 1/2 ks(r-r0)2 … + C3(r-r0)3 + C4(r-r0)4 bond stretch E - True .. modeled via (r-r0)2 r0 r

  7. C C C C E Force field parameters: bond lengths (Dreiding) Es = 1/2 ks(r-r0)2 Bond type r0 (Å) ks (kcal/mol.A2) C(sp3)--C(sp3) 1.53 700 C(sp3)--C(sp2) 1.43 700 C(sp2)--C(sp2) 1.33 1400 C(sp3)--H 1.09 700 kS=700: E=3 kcal ~ r=0.09Å

  8. MM: interactions via bonds bending Eb = 1/2 kb(-0)2  E  0

  9. E Force field parameters: bond angles (Dreiding) Angle type 0 (°) kb(kcal/mol.rad2) X--C(sp3)--X 109.471 100 X--O(sp3)--X 104.510 100 C O H E=3 kcal ~ =14°

  10. C C Force field parameters: torsion angles (Dreiding) Etor = V1[1 - cos (-01) ] V2[1 - cos 2(-02)] V3[1 - cos 3(-03)] E V3 0 60 120 180

  11. Force field parameters: torsion angles (Dreiding) C C torsion type n V (kcal/mol) 0 (°) X--C(sp3)--C(sp3)--X 3 1.0 180 X--C(sp2)--C(sp2)--X 2 22.5 0 C C

  12. Non-bonded interactions: Van der Waals repulsive: ~r-10 attractive: ~r-6 E=D0[(r0/r)12-2(r0/r)6] (Lennard-Jones) E=D0{exp[a(r0/r)]-b(r0/r)6} (Buckingham; “exp-6”)

  13. Non-bonded interactions: Coulomb (electrostatic) atomic partial charges: Eij=(qixqj)/(rij) atomic/molecular multipoles: E=ixj/Dr3 + - + - + +

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

  15. MM energy calculation EMM = Estretch + Ebend + Etorsion + Evdw + Ecoul + ... bonded non-bonded bonded non-bonded 1…2 1…3 1…4 1…4: scaled 1…5 1…6/7/8 5 1 2 3 4 8 6 7

  16. 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

  17. 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.

  18. 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’

  19. 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. - ?

  20. Atomic charges Why? To include the effect of the charge distribution over the system. Some sp2 oxygens are more negative than others. How? Assign a small charge to each atom. Caveat: interaction with other force field parameters (e.g. VdW).

  21. 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.

  22. 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

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

  24. H sample point O H Atomic charges:charges fitted to the ElectroStatic Potential (ESP) molecule QM wave function electron density sample true ESP mathematical fit for each sample point: atomsq/r= ESPQM * atomic q as variables atomic charges that reproduce the true ESP

  25. Atomic charges:charges fitted to the ElectroStatic Potential (ESP) Properties and features of different fittingschemes: * 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: ChelpG RESP

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