1. MODELING MATTER AT NANOSCALES 3. Empirical classical PES and typical procedures of optimization 3.01. Classical potentials

2. Molecular mechanics

3. The energy of the hypersurface or potential energy surface (PES) corresponding to a nanoscopic system can be simulated with any function that fits, or approximate, the behavior of the real system.

4. Molecular mechanics In order to model the very complex supramolecular systems and many other of biological origin, different functions have been developed consisting in the more influencing physical terms on the total energy in an approximate and additive way.

5. Molecular mechanics molecular mechanics The theory of molecular mechanics* was one of the first ways to simulate hypersurfaces on the grounds of classical potential functions. Under such approach, several sets of formulas and procedures were developed on the grounds of expected classical behaviors of interactions among nano and picoscopic bodies. * See: Simonetta, M., Structural investigations in organic molecules and crystals by means of molecular mechanics and x- ray diffraction. Acc. Chem. Res. 1974, 7 (10), 345-350.

6. Basis considerations of classical potentials 1.Considering a polyatomic system as being sets of spheres with a mass depending, approximately, on the element of their corresponding nuclei, and being linked together by virtual springs in a convenient way according their chemical structure.

7. Basis considerations of classical potentials 2.Considering an extended composition, that not only takes into account the elements in the periodic table, but also those more frequent forms of participation of each element in typical molecular environments. It deals with type atoms and not with chemical element atoms.

8. A typical formulation: MM2

9. MM2 Several formulations exist of classical potentials (or hypersurfaces). Those developed by Norman Allinger can be taken as paradigmatic, and particularly the so – called MM2 [ Allinger, N. L., Conformational analysis. 130. MM2. A hydrocarbon force field utilizing V1 and V2 torsional terms. J. Am. Chem. Soc. 1977, 99, (25), 8127-8134 ]

10. MM2: Energy between two centers Bonding interactions between two atoms in a molecule, or between two of them, are fitted by an armonic potential function, corresponding to an ideal oscillator. However, the best fit is given by the Morse’s potential that has been obtained long ago for calculations of diatomic molecules:

11. MM2: Energy between two centers In the case of most classic potentials, including MM2, bonding interactions of this type use to be simulated according an empirical harmonic potential or a modified Hooke’s law: where. Here and are specific parameters for each kind of atom type pair AB

12. MM2: Angle of tree centers The function for angles of tree centers is a kind of modified harmonic potential: where and are specific for each ordered trio of atom types.

13. MM2: Angle four centers: torsions or dihedral angles For 3D angles (dihedral) Fourier series are employed: where,, and are specific for each ordered four member set of atom types.

14. MM2: Electrostatics The electrostatic potential is Coulombic with parameterized charges: where q­­ A and q B are fixed parameters for each atom type.

15. MM2: Non – bonding interactions The non – bonding potential (van der Waals) in MM2 parameterization cis given by the Buckingham formulation: Where and. The values of and r A are optimized parameters for each type atom. In addition, if r AB /r > 3.311, then

16. MM2: Interactions between distances and angular deformations The energy between two atoms is NOT independent of that existing among them and a third one in the neighborhood. It means that each term in these potentials is not necessarily orthogonal with respect to others. It is used the term “coupling” between distances AB and BC with bendings ABC (stretch- bending coupling) if the triad has a common atom in the vertix: where when B is an atom in the first row of the periodic table, when B is an atom in the second row, when B is an atom in the first row and C is hydrogen, when B is an atom in the second row and C is hydrogen Atom A can be indistinctly in the first or the second row. In the case when C is hydrogen the term

17. MM2: Dipole – dipole interactions Local dipole moments (diatomic) can generate interaction energies in molecules, and can be calculated from the expression: where dipole moments are oriented from + to – and are considered centered as points in their middle length. This interaction could not be considered when both bond dipoles have a common center.

18. MM2: Aromatic and conjugated systems In cases of electronic conjugation or aromaticity were there two ways to afford the problem: 1.Doing local calculations of the conjugated system by the SCF – PPP approximation for the p type component of each bond. 2.Considering that atoms involved in conjugation are a separate type atom with respect to other similar.

19. MM2: Total energy The final solution of a molecular mechanical calculation is then the sum of all different contributions to the energy of the system of each and all atoms or atom subsets and thus obtaining the total energy.

20. Some examples

21. MM2: Examples

22. Case of the bonding potential between two centers:

23. MM2: Examples Case of the angular potential among tree centers:

24. MM2: Examples Angular potential among four centers: torsions or dihedral angles:

25. MM2: Examples Potential for non - bonding interactions at intermediate distances:

26. Recommended references: Cramer, C. J., Essentials of Computational Chemistry. Theories and Models. 2nd. ed.; John Wiley & Sons Ltd: Chichester, 2004; p 596. Allinger, N. L., Conformational analysis. 130. MM2. A hydrocarbon force field utilizing V1 and V2 torsional terms. J. Am. Chem. Soc. 1977, 99 (25), 8127-8134.