Theoretical Models “The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.” Paul Dirac 1929 (Nobel Prize 1933)

Requirements for Theoretical Models The exact solution of the Schrödinger equation is impractical for real systems  We need to devise approximate solutions (models)

What Tools Can We Use? Molecular Mechanics, Force Fields Semi-Empirical Methods Density Functional Theory Ab Initio Methods (from the beginning) Increasing Accuracy Increasing Computational Expense/ Smaller Systems

Assessment Golden Rule: Before applying a particular level of theory to an experimentally unknown situation it is essential to apply the same level of theory to situations where experimental information (for similar systems) is available Clearly unless the theory performs satisfactorily in cases where we know the answer, there is little point in using it to probe the unknown Conversely, if the theory does work well in known situations this lends confidence to the results obtained in the unknown case.

What Tools Can We Use? Molecular Mechanics, Force Fields easy to comprehend quickly programmed extremely fast  no electrons: limited interpretability  many properties may not be well defined  may not be sufficiently accurate

Molecular Mechanics, Force Fields What is a force field? “A mathematical expression describing the dependence of the energy of a molecule on all atomic coordinates” Eg A Simple Harmonic Oscillator, V= ½ k( r-r 0 ) 2 where r is the bond length, r 0 a reference bond length and k is the harmonic force constant. r V r=r 0

Molecular Mechanics, Force Fields More generally: Assumption: the definitions of bond and/or atom types are transferable from one molecule to another ie all atoms of the same type behave in the same way, regardless of their environment (never really true - be very careful!) Stretches normally harmonic not Morse Dispersion: form usually used (Lennard-Jones potential) is simple but wrong at short inter-atomic distances Electrostatic term: often dominant long range interaction, often just atom-centred charges… Cross Terms: large variety of forms in analogy to other terms, empirical. E = E bond + E bend + E torsion + E vdW + E elec + E cross

Molecular Mechanics, Force Fields More generally: Allows modelling of enormous molecules (e.g. proteins DNA) making it the primary tool of computational biochemists Structure Dynamics Together with more accurate methods in QM/MM It is not necessarily accurate It is limited to what it has been parametrized from E = E bond + E bend + E torsion + E vdW + E elec + E cross

Molecular Mechanics, Force Fields Typical Accuracy (Eg MM4 force field of Allinger et al.): ~0.03 Å Bond lengths ~5º bond angles few degrees for torsional angles conformational energies: accurate to 1 kcal/mol (at best – for parametrized molecules!!) more likely to be several kcal/mol vibrational frequencies: cm –1 (at best!!), sometimes several hundred cm –1 configurational sampling (in MD simulations): few kcal/mol NB These are what he quotes…

Molecular Mechanics, Force Fields Some Available Force Fields: CFF: (Consistent Force Field) Warshel, Lifson et al.; wide variety of experimental data, software for fitting force field parameters, parametrised to organic compounds, polymers, metals. MMFF: derived from both experimental and ab initio data, including HF and MP2 energies of torsion sampled structures and conformations. MM2/MM3/MM4: Allinger et al.; parametrised to heats of formations and small molecule gas phase data (particularly structures and conformational energies). Primarily for geometry optimization and prediction of thermodynamic values and IR spectral. MM3 and MM4 include hydrogen bonding.

Molecular Mechanics, Force Fields Some Available Force Fields: AMBER (Assisted Model Building with Energy Refinement) is the name of both a family of force fields developed for biomolecules by Peter Kollman, and a program for implementing them. AMBER uses harmonic stretches and bends, a cosine function for torsions, a Coulomb electrostatic interaction and a 12-6 Lennard-Jones van der Waals interaction. AMBER has been designed primarily for proteins and nucleic acids. CHARMM (Chemistry at HARvard Macromolecular Mechanics) is also a family of force fields and a program. CHARMM has all-atom and united atom variants and is widely used for drug molecules and macromolecules. One variant also includes the TIP3P force field for water, allowing it to be used as an explicit solvent.

Molecular Mechanics, Force Fields Some Available Force Fields: The GROMOS (GROningen MOlecular Simulation computer program package) force field and package were developed for biomolecular systems at the University of Groningen and at ETH in Zurich. GROMOS uses a united atom approach to fragments within biomolecules. There are both aqueous and gas phase versions. GROMACS (GROningen MAchine for Chemical Simulations) is the free molecular simulation “engine” that has grown out of GROMOS and can also support most of the other available force fields. Indeed AMBER, CHARM and GROMOS were all primarily developed for molecular dynamics.

Molecular Mechanics, Force Fields

What Tools Can We Use? Semi-Empirical Methods quantum method valence electrons only fast Parametrized: for molecules similar to those in the data set results may be very good Can be used to describe large sytems, up to ~10000 heavy (=non-hydrogen) atoms  limited accuracy  Parametrized

Semi-Empirical Methods Based on general structure of simple quantum calculations but instead of actually calculating difficult integrals they are either approximated or completely omitted. To overcome the errors introduced the method is parametrized to experimental data Much faster than ab initio calculations has been quite useful in organic chemistry results can be erratic limited to the parametrization set

Semi-Empirical Methods Some Available Methods Hückel Theory CNDO - Complete Neglect of Differential Overlap: spherically symmetric orbitals only INDO - Intermediate Neglect of Differential Overlap: one centre repulsion integrals between orbitals on the same atom. MINDO - Modified Intermediate Neglect of Differential Overlap: empirical data to parameterize the repulsion integrals rather than analytic solutions NDDO - Neglect of Diatomic Differential Overlap: includes directionality of orbitals on the same atom for repulsion integrals MNDO - Modified Neglect of Differential Overlap: better determination for multi-centre repulsion integrals

Semi-Empirical Methods Some Available Methods The MNDO methods: AM1, PM3 - PM6 were designed to reproduce heats of formation and structures of a large number of organic molecules. They describe ground electronic states only; they were not designed for electronic states or transition states Other semi-empirical methods are specifically optimized for spectroscopy, eg ZINDO/S or CNDO/S, which involve CI calculations and are quite good at prediction of electronic transitions in the UV/VIS spectral region

Semi-Empirical Methods Accuracy Depends… In general go for the most recent method: PM6 QuantityPM6PM5PM3AM1Units HfHf kcal/mol Bond Length Angstrom Angles Degrees Dipoles Debye Ionisation Potential eV Average Unsigned Errors wrt database of ~9000 species: openmopac.net

Semi-Empirical Methods Advantages and disadvantages MethodNo. of elements Advantages over other methodsDisadvantages MNDO/d9NoneUsed MNDO for non-MNDO/d elements AM137Lanthanides as sparkles – pure ionic charges PM337Lanthanides as sparkles – pure ionic charges PM551Good torsion angles in biphenylsNot published. No good statistics on accuracy RM110Most accurate dipoles and I.P.sLimited range of elements PM670 Most accurate  H f, geometries, good H-bonds; Under active development. Zwitterions too stable, dipoles of low accuracy, non-bonded interactions often too strong openmopac.net

Semi-Empirical Methods