theoretical models “the underlying physical laws necessary for the mathematical theory of a large...

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.


• 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-r0)2

where r is the bond length, r0 a reference bond length and k is the harmonic force constant.

r

V r=r0


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 = Ebond + Ebend + Etorsion + EvdW + Eelec + Ecross


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 = Ebond + Ebend + Etorsion + EvdW + Eelec + Ecross


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: 20-35 cm–1 (at best!!), sometimes several hundred cm–1

• configurational sampling (in MD simulations): few kcal/mol

NB These are what he quotes…


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.



• 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.



• 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.


• 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


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


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


Accuracy

Depends…In general go for the most recent method: PM6

Quantity PM6 PM5 PM3 AM1 Units

Hf 8.01 22.19 18.20 22.86 kcal/mol

Bond Length 0.091 0.123 0.104 0.130 Angstrom

Angles 7.86 9.58 8.50 8.77 Degrees

Dipoles 0.85 1.12 0.72 0.67 Debye

Ionisation Potential

0.50 0.50 0.68 0.63 eV

Average Unsigned Errors wrt database of ~9000 species:

openmopac.net


Advantages and disadvantages

Method No. ofelements

Advantages over other methods

Disadvantages

MNDO/d 9 None Used MNDO for non-MNDO/d elements

AM1 37 Lanthanides as sparkles – pure ionic charges

PM3 37 Lanthanides as sparkles – pure ionic charges

PM5 51 Good torsion angles in biphenyls

Not published. No good statistics on accuracy

RM1 10 Most accurate dipoles and I.P.s

Limited range of elements

PM6 70 Most accurate Hf, geometries, good H-bonds;Under active development.

Zwitterions too stable, dipoles of low accuracy, non-bonded interactions often too strongopenmopac.net

theoretical models “the underlying physical laws necessary for the mathematical theory of a large...

Documents

accurate slide

molecular mechanics

force fields easy

force fields typical

harmonic force constant

bond lengths

mathematical theory

accurate methods