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
V= ½ k( r-r0)2
r=r0
r
where r is the bond length, r0 a reference bond length and k is the
harmonic force constant.
Molecular Mechanics, Force Fields
More generally:
E = Ebond + Ebend + Etorsion + EvdW + Eelec + Ecross
• 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.
Molecular Mechanics, Force Fields
More generally:
E = Ebond + Ebend + Etorsion + EvdW + Eelec + Ecross
• 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
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: 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…
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
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
Average Unsigned Errors wrt database of ~9000 species:
Quantity
PM6
PM5
PM3
AM1
Units
DHf
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
openmopac.net
Semi-Empirical Methods
Advantages and disadvantages
Method
No. of
elements
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 DHf, 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