Introduction to
Computational Chemistry
Meredith J. T. Jordan
m.jordan@chem.usyd.edu.au
School of Chemistry, University of Sydney
Format
Introductory lectures from me
Master Classes from people who know what they are doing:
Haibo Yu
Peter Gill
Brian Yates
Tim Clark
Michelle Coote
Concurrent introductory structured workshops to introduce you to
computational chemistry (me)
Overview
Computational Chemistry:
• What is Computational Chemistry
• Overview
• What kinds of problems can we solve?
• What kinds of tools can we use?
• Some examples
What is Computational Chemistry?
Chemistry in the computer instead of in the laboratory
Use computer calculations to predict the structures, reactivities
and other properties of molecules
Computational chemistry has become widely used because of
• Dramatic increase in computer speed and the
• Design of efficient quantum chemical algorithms
The computer calculations enable us to
• explain and rationalize known chemistry
• explore new or unknown chemistry
Why do Chemistry on a Computer?
• Calculations are easy to perform whereas experiments are
difficult
• Calculations are safe whereas many experiments are
dangerous
• Calculations are becoming less costly while experiments are
becoming more expensive
• Calculations can be performed on any chemical system,
whereas experiments are relatively limited
• Calculations give direct information whereas there is often
uncertain in interpreting experimental observation
• Calculations give fundamental information about isolated
molecules without the complicating solvent effects
What Properties can be Calculated?
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Equilibrium structures
Transition State structures
Microwave, NMR spectra
Reaction energies
Reaction barriers
Dissociation energies
Charge distributions
Reaction Rates
Reaction Free Energies
Circular Dichroism (optical, magnetic, vibrational)
Spin-orbit couplings
Full relativistic energies
Excited States (vertical)
Solvent Effects
pKa’s
Density matrix methods/geminals
Linear Scaling (ie of the methods with number of electrons/basis functions)
Local correlation methods
Accurate enzyme-substrate interactions
Crystal structures (prediction)
Melting points
Protein folding
Full reaction dynamics
Molecular dynamics
Solvent dynamics
Systematic improvements of DFT
Excited States (adiabatic)…
What Properties can be Calculated?
In order of difficulty:
• Molecular Structures (+/– 1%)
• Reaction Enthalpies (+/– 2 kcal/mol)
• Vibrational Frequencies (+/– 10%)
• Reaction Free Energies (+/– 5 kcal/mol)
• Infrared Intensities (normally not too bad for fundamentals)
• Dipole Moments (depends…)
• Reaction Rates (errors vary enormously)
Conceptual Approach
• Validation
• Interpretation
• Prediction
“…give us insight, not numbers” C. A. Coulson
• It is absolutely essential that we know how accurate our
computed results are to be if they are to be of any use: we
want to get the right answer for the right reason.
• A celebrated target accuracy is “Chemical Accuracy”
ie to within 1 kcal/mol (~4 kJ/mol) in energy.
A Computational Research Project
What do you want to know? How accurately? Why?
This is your research project
How accurate do you predict the answer will be?
What is an appropriate method to use
How long do you expect it to take?
What method can you feasibly use
What approximations are being made? Which are significant?
Can you actually answer your questions
Once you have finally answered all of these questions, you must
determine what software is available, what it costs and how to
use it.
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 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.
Flow Chart for a Calculation
Molecule
Supplied
• Graphically
• by hand
Coordinates
human input:
• choice!
• Difficult
Program
Molecular Properties
Interpretation
• Cartesian
• internal
different types for
different purposes
many different ones:
• AMBER, CHARMM,
• GROMOS, Sybyl…
• AMPAC, MOPAC, VAMP…
• Gaussian, Gamess,
MOLPRO…
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structures
energies
molecular orbitals
IR, NMR, UV
Overview of Methods
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Molecular mechanics,
force fields
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– full quantum method
– only experimental fundamental
constants
– in principle very high accuracy
– complete (all interactions are
included)
– very time consuming (“expensive”)
– systematic improvement possible
easy to comprehend
quickly programmed
extremely fast
no electrons: limited interpretability
Semiempirical methods
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quantum method
valence electrons only
fast
limited accuracy
ab initio methods
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Density Functional Theory
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quantum method
in principle “exact”
faster than traditional ab initio
variable accuracy
no systematic improvement
Atomic Units
Quantity
Name
Physical Significance
Value in SI units
Energy
Hartree
2  ionization energy of H
4.3597482  10–18 J
Length
Bohr
Bohr radius of H 1s orbital
0.529177249  10–10 m
Charge
Electron’s
charge
Electron’s
mass
Mass
Velocity
Time
1.60217733  10–19 C
9.1093898  10–31 kg
Electron velocity in 1 Bohr orbit
2.1876914  106 m/s
time for electron to travel 1 Bohr
radius
2.4188843  10–17 s
Starting Point: The Schrödinger Equation
• Foundation is the Schrödinger Equation of quantum mechanics
H=E
• E
• 
• H
is the energy of the system
is the molecular wavefunction.  has no simple physical
meaning but 2 represents a probability distribution
is the Hamiltonian operator (a set of mathematical
operations) describing the kinetic energy (T) and the
potential energy (V) of the electrons and the nuclei
In principle we need to consider the electrons and nuclei in a
molecule together, in practice, nuclei move much slower and we
separate out electronic and nuclear motion
(the Born-Oppenheimer approximation)
Predicting the Structure of a Molecule
• The Schrödinger equation allows us to calculate the energy (E)
of a system as a function of geometry
H=E
Predict the structure of
carbon monoxide (CO)
Energy
Re
R(C…O)
Potential Energy Surfaces
The Born-Oppenheimer approximation lets us consider how
electronic energy changes with the nuclear geometry, giving a
Molecular Potential Energy Surface
• multidimensional (3N-6 dimensions)
• describes how energy varies as the atoms in the system move,
ie energy as a function of molecular displacement
• principally determined by what the bonding electrons
(the valence electrons) are doing
Potential Energy Surfaces
E
0
ri
• For any stationary point:
• For equilibrium structures:


2E
0
rirj

Local Minima
Equilibrium Molecular Structures
• Stable structures are minima
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energy curves upwards in all directions
curvature is positive: all vibrational frequencies are real
often there are lots of minima
the most stable structure is the global minimum
2E

rirj

Finding Minimum Energy Structures
• Gradient methods
– Steepest descents
– Conjugate gradient
• Second derivative methods
– Newton-Rhapson
– Quasi-Newton
• Fletcher Powell
• Rational Function Optimisation

Finding Minimum Energy Structures
• Monte Carlo Methods
– Metropolis sampling
– Simulated annealing
• Divide and Conquer
– Break the problem
into smaller,
more tractable chunks
http://www.cs.gmu.edu/~ashehu/?q=ProjectionGuidedExploration
Transition State Structures
• For transition state structures:


E
0
ri
2E
one
0

r
rj
i

Transition State
structure
Transition State Structures
• The maximum energy configuration along the reaction path
is called the transition state
– energy curves downwards in one direction only
– There is one imaginary vibrational frequency, all other
vibrational frequencies are real


2E

rirj

Finding Transition State Structures
• Newton-Rhapson type method
– Start with a good guess structure
– Start with accurate second derivatives
– Walk uphill following the least steep route


2E

rirj

Chemical Reactivity
• Reactions are paths on the surface
– the lowest energy path between reactants and products is called
the intrinsic reaction path
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

Vibrational Frequencies
• Indicate if the structure is a minimum (equilibrium structure all real frequencies – or a saddle point (transition state) – one
imaginary frequency – on the potential energy surface
• Allow us to calculate:
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IR and Raman spectra
zero-point vibrational energy (ZPVE)
useful thermochemical quantities
Reaction rate coefficients
Isotopic substitution effects
Tunneling corrections
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)