Introduction to Computational Chemistry

1
Introduction to Computational Chemistry
Meredith J. T. Jordan m.jordan_at_chem.usyd.edu.au S
chool of Chemistry, University of Sydney
2
Format

• Introductory lectures from me
• Master Classes from people who know what they are
  doing
• Concurrent introductory structured workshops to
  introduce you to computational chemistry (me)

Peter Gill
Michelle Coote
Haibo Yu
Brian Yates
Tim Clark
3
Overview

• Computational Chemistry
• What is Computational Chemistry
• Overview
• What kinds of problems can we solve?
• What kinds of tools can we use?
• Some examples

4
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

5
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

6
What Properties can be Calculated?

• 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
• pKas
• Density matrix methods/geminals
• Linear Scaling (ie of the methods with number of
  electrons/basis functions)
• Local correlation methods
• Accurate enzyme-substrate interactions

7
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)

8
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.

9
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.

10
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.

11
Flow Chart for a Calculation

• Molecule
• Coordinates
• Program
• Molecular Properties
• Interpretation

• Cartesian
• internal
• different types for different purposes

• Supplied
• Graphically
• by hand

• many different ones
• AMBER, CHARMM,
• GROMOS, Sybyl
• AMPAC, MOPAC, VAMP
• Gaussian, Gamess, MOLPRO

• human input
• choice!
• Difficult

• structures
• energies
• molecular orbitals
• IR, NMR, UV

12
Overview of Methods

• Molecular mechanics, force fields
• easy to comprehend
• quickly programmed
• extremely fast
• no electrons limited interpretability
•  
• Semiempirical methods
• quantum method
• valence electrons only
• fast
• limited accuracy
• ab initio methods
• full quantum method

13
Atomic Units
Quantity Name Physical Significance Value in SI units
Energy Hartree 2 ? ionization energy of H 4.3597482 ? 1018 J
Length Bohr Bohr radius of H 1s orbital 0.529177249 ? 1010 m
Charge Electrons charge 1.60217733 ? 1019 C
Mass Electrons mass 9.1093898 ? 1031 kg
Velocity Electron velocity in 1 Bohr orbit 2.1876914 ? 106 m/s
Time time for electron to travel 1 Bohr radius 2.4188843 ? 1017 s
14
Starting Point The Schrödinger Equation

• Foundation is the Schrödinger Equation of quantum
  mechanics

H ? E ?

• E is the energy of the system
• ? is the molecular wavefunction. ? has no simple
  physical meaning but ?2 represents a probability
  distribution
• H 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)

15
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 ?
Re
16
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

17
Potential Energy Surfaces

• For any stationary point

18
Equilibrium Molecular Structures

• Stable structures are minima
• 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

19
Finding Minimum Energy Structures

• Gradient methods
• Steepest descents
• Conjugate gradient
• Second derivative methods
• Newton-Rhapson
• Quasi-Newton
• Fletcher Powell
• Rational Function Optimisation

?
20
Finding Minimum Energy Structures

• Monte Carlo Methods
• Metropolis sampling
• Simulated annealing
• Divide and Conquer
• Break the probleminto smaller, more tractable
  chunks

http//www.cs.gmu.edu/ashehu/?qProjectionGuidedE
xploration
21
Transition State Structures

• For transition state structures

?
22
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

23
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

24
Chemical Reactivity

• Reactions are paths on the surface
• the lowest energy path between reactants and
  products is called the intrinsic reaction path

25
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
• IR and Raman spectra
• zero-point vibrational energy (ZPVE)
• useful thermochemical quantities
• Reaction rate coefficients
• Isotopic substitution effects
• Tunneling corrections

26
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)