An Introduction to Molecular Orbital Theory

1
An Introduction to Molecular Orbital Theory
2
Levels of Calculation

• Classical (Molecular) Mechanics (previous 2
  lectures)
• quick, simple accuracy depends on
  parameterization no consideration of orbital
  interaction not MO theory)
• Molecular Orbital Theory (Quantum Mechanics)
• Ab initio molecular orbital methods...much more
  demanding computationally, generally more
  accurate.
• Semi-empirical molecular orbital methods
  ...computationally less demanding than ab initio,
  possible on a pc for moderate sized molecules,
  but generally less accurate than ab initio,
  especially for energies.

3
Relative Computation Cost

• Molecular mechanics...cpu time scales as square
  of the number of atoms...
• Calculations can be performed on a compound of
  MW 300 in a minute on a Pentium computer, or in
  a few seconds on a high performance computer.
• This means that larger molecules (even peptides)
  can be modeled readily by MM methods.

4
Relative Computation Cost

• Semi-empirical and ab initio molecular orbital
  methods...cpu time scales as the cube (or fourth
  power) of the number of orbitals (called basis
  functions) in the basis set.
• Semi-empirical calculations on MW 300 compound
  take a few minutes on a Pentium pc, or several
  seconds on a high performance computer.
• Ab initio calculations (to be discussed later) of
  such molecules can take hours.

5
Semi-Empirical Molecular Orbital Theory

• Uses simplifications of the Schrödinger equation
  E ? H ? to estimate the energy of a system
  (molecule) as a function of the geometry and
  electron distribution.
• The simplifications require empirically derived
  (not theoretical) parameters (fudge factors) to
  bring calculated values in agreement with
  observed values, hence the term semi-empirical.

6
Properties Calculated by Molecular Orbital Theory

• Geometry (bond lengths, angles, dihedrals)
• Energy (enthalpy of formation, free energy)
• Vibrational frequencies, UV-Vis spectra
• NMR chemical shifts
• IP, Electron affinity (Koopmans theorem)
• Atomic charge distribution (...but charge is
  poorly defined)
• Electrostatic potential (interaction w/ point )
• Dipole moment.

7
History of Semi-Empirical Molecular Orbital
Theory

• 1930s Hückel treated ? systems only
• 1952 Dewar PMO first semi- quantitative
  application
• 1960s Hoffmann Extended Huckel included
  ??bonds
• 1965 Pople CNDO first useful MO program
• 1967 Pople INDO

8
History...

• 1975 Dewar MINDO/3 was widely used
• 1977 Dewar MNDO
• 1985 Dewar AM1 added vdW attraction
  H-bonding
• 1989 Stewart PM3 larger training set
• 1970s Zerner ZINDO includes transition
  metals, parameterized for calculating
  UV-Vis spectra

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Basis of Molecular Orbital Theory

• Schrödinger equation
• E ? H ?
• (can be solved exactly ONLY for the Hydrogen
  atom,
• but nothing larger!!)
• P.A.M. Dirac, 1929 The underlying physical
  laws necessary for the mathematical theory of a
  large part of physics and the whole of chemistry
  are thus completely known.

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Basis of M.O. Theory...

• Three Simplifying assumptions are employed to
  solve the Schrödinger equation approximately
• Born-Oppenheimer approximation allows separate
  treatment of nuclei and electrons
• Hartree-Fock independent electron approximation
  allows each electron to be considered as being
  affected by the sum (field) of all other
  electrons.
• LCAO Approximation
• Variational Principle

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Born-Oppenheimer Approx.

• States that electron motion is independent of
  nuclear motion, thus the energies of the two are
  uncoupled and can be calculated separately.
• Derives from the large difference in the mass of
  nuclei and electrons, and the assumption that the
  motion of nuclei can be ignored because they move
  very slowly compared to electrons
• Htot a (Tn) Te Vne Vn Ve
• Kinetic energy
  Potential energy
• (Tn is omitted this ignores relativistic
  effects, yielding the electronic Schrödinger
  equation.)

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Hartree-Fock Approximation

• Assumes that each electron experiences all the
  others only as a whole (field of charge) rather
  than individual electron-electron interactions.
• Introduces a Fock operator F
• F????????
• which is the sum of the kinetic energy of an
  electron, a potential that one electron would
  experience for a fixed nucleus, and an average of
  the effects of the other electrons.

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LCAO Approximation

• Electron positions in molecular orbitals can be
  approximated by a Linear Combination of Atomic
  Orbitals.
• This reduces the problem of finding the best
  functional form for the molecular orbitals to the
  much simpler one of optimizing a set of
  coefficients (cn) in a linear equation
• ? c1 f1 c2 f2 c3 f3 c4 f4
• where ? is the molecular orbital wavefunction
  and fn represent atomic orbital wavefunctions.

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Variational Principle

• The energy calculated from any approximation of
  the wavefunction will be higher than the true
  energy.
• The better the wavefunction, the lower the energy
  (the more closely it approximates reality).
• Changes are made systematically to minimize the
  calculated energy.
• At the energy minimum (which approximates the
  true energy of the system), dEcalc-real 0.

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Basis sets

• A basis set is a set of mathematical equations
  used to represent the shapes of spaces (orbitals)
  occupied by the electrons and their energies.
• Basis sets in common use have a simple
  mathematical form for representing the radial
  distribution of electron density.
• Most commonly used are Gaussian type basis sets,
  which approximate the better, but more
  complicated Slater-Type orbitals (STO).

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Slater-type orbitals (STO)

• Slater-type orbitals describe the electron
  distribution quite well, but they are not simple
  enough to manipulate mathematically.
• Several Gaussian-type orbitals can be added
  together to approximate the STO. Here 4 GTOs
  mimic 1 STO fairly well.

17
Basis Sets

• STO-3G (Slater-type orbitals approximated by 3
  Gaussian functions) a minimal basis set,
  commonly used in Semi-Empirical MO calculations.

(L-click here)
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Hartree-Fock Self-Consistent Field (SCF) Method...

• Computational methodology
• guess the wavefunction (LCAO orbital
  coefficients) of all occupied orbitals
• compute the potential (repulsion) each electron
  would experience from all other electrons (taken
  as a group in the H-F approximation)
• solve for Fock operators to generate a new,
  improved wavefunction (orbital coefficients)
• repeat above two steps until the new wavefunction
  is not much improved at this point the field is
  called self-consistent. (SCF theory)

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Semi-empirical MO CalculationsFurther
Simplifications

• Neglect core (1s) electrons replace integral for
  Hcore by an empirical or calculated parameter
• Neglect various other interactions between
  electrons on adjacent atoms CNDO, INDO, MINDO/3,
  MNDO, etc.
• Add parameters so as to make the simplified
  calculation give results in agreement with
  observables (spectra or molecular properties).

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Steps in Performing a Semi-empirical MO
Calculation

• Construct a model or input structure from MM
  calculation, X-ray file, or other source
  (database)
• optimize structure using MM method to obtain a
  good starting geometry
• select MO method (usually AM1 or PM3)
• specify charge and spin multiplicity (s n 1),
  where n unpaired electrons, usually 0, so s
  usually is 1.
• select single point or geometry optimization
• set termination condition (time, cycles,
  gradient)
• select keywords (from list of gt100) if desired.

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Comparison of Results

• Mean errors relative to experimental
  measurements
• MINDO/3 MNDO AM1 PM3
• ?Hf, kcal/mol 11.7 6.6 5.9 --
• IP, eV -- 0.69 0.52 0.58
• ?, Debyes -- 0.33 0.24 0.28
• r, Angstroms -- 0.054 0.050 0.036
• ???degrees -- 4.3 3.3 3.9

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More results...

• Enthalpy of Formation, kcal/mol
• MM3 PM3 Expt
• ethane -19.66 -18.14 --
• propane -25.32 -23.62 -24.8
• cyclopropane 12.95 16.27 12.7
• cyclopentane -18.87 -23.89 -18.4
• cyclohexane -29.95 -31.03 -29.5

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Some Applications...

• Calculation of reaction pathways (mechanisms)
• Determination of reaction intermediates and
  transition structures
• Visualization of orbital interactions (formation
  of new bonds, breaking bonds as a reaction
  proceeds)
• Shapes of molecules including their charge
  distribution (electron density)

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more Applications

• QSAR (Quantitative Structure-Activity
  Relationships)
• CoMFA (Comparative Molecular Field Analysis)
• Remote interactions (those beyond normal
  covalent bonding distance)
• Docking (interaction of molecules, such as
  pharmaceuticals with biomolecules)
• NMR chemical shift prediction.
• As we will see, ab initio and DFT calculations
  generally give better results than Semi-empirical
  MO calculations