An Introduction to
Molecular Orbital
Theory
Levels of Calculation
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
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.
Relative Computation “Cost”
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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.
Relative Computation “Cost”
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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.
Semi-Empirical
Molecular Orbital Theory
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
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.
Properties Calculated by
Molecular Orbital Theory
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Geometry (bond lengths, angles, dihedrals)
Energy (enthalpy of formation, free energy)
Vibrational frequencies, UV-Vis spectra
NMR chemical shifts
IP, Electron affinity (Koopman’s theorem)
Atomic charge distribution (...but charge is
poorly defined)
Electrostatic potential (interaction w/ point +)
Dipole moment.
History of Semi-Empirical
Molecular Orbital Theory

1930’s
1952
Hückel
Dewar

1960’s
Hoffmann
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1965
Pople
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1967
Pople
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treated  systems only
PMO; first semiquantitative application
Extended Huckel;
included bonds
CNDO; first useful MO
program
INDO
History...
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1975
1977
1985
Dewar
Dewar
Dewar
1989
1970’s
Stewart
Zerner
MINDO/3; was widely used
MNDO
AM1; added vdW
attraction & H-bonding
PM3; larger training set
ZINDO; includes transition
metals, parameterized for
calculating UV-Vis spectra
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.”
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
Born-Oppenheimer Approx.
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
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.)
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.
LCAO Approximation
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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.
Variational Principle
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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.
Basis sets
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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).
Slater-type orbitals (STO)
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
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 GTO’s
mimic 1 STO fairly well.
Basis Sets
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STO-3G (Slater-type orbitals approximated by 3
Gaussian functions)… a minimal basis set,
commonly used in Semi-Empirical MO
calculations.
(L-click here)
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)
Semi-empirical MO Calculations:
Further Simplifications
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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).
Steps in Performing a Semiempirical MO Calculation
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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 >100) if desired.
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
adegrees
--
4.3
3.3
3.9
More results...
Enthalpy of Formation, kcal/mol
ethane
propane
cyclopropane
cyclopentane
cyclohexane
MM3
-19.66
-25.32
12.95
-18.87
-29.95
PM3
-18.14
-23.62
16.27
-23.89
-31.03
Exp’t
--24.8
12.7
-18.4
-29.5
Some Applications...
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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)
…more Applications
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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