Title: An Introduction to Molecular Orbital Theory
1An Introduction to Molecular Orbital Theory
2Levels 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.
3Relative 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.
4Relative 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.
5Semi-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.
6Properties 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.
7History 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
8History...
- 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
9Basis 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.
10Basis 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
11Born-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.)
12Hartree-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.
13LCAO 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.
14Variational 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.
15Basis 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).
16Slater-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.
17Basis 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)
18Hartree-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)
19Semi-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).
20Steps 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.
21Comparison 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
22More 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
23Some 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)
24more 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