Title: An Introduction to Molecular Orbital Theory
1An Introduction to Molecular Orbital Theory
2Levels of Calculation
- Classical (Molecular) Mechanics
- 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 few minutes on a Pentium computer,
or in a few seconds on the SGI. - This means that larger molecules (even peptides)
and be modeled 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 10 minutes on a Pentium pc, less
than one minute on our SGIs, a second on a
supercomputer.
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 (...ill 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 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...
- 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), dE 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 basis sets, which
approximate the better, but more complicated
Slater-Type orbitals (STO).
16Slater-type orbitals (STO)
- Slater-type orbitals describe the electron
distribution fairly well, but they are not simple
enough to manipulate mathematically. - Several Gaussian-type orbitals can be added to
approximate the STO. Four GTOs mimic one STO.
17Basis Sets
- STO-3G (Slater-type orbitals approximated by 3
Gaussian functions)minimal basis set, commonly
used in Semi-Empirical MO calculations.
(L-click here)
18Hartree-Fock Self-Consistent Field (SCF) Method...
- Computational methodology
- guess the orbital occupation (position) of an
electron - guess the potential each electron would
experience from all other electrons (taken as a
group) - solve for Fock operators to generate a new,
improved guess at the positions of the electrons - repeat above two steps until the wavefunction for
the electrons is consistent with the field that
it and the other electrons produce (SCF).
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 M O
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)
- select single point or geometry optimization
- set termination condition (time, cycles,
gradient) - select keywords (from list of gt100).
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.