Lecture 16 Molecular Structure and
Thermochemistry
Structure = min[f(x1,y1,z1,…xN,yN,zN)]
©2013, Jordan, Schmidt & Kable
Lecture 16
Copyright Notice
Some images used in these lectures are taken, with permission, from
“Physical Chemistry”, T. Engel and P. Reid, (Pearson, Sydney, 2006);
denoted “ER” throughout the lectures
and other sources as indicated, in accordance with
the Australian copyright regulations.
©2013, Jordan, Schmidt & Kable
Lecture 16
Learning outcomes
16.1 Recall that equilibrium geometries can be calculated by
minimizing the energy, often using Newton’s method.
16.2 Know that the calculated geometries and energies depend
on the basis sets used and the level of correlation.
16.3 Recognize MP2 as a method that can recover correlation
for ground states, but is not variational.
16.4 Recognize G1, G2 and G3 as composite methods which try
to get to “chemical accuracy” – 1 kcal/mol
©2013, Jordan, Schmidt & Kable
Lecture 16
The adiabatic Born-Oppenheimer approximation
Curves like this form a landscape on
which molecules vibrate and react.
-0.90
-0.92
-0.94
-0.96
Implicitly, we assume that the
electrons instantaneously adapt to
the positions of the nuclei. This is the
adiabatic Born-Oppenheimer
approximation.
Energy (hartree)
-0.98
-1.00
-1.02
-1.04
-1.06
-1.08
-1.10
-1.12
-1.14
-1.16
-1.18
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
H-H distance (Å)
This approximation is the foundation of chemical physics, and only breaks
under certain circumstances, for instance where two electronic states are
close in energy.
Internal conversion is an example of Born-Oppenheimer breakdown.
©2013, Jordan, Schmidt & Kable
Lecture 16
Potential energy surfaces – electronic states
Here are pictured the various lowlying electronic states of C2. Each
has its own potential energy curve
and is described well by one
configuration at equilibrium.
However, as the molecule
dissociates, there are many
instances where CI is important.
The emission from d to a is
responsible for the blue coloration
of lean flames and was observed in
comets since the 19th century. Scott
will teach you all about that stuff.
©2013, Jordan, Schmidt & Kable
Lecture 16
Potential energy surfaces – reactions
Here is pictured the classic potential
energy surface (PES) for a linear
harpoon reaction with a barrier:
A + BC→AB + C.
The reaction barrier appears as a
mountain pass, and is called a saddle
point – it has one direction with
negative curvature, like a saddle.
If the PES is known with sufficient
accuracy, then dynamical calculations
can be made which predict the
outcome of chemical reactions.
Meredith like this. Scott likes
Meredith doing this.
©2013, Jordan, Schmidt & Kable
Lecture 16
Contour Plots of acetylene H-C☰C-H
Ground state acetylene
(draw yourself)
Excited state acetylene
©2013, Jordan, Schmidt & Kable
Lecture 16
Very strange world of Jahn-Teller
In the E” excited state of phenalenyl radical, there are e’ vibrational modes
which lift the electronic degeneracy. The potential energy surface splits into
two at a conical intersection. An adiabatic walk around the conical
intersection accompanies a change in sign of the electronic wavefunction…
BO approximation is completely broken here. (Tim likes doing this…)
©2013, Jordan, Schmidt & Kable
Lecture 16
Energy Minimization
Classically speaking, the structure of a molecule at 0K can be described by
the minimum energy on the electronic ground state potential energy
surface.
This is the minimum of the electronic energy PLUS the nuclear repulsion
energy.
angle
Minimum energy for water
is here:
ROH
©2013, Jordan, Schmidt & Kable
Lecture 16
Energy Minimization
If gradients are available, then the minimum can be found quickly using
Newton’s method, solving for dV/dq = 0
NSERCH=
NSERCH=
NSERCH=
NSERCH=
NSERCH=
NSERCH=
NSERCH=
0
1
2
3
4
5
6
ENERGY=
ENERGY=
ENERGY=
ENERGY=
ENERGY=
ENERGY=
ENERGY=
-75.8864335
-75.9452753
-76.0012472
-76.0060757
-76.0109427
-76.0109543
-76.0109546
----------------------GRADIENT (HARTREE/BOHR)
----------------------ATOM
ZNUC
DE/DX
DE/DY
DE/DZ
-------------------------------------------------------------1 O
8.0
0.0000000
0.0000000
-0.0000514
2 H
1.0
0.0000000
0.0000176
0.0000257
3 H
1.0
0.0000000
-0.0000176
0.0000257
1
MAXIMUM GRADIENT = 0.0000514
RMS GRADIENT = 0.0000225
***** EQUILIBRIUM GEOMETRY LOCATED *****
COORDINATES OF ALL ATOMS ARE (ANGS)
ATOM
CHARGE
X
Y
Z
-----------------------------------------------------------O
8.0
0.0000000000
0.0000000000 -0.0470528019
H
1.0
0.0000000000 -0.7832785007 -0.5764735991
H
1.0
0.0000000000
0.7832785007 -0.5764735991
Starting at wrong geometry, convergence
in just 6 steps. (RHF/6-311G)
©2013, Jordan, Schmidt & Kable
Lecture 16
Geometry depends on basis set
Since the energy depends on the basis set, so does the potential energy
surface and thus so does the equilibrium geometry.
Method
Bond
length
angle
Energy (Eh)
RHF/STO-3G
0.9894
100°
-74.9659
RHF/3-21G
0.9667
108°
-75.5860
RHF/6-31G
0.9497
112°
-75.9852
RHF/6-311G(d,p)
0.9411
105°
-76.0471
MP2/6-311G(d,p)
0.959
102°
-76.2679
©2013, Jordan, Schmidt & Kable
Spectroscopic determination
Lecture 16
What is MP2?
MP2 is a non-variational theory which takes account of excited determinants
(correlation) in the ground state wavefunction using second order
perturbation theory.
It is named for Moller and Plesset who published the original idea.
It is based on Raleigh-Schrödinger perturbation theory (RS-PT). This theory
makes use of the fact that corrections to the wavefunction are small in order
to make a first order correction to the wavefunction, which gives a second
order correction to the energy.
The calculation will fail if the ground state is not dominated by a single
configuration.
©2013, Jordan, Schmidt & Kable
Lecture 16
Geometry depends on level of theory
Since the energy depends on the level of theory, so does the potential
energy surface and thus so does the equilibrium geometry.
Complete
basis set
Data from
Jonathon Tennyson
©2013, Jordan, Schmidt & Kable
aug-cc-pV5Z – quintuple-zeta basis, 5 functions
to describe atomic orbitals in the valence shell
ICMRCI – internally contracted multireference
configuration interaction
Lecture 16
Chemistry
There are a large number of people who calculate important points on PESs
– transition states and equilibrium structures – to determine the energetics
of reactions.
Optimizing to a transition state finds one imaginary frequency
(negative force constant, corresponding to the barrier).
TS
eq
©2013, Jordan, Schmidt & Kable
eq
Lecture 16
Chemistry
Complicated reactions such as propargyl radical recombination to make
benzene in flames consist of many transition states and minima.
©2013, Jordan, Schmidt & Kable
Lecture 16
Chemistry
You can go crazy.
©2013, Jordan, Schmidt & Kable
Lecture 16
Chemistry
©2013, Jordan, Schmidt & Kable
Lecture 16
How to be chemically accurate?
For chemistry, we would like energies to be accurate to within about 4
kJ/mol. So we need to use a very good method, but it would be very
expensive to do everything with configuration interaction at the complete
basis set limit.
Geometry optimizations and frequency calculations (for zero point energies)
take a lot of effort, and cannot be done with high levels of theory.
We could calculate geometries at one level, and energies at another, e.g.
CCSD/6-31G(d)//B3LYP/6-31G(d). Here we get the density function theory
geometry and then use a so-called coupled-cluster calculation to get the
energy.
But, smart people (Curtiss, Pople) have developed combinations of methods
which end up being chemically accurate, with more modest cost.
©2013, Jordan, Schmidt & Kable
Lecture 16
Model Chemistries
The Gaussian-1 method, for instance
©2013, Jordan, Schmidt & Kable
Lecture 16
Model Chemistries
©2013, Jordan, Schmidt & Kable
Lecture 16
Model Chemistries
©2013, Jordan, Schmidt & Kable
Lecture 16
Model Chemistries: G1, G2
1989
©2013, Jordan, Schmidt & Kable
1991
Lecture 16
G3 Theory 1998
J. Chem. Phys. 109, 7764 (1998);
Gaussian-3 (G3) theory for molecules containing first and second-row atoms
Larry A. Curtiss, Krishnan Raghavachari, Paul C. Redfern, Vitaly Rassolov, and John A. Pople
The Nobel Prize in Chemistry 1998 was divided equally between Walter Kohn "for his
development of the density-functional theory" and John A. Pople "for his development of
computational methods in quantum chemistry".
©2013, Jordan, Schmidt & Kable
Lecture 16