In this lab we will consider two separate topics.
The first topic that we will consider this lab is the influence of special
relativity on chemical properties. The Schrödinger equation is not
relativistic. In turns out that this leads to significant errors
in some calculations. We will look at an approximate way to include
relativity in our calculations. Further, we will consider when relativity is important and when it is not.
In the second topic we will study how to include the influence of a solvent
in our calculations. We will describe the
solvent first by discrete molecules and then later through a continuum model.
The advantages and disadvantages of the two methods will be compared.
A combination of discrete molecules and a continuum model is possible.
Along with quantum mechanics, Einstein's theories of
relativity form the basis for much of the new
physics developed in the 20th centrury. Relativity comes in
two flavours, general and special. General relativity usually
applies to large masses and the theory of gravity. Gravity
is so unimportant to the properties of molecules that we
almost always neglect it and the influence of general
relativity in computational chemistry calculations.
Special relativity on the other hand can
be very important.
Special relativity has two basic postulates: that the speed
of light is constant in any given reference frame and that
the laws of physics are invariant to choice of reference frame.
The Shrödinger equation does not
follow these postulates and as such is a nonrelativistic equation.
An equation equivalent to the Schrödinger equation but also
following the postulates of relativity was derived not long after
the Schrödinger equation. This second equation is called
the Dirac equation.
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(8.1)
#1The Dirac equation of a molecule with the electron-electron
interaction of Breit.
What happens if we describe a molecule with the Dirac equation
rather than the Schrödinger equation? If we consider a very simple
system, a nucleus with one electron we find that the energy of that electron
that we get with the Dirac equation is lower than what we get with the
Schrödinger equation. Furthermore, the difference the relativistic and
nonrelativistic
energies becomes bigger as the charge on the nucleus
becomes bigger.
If we go to many-electron atoms things are a bit more complicated.
The lowest energy electrons (the 1s electrons) are stabilized by relativity. However,
now we also have the electrons interacting with each other. If an electron
is stabilized its orbital is smaller and it shields the nucleus better. This
means that other electrons feel the nucleus' charge less and they are destabilized.
So, some electrons are stabilized and some are destabilized. Which electrons do what?
It turns out that all s electrons are stabilized because their orbitals have some amplitude
very near the nucleus and so they feel
the nuclear charge at its strongest. p orbitals are not stabilized or destabilized much and d and f
orbitals are destabilized. Thus, in comparison to a nonrelativistic calculation
s orbitals are stabilized and contracted (smaller), p orbitals are not changed so much
and d and f orbitals are destabilized and expanded. Once we go
to molecules things become even more complicated but they can often
be understood in terms of what happens to the atomic orbitals.
Relativity also has a second influence beyond stabilizing electrons. This
second influence is spin-orbit coupling which does not exist in nonrelativistic
descriptions of the world. As we will see in the next lab, spin-orbit coupling has
an influence in NMR and EPR. Spin-orbit coupling appears in a lot of places because it
breaks symmetry. Some forbidden electronic transitions become allowed once
spin-orbit coupling is included.
These contractions and decontractions, stabilizations and destabilizations and spin-orbit splittings are
called relativistic effects. Relativistic effects in chemistry are the term given to
anything that arises because of special relativity. Relativity is always present in
reality but we can calculate relativistic effects by simply calculating the same thing
twice, once with a nonrelativistic approach and once with a relativistic approach.
Just like the one-electron atoms, the relativistic effects on the properties of
a molecule become larger as the atoms in that molecule become heavier. It turns
out that the errors created by neglecting relativity become unacceptably large
once one of the atoms in a molecule is a 4d transition metal or
heavier. Whether an error is unacceptably large or not will depend on the
required accuracy but the cutoff described above should work in
most cases where chemical phenomena are considered. If the
phenomena of interest is caused by the presence of spin-orbit coupling
(for example, g-factors) then of course spin-orbit coupling
must be considered somehow.
Figure 8.1: Choose to run a relativistic calculation.
In turns out that the Dirac equation is pretty horrible to solve so
we want to approximate it somehow. There are many ways to
approximate the Dirac equation. Two are implemented in ADF.
The first is called the zero-order regularized approximation or ZORA for
short. The other, somewhat older, method makes use of the Pauli Hamiltonian.
To choose to perform a relativistic calculation, use the Model: Relativity
menu option. This will produce just two buttons (figure 8.2). The
first chooses what kind of relativity to include: None (the default)
Scalar only or Scalar and spin-orbit coupling. The second button chooses
the method of including relativity, ZORA or Pauli.
In general, we would recommend using the ZORA form, especially when very heavy
elements (5d transition metals and heavier) are involved but there are
occasions when the Pauli method is appropriate.
Exercise 8.1
Calculate the energy of the Au atom with and without scalar relativistic
corrections.
The valence configuration of
gold is 5d106s1 so don't forget to choose an unrestricted calculation
and the correct spin polarization. Use a QZ3P basis set and the BP functional.
Compare the energies of the 5d and 6s electrons and the 6p virtual orbital
of Au in the two calculations.
Which are stabilized, which are destabilized by relativity and how much?
Gold is gold coloured (metallic yellow) because it absorbs violet light.
The transition responsible for the colour of gold is 5d ® 6s. If we
make the rough approximation that the energy of this transition is equal to
the difference in the eigenvalues of the 6s and 5d orbitals, what colour
is nonrelativistic gold?
In all of the calculations that we have done up to this point the molecules and atoms
were isolated. We were studying molecules placed in a vacuum. This approach is a good approximation to
a molecule in the gas phase. It is sometimes a good approximation to a
solvated molecule. Often it is not. The solvent may play an important
role in structure, reaction energies and many other issues
of chemical importance. The question is, how to include the influence of the solvent?
Including millions of solvent molecules in a calculation is not an option. The wide range of
ways that have been tried to approximately include the influence
of solvation in molecular calculations usually fall into one of two
main classes: explicit solvent molecules and contiuum models.
Figure 8.2: A molecule of H2O shuttling a proton. The
solvent molecule must be included in any calculation of
this reaction.
While it is not possible to include a million molecules in a calculation, a
few can be included. "A few" could mean 1 or 2 or 100 or 200 depending
on how big the solvent molecules are and how big your computer is.
Sometimes a small number of molecules is a suitable replacement for the bulk
solvent.
Sometimes a solvent molecule is of crucial importance to
the property of interest. It may act as a ligand in a complex or actually
take part in the reaction under investigation. Sometimes what the solvent
molecules are doing is the topic that you are interested in. In these cases, solvent
molecules must be included in a calculation.
Aside: These days, it is actually possible to include a million solvent molecules
in your calculation if you describe those molecules in a very simple, non-quantum mechanical way. We
will not consider this option here and will limit ourselves to describing
explicit molecules quantum mechanically.
Exercise 8.2
Optimize the geometry of the ScCl3 molecule.
Repeat the calculation but this time add a molecule of H2O.
Repeat both of these calculations but with CH4 in place of ScCl3. Compare
the geometry of ScCl4 and CH4 in the gas phase
with that of the same molecules "solvated" by a single H2O
molecule. Use the LDA functional and a
a DZP basis set. These calculations may require
a fairly large number of geometry optimization steps. Change the
default maximum value from 30 to a larger number of your choice
in the Details: Task: GeometryOptimization
option. It's not obvious ahead of time exactly where the
H2O molecule should go. Make a good guess. The calculations including a
solvent molecule might take a while (10-30 minutes on one processor) so
it is recommended that they be run on 2 processors.
Figure 8.3: A molecule solvated by a continuum model of the solvent.
Often a solvent molecule will not bond directly to the molecule of interest or
participate in a reaction that is under investigation. The solvent
can still be influential however by stabilizing a product, reactant
or transition state. Solvent molecules will have a dipole moment or at
least be polarizable. These dipolar or polarizable species will
interact with charge build up in a molecule and stabilize it.
A solvent may then affect the structure and properties of a molecule
through this interaction especially if the solvent and/or molecule are
especially polar. How a solvent will influence the progress of a reaction is
difficult to predict ahead of time because the solvent will interact differently
with the reactants, products, intermediates and transition
states of that reaction.
Including this "dipole/polarizable medium" interaction in a calculation through the
addition of discrete solvent molecule is difficult because in order
to do a reasonable job a very large number of solvent molecules
must be included.
An alternative approach is to use a "continuum model" of the solvent. What
is meant by "continuum model" is that we describe the solvent
as a single continuous blob with appropriate electrostatic properties
rather than as isolated molecules. The solvation
is achieved by digging an appropriately sized hole out of the blob,
putting the molecule into the hole and allowing the electrons and
nuclei of the molecule to interact with the blob (figure 8.4).
Numerous different continuum models exist. The differences
between each of these models will depend on the technical
details of the calculation. A few choices
can be made about how the blob will behave. For example, it
can be treated as a dietlectric or a conductor. Also, a number of
choices about how to define the hole in the blob can be made.
Several
other choices can be made.
Each continuum model contains some parameters. It could be many or few.
These parameters must come from somewhere. Often they fitted to
experimental data. An experimental parameter that describes the
dipole moment and polarizability of a solvent is the
dielectric constant (see http://en.wikipedia.org/wiki/Dielectric_constant
for more information on dielectric constants). This parameter is
often used in a continuum model. Other parameters that can be applied
include the size of the solvent and solute and the surface tension of
the solvent.
Coninuum solvation models are still an area of active research. See
http://www.cosmologic.de/data/DOK_HTML/node259.html,
http://en.wikibooks.org/wiki/Computational_chemistry/Continuum_solvation_models,
http://www.cup.uni-muenchen.de/oc/zipse/the-polarizable-continuum-model-pcm.html,
http://www.cup.uni-muenchen.de/oc/zipse/onsager-reaction-field-theory.html and
http://www.cfs.dl.ac.uk/docs/html/part8/node11.html
for more details on four different approaches. The first of these approaches, the
COSMO method, is the one implemented in ADF.
In order to include a continuum solvent into a calculation
using the COSMO method, first open the solvation option
through the Model: Solvation menu option (figure 8.5).
The window to the right will display some of the parameters and options of
a COSMO calculation. Entries include the solvent epsilon (dielectric contant)
solvent radius, solute atom radii and solvent. The solvent option
gives a list of possible solvents to choose from. The default option is
of course None. By choosing a predefined solvent
a set of default parameters will be chosen.
Sometimes the desired solvent is not listed.
Further, parameterization of continuum models
is an arcane subject and it may be that you have your own set of
parameters that you believe to be better.
The UserDefined option from the solvent list
can be chosen and your new/alternative parameters entered.
Many other options and parameters of a continuum model can
be selected and/or modified through the Details: Solvation Details
menu option (figure 8.6.
Figure 8.5: A couple of other options of the continuum model that you might like to play
with.
In this lab course we not deal with all these parameters and will only make use of the list of solvents from the
Model: Solvation menu option. If you would like to understand
some of the parameters and options of COSMO in ADF consult the paper
by Pye and Ziegler that describes how COSMO is implemented
in the ADF program (C. C. Pye and T. Ziegler, Theor. Chem. Acc.101, 396 (1999))
Exercise 8.3
Optimize the geometry of ScCl3 and CH4 in water using a
continuum model to describe the solvent. Compare the resulting geometry with those
obtained for a molecule in a vacuum and in the presence of
a single water molecule. Also compare the length of time each calculation took.
Use the same basis set and functional
as was used in the previous exercise.
Figure 8.6: A system including a solvent molecule along
with a continuum solvent model.
A continuum model can include the influence of the many solvent molecules that surround a
solute in a calculation but it cannot include the influence of a solvent molecule
participating directly in a reaction or bonding directly to a solute
molecule. On the other hand, including solvation effects through explicit solvent molecules
will be able to reproduce the direct influence of a solvent molecule but
will generally not be able to simulate the effect of the many surrounding
solvent molecules.
The two differing approaches to solvation clearly each have their own advantages and
disadvantages. There is nothing to say that both can't be used at the same
time however. In this way, we can take advantage of the good points of both methods while
reducing the influence of their less attractive aspects.
In order to use both methods at the same time we would first build the molecule
we are interested in, add in the explicit solvent molecules that are
required and then surround the whole set of molecules by our continuum model
for the solvent (figure 8.7).
Exercise 8.4
Optimize the geometry of ScCl3 and CH4
in the presence of a single molecule of H2O and surrounded by a
continuum model of H2O. Use the same basis set and functional as
you have applied to the previous exercises in this lab. Compare the geometries
and calculation times obtained in the combination calculation
with the previous results.
8.7.1 The Influence of Scalar Relativistic Corrections on the
Bond Lengths of Diatomic Molecules
Calculate the bond lengths of the following group 1 diatomics with and without
the inclusion of scalar relativistic corrections: K2, Rb2 and Cs2. Perform the
same calculations but this time consider three group 11 diatomics
Cu2, Ag2 and Au2. Perform the same calculations but this time
consider three group 17 diatomics, Br2, I2 and At2. Make a table of
your results and include in your table the difference between the relativistic and
nonrelativistic results. How do the relativistic effects change going down the periodic table?
How do they change going across the periodic table?
Use a TZ2P basis set and the BP functional. The ZORA option is the best choice
for relativistic contributions here.
