Ethane Bond Rotation and Vibrational Energies
Introduction to Molecular Modeling and Spartan
In quantum chemistry, the goal is to solve for the energy and/or geometry of a molecular
system. A molecular system can mean molecules, atoms, or small aggregates of
molecules having definite characteristics. Two basic approaches exist:
• solve the Schrödinger equation exactly
• model the system using classical physics.
The first approach is a quantum chemical method while the second is called molecular
mechanics.
Sometimes it is not possible to solve the Schrödinger equation exactly and we must use
approximate methods. In quantum chemistry, there are a number of approximation
methods available. These methods are separated into categories depending on the
approach. Methods in which the Schrödinger equation is solved by computing all of the
integrals are referred to as ab initio (meaning “from first principles”). Ab initio methods
are the most rigorous and exacting, but are also the most time consuming. When some
integrals are disregarded or are set equal to numerical values obtained from experiment,
the method is termed a semi-empirical approximation. Semi-empirical methods can also
be extremely accurate; it depends on the effort made to obtain good parameters.
Molecular mechanics methods do not calculate any integrals or molecular energies.
Instead, simple physical models are used to describe stretching, bending, and other strains
in a molecule. These strains are modeled as Hooke’s law forces and the method
calculates the total strain energy.
Spartan™ is a software tool that allows us to make these calculations. It uses the BornOppenheimer Approximation, which says that nuclei move much slower than electrons
due to their greater mass. Therefore one can locate nuclei at specified positions, then
calculate the lowest energy distribution of electrons around those nuclei. One then
iteratively modifies the positions of the nuclei until the lowest possible energy is found.
Molecular Mechanics
Full quantum-mechanical calculations of molecular motion can become quite involved.
A vastly simplified model, meaning not very accurate but easy to do on the computer, is
to use classical equations of physics to describe the potential energy surface of molecules.
These equations are written in terms of force fields such as the Hooke’s law force
(harmonic oscillator) discussed in class. The molecule is constructed as if the individual
atoms are held together by ideal springs. These force fields can then be used to calculate
the motion of the atoms in a molecule. A different parameter (force constant) is used for
each atom and for each type of motion (stretching, bending, torsional, and electrostatic
and van der Waals). Molecular mechanics does not treat electron motion explicitly and
therefore the method cannot account for the formation of bonds, predict bond properties
such as dipole moments, or charge distributions. The method is useful, however, for
2015
determining the geometry of molecules and is useful in comparing energies at different
geometrical arrangements.
There are many molecular mechanics methods; MM+, AMBER, and SYBYL and Merck
Force Field (MMFF) are some examples. MM+ is a variation of the MM2 method
proposed by Dr. Norman Allinger (J. Amer. Chem. Soc., 99, 8127, 1977). Molecular
mechanics methods are very fast but they require parameters for each atom in the
chemical environment of interest. For example, if you were to study NeF2, which does
not exist, you could not get MM+ parameters for either neon or fluorine in this particular
environment. You could not complete the calculation. You could not look up the
parameters for fluorine in, for example, HF and use them in NeF2 because the bonds are
different.
Ab Initio Methods
A more exact approach to understanding molecular bonding and molecular motion is to
explicitly account for the multiple interactions of all the electrons and all the nuclei with
one another. This is impossible to do exactly since we know from Heisenberg that we
never know where all of the electrons are located at any one instant in time. We can,
however, use different levels of approximation to approach a correct answer.
At the highest level of accuaracy, we have the Hartree - Fock Configuration
Interaction method (HF-CI). The HF-CI approach uses a large number of functions
(orbitals) and includes correlation of the electronic motion. The approach is quite
complex and is included here only as a reference point. All other methods only approach
some level of approximation to the HF-CI result.
The Hartree - Fock method is the next level of approximation; it does not include
instantaneous correlation of electron motion but instead uses an average electron motion.
HF calculations typically build the molecular orbitals from linear combinations of the
original atomic orbitals on each atom in the molecule. Hartree - Fock calculations come
in various ‘sizes’.
- A minimal basis set would be based on only the occupied atomic orbitals on each
atom such as the 1s, 2s, 2p on a carbon.
- A split basis would include two functions for every atomic orbital.
- Another type of basis set is a split basis with polarization. Polarization functions are
higher-energy orbitals NOT normally occupied in the ground state of an atom
(e.g., d functions on carbon). In polar molecules, polarization functions are a
significant help in describing bond formation.
To perform any of these ab-initio calculations you must solve a large number of integrals.
The number of integrals is equal to N4, where N is the number of functions used. These
integrals are recalculated every time the geometry of the molecule changes. The more
basis orbitals that are included in the calculation the better the answer… but the longer
the computation time required.
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Semi-empirical Methods
Considering the large amount of computational time required for a medium to large sized
molecule, it is not surprising that considerable effort has been expended to devise
techniques that will produce reasonably accurate results in less time. These techniques try
to reduce or eliminate some of the integrals required to calculate molecular energies since
it is integral calculation that takes the most time. There is a trade-off here: we want to
eliminate integral calculation but we also want to achieve accurate results.
The simplest approach was devised by Hückel in the early days of quantum chemistry
and was extended in the 1970s by Roald Hofmann, the Nobel Laureate at Cornell. No
electron interaction is included and only the integrals related to overlap of atomic orbitals
on bonded atoms are calculated. The Extended Hückel method is used only as a rough
estimate to begin calculations that are more sophisticated.
All other methods vary in their treatment of electron repulsion integrals. These integrals
are classified as Coulomb integrals or Exchange integrals. Coulomb integrals represent
the repulsion between two electrons (as in Coulomb’s law). The Exchange integrals are a
result of the Pauli Principle that states that electron wavefunctions must be
antisymmetric. The Exchange integrals give rise to different spin states such as singlets
and triplets. In all cases, the integrals are never actually computed; instead they are
represented by some atomic property such as ionization energy or electron affinity.
Each semi-empirical method has an acronym: CNDO, INDO, MINDO, ZINDO AM1,
and PM3.
CNDO (Complete Neglect of Differential Overlap) is the simplest of these methods. It
states that the repulsions in different atomic orbitals depend only on the nature of the
atoms and not on the type of orbital. It treats most energy gaps poorly but predicts
reasonable geometries.
INDO (Intermediate Neglect of Differential Overlap) allows for differences between
atomic orbitals and is therefore superior to CNDO. MINDO/3 is the same as INDO but
with a different parameter set for the integrals.
ZINDO/1 and ZINDO/S are INDO versions that have been parameterized for transition
metals. ZINDO/1 predicts better geometries while ZINDO/S is parameterized for UV
spectra.
MNDO, AM1, and PM3 include another set of repulsion integrals that the previous
methods ignore; thus, they incorporate more electron interaction terms and should be
more accurate. Since they include more integrals, computation time will be slightly
longer. The three methods are very closely related and they differ only in the choice of
parameters. AM1 and PM3 are generally the most accurate methods.
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Internal Vibrations and Rotations of Ethane
- An Introduction to Spartan In this experiment, you will use Spartan to look at the energies associated with molecular
vibrations and internal rotations. These calculations are based on locating the atomic
nuclei at assigned positions, then distributing the electrons so as to minimize the total
energy of the system. Equilibrium geometries can be found by iteratively moving the
nuclei to a series of modified positions until the lowest possible energy is reached.
You will use Spartan to measure the energy barrier for rotation about the C-C bond in
ethane. You will also measure the vibrational energy needed for the C-C bond stretch.
EQUILIBRIUM GEOMETRY OF ETHANE
•
Build an ethane molecule using Spartan and then perform “equilibrium geometry”
calculations with 3 different methods: Semi-Empirical PM3, HartreeFock STO-3G, and Hartree-Fock 321-G. These methods calculate the lowest energy
conformer of the molecule based on the different assumptions in each model.
•
Record the energy of the molecule that is listed at the end of the “Optimization”
iterations. This will be given in units of kJ/mol. In addition, use the “Measure
Distance” and “Measure Dihedral” tools to obtain and record the bond length between
the two carbons, and the rotational dihedral H-C-C-H angle. (The dihedral angle
measures whether the two methane groups at each end are anti or gauche relative to
each other.)
•
Save the output by doing a Save-As and choosing .txt format.
Data Analysis and Discussion for Lab Report
•
Report the energies and equilibrium geometries for the different calculation methods.
According to the Variational Principle, the lower (or more negative) the calculated
energy, the closer it approaches the actual value. According to this principle, which
calculation method gave the best result?
•
How many “basis functions” are used in the calculation for the STO-3G and 321-G
calculations? (see Spartan output)
•
What would be the approximate number of integrals required to calculate the STO-3G
and 321-G results? (See discussion preceding these procedures)
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ENERGY BARRIER BETWEEN ANTI AND GAUCHE C-C BOND ROTATIONS
Spartan can be used to calculate the energy needed for the two methane end-groups to
rotate about their common C-C bond. As discussed in your organic chemistry course,
steric hindrance from the hydrogens creates a barrier to free rotation about the C-C bond.
•
Calculate the energy profile for ethane based on varying the rotational (dihedral)
angle by 10º increments between 0º and 180º. Remember to include the starting point
when deciding how many “steps” you need when specifying the dynamic range.
You only need to do the energy profile for the HF/STO-3G computational model.
See the Spartan primer for a guide on how to calculate angle constraints.
Data Analysis and Discussion for Lab Report
•
Graph your data in Excel and be sure to connect the points with a smooth line (there
is a checkbox for this).
•
Use the resulting spreadsheet and plot to determine the energy barrier for rotation
about the C-C bond (i.e. difference in energy between anti- and gauche- conformers).
Report the barrier in units of kcal/mole as well as cm-1/molecule.
•
Use your organic text or other reference to find a literature value for the rotational
barrier for ethane. How does the calculated barrier compare with the literature value?
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ENERGY PROFILE FOR THE C-C STRETCH VIBRATION IN ETHANE
Spartan can also generate a vibrational energy profile based on varying the C-C bond
length rather than bond rotation. This calculation is performed in the same way as the
rotation, except instead of constraining the dihedral angle one constrains the bond length.
1. First rebuild the molecule and using Calculate / HF/ STO-3G doing an “Equilibrium
Geometry” calculation. (This resets the molecule to the best starting point.)
2. Using the optimized molecule from above, do an “Energy Profile” calculation.
Continue to use the / HF/ STO-3G model for your calculations. Constrain the C-C
bond length from 1.41 to 1.67Å, going in increments of 0.01Å. (The default units for
bond length in Spartan are angstroms, Å, which are 10-10m.)
Review the steps for doing constrained calculations if you need to. Use Spartan to
create a spreadsheet and plot the data. Transfer the spreadsheet data to Excel for
additional analysis later.
3. Repeat the above but for bond lengths ranging from 1.0 to 6.0Å, going in increments
of 0.1Å. Copy the spreadsheet data to an Excel spreadsheet.
[HINT: it will be hard to click on the “constraint ribbon” once you have done the
profile calculations since the carbons will be close to each other. You can use the
scroll wheel on the mouse to expand or shrink the displayed molecule.
4. Return to the initial Ethane model in Spartan and redo the Equilibrium Geometry
calculation. This time, however, tell Spartan to calculate all of the vibrations.
- Select Equilibrium Geometry in the Calculation pull-down menu
- Check the Infrared Spectra box in the Calculate area of the dialog box.
- Submit
When the program is done, go to Display / Spectra. This will give a list of
frequencies (in units of cm-1 energy). Move the display list to the side so that you
can see the molecule underneath. Click on the different frequencies and watch the
motions for the molecule model on the screen.
While in the Spectra display, you can also click on Show Calculated to see what the
IR spectrum will look like.
For your Report
Which vibrational energy corresponds to a C-C bond stretch? Record the energy
and compare it to the energy that you calculate on the next page.
Which vibrational energy corresponds to a vibrating C-C bond rotation?
How does this vibrational energy compare to the barrier of rotation?
Is there enough energy to get over the barrier?
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Data Analysis and Discussion for Lab Report
For the 1.41-1.67Å data,
-
Create new columns on your Excel spreadsheet and convert your bond distances
and molecular energies to SI units (meters and J/molecule).
-
Plot the energy vs. distance data and fit to a quadratic equation. What is the
coefficient in front of x2 for your calculated curve? [The other terms in the
equation move the curve up or down or to the right or left.]
-
Since an ideal harmonic oscillator changes its potential energy according to ½kx2,
what is the value of k from your Spartan calculations?
-
Calculate the reduced mass (in SI units) for the C-C stretch assuming that the CH3
groups at the two ends of the bond each move as single mass units.
-
You should now be able to calculate the frequency of vibration, the energy of transition
from n=0 to n=1 in energy units of J and in units of cm-1. Report these values.
Comment on the goodness of the fit for approximating the calculated energies to the
energy equation used for an ideal harmonic oscillator.
How well did your calculation of the vibrational energy of transition compare to the
number computed by Spartan?
Now that you know the energy of the energy states and you know the spring constant,
you are in position to go back to the V(x) = ½ kx2 equation and calculate how far the
atoms actually move during a vibration.
-
Given that at maximum displacement, V(x) is equal to the vibrational energy for a
given quantum state, what is the range of motion (± x) expected for the lowest
vibrational state?
-
Does this range of motion fall within the range that you looked at in your first plot
that appeared to be mostly quadratic? Or does your range of motion fall more
within your plot showing the anharmonic Morse potential behavior? That is to
say, is the first vibrational level primarily in the ideal quadratic harmonic region
or is it in the anharmonic regime?
-
What is the amount of displacement expressed as a percentage of the equilibrium
bond length? (The equilibrium bond length is the low point on your Spartan curve).
For the 1.0-6.0Å data,
-
Convert your data to meters and J/molecule and plot.
-
Use your previous quadratic-fit line and generate ideal harmonic oscillator points
for bond lengths ranging between 1.0-2.1Å. Plot these points as a smoothed line
on the same graph as your 1.0-6.0Å data.
NOTE: You will need to set Excel to give more significant digits on your trendline
equation. Do this by clicking on the trendline equation on the earlier plot, then
right-click and go to “Format Trendline Label”. Select Scientific Notation and
include 6 digits past the decimal point.
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Do your data continue to conform to a symmetric quadratic curve when the bond length
is extended to larger displacements?
Use your calculated data to determine the bond dissociation energy, De. (This is the energy
from the bottom of the energy well to the plateau where the molecule breaks apart.)
Using your earlier harmonic oscillator calculations, what is the value of the bond energy
Do? (Remember that Do is measured from the ground vibrational energy level, ½hν,
rather than from the absolute bottom of the energy well.)
What is the ratio of the first vibrational transition energy for n=0 to n=1 compared to the
total bond energy? How far along does the first vibrational transition energy take the
molecule towards the energy needed to break the bond?
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