For each of the following molecules…

CHM111L - Computational Chemistry Exercise

Using SPARTAN to Create Potential Curves for Li2, Li22-, O22+, and O22-.

Today we will carry out Molecular Orbital calculations in order to understand the bonding (or lack thereof) in several homonuclear diatomic molecules and ions. We will construct a potential curve showing how the energy changes with internuclear distance and thereby be able to estimate the equilibrium internuclear distance as well as the bond strength. We will also determine the energy-ordering of the molecular orbitals, and consequently will be able to predict which MO’s are occupied (the electron configuration). Finally, a comparison of orbital symmetries will enable us to deduce how the occupation of certain orbitals impacts the length and strength of the bond.

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Introduction:

What goes on in the chemical world is dictated by the behavior of electrons and nuclei; and what dictates the behavior of electrons and nuclei is their physics, i.e. their potential and kinetic energies and their distribution in space. One of the main goals of Computational Chemistry is to calculate how the electrons and nuclei might be arranged in atoms and molecules and to estimate the energies associated with those arrangements. Specifically with molecules, one can carry out calculations looking for the lowest energy arrangements of the nuclei and electron waves, then predict that those are the ones likely to be seen in the real world. The good news is that we can use sophisticated computer programs to do the energy calculations for us; all we have to do is tell the program what atoms are involved and what sort of calculation we want do.

Review:

You have already studied the behavior of electrons in atoms and atomic ions. You learned four key aspects of electrons in atoms:

1) they are described by waves (orbitals) that have distinct shapes and sizes,

2) they have specific energies,

3) the ground state involves filling the lowest energy orbitals, and

4) no more than 2 electrons can be “in” an orbital at one time (Pauli Principle).

These same four principles are pertinent for electrons in molecules. The only real difference is that atomic orbitals (AOs) are localized on one atom while molecular orbitals (MOs) can be spread over the entire molecule. Consequently, MOs aren’t characterized by the same 3 quantum numbers used for AOs, and they rarely have any meaningful quantum numbers. The good news is that there are some simple ways to describe the MOs. MOs in homonuclear diatomic molecules are among the simplest MOs, and you will gain some experience with them today.

1) With homonuclear diatomic molecules the MOs are spread evenly between the two atoms, and they resemble combinations of the same AO on each atom. These combinations of AOs can be characterized by their symmetry along the internuclear axis and their symmetry around the internuclear axis. MOs that lie on the internuclear axis are referred to as sigma orbitals (σ), and those that lie on either side of the axis are referred to as pi orbitals (π). MOs that are symmetric along the internuclear axis are referred to as bonding and those that are asymmetric (i.e., change sign around the center) are referred to as antibonding and are identified with an aterix *.

These various distinctions lead to the following 6 homonuclear diatomic MOs:

[red lobes represent one sign (say, +) and blue the other (say, -).]

σ2s

σ2s*

σ2p

σ2p*

π2p

π2p*

[The π MOs actually represent 2 MOs, one in the plane of the paper and one coming out at you.]

2) The relative energies of these MOs follows the general notion of “more nodes means higher energy”: σ2s < σ2s* < σ2p < σ2p* and π2p < π2p*. Just where the π MO energies overlap with the σs differes for different molecules – they are usually near the σ2p but less than the σ2p*.

3) & 4) Once the energy ordering is known, one can write a valence electron configuration by listing the occupied MOs and their occupations.

For O2, in which there are 12 valence electrons, the valence electron configuration would be …

σ2s2 σ2s*2 σ2p2 π2p4 π2p*2

Calculations:

One of a molecule’s most important properties is its geometry or shape, and it can be determined and understood through the use of these energy calculations. The program …

…starts with some initial configuration of the nuclei (a guess made by you or the program),

…then it evaluates the energy the molecule would have in that configuration,

…then examines the forces on the nuclei in that configuration, and allows them to move slightly in response to those forces to make a new configuration,

…then repeats the energy calculation to see if the overall energy is any lower.

This process is repeated until a lowest energy geometry is identified, and that minimum-energy configuration is usually a pretty good approximation to the true configuration found in the real world.

One could say that the energies of all the possible arrangements of the nuclei describe a potential energy function, and the actual arrangement of the nuclei in real life tends to find its way to the lowest energy part of that function.

In today’s exercise we will “brute-force” our way through the energy optimization process outlined above.. Specifically, we will attempt to predict the equilibrium internuclear distance and bond strengths for O22+, O22-, Li2, and Li22 . To do this, we will calculate the energies of the nuclei and electrons if the nuclei are held at a wide range of internuclear distances, from very far apart to very close together. The internuclear distance corresponding to the minimum energy will be our prediction of the equilibrium internuclear distance, and the energy difference between when the nuclei are very far apart and when they are at the equilibrium internuclear distance will be our approximation of the bond strength.

An example of what we will be doing is shown in the figure below for HCl. The points on the curve represent the calculated energies of an arbitrarily chosen set of 10 internuclear distances. The line is just a smooth line drawn through the points to show our expectation for the energies of other internuclear distances if we bothered to calculate them. This is called a potential energy diagram for HCl . The lowest energy separation is expected to be the equilibrium internuclear distance, i.e. the true bond distance you’d see for HCl molecules flying around in the gas phase.

Furthermore, the difference in energy between the energy at the equilibrium internuclear distance and the energy of the fully separated nuclei represents the energy needed to pull the two atoms totally apart, i.e. the bond strength or dissociation energy of the H-Cl bond.

The calculations used in computational chemistry cover a large number of different approaches, some faster than others, and some more reliable than others. The complexity of the molecules involved prohibits any calculation involving more than two particles (one nuclei and one electron) from being calculated exactly, so approximations are used.

- MOLECULAR MECHANICS calculations use empirical observation as well as high quality calculations on small molecules to approximate “force fields” around atoms. The program is told how the atoms are bound to each other, and then it uses the force fields to find how they will arrange themselves to reach the most stable configuration. These are among the fastest but least precise calculations. The VSEPR model discussed in lecture is a qualitative approach to a molecular mechanics technique; it uses a force field which simply indicates that groups of electrons (bonding and non-bonding) repel one another and get as far apart as possible without breaking bonds.

- MOLECULAR ORBITAL calculations are probably the most commonly used calculation in Computational Chemistry. In these calculations one assumes that the electrons are paired in molecular orbitals (MO’s) spread over a molecule, in much the same way we think about them being paired in atomic orbitals (AO’s) in atoms. The energy of the system can be calculated using this assumption of the arrangement of the electrons. In addition to the total energy, another interesting feature of a MO calculation is seeing what the MO’s look like. Most MO calculations attempt to approximate the MO’s as combinations of AO’s. The set of AO’s used to describe the MO’s is called the “basis set”. The more AO’s in the basis set, the more accurate the calculation is likely to be. In addition to the system energy, an MO calculation determines (1) which AO’s are used to describe the various MO’s, and (2) the energy of an electron in each MO; both of which are important in developing a qualitative feel for the chemistry surrounding the molecule. In our exercise we will want to see how some orbital energies go up in energy as the atoms come together, and how some go down.

- SEMI-EMPIRCAL MO Calculations are a subset of MO calculations in

which some of the more rigorous applications of quantum physics are

only approximated – usually using some sort of empirical observations.

The program calculates MO shapes and energies, just like other MO

calculations, but these calculations tend to be quite fast and not always

reliable.

- Ab Initio MO calculations only assume that doubly occupied MO’s

exist; then it uses rigorous quantum physics to calculate MO shapes

and energies. These are the sorts of MO calculations we will use

today, and you will see them referred to as “Hartree Fock calculations”.

- ELECTRON CORRELATION calculations recognize the fact that electrons in molecules don’t actually occupy MO’s, but that their motions are correlated with one another (i.e., when one electron moves, others will adjust their distributions accordingly). These calculations describe the motion of the electrons as a combination of occupations of various MO’s while other electrons are in other combinations of other MO’s. These calculations are quite involved computationally, and have the potential to be exactly correct, but they, consequently, take quite long to complete. We won’t do any of these in this experiment.

Today you will be working with ab initio MO calculations on homonuclear diatomic molecules and molecular ions.

Before you begin, you should be aware of the existence of the four most common types of MO’s: σ bonding MO’s, σ antibonding MO’s, π bonding MO’s, and π antibonding MO’s, and how their occupation affects bond strength. You should also be familiar with the concepts of total energy as opposed to orbital energies. They can be reviewed in Chapter 9 (section 9.7) of Gilbert, etal, Chemistry, 3e.

Also, before you come to lab, you need to install the program on your computer. The program name is SPARTAN, and it can be installed by CAREFULLY following the instructions found at the Chemistry department website.

http://college.wfu.edu/chemistry/course-materials/course-software/wavefunction-spartan-student 

You will need to get a code from your instructor before you can download and install the program.

General Procedure:

Your goal is to produce potential energy diagrams for one of the pairs of molecules and/or ions, O22+ and O22-, or Li2 and Li22-. You will also examine the differences in the electron configurations of these species, specifically showing how the orbital energies change with internuclear distance for the underlined molecules. To do this, you will fill-in the following table and plot the data accordingly.

Starting with your underlined molecule, you will set an internuclear distance, do the calculation, then record the orbital symmetries (names) and their energies and the total energy. You’ll set a new internuclear distance and repeat the calculation… and repeat… and repeat…

Next, you’ll set up the calculation for your other molecule or ion and repeat the calculations, though you don’t have to pay attention to the orbital names or energies any more, just the total energies.

You will begin by doing calculations at a set of prescribed internuclear distances, but you may supplement the table with additional internuclear distances if you would like to clarify the MO or total energy behavior in certain regions.

Use Excel to plot the energy of each orbital symmetry for the underlined molecule as a function of internuclear distance. Since two pairs of the eight orbitals are degenerate, the π2px or 2py, and π*2px or 2py, that plot will have six lines on it. You will also make a plot of the total energies of the two species as a function of the internuclear distance. From these plots you should be able to make general conclusions about how bonding and antibonding MO energies change as the internuclear distance decreases. You should also be able to estimate the equilibrium internuclear distance and the bond strength. Furthermore, by relating the differences in your two potential curves to the differences in electron configuration of your two species, you should be able to conclude how occupation of different types of orbitals affects bond strength.

Be sure your lab report includes…

… an abstract containing an objective, the results, and a brief mention of the technique used.

… one plot showing MO energies for your underlined molecule as a function of

internuclear separation. [You do not need to include the completed table.]

… a general statement of the way the bonding and antibonding MO energies change with internuclear distance.

… one plot showing the total energy vs internuclear separation for both of your molecules.

… the electron configuration for each molecule at the equilibrium internuclear distance (or some other appropriate distance if there is no equilibrium).

… identification of the bond lengths and strengths, and an explanation of how the electron configurations affect them.

Results for _Li2 / Li22-___:

MO #

1.5Ǻ

OrbE Sym

2.0Ǻ

OrbE Sym

2.3Ǻ

OrbE Sym

2.6Ǻ

OrbE Sym

3.0 Ǻ

OrbE Sym

4.0Ǻ

OrbE Sym

6.00Ǻ

OrbE Sym

8.00 Ǻ

OrbE Sym

3

4

5

6

7

8

9

10

ETot Li2

ETot Li22-

Results for _____O22+_/ O22-____:

MO #

1.00Ǻ

OrbE Sym

1.10Ǻ

OrbE Sym

1.25Ǻ

OrbE Sym

1.5Ǻ

OrbE Sym

1.75Ǻ

OrbE Sym

2.00Ǻ

OrbE Sym

3.00Ǻ

OrbE Sym

6.00 Ǻ

OrbE Sym

3

4

5

6

7

8

9

10

ETot O22-

ETot O22+

How to Use SPARTAN to Generate of Potential Curve:

To construct the potential curve for a diatomic molecule, your strategy is to carry out single-point calculations at a representative number of internuclear separations, filling in the table as you go. Then plot the total energy and the various orbital energies vs the separation.

To do this, you will…

…1) Build the molecule, Li2 or O22-.

…2) Constrain a particular internuclear separation,

…3) Setup the surfaces,

…4) Setup and Submit the calculation,

…5) Display the surfaces to help identify the symmetry of the orbitals,

…6) Display and record the Total Energy and the MO Energies found in the OUTPUT, and

…7) plot the points.

All of the listed activities are discussed one-by-one below.

1) Build the molecule.

Once the program has been started, make a file for the desired molecule by first clicking the FILE menu item and selecting NEW. [If you are building a new molecule after finishing with another, you’ll want to CLOSE the old FILE before starting to build the new molecule.]

If the atom/bond selection template is not showing on the right side of the screen, click on the BUILD and ADD FRAGMENT to bring it up. In the “Model Kit” window on the right side of the page, click on INORGANIC, then select an atom to add from the drop down periodic chart.

Molecule building template.

Next, select a bonding scheme (don’t count lone pairs). Since we are only dealing with diatomic molecules today, you want to select the lone single bond.

Bonding Scheme Selection

Next click on the main screen, and the atom and bond will be deposited there. To attach another atom to the first, select the atom and bonding scheme then click on the dangling bond of the first atom. Repeat this process until all the atoms are present. Bonds that didn’t quite match up can be joined with BUILD>MAKE BOND selection. Then click sequentially on the two dangling bonds that need to be joined. For this exercise, one does not need to worry about specifying multiple bonds; the Hatree-Fock calculation decides whether they are needed and where they should be located.

Errors can be fixed by Deleting the mistaken atoms or bonds. To do this, first click on BUILD>DELETE, then click on the feature you want to get rid of.

In order to view all aspects of the molecule and make sure you have everything correct, you may want to rotate or move the molecule. Left clicking on the main screen and moving the mouse will rotate the molecule, and Right clicking on the main screen and moving the mouse will translate the molecule without rotation.

2) Constrain a bond length.

Click on GEOMETRY >CONTRAIN DISTANCE from the top menu line. Next click on the bond you want to constrain. If all has gone well, there will be an input box in the bottom right corner of the screen that is labeled “CONSTRAINT = ”. A pink lock icon next to the box indicates if the value is among the variables that can be optimized (unlocked) or if it is to be locked to a certain value (locked). To constrain a bond length, click on the lock icon until it is locked. Then type the desired value in the input box and hit return. The program assumes the units are Ǻ.

Parameter input box in lower right corner.

You will need to click on the E↓ button (minimize) on the top icon line to institute the constraint and show it on the screen.

3) Setup the surfaces

Click on SETUP>SURFACES. Assuming the list is blank, select MORE SURFACES, then from the Surfaces drop down, select HOMO, LUMO, HOMO(-), or LUMO(+) depending on what orbital you want to view. [ HOMO stands for Highest Occupied MO, while LUMO stands for Lowest Unoccupied MO.] Click OK. Repeat the process until you have added all the surfaces you want to view. Close the Surfaces window by clicking the “X”.

[For Li2, you will want to include HOMO, LUMO, LUMO+1,…,LUMO+6; and forO22- you will want to include HOMO-6,…,HOMO-1,HOMO, and LUMO.]

Selecting and Setting Up Surfaces.

After you complete a calculation you will go back and look at these surfaces to see what sort of MO’s they represent σ, σ*, π, or π*. Unfortunately, the way they are designated in the output is by number rather than name. The following table shows how the names of the surfaces correspond to the number of the MOs in the output.

MO #

Surface in Li2

Surface in O22-

10

LUMO+6

LUMO

9

LUMO+5

HOMO

8

LUMO+4

HOMO-1

7

LUMO+3

HOMO-2

6

LUMO+2

HOMO-3

5

LUMO+1

HOMO-4

4

LUMO

HOMO-5

3

HOMO

HOMO-6

2

Core – not named

Core – not named

1

Core – not named

Core – not named

4) Setup and Submit the calculation .

Click on SETUP>CALCULATION to perform an ENERGY calculation (not the default Equilibrium Energy calculation). The calculation will be done with the HARTREE-FOCK or ab initio molecular orbital method. Select the STO-3G basis set in the box which defaults to 3-21G. Make sure the total charge is appropriate and that there are no unpaired electrons. The Orbitals & Energies should be checked to Print. The Total Charge should be selected accordingly (Neutral for Li2 or Dianion for O22-).

Setup the Calculation

Finally, click on SUBMIT and wait for the calculation to be completed. It will tell you that it started and also that it finished. You simply click your acknowledgement when those windows come up.

5) View surfaces

Click on DISPLAY>SURFACES, and the window with the list of surfaces comes up. Click on the surface you want to see. Remember to click it off before trying to look at another surface.

Sample σ orbital.

Sample π orbital.

Sample π* orbital.

Sample σ* orbital.

Identify and record the symmetry of the orbitals, i.e. σ, σ*, π, and π*. Remember, any σ or σ* orbitals have probability on the axis running between two bonded atoms, with σ being concentrated along the bond, and σ* changing sign near the center of the bond. The π and π* orbitals have all their probability off the axis, with π showing no change of sign between the bound atoms, and and π* having such a change about midway.

6) Display Output

When you DISPLAY > OUTPUT, a text file comes up that has a bunch of information in it (see below).

Near the top is an entry labeld “SCF Total Energy” and it has a value with units of hartrees, that is the total energy of the electrons and nuclei in the molecule, i.e. the “Total energy”. Midway through the file is a table of numbers with the headings MO, Eigenvalues, and (eV). The MO number simply identifies the resulting MO’s in order by energy, lowest energy orbitals first. In a molecule with n electrons, the first n/2 MO’s will have 2 electrons. The higher energy orbitals will typically be empty.

Eigenvalues are the exact same thing as orbital energies. The units for the Eigenvalues are hartree’s, (1 hartree = potential energy of the 1s electron in the Hydrogen atom). The (eV) entries present the orbital energies in the units of “electron Volts,” the energy associated with accelerating an electron through 1 volt. The conversion between these two energy units is

1 hartree = 27.3 eV.

The array of numbers below the orbital energies is the list of coefficients, ci, for the MO’s in terms of the AO’s in the basis set,

MO = Σ ci AOi .

[It is interesting to look at the column of coefficients pertaining to the different MO’s to see which AO’s correspond to the largest coefficients. The MO’s resemble the AO’s with the largest coefficients. MO’s 1 and 2, for instance, will have large contributions from the 1s orbitals because they are essentially just those AO’s. Don’t spend too much time looking at these coefficients today since they will not be used in today’s exercise.]

Portion of the DISPLAY>OUTPUT file.

Fill in the appropriate table with the orbital energies (pick one unit and stick with it) for MO’s 3 through 8 for all the calculations at the different internuclear distances.

After you finish the first set of calculations and data collection, you need to repeat the process with your other molecule, Li22- or O22+. This time, however, you only need to record the total energies – we presume the orbital energies behave similarly in both species. You can figure out how to Setup this new Calculation.

[One minor piece of advice: Before doing these last Total Energy Calculations, delete the surfaces in the Setup Surfaces list. If the program tries to evaluate a surface that is no longer there, you’ll get an error.]

7) Some notes about the plotting:

For your plot of the MO energies vs internuclear distance, you should recognize that MO#5, for instance, is not always the same symmetry (i.e. π2p); but the lines you plot should show how each type of orbital changes in energy with internuclear distance. In other words, don’t simply plot the energies of MO#5 vs internuclear distance. When you transcribe the orbital energies to EXCEL, you need to make sure the orbital energies in each y-series represent the same type of orbital.

When plotting the total energies vs internuclear distance, you are going to be comparing the energies at the equilibrium distance to the energies when the nuclei are far apart. For good, but presently unimportant, computational reasons your calculations will not dissociate to the same energies if they have different numbers of electrons. You should approximate the desired energies by shifting your tabulated total energies up or down so that the energies of the largest separations are zero. In other words, if the total energy of the largest separation for one species is -12.876a.u., then add 12.876 to all the energies in that series. The result will be a plot with two total energy vs internuclear distance lines, and the total energies are quite different when the nuclei are close but become the same when they are far apart.

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