Computational Study of Atomic & Molecular Orbitals Revised: 10/21/15

1

COMPUTATIONAL STUDY OF ATOMIC & MOLECULAR ORBITALS

REPORT INSTRUCTIONS

All work for this experiment must be recorded, attached, or answered in the ELN. Create a Pre/InLab page in this week’s Experiment folder. An objectives and observations (data) sections are required, chemical/equipment tables and procedures are NOT required. All Spartan or Excel data collected or files created should be recorded or attached in the observations section. The postlab questions at the end of this document should be answered on the Post Lab page in the same folder.

INTRODUCTION

Electrons are responsible for most chemical and physical properties. So, to understand molecules and polyatomic ions, we must first understand electrons. What volume of space do they occupy? Where are they most likely to be found? What is their energy? After Niels Bohr’s concept of an orbit, the theories about electrons became increasingly more mathematically complex. The advent of computers in the 1950s enabled scientists to begin to find solutions. Now many computational molecular modeling programs are available. In this course we will use Spartan. (Note: Computational Chemistry is one of the fastest growing fields in chemistry. Many Chemistry Professors here at UC Irvine do research in this field.) The fundamental theory used in computational chemistry is from Erwin Schrödinger. Schrödinger postulated that the tiny electron, while a particle with mass, also behaves like a wave. He proposed the idea of wave functions (orbitals), a mathematical description of an electron’s three-dimensional location in space. Wave functions are incredibly complex formulae, in fact only the wave function for a 1 electron atom or ion can be completely solved. (All other wave functions are approximated the use of simplifying assumptions.) Here is the equation for that one electron atom (or ion):

ψnlm(r,θ,φ) = Rnl(r) • Ylm(θ,φ) radial spherical Rnl(r) = − 4(n – l –1)!½ (2r)l L2l+1 (2r)-r/n n4[(n+1)!]3 (n) n-l-1 (n) Ylm(θ,φ) = [(2l + 1)(l – |ml |)!]½ P|m

l| (cosθ) + (2π)-½ eim

lφ [2(l + |ml |)!]½ l

The above equations contain quantum numbers (n, l, and ml) that take different values depending on the desired orbital of the 1 e- atom. (For example, if n = 1, l = 0, and ml = 0, ψ1s will be calculated; if n = 2, l = 1, ml = 0, ψ2px will be calculated.) Spherical (r,θ,φ) (instead of Cartesian

Computational Study of Atomic & Molecular Orbitals Revised: 10/21/15

2

(x,y,z)) coordinates are used to describe the motion of the electrons around a center nucleus. The connection between the two types of coordinates is demonstrated below (Figure 1).

r

z

y

x

θ

φ

rcosθ

rsinθ

electron

nucleus

Figure 1. Relationship between Spherical & Cartesian Coordinates.

Look at the very end of the spherical (Ylm) component (the last equation) of ψ1s. Do you see the "i" in the eim

lφ term? This complex number (i = ) is the mathematical indication of phase. What is phase? Waves have phase: if two waves come together in phase (both at the crest or both at the trough), the wave grows larger; if two waves come together out of phase (one at the crest and one at the trough), the waves are cancelled out. Analogously, two atomic wave functions (orbitals) come together to form two molecular wavefunction (orbitals). When two atomic wave functions (ψ, atomic orbitals, AOs) come together in phase a bonding molecular wave function (Ψ, molecular orbital, MO) (a larger wave) is created (Figure 2). However, if the same two AOs come together out of phase an antibonding MO (no wave) is created (Figure 3).

Figure 2. In phase overlap of two s AOs (ψa + ψb) to create a σ bonding MO (Ψ+).

Computational Study of Atomic & Molecular Orbitals Revised: 10/21/15

3

Figure 3. Out of phase overlap of two s AOs (ψa - ψb) to create a σ* antibonding MO (Ψ-).

Spartan represents phase by coloring orbitals (or lobes of orbitals) red or blue. Red represents the negative phase (or trough); blue represents the positive phase (or crest). The point or plane that an orbital changes phase (color) is called a node. If a node is present in an orbital no electrons are found in that area of space. (In other words, the probability of finding an electron there is zero.) The s atomic orbitals and bonding σ molecular orbital in figures above do not change colors, and, therefore, do not have any nodes. The σ* MO contains 1 node: a vertical plane perpendicular to the bond axis (the silver "bond") directly between the nuclei (the white balls) (Figure 4).

Figure 4. Nodal Plane of σ*

Electrons of a bonding molecular orbital occupy a larger space between the two nuclei when compared to the electrons localized around the nuclei of two separate atomic orbitals. This delocalization of the electrons lowers their kinetic energy (the electrons slow down), resulting in a molecular orbital that is more stable than the atomic orbitals that formed it. Conversely, electrons in antibonding orbitals move faster as they are more localized than the atomic orbitals from which they came. The molecular orbitals in Figures 2 & 3 require complicated calculations and assumptions that far exceed the equation shown for ψnlm on page 1. Spartan calculates and creates red and/or blue images of these orbitals as HOMOs or LUMOs. • The HOMO (highest occupied molecular orbital) is the valence orbital that received the last

valence electron(s). The orbital directly beneath the HOMO in energy is labeled HOMO(-1). The "1" can be changed to 2, 3, 4, ... to create a complete set of all MOs in the molecule.

• The LUMO (lowest unoccupied molecular orbital) is the empty orbital just above the HOMO. The orbital just above the LUMO is labeled LUMO(+1). Again, the "1" can be changed to 2, 3, 4, ... to create a complete set of all MOs in the molecule.

The HOMO and LUMO images (surfaces) created by Spartan can be correlated/represented with the lines on a molecular orbital (MO) diagram. Molecular orbitals in a molecular orbital diagram are the horizontal lines in the middle. The atomic orbitals (which create the molecular orbitals) are represented by the horizontal lines on each side. In this experiment, you will be asked to create the MO diagrams complete with Spartan orbital surfaces for H2

+, H2, and H2-.

Find the H2 MO diagram in your lecture textbook now.

Computational Study of Atomic & Molecular Orbitals Revised: 10/21/15

4

Spartan can also generate electrostatic potential maps. The volume of these images represents the relative size of a molecule. The gradient of colors on these images represents electron distribution. Areas with a high probability of finding an electron are shaded red. As the probability of finding an electron decreases the color changes to orange, yellow, green, and then finally blue where the probability is smallest. Typically, a strong single covalent bond will have a red band in between the two nuclei of that bond. In this experiment you will calculate the electrostatic potential maps for H2 at 6 different bond lengths. These images will help visualize the lengthening bond. Internuclear separation (bond length) effect on HOMO, LUMO, and molecular energies will also be investigated. The following trends will be observed: • For a very long bond length the molecular energy equals the atomic energy of the separated

atoms. In other words, there is no molecule. There is no HOMO and LUMO for the molecule because the atoms exist as separate entities, there are only atomic orbitals (AOs)).

• Initially, as the atoms begin to move toward each other, the molecular energy will be minimal and the difference between HOMO and LUMO energies is very small.

• As the atoms move toward each other, the molecular energy grows in magnitude, becoming negative: a molecule that is more stable than the separate atoms begins to form. The stabilization results from the electrostatic attraction ("!, Figure 5) of both electrons to both nuclei.

Figure 5. Electrostatic Interactions in H2.

When the distance between the two atoms enters an optimum range, molecular energy is negative. The difference between HOMO and LUMO energies increases as the atoms move together: bonding MOs lower in energy and become more stable, antibonding MOs increase in energy becoming less stable.

• When the molecular energy is at its most negative, the distance between the nuclei of the atoms is called the equilibrium bond length, R. The molecular energy is the negative of the bond dissociation energy, -De.

• For a separation smaller than R, nucleus-nucleus repulsion (1 , Figure 5) becomes very prominent and the kinetic energy increases again because the space available for the electrons is much smaller. The electron-nuclei attraction does not compensate for these two effects, and the molecular energy becomes positive. The difference in HOMO and LUMO energies increases further.

Computational Study of Atomic & Molecular Orbitals Revised: 10/21/15

5

PROCEDURES

Note: Holding down the mouse button while over the drawing allows for the rotation and translation of the molecule. A. Finding ψH: Monoatomic Hydrogen Atom (H·)

1. Create the following table using the Rich Text Editor. Take screen shots of Spartan, crop, and attach the image file into the table by clicking on . Table 1 H· H:- Density Electrostatic Potential

HOMO LUMO

2. Open Spartan Student. On the menu bar, click File and then New. 3. In the Model Kit in the right panel, click the Organic tab. Click –H to select hydrogen.

In the teal Workspace area, click anywhere to place hydrogen. 4. On the menu bar, click Build and then Delete. Click on the yellow bond on the hydrogen

so that it disappears. (The hydrogen atom (white ball) should remain on screen.) 5. On the menu bar, click Setup, and then Surfaces. In the Surfaces window, click the Add

button. Choose HOMO, LUMO, Density, and Electrostatic Potential Map. Close the Surfaces window.

6. On the menu bar, click Setup, and then Calculations. Select Calculate: Equilibrium Geometry with Hartree-Fock 6-311+G**. Set the Total Charge to Neutral and the Unpaired Electrons to 1. (A neutral hydrogen atom (H·) with one electron is being calculated.) Click Submit to exit Calculations window.

7. When the Save As window appears name the file and click on Save. Click OK button to start the calculations. Almost immediately, a dialog will appear indicating that the calculations are complete. Click the OK button to clear the dialog. Close this file.

8. Reopen the Surfaces window, click on the gray box next to the HOMO surface. Click on the image. The Style field should appear in the lower right hand corner of the screen. Change the Style from Solid to Mesh. Double click on the other image and bring up its density image as Mesh. You can use this image on the MO Diagrams you are asked to create.

9. Repeat the above steps for the hydrogen anion (H-). Exception: Set the Total Charge to Anion and the Unpaired Electrons to 0.

10. Repeat the above steps for the hydrogen cation (H+). Exception: Set the Total Charge to Cation and the Unpaired Electrons to 0. Record the error message you receive. Close this file.

Computational Study of Atomic & Molecular Orbitals Revised: 10/21/15

6

B. Finding ΨH2: Diatomic Hydrogen (H2) Calculations 1. Determine the Lewis structure and sketch the molecular orbital (MO) diagram of H2

using the 1s and 2s atomic orbitals. 2. On the menu bar, click File and then New. In the Model Kit in the right panel, click the

Organic tab. Click –H to select hydrogen. In the Workspace area, click anywhere to place hydrogen. Click on the yellow bond to add an additional hydrogen.

3. On the menu bar, click Setup, and then Surfaces. In the Surfaces window, click the Add button. Choose HOMO, then LUMO, and finally Electrostatic Potential Map.

4. Click the More Surfaces… button. Change the Surface field to LUMO (+1). (Do not change the Property and Resolution fields.) Click Apply. Change the Surface field to HOMO (-1). Click Apply. Click OK to close the More Surfaces window. Leave the Surfaces window open.

5. On the menu bar, click Setup, and then Calculations. Select Calculate: Equilibrium Geometry with Hartree-Fock 6-311+G**. Click Submit to exit Calculations window. When the Save As window appears name the file and click on Save. Click OK to start the calculation.

6. Create the following table using the Rich Text Editor. Paste the images from Spartan into the ELN. Table 2. Species & Bond Length (Å)

HOMO (-1)

HOMO & E HOMO

(kJ/mol)

LUMO & E LUMO

(kJ/mol)

LUMO (+1)

Electrostatic Potential Map & E (kJ/mol)

H2+

H2 H2

-

7. In the Workspace Area, click on the bond between the hydrogen atoms so that it is highlighted. In the first column of Table 2, record the equilibrium bond length displayed in the lower right hand corner of the screen.

8. Arrange the molecule so the bond is horizontal. (All images should be copied with the bond in the same orientation to make comparison easy.) Only Correct Orientation:

Many Incorrect Orientations: , , , , et cetera.

9. In the Surfaces window, click on the box next to any of the surfaces to display an image. Fill in Table 2 with the indicated images for HOMO (-1), HOMO, LUMO, and LUMO (+1). (All images should be in the Transparent style.)

10. In the Surfaces window, click on the box next to electrostatic potential map. Go to the Display menu and click on Properties. A window will appear that is named Molecule Properties. Click on the electrostatic potential map green surface, the name of the window should change to Surface Properties. Change the values for the Property Range

Computational Study of Atomic & Molecular Orbitals Revised: 10/21/15

7

(kJ/mol) to -14.65 and 57.96. The electrostatic potential map surface should now have a rainbow of colors on it.

11. To find the energies indicated on Table 2, go to the Display menu and choose

Spreadsheet. Add E, E HOMO, and E LUMO. Click Add and then OK. Record the energy values (in kJ/mol) indicated under the images on Table 1. Close this file when finished.

C. Diatomic Hydrogen Cation (H2

+) Calculations Repeat the steps in Part B. Exceptions: Set Total Charge to Cation and Unpaired Electrons to 1. Also, in step 10 from Part B, change the Property Range values to 1021.61 and 1166.49 kJ/mol.

D. Diatomic Hydrogen Anion (H2

-) Calculations Repeat the steps in Part B. Exceptions: Set Total Charge to Anion and Unpaired Electrons to 1. Also, in step 10 from Part B, change the Property Range values to -722.73 and -637.88 kJ/mol.

E. Variation of H2's Bond Length

1. Create the following table using the Rich Text Editor. Follow the instructions below to copy and paste the images from Spartan directly into the ELN.

Table 3. Internuclear Distance (R, Å)

Electrostatic Potential Maps

2. Create another H2 molecule. 3. On the menu bar, click Setup, and then Surfaces. Click the Add button and choose

Electrostatic Potential Map. Close the Surfaces window. 4. On the menu bar, click Geometry and then Measure Distance. 5. In the Workspace Area, select the bond by clicking on the bond between the hydrogen

atoms or by clicking on one hydrogen atom and then the other. (When the bond is selected it will be covered with a yellow-gray mask.

6. In the lower right corner of the screen, click inside the field indicating the distance. Change the value displayed to 0.400 and press Enter.

7. On the menu bar, click Setup, and then Calculations. Select Calculate: Energy with B3LYP and 6-31G*. Click the Submit button. Why the change from Equilibrium Geometry to Energy? In this section we are not interested in the "equilibrium" or most stable molecular form in Spartan. Bond lengths will be fixed to values other than the best (or equilibrium) bond length. Spartan does not

Computational Study of Atomic & Molecular Orbitals Revised: 10/21/15

8

change bond lengths or bond angles (i.e. molecular structure) when an "Energy" calculation is performed. Why the change from Hartree-Fock to B3LYP? The previous method (Hartee-Fock, 6-311+G**) was chosen because it created the MO surfaces correctly. This new method (B3LYP, 6-31G*) calculates an equilibrium bond length closer to the accepted experimental value. Since we are altering bond length in this part, we will switch to a method that does a better job with bond length estimation.

8. Enter a name for the file in the Save As window. Click OK button to start the calculations. When the dialog appears indicating that the calculations are complete, click OK.

9. Open the electrostatic potential map image. Change to Transparent. Repeat step 10 from Part B to reset the Property Range. Copy and paste this image into Table 3.

10. Repeat steps 1 - 8 (closing the preceding file and opening a new file for each bond length, using bond lengths 0.580, 1.25, 3.20, and 5.00.

11. Repeat steps 1 - 3. On the menu bar, click Setup, and then Calculations. Select Calculate: Equilibrium Geometry with B3LYP and 6-31G*. Click the Submit button. When the calculation has completed, click on the bond between the hydrogen atoms so that it is highlighted. Record the bond length displayed in the lower right hand corner of Table 2 in the cell labeled equilibrium bond length and paste in the transparent Electrostatic Potential Map image.

F. Bond Length's Effect on Total Energy and ΔE (HOMO/LUMO Gap)

1. Open the file for the H2 molecule with the smallest bond length (0.400 Å). Go to the Display menu and choose Spreadsheet. A spreadsheet window should open.

2. Go to the Geometry menu and choose Measure Distance. Click on both of the hydrogen atoms in the H2 molecule. (Both atoms should now have a yellow-gray mask.) Click on the P in the bottom right hand corner of the screen (next to the distance value). The bond length value should show up under the Distance heading. You may not see a significant difference, this is okay.

3. Click Add in the Spreadsheet window. Choose the following: E, E HOMO, and E LUMO. (Your choices will be highlighted once selected.) Click Apply then OK. Four quantities should now be visible on the spreadsheet: Distance, E, E HOMO, and E LUMO. The spreadsheet window may need to be enlarged to see all values. The definition of these four quantities: • Distance: Bond length. These values are the same as the values set in Part E. • E: Molecular Energy. This is the energy required to create the molecule either from

its elements in their standard state (enthalpy of formation, ΔHf) or from a collection of separated nuclei and electrons.

• E HOMO: Energy of the highest occupied MO. This energy value is implied by the placement of the MO on a molecular orbital diagram.

• E LUMO: Energy of the lowest unoccupied MO. This energy value is implied by the placement of the MO on a molecular orbital diagram.

Computational Study of Atomic & Molecular Orbitals Revised: 10/21/15

9

4. From the File menu choose Append Molecule(s)… Open the file for the H2 molecule with the next largest bond length (0.580 Å). (Make sure you put in the molecule created in Part E. #12.) The Distance, E, E HOMO, and E LUMO values should have appeared in the next row of the Spreadsheet. Repeat this step until all of the molecules created in Part E have been appended to the Spreadsheet.

5. Copy the four columns (Distance, E, E HOMO, and E LUMO) by dragging the cursor across to highlight the cells to be copied. Copy the cells, then open Excel and paste the table into a worksheet. (Save the changes to the Spartan file you are working on.)

6. Create 2 Excel plots (change axis max and minima so that curves fill up entire plot window and be sure to put the independent variable on the x-axis): • Potential Energy Curve (E vs. Distance): This plot will demonstrate how bond length

affects the total energy (E) of a molecule. In Excel, choose the XY (Scatter) plot with Smooth Lines and Markers. This will give you a computer generated curve through the points.

• Effect of Bond Length on ΔE (E HOMO and E LUMO vs. Distance): This plot will demonstrate what happens to the HOMO/LUMO energy gap as bond length is increased. Make sure you fit two curves on the same plot – one to the E HOMO data the other to the E LUMO data. (The same scatter plot option used for the first data works well.)

DISCUSSION

Using the previous lab’s entry into the Post Lab Results & Discussion as a template, create a page answering the following questions. There should be at least 5 headings (one for each question). The TA should not need to refer to the lab manual to see the question.

1. Using the data collected in Part A, answer the following questions: a. What are the principal and angular quantum numbers for the HOMO? Provide the

number and letter designation for the atomic orbital. b. What are the principal and angular quantum numbers for the LUMO? Provide the

number and letter designation for the atomic orbital. c. Explain why the calculation for H+ failed.

2. Using both 1s and 2s atomic orbitals, create MO diagrams for H2

+, H2, and H2-. Shrink and

then cut out the HOMO(-), HOMO, LUMO, and LUMO(+) images from Table 1. Paste each image next to its energy level on the MO diagram. Provide the molecular electron configuration and bond order for each. Which species is the most stable? Why?

3. Answer the following questions for H2+, H2, and H2

-: a. What is density, HOMO, & LUMO in terms of the molecular wave function? b. What is indicated if two colors are present on a HOMO or LUMO plot? c. Is the electron more delocalized in the HOMO or LUMO plots of H2

+ and H2? d. Is there a relationship between HOMO & LUMO and bonding & antibonding? Explain. e. Which of the three species (H2

+, H2, and H2-) has a HOMO with a node?

Computational Study of Atomic & Molecular Orbitals Revised: 10/21/15

10

f. What atomic orbitals create the LUMO for H2-? What orbital for H2

+ and/or H2 more closely resembles the HOMO in H2

-? g. What do the electrostatic potential maps demonstrate about bond order? (Provide a

specific answer for H2+, H2, and H2

-.) h. Which two images on Table 2 didn't give normal or any surface images? Why? i. Two more images need to be calculated so that all of the molecular orbital energy levels

drawn in question #2 have images. What are they?

4. Sketch the Molecular Energy curve shown below. What part of the curve below is the 1st Excel plot created in Part F #6? (Use a different color to indicate that part of the plot.) Label the 6 bond lengths (0.400, 0.580, equilibrium bond length, 1.25, 3.20, 5.00) on your sketch.

5. The energy of the HOMOs and LUMOs (E HOMO and E LUMO) are reported relative to the

atomic orbitals (AOs) used to create the molecule. For the 2nd Excel plot created in Part F, #6, draw a horizontal line parallel with the x-axis that does not cross the data lines for the E HOMO or E LUMO data. This line represents the energy of the two 1s atomic orbitals that created the HOMO (σ1s) and LUMO (σ1s

*). Why do the E HOMO and E LUMO curves appear that they will converge at longer distances (bond lengths)?

Re

Internuclear Distance, R (Å)

-De

Ener

gy (k

J/m

ol)