QUANTUM CHEMISTRY PROJECT 3: ATOMIC AND MOLECULAR STRUCTURE

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1 Chemistry 460 Fall 2017 Dr. Jean M. Standard November 1, 2017 QUANTUM CHEMISTRY PROJECT 3: ATOMIC AND MOLECULAR STRUCTURE OUTLINE In this project, you will carry out quantum mechanical calculations of atomic and molecular structure. In addition to structure and properties of atoms and molecules, some of the things you will explore are the effects of level of theory and basis set size on the results. The computer programs that you will use for these simulations include a state-of-the-art software package called Gaussian for the quantum mechanical calculations and a software package called Avogadro for visualization. You will be able to access Gaussian and Avogadro in the computer lab located in Julian Hall Room 216. Please note that the computers in the JH 216 are usually available at any time during the day; however, they are occasionally reserved for classes as noted on the schedule posted on the door. In the event that the room is busy during a time you want to use the Macintoshes, you may use the Macintosh in Julian 222 (outside my office). You may use any of the Macintosh computers in JH 216 to access the Avogadro/Gaussian programs (instructions are provided later in this handout). In fact, you can use one Macintosh for one part of the project, and another Macintosh for another part of the project since none of the data is stored on the Macs. All the calculations actually will be carried out on Linux workstations in SLB 220 that you will connect to from the Macintoshes via the internet. DUE DATE The project is due no later than 6 PM on MONDAY, NOVEMBER 27, GRADING Project 3 is worth 50 points and consists of three parts. Part A is worth 15 points, Part B is worth 20 points, and Part C is worth 15 points.

2 2 OVERVIEW: QUANTUM CHEMISTRY METHODS Quantum Mechanics of Many-electron Systems Because the Schrödinger equation cannot be solved exactly for atoms and molecules containing more than one electron, the goal in most cases is to obtain the most accurate approximate solution possible within the constraints of time and available computational resources. Improving the Accuracy of the Results There are two primary ways in which the accuracy of the approximate solution may be improved. One approach is to increase the size of the basis set so that it is closer to the limit of a complete set. Since any arbitrary function (including the true wavefunction) may be represented exactly using a complete set of functions, increasing the number of basis functions generally leads to an improved representation of the wavefunction. Another approach to improve the accuracy of the approximate solution is to increase the level of approximation in the methodology (or level of theory, as it is commonly stated) employed to obtain the approximate energy and wavefuntion. The two approaches to improving the accuracy of the approximate solution often are utilized in combination, as illustrated in Figure 1. Full CI CCSD(T) Level of Theory QCISD MP4 MP2 Accuracy DFT HF Minimal DZ TZ QZ Basis Set Size Figure 1. The two key factors that control the accuracy of quantum chemical calculations, level of theory and basis set size. Levels of Theory The lowest level of ab initio quantum theory for atomic and molecular systems is the Hartree-Fock model. This method is often denoted HF, more specifically RHF for closed shell systems (Restricted Hartree-Fock) and UHF for open shell systems (Unrestricted Hartree-Fock). This method is characterized by approxmations that lead to the neglect of electron correlation; that is, electron-electron repulsion is treated only in an average sense. To improve upon the Hartree-Fock model, one approach for the inclusion of electron correlation is the use of perturbation theory, which determines a correction to the Hartree-Fock energy. The most common form of perturbative correction is known as MØller-Plesset Second-Order Perturbation Theory, or MP2. Higher orders of perturbation theory, such as MP4, also may be used. The MP2 method is the least expensive ab initio method for the correction of the Hartree-Fock model and inclusion of a significant portion of the electron correlation in the system. However, the use of the MP2 or MP4 methods comes with additional cost in computing time.

3 3 Beyond perturbative methods such as MP2 and MP4, the primary methods to improve the approximate solution of the Schrödinger equation include Coupled-Cluster (CC) theory and Configuration Interaction (CI) methods. Both of these types of methods move beyond adding corrections to the Hartree-Fock solution and include electron correlation effects directly into the wavefunction. The wavefunction of the Hartree-Fock method is based upon a single Slater Determinant. The wavefunctions of the Coupled Cluster and Configuration Interaction methods are based upon multiple Slater Determinants. There are many varieties of Coupled Cluster theory and Configuration Interaction; they differ by how many additional Slater Determinants are included in the calculation, which are based upon excitations from a single reference Slater Determinant (or configuration). The most common Coupled Cluster methods include CCSD, which adds single and double excitations from the reference, and CCSD(T), which adds single, double, and a selection of triple excitations. Similary, for Configuration Interaction, common methods are CISD and CISD(T), which refers to methods which include single, double, and in the latter case, a selection of triple excitations. Also used frequently are quadratically convergent related methods, QCISD and QCISD(T). In all these cases, the accuracy improves as the level of excitation increases; hence CISD(T) would be expected to be more accurate than CISD. The computational cost for the CC and CI methods is prohibitive; thus, these methods can only be applied to small molecules (generally, systems with fewer than 10 atoms). Basis Sets The basis set refers to the collection of functions that is used to represent the approximate atomic or molecular wavefunction. For molecular systems, the LCAO-MO (linear combination of atomic orbitals to form molecular orbitals) approximation is generally employed; thus, the basis set consist of a set of atomic functions used to construct the molecular orbitals. For atoms, the atomic functions are used directly. The smallest set of atomic functions that can be used in an atomic or molecular calculation is referred to as a minimal basis set. This set is comprised of atomic functions for all the atomic orbitals occupied in the separated atoms, and must include all orbitals of a given angular momentum even if the subshell is only partially occupied. For example, the minimal basis set for carbon atom with electron configuration 1s 2 2s 2 2p 2 includes 1s and 2s atomic functions, along with all three 2p-type atomic functions (2p x, 2p y, and 2p z ), even though the 2p subshell is only partially filled. In order to improve the approximate solution of the Schrödinger equation, the basis set size may be increased in three ways: (1) including more functions with the same angular momentum as those in the minimal basis set but with different spatial extent; (2) adding polarization functions to better account for the shifts in electron distributions when atoms are involved in chemical bonding; and (3) adding diffuse functions to account for electron density far from the atomic nuclei which may be involved in weak interactions such as hydrogen bonding or van der Waals interactions. Often, the first two approaches for increasing the basis set are employed together, and in some cases, all three approaches are employed. If the number of functions employed in the minimal basis set is doubled, we refer to this as a double zeta (DZ) basis set. If the number of functions in the minimal set is increased by 3, 4, or 5, the basis sets are referred to as triple zeta (TZ), quadruple zeta (QZ), and quintuple zeta (5Z), respectively. A common double-zeta quality basis set is 6-31G and a common triple-zeta quality basis set is 6-311G; these basis sets are referred to as Pople-style basis sets. Other common basis sets are referred to as correlation-consistent basis sets; examples include cc-pvdz, cc-pvtz, cc-pvqz, and cc-pv5x. Polarization functions in Pople-style basis sets are denoted by including the angular momentum of the basis functions being added at the end of the basis set name. Examples include 6-31G(d) and 6-311G(d,p). The correlation-consistent basis functions automatically include polarization functions, and the number and highest angular momentum of the polarization functions increase as the basis set increases from DZ to 5Z and higher. Diffuse functions in Pople-style basis sets are denoted by adding '+' after the G in the basis set name. Examples include 6-31+G(d) and G(d,p). The inclusion of diffuse functions in correlation-consistent basis sets is denoted by adding 'aug' to the name. Examples include aug-cc-pvdz and aug-cc-pv5z.

4 4 USING THE COMPUTERS AND THE AVOGADRO SOFTWARE PACKAGE This section contains instructions for logging in to the Macintosh computers in JH 216 and accessing the Linux computers on which the calculations will be performed. In addition, it provides instructions on how to start the Avogadro software package. The use of the Gaussian software package will be described later. Logging In To log in to one of the Macintosh computers in JH 216, enter your ISU ULID and password. Starting X11 In order to access the Linux workstations to run the Avogadro/Gaussian software packages, you must run a software package on the Mac called X11. To start the X11 application, go to the Applications folder on the Mac hard drive and click on the "X" icon. A single white window should appear on the screen; this is called an "xterm" window, or simply a terminal window. Logging on to the Linux Computers You may use any of the six available Linux computers for the project. To log on to a Linux computer, type the following command in the xterm window: ssh Y che460@hostname.che.ilstu.edu Here, "ssh" stands for secure shell; this application provides a secure connection to the remote Linux workstation. Also, hostname is the name of a Linux computer [select one of: frodo, samwise, gandalf, aragorn, gimli, or legolas]. The password is "erwin". Enter the password as prompted. Once you have done this, you are connected to the Linux computer. Starting the Avogadro Program To start the Avogadro program, type "avogadro" in the xterm window and hit the return key. A window for the Avogadro software package should appear on your screen. At this point, you are ready to begin the project.

5 5 PART A: NOBLE GAS IONIZATION POTENTIALS (15 points) Introduction In this part of the project, you will use quantum mechanical calculations to predict the first ionization potentials for the series of noble gas atoms, investigate trends, and compare the results to experiment. For an atom A, the ionization potential may be defined as the energy required to remove an electron in the process A A + + e. (1) In a many-electron atom, Koopmans' theorem states that the atomic orbital energy of the highest occupied orbital ε HO can be related approximately to the atom's first ionization potential, IP K = ε HO, (2) where IP K is the ionization potential (from Koopmans' theorem). Koopmans' theorem gives an approximation to the ionization potential because it assumes that the orbital from which the electron is removed has the same shape and energy in the neutral species and the cation; however, some relaxation of the orbital usually occurs in the real system. Alternately, the ionization potential may be determined from calculations of the total electronic energies of the neutral atom and cation, E cation and E neutral, respectively as IP = E cation E neutral, (3) where IP is the ionization potential. In this case, the relaxation of the cation orbital is included in the calculation. Procedure 1.) The first atom that you will perform a calculation on using the Avogadro/Gaussian software packages is helium. Following the instructions above to start the Avogadro software package should lead to a window similar to the one shown in Figure 1 appearing on your screen. Figure 1. The main Avogadro window.

6 6 The Avogadro window that you see may have a black background. If you would like to change the background color, you may do so by selecting "View Set Background Color." In addition, if the "Settings" area on the left side of the window does not appear, you may toggle it on by clicking on the "Tool Settings" button. To construct the He atom, click on the Drawing Tool,. Set the Element to Other and then select He from the periodic table, Bond Order to Single, and make sure that Adjust Hydrogens is clicked off. Then, click with the left mouse button in the Avogadro window and a He atom should be displayed in a manner similar to that shown in Figure 2. Figure 2. A helium atom in the drawing window. The next step is to perform a quantum mechanical calculation of the He atom using the Gaussian software package. The total electronic energy and atomic orbital energies of the atom will be determined. From the Avogadro menus at the top of the window, select "Extensions Gaussian". You should see a popup window similar to that shown in Figure 3. Make sure that the following settings are entered for the calculation: Calculation: Single Point Energy Processors: 4 Theory: RHF Basis: 6-31G(d) Charge: 0 Multiplicity: 1

7 7 Figure 3. Setting up a Gaussian calculation for a noble gas atom. Once the settings are correct, click "Compute". You will be prompted to enter a filename. Use a filename something like 'he-xy.com', where 'XY' corresponds to your initials. After the filename is entered, click "Save". The calculation will begin and you should see a popup window appear that says "Running Gaussian calculation...". When the popup window disappears, your calculation of the He atom should be complete. [Note: this calculation is very fast; sometimes you don't see the popup window. If you don't see the popup window after a minute or two, go on to the next paragraph to check the results file.] To retrieve the computed value of the highest occupied orbital energy, close Avogadro by selecting "File Quit". You may be asked to save changes, but you can safely discard any changes because the file you need is already saved. Next, in the terminal window, type "gedit he-xy.log" (where 'XY' corresponds to your initials). You will see a window open on the screen similar to that shown in Figure 4. The file you are viewing contains the results from the Gaussian calculation that you ran for helium.

8 8 Figure 4. Results file from a Gaussian calculation of the He atom. To find the computed value of the orbital energies, search the file for the string 'Alpha occ. eigenvalues' by selecting "Search Find" from the menu. The listing you find gives the orbital energies of the occupied and virtual (unoccupied) atomic orbitals in atomic units (hartrees). For example, for He, the listing given is: Alpha occ. eigenvalues Alpha virt. eigenvalues For He, there is only one occupied orbital (1s) and its computed energy is hartrees. For each atom, you should record the energy of the highest occupied orbital. When you are finished, select "File Quit" to close the results file. 2.) Carry out similar quantum mechanical calculations for the Ne, Ar, and Kr atoms using Avogadro/Gaussian and the same procedure described in step #1. You should also perform a similar calculation for Xe; however, in this case, the basis set selected should be LANL2DZ (all the other settings should be the same). Once the calculation of each atom completes, find the highest occupied orbital energy in the results file and record its value. 3.) The next step is to carry out some additional calculations on neon. The first thing you need is the total electronic energy of Ne from the HF/6-31G(d) calculation you performed in step #2. Use the 'gedit' command to open the log file for this calculation again.

9 9 To obtain the total electronic energy, scroll to the bottom of the file, and look for a block of text similar to the following: 1\1\GINC-FRODO\SP\RHF\6-31G(d)\Ne1\STANDARD\22-Mar-2015\0\\#n RHF/6-31G(d) SP\\Title\\0,1\Ne,0, , ,0.\\Version=AS64L-G09RevD.01\State=1- A1G\HF= \RMSD=2.635e-12\Dipole=0.,0.,0.\Quadrupole=0.,0.,0., 0.,0.,0.\PG=OH [O(Ne1)]\\@ The total electronic energy is listed in bold above (HF=xxxxx) and is given in hartrees. Record this value. 4.) Now repeat the calculation of the neon atom using different levels of theory and basis sets. You will probably want to do this by building new copies of the neon atom with Avogadro and changing the settings for the Gaussian calculation as appropriate. It is also helpful to save each calculation with a different filename in case you need to refer back to them. Once you have selected "Extensions Gaussian", use the following settings for the calculations: Calculation: Single Point Energy Processors: 4 Theory: RHF Basis: 6-31G(d) Charge: 0 Multiplicity: 1 You should see on the second line in the white input box the following text: #n RHF/6-31G(d) SP Edit the line so that it has the following format: #n RHF/aug-cc-pVDZ SP Note that here you are changing the basis set from 6-31G(d) to aug-cc-pvdz. In addition, you should add a line at the beginning of the white input box that has the form: %mem=1gb Once you have made all the changes in the settings, click "Compute" to submit the calculation. When the calculation is complete, follow the instructions given in step #3 to retrieve and record the total electronic energy from the results file. Make sure to note the level of theory and basis set of the calculation. Next, repeat the calculation for additional times, changing the level of theory and basis set to those shown in the table below. Open the results file for each calculation using the 'gedit' command and record the total electronic energy of neon for each calculation. When you are finished, you should have results for the following six calculations of the neon atom: Run Theory Basis Set A (original, step #1) RHF 6-31G(d) B (above) RHF aug-cc-pvdz C MP2 aug-cc-pvdz D CCSD(T) aug-cc-pvdz E CCSD(T) aug-cc-pvtz F CCSD(T) aug-cc-pvqz

10 10 5.) The next step is to repeat the six calculations for the neon cation. You should build each neon cation using Avogadro as you did in step #1, but build them as if you were making a neutral atom because the charge of +1 will be set before the calculations are performed. From the Avogadro menus at the top of the window, select "Extensions Gaussian". You should see a popup window similar to that shown in Figure 5. Figure 5. Setting up a Gaussian calculation for neon cation. For the neon cation calculations, make sure that the following settings are selected from the pull-down menus: Calculation: Single Point Energy Processors: 4 Theory: RHF Basis: 6-31G(d) Charge: 1 Multiplicity: 2 You should see on the second line in the white input box the following text: #n RHF/6-31G(d) SP Edit the line so that it has the following format: #n UHF/6-31G(d) SP Note that here you are changing the level of theory from RHF (Hartree-Fock theory for closed shells) to UHF (Hartree-Fock theory for open shells). In addition, you should add a line at the beginning of the white input box that has the form: %mem=1gb Once you have made all the changes in the settings, click "Compute" to submit the calculation. When the calculation is complete, follow the instructions given in step #3 to retrieve and record the total electronic energy from the results file. Make sure to note the corresponding level of theory and basis set.

11 11 Next, repeat the calculation of the neon cation, changing the level of theory and basis set, in order to complete five additional calculations as shown below. Open the results file with 'gedit' and record the total electronic energy of the neon cation for each calculation. When you are finished, you should have results for the following calculations of the neon cation: Run Theory Basis Set A (above) UHF 6-31G(d) B UHF aug-cc-pvdz C MP2 aug-cc-pvdz D CCSD(T) aug-cc-pvdz E CCSD(T) aug-cc-pvtz F CCSD(T) aug-cc-pvqz Results, Analysis, and Discussion 1.) Use Koopmans' theorem to predict the first ionization potential for each noble gas atom using Equation (2) and the RHF/6-31G(d) calculations that you completed in step #1. Report your results in both hartrees and ev. 2.) Obtain literature values of the experimental first ionization potentials of He, Ne, Ar, Kr, and Xe. Remember to cite your source; you should also record the ionization potential for radon because you will need it in #5. Compare the experimental values to the calculated values. What are the percent errors? 3.) Construct a graph in which your calculated ionization potentials from Koopmans' theorem are plotted on the y- axis and the atomic number of each atom is plotted on the x-axis. Include this plot in your project. Discuss qualitatively any trends observed in first ionization potentials as a function of atomic number. Provide physical reasons for the observed behavior. 4.) Construct another graph in which your calculated ionization potentials from Koopmans' theorem are plotted on the y-axis and a literature value of the atomic radius of each atom is plotted on the x-axis. Again, remember to cite your source for the atomic radii; you should also record the atomic radius for radon as you will need it in #5. Include this plot in your project. Discuss qualitatively any trends observed in first ionization potentials as a function of atomic radius. Is the trend linear? Fit the observed trend to a polynomial or power series and give the equation of the trendline. 5.) Obtain the atomic radius of radon from the literature. Use the best-fit equation obtained in #4 along with the atomic radius of radon to obtain a prediction for the ionization potential of radon. Compare to a literature value and discuss the agreement. 6.) Use the results from steps #4 and #5 of the procedure to determine the ionization potential of neon at different levels of theory. Use Equation (3) to perform the calculations. Tabulate your results along with percent errors compared to the experimental value. 7.) Discuss your findings with respect to variations in level of theory with a fixed basis set of aug-cc-pvdz. Are there trends in the percent errors as the level of theory increases from RHF/UHF to MP2 to CCSD(T)? 8.) Discuss your findings with respect to variations in basis set with a fixed level of theory, CCSD(T). Are there trends in the percent errors as the basis set increases from aug-cc-pvdz to aug-cc-pvtz to aug-cc-pvqz?