QUANTUM CHEMISTRY PROJECT 3: PARTS B AND C

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1 Chemistry 460 Fall 2017 Dr. Jean M. Standard November 6, 2017 QUANTUM CHEMISTRY PROJECT 3: PARTS B AND C PART B: POTENTIAL CURVE, SPECTROSCOPIC CONSTANTS, AND DISSOCIATION ENERGY OF DIATOMIC HYDROGEN (20 points) In this part of the project, you will explore the ground potential energy curve of the diatomic molecule H 2, extract vibration-rotation spectroscopic constants, and determine the dissociation energy. The chemical reaction for dissociation of a homonuclear diatomic molecule X 2 into its constituent atoms in the gas phase is given by The dissociation energy D e X 2 (g) 2 X (g). (1) is defined as the energy of the products minus the reactants in the reaction above, D e = 2E ( X) E ( X 2 ), (2) where E is the total electronic energy of the atom or molecule. A typical potential energy curve of a diatomic molecule X 2 which exhibits the bond dissociation into constituent X atoms is shown in Figure 1. At low energies, the curve is approximately parabolic (i.e., the harmonic oscillator approximation is valid), but as the energy increases the anharmonicity becomes more important. Figure 1. Typical potential curve for a diatomic molecule, where R R e is the bond displacement.

2 2 Beyond the harmonic oscillator, rigid rotor approximation, the quantized energy levels E vj of a vibrating, rotating diatomic molecule may be fit to a double polynomial expansion in the vibrational and rotational quantum numbers, v and J. This is known as a Dunham expansion, which has the following form in wavenumbers, ω vj = E vj hc! =!ν 0 v + 1 $ # & + B e J J +1 " 2 % ( ) α v + 1 e # " 2! $ &J J +1 % ( )!ν x v e # " 2! 2 $ & % 2 D() J ( J +1) * + +. (3) Here, ω vj is the vibration-rotation energy expressed in units of wavenumbers,!ν 0 is the harmonic vibrational frequency in wavenumbers, B e is the rotational constant, α e is the vibration-rotation coupling constant, x e is the anharmonicity constant, and D is the centrifugal distortion constant. The spectroscopic constants in the Dunham expansion of the vibration-rotation energy may be determined from a high-accuracy potential energy curve. Such a procedure can be found in Halpern, A. M. J. Chem. Ed. 2010, 87, , for example. In this part of the project, you will obtain a potential energy curve for H 2 from high-level quantum mechanical calculations and fit it to a 6th order polynomial in the bond displacement in order to extract the vibration-rotation spectroscopic constants as well as the dissociation energy. The level of theory you will employ is QCISD (quadratically-convergent Configuration Interaction with double excitations); because H 2 has only two electrons, in this case QICSD is equivalent to full CI. Procedure 1.) The first step in the analysis is obtain the equilibrium bond length and total electronic energy of H 2 from a geometry optimization calculation. Using Avogadro, build the H 2 molecule. To begin, start Avogadro and click on the Drawing Tool,. Set the Element to Hydrogen, Bond Order to Single, and make sure that "Adjust Hydrogens" is turned OFF. Click with your mouse in the drawing window to place a single hydrogen atom. Then click again to place another hydrogen atom nearby (the H 2 bond length is less than 1 Å). Use the mouse to draw a bond between the hydrogen atoms in order to complete the construction of the H 2 molecule. Next, select "Extensions Gaussian", and set up the calculation as follows: Calculation: Geometry Optimization Processors: 4 Theory: RHF Basis: 6-31G(d) Charge: 0 Multiplicity: 1 Format: Z-matrix You should see on the second line in the white input box the following text: #n RHF/6-31G(d) Opt Edit the line so that it has the following form (you are replacing the level of theory and basis set and adding '=Z-matrix' to the Opt keyword): #n QCISD/aug-cc-pVQZ Opt=Z-matrix In addition, you should add a line at the beginning of the white input box that has the form: %mem=2gb

3 3 Once you have made all the changes in the settings, click "Compute" to submit the calculation. Once the calculation is complete, close Avogadro and open the H 2 results file with the 'gedit 'command. Scroll to the bottom of the file, and look for a block of text similar to the following: 1\1\GINC-FRODO\FOpt\RQCISD-FC\Aug-CC-pVTZ\H2\STANDARD\26-Mar-2015\1\\#n QCISD aug-cc-pvtz OPT=Z-matrix\\H2\\0,1\H\H,1,R\\R= \\Version=AS64L- G09RevD.01\State=1-SGG\HF= \MP2= \MP3= \MP4D= \MP4DQ= \MP4SDQ= \QCISD= \RMSD=9.864e- 09\RMSF=4.652e-05\Dipole=0.,0.,0.\PG=D*H [C*(H1.H1)]\\@ The total electronic energy is listed in bold above (QCISD=xxxxx) and is given in hartrees. Record this value (your results will be slightly different than those listed above). To obtain the equilibrium bond distance, search in the Gaussian results file for the string 'Stationary point found'. The equilibrium value of the H 2 bond distance is reported in angstroms a few lines below this string, and just below that with more significant digits as part of the Z-matrix listing. 2.) The next step is to generate a portion of the potential energy curve of H 2, which you will use to obtain spectroscopic constants. Using Avogadro, build another copy of the H 2 molecule. Select "Extensions Gaussian", and set up the calculation as follows: Calculation: Single Point Energy Processors: 4 Theory: RHF Basis: 6-31G(d) Charge: 0 Multiplicity: 1 Format: Z-matrix You should see on the second line in the white input box the following text: #n RHF/6-31G(d) Opt Edit the line so that it has the following form (you are replacing the level of theory and basis set and changing the calculation type to SCAN): #n QCISD/aug-cc-pVQZ SCAN In addition, you should add a line at the beginning of the white input box that has the form: %mem=2gb Finally, the last line in the white input box should look something like the following: B The numerical value may be different because it corresponds to the H-H bond length in the initial structure that you built. Edit the line so that it has the following form: B This sets up a scan of the potential energy curve of H 2 which starts with a bond distance of 0.50 Å and computes the total electronic energy of H 2 at 14 additional distances using an increment of 0.05 Å.

4 4 Once you have made all the changes in the settings, click "Compute" to submit the calculation. When the calculation completes, close Avogadro and open the results file using 'gedit'. Search for the string 'Summary of the potential surface scan'. You should see a table of results that looks similar to those shown in Table 1. The second column gives the H-H bond distance in Å and note that the desired QCISD energies are listed in the last column of the results. Table 1. Example of results from potential surface scan of H 2. Note that these results were obtained at the QCISD/aug-cc-pVTZ level, so they will be slightly different than the results from the QCISD/aug-cc-pVQZ level. Summary of the potential surface scan: N B1 SCF MP2 MP3 MP4DQ QCISD You will need to copy the table from the Gaussian results file into a separate text file which will then be used to import the results into Microsoft Excel. To do this, start Microsoft Word on the Macintosh computer and open a new blank document. Copy the rows corresponding to the potential surface scan results from the Gaussian file (like those above) into this blank document. [To copy from the results file, select the desired rows of data with the mouse and use Control-C. Copy into Word using Command-V.] Save the Word document on the Desktop as a Plain Text File using "Save As", and give the file a descriptive name with the file extension '.txt' (the default settings for Text Encoding and Line Breaks are fine). You should then copy the plain text file to a removable drive for later analysis. 3.) Next, to determine the dissociation energy of H 2, you will need to calculate the energy of the H atom using Avogadro and Gaussian. To do this, follow the procedure from Part A#1 to construct the H atom. Select "Extensions Gaussian", and 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: 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 form (you are replacing the level of theory and basis set): #n QCISD/aug-cc-pVQZ SP Submit the calculation, and when it completes close Avogadro and open the results file using 'gedit'. Locate and record the total electronic energy of the H atom.

5 5 Results, Analysis, and Discussion 1.) From the total electronic energies of the H 2 molecule and H atom from steps #1 and #3 of the procedure, compute the bond dissociation energy of H 2 at the QCISD/aug-cc-pVQZ level of theory using Equation (2). Report your result in both atomic units and cm 1 [the conversion factor is 1 a.u. = cm 1 ]. 2.) Compare your results for the equilibrium bond length and dissociation energy of H 2 at the QCISD/aug-cc-pVQZ level of theory with experimental literature values. Compute percent differences. Which agrees better with experiment, the equilibrium bond length or the dissociation energy? You also should compare your QCISD results to results from other lower levels of theory; some results may be found in Table 11-9 of Lowe and Peterson. Discuss the degree of variation in the bond length and dissociation energy as the level of theory is improved. 3.) Import your H 2 potential energy curve data at the QCISD/aug-cc-pVQZ level from step #2 of the procedure into Microsoft Excel. Include a table listing this data in your project. Next, calculate bond displacements by subtracting the equilibrium bond distance from each value of the bond distance; convert these bond displacement values into units of bohr. Produce a graph with the QCISD energy (in hartrees) on the y-axis and bond displacement (in bohr) on the x-axis. Include this graph in your project. Fit the data in the graph to a 6th-order polynomial using Microsoft Excel. Include a plot with this trendline in your project and make sure that the equation and R 2 value are displayed on the chart. It is also helpful to modify the trendline format to display the parameters of the fit to about 5 or 6 decimal places. The 6th-order polynomial representing the potential energy V of H 2 has a functional form given by V(R) = A 0 + A 1 ( R R e ) + A 2 R R e + A 5 ( R R e ) 5 + A 6 ( R R e ) 6. ( ) 2 + A 3 ( R R e ) 3 + A 4 ( R R e ) 4 (4) In this equation, R is the bond distance, R e is the equilibrium bond distance (and thus R R e is the bond displacement), and A 0 through A 6 are the parameters of the fit. Note that since the polynomial expansion is centered about the equilibrium bond distance (i.e., the minimum of the potential curve), the parameter A 1 should be close to zero, though usually not exactly zero due to round-off errors and such. The dissociation energy D e for H 2 may be determined from Equation (2) using the total electronic energy of the H atom from step #3 of the procedure, and the total electronic energy of H 2 as parameter A 0, which should closely match the total electronic obtained in step #1 of the procedure. Compute and report the dissociation energy D e of H 2. Note that the value of D e obtained here should closely match the value obtained in #1, though there may be small differences due to the fitting procedure. The harmonic vibrational frequency ν 0 is given by ν 0 = 1 k 2π µ 1/2, (5) where k is the harmonic force constant and µ is the reduced mass of the diatomic molecule, µ = m 1 m 2 m 1 + m 2, (6)

6 6 where m 1 and m 2 are the masses of the two atoms (in this case, both are hydrogen atoms). We showed in class that when the potential energy function V(R) is expanded in a polynomial series in terms of the bond displacement, the harmonic force constant equals two times the parameter of the second-order term (i.e., the term has the form 1 2 kx2, where x is the bond displacement). In this case, therefore, the force constant k is defined as k = 2A 2. (7) Use this relationship to obtain the force constant k from your fit, (note that A 2 has units of hartree bohr 2 ) and then use Equation (5) to determine the harmonic vibrational frequency ν 0 in s 1. Make sure to use a reduced mass with at least five significant figures so that you get an accurate value for the harmonic frequency. Convert into units of cm 1 and report the results for the harmonic frequency using the relation where ν 0 is the harmonic frequency in units of cm 1. ν 0 = ν 0 c, (8) The rotational constant B e can be determined from the equilibrium bond length and reduced mass using the relation B e = h 8π 2 cµr e 2. (9) Compute the rotational constant B e for H 2 and report its value in cm 1. The centrifugal distortion constant D (not to be confused with the dissociation energy D e ) is defined by the equation, D = 4B 3 e!ν. (10) 2 0 Using your calculated values of the harmonic vibrational frequency!ν 0 and rotational constant B e in cm 1, determine and report the centrifugal distortion constant D in cm 1. The vibration-rotation coupling constant α e may be determined from the following relation, α e = 6B 2 " e $ 1 + A 3R e!ν 0 # A 2 % '. (11) & Using R e in bohr, the previously-calculated values of the harmonic vibrational frequency!ν 0 and rotational constant B e in cm 1, along with the parameters A 2 and A 3 from the 6th-order fit, calculate and report the vibration-rotation coupling constant α e in cm 1.

7 7 Finally, the anharmonicity constant x e may be determined using the equation, x e = ( B! e α!ν $ * e 0 # 2 & 8!ν 0 ) * " 6B e % 2 12 A 4 R 2 e A 2 + -, -. (12) In this equation, if α e, B e, and!ν 0 are expressed in cm 1, with R e in bohr, then x e will be unitless. Using the previously-calculated spectroscopic constants, along with the fit parameters A 2 and A 4, calculate and report the anharmonicity constant x e. In summary, when you have completed this section, you should have a graph of the energy vs. bond displacement for H 2 along with a 6th-order polynomial fit of the data. In addition, you should have determined the dissociation energy D e along with the following five vibration-rotation spectroscopic parameters for H 2 : (1) the harmonic vibrational frequency!ν 0, (2) the rotational constant B e, (3) the centrifugal distortion constant D, (4) the vibration-rotation coupling constant α e, and (5) the anharmonicity constant x e. These items all should be included in your project. It might be helpful to show your work for any calculations in order to receive partial credit in the case that something is in error. 4.) Obtain experimental literature values for each of the five vibration-rotation spectroscopic constants. Cite your source(s). Compare your results to the literature, being quantitative in your comparisons. Can you suggest an approach to obtain even better agreement between the calculations and experiment?

8 8 PART C: COMPLETE BASIS SET EXTRAPOLATION OF THE ENERGY AND PROPERTIES OF WATER (15 points) In this part of the project, you will explore quantum mechanical calculations on the water molecule, H 2 O. Because water is a small molecule with only one non-hydrogen atom, very high-level calculations with large basis sets may be performed in order to obtain highly accurate energies and properties. The calculations will employ the CCSD(T) level of theory, which has been referred to as the "gold standard" of quantum chemistry for molecular systems [Cramer, C. J. Essentials of Computational Chemistry, Wiley, New York, 2002, pp ]. This refers to the fact that when the basis set size is increased, energies and many properties computed at the CCSD(T) level of theory appear to converge to asymptotic values, known as the complete basis set (CBS) limit. And for many systems, the CBS extrapolation of the CCSD(T) results provides excellent agreement with experimental results. Please refer to the many examples of CBS extrapolation found in Dunning, T. H. J. Phys. Chem. A 2000, 104, There are a variety of functional forms used to obtain the asymptotic limit from the CBS extrapolation. One of the most comment is the X 3 fit, in which the property of interest P is calculated using correlation-consistent basis sets, aug-cc-pvxz, where X=D, T, Q, 5, etc. The values of the property P for each basis set level, P X ( ), is then fit to the following functional form, P( X) = P CBS + A X 3. (13) In this equation, X=2, 3, 4, 5. Also, P CBS is the value of property P in the limit of a complete basis set (i.e., as X ) and A is a fitting parameter. A typical CBS extrapolation is shown in Figure 7 for the determination of the dissociation energy of N 2 at the MP2, MP3, and MP4 levels of theory with cc-pvxz (X=D, T, Q, 5) basis sets. Figure 7. CBS extrapolation of the dissociation energy of N 2 at various levels of theory [from Dunning, T. H. J. Phys. Chem. A 2000, 104, ]. In this part of the project, you will perform CBS extrapolations of the energy, geometry, and dipole moment of the water molecule in order to investigate their asympototic convergence and obtain these properties to high accuracy. The agreement between the CBS-extrapolated results and experimental values will be explored.

9 9 Procedure 1.) The first thing you need to do is build the H 2 O molecule using Avogadro. To construct an initial structure for the H 2 O molecule, start Avogadro and click on the Drawing Tool,. Set the Element to Oxygen, Bond Order to Single, and make sure that "Adjust Hydrogens" is turned ON. Click with your mouse in the drawing window to place a single oxygen atom. This should actually place an oxygen atom along with two attached hydrogens in the drawing window (i.e., you have built the H 2 O molecule). Next, perform your first quantum mechanical calculation of the H 2 O molecule using the Gaussian software package. Select "Extensions Gaussian". Use the following settings: Calculation: Geometry Optimization Processors: 4 Theory: RHF Basis: 6-31G(d) Charge: 0 Multiplicity: 1 Format: Z-matrix You should see on the second line in the white input box the following text: #n RHF/6-31G(d) Opt Edit the line so that it has the following form (you are replacing the level of theory and basis set and adding '=Z-matrix' to the Opt keyword): #n CCSD(T)/aug-cc-pVDZ Opt=Z-matrix In addition, you should add the following line to the beginning of the white input box: %mem=4gb Once you have made all the changes in the settings, click "Compute" to submit the calculation. When the calculation is complete, close Avogadro and open the H 2 O results file with the 'gedit 'command. You will need to extract a variety of information from the results file, including the total electronic energy, CPU time, number of basis functions, and equilibrium geometry. 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\POpt\RCCSD(T)-FC\Aug-CC-pVDZ\H2O1\STANDARD\26-Mar-2015\1\\#n CCSD(T) aug-cc-pvdz OPT=Z-matrix\\H2O\\0,1\H\O,1,B1\H,2,B2,1,A1\ \B1= \B2= \A1= \\Version=AS64L-G09RevD.01\State=1- A'\HF= \MP2= \MP3= \MP4D= \MP4DQ= \MP4SDQ= \CCSD= \CCSD(T)= \RMSD=2.317e- 09\RMSF=1.559e-04\PG=CS [SG(H2O1)]\\@ The total electronic energy is listed in bold above (CCSD(T)=xxxxx) and is given in hartrees. Record this value. You should also record the value of the energy computed at the Hartree-Fock level (HF=xxxxx) for later use in exploring the electron correlation effects. To obtain the dipole moment, look in the results file just above the block of text with the energy. You should see some lines listing the dipole moment, such as shown below: Dipole moment (field-independent basis, Debye): X= Y= Z= Tot=

10 10 The value you want is the total, listed at the end of the line (in this case, ); the units are Debye. Before you close the results file, you also should record the CPU time to the nearest minute, which is listed at the very end of the file after the random quote. And finally, you should record the number of basis functions, which may be found by searching for the string "Standard basis". You should find a block of text similar to the following: Standard basis: Aug-CC-pVDZ (5D, 7F) There are 20 symmetry adapted cartesian basis functions of A1 symmetry. There are 4 symmetry adapted cartesian basis functions of A2 symmetry. There are 7 symmetry adapted cartesian basis functions of B1 symmetry. There are 12 symmetry adapted cartesian basis functions of B2 symmetry. There are 18 symmetry adapted basis functions of A1 symmetry. There are 4 symmetry adapted basis functions of A2 symmetry. There are 7 symmetry adapted basis functions of B1 symmetry. There are 12 symmetry adapted basis functions of B2 symmetry. 41 basis functions, 65 primitive gaussians, 43 cartesian basis functions The total number of basis functions is listed in bold in the example. Record this value for each calculation. To obtain the equilibrium bond distances and bond angle, seach in the Gaussian results file for the string 'CONVERGENCE CRITERIA'. The equilibrium values of the H 2 O bond distances (in angstroms) and bond angle (in degrees) are reported a few lines below this string under 'Optimized Parameters', and just below that with more significant digits as part of the Z-matrix listing. Record the bond distances to 5 decimal places (they should be equivalent to that number of digits) and the bond angle to 4 decimal places. 2.) Next, you should repeat the calculation of water for the TZ, QZ, and 5Z basis sets. Follow the same procedure as in step #1, except perform three more calculations in which you change the basis set to the following: a) aug-cc-pvtz b) aug-cc-pvqz c) aug-cc-pv5z. All the other settings should be as in #1. The aug-cc-pvtz calculation may be run as usual by clicking "Compute" from the Avogadro window to submit the calculation. The aug-cc-pvqz and aug-cc-pv5z calculations are more lengthy and hence you should submit these using a different method so that you can have the calculations run while you are off doing other things. For the QZ and 5Z calculations, from the Avogadro setup window for Gaussian, click "Generate". You will be prompted to enter a filename. Make sure to record the name of the file for future reference; it should end with a '.com' extension. To submit the calculation, close Avogadro, and in the terminal window type the following command: g09 your-h2o-file.com your-h2o-file.log & Here, 'your-h2o-file.com' corresponds to the name of the file that you saved with the "Generate" command. The command above submits your Gaussian calculation. Run one calculation at a time. Do not try to run both the QZ and 5Z calculations simultaneously; it will bog down the computer significantly. The QZ calculation should take about 30 minutes or so of real time, while the 5Z calculation should take about 6 hours of real time (note that since you are using 4 processors for the calculations, the amount of CPU time is roughly 4 the real time). Once you have submitted a calculation, you may log off and return later to get the results. Make sure that you log onto the same Linux computer where you submitted the calculation (i.e., if you submitted the QZ calculation on frodo, make sure to go back to frodo to get the results).

11 11 Once the TZ, QZ, and 5Z calculations are finished, follow the instructions in step #1 of the procedure to record the HF and CCSD(T) total electronic energies, dipole moment, CPU time, number of basis functions, and equilibrium bond distances and angle for each calculation. Results, Analysis, and Discussion 1.) Include in your project a table listing the basis set, number of basis functions, CPU time, HF energy, CCSD(T) energy, dipole moment, bond distance, and bond angle for H 2 O at the CCSD(T) level with the aug-cc-pvxz (X=D, T, Q, and 5) basis sets. 2.) For the CCSD(T) energy, dipole moment, bond distance, and bond angle, make a graph with the property on the y-axis and the ordinal number of the basis set (X=2, 3, 4, 5) on the x-axis. You should have four graphs total; turn these in with your project. Discuss whether or not each of the properties appears to be approaching an asymptotic value. 3.) To fit the properties to the asymptotic form given in Equation (13), for those properties studied in #2 above create a graph with the property on the y-axis and 1/X 3 on the x-axis. Only do this for those properties that appear to go to an asympotote. You should have a minimim of two graphs total for this part; turn these in with your project. The DZ results often deviate somewhat from the trend and are therefore usually left out of the fit. Therefore, you should fit only the X=3,4, and 5 data points to a linear trendline (this is referred to as a TQ5 fit); make sure to include the equation of the line and the R 2 value on the graph. Also, even though you are not including the X=2 data points in the fit of the trendline, make sure to show them on the graphs; you may include the X=2 data point as a separate series on each graph if necessary. 4.) From the trendlines for each graph in #3, tabulate and report the values of P CBS and A for each of the properties with trendlines. While there is no experimental value of the total electronic energy at the CCSD(T) level, you should be able to find experimental literature values of the gas phase dipole moment, bond distances, and bond angle of H 2 O (cite your sources). Prepare a table of results listing the values of the properties (dipole moment, bond distance, and bond angle) for each basis set and at the CBS limit if available. Compare your results at each basis set level and at the CBS limit with the literature value; use percent differences. Discuss your findings. Are there properties which show little or no improvement as the basis set increases? Are there others that show improvement? For those properties for which you have a CBS limit, discuss the agreement between the CBS results and experiment. Do you think that the CCSD(T)/CBS method holds up as a "gold standard" in these cases? 5.) From your CPU times and basis set sizes, make a graph with the natural log of CPU time (in minutes) on the y- axis and natural log of the basis set size on the x-axis. Fit the data to a linear trendline and display the equation and R 2 value on the chart; include this graph in your project. The graph allows you to determine the approximate scaling of the CCSD(T) method with basis set size. In general, the CPU time of a method may be given by the following relation, CPU time = ck S, (14) where K is the number of basis functions, c is a proportionality constant, and S is the scaling of the method. This relationship may be converted into a linear one by taking the natural log of both sides to yield, ln( CPU time) = ln( c) + S ln( K). (15)

12 12 Here we see that a graph like the one you created should be linear with a slope corresponding to the scaling factor S. For the CCSD(T) method in the Gaussian software package, what do you find to be the scaling? Report S. Based on your computed scaling factor S, estimate (and report) how much CPU time (in minutes, then convert to hours and days) it would take to perform a CCSD(T)/aug-cc-pV5Z calculation on ethanol, CH 3 CH 2 OH. Using the aug-cc-pv5z basis set, the basis set size for ethanol is 861 functions.