Introducing Quantum Calculations into the Physical Chemistry

Chapter 9

Introducing Quantum Calculations into the Physical Chemistry Laboratory Downloaded via AUBURN UNIV on August 21, 2019 at 06:22:14 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

Thomas C. DeVore* Deparment of Chemistry and Biochemistry, James Madison University, Harrisonburg, Virginia 22807, United States *E-mail: [email protected].

This chapter contains computational chemistry exercises to explore topics often presented in upper level undergraduate courses. It is used to supplement the classic HCl/ DCl infrared spectroscopy experiment, to determine the IR spectra of the isoelectronic series BO2-, CO2, and NO2+ and to assign the IR and Raman spectra of benzene. Computational chemistry can be used to determine the ground state term symbols for atoms and compare them to the predictions made using the aufbau principle and Hund’s rules. The final project investigates possible reaction pathways to form the ions found in the mass spectrum of methane.

Introduction Computational chemistry was largely left to the experts until the 1990’s when more powerful computer systems started becoming common on campuses and ways to integrate the computational chemistry into the curriculum began to appear in the chemical education literature (1–7). Rapid advances in computer technology since then have led to the development of high level computational software that can be run on personal computers (8, 9). This has led to many publications devoted to incorporating computational chemistry into lecture and laboratory courses throughout the chemistry undergraduate curriculum as indicated by a small sampling of suggested exercises published since 1995 given in the references (1–7, 10–30). Much of the software needed to do these exercises can now be operated using the personal computers found on most campuses. By acquiring a site license, this software can be used by multiple users making it easy to add this tool to an undergraduate laboratory courses. Computational chemistry has the advantage of requiring no chemicals, requiring no instrumentation other than the computer/software needed to do the exercise, and always producing results. The exercises presented in this chapter are focused on investigating the concepts rather than of presenting computational methods that will produce the best answers possible. While these exercises were designed for the physical chemistry laboratory or similar upper level undergraduate laboratory, some, such as the atomic investigations presented, © 2019 American Chemical Society

Grushow and Reeves; Using Computational Methods To Teach Chemical Principles ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

explore topics presented in lower level courses and could be modified for use in these laboratories if desired. A convenient way to introduce computational chemistry into the laboratory is to compare the results calculated using different methods and/ or basis sets to those measured experimentally. This lets students investigate the calculations and discover the advantages and limitations of the various computational methods available while they are learning to use the software. This approach is illustrated using the calculations done for HCl and CO2 given below. Computational chemistry can also be used to produce quantities that cannot be obtained because of lack of instrumentation, expense, or safety concerns. This enables students to gain experience with topics that normally would not be covered in the lab. Examples of this are using the investigation of the dissociation energy with and without zero point corrections done as part of the HCl/DCl project and the exercise comparing the IR spectra of CO2 to the spectra of the isoelectronic BO2- and NO2+ ions. Computational chemistry can be used to support the measurements as shown for the project presented where students assign the IR and Raman spectra of benzene. The exercises investigating the aufbau principle and Hund’s rule and investigating the ions formed in the mass spectrum of methane are computational only exercises. These exercises could also be done as a homework assignment as part of a lecture course.

Computational Methods Most of the calculations presented were originally obtained using Gaussian 03 on Dell personal computers (8, 9). While Gaussian 09 on newer Dell PC’s is now being used, the original system would still be adequate for the exercises presented in this chapter. Calculations are generally done by first building the molecule in GaussView and sending it to Gaussian where the energy is minimized. The bond lengths, bond angles, rotational constants, the molecular energy, and Mulliken populations are determined by this program during this calculation. The vibrational frequencies are calculated to confirm that the minimized structure is at least a local minimum on the energy surface by checking for imaginary frequencies (these are given as negative frequencies in the Gaussian output file). One imaginary frequency often indicates that the structure is a transition state while two or more imaginary frequencies indicates the program is trying to determine an unrealistic structure. While multiple imaginary frequencies are seldom found for the small molecules presented here, they can occur when trying to compute the structures of larger molecules. The zero point energy, temperature corrections for the enthalpy, heat capacity, and the entropy are also determined by this program as part of the vibrational calculation. I. Integrating Calculations with IR Spectroscopy Infrared spectroscopy is typically introduced in organic chemistry, where student are taught about characteristic vibrations of functional groups for use in qualitative analysis (10, 24). The visualization software in GaussView lets students examine the normal modes of vibration and discover that the characteristic frequencies presented in the frequency tables are usually coupled with other modes in the molecule (8, 9). This shows why a range rather than isolated frequencies are given in the frequency tables.

110 Grushow and Reeves; Using Computational Methods To Teach Chemical Principles ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

A. Ro-vibrational IR spectra of HCl/DCl Learning objectives: The objectives for the experimental part of this project are to use the IR spectrum of a mixture of gaseous HCl and DCl to determine the vibrational frequencies, the rotational constants, and the bond lengths of these molecules (4, 7, 31–35). The objectives for the computational part are to learn to use the software and to evaluate the calculations by comparing the results calculated for 1H35Cl using Hartree-Fock (HF), MP2, and DFT-B3LYP with one small and one larger basis set to the measured values. One of the oldest and still most commonly used physical chemistry laboratory exercises is the analysis of the ro-vibrational IR spectra of the four isotopes of HCl (See Figure 1) (4, 7, 31–35). Students discover that while some properties like the rotational constants and the vibrational frequencies depend on the reduced mass, others like the bond length and the force constant do not. In the computational part of this exercise, students learn to use the software to compute the quantities that are measured in the lab and to determine some quantities that were not measured to gain experience using the software. Since the bond length, the vibrational frequency and the rotational constant were determined from the IR spectra, these values can be compared to the calculated values. By changing the method (we use HF, MP2 and DFT-B3LYP) and the basis set (we use 6-31+G (d) and 6-311+G (2p,d)) the accuracy of the results obtained from each method can be evaluated. One discovery is that the HF method consistently gives values that are ~ 10 % too large and must be scaled to give values in reasonable agreement with the experimental values. As shown in the Tables below, the DFT calculations are generally in better agreement with the measured values and the errors tend to be more random. As a result, we do not have the students scale their DFT results. Typical results obtained for 1H35Cl using the DFT-B3LYP method with the 6-31+G (d) and the 6-311+G (2d,p) basis sets are compared to the measured and the literature values in Table 1 (36). Students quickly discover that the calculations using the larger basis set take longer to finish but often produce results in better, but still not exact, agreement with experiment. They also discover that for some applications, the smaller basis set works as well as the larger one. Most applications will involve a trade-off between speed and accuracy. The animated vibrations also provide useful insights about the atomic displacements occurring during the vibration. This is especially useful for larger molecules where the modes are more coupled. The main discovery for HCl is that most of the vibration comes from the movement of the hydrogen. Students also learn to calculate the dissociation energies (De and Do), the enthalpy of formation (ΔfH), the entropy (S) and the heat capacity at constant volume (Cv) during this exercise (37, 38). The accuracy of these calculations is established by comparing the results to the values given in the literature (36). We usually have the students determine the dissociation energy relative to the bottom of the well (De) by subtracting the energy calculated for each atom from the energy calculated for the molecule using the same method and basis set. While this approach does not produce research quality results because it ignores a detailed analysis of electronic states of the molecule dissociation pathway, it reinforces the description of De given in physical chemistry textbooks (37). As shown in Table 1, the values produced are acceptable for this laboratory exercise. Attempting to calculate the Morse potential by doing single point calculations at several bond lengths clearly shows the effect of using a limited basis set since HCl dissociates into the ions rather than the atoms. The dissociation energy from the first vibrational level (Do) is determined by subtracting the zero point energy from De. The enthalpy of formation (ΔfH) is determined from the energies calculated for 111 Grushow and Reeves; Using Computational Methods To Teach Chemical Principles ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

H2, Cl2, and HCl. The value at absolute zero is given by E(HCl) – ½ E(H2) – ½ (Cl2) (38). ΔfH at 298 K can be found using the temperature corrections calculated by the program. The entropy and heat capacities are determined from statistical thermodynamics by the program. The translational, rotational, vibrational, and electronic contributions of each are also determined, so students studying statistical thermodynamics can investigate this in detail if desired. Since most of the contribution is from translation and rotation which are determined by the high temperature approximation of the partition functions, the entropy and heat capacity are largely independent of the method used (9, 38).

Figure 1. The ro-vibronic IR spectrum of HCl (A) and DCl (B) obtained using 128 scans at 0.125 cm-1 resolution Based on the Morse potential, the anharmonicity constant (xe) can be determined from the vibrational frequency and the Morse dissociation energy (De) using the equation given in Table 1 (4, 37). Our students use these calculated values to estimate ωexe as shown in Table 1. Once the value for ωexe is established, ωe can be determined from the measured ωo since ω0 = ωe - 2ωexe. This exercise helps the students understand the difference between harmonic and anharmonic frequencies and between De and Do.


Table 1. Comparison of the measured molecular constants for 1H35Cl to the values calculated using the DFT- B3LYP method with the 6-31+G (d) and the 6-311+G (2d,p) basis sets Constant

Measured

6-31G+ (d)

6-311G+ (2d,p)

Literaturea

ω0 (cm-1)

2885.2

2802.9

2812.4

2885.309

Be (cm-1)

10.59

10.33

10.45

10.59342

re (pm)

127.5

129.0

128.25

127.455

αe (cm-1)

0.303

-------

-------

0.3071

ωexe (cm-1)

-------

61.16b

58.88b

52.8186

De (ev)

-------

4.34c

4.52c

4.574

Do (eV)

-------

4.16d

4.34d

4.432

ΔfH (kJ/ mol)

------

-62.75e

-95.39e

-92.31

S (J/K/mol)

------

186.7

186.6

186.9

Cv (J/K/mol)

------

20.786

20.786

20.788

a from

ref. (36).

b Calculated

using ωexe = ωe2 / 4 De.

c Calculated

from E(HCl) – E(H) - E(Cl) without

d Calculated from E(HCl) – E(H) - E(Cl) with zero point corrections.

zero point corrections. from E(HCl) – ½ E(H2) - ½ E(Cl2) with thermal corrections.

e Calculated

B. IR of Isoelectronic XO2 Molecules Learning objectives: The experimental objectives are to use the ro-vibronic spectrum to determine the vibrational frequencies, the rotational constant and the bond length of CO2. The computational objective is to compute these quantities for BO2-, CO2 and NO2+ to investigate similarities and differences in this isoelectronic series. As shown in Figure 2, the ro-vibrational spectrum of the ν3 band of CO2 is sufficiently resolved at modest resolution to permit rotational analysis and resolved well enough to also determine these quantities for 13CO2 at 0.125 cm-1 resolution. The procedure for determining the vibrational frequencies and the rotational constants from the IR spectrum of CO2 and a method for using the IR spectrum of this molecule to determine, the rotational temperature, Plank’s constant, the Boltzmann constant, or the speed of light from the measured intensities have been published previously (39–41). Including computational chemistry permits investigating concepts similar to those done for HCl. Typical results for 12C16O2 using DFT-B3LYP/6-311+G (2p,d) are compared to those measured and from Herzberg in Table 2 (42, 43).


Figure 2. The IR spectrum of the ω3 band of CO2 at 1 cm-1(A), 0.5 cm-1(B), 0.25 cm-1(C) and 0.125 cm1 (D) resolution.

Table 2. Comparison of the molecular constants determined for 12C16O2 using DFT-B 3LYP/ 6-311+G (2p, d) to the experimental values Constant

Measured

Calculated

Literaturea

Bo (cm-1)

0.3903b

0.39114

0.39021

B1 (cm-1)

0.3872 b

-----------

0.3871

ro (pm)

116.22

116.03

116.21

ω2 (cm-1)

667.39b

675.64

667.40

ω1 (cm-1)

--------

1364.47

1388.17c

ω3 (cm-1)

2349.2b

2400.49

2349.16

α2 (cm-1)

-0.00041b

----------

-0.00072

α3 (cm-1)

0.0031b

----------

0.00309

(42, 43). b Determined using ωi = ωio + (B1 + B0) m + (B1 - B0) m2 (39–41). Resonance with 2w2 (42, 43). a From

c Shifted

by Fermi

BO2-, CO2, and NO2+ form an isoelectronic series and the Walsh diagram predicts that each of these 16 electron species should be linear (43). Relying on the familiar Lewis dot structures often done in general chemistry, students confidently predict that each bond will be a double bond (44, 45). Hence, similar IR spectra are expected for each species. Although the IR spectra of the gaseous ions cannot be investigated at most institutions, computational chemistry can be used to compare the IR spectra of BO2- CO2 and NO2+ obtained at the same level of theory. B3LYP-6-311+G(2d, p) calculations confirm that each molecule is linear. The molecular constants produced are given in Table 3. 114 Grushow and Reeves; Using Computational Methods To Teach Chemical Principles ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

Table 3. Molecular constants determined for 11B16O2-, 12C16O2 and 14N16O2+ using DFTB3LYP- 6-311+G (2p, d) Constant

11B16O -

12C16O

14N16O +

re (pm)

126.16

116.03

111.83

θ (degrees)

180.0

180.0

180.0

Be (cm-1)

0.3316

0.3911

0.4211

Πu (cm-1)

598.70

675.64

653.78

Σg (cm-1)

1101.53

1364.47

1434.91

Σu (cm-1)

1963.24

2400.50

2405.95

fr (N/m)

1036

1617

1800

frr (N/m)

107

136.7

194.0

2

2

2

The trend in the bond lengths, the vibrational frequencies and the stretching force constants indicate that the bonds are strongest in NO2+ and weakest in BO2-. Most students hypothesize that the charge of the ion is a possible reason for this. Since Lewis dot theory indicates that the formal charges expected for B in BO2-, C in CO2 and N in NO2+ are -1, 0, and + 1 respectively, they conclude that the interaction of this charge with the slightly negative charges expected for the more electronegative oxygen are responsible for the observed trends (44, 45). However, the Mulliken populations produce a different picture for the bonding in these species (9, 46). The calculated charge on the boron in BO2- is +0.3418 with the charge on each oxygen being -0.6709 showing that the negative charge is not on the boron. The charges on NO2+ are +0.5458 on N and +0.2271 on each O indicating that the positive charge is spread over the molecule and not isolated on the N. Even CO2 is calculated to have a charge separation with the charge on C being 0.4318 and the charge on O is -0.2159. The stronger bonds in NO2+ are from increased covalent bond strengths rather than ionic charge stabilization. While there are other, more sophisticated ways to investigate the charge separations in these species, we use Mulliken populations because they are included in the out-put file for the program. Students can examine them without having to do additional calculations reducing the time needed for this exercise. C. IR and Raman of Benzene Learning objectives: This project investigates the relationship between the IR and Raman frequencies for a molecule that has an inversion center to confirm the exclusion rule. Students discover that degeneracies lead to fewer unique frequencies than the 30 frequencies predicted using the 3N-6 rule (37, 42, 43). The role computation plays in this exercise depends upon the available instrumentation. If IR and Raman are available, then the calculations are used to assist in making the spectral assignments. The calculations can also provide the data for any part of this exercise that can’t be measured because a needed instrument is not available. Although benzene is a known carcinogen and must be handled with care, measuring the IR and Raman spectra of this molecule reinforces several concepts about vibrational spectroscopy. While it 115 Grushow and Reeves; Using Computational Methods To Teach Chemical Principles ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

is easy to obtain the IR spectrum of benzene vapor by placing a few drops of the liquid in a gas cell and/or of benzene liquid by placing a drop of the liquid between two salt plates and to obtain the Raman spectrum of benzene liquid if a Raman spectrometer is available, this exercise can be done totally computationally if avoiding using benzene in the lab is desirable. Computational chemistry can also be used to provide data if one of the instruments needed to collect the data is not available. Results obtained from the calculations for the vibrational frequencies sorted by symmetry are presented in Tables 4 and 5. Fifteen (15) of the 30 normal modes of vibration are symmetric with respect to inversion (given as gerade (g) in the output) and 15 change sign on inversion (given as ungerade (u)). All of the normal modes with calculated Raman intensities greater than zero have gerade symmetry, while those with calculated IR intensities greater than zero have ungerade symmetry confirming the exclusion rule. Some normal modes are calculated to have zero intensity in both the IR and Raman spectra establishing that not every normal mode is spectroscopically active. Students often ask whether it is possible for a normal mode not to be observed. This exercise clearly shows that it is. Table 4. Calculated symmetries, frequencies, IR and Raman intensities for the normal modes of benzene with gerade symmetry. Frequencies and intensities were calculated using DFTB3LYP/ 6-311+G (2d, p). Symmetry

cm-1

IIR

IRaman

A1g

3190.2351

0.0000

362.3573

A1g

1010.4558

0.0000

54.7730

A2g

1389.4582

0.0000

0.0000

B2g

1002.0778

0.0000

0.0000

B2g

713.2584

0.0000

0.0000

E1g

857.5413

0.0000

3.2437

E2g

3165.4700

0.0000

130.5507

E2g

1632.1368

0.0000

12.8463

E2g

1198.8029

0.0000

7.1596

E2g

624.5751

0.0000

3.8623

Table 5. Calculated symmetries, frequencies, IR and Raman intensities for the normal modes of benzene with ungerade symmetry. Frequencies and intensities were calculated using DFTB3LYP/ 6-311+G (2d, p). Symmetry

cm-1

IIR

IRaman

A2u

681.8153

107.2564

0.0000

B1u

3156.1551

0.0000

0.0000

B1u

1030.4782

0.0000

0.0000

B2u

1329.0925

0.0000

0.0000

B2u

1175.4644

0.0000

0.0000


Table 5. (Continued). Calculated symmetries, frequencies, IR and Raman intensities for the normal modes of benzene with ungerade symmetry. Frequencies and intensities were calculated using DFT-B3LYP/ 6-311+G (2d, p). Symmetry

cm-1

IIR

IRaman

E1u

3180.4998

62.6327

0.0000

E1u

1517.0657

11. 6090

0.0000

E1u

1058.2068

2.8933

0.0000

E2u

975.6244

0.0000

0.0000

E2u

411.1954

0.0000

0.0000

II. Investigating the Properties of Atoms While the electronic structure of atoms is a common topic in physical and general chemistry, there are few laboratory exercises designed to investigate this topic (37, 44, 45). This section presents a way to use computational chemistry to investigate the electronic structures of atoms. Aufbau Principle and Hund’s Rules Learning objectives: This computational exercise is used to confirm that the aufbau principle and Hund’s rules predict the most stable ground state term symbols for atoms and for many ions. The aufbau principle and Hund’s rules are usually introduced as part of the discussion of electron configurations in general chemistry texts (44, 45). These configurations are used to determine term symbols in more advanced courses (37, 42, 43). With some knowledge about wave functions and electron configurations coupled with the realization that larger negative energies indicate the more stable state, students can use computational chemistry to investigate these concepts. For example, knowing that an s orbital can hold two electrons, it is easy to show that a pair of electrons in an s orbital with opposite spins produces a 1S state while a pair of electrons with the same spin would produce a 3S. Since this would violate the Pauli Exclusion Principle, the lowest energy triplet state corresponds to the promotion of an electron into the next available orbital. Hence the lowest 3S state for He corresponds to the 1s12s1 configuration, the lowest 3P for Be corresponds to the 2s12p1 configuration, and the lowest 3D state for Ca corresponds to the 4s13d1 configuration. By calculating the triplet – singlet energy differences for helium and the group 2 metals, the more stable state can be identified. As shown in Table 6, this value is positive in all cases confirming that the singlet states are more stable than the triplet states confirming that the electrons are paired in the s orbital as predicted by the aufbau principle. Computing the quartet – doublet energy difference for the group 13 elements for the term symbols determined for the ns2np1 (2P) and the ns1np2 (4P) configurations confirms that two electrons remain paired in the s orbitals after an electron is added to the p orbital. A similar result is obtained for Sc after an electron is added to the 3d orbital. As shown by the relative energies in in Table 7, the configurations where the low spin state where the s electrons remain paired are more stable in all cases, again confirming the predictions made using the aufbau principle.


Table 6. The calculated triplet – singlet energy differences using DFT-B3LYP/ 6-311+G (2d, p). All values are in atomic units. Atom

Esinglet

Etriplet

Etriplet – Esinglet

He

- 2.9135 a

- 2.1525 b

0.7611

Be

-14. 6713 a

-14.5811 c

0.0902

Mg

-200.093 a

-199.991c

0.1021

Ca

-677.576 a

-577.505d

0.0706

a ns2.

b 1s1 2s1.

c ns1 np1.

d ns1 (n-1)d1.

Table 7. Calculated quartet – doublet energy differences using DFT-B3LYP/ 6-311+G (2d, p). All values are in atomic units. Atom

Edoublet

Equartet

Equartet − Edoublet

B

- 24.6626 a

- 24.5300 b

0.1326

Al

-242.3867 a

-242.2472 b

0.1394

Ga

-1924.821 a

-1924.640 b

0.1915

Sc

-760.6207 c

-760.5866 d

0.0341

a ns2 np1.

b ns1 np2.

c 4s2 3d1.

d 4s13d2.

Adding additional p electrons increases the number of possible states. Using carbon as an example, there are possible states corresponding to the 2s12p3(5S), 2s22p2 with the p electrons unpaired (3P), and 2s22p2 with the p electrons paired (1D). Similar configurations, albeit with different term symbols, can be determined for any atom expected to have half-filled or less orbital (p1-3 or d1-5). Some examples of the calculated energies for several second period elements and ions and a few first row transition metals are presented in Table 8. Since the multiplicities depend on the number of unpaired electrons, the states are listed as high (2s12pn or 4s13dn), middle (2s22pn or 4s23dn with the p, d spins unpaired) and low (2s22pn or 4s23dn with the p, d spins paired). In all cases but chromium, the middle spin state has the lowest energy confirming the s electrons are paired and that the p or d electrons are unpaired as predicted by the aufbau principle and Hund’s rules. The results for Cr are consistent with the discussion of the stability of the half-filled d orbitals often presented in general chemistry textbooks (44, 45). Atoms with half-filled shells or higher can also be investigated, but the highest spin states will no longer automatically correspond to the promotion of an s electrons. To avoid confusion, these atoms are not investigated at JMU. One topic related to the aufbau principal that gives our students trouble is electron configurations for the 3d transition metal ions. Removing a paired 4s electron will produce a higher multiplicity while removal of an unpaired 3d electron will produce a lower multiplicity for the Sc, Ti, V, and Mn ions. As shown in Table 9, DFT calculations indicate the high spin state is more stable for these ions showing that removal of the s orbital is energetically more favorable. Once the d electrons are paired or for Cr, removal of either a d or an s electron produces the same multiplicities so term symbols must be determined to extend the analysis to these atoms. 118 Grushow and Reeves; Using Computational Methods To Teach Chemical Principles ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

Table 8. Calculated high, middle, and low spin energies for atoms and ions using MP2/ 6311+G (2d, p). All values are in atomic units. Element

ELow spin

EMiddle spin

EHigh spin

C

-37.67709 a

-37.75083 b

-37.62003c

C-

-37.71500 a

-37.79369 b

-37.53259 d

N+

-53.84417 a

-53.95151 b

-53.76551c

N

-54.36075 a

-54.48405 b

-53.89342d

N-

-54.38674 a

-54.45746 b

-53.85890e

O+

-74.28278 a

-74.45263 b

-73.36128d

O

-74.82509 a

-75.93517 b

-73.50158e

O-

-74.97782 a

-74.76729d

--------------

Ti

-849.2908 f

-849.3455 g

-849.3443h

V

-942.8066 f

-942.9014g

-942.8943h

Cr

-1043.167 f

-1043.341g

-1044.388h

Mn

-1149.638 f

-1149.831 g

-1149.842 i

Fe

-1262.374f

-1262.539 g

-1262.462 i

a 2s2 2pn p

spins paired. b 2s2 2pn p spins unpaired. c 2s1 2pn p spins unpaired. unpaired. e 2s2 2pn-13s1 p spins unpaired. f 4s2 3dn d spins paired. h 1 n i 1 n 1 unpaired. 4s 3d d spins unpaired. 4s 3d 4p d spins unpaired.

d 2s1 2pn-13s1 p g 4s2 3dn d

spins spins

Table 9. The calculated high and low spin energies for selected first row transition metal 1+ ions Using DFT-B3LYP/ 6-311+G (2d, p). All values are in atomic units. Element

ELow spina

EHigh spinb

ELow spint − EHigh spin

Sc

-760.3227

-760.3796

0.0569

Ti

-849.0873

-849.1099

0.0206

V

-943.6142

-943.6521

0.0379

Mn

-1149.586

-1149.657

0.0710

a 4s23dn.

b 4s13dn.

III. Mass Spectrometry Illustrated with Methane Gas chromatography-mass spectrometers (GC-MS) have become more common in organic and instrumental laboratories (47, 48). While students usually know or soon discover that a molecule will fragment into several different ions in the ion source, they have little understanding of the pathways that produce these ions. Computational chemistry can be used to investigate the energy needed to produce ions in the ionization source of the mass spectrometer. These exercises provide insight into one method of determining the ionization potential for an atom or a molecule. 119 Grushow and Reeves; Using Computational Methods To Teach Chemical Principles ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

Learning objectives: This exercise is used to investigate the chemical reactions that produce ions in the ionization source of a mass spectrometer. The minimum energy needed to produce each ion is determined and compared the values measured (appearance potentials) from the literature. A gas chromatography-mass spectrometers (GC-MS) are used in the JMU Applied Physical Chemistry Laboratory to measure the isotopic abundance of the chlorine using chloroform as the source molecule. As part of this exercise students use computational chemistry to investigate the ions observed in the mass spectrum. Since methane is the simplest hydrocarbon and can be used as a model hydrocarbon, it is used to investigate some possible ionization pathways that could produce the ions observed in the mass spectrum of this molecule. In an EI-MS, the ions are formed from a collision between an accelerated electron and the molecule in the ion source. This collision removes an electron and may also fragment the molecule to create positive ions that are detected by the mass spectrometer. Most EI-MS are done with a fixed voltage of 70 eV (47–50). Historically, variable electron energies were often used to produce the ions (49, 50). The ionization energy for the molecule could be determined by measuring the minimum energy needed to observe the ion (the appearance potential). Bond energies could also be determined from the difference in the appearance potentials for various ions produced as the electron energy was increased. It is assumed that all ions are produced from one collision between methane and the electron in this exercise. Students estimate the appearance potentials by calculating the energy needed to form an ion for several possible reaction pathways. The mass spectrum of methane taken with a 70 eV ionization voltage has mass/ charge signals at 16, 15, 14, 13, and 12 amu (36, 50). Assuming only species containing 12C and 1H are observed, these signals are readily assigned as CH4+, CH3+, CH2+, CH+ and C+ respectively. The appearance potentials and possible chemical reactions that could produce these ions have been reported by Stano et al. (50) The literature values reported below are all from this source. The simplist reaction for forming CH4+ is from the collision of methane with the energetic electron (50).

The minimum energy needed to observe the CH4+ is the ionization energy for CH4 and can be approximated from equation (2) (9).

The value determined using DFT-B3LYP/ 6-311+G (2d,p) is 12.70 eV (Lit. = 12.65 eV) when calculated using the energy minimized ground states of methane and the methane +1 ion. As shown in Figure 3, CH4+ has D2d symmetry rather than the Td structure most students predict. While this is expected from the Jahn – Teller effect, this topic is generally not discussed in most undergraduate physical chemistry textbooks. Fragmentation occurs at higher excitation energies. Two pathways that produce CH3+ from the interaction with one electron are possible (50).


Figure 3. Energy minimized structures for methane and the methane+1 ion Computational chemistry can be used to explore each of these pathways and to investigate the CH3+ ion. The appearance potential for reaction (3) determined using equation (5) is 14.52 eV (lit. = 13.58 eV).

A similar calculation for equation (4) gives14.72 eV (lit = 14.34 eV). Better agreement with the literature can be obtained by adding zero point energies and temperature corrections for the enthalpy to reaction (5). The lowest energy pathway to form CH2+, CH+ and C+ involve forming hydrogen during the electron collision (50).

The appearance potentials calculated with zero point corrections are 15.05 eV (lit 15.1 eV), 19.78 eV (lit. 19.8 eV) and 19.45 eV (lit. 20.5eV) for reactions 6-8 respectively.

Conclusions Most physical chemistry laboratories do not have access to every instrument needed to let students to investigate all of the topics introduced in a typical physical chemistry lecture course. Adding calculations compliments the experiments that can be done by allowing students to visualize concepts like molecular structures, molecular vibrations, dissociation pathways, etc. This manuscript presented some of the computational exercises that have been done at JMU, but it certainly is not a complete list of things that could be done. Since the goal of this chapter was to present applications that investigate topics normally discussed in the physical chemistry curriculum that could be done using modest systems, the systems presented use small molecules that can be calculated quickly at 121 Grushow and Reeves; Using Computational Methods To Teach Chemical Principles ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

a high enough level of theory to produce reasonable (though not exact) agreement between the calculated and the experimentally measured values. The focus is on exploring the chemical concepts rather than obtaining the highest precision possible using the best theory available. Our experience shows that these exercises often clear up misconceptions that students have about the topic and produce a better understanding of it.

Acknowledgments The author is grateful to the NSF- REU- 1062629, the NSF- REU – 1461175, and the Research Corporation Departmental Development Grant #7957 for providing the software and the summer support to develop these exercises.

References 1. 2. 3. 4.

5. 6. 7.

8.

9.

Jonnsson, H. Research Level Computers and Software in the Undergraduate Curriculum. J. Chem. Educ. 1995, 72, 332–336. Foreman, J. B.; Frish, A. E. Exploring Chemistry with Electronic Structure Methods; Gaussian: Pittsburg, PA, 1996. Hehre, W. J.; Shusterman, A. J.; Nelson, J. E. The Molecular Modeling Workbook for Organic Chemists; Wavefunction, Inc.: Irvine, CA, 1998. Sorkhabi, O.; Jackson, W. M.; Daizadeh, I. Extending the Diatomic FTIR experiment: A Computational Exercise to Calculate Potential Energy Curves. J. Chem. Educ. 1998, 75, 238–240. Martin, N. H. Integration of Computational Chemistry into the Chemistry Curriculum. J. Chem. Educ. 1998, 75, 241–244. Whisnant, D. M.; Lever, L. S.; Howe, J. J. Cl2O2 in the Stratosphere: A Collaborative Computational Chemistry Project. J. Chem. Educ. 2000, 75, 1648–1649. Karpen, M. E.; Henderleiter, J.; Schaertel, S. A. Integrating Computational Chemistry into the Physical Chemistry Laboratory Curriculum: A Wet Lab/ Dry Lab Approach. J. Chem. Educ. 2004, 81, 475–477. Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery, Jr., J. A.; Vreven, T.; Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; Tomasi, J.; Barone, V.; Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Petersson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hratchian, H. P.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C.; and Pople, J. A.; Gaussian 03, Revision C.02; Gaussian, Inc.: Wallingford, CT, 2004. Gaussian. http://gaussian.com (accessed 6/1/2018). 122 Grushow and Reeves; Using Computational Methods To Teach Chemical Principles ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

10. McClain, B. L.; Clark, S. M.; Gabriel, R. L.; Ben-Amotz, D. Educational Applications of IR and Raman Spectroscopy: A Comparison of Experiment and Theory. J. Chem. Educ. 2000, 77, 654–660. 11. Matta, C. J.; Gillespie, R. J. Understanding and Interpreting Molecular Electron Density Distributions. J. Chem. Educ. 2002, 79, 1141–1147. 12. Watson, L.; Eisenstein, O. Entropy Explained: The Origin of Some Simple Trends. J. Chem. Educ. 2002, 79, 1269–1277. 13. Cramer, C. J.; Bumpus, J. A.; Lewis, A.; Stotts, C. Characterization of High Explosives and Other Energetic Compounds by Computational Chemistry and Molecular Modeling. J. Chem. Educ. 2007, 84, 329–332. 14. Rowley, C. N.; Woo, T. K.; Mosey, N. J. A Computational Experiment of the Endo Versus Exo Preference in a Diels–Alder Reaction. J. Chem. Educ. 2009, 86, 199–201. 15. Marzzacco, C. J.; Baum, J. C. Computational Chemistry Studies on the Carbene Hydroxymethylene. J. Chem. Educ. 2011, 88, 1667–1671. 16. Pernicone, N. C.; Geri, J. B.; York, J. T. Using a Combination of Experimental and Computational Methods to Explore the Impact of Metal Identity and Ligand Field Strength on the Electronic Structure of Metal Ions. J. Chem. Educ. 2011, 88, 1323–1327. 17. Wang, L. Using Molecular Modeling in Teaching Group Theory Analysis of the Infrared Spectra of Organometallic Compounds. J. Chem. Educ. 2012, 89, 360–364. 18. Simpson, S.; Autschbach, J.; Eva Zurek, E. Computational Modeling of the Optical Rotation of Amino Acids: An ‘in Silico’ Experiment for Physical Chemistry. J. Chem. Educ. 2013, 90, 656–660. 19. Freitag, R.; Conradie, J. View of the Mn(β-diketonato)3 Complex. J. Chem. Educ. 2013, 90, 1692–1696. 20. Pritchard, B. P.; Simpson, S.; Zurek, E.; Autscbach, J. Computation of Chemical Shifts for Paramagnetic Molecules: A Laboratory Experiment for the Undergraduate Curriculum. J. Chem. Educ. 2014, 91, 1058–1063. 21. Jeanmairet, G.; Levy, N.; Levesque, M.; Borgis, D. Introduction to Classical Density Functional Theory by a Computational Experiment. J. Chem. Educ. 2014, 91, 2112–2115. 22. Baseden, A.; Tye, J. W. Introduction to Density Functional Theory: Calculations by Hand on the Helium Atom. J. Chem. Educ. 2014, 91, 2116–2123. 23. Reeves, M. S.; Whitnell, R. M. New Computational Physical Chemistry Experiments: Using POGIL Techniques with ab-initio and Molecular Dynamics Calculations. Addressing the Millennial Student in Undergraduate Chemistry, ACS Symposium Series 2014, 1180, 71–90. 24. Esselman, B. J.; Hill, N. J. Integration of Computational Chemistry into the Undergraduate Organic Chemistry Laboratory Curriculum. J. Chem. Educ. 2016, 93, 932–936. 25. Miorelli, J.; Caster, A.; Eberhart, M. E. Using Computational Visualizations of the Charge Density to Guide First-year Chemistry Students Through the Chemical Bond. J. Chem. Educ. 2017, 94, 67–71. 26. Rodríguez González, M. C.; Hernández Creus, A.; Carro, P. Combining Electrochemistry and Computational Chemistry to Understand Aryl-radical Formation in Electrografting Processes. J. Chem. Educ. 2018, 95, 1386–1391.


27. Bagley, A. C.; White, C. C.; Mihay, M.; DeVore, T. C. Physical Chemistry Laboratory Projects Using NMR and DFT-B3LYP Calculations. ACS Symposium Series 2013, 1128, 229–243. 28. Baglie, A. C.; AbuNada, I.; Yin, J.; DeVore, T. C. Investigation of NMR Chemical Shifts Using DFT-B3LYP Calculations. ACS Symposium Series 2016, 1225, 67–77. 29. Wolff, S. K.; Ziegler, T. Calculation of DFT-GIAO NMR Shifts With the Inclusion of Spinorbit Coupling. J. Chem. Phys. 1998, 109, 895–905. 30. Jain, R.; Bally, T.; Rablen, P. R. Calculating Accurate Proton Chemical Shifts of Organic Molecules with Density Functional Methods and Modest Basis Sets. J. Org. Chem. 2009, 74, 4017–4023. 31. Stafford, F. E.; Holt, C. W.; Paulson, G. L. Vibration-rotational Spectrum of HCl: A Physical Chemistry Experiment. J. Chem. Educ. 1963, 40, 245–249. 32. Richards, L. W. The Infrared Spectra of Four Isotopes in HCl: A Molecular Structure Experiment. J. Chem. Educ. 1966, 43, 552–554. 33. Feller, S. E.; Blaich, C. F. Error Estimates for Fitted Parameters: Application to HCl/ DCl Vibrational-Rotational Spectroscopy. J. Chem. Ed. 2001, 78, 409–412. 34. Furlong, W. R.; Grubbs, W. T. Safe Preparation of HCl and DCl for IR Spectroscopy. J. Chem. Educ. 2005, 82, 124. 35. Mayer, S. G.; Bard, R. R.; Cantrell, K. A Safe and Efficient Technique for the Production of HCl/ DCl Gas. J. Chem. Educ. 2008, 85, 847–848. 36. NIST Chemistry Webbook. https://webbook.nist.gov/chemistry (accessed 10/22/2018). 37. Atkins, P.; dePaula, J.; Keeler, J. Atkins’ Physical Chemistry, 11 ed.; Oxford, 2018. 38. Thermochemistry in Gaussian. http://www.gaussian.com/g_whitepap/thermo.htm (accessed 6/1/2018). 39. Ogren, P. J Using the Asymmetric Stretch Band of Atmospheric CO2 to Obtain the C=O Bond Length. J. Chem. Educ. 2002, 79, 117–119. 40. Gonzalez-Gaitano, G.; Isasi, J. R. Analysis of the Rotational Structure of CO2 by FTIR Spectroscopy. Chemical Educator 2001, 6, 362–364. 41. Campbell, H. M.; Boardman, B. M.; DeVore, T. C.; Havey, D. K. Experimental Determination of the Boltzmann Constant: An Undergraduate Laboratory Exercise for Molecular Physics or Physical Chemistry. Am. J. Phys. 2012, 80, 1045–1050. 42. Herzberg, G. Electronic Spectra of Polyatomic Molecules; D. Van Nostrand Co.: Princeton, NJ, 1966. 43. Herzberg, G. Infrared and Raman Spectroscopy; D. Van Nostrand Co.: Princeton, NJ, 1945. 44. Bretz, S. L.; Davies,G.; Foster,N.; Gilbert,T. R.; Kirss, R. V. Chemistry: The Science in Context, 5th ed.; W.W. Norton & Co. 2017. 45. Silberberg, M. S. Principles of General Chemistry, Third Edition; McGraw Hill, 2013. 46. Carbo-Dorca, R.; Bultinck, P. Quantum Mechanical Basis for Mulliken Population Analysis. J. Math. Chem. 2004, 36, 231–239. 47. Klein, D. Organic Chemistry, 3rd ed.; Wiley, 2017. 48. Robinson, J. W.; Skelly-Frame, E. M.; Frame II, G. M Undergraduate Instrumental Analysis, 7th ed.; CRC Press, 2014. 49. Nicholson, A. J. C. Measurement of Ionization Potentials by Electron Impact. J. Chem. Phys. 1958, 29, 1312–1318. 124 Grushow and Reeves; Using Computational Methods To Teach Chemical Principles ACS Symposium Series; American Chemical Society: Washington, DC, 2019.

50. Stano, M.; Matejcik, S.; Skalny, J. D.; Mark, T. D. Electron Impact Ionization of CH4: Ionizatiom Energies and Temperature Effects. J. Phys. B: At. Mol. Opt. Phys. 2013, 36, 261–273.


Introducing Quantum Calculations into the Physical Chemistry

Recommend Documents