ed500642y

Using an Advanced Computational Laboratory Experiment ToExtend and Deepen Physical Chemistry Students Understanding ofAtomic StructureGary G. Homan*

Department of Chemistry & Biochemistry, Elizabethtown College, Elizabethtown, Pennsylvania 17022, United States

*S Supporting Information

ABSTRACT: A computational laboratory experiment is described, which involvesthe advanced study of an atomic system. The students use concepts and techniquestypically covered in a physical chemistry course but extend those concepts andtechniques to more complex situations. The students get a chance to explore thestudy of atomic states and perform calculations at a high enough level to providepredicted emission lines in good agreement with literature results. The specicexercise described applies these methods to fth period transition metals (ground-state systems) and to various congurations of several period two representativeelements (both ground and excited states). The computations are performed withprograms written by Froese Fisher and co-workers, which perform HartreeFock(HF), multiconguration HF, and conguration interaction calculations. Sincethese are atomic systems, the angular dependence of the orbital functions can bedescribed by spherical harmonics (or combinations thereof), which leaves adierential equation in the radial coordinate that can be solved by numerical methods. The details involved in this exercise arepresented along with some typical results.

KEYWORDS: Upper-Division Undergraduate, Physical Chemistry, Laboratory Instruction, Computer-Based Learning,Hands-On Learning/Manipulatives, Atomic Properties/Structure, Computational Chemistry, Quantum Chemistry

INTRODUCTIONElectronic structure calculations are performed in all areas ofchemistry research, and for this reason, it is important thatundergraduate students gain some experience in this area. Theformalism is taught in the traditional physical chemistry coursewhere simple exercises are provided to illustrate some of theimportant concepts. However, a true appreciation of the eldcomes from laboratory work where the student can experimentwith the parameters and work toward obtaining reliable resultsfor the problem at hand. It is not uncommon to incorporateelectronic structure calculations into physical chemistrylaboratory exercises. The results of such calculations may helpto guide the experimental work, or they may be used to explainthe results obtained in the lab. Alternatively, the experimentmay focus directly on the computational procedure itself andthe value of the results gauged by comparison withexperimental data.Atomic systems are an attractive target for computational

work because they allow some considerable simplications inthe equations of motion. The orbital functions are all centeredon the single nucleus, and the angular portions of thesefunctions can be modeled well with spherical harmonics. Thesolution of the HartreeFock (HF) equations for the heliumatom ground state is particularly simple in that both electronsare described by the same orbital function, and these functionsare spherically symmetric. There results a single dierential

equation in a single variable. Exercises can be found in thechemical education literature that guide students through thesolution of this equation using basis sets1,2 or numericalintegration methods.3 A recent example4 investigates thesolution of the analogous equation that arises with densityfunctional calculations. Larger atoms are more complicated, butconsiderable simplications are still possible. Sphericalharmonics can still be introduced to describe the angulardegrees of freedom, leaving dierential equations in theremaining radial coordinate. Powerful numerical methods canbe applied to solve these equations (with no need to introducebasis functions), and code to perform these calculations hasbeen generated by Froese Fisher and co-workers.5

The exercise presented in this paper involves theinvestigation of an atomic system using accurate computationalmethods. To gauge the quality of the results, the studentspredict atomic emission spectrum lines and compare theirpredictions with literature values. Briey, this experiment hasstudents generate a set of acceptable electron congurations fora many-electron atom, generate the associated term symbols forthose congurations, perform electronic structure calculationson the generated terms, use the computed energies to calculatewavelengths for the possible transitions, use the spectroscopicselection rules to predict the lines that should appear in an

Published: February 26, 2015

Laboratory Experiment

pubs.acs.org/jchemeduc

2015 American Chemical Society andDivision of Chemical Education, Inc. 1076 DOI: 10.1021/ed500642y

J. Chem. Educ. 2015, 92, 10761080

atomic spectrum, and compare those predictions withexperimental data.The exercise builds on concepts and techniques already

covered in a traditional physical chemistry course but goes wellbeyond the simple cases typically presented in the lecturecourse. For example, the student should be familiar with theidea of electron congurations. The focus in the classroom,however, is with ground-state congurations, and the studentmay not fully appreciate the construction of electroncongurations for excited states as well. Further, the studentshould be familiar with generating the term symbols associatedwith an electron conguration,69 but the lecture course againfocuses on the ground-state conguration. Situations wherethere is more than one open shell typically do not arise in thelecture course, and this exercise provides a learning experiencein this respect. It is also common in the undergraduate physicalchemistry course to present atomic spectroscopy and theassociated selection rules.6,8,10,11 Some simple cases may beposed as problems, and the student can predict whichtransitions may or may not occur. This exercise goes wellbeyond these simple cases and illustrates several complicatingfactors; another learning experience is gained. Finally, thestudent is most likely exposed to the HF approximation and thenotion that corrections are necessary for accurate work. Thespecics associated with multicongurational HF (MCHF) andconguration interaction (CI) may not be as familiar, and thestudent will learn about these techniques rst-hand. Eventhough the concepts used in this exercise may be alreadyfamiliar to the student, the situations encountered will beconsiderably more in-depth. The student will have to wrestlewith the concepts and, it is hoped, gain a deeper understandingand appreciation of them. There is also the chance ofexperimenting with the input parameters to see how theyaect the results obtained.This paper will present a particular way in which the exercise

may be performed. There are many possible variations, though.Such variations will be discussed later.

THE EXERCISEOverview

This exercise was originally designed to be covered in fourclasses. One may note, though, that this may be modied. Forinstance, part (or all) of the lecture portion of this exercise maybe transferred to the lecture portion of the course, which wouldallow the laboratory portion to be covered in fewer classmeetings, if desired. Further, some material may be left out orstreamlined, which would also lead to fewer classes. Beforegetting to the specics of the assignment, an overview of theoriginal exercise will be presented. The exercise can be brokenup into two main tasks, and it is useful to rst present a generaldescription of these tasks. This overview will be followed upwith the technical details that are associated with each of thosetasks.The rst task is the study of a transition metal element, the

objective being to predict the appropriate ground state electronconguration. This task gives the student a chance to learn howto use the electronic structure programs on a relatively simplesystem. To make it interesting, there is a specic question toanswer. Electronic structure calculations are performed for thecongurations in which two, one, or zero electrons are in thevalence s orbitals of the transition metal element, and the job isto determine which conguration yields the lowest energy. At

the start, the student will need to identify the lowest energyterm symbol for each conguration and then perform theassociated electronic structure calculations. Each student isassigned a dierent element. The fth period has a number ofanomalous congurations, and elements in this row havetypically been assigned for study. Elements in other rows, ofcourse, are equally appropriate. This is a straightforward andquickly achieved task and serves as a vehicle to get the studentsexposed to the necessary tools for studying atomic systems.They will encounter some diculties but should be able toovercome them with minimal eort and achieve some positiveresults before they move on to the more challenging secondtask.The second task is the study of a set of states for a

representative element and the possible emission transitionsamong those states. In addition to the ground-stateconguration, several low-lying excited-state congurationsare assigned for which the student must generate all of theassociated term symbols. There is little exposure to excitedstates for many-electron atoms in the typical physical chemistrycourse, and this exercise is a good remedy for this. For all of theterms, electronic structure calculations (to be described below)are performed to obtain the associated energies. By applyingthe selection rules, the allowed spectral transitions among theassigned congurations are predicted and compared with datafrom the literature. Each student is assigned a dierent atom forstudy. Because of the large number of experimentalspectroscopic lines available and the relative simplicity of theassociated calculations, representative elements in the secondperiod have typically been assigned. The neutral atoms of theelements carbon, nitrogen, oxygen, and uorine are sucientlychallenging. In addition, numerous cations with the sameelectronic congurations as these four are appropriate targetsfor study. The other elements, of course, and many otherelectron congurations may be chosen as well.The exercise is carried out in stages, the necessary tools are

presented one at a time, and the students generate their data asthey go. Diculties arise at various points, and the studentslearn how to deal with them as they move on. At the end, theywrite up a formal report on the results of their calculations.

Technical Details

The generation of the congurations and term symbols requiresjust pencil and paper (and patience). An industrious studentmay be able to write a program to generate the appropriateterm symbols, and such a solution is encouraged. For theatomic structure calculations, the numerical programs devel-oped by Froese Fisher and co-workers have been used. Theseprograms are described in a book5 and may be downloadedfrom the NIST Web site.12 These programs are written inFortran 77 and must be compiled before use. These programswere installed on an Alpha workstation running Tru64 UNIXto which the students connect over the Web. They must learnsome basic UNIX commands and some detailed instructionsfor running the programs and reading the output. This has notbeen a hindrance, however. Some students are actuallyenthusiastic about learning these things.Specic details associated with the tasks are contained in two

handouts, which are contained in the Supporting Informationfor this paper. The following sections present a detailedstructure for a four-class sequence of lectures and tasks. Asmentioned previously, this can be revised by presenting lecturesin another class or by streamlining (or augmenting) the tasks.

Journal of Chemical Education Laboratory Experiment

DOI: 10.1021/ed500642yJ. Chem. Educ. 2015, 92, 10761080

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First Class

A handout (Atomic states and spectroscopic transitionsbetween them, see the Supporting Information) is providedto the students at or before the rst class. This handoutcontains much of the detail covered in the lecture for the rstclass and is provided to make sure the students have the detailsright when they try to apply the associated material to theiratoms. The lecture component is a prominent part of this class.The particular material covered is spectroscopy in general,atomic structure, electron congurations for many-electronatoms, and the determination of term symbols. The studentsare assigned their transition metal element and theirrepresentative element atom or ion. The tasks to be completedfor the next class are as follows.

(1) The generation of the lowest energy term symbols forthe transition metal element in the 5s24dn, 5s4dn+1, and4dn+2 congurations (assuming a fth period element ischosen). It is important to stress that only the lowestenergy term symbol is needed for this task. Generation ofall the term symbols for a transition metal is ratherforbidding, and there are shortcuts when only the lowestenergy term is needed.

(2) The generation of all possible term symbols for therepresentative element atom or ion (assuming a secondrow element) in

(2a) the ground-state conguration,(2b) the excited-state conguration obtained by promoting a

2s electron in the ground state to a 2p orbital,(2c) the excited-state conguration obtained by promoting a

2p electron in the ground state to a 3s orbital, and(2d) the excited-state conguration obtained by promoting a

2p electron in the ground state to a 3p orbital.

Second Class

The second class begins with the students presenting the resultsof the assignment. After clearing up any confusion that mighthave arisen, there is a lecture on atomic spectroscopy and theassociated selection rules. The students are given the task ofidentifying all of the allowed transitions among thecongurations generated for their representative element.They are also asked to look up all the experimental linesassociated with these congurations. The NIST Web site isrecommended for nding these data.13 It is pointed out thatsome of the allowed transitions may be missing from theexperimental data and that some forbidden transitions may bepresent.In carrying out this part of the assignment, many students

will try a brute force approach, which works but can be rathertime-consuming. It is possible to organize the task to eliminatelarge sets of transitions at a time. For instance, all transitionsbetween cases 2b and 2c above can be eliminated with theappropriate selection rule. The students also may need somehelp downloading the data eciently from the NIST Web site.The instructor provides guidance, as needed, for transferringthe data and organizing it in an Excel spreadsheet.At this class, a second handout is distributed (Electronic

structure calculations on atoms, see the Supporting Informa-tion). This handout covers much of the material to bepresented at the next class. It also describes, in full detail, howto run the programs for performing the atomic electronicstructure calculations.

Third Class

The third class starts with a discussion of the results obtainedfrom the last assignment, and a lecture on electronic structuretheory is now given. This topic is covered briey in physicalchemistry lecture but will most likely need to be augmented forthe task at hand. Relativistic contributions are presented (verybriey) as well as spinorbit coupling. The HF approximationis presented (which should be review of material presented inthe physical chemistry lecture course) and the concept ofdynamical correlation introduced (which may or may not be areview). The construction of Slater determinant congurationsand the expansion of the wave function in terms of thesequantities is presented. The application of the variationalprinciple leading to the (a) MCHF and (b) CI methods is thendescribed. The concept of an active space and its choice is alsodiscussed. In the end, the students should understand that theywill be performing HF, MCHF, and CI calculations on theirsystems. The role of the quantum numbers L, S, and J in thecalculations is discussed and where they need to be specied inthe electronic structure calculations. The programs usedperform the calculations by numerical quadrature, so there isno need to introduce basis sets.The students are given a walkthrough of the computational

procedure (it is fully presented in the handout). HFcalculations are performed in a straightforward way with aprogram called HF. These calculations require specication ofonly the electron conguration and the L and S quantumnumbers. The program presents both nonrelativistic andrelativistic results (including corrections up to order 1/c2).The students are asked to record both values. After this, thehigher-order calculations are set up. The rst step is to generatethe set of congurations needed for the remaining calculations.A program called Gencl is run that takes as input the referenceconguration, the values of L and S, and the active space. Thechoice of active space is discussed to some extent, and themessage is given that a good choice for the representativeelements is the n = 2 and n = 3 orbitals (corresponding to 2s,2p, 3s, 3p, and 3d). It is also noted that they may have toexperiment with this for some cases. The next program, Breit,computes a set of integrals associated with the congurationsand writes them to a le. By using the orbitals generated by theHF calculation, then, an MCHF calculation is performed with aprogram called MCHF. The input for this is pretty much thesame as for the HF calculation. The conguration list andintegral values are read in by the program for this calculation.The main purpose of this computation is to generate a set oforbital functions to be used in the subsequent CI calculations.Even so, the students are asked to record the MCHF energy foreach of the states. Finally, a CI calculation is performed withthe program CI, which generates the states and energies for thedierent J values associated with the L and S values of interest.After the walkthrough, the students are directed to perform

calculations on the terms they generated for both the transitionmetal and the representative element. The transition metalcalculations are to be presented at the next class.

Fourth Class

At the start of the fourth class, the students write their resultsfor the transition metal on the board. The students are asked topredict the ground-state conguration and term symbol fortheir element at the (1) HF nonrelativistic level, (2) HFrelativistic level, (3) MCHF level, and (4) CI level. With thisactivity, it is possible to identify problems that students are



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having before they get too far on the more involved project.The class is then turned over to a discussion of thecomputations performed so far. Students are asked about anyspecic diculties they encountered. Some problem situationsare typically mentioned. This provides the instructor a chanceto discuss these situations and how they are to be dealt with.The problem cases and how to deal with them are actuallydescribed in the handout, and this is enough for the betterstudents. However, it is worthwhile to explicitly mention thoseissues in class.There are several specic diculties that arise. First,

nonconvergence is sometimes encountered in the HF andMCHF calculations. The procedure for continuing thecalculation to achieve convergence is described. Second,congurations that have more than one of the same termdesignation (for instance, the 2s22p33p conguration of oxygenhas three distinct sets of 3P terms) require a distinction of theseterms. This is done by specifying the parent term, and a meansof generating the possible parent terms is described. Third, theneed is emphasized for checking the CI output to make surethe energy for the desired state is extracted. Finally, the possiblerequirement for modifying the active space for some cases isnoted, and a procedure for this is described. The students aredirected to nish the calculations, compare their results withexperimental data, and discuss those results in a formal report.

TYPICAL RESULTSOnce the procedure for performing the atomic structurecalculations has been mastered, the calculations go fairlyquickly. Many students end up being able to perform acomputation with a rapid series of memorized keystrokes. Thetransition metal calculations predict the accepted electronconguration at the CI level in all cases studied (the fth periodtransition metals from Mo to Ag) except for Pd. The predictionbased on HF (nonrelativistic), HF (relativistic), and MCHFcalculations almost always agrees with the CI results. Curiously,for Pd, all the levels of calculations besides CI predict theaccepted electron conguration.The representative element calculations require more eort.

Ignoring J values, there are 17 LS terms for carbon, 21 fornitrogen, 25 for oxygen, and 17 for uorine. Each of these LSterms requires a separate set of calculations. Without anycomplications, these can all be performed in 23 h. Very often,however, problems arise, and the student will have to spendsome extra time wrestling with nding solutions for these cases.The calculations will normally take more than one laboratoryperiod but not more than two. A lot of this depends on thestudent. In the end, as long as a reasonable active space ischosen, the predicted spectral lines agree rather well with theliterature data, dierences in wavelengths mostly being less than2%. Occasionally, there are some considerable dierences, andthese can usually be associated with one or maybe a fewparticular states. For instance, the transitions for the carbonatom having wavelengths less than 5000 are presented in

Figure 1. Comparison of experimental (top) and calculated (bottom) spectral lines at wavelengths below 5000 for carbon. The red linescorrespond to transitions from the 1s22s22p2 states to the 1s22s2p3 states, the blue lines to transitions from the 1s22s22p2 states to the 1s22s22p3pstates, the gold lines to the forbidden transitions from the 1s22s22p2 states to the 1s22s22p2 states, and the green lines to the forbidden transitions fromthe 1s22s22p3p states to the 1s22s2p3 states.



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Figure 1. These calculations used an active space consisting ofthe 2s, 2p, 3s, 3p, and 3d orbitals. For the most part, theobserved and predicted lines agree well. However, there is a setof three closely spaced lines that are considerably dierent.They are calculated to be at about 1657 but observed to be atabout 2905 . These three lines involve transitions between the3P terms of the 1s22s22p3p conguration and the 3S1 term of the1s22s2p3 conguration for this atom. By probing a bit moredeeply, it is noted that other transitions that involve the 3S1term are found to dier noticeably from the experimental data,while those of the 3P terms do not. The conclusion can bedrawn that the calculation for the 3S1 term of the 1s

22s2p3

conguration for some reason falls short. This is the type ofanalysis expected of the students and the type of conclusion tobe drawn. It would be unreasonable to expect anything deeperthan this (such as the need for a second referenceconguration) from an undergraduate, but some exceptionalstudents may make the leap.There is a lot of bookkeeping in this exercise. The energies

for all of the states must be computed and used to compute thewavelengths for all the transitions. For the exercise outlinedabove, there are 44 allowed transitions and 60 forbiddentransitions found for carbon on the NIST Web site. Similarly,there are 103 allowed transitions and 43 forbidden transitionsfor nitrogen, 120 allowed transitions and 27 forbiddentransitions for oxygen, and 27 allowed transitions and 1forbidden transition for uorine. The use of a spreadsheetprogram is virtually required. While students may grumbleabout the tedium involved, they perform the tasks and someeven exhibit a certain amount of pride in successfully tabulatingthe data and generating spectra that agree favorably with theexperiment.

VARIATIONS AND EXTENSIONSThere are many variations that can be applied to this exercise. Itmay be applied to most of the elements. It may be applied toneutral or charged species. It may be applied to a large numberof possible electron congurations. Each student may beassigned a dierent atom or ion, or a group of students may beassigned to the same element but dierent congurations. It ispossible, with a large enough class, to try to predict a spectrumfor a single element, boron for instance. One may assess theeect of modifying the active space on the calculations. Howmuch of an eect does the addition of certain orbital subshellsactually have? It may also be possible to generate an atomicspectrum in the laboratory and try to assign the lines observed.

CONCLUSIONSA computational laboratory experiment has been described thatallows students to apply techniques and concepts learned in thetypical physical chemistry course to a system of considerablecomplexity. They perform electronic structure calculations ofsome sophistication and compare their results to literature data.For the most part, the comparison with the literature is rathergood and leads to a feeling of condence in performingelectronic structure calculations and their use in laboratorywork. A number of possible extensions have also beensuggested.

ASSOCIATED CONTENT*S Supporting InformationTwo laboratory handouts: atomic states and spectroscopictransitions between them, and electronic structure calculationson atoms. This material is available via the Internet at http://pubs.acs.org.

AUTHOR INFORMATIONCorresponding Author

*E-mail: [email protected]

The authors declare no competing nancial interest.

REFERENCES(1) Snow, R. L.; Bills, J. L. A Simple Illustration of the SCF-LCAOMO Method. J. Chem. Educ. 1975, 52, 506509.(2) Rioux, F. An Interactive SCF Calculation. J. Chem. Educ. 1994,71, 781782.(3) Hoffman, G. G. Self-Consistent Field Calculations on AtomsUsing Excel. J. Chem. Educ. 2005, 82, 14181422.(4) Baseden, K. A.; Tye, J. W. Introduction to Density FunctionalTheory: Calculations by Hand on the Helium Atom. J. Chem. Educ.2014, 91, 21162123.(5) Froese Fischer, C.; Brage, T.; Jonsson, P. Computational AtomicStructure: An MCHF Approach; Institute of Physics Publishing: Bristol,UK, 1997.(6) Dykstra, C. E. Physical Chemistry: A Modern Introduction; PrenticeHall: Upper Saddle River, NJ, 1997.(7) Engel, T. Quantum Chemistry and Spectroscopy; Pearson,Benjamin Cummings: San Francisco, CA, 2006.(8) McQuarrie, D. A.; Simon, J. D. Physical Chemistry: A MolecularApproach; University Science Books: Sausalito, CA, 1997.(9) Ra, L. M. Principles of Physical Chemistry; Prentice Hall: UpperSaddle River, NJ, 2001.(10) Mortimer, R. G. Physical Chemistry, 2nd ed.; Harcourt,Academic Press: San Diego, CA, 2000.(11) Silbey, R. J.; Alberty, R. A. Physical Chemistry, 3rd ed.; JohnWiley & Sons, Inc.: New York, 2001.(12) See, in particular, the following URL. It is necessary to accept alicense agreement before proceeding to the page for downloads. Afterproceeding to the new page, go to the bottom and press the linklabeled Download ATSP. This will download a gzipped tar-le thatcontains all the necessary programs. http://nlte.nist.gov/cgi-bin/MCHF/download.pl?d=ATSP, accessed Feb. 25, 2015.(13) NIST Atomic Spectra Database; The National Institute ofStandards and Technology: Gaithersburg, MD, 2014. http://nist.gov/pml/data/asd.cfm (accessed January 2015).



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