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Steven D. Gammon University of Idaho Moscow, ID 83844
Electronic States and Configurations: Visualizing the Difference Igor Novak* Department of Chemistry, National University of Singapore, Singapore 119260
Electron configuration (EC) is a ubiquitous concept in the teaching of chemistry, since it underpins many fundamental concepts such as bonding and reactivity. The notion of electron configuration arises from consideration of the arrangement of elements in the periodic table and from ionization energy values for the elements (1). However, electron configurations are not observables, but are a part of the independent electron approximation (orbital model). Another fundamental concept related to electronic structure is electronic state (ES). ES has properties that are observables: energy, electron distribution, magnetic moment, etc. Teaching has for a long time focused on EC rather than ES. A recent article in this Journal (2) discussed in some detail the pedagogical advantages of utilizing electron distributions (densities) rather than electron configurations.
Discussion
The Problem While both concepts (ES and EC) are useful, many students and even teachers do not appreciate the difference between them. It is the purpose of this article to describe the difference on a level accessible to all undergraduates and not just to those taking physical chemistry courses. In courses on organic chemistry, photochemistry, and environmental chemistry reactivities of atoms and molecules in their excited states are discussed. In photochemistry, for example, the chemical behavior of singlet and triplet states is very different and can be grasped only if one comprehends the EC–ES differences. Graphical presentation of electron configuration given in textbooks and lectures relies on orbital level diagrams with “up” and “down” arrows. Such diagrams are used to describe both electronic states and configurations. The typical orbital diagram drawn below refers to the helium atom in the ground and excited states: 2s
2s
1s
1s 1s 2
1s 12s 1
although simple, is neither pedagogically nor scientifically sound. The conclusion that becomes imprinted strongly in students’ minds is that the two states arising from the 1s 12s1 configuration differ only in their total spin. This seductively easy conclusion (stimulated by the diagram) is wrong and makes it very difficult to grasp the fact that singlet and triplet states have different energies and other properties. Subsequent explanations usually fail to clarify the singlet–triplet energy difference in particular or the differences between electronic configurations and states in general. The distinction between electronic states and configurations is pertinent not only for excited states, but also for ground states of many atoms (e.g., transition metals), molecules (e.g., ozone), and ions that are described by open-shell configurations.
1s 1s 12s 1
1s 12s 1
2s
2s
1s
1s 1s 12s 1
The diagram correctly predicts the existence of a single ground state (1 1S) based on the single closed-shell configuration 1s2. However, for excited states the problem of relating configurations and states arises, because “one configuration–one state” correspondence is no longer valid. This type of diagram,
The distinction between electronic states (described via electron distributions) and electron configurations has, of course, been recognized for a long time. In early textbooks (3, 4 ) and publications (5) dealing with atomic physics, total radial electron distributions (densities) as well as orbital densities were plotted. Herzberg in his classical textbook (4 ) compares electron distributions for ground and excited states of the hydrogen atom. However, the pedagogically important point that different states have different electron distributions has not been emphasized in the chemical education literature. One possible reason for this is the tedious mathematical manipulations of wave functions that are necessary to describe excited states. Such wave functions have been reported for excited states of atoms at the SCF/HF level (6 ). Better wave functions, which go beyond Hartree–Fock approximation (HF), are available for ground and excited states of He–Be atoms (7, 8). In this work we wish to clarify the ES–EC difference, for pedagogical purposes, for the simplest case of the helium atom. Eigenfunctions for helium atom have a compact mathematical form and thus permit a general discussion of the EC–ES difference without undue mathematical complexity. The electronic wave functions for the 2 3S and 2 1S excited states of helium, described in the textbook by Bransden and Joachain (9), will be used in this work. These variational wave functions were derived in such a way as to keep them orthogonal to the ground-state wave function. Electronic states 2 3S and 2 1S correspond to the same electron configuration 1s12s1. The explicit forms of wave functions for singlet and triplet states are given by eqs 1 and 2 in atomic
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JChemEd.chem.wisc.edu • Vol. 76 No. 1 January 1999 • Journal of Chemical Education
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units and spherical coordinates: for 2 3S: ψ t =NT [(1 – 0.765r2) e᎑(2.01r +0.765r ) – (1 – 0.765r1) e᎑(2.01r +0.765r )] 1
2
2
(1)
1
for 2 1S: ψ s = NS [e᎑2r (e᎑0.865r – 0.432784r2 e᎑0.522r ) + 1
2
2
e᎑2r (e ᎑0.865r – 0.432784r1e᎑0.522r )] 2
1
(2)
1
where NS and NT are normalization constants and r1 and r2 are positions of the two electrons. The quality of these wave functions is satisfactory for the purposes of present discussion. We do not expect that electron densities would be qualitatively different if we used better wave functions. The wave function plots are shown in Figures 1 and 2. Modern computer algebra software (Mathematica, Maple, Matlab, MathCad) can easily perform calculations of electron distributions and their graphical representation. The normalization constants were calculated from eqs 3 and 4 with Mathematica software (10) and amount to 2.470 and 5.435 for 2 1S and 2 3S states, respectively. ∞ ∞
NT = [ ∫ 0 ∫ 0 r12 r22 ψ t2dr 1dr2 ]᎑1/2
(3)
∞ ∞ ∫ 0 ∫ 0 r12 r22 ψs 2dr1dr2 ᎑1/2
(4)
NS = [
]
The radial electron distributions R for singlet and triplet states were calculated from eqs 5 and 6: ∞
RT = r12 ∫ 0 r22 ψ t2dr 2
(5)
RS =
(6)
∞ r12 ∫ 0 r22 ψs 2dr2
The Mathematica algorithms used for calculating and plotting are given in Appendices A and B. The results shown in Figure 3 reveal several interesting features. The electron distributions in singlet and triplet states are different (albeit only in the valence shell), which explains why the two states have different energies and other properties. Figure 3 also provides ready illustration of the core and valence electron concepts; core electron distributions are the same in the singlet and triplet states. The orbital diagram, on the other hand, suggests that the electrons in 2 1S and 2 3S states occupy “the same orbitals” (and hence have similar spatial distributions), which is clearly incorrect. The visualization can thus be used as a convenient starting point for discussing electron correlation and spinorbit coupling. The 2 1S state retains more of a “shell structure”; that is, the 1s and 2s electrons seem to keep (on average) further apart than in the 2 3S state. This is evident in the nodal minimum, which is deeper in the 2 1S than in the 2 3S state. If one uses concepts of radial and angular electron correlation in lectures, then Figure 3 shows that the angular correlation is more important in the 3S state, whereas radial is significant in the 1 S state. Radial distribution (in the 2s shell) is broader in the 2 1S state and reaches its maximum value at larger r, which indicates that the helium atom in the 2 1S state has a larger effective radius than in 2 3S. Once again, this observation may be useful when discussing atomic collisions and the influence of the electronic states on collision probabilities. Finally, this article shows how computer algebra packages make plotting radial
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Figure 1. 1s and 2s wave functions for the singlet state of He (1s12s 1).
Figure 2. 1s and 2s wave functions for the triplet state of He (1s12s 1).
Figure 3. Superimposed radial distributions for 2 3S (dashed line) and 2 1S (thick line).
distributions accessible to the majority of undergraduates. This computer visualization is sufficiently simple to enable students with average mathematical ability to grasp the fundamental concepts involved. This work belongs to a series of articles published in this Journal (11), which discuss theoretical and experimental aspects of ES and EC concepts and their application in chemistry. Literature Cited 1. Gillespie, R. J.; Spencer, J. N.; Moog, R. S. J. Chem. Educ. 1996, 73, 617. 2. Shusterman, G. P.; Shusterman, A. J. J. Chem. Educ. 1997, 74, 771. 3. White, H. E. Introduction to Atomic Spectra; McGraw-Hill: New York, 1934; p 100.
Journal of Chemical Education • Vol. 76 No. 1 January 1999 • JChemEd.chem.wisc.edu
Information • Textbooks • Media • Resources 4. Herzberg, G. Atomic Spectra and Structure; Dover: New York, 1944; p 136 and references cited therein. 5. Hartree, D. R. Proc. Camb. Philos. Soc. 1928, 24, 89. Hartree, D. R. Proc. R. Soc. London 1933, 141, 282. 6. Clementi, E.; Roothan, C. C. J.; Yoshimine, M. Phys. Rev. 1963, 127, 1618. 7. Bethe, H. A.; Salpeter, E. E. Quantum Mechanics of One- and TwoElectron Atoms; Academic: New York, 1957. 8. Alexander, S. A.; Coldwell, R. L. Int. J. Quantum Chem. 1997, 63, 1002. 9. Bransden B. H.; Joachain C. J. Physics of Atoms and Molecules, Longman Scientific & Technical: Harlow, UK, 1983; p 284. 10. Mathematica 2.2.3; Wolfram Research: Champaign, IL 618207237. 11. Eichinger, J. W. J. Chem. Educ. 1957, 34, 504. Gregory, N. W. J. Chem. Educ. 1956, 33, 144. Hochstrasser, R. M. J. Chem. Educ. 1965, 42, 154. Scerri, E. R. J. Chem. Educ. 1989, 66, 481. Mazo, R. M. J. Chem. Educ. 1990, 67, 135. Suzer, S. J. Chem. Educ. 1982, 59, 814. Shenkuan, N. J. Chem. Educ. 1992, 69, 800.
Appendix A Mathematica algorithms for calculating normalization constants of 2 1S (a) and 2 3S (b) wave functions for excited helium atom. The integrands obtained from a and b are equal to NS-1/2 and NT-1/2 , respectively. a. NIntegrate[r1^2*r2^2*(Exp[-(2*r1+0.865*r2)]-0.432784*r2* Exp[-(2*r1+0.522*r2)]+Exp[-(2*r2+0.865*r1)]-0.432784*r1* Exp[-(2*r2+0.522*r1)])^2, {r1,0,Infinity},{r2,0,Infinity}] b. NIntegrate[r1^2*r2^2*(Exp[-(2.01*r1+0.765*r2)]-0.765*r2* Exp[-(2.01*r1+0.765*r2)]-Exp[-(2.01*r2+0.765*r1)]+0.765*r1* Exp[-(2.01*r2+0.765*r1)])^2,{r1,0,Infinity},{r2,0,Infinity}]
Appendix B Mathematica algorithm for calculating and plotting radial electron distributions for 2 1S and 2 3S states of helium atom: ps1[r1_]:=NIntegrate[r2^2*(Exp[-(2*r1+0.865*r2)]-0.432784*r2* Exp[-(0.522*r2+2*r1)]+Exp[-(2*r2+0.865*r1)]-0.432784*r1* Exp[-(0.522*r1+2*r2)])^2,{r2,0,Infinity}] ps2[r1_]:=NIntegrate[r2^2*(Exp[-(2.01*r1+0.765*r2)]-0.765*r2* Exp[-(2.01*r1+0.765*r2)]+0.765*r1*Exp[-(2.01*r2+0.765*r1)]Exp[-(2.01*r2+0.765*r1)])^2,{r2,0,Infinity}] p1=Plot[Evaluate[r1^2*ps1[r1]*N S 2 ],{r1,0,14},PlotStyle-> {Thickness[0.01]}] p2=Plot[Evaluate[r1^2*ps2[r1]*NT 2 ],{r1,0,14}, PlotStyle-> {Dashing[{0.01,0.01}]}] Show[p1,p2,PlotRange->{0,0.6},AxesLabel->{“r/au”,”R”}]
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