The Correct Interpretation of Hund's Rule as Applied to "Uncoupled States" Orbital Diagrams Mark L. Campbell United States Naval Academy, Annapolis, MD 21402
Electron configurations, quantum numbers, and Hund's rule are typically introduced very early in a first-year chemistry course. Many general chemistry texts (I) use orbital diagrams to illustrate the occupation of atomic orbitals by electrons. In orbital diagrams the electrons are represented by arrows either pointing up (representing an electron in a m." = state with m. = +lh) or oointine down (reoresentine . -5/2) placed & boxiswhich correspond to a particular magnetic quantum number (ml) for a particular type of orbital (s, p, d, etc.). Thus, s orbitals have one box corresponding tom, = 0 in which two electrons (spin up and spin down) can be accommodated, p orbitals have three boxes corresponding to ml = +I, 0, and -1 in which six electrons can be accommodated, etc. Orbital diagrams have the advantage of giving a visual representation of the occupied orbitals along with giving an indication of the quantum numbers associated with the electrons. For example, conforming to the convention advocated in this Journal (2) . . in which the left box represents the maximum posirive value of ml,two electrons in the 2p suhshell are represented by the orbital diagram
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~
~
~
Journal of Chemical Education
Term Symbol
's,
Cou~ledState' IL S J M,)
lO000) =
Orbital Diagramsb 8-7,~-
+
+
1
~~~~
which represents a 2p2 electron configuration in which the arrow oointine un in the left box remesent8 the sdn-orbital state 15 11 q2)-and the arrow in the middle box;epresents the spin-orbital state 12 10 '/2 %) where the ket notation in 1 ml s m,) is used to designate the principal, angular momentum, magnetic, spin, and z component of the spin quantum numbers. Normally when drawing orbital diagrams, for brevity, only the outer partially filled subshell(s) is(are) explicitly shown although the lower energy filled subshells are understood to be present. Orbital diaerams are freouentlv used to demonstrate the building up ofground-stateklectron configurations of atoms in accordance with the Aufiau principle. The appropriate number of electrons are inserted into the lowest energy orbitals within the constraints of the Pauli exclusion nrincinle. Thus, hydrogen is represented by one arrow placed in the Is orbital box, helium has two arrows pointed in opposite directions placed in the I s orbital box, etc. As the electron configurations are sequentially built up, one encounters the dilemma of what to do with the two 2p electrons for the carbon atom. Since there are three 2p orbitals available, there are six ~ m s i b l enlaces for one electron and concomitantlv five possibilities ior the second electron resulting in 15 different orbital diagrams that can be drawn for a pair of indistinguishable 2p electrons. At this point the common practice ( I ) is to invoke Hund's rule to designate a particular orbital diagram as "the" ground state. This procedure is not technicallv correct since the actual ground state of the carbon atom cannot be correctly represented by any one single orbital diagram but really corresponds to a linear combination of several orbital diagrams. The mixing of orbital diagrams to give the actual electronic state of the atom is analogous to the employment of resonance in chemical bonding. Just as 134
Relatlonshlp between L-S Coupled States and Uncoupled States Orblfal Dlaararns for an mZElectron Conflauratlon
+I.
"chat me three p orbitals is repre&nted by a box. me ien box has m = the middle box has n$ = 0 , and me right box has m = -1. An anow in a box poiming up represents an decbon occupying mat orbltsl with 4 = +%; an anow pointing down repre3ents an eleofmn with mi = -%.
the actual structure of a molecule such as sulfur trioxide is represented by amixture of several Lewis dot structures, the actual electronic state of an atom with an unfilled subshell must be represented with a mixture of several orbital diagrams. Although not normally explicitly stated, orbital diagrams represent an "uncoupled" representation of the electrons in an atom in which each electron's spin and orbital angular momenta act independently. In a "real" atom the electrons interact with one another to correlate their motions so that the electrons' motions are "coupled". I t is beyond the scope of this paper to go into detail about the different coupling schemes and when each applies. The interested reader may consult the book by Condon and Shortley (3)for a thorough treatment of coupling. The most frequently encountered coupling scheme and the case that is applicable with Hund's rule is RussellSaunders or L-S coupling. In L-S coupling, the valence electrons' individual angular momenta, lj, couple to yield the total orbital angular momentum, L, which is a constant of the motion. Similarly, the individual spins, si, couple to yield the total spin angular momentum, S, which is also a constant of the motion. A state reoresentine a narticular combination of L and S is called a term. The spectroscopicnotationfora termisobtained hy representing the valueof
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the total orbital angular momentum quantum number L by a capital letter S, P, D,. . .for L = 0, 1, 2,. . .and using a numerical sunerscrint to indicate the value of the multinlicl.'The determination of uss sell-~aunhers ity given by term svrnbols for different electron confieurations has been coverld thoroughly in this Journal (4-6).-1n general, several G S terms will result from any one electron confieuration. Hund's rule is an empirically determined rule that applies to the L S couoled re~resentationof an atom. The rule as formulated hy ~ u n finds d application in determining the around state of an atom in which the mound-state electron eonfiguration yields more than one L% term and/or level. Hund (7)concluded from atomic spectra that:
+
For terms arising from a given configuration,those with the highest multiplicity lie deepest, and of these, the lowest is that with the greatest total electronic orbital angular momentum quantum number L;for a subshell which is less than half-filled, the level with the lowest J value will lie deepest. Hund's rule is typically stated in general chemistry texts in a form similar to that in Ebbing (Ii): when electrons fill a suhshell, every orbital in the suhshell is occupied by a single electron before any orbital is doubly occupied, and all electrons in singly occupied orbitals have their spins In the same direction. Unfortunatelv. this internretation is not consistent with ~ u n d ' origi&l s formulati&. T o illustrate the inconsistency, let us look a t the nreviouslv described situation for carbon. The p2 electron Eonfiguraiion of carbon splits into three terms, a 3Pterm, which consists of three levels (J= 0,1, and 2), a lD term, and a 'S term. The representation in which L and S are good auantum numbers is obtained by forming suitable co&bina;ions of the uncoupled states forthe given electron confieuration. The methodology used to derive the correct combinations of the 15 uncoupied states to form the coupledstates is beyond the scope of this paper. The book by Eyringet al. (8)is anexcellent source for the general method along with an explicit derivation of the results for the np2 electron confieuration. An extensive tabulation of the linear - - ~ combinationsassociated with pn and dn electron configurations can be found in the book by Slater (9).The coupled states written in t e r m of linear combinations of uncoupled states for the carbon atom are shown in the table. Using Hund's rule, one determines that the coupled statea in the 3~ manifold reoresent the lowest energy states, while the 'D ~ the J = 0 and 'S terms are excited terms. In t b e s manifold, level represents the ground state. One immediately notices from the linear combinations in the table that the 3Postate has contributions from two uncoupled states for which the individual electrons'spins arenot aligned ( m , , # rn,,). Thus, there is no fundamental law that requires electrons to align their spins parallel to one another in an uncoupled representation of the ground state. I t is interesting to note that the three uncouoled states that have the two o electrons in the same orbitai all correspond to excited terms and do not contribute to the lowest energy aPterm. This result is in fact valid for any electron configuration with two or more electrons in an unfilled subshell; i.e., the observation that electrons tend to occupy orbitals singly to the maximum extent ~~
~
possible is accurate and is in fact the correct interpretation of applying Hund's rule to uncoupled states. The Role of Hund's Rule In General Chemistry The prediction of magnetic behavior for atoms and simple molecu~lesby general chemistry students is typically accomplished by applying Hund's rule to an uncoupled representation of the electronic confirurntions. Even thoueh " Hund's rule does not rigorously apply to uncoupled systems, the annlication of Hund's rule hv. .eeneral chemistrv students to , determine magnetic behavior is appropriate as long as Hund's rule is interoreted correctlv. 'J'he difficultv with the current usage of ~ L n d ' srule is n i t with its validity with reeard to uncou~ledre~resentationsof electronic states. T K ~problem lies with the fact that many chemists have misinterpreted Hund's rule as to what it actually means. The currenG accepted interpretation that requires the electron spins to align themselves parallel to one another is not a valid interpretation of Hund's rule, and this interpretation should he discarded. A formulation such as "the most stahle arrangements of electrons in a subshell are the ones that yield the maximum number of unpaired electrons", while not a rieorous internretation of Hund's rule. is. however. consisteLt with atomic structure. This formulation still a11 lows one to specify the number of unpaired electrons for the ground state of an electron configuration without requiring the snins of the electrons to he directed narallel to one another. Again, there is no requirement for electrons in atoms to align themselves parallel to one another in an uncoupled representation of the ground state, andit is this misconception that needs to be corrected.
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Literature Clted (1) la) Atkina. P. W. Ganeml Chemistry; Freeman: New York. 1989: pp 250-252. (b) Bodner, G. M.; Pardue.H. L. Chemistry An +perimantolScisnee: Wiley: New York, 1989; pp 1%-137.(el Brenica. F.: Arenfs. J.; M~islich,H.; Turk. A. Fundamenlola of Chrmisfry.5th ed.: Acsdemie: New York, 1988: p 153. (dl Brown. T. L.;Lomay. H. E. Chemistry The Cenlroi Science. 4th ed.; Prentiee-Hall: Englewoad Cliffs, NJ. 1968; pp 198-201. (e) Brady, J. E.: Humiston, G. E. General Chemistly Principler and Struclure.3rded.:Wilev:NewYork, 1982;pp89-92.(0Chang,R. Chamirlry.3rded.: RandomHouae: New York, 1988:pp247-254. (gl Daui8.R. E.: 0ailey.K. D.; Whiffen, K.W. Principles of Chemisfry:Ssunders: Philsdelphla, l934:pp 158-162. (hlDiekereon, R. E.; Gray, H. B.: Haight, G. P. Ch~mieolPlinciples. 3rd ed.; Bemamini Cummings: Menlopark, CA, 1979; pp 333-345. (i) Ebbing, D. 0. Gonerd Chsmialry, 2nd ed.:Houghton-Mifnin:Boston, 1987;pp215-227.0) Gillespie. R. J.;Humphreys, D. A,: Baird. N. C.: Robinson. E. A. Chemistry, 2nd ed.; Ailyn and Bacon: Boston, 1989:pp357-361. (k) KeenqC. W.;Kleinfe!fer.D. C.: Wood, J. H. GenamlCollegs ChemisIry, 6th ed.; Hewer & Row: New York, 1980; p 113. (1) MeQuarrie, D. A.; Rack, P. A. Gsnerd Chemistry; Freeman: New York, 1981: pp 292-303. (m) Masteram. W . L.: Slowinski, E. 3.: Sfanitski, C. L. Chsmirnl P d n c i p l ~ awith Quolitotim Analy~i8,6thed.:Seunder8:Philadelphia,1968:pp 215.218 (n)Maeller,T.:Bailar,J. C.; Kleinberg, J.; Gum, C. 0.; Clutellion, M. E.; Metz, C. Chemistry uith Inorganic Quditatiua Anolyais, 2nd ed.:Academic: Orlando. FL. 1984; p 221. (a) Mortimer, C. E. Chemistry, 6th ed.: Wadsworth: Belmont. CA, 1988; p 134. (PI Oxtoby. D. W.; Nachtrieb, N. H. P~inciplrao/ Modern Chemistry; Saundem: Philadelphia. 1988: p 434. (ql P~trueei,R. H. General Chomialry, 5th ed.:Maemi1kn:Nes York, 1989; pp 263-266. (PI Seagel, S. L.:Slabaugh, M.R.:Chsmirtry for Todoy: Weat: St.Psu1,MN, 1987:pp 71-76.1%) Whitten,K.W.:Gailey,P.D.:Dauis,R.E. Genera! Chemislry.3rd ed.:Saundem: Philadelphia, 1984:pp 113-117. (ti Zumdshl, S. S. Chomi~try.2nd ed.; Heath: Lexington, MA, 1989: p 292.
(2) Strong, J. A.J . Chrm. Educ. 1986,63,834. (3) Candon, E. U.;Shortley. G. H. The Theory ofAtomie Specfm; CarnbridgeUnivenity: New York, 1935. (4) G0rmsn.M. J. Chm".Ed"~.1973,5O, 169. (6) Hyde,K.E.J. ChsmEduc. 1975.52,87. (6) Vieents. J. J . Chem. Edur. 1383.60.561. (7) Hund, F. Linien~paklrenund Periodisches System dsr Elemente: Springer: Bedin. 1827. (8) Eyring.H.; Waltm. 3.: Kimbsll, G.E. Quantvm Chemialry; Wiley: New York,1944,pp 140.143,
(#I Slater. J. C. Buonhrm Theory @Atomic Strumre: MeCrew-Hill: New York, 1980; Vol. 11. pp 335-337.
Volume 68
Number 2 February 1991
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