Systematics in the Assignment of Electronic and ~ h r o n i cStates for linear Molecules A. Maitland Physics Department, St. Andrews University, St. Andrews, Scotland KY16 9SS R. D. H. Brown West Glamorgan Institute of Higher Education, Mount Pleasant, Swansea, Wales SAT 6ED In gas discharges and many chemical reactions, excited atoms, ions, and molecular dissociation products are present. Many collision processes can he regarded as two species in particular states uniting to form a pseudomolecule of short life which dissociates into two species of different states. Short lived species such as free radicals have long been important to the reaction kineticist and are now under extensive investigation by laser physicists seeking new and more efficient gas lasers. Wigner and Witmer (1) determined the theoretically possible molecular states resulting from the union of two species, and their work is summarized in the Wigner Witmer rules. The states predicted are not necessarily all observed, but all observed states comply with the predictions. Information presented in the various texts summarizing the rules is rather fragmented. The most complete account is to be found in the two texts by Herzberg (2, 3) for diatomic and polyatomic molecules, respectively. While the tabulations of Herzberg for linear molecules cover most cases which have arisen in practice to date, the general pattern to which the molecular states conform is not obvious from the tables and the data, and therefore, cannot be conveniently extrapolated to discover other states. The matrix tabulations which we The matrices cover the common coupling cases and molecular formation according to the models based on union of separate species, disruption of the united atom, and electron configuration. Spin Multiplicity Matrix Multiplicity is given by ZS 1where S is the resultant spin quantum number. The matrix shown in Table 1 gives sequences of numbers which represent the multiplicities of the molecular states resulting from the union of species A and B which have multiplicities M A and M B , respectively. Species A and B are designated so that M A 2 MB. The maximum multiplicity from the combination of M A and M B is M A + M B - 1 and the minimum multiplicity is M A - M B + 1. If M A M B - 1is odd, then the sequence of possible multiplicities is in odd integers; if M A M B - 1is even, then the sequence is in even integers. Multiplicities for ground state atoms may be up to 9 (Cm, 9D;).
+
+
+
Atom-atom Matrix Total orbital angular momentum L of the electrons in an atom is the sum of the angular momenta of the individual electrons 1, thus L = 21. The state of an atom for which Russell Saunders ( L S ) coupling is appropriate is usually described hy one of the letters S , P , D, F, G , H , I , K , L, . . .according to the L value O,1,2,3, . . . Ground states of atoms are known for all letters up to L (U, "3. The letters give a distinctive nomenclature but were adopted for reasons associated with the early history of spectroscopy. By returning to numbers and displaying them in matrix form as in Table 2, we see sequences behaving so systematically that extrapolation is quite obvious. States resulting from j j coupling are numerically specified hv their J values where J = Z i and i = 1 s. These states are aiso represented in the matrix: The sequence displayed by the members of the matrix reoresent the ~ossiblemolecular states Z, 11, A, a, I'correspond& too, 1,2;3,4,. . . for LS coupling or, if j j coupling is appropriate, 0,1,2,3,. . . or '12, Y2, %, 7/2,. . . . Several states of the same type may occur and the number of these for a particular state is given by the smaller sized numher immediately above the number describing the state. The sequences in the matrix can be constructed and interpreted as described in the following steps. 1) For LS coupling, we express the atomic states in terms of integers according to the usual scheme given by
+
SPDFGHI
...
For j j coupling we obtain J for each atom as usual 2) Define atoms A and B so that L A 3 Lg for LS coupling,
J A 3 J B for j j coupling.
3) The states of the combining atoms A and B define the columns and rows of the matrix. 4) The non-2 (or non-0) states of the resultant molecule AB are given by the sequences
Table 1. Spin Multiplicity Matrix (Allowed spin multiplicities resulting from the union of two species o f given multiplicity MA and MB w i t h MA
MA + M a + M n c l
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> MB)
( L A + L B ) , ( L A + L H - ~ ) , ( L A + L B - ,..., ~) 3,2,1 for LS coupling,
or
+ Jd, (JA + JB - 11, (JA + JB
(JA
- 21.
. .,%,I12
or
I
for11 coupling.
(JA + JB - 11,(JA+ JB - 21,. . . , 3 , 2 , l The numher of each type of state present is given by placing the natural numbers from 1in order ahove the states (LA LB), etc. until the number ~ L B1 (or 2 5 ~ 1) is reached. This numher is then repeated over each member of the sequence representing the states until the end of the sequence is reached. 51 The numher of Z states is ( ~ L B 1) made up of LB states of type Z+, LB states of type 2-, and one state which is either Z + or 2- according to whether LA LB ZIA 21B is even ( + ) o r odd (4.Of course, in and 1~refer to electrons in incomplete shells. Likewise, the number of 0 states is (2JB + 1) made up of JBstates O+, JBstates 0-, and one state which is either O+ or 0-according to whether JA JB 21A ZIB is even or odd, respectively. If JA J g is half-integral, there is no 0 state; the lowest state is %. If JA and JBare both halfintegral, the 0+ and 0- are of equal number to give a total of 258 1. An atomic state is designated as odd (subscript u) or even (subscriptg) depending on the odd or even character of Z1 for the given state. (Sometimes a superscript O is used after the term symbol to designate an odd state.) This being so, to determine whether the extra Z (or 0) state is 2+(or 0+) or 2(or 0-1 we merely obtain L A L B (or JA J B ) and add 1for u g or 0 (zero) for u u andg g. Examples 1and 2 below illustrate some of the above steps. 61 If the atoms producing the molecule are of the same element hut in different states, each of the molecular states (JA
+ JB),
+
+
+
+
+
+ +
+
+ +
+
+
+
+
+
+
found in molecules formed by combination of unlike atoms occurs twice, once with g symmetry and once with u symmetry. If the two atoms are of the same element and in the same
+
each molecular state occurs withg or u symmetry depending on its resultant spin (multiplicity). About half the possible numher of each state hasg symmetry, and half has u. Where the possible numher of a given state is even, g- and u-type symmetries for that state are of equal numher. Where a given molecular state occurs an odd numher of times in the case of LS coupling, the extra state is o f g or u symmetry depending on the resultant spin S of the electrons in the molecule. We find the resultant S of the electrons in the molecule by ohtaining the multiplicity ( 2 s + 1) from Tahle 1and noting step 7 below. If S is even, the extra state is g, if odd, it is u. (See example 3 below.) Where a given molecular state occurs an odd uumber of times in the case of jj coupling, the extra state is g if the combining atoms have integral J , hut i t is u if the atoms have a half-integral J value. (See example 4 below.) 7) Each resultant state given in Tahle 2 occurs with each of the multiplicities derived from Table 1. Examples
+ +
(1) Unlike atoms, LS-coupling Pp + Du (or Py Dg). Table 2 gives 03, 13,22,31. In this case we have L A LB = 1 2 = 3 and to this we add 1t o give 4 because of u g. Since 4 is even, the extra 2 state is Z+. Thus, we obtain the following molecular states
+
+
(2) Unlike atoms, jj-coupling Zg + 2,. Table 2 gives 0" 14,23, 3'" 44 In this case we have JA+ JB= 2 2 = 4. T o this we add 1 (because of u g). The resulting odd number (5)
+
+
Table 2. Possible Molecular States Resulting From the Union of Two Atoms in States Described by LS o r j j Coupling
B s u p e r ~ ~ r i refer p t ~ t o the number of occurrences of a Dartlcular state
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means the extra 0 state is 0-. The possible molecular states obtained are thus 0-(3), O W , 1(4),2(3), 3i2), 4
+
(3) Like atoms in the same state, LS-couplmg. 3P 3P.Tahle 1shows that the multiplicities are 1,3,5. The total molecular spin S associated with these multiplicities are 0, 1, and 2, respectively. Table 2 gives 03, 12, 2l. According to step 7 above, each state occurs with multiplicity given by Table 1so we have, finally
(4) Like atoms in the same state, jj-coupling. From Table 2 and step 6 we get, finally for 2 2
+
tom-Molecule Matrix
The resultinr electronic states of a linear molecule which
the rows, and the states of the separated molecule label the columns. Again, the numbers are used to represent the states, and the matrix can be used for cases of L S and jj coupling. The particular 2-state (i.e., 2+or 2-1 resulting from combination of an atom and a molecule in a 2-state can be determined using the boxes in the first column, e.g., an atom in state F, combined with a molecule in state 2+can give a molecule in state 2-. Likewise for O-states. Molecule-molecule matrix In Table 4 the rows and columns refer to molecular elec-
tronic states of linear molecules which combine to form a linear molecule. With regard to u and g symmetries arising from combination of identical molecules, if the multiplicity of a state other than 2 (or 0) is such that the total spin S is an odd number, the symmetry is u, and if even, g. In assigning u or g to 2 (or 0) states, the odd or even number given by the S value is added to 0 (zero) for the (+) of 2+or 1for the (-) of 2.If the sum is an even number, the state isg, and if an r . stare i: I . . Thc ~.,llua.ingL ' X ~ ~ ~ I I I P.I. Ph i r ~ v ithe utld n ~ ~ m h cthe rI~.a.lnenicd a t e i 4 ;t 1int.w inL!.:~tumi; ~n~,lrcule arLinh, r'r t m separated identical molecules in 3A states
U + i-A gives Table 3. Possible Molecular States Resulting From the Union of an Atom and a Molecule
aEach state occurs only once unless the wperrcript 2 1% prerenf
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I + 1 P'
32:,
" 2
'r8
8z;, T"
5 q , 5 2 ; , 5rg
Table 4 can also he used to determine molecular electronic states resulting from given configurations of non-equivalent electrons outside closed shells. For a single electron with orbital angular momentum h about the internuclear axis, the molecular electronic term is 2A, e.g., an electron described by o gives the molecular electronic term 22+,a ?r-electron gives a 6-electron gives ZA, and so on. T o determine the molecular electronic terms corresponding to an electron configuration such as o n 6 we use Table 4 repeatedly. The configuration u ?iyields 'II, 3 I I which when combined with 2A gives 211(2), 4 n , 2@(2),44, for the molecular electronic states resulting from the configuration u n 6. In the case of equivalent electrons, the Pauli principle reduces the number of possible states so that some of the terms given in Table 4 are missing. For two equivalent electrons, the possible states are given by the leading diagonal without the 2- state for the singlets and with 2- as sole trinlet. Three eauivalent electrons result in the one molecular electronic statk (a doublet) corresponding Linear Molecule
Four to the state of one of the electrons (e.g., $3 results in equivalent electrons close the shell and thus form the state 'P. For molecules with g or u symmetry, all terms corresponding to a particular electronic configuration have the same eigenfunction and hence the same g or u symmetry. The states are allg if the number of u electrons (o,, nu,. . . etc.) in the molecule is even, but if the number of u electrons is odd, then all the states are of u symmetry. Perhaps, in the context of this paper, it should be stressed that molecules can he linear (straight) or bent. Gimarc ( 4 ) gives a nice discussion of how molecular orbital theory can he applied qualitatively to indicate whether a given molecule (AH2) is linear or bent. Herzherg ( 5 ) discusses the bent and linear low-lying states of CH2 and gives a correlation diagram for them. T o illustrate the working of some of the rules given above we shall consider the Rydherg states of linear CH2 as predicted by Herzberg (Table 36 and page 492 of reference ( 3 ) ) which arise from the electron configuration
. . . (20,)~( I d (1n,A2(nsua) '2:. '4,la,, 3Z:. 32;(2),aA,,, 52; T o see how these states are derived we note first that the number of u electrons is odd, and therefore, all the final states ~ )a closed ~ shell will have u symmetry. The electrons ( 2 ~form for which the state is always 'Z+ and this need not be considered. Considering the two equivalent n electrons, the states produced are '2+, 32-, 'A. Combining each of these terms in turn with the 22+term resulting from a #-electron we get 22+, 42-, 2A. Combining these with 22+again (due to the other o-electron) we pet the result given above. Each vibration of alinear molecule may have a vihrational aneular momentum about the internuclear axis and the total viLrationa1 angular momentum is lh where 1is the sum of the individual vibrations. The vibrational states with 1 = 0,1,2, . . .are designated 2 , II,A,. . . states. As we have noted above, the electronic orhital angular momentum about the internuclear axis is Ah with A = 0, 1,2, . . .also designated by 2, I I , A,. . . . Coupling of these angular momenta give the resultant vihronic angular momentum Kh where we have
a-,
and K = 0, 1, 2 , . . . corresponds to 2, 11, A vibronic states. Table 4 can he used to determine the vihronic states of linear molecules. For example, if a r vihration is excited in a molecule in a A electronic state, the resulting vihronic species are A I. If electron spin and orhital angular momentum are strongly coupled, the total electronic angular momentum is Rh where we have
+
and 2 h is the component of spin along the internuclear axis. The resultant vihronic angular momentum including spin is then P h where P is given by T o see how Table 4 may be used for this case, consider a linear molecule in electronic,state with a single vibrational state (1 = 1) excited. The vihronic states of this molecule have P values of '/2,3/2, 1/2,% obtained by noting that Q = '12, 11/2 may indicate the columns and 1 = 1,the row. Literature Cited (11 Wigner,E.. and Witmer.E. R . . Z Physik, 51,R69 (19281. (21 Herzber~,G., "Molecular Spectra and Molecuiai Structure. 1 Spectra of Diatomic Molecules," 2nd Ed., V m Nosoand, New York, 1950. (31 Herrberg. G., '"MnleculsrSpectra and Molecular Structure. 111Electnmics Spectia and Eiectrnnic Structure of Polystomic Mnlecules," Van Nostrand, Princeton. 1966. (4) Gimarc,R. M., Account8 olCiirrnicol Rmeoich, 7,384 0974). ( 5 ) Herzheig. G.. Pmc. R. Sui. London, Ssr. A 282,291 (1961).
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