SOME PROPOSALS C O S C E R S I X G SONENCLATURE AND SYMBOLS FOR POLYATOIIIC MOLECULES‘ ROBERT S. JIULLIKES R y e r s o n P h y s i c a l Laboratory, C n i v e r s i t y of Chicago, Chicago, Illanois Received October 92, 1936 I. INTRODUCTIOK
The desirability of a more or less standardized nomenclature for describing the spectra and especially the energy levels of polyatomic molecules hardly needs much argument. If different writers agree in their language and symbols, clarity and economy of effort result both in the writing and in the reading of papers. Further, a properly chosen systematic nomenclature for molecular energy levels greatly facilitates the understanding and discussion of their combining properties. By “combining properties’’ are here meant the selection and polarization rules governing transitions involving radiation (both ordinary a i d Raman spectra), the selection rules governing perturbations and predissociation, and such correlation rules as may apply to dissociation and other processes. Finally, an orderly nomenclature assists in the proper comprehension of the subject as a whole. One might, hon-ever, question whether the time is yet ripe for a stabilization of nomenclature. One might ask: do not most of the questions of nomenclature come u p only in connection with higher states of vibration, and eqxcially n.ith excited electronic states, which are of relatively little interest to most people as get, and about which n o one yet knows very much? This question has force, but nevertheless the n-riter is inclined to think that it is not too soon to begin at least a discu on of the subject. Spectra involving higher states of vibration, and excited electronic states, are now being increasingly investigated, especially in the photographic infra-red, and in the ultra-violet. Further, an understanding of these higher states is important in connection with activation energies and with photochemical reactions. Hence, a number of proposals are here submitted for discussion. I t is hoped that the present paper may also have some value in facilitating the presentation of certain topics in the theory 1 Presented a t the Symposium o n Molecular Structure, held a t Princeton University, Princeton, New Jersey, December 31, 1936 t o January 2, 1937, under t h e auspices of the Division of Ph? qical and Inorganir C’hemistry of the AmericanChemical Society. 139
160
ROBERT S. RKLLIKES
of polyatomic spectra, and in the discussion of the energy levels of certain molecules. As compared with diatomic molecules, the nonieiiclature problem for polyatomic molecules is inherently much more complicated. The diff erences between the two cases may be summarized as follows. (1) Polyatomic molecules include molecules of a great many different types of symmetry, with the problem of nonienclature in some respects a new one for each type. Diatomic molecules really comprise merely two special cases of this general problem (with symmetries C,, and DaJd). (2) In polyatomic molecules, classification and nomenclature are needed for vibrational levels and for electronic-vibrational levels, whereas in diatomic molecules this is unnecessary, since there is only one possible type of vibration. (The vibrational waye functions are all of the type 2+ for heteropolar, 2YLgfor homopolar diatomic molecules.) (3) I n lesser measure, there is a similar increase in coniplexity of type for rotational levels. (4) I n polyatomic molecules, intermediate and generalized cases, including cases where classification according to an equilibrium type of symmetry breaks down more or less, are common. In the present paper, attention will be confined mainly to vibrational and electronic properties of molecules, largely leaving aside questions of rotational and nuclear-permutation properties for the time being.2 Just one or two simple proposals concerning rotational states will be made. Further, no attempt will be made to cover exhaustively all aspects and special cases of vibrational and electronic classification. 11. COMPLETE WAVE F U S C T I O S
For the complete wave function of every molecule, tx-0 properties always exist which can be easily designated: (1) the property now universally designated in atoms and diatomic molecules by assigning a quantum number J ; (2) the property according to which every wave function (at least if non-degenerate) can be classified as either w e n or odd with respect t o a n inversion of all coordinates a t the center of gravity. I n atoms this second property is called parity, and the two kinds of levels are called even and odd, x-hile in diatomic molecules they are called and - (the terms even and odd then being reserved for another property, of electronic levels alone). There seems to be no reason why the symbol J should not he universally used for the resultant angular momentum quantum numbcr, and the symbols and - to indicate the parity of the complete wax-c function, for all polyatomic niolecde~. Further, it would be logical to extend the diatoniic use of the symbol K t o polyatomic molecules. In diatomic moleculeq, K is the Tame thing as
+
+
* In this connec:tion
cf. especially reference 4.
SOMESCLATURE FOR POLYSTORZIC MOLECULES
161
J in case the resultant electronic spin X is zero, and the two symbols can then be used interchangeably. K h e n , however, X is greater than zero, but is loosely coupled to the remaining edifice of nuclear and electronic orbital angular momentum vectors, the resultant of the latter is designated by K ; K and X then combine vectorially to give J . This use of K might well be extended to all polyatomic molecules in which the spin is zero or loosely coupled. I n this connection, it may be noted that loose spin coupling is expected to be much more the usual rule in the polyatomic than in the diatomic case. On the other hand, the case S = 0, where J can be used just as well as K , has hitherto been much more common in practise in polyatomic than in diatomic spectra,- largely because of the relatively greater emphasis on infra-red and Ranian work. The only objection to the use of K here proposed is that K is now often used (e.g., in “3) to designate the component of nuclear angular moment u m around a n axis of symmetry. It would seem, however, that some other symbol, e.g. P or Q, could be substituted for the latter purpose,perhaps P for ordinary rotational angular momentum, Q for angular momentum associated with a degenerate vibration. 111. ELECTROSIC WSVE FUSCTIOSS A S D ORBITALS
As an approximation n-hich in ordinary cases is useful, the complete wave function ) I of a polyatomic molecule can be written as a product of factors, as follows (a final factor fiCgfor the motion of the center of gravity is omitted): 1c,
=
$ellClv+r$nu
sp
(1)
The factors are respectively called the electronic, vibrational, rotational, and nuclear spin wave functions. More accurately, one may m i t e
Here + l e v and + I e u r , respectively, represent corrections for mutual interactions between electronic and vibrational motions, or between electronicvibrational motions and nuclear rotational motions. The factor + e v niay be called the vibroriic n-ave function, the factor $e,,T the rovibronic wave function. Khile “vibronic” wave function may sound disagreeable a t first, it is important to have a name for qeV(see section I-),and the writer has sought in vain for a better one of reasonable simplicity. The classification of the complete electronic wave functions of atoms and of diatomic molecules into types is well known. Instead of “types” it will be conT7enient to refer to species (cf. the German “Rasse”, often applied to a class of electronic states of a molecule).
162
ROBERT S . MULI.II(ES
The nature of the claisificatioii into electronic species, in diatomic molecules as well as in atoms, depends on the nature of the spin-orbit coupling. It will be convenient to refer to the usual types of coupling as normal coupling. Sormal coupling in atoinq is usually called RussellSaunders, or L, X coupling. In diatomic molecules it includes Hund's cases a and b and intermediate cases.3 In most polyatomic moleculeq normal coupling will resemble Hund's diatomic case b, but in some (especially linear) molecules, n-ill include also cases like Hund's diatomic case a. I n atoms and diatomic molecules, for normal coupling, each electronic species is denoted by a species symbol (lS, 'So, 2Po,'0,300, etc. for atoms; l 2 + , '8-, *8+, 'II, etc., for heteropolar diatomic molecules; 'Z+,, 1 8 ~ 2Hu, u, aZ-,, etc., for homopolar diatomic molecules), Each species symbol implies a particular value of each of the angular momentum quantum numbers L and S (atoms), or I and S (diatomic molecules) ; but it also implies certain other properties (the even or odd property, in atoms; the property denoted by or - in diatomic 2 states; the even or odd ( B or J property in states of homopolar diatomic molecules). Fundamentally, each species symbol can be taken to indicate certain properties of form and symmetry of the wave function G e l . In addition t o electronic species, one may often distinguish electronic sub-species, which are indicated by final subscripts. The latter refer to different possible relative modes of orientation of spin and orbital angular momenta. For polyatomic molecules, classification of electronic states into species can also be carried out, and symbols for the species for normal coupling have been devised4 Examples of species symbols are: 'il, l-41, ' B z , 3B1, 2A1', 'A:', 'E, 3Ex, * T , 3T1. Hitherto no good sub-species symbols have been developed; they are not likely to be needed as often as in atoms and diatomic molecules, because in polyatomic molecules the spin usually is only loosely coupled to the rest of the molecule. In linear polyatomic molecules, to be sure, they will often be needed; but for such molecule^, the electronic species and sub-species and their symbols are exactly the same as for diatomic molecules. It should be noted that the numerical and other subscripts appearing in such synibols as lA1, lB2, 3E*,etc., do not denote sub-species. They are parts of the species symbols, which are
+
3 See t h e writer's paper in Section I of this Symposium in regard to diatomic molecules. T h e electronic species symbols there introduced are nearly 4 See reference 2 . t h e same, except for addition of superscript prefix to denote t h e multiplicity, as t h e vibratzonal species symbols previous11 developed by G Placzrk (Mar-? Nandbuch der Radiologze, Yo1 VI/2) The principal change has been t h e substitution of 2' where Placzek used F , a i t h the idea of avoiding confusion Tvith the atomic species symbol F . Another change, of doubtful value, has been t h e substitution of E+ a n d E**, for Placzek's E + a n d E - See also L. Tisza (3) in regard t o t h e classification of vibrations a n d vibrational species.
SOMENCLATVRE FOR POLTATOMIC MOLECULES
183
more coniplicated here than for atoms and diatomic molecules; in regard t o the reasons for this, see below.. Although in atonis and diatoniic molecules angular iiionieiituni properticv can be used t o a large extent in characterizing elect,ronic species, this is hardly true at all in polyatoniic ~iiolecules,other than linear ones. As a result, the language of group theory is particularly useful in characterizing ,species. I n polyatoniic niolccules as well as for atoms and diatomic molecules, the group theory approach, introduced particularly by Wigner, appears to he the most natural and penetrating one. Although t o sonic' people this may make the matter seem formidable, the writer k n o w no simpler way to gain a good understanding of the necessarily ra.ther great variety of wave-funct'ion types which exists in the field of polyatoniic niolecules. In judging the suitability of the species symbols n o x in use for polyatomic molecules, one should bear in mind that, a new classification and, in principle, a new set of symbols must be set up for each different type of molecular symmetry. (Fortunately, however, the number of distinguisliable species decrea,ses with diminishing symmetry, being finite except for at'onis and diatomic and linear molecules.) In practise, it would Ijc too great a hurden on the memory, and would require too many alpha,bcts, if we tried t o develop a distinctive set of simple symbols for each polyatomic> symmetry. Iiist'ead, it ha.: seemed wise t o .use a restricted numljer of sonien-hat cunihersome h i t more or less self-explanatory symbols, and frequently to permit thc same fornial symbol t o appear for species lirlonpiiig t o different' symmetries. For instance, Idlis used t o denote, for each of several different syrnnictries, the respective niost, symmetrical specics of $ e l . If we extended the same system t o include atoms and diatomic niolecules, the ' S spwien of atoms and the 2' and lz+gspecies of diatomic Such a system tends t o cause conmolecules would also lir labelled fusion n-hen, but only n-hen, moleci1les of different symnietry are discus.setl siniultaneously ; in this cvcntua,lity, a n extra index might be attached t o the species q m h o l to indicatcx thcl ~synimetry. For atoms and diatomic nioleciiles, howvcr, l)ccaiih(' of thc>ir predoiiiinarit importance, it is certainly good that nc havc tlistiiic.ti~c~ of symbols. In general, and I3 huvc l ) c ~ iisctl ~ i for orhitally non-degenerate specics, E for -pcrics having tn-ofold, T for those having threefold, orbital degeneracy while subscript and siipcrscript suffixes have been used to designat(, ot'her characteristics of the forni of $ e l . h perhaps desirable improvement in the species symbols would be the replacement of the ~iuniericalsubscripts (1, 2, and rarely 3) by literal subscripts ( a , b, and c ) . That n-oultl m a k it less likely that the subscripts n-oultl he confused with the s~rb-.spr~cI'rs subscript s nscd for atonis and for diatomic and linear ndeciiles. Xsiclc froin this and a few other possihle minor chaiiges, the writer is inclinc.cl t o
164
ROBERT S. MUI.I,IKEN
think that t h e prwent system of electronic species symbols for polyatomir niolecules is a satisfactory one. \\'hen electroil-c.oiififfuratioli ap~)roxiiiia~tions are userl-a,< is nearly always the case-in dincwsing rlectroiiic structures of atoms or molecules, a symbol i,s iieeded for each of the orbitals used in making up the electron configuration. Thus for atoms we h a w Is, 2s, 2 p , aiid so 011, and for diatomic molecule,q , ~ c symbols h as lsu, 2su, 2 p n , arid so 011, u g l s , u,,2p, r t c . , or u l s , 8 1 8 , u2p, ete. Each such symbol is made up of a parficulnior indiridiia.1 and N s p c c i ~ ssymbol. For atomic orbitals the possiblc ,q)ecies arc symholiz~clhy *s, p , (1, j , a i d so on, for heteropolar dia.toiiiic molecular orllitals hy u, T , 6, 4, ant1 .so on, arid for homopolar diatoniic orbitals by uQ,u z , ,K , ~ T, ~ 6,, , and so on. The species symbols ought t o lie, a i d are, definitely fixed arid staiidardized, but (for. diatomic molecules) t h r particular symlmls are more varied and less standardized. The species symbols for orbitals arc. the same as those for electronic states except t h a t ( 1 ) small letters are iimd in placc of capitals, ( 2 ) the miiltiplicity suprrscript is omitted, a,iitl (3) +omctinies othcr superscripts are omitted I)ecawe iuiambiguous (e.g., thc is omitted in u+, since u- is impossible). For polyatoniic molecules, the relation of species symbols for orbitals to those for electronic state.: i:: jiist t h e same as iii the case of atoms n i i t l diatomic molccules. A s has I)crn iiotcd already, the species sym1)ols f o r polyatoniic moIeciilr.+arc more ciimlmsonic than for atoms or diatomic ~iioleciiles,h i t it liartlly semi.< clrsirable to attempt t o simplify tlicm, miless possibly for a fcn- special ca.ws of particularly important typw of syliinietry. Bccaii,+ of t h c raricty and complexity of polyatomic molcculcs, relatircly c.om~)licatctland varied pariiciilar symbols h a r e hecn u s c d in descrihing their orbitals. 'I'here appcars to he no point in trying to staiidardize these at present. +
I V . TIBRATIOh AI, WAVE, F U S C T I O S S AND 1 IBRATIOS MODES
Although the Schrodingcr rquatioiis for of equations 1 aiid 2 arc entirely diffcrcnt from tho-c for ne1 ertheless both are characterized by the same foriiial typei of gcoinrtrical syiiiinetry. As a consequPncr, can be clacsificd accortliiig to exactly the >amp formal system a i $ e l , and the ymbolc can lie iiicd for both, except that for the v26idionnl species, no miiltiplicity \iiprr-cript i, included. It might of coiir+ 1)r argiicd that, 4 i c c tht. x ihratiorial and electronic n a v e function5 arc entirely diffcrciit thing-, they yhoiild be cla-qificd hy differeiit symbol.. The nriter 1s of the opinion that thiy is iiniieccwiry, and that it would he undewable for the folloniiig reasoils: (1) the miiltiplicity super-cript i> atlcquatc to di+tiiigui+han electronic from a r i 1 ) i ~ tioiial s p e c k qyiiihol ; ( 2 ) tlic ii\c of iiearly tlie iame iynilml.; her(>promot('iindcrstantiiug miich morc than it caiices confusion; (3) widerstanding :LI)(~
%$
%$ SOMENCLA4TURE FOR POLTATOJIIC MOLECULES
165
c~las~ificationof vibronic wave functions is greatly facilitated (cf. .vetion TT) b y having a coninion ha of symbolism for +v, and Gee. It sinetinier happens, of course, that one wishes to speak of electronic .pecieb without specifying multiplicity. I n that event, they can, I f ncce>sary, be distinguished from vibrational species b y adding a subscript prefix, as, for example, ,iil for an electronic -4,species, ,A1 for a vibrational A41qpecies. At present, the vibrational species symbols introduced b y Placzek and iihed b y Tisza differ slightly4 from the electronic species synibolq used by the writer, In the writer’s opinion, it would be desirable that exactly thr mme hymbols, aside from the multiplicity index of course, ihould be u w l in the tlTo cases. If this idea is carried out, the whole set of symbols -honld be gone over carefully at the same time for possible niinor imlmnwnents. Follon ing out the foriiial nnalogicb betwecn + b l and it is coiii.ciiicTiit t o clcfine the term vzbrational conjiguratio?z in analogy to “electronic coiifiguration.” By vibrational configuration is nieant the detailed .tare of \ibration of the molecule, in >o far a> it can be de5cribed by stating the iiuiiibrr of quanta (Le., the quantum number) with which each of 1 arious normal modes of vibration is excited. Such a description is of course only an approximate one, and often fails badly, being defective to the extent that iiorinal coordinates are unsuitable for describing large vibration.. These defect.. are sometimes less, sometimes more, serious than the rather similar liniitatioiir to which electron configuration descriptions are subject. We may now proceed to set up vibrational configuration symbols in analogy to electron configuration symbols, ab follon-s: let each normalmode-of-vibration symbol in the former correspond to an orbital syinbol in the latter, and let the number of quanta with nhich a given vibration mode is excited be indexed in the .amp n a y a. is the niinibw of electrons nhich occupy a given orbital. Different nornial modes of vibration, or the correhpondiiig normal coordinates, can be clawified according to species, juit as orbital. are. -4s n as shonn by W ~ g n e r the , ~ possible niodal speczes (i.e., vibration-mode >pecies) are formally of exactly the .anie kinds as are the 1-ibrational (i.e., the vilmtional n ave function) species, even though t hc cladicatioii of modal species is based on clasbieal mechanics, that of vibrational species 011 quantum mechanic, w t up nlw among the recpectir v -yniholh. IYv thcsn ] l o t ( > , among ot1ic.r tliingi, that 1% here vibrational and electronic hpecic,i arc tlciiotcd by capital letters, niodal and orbital species are denoted by corrczponding small letters. +%,
1Mj
HOBEItT S. MULLIKES
Something iiiust now be said about the relations between a vibrational coiifiguration aiid the corresponding vibrational a t ate
