R. L. Flurry, Jr. University of New Orleans Lake Front, New Orleans, Louisiana 70122
Site Symmetry and Hybridized Orbitals
The mathematical principles of group theory were first set down in a letter from Evarist Galois to August Chevalier the night before Galois was fatally wounded in a duel in 1832 ( 1 ). The earliest practical applications of symmetry groups to chemistry were in the field of crystallography (2). The first applications of group theory to quantum mechanics were in the field of atomic spectroscopy (3). These were followed, in short order, by applications to infrared spectroscopy (4) and to molecular orbital theory (5).The first extensive review of group theory in the form most commonly employed by chemists was by Rosenthal and Murphy ( 6 ) .The most spectacular achievements of gr&p theory have been outside of chemistry, in the prediction of the existence of elementary particles (7). The concept of hyhridization was introduced by Linus Pauling ( 8 ) when he noticed that if he took certain linear combinations of atomic valence orbitals, the resulting "hybridized" orbitals assumed the same geometrical orientation as did the chemical bonds around that atom. Hvhridization thus l~rovidedan early explanation for the shapes of moleculrs. The use of hylmdized orhirals is nerrssary for valence bond calcularions and for certain tvpes of molecular orbital cnlrulations and is useful for detcrmining.shapes . of molerules and for other applications. Group theory provides the solution for two problems in hyhridization: the determination of the hyhridization associated with a certain shape and the construction of the appropriate hybrid orbitals from a given basis set. There are several approaches to each of these problems. We will present here an approach hased upon site symmetry (91. Frrr atnms, in the at~senced a n y external perrurhations, have sohericnl svmmetrv. 'l'hev are dcscrihed hv the full spheriEal poiot group ~ ( 3 () 3 ) . ~ natom in a field of lower symmetry is subjected to a perturbation having the symmetry of the perturbing field. This lower symmetry, or site symmetrv. describes a svmmetrv group. which is a suharouo .. . of the ~ i 3 point , gnup..'l'he idea d u c h a symmetry reduction was first exoloited hv llethe r 10, in his classic paoer whirh i n t m duced crystal field theory. The concept if site group was exploited in a different context by Halford (11) for deducing the spectroscopic selection rules for molecules a t specific sites in crystals.
with all elements of the other. A semidirect product is when one subgroup, the invariant subgroup, commutes with all of the elements of the other. A weak direct product occurs when neither of the subgroups is invariant with respect to the other.) If a semidirect product is involved, either GI or Gs may he the invariant subgroup. In constructing hybridized orbitals, the nature of the atoms attached to the considered atom is usually ignored. Thus, the symmetry defined by the hybrid orbitals is the highest symmetry consistent with the geometric structure. The site symmetry of the hybrid orbitals defines Gs while the subgroup interchanging them defines GI. Table I lists Gs and GI for some selected systems. Determinatlon of HybridIration Once the site symmetry of the hybrid orbitals has been found, the representations in the complete group which are spanned by these can be found immediately by use of the correlation theorem (13). The hyhridization is determined directly from the representations by consulting the right-hand side of the character tables to see what basis functions span these representations. The correlation theorem states that the
a
Site Symmetry and Interchanges The site symmetry of a point in a system can be defined by the group of symmetry elements of the symmetry group of the complete system which pass through the poiot under consideration. For examde. . . a cuhe possesses octahedral svmmrtry as expressed in the 0, point group. The site symmetq of one of the wrners o t a cuhe is defined 11v the identitv, the single C S axis and the three planes of symrnetry which pass (Fig.). through the corner and is thus CQ,, Groups can he constructed as products of subgroups. One particularly useful construction is to use the group of the site symmetry Gs of a particular function, atom, etc., and the group of interchanges of the equivalent sites, Gr (eqn. (1)) (9). C=Gr.Gs (1) In eqn. (1)a "dot" has heen used to indicate the product since, depending upon the specific case, either a direct, a semidirect (12), or a weak direct product may be involved. (A direct product occurs when all elements of each subgroup commute 554 / Journal of Chemical Education
View
along the bcdy diagonal of a cube illustrating the G,site symmetry of
a corner.
Table 1. Site Symmetry and Interchange Groups far Some
Selected Svsterns
cw
AX AX:V
X
'A"
AX AX~VZ
X
AX, AX%v%Zz AX%V% AX, AXsV9
X
cn " a h
Did D,h Dsd ='ah
Ax
6
v v
IZI
C
c",,
C,
2" cs
C
c, c:
X W.Zl X IVI
Cxv
cn ca
cs
D,
x
Cw
v X X
v
c 3
"
c 3
c,v
C."
c,
c, s, ca C,
sidered to he sp3d, although it could equally well he (ns) p3(n's), (ns)p3(Xn's + +d) or most generally (ns)(Xp ~ d (an's + wd) where n and n'represent two different principal quantum numbers. In this case variations of h and p or a and w do not change the shape of the molecule.
Table 2. Correlation of C,.,with T,+
+
Construction of Hybrid Orbitals Let us now consider the actual construction of the hvhrid orhitals from the basis set. If we symbolically denote the hybrid orhitals as 4; and the basis orhitals as x k then we have
irreducible reoresentations in a erouo bv a function - .G manned . (or set of funltions) located at a site of lower symmetry Gs can he found by finding the irreducihle representations spanned by the function (or functions) in Gs and then correlating these The problem is to find the cjk. Since the basis set and the set with the irreducihle representations of G. Each function of hybrids span the same representations, either can he exhaving symmetry r' in Gs will contribute to every r in G pressed as a linear combination of the other. Thus, we can which correlates with r'. Correlation tables can he c o n s t ~ ~ t e d validly write the inverse of the transformation of eqn. (2) hy mapping the representations of the full group onto the xa = Z.c'atdL (3) subgroup, and reducing the resulting representations which are hot irreducible. ~ h k s ecan he foundin many texts (14). Since the individual basis orbitals form independent hases Tahle 2 shows the C3,, - Td correlation table. From this table for their corresponding irreducible representations, the we see that a function belonging to the A1 representation in coefficients of eqn. (3) can he determined directly by the use C a would contribute to functions having A1 and TZsymmetry of projection operations (15). (The appropriate projection in Td; a function transforming as E in C3, would contribute operator operating on x k gives it back unchanged.) Equation t o E , TI, and TZin Td; etc. (3) can be written in the form of a matrix transformation of For the present purposes, a correlation table is not really a vector needed. The hyhrid orbitals transform as the totally symx = O (4) merric representation within C s . The totally symmetric repwhere the set of x k and 4, are represented as row vectors and resentation ot a subgroup correlates with those represents. each column of T is one of the sets of coefficients. If the coltions from the full group which have non-negative chararrers umns of T are individgally normalized, T becomes unitary and for the symmetry operations of rhe subgroup. Consider, fur right multiplying by T yields example. tetrahedral hybridization. The hyhrid orbital5 have CR.site symmetry and transform as A , within C:U.The E, C g i+ = 4 (5) and n. perat at ions of C:,,, correspond to the E, CR.and od opThus, the form of the hybrid orbitals can he found simdv erations of 7h. The only representations of fi hwing now . . bv . determining the columns of the vector T by applying the apnegative entries fur these are A , and 7'2. Thus these are the propriate projection operators to the set of 4; and then reprrsentations spnnned by the hyhrid orhirals. (Note that carrying out the inversetransformation of eqn. (5).Of course, the results are the same as would have heen obtaini,d from the xi must be ordered in x in the order corresponding to the 'I'shle 2.) An .< orbital always furms a hasis for the totally order of the irreducible representations in T. svmmetric representation (A, in thiscase,. Consultine the T,r The use of the group of interchanges GI, rather than the full cjlaracter table we see that ~2 is spanned by the set ix, y, 2) point group, greatly simplifies the construction of T. The set or by the set (xy, xz, yz). If only an s and p atomic basic set is of hybrid orbitals spans the regular representation of GI. The heing used, we see immediately that the hybridization is sp3 regular representation TIR)of a group is defined as (the set p,, py, pz corresponding to the set x, y, 2). If d orbitals were included in the basis set, the hybridization would correspond tos(Xp pd)3 where A and @aremixing coefficients where the r are the irreducible representations of the group which would have to be determined by energy considerations and mr is the dimension of the irreducible representation r. rather than by symmetry. Thus, if the rnr-dimensional representations are resolved into As a second example, consider the hybridization of a trim r independent representations, there is a one to one corregonal pyramid such as NH3 or PH3 with an s and p atomic snondence between the hvbrid orbital combinations and basis set. The full uoint svmmetrv is C?,,while the site s v n representations of GI. The combinations can he determined metry of the hybrid bond orbitals is C,. The correlation of"the immediatelv from the character tahle for GI. In doine - so. , two totallv svmmetric A' representation of C. with Ca..vields A, words of caution are in order. First, either the correlation and E representations for the hybrid orbitals. B ~ and Xp i~ bles or the transformation properties of the hases within taGI orbitals man A1 of Cn,, while D, and D., soan E. The resultinn should be used to properly order the xi with the corresponding hybridization is (As +-pp,)pi. he X&dp must again be dey representations Secondly, the orientation of the reference axis termined by energy considerations. If X equals p (= 1 1 4 the system will be determined by GI. For all common symmetries hyhridization becomes sp3 and the bond angles become other than octahedral, the results are the usual set of simplest 109"28', near the value for NHn. If X eauals zero. the orbitals hybrid orhitals; however, for octahedral hyhridization the become pure p in character leading to predicted bond angle reference z-axis will he a Ca axis (the nrincipal axis of the SF. of 90°,very near the value for PH3. If p equals zero, the hysubgroup) rather than the usual orientation along a Cq axis. bridization becomes sp2,the hybrids are coplanar with a bond Consider again tetrahedral hybridization. In this case GI angle of 120°,the structure of the excited electronic states of is D2. In 0 2 an s orbital transforms as A while p,, p,, and p, NH3. transform as BI, Bz, and B3, respectively. Thus, normalizing As a final example, consider a trigonal hipyramid such as the columns of T we have PCk. The full point symmetry is D3h. There are, however, two site symmetries for the hyhrid orbitals; C* for the axial bonds and Cz, for the equatorial bonds. The totally symmetric A1 representation of CQ"correlates with the Al' and Az" representations of D3h while the A1 representation of C b correlates with A,' and E' in D3h. From the Dgh character table it is seen that the A,' representation is spanned bv d,2 as well as s. The A2" representation is s p a n n e d b ~ p ,and the E'by p, and py as well as dZ252and dly. The hyhridization is normally con~
~
+
~~~~~
~
~
~
a
Volume 53. Number 9, September 1976 / 555
)
~
Finally, consider the construction of the hybrid orbitals of PC15 assuming sp3d hybridization. Here there are two interchange groups; C p for the axial bonds and Ca for the equatorial bonds. For the most general case the A 1 ' contrihutions to both the axial and equatorial bonds should be of the form
and
c=Xs+rdr2
(21)
For the axial orhitals we have
m2 = 1 / 4 0- l/&p,
(24)
For the equatorial orbitals Note that in eqn. (8) the matrix is the character table for GI with the rows normalized. This will be the general situation, exceDt that deaenerate re~resentationswill have to be exprcised as sets of independenr representations. Consider the construc~ionof the hyhrid orhiulls fur NH3. The group of interchanges is C 3 . Let o = As
+ up,
(13)
We have
63 =
+
1 / 4 0 2/V%pz
44 = 1143~ - ll&p,
+l/4p,
ms = l/Gc- l/V%p, - ll&py
(26) (27) (28)
Note that eqns. (25-28) are completely equivalent to eqns. (1&17).
41 = 1/&c + 2 / 4 p , (15) b2 = 114s - ll&p, + ll&py (16) 4 3 = 114s - 1 1 4 -~ll&py ~ (17) In eqn. (14) we have constructed two independent real forms for the E representation of CQ.In the character table, the E representation of CQis written
Adding these gives the real component ( E J=~(2 2 ~ 0 ~ 1 2 20 ~~0 s 1 2 0 ~ 1 = (2 -1
-11
(19)
Normalization leads to the second row of the matrix in eqn. (14). Subtracting the two representations of eqn. (18) gives the pure imaginary component ( E l 1 = (0
2isin120° -2isin120°1
&i -ail (20) Normalizing this and dividing through by i leads to the last row of the matrix in eqn. (14). ={O
556 / Journal of Chemical Education
The construction of any other hybridization scheme is equally simple. The previously mentioned word of caution about the orientation of the reference axis system for an octahedral system should, however, he remembered. In that case, to get the usual orientation of hybrids, one should directly construct the matrix T from the projection operators of the 0 or Oh point group. Literature Cited
.~.
i l l N e m a n . J. R.. "The World of Mathematia."Simonand Schuster. New York. 1956. VOI.11i.~.1534. s their Rem.entatians." Aeedemie prenn, New 121 see Kortcr. G. F.."Soace G r o u ~ and Ywk, 1957 for; hi~totorieslsubmary. (3) This culminated in the now claric. Wigner,E.,"GroupTh~aryandits Applicationto the Quantum Mechanin of AtomieSpedra," Academic Press, New Yark. 1959.The fimtcerrnan edition ofthiswlupublishd in 1931,~heothernearcl-ic, weyl, H., "Th. Theory of Groups and Quantum Mechanics." Dover Publications, In=. Now York. 1950,suffenfrornthefactthatthe SrstGermanditionwar published in 1928, and wasapparently written before Weyl had digested the 1916 papenof Hei~enbew and schfidinger. . (4) Wigmr, E;'Nochr. Aknd. Wlrs Dotingen.). 1 9 3 0 , ~133. ( 5 ) Mul1iken.R. S.,Phys. R O U ,43.279 (19931. 1s Rosenthal, J. E., and Murphy, G. M., Reu. Mod Phya.. 8.317 11936). (71 Geil-Mann. M.. and No'ernan. Y.. "The Ethtfold Way." Benjamin. New York, 1964. (8) Paulin., L..Pcc.Noll Acod Sci. US., 14,359 11928). I91 Rwry, R L..Jr. Inf. J . Qvnnrum Chem.. S6.455 (1972);Theoret. Chim. Acto (Berlin). 31.221 119731. (10) Bethe, H.,Ann. Phys.. 5,133 (1919). (111 Ha1ford.R. S . , J Chem.Phys., 14.8 (1946). (12) A1tmann.S. L.,Phil. Tmm.Roy Soe, 225&216 (1961):Reu. Mod. Phys.. 35.M111963). (13) Wilnm, E. B.. Decius. J. C..andCn%,P. C.;'Molecular Vibrations? McGraw-Hill Bmk Co.. New York. 1955.p. 113. (14) Hall, L. H.,"GmupThwry and Symmetry in C h e m k t v . ) ) M ~ G r ~ ~ - H Boak i U CC.,New Ymk. 1969,Appendir II;Henbe=, G.,"EledronicSpedraof PolyatornicMoleculos." D. Van Nostrand Co. Ine.. Princeton. 1966,Appendir 11: Ref. 113J.p.833. 116) McWeeny, R.."Symmetn--An Introduction toGmupTheary,"PorgsrnonPress. New York, 1963.p. 197. ~
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