Roger L. DeKock' I The Symmetry of Atomic States and

~lb ionJ . Kromminga2 and Timothy S. Zwier3

Calvin Colleqe I Atomic Orbitals

Grand Rapids. MI 49566 I In aysfalline and molecolar environments

The use of the valence bond, molecular orbital, and ligand field descriptions of chemical bonding has been aided greatly by the application of group theoretical principles (1,2). In any molecule that has an atom located at the center of the point group it is necessary first to calculate the character x of a re- ducible representation in spherical symmetry. This reducible representation arises from the atomic states or atomic orbitals on the central atom. The component irreducible representa- tions generated by this reducible representation are then obtained by standard group theoretical methods.

The purpose of this correspondence is to delineate the five iormulnr &at are required 10 ralrultue the character x of a reducihle representation in spherical symmetry. These are

x(i) = *(Zj + 1) ( 5 )

In these formulas E, C, S, u, and i refer to the identity, proper rotation, improper rotation, reflection, and inversion opera- tions, respectively. The angle of rotation is 6. The factor f is +1 for a gerade state and -1 for an ungerade state. The symbol j refers to the angular momentum quantum number of the state under consideration. In the Russell-Saunders scheme, j can he replaced by L when considering the spatial wave function and by S when considering the spin wave function. where L and S refer to the total orbital and spin angular momentllm quantum numhers, respecti\,ely.

llcruations r l ) and (21 an. found in mans texthnoks relatma the application of group theory to physics and chemistry ( ~ ~ 3 ) . Equations (31, (4), and (5) are easily derived from eqn. (1) (see Appendix).

As an example of the application of these equations, con- sider a 2Dp electronic state Cj = L = 2) in D3h symmetry. The characters for the reducible representation r are then

The representation r can be reduced by the standard for- mula

In this equation a, denotesthe number of times the ith irre- ducible representation occurs in the reducible representation r; h is the order of the group; p is the number of classes in the group; gp refers to the g operations in the pth class and Rp refers to any operation R in the p th class. The characters x*i(R,) and x(R) refer to the character of the ith irreducible representation and the reducible representation, respectively.

Presented at the 174th ACS National Meeting, August 2LLSep- tember 2, 1917 in Chicago, Illinois.

'Author to whom correspondence should be addressed at the De- partment of Chemistry.

'Department of Physics. Vtesent address: Department of Chemistry, University of Colo-

rado, Boulder, Colorado 80309.

We then obtain

D, - A',, E', E"

Equations (1)-(5) also can he applied to the determination of the symmetry property of the spin function. Then by taking the direct product of the spin and orbital symmetry, we can determine the total symmetry of the wave function or atomic state (4).

To determine the symmetry of the spin function for the 2D, electronic state in D3h symmetry we recognize that the total spin angular momentum quantum number S is 112 and therefore] = 112. The term f is +I since electron spin func- tions have intrinsic gerade symmetry. By applying eqns. (1)-(6) with j = 112 we find that the spin function transforms as Ellz in D3h symmetry. The final results are

Orbital symmetry: D, -A',, E', E" Spin symmetry: S L El/; Total symmetry: A', X Ella- Elir

E' X El12 - Ewz. ESIZ E" X El12 - Em. E m

Consequently, we see that upon inclusion of spin-orbit cou- pling the 2D, electronic state splits into five electronic states in symmetry

'D, -Em(2). E312(2), E u ~

This compares to only three electronic states in the absence of spin-orbit coupling PA'1, 2E', 2E). This situation is anal- ogous to the free atom case which splits in the presence of Russell-Saunders coupling:

'D, - 2D5~z,2 2D3~~ ,

Appendix

Equations (3) and (4) can he derived from eqn. (1) and some simple group theoretical considerations. We recognize first that a one-electron function @,,r,,, is an eigenfuuctiou of the inversion operator i with eigenvalue (-1)'.

One-electron functions with odd 1 change their sign and are called un~erade whereas functions with even 1 transform into themselv& and are calledgerade. Similarly, for a wave func- tion 'P describing a multi-electron electronic state we ob- tain

i4' = ( - l ) 'k4 ' k

where the product is over the k electrons in the electronic state (6). The factor &(-l)'h is +1 for agerade state and -1 for an ungerade state.

Next we recognize that any reflection can he represented as the product of an inversion with a rotation by a. That is

a = i . C ( s )

where the axis of rotation is perpendicular to the reflection plane. Since eqn. (1) gives the character for rotation by any angle 6, we obtain eqn. (3)

sin ti + 112)s - x(" = g (-I)'b sin s,2

- n (-1)Ih sin ti + 112)s k

510 / Journal of ChemicaiEducation

Notice that

X(U) = +1 for S,, P., Dg, Fur Gg,. . . x(o) = -1 for S., P,, D,, F,, G., . . .

~ ( o ) = 0 for half-integer j

that a net of one such reflection will have taken place. Hence in the D a b point group we can apply eqn. (4) for any rotation Snm where 4 = 2sln and m is odd. When m is even the oper- ation does not correspond to an improper rotation since the net number of reflections is zero. For example, S$ = C32. S34

= Cn', S n 6 = E. For odd m such as S x 3 - oh, ems. (3) and (4) Next we derive the equation which denotes the character give the same result.

for an improper rotation by angle 4. Recall Literature Cited

S(b) = o. C($) = i . C(r). c(4) = i . c(b + r) I11 (a1 Liehr,A.D.,J.CHEM.EDUC.,39,13511962):(b)Cotton,F.A.,J.CHEM.EDUC., 11,466 11964): (cl 0rchin.M.andJaffC.J. CHEM. EDUC.. 53,554 119761.

(21 Cotton, F. A , '"Chemical Applications of GroupTheory." 2nd ed.. Wiley-lnteraeienee. where the rotations by ?r and 4 occur about the same axis, N ~ V YWL, 1971.

namely an axis which is perpendicular to the reflection plane 1" I d Hammermerh, M., "Gmup Theory;'Addisun-Wesley Publ. Cu., lnc., Reading, MA. U.S.A.. 1962. p. 316: (bl Msrgenau, H., and Murphy, C. M.,"The Mathematics ol

u. We then readily obtain eqn. (4). From eqn. (4) we can obtain physics and chemistry: 2nd ed., D. Van strand CO., I""., ~ e w YO& 1956, p. 565.

the character for an inversion since S2 (1800) = (I1 Horzhorg,G.,"MolecularSpeetrssnd MolaeularStruetureIII.Ele~tronieSpctraand i. By applying 1'Hospital's rule twice we obtain eqn. (5). medronic structure 01 ~ ~ i ~ ~ t ~ ~ i ~ M O I ~ ~ U I ~ S , - vsn ~ ~ ~ t ~ ~ ~ d ~ ~ i ~ h d d CO.. NW An improper rotation S(4) corresponds to rotation by an Y O ~ X . 1966. p. 337.

161 Reference 2, p. 289. 6 reflection in a plane perpendicular to the I61 Tinkham, M.."GruupTheon.and Quantum Meehsnia."McGiau-Hill. NeaYork. 1W. axis of rotation. It is important to notice that eqn. (4) assumes p. 8s.

Volume 56, Number 8, August 1979 / 51 1