Donald W. Rogers The Brooklyn Center Long Island University Brooklyn, New York 11201
sylnllletry and Degeneracy The particle in a "squeezed box"
Recently, Schreiber and Spencer described the .pedagogical nse of the particle in a threedimensional box to demonstrate the directional and nodal properties of orbitals and to give a better feeling to the student for the meaning of probability density contour diagrams (1). We would like to suggest some ways in which superimposition of an external force field (perturbation) on the simple system of the particle in a cubic box can lead to chemically results . significant (2).
The particle in a stretched or rectangular box bears the same analogy to u and rr molecular orbitals that the particle in a cubic box bears to atomic s and p orbitals. These orbitals have been discussed by Schreiber and Spencer ( I ) . Reduction of symmetry of a central force field by an external field reduces degeneracy of orbitals within the central field. Multiplet splitting of p, d, etc., orbitals due to the imposition of an external magnetic field (Zeeman effect)or electrical field (Stark effect)is a wellknown example of splitting due to external perturbation. The concept of degeneracy decrease and energylevel splitting in an unsymmetrical field makes a useful introduction to nmr and esr spectral splitting. The particle in a cubic potential well is constrained by a force field which has high symmetry and leads to the energy
where the lowest energy solution, 3hP/8mahas all three quantum numbers n,, n,, and n, equal to one. We shall call this the (1,1,1) state in accordance with the usual convention. The mass of the particle is m, h is Planck's constant, and a is the length, in cm, of one edge of the cube. The second highest energy level has three-fold degeneracy, the (2,1,1) solution having the same energy as the (1,2,1) and the (1,1,2) s$utions. In particular, consider a cubic box 1.0 A on an edge containing an electron. Three-fold degeneracy exists and the total energy of the system is
for the (2,1,1) solution and those degenerate with it. If the potential well is a rectangular parallelopiped, a, b, and c, its dimensions in the z, y, and z directions are not the same and
causing degeneracy to disappear. If an external force field distorts the 1-A cube previously referred to so that it is squeezed in the z dimen-
Figure 1.
Perlurbolion of
0
cubic potentid well b y on e r h r n d force
Reld.
sion by 0.1 but is expanded in the z and y dimensions by 0.1 A as shown in Figure 1, the degeneracy of the system is reduced. The orbital with n, = 2 is constrained by the external force field and its energy increases. Constraint by the potential well in the x and y dimensions of the squeezed box has been relieved, causing the remaining two orbitals, the (2,1,1) and (1,2,1) orbitals to be reduced in energy. A kind of "field splitting" has taken place between the high energy orbital and the two low energy orbitals which remain degenerate with each other. Numerical calculations using the new dimensions of the box show the high energy orbital to have E = 3.970 X 10-lo erg/ electron while the low lying levels have E = 3.232 X 10-'0 erg/electron leading to a splitting of 0.738 X erg/electron. Analogy may then be drawn to the ligand field splittin= of the five d orbitals brought about bv su~erimuosition of octahedrally oriented ligands about a cehtral metal ion (5). The daZand d,.,. orbitals are more severely constrained by juxtaposed ligand fields than the remaining three d orbitals. Field symmetry leads to the familiar ligand field splitting. Repeating this calculation for a box which has been squeezed more or less than 0.1 A demonstrates the dependence of magnitude of splitting on the magnitude of distortion and leads to a discussion of field strength, high-spin and low-spin splitting and color of transition metal ions. Substitution of ligand fields with geometries other than octahedral leads to diierent splitting patterns (4) but always to loss of degeneracy as compared to the uncomplexed ion. Slight distortion of an octahedral field reduces the symmetry of an octahedral complex, further decreasing degeneracy and causing spectral splitting known as Jahn-Teller splitting. In 1937, Jahn and Teller proved a rather remarkable theorem which states thst any nonlinear molecular system in a degenerate electronic state will be unstable and will undergo some kind of
Volume 49, Number 7, July 1972
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the system with three electrons, one in each energy level. The distorted box with three electrons has a total energy of 10.434 X 1 0 - l o erg, lower than the system with cubic symmetry. Energy reduction of this kind provides the driving force for Jahn-Teller distortion and the lowest energy state determines what kind and how much Jahn-Teller distortion takes place (Fig. 2). Acknowledgment
0
0.2
0.1 0
The author wishes to acknowledge the sabbatical leave granted by The Brooklyn Center of Long Island University and the valued communication of Mr. F. J. McLafferty during the completion of this work.
Distortion, A Figure 2. Energy of the higher and lower levalr of on electron in a squeezed box or a function of deformation.
distortion which will lower its symmetry and split the degenerate state (6).
We have seen that the total energy of the cubic box is 3.614 X 1 0 - l o erg/electron or 10.852 X 1 0 - l o erg for
502 / lournof of Chemicol Education
Literature Cited (1) S c x ~ m e ~H. n . D..ANDSPENCER, J. N..J . CHEM.~onc.,48, 185 (1971). (2) Abstraoted in part from Ro(isls, D. W.. "Theory and Problems in Ph~siealChemistry,"College Notes I n e . New York. 1971. (3) MOORE.W. J.. "Physied Chemistry," Prentioe Hall he., Englewood Cliffs,N.S., 1962,pp. 547-9. (4) COMP*NION, A. L., "Chemical Bonding," MoGraw-Hill Rook Co.. New York. 1964, pp. 1 2 6 8 . F. A,. AND WILKINBON.0.. "Advhneed Inorganic Chemistry." (5) COTTON, J . WileyBSons, Ino., NewYork, 1962.p.582.
