A Simple Quantum Me the Jahn-Teller Effect Peter Senn
ETH-Zentrum, CHN H32, 8092 Zurich, Switzerland When a set of noninteracting particles is put in a two-dimensional rectangular box with idmite walls, according to the exclusion principle for fermions, total energies arise that are lowest for a square box if the corresponding quantum mechanical state is nondegenerate. In accordance with the Jahn-Teller theorem, states that are degenerate for a square well decrease in energy if the well is changed from square to rectangular when the change of shape is assumed to be subject to a constraint. The area of the well was kept constant.
The Parameters a ~ a n d p ~ f o Different r Numbers (N) of "Fermions"
The Energy Levels
The energy levels of a particle on a plane confined to a rectangular box with infinite walls and with sides a and b are described by the following equation ( I ) :
Let us assume that the lengths of the sides of the rectangle can be changed. However, the changes are subject to the constraint that the area A of the rectangle remains constant. Since A = ab we can write the following equation for the energy:
Tne parameten m a n o I ) ~ t o r o t l e r e nn-mbers l [M of Iermons'pacea in the rectangd ar oox accord ng to the e x c l ~ s a npnncpe . Tnese fwo parame. ters detme the total e n e m F & wnere c h = O A Y , IOM,. Tne %ant iv.trn7,M . . . . oenotes the ratio &for whTih the total energy eNis a rn'inimum When ymi,(N) = 1 the equilibrium geometry is a square.
At y = ymi. we get the following: Minimizing the Total Energy
The formulas for y- and the corresponding value of the minimum energy can readily be obtained from eq 4 by noting that the first derivative of EN with respect to y vanishes at y = ymin. When
where
Without loss of generality, we can take atb
such that
the square that represents the energetically most favorable shape will be described by the following equation:
ytl
Let us assume that the energy levels in the rectangular well can be filled with fermions. In other words, we assume that each energy level can hold at most two particles. Let Ej$ denote the total energy obtained as a sum of oneparticle energies E,,, by placing N 'fermions" into the lowest energy levels according to the above "exclusion principle". Then EN denotes the total energy expressed in units of h2/8mAas follows:
When
Y- > l a rectangular well is more favorable. When y=( 8 1 3 ) ~
the energies Elz and Eta cross where E12 is lower for small values of y. When y > (813)"
E13is the smaller of the two. The dimensionless quantity EN is then a function of the ratio y of the sides a and b only.
In the table, the parameters a and P are both tabulated for N S 6. The quantity EN goes through a minimum at
Counting the Degeneracies
Thus, depending on the geometry, there are two different ground states for the rectangular well that is filled with five or six "fermions". For
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In Figure 1the broken lines indicate what would be obtained for N = 5 and N = 6 if the uppermost electrons are put in the energy level Ezl instead ofElz. When 3 1 are more favorable. The exact location of the minima can readily be computed with paper and pencil as indicated in the introduction. The Jahn-Teller Distortion In 1937Jahn and Teller proved a rather remarkable theorem. Essentially it states the following: Any nonlinear molecule in a degenerate ground state will undergo a distortion that lowers its symmetry, thereby splitting the degenerate state (3,4).
Figure 1. The total energy of N"fermions"in a rectangular hole as a function of the ratio of the lengths of the sides of the rectangle: ah. The arrows that point upward show the energy minima when a rectangle is energetically favored over a square. The broken lines show the total energy for N = 5 and N = 6 when the uppermost electrons are placed in the level with energy E3, instead of El,. the energy levels are at least doubly degenerate if "Z
+ n,
For example, for a square with we always have This can readily be verified with eq 2 using y = 1. There are some accidental degeneracies of higher order if y = 1.For example, Recently Hollingsworth (2)has discussed in this Journal the accidental degeneracies of a particle in a three-dimensional box. Some are relevant to the present problem. Results Figure 1depicts the results obtained for EN as a function of y for different values of N. If there are one or two fermions, then they are put in the energy level En. Then the shape with the lowest energy is the one with y = 1, which is a square. The next four electrons are put in the energy levels Elz and Ezl, which are degenerate a t y = 1. When the following relation holds: Ez1 1. The iifth and the sixth fermion are put in the energy level Elz. However, when y >(813)~
the following relation holds.
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The molecule may osrillate among two or more equivalent structures. showine a dvnamic Jahn-Teller effect. ~ l t e r n a t i v e l if-the ~, barriers separating the different equivalent geometries are too high, the molecule may be observed (for example, by X-ray diffraction) in a particular structure. Then the molecule is said to have undergone a Jahn-Teller distortion. The Jahn-Teller distortion removes the degeneracy by raising one or more of the degenerate levels, while lowering others. For linear molecules a distortion that is analogous to the Jahn-Teller distortion can be observed: distortion from a linear to a bent shape. This is called Renner-Teller distortion (5,6).An effect somewhat similar to the Jahn-Teller distortion can also be observed in crystalline solids in which the crystal lattice is undergoing so-called Peierls distortions (7). The Jahn-Teller Distortion of Cyclobutadiene Let us look at a specific example of a Jahn-Teller distortion of a molecule. Cyclobutadienein a hypothetical square geometry would be expected to have a degenerate triplet ground state. A Jahn-Teller distortion to a Du shape removes this degeneracy so that a singlet ground state is observed. Cycobutadiene is a metastable molecule that almost certainly has a D u singlet ground state (8,9). Stability in the Cavity of a Host Recently Cram et al. repoked a photochemical reaction in which cyclobutadiene is formed in the cavity of a large organic molecule (10). Inside the cavity of its host, cvclob&adiene is reported as stable at room temperature as l o w as it is not exposed to oxygen. Several accurate ah initio-calculations duggest t h a t the harrier to conversion among the equivalent D u shapes with rectangular carbon skeletons measures roughly 10 kcaVmol(8). Pseudo-Jahn-Teller Effects Jahn-Teller distortions are observed not only for degenerate states but also for near-degenerate states. Then the distortion is considered due to a Jahn-Teller effect of the second-order, called a "pseudo-Jahn-Teller" effect. Accurate ah initio calculations with electron correlations indicate that the singlet state of cyclobutadiene is lower in energy than the triplet state, even with square geometry (11).The lowest triplet state has aD4hequilibrium symmetry. Its minimum is located roughly 10 kcalimol above the maximum of the barrier to conversion among the equivalent Dzhconfigurations of the singlet state (8). This means that a pseudo-Jahn-Teller effect is responsible for the ob~ served distortion of cyclobutadienefrom DG to D z symmetry.
Figure 2. Energy-level diagrams for cyclobutadiene.The diagram in the middle is for D4hsymmetry,while the diagrams to its right and left are for kn symmetries. The arrows above the pair of vertical dashed lines that separate the energy level diagrams show the delocalized structures of cyclobutadiene. They represent the normal mode that lowers the symmetry from DahtoDZh.The distortion lowers one of the degenerate energy levels and raises the other by roughly the same amount of energy. Symmetry Avoidance Heilbronner recently discussed in this Journal the reasons underlying what he called "symmetry avoidance" of cyclobutadiene, among other molecules (12).The ratio of single to double bond lengths in cyclobutadiene (8)i s roughly 1.2. For the rectangular hole the ratio alb for N = 4 for the lowest energy is roughly 1.6. Heilbronner has pointed out that many molecules with conjugated double bonds have shapes that differfmm what would be expected from an analysis of the a bonds alone. The efects of the o bonds are obviously completely ignored in the model with the rectangular hole. The Jahn-Teller theorem applies also to excited states of molecules. Distorted Equilibrium Geometery Figure 2 shows energy level diagrams of cyclobutadiene. With any single determinantal theory Le., if electron correlations are ignored), the Jahn-Teller theorem must be used to explain why the equilibrium geometry of this molecule is distorted from a Ddhshape to a shape of lower Du, svmmetrv. The "real" oroblem with the "distorted" eauilibrium geometry ofthe cyclobutadiene molecule is somewhat reminiscent of the Dresent treatment of a "Fermi -gas" in a rectangular hole. However, we must ask just how much this simple model has in common with the real thing. What kind of interactions are responsible for the distortion in the rectangular hole? What is the justification for assuming that the change in shape is from a square to a rectangle. Why introduce the constraint that the area of the rectangle remain constant in the deformations? In the model the interactions that drive the distortion are left open. The change from a square to a redangular shape for the hole is largely arbitrary, except that it comespends to a distortion from a shape of high symmetry to a shape of lower symmetry This is predicted by the Jahn-
Teller theorem for molecules with a degenerate ground state. The constraint that the area ofthe rectanele remain constant in the deformations is motivated simgy by the desire to make the mathematical treatment of the model a s straightforward a s possible. In real molecules nonadiabatic couolines. that is. interactions between electronic and nucl& motions, ark responsible for the distortion of the equilibrium geometry of the nuclear frame. The JahnTeller theorem itself does not indicate the extent or kind of distortion that will occur in the molecule. However, any distortion can be represented by a linear combination of the set of normal modes of the molecule. Symmetry considerations will indicate that this linear combination of normal modes can contain only a limited number of the normal modes of the molecule. Thus, the change in shape of the molecule is not entirely unpredictable. Figure 2 shows a normal mode, which is indicated by ~ Du, symmetry, and arrows, that changes shape from D 4 to back. Summary The modcl introduced here illustrates some phenomenolo~lcalasoccis of the Jahn-Teller effect. Its main advant G e is that it can be analyzed quantitatively quite readily. Literature Cited 1. Brenunan, G. L. J Chcm. Educ. 1990,67,866-868. 2, Hollingslwrth,C,A,J,Chrm.Educ, 1m0,67,999, ,, Jahn,H.A.;,er, E.,cR,. . , 198,,Al,1, 4 . h h n , H . A. ~m~ a s ym . ondo don) ~ ~ S S , A I117-131. M, 5. He~zbe%G.;TeUer,E.Z. ~ h y s * o l . c h r m 1 ~ 8 8 , ~ 2 1 , 4 1 0 - ~ 6 . 6 . Renner,R.Z Physik 1984,92,172-193. 7 . ~eierls,R E ~ v a n t v m~ h e o r yofsoiids; university ~ r e s a ~ : d 1972:, pp 10% 114. Miuer,G,Aww, Ckm d,in ngbsh, 1988,27, w32, 9. , h P;DW,, T.; pettit, R. J A ~them. . SW. 1 9 ~ 991.5890-5891. , l o . Cram, D.J.; Tanner, M . E . ; mornas, R.~ n g e wc. k m . (~nternat.ed. in ~ n g l i a h ) 1901,30,1024-1027. chm 8w, i z . ~ e i i b m n n e rE. , J . c k m . E ~ WlB89,66,471478. .
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