Solid StateCommunications,Vol.11, pp. 601—603, 1972.PergamonPress. Printed in GreatBritain

A VARIATIONAL JAHN—TELLER GROUNDSTATE: COMPARISON WITH EXACT RESULTS

J.R. Fletcher

Departmentof Physics,University of Nottingham,NG7 2RD

(Received11 April 1972 by R.A. Cowley)

A variational method recently proposedfor the Jahn—Tellerinter-action in a lattice is applied to the clustermodel. Comparisonofthe resulting energy levels andmatrix elementswith the knownexact result gives good agreementandsupportsthe useof themethod for the interacting lattice.

INTRODUCTION

IN A RECENT paper1a variational groundstate or in phononoperatorswasproposedfor the dynamicinteraction of an H = hw(c* c÷d~d+ 1) +- G11.(c -~-d) + T_ (c~± a))E symmetrydefectwith thevibrations of a cubic whereharmoniccrystal lattice. This calculation predictsrotational energy levels for the systemwhich G = ~ [T

1 ~ I = iT3correspondin the strongcoupling limit to those

= (4mha.)~[mw(Q3+ iQ2) — i(P3 + iP2 ))predictedby the simpler cluster model. As noexactcalculationshavebeenmadeof these d~= (4mhw)~[mwXQ3— iQ2) — i(P3 — iP2 )]energylevels for the J ahn—Tellerinteraction ina lattice, no direct checkon the variational The states T3 = ±~describethe two states

statesis possible,but the samevariational of the defect. The Hamiltonian andthe commutationmethodis also applicableto the cluster model rules are invariant underthe symmetry operationin which the defect interacts with two vibrational R~modesonly. Longuet-Higginset at.

2 andChild R~c~= ~ R~d~= e~’~dand Longuet-Higgins3havegiven exactnumericalevaluationof energylevels andperturbation Rc~,T÷= e”~T~,R~T

3=

matrix elementsfor thecluster model, andso acomparisonwith thesevalueswill give a useful The variational groundstateis constructedcheckon the accuracyof the variational method. by operatingon the two-dimensionaloscillator

grouridstate 0,0> with the translationoperator

VARIATIONAL METHOD FOR THE CLUSTER exp p(c~±d~- c - d) = exp - 2i p ~ (mhw)~

The cluster model considersthe interactions This vibrational state is combinedwith aof two vibrational modes,with coordinatesQ stateof the fictitious spin and thepart trans-and Q3 andmomentaP2 andP3 with a two-fold forming irreducibly underthe rotation R~isdegeneratedefect. The Hamiltonian in the projected out giving

harmonicapproximation is

rm> = ~JeiMøR~texp p(c~±d~_c_d)l0,0>

2mH = I (P ÷P~)÷~mw~(Q2 ~ Q2~ 1~os3÷sin~ ÷~>_ ~cos~~sin~)- ~

÷2e(TQ3÷T2 Q2) 2

601

602 A VARIATIONAL JAHN—TELLER GROIJNDSTATE: Vol. 11, No. 5

For details seeFletcher (1972). The energy For ~! = ~ this wavefunctionshows some

of the state is minimised by varying the para- similarity to that proposedby Englrnanandmeters P and 6, and in contrastwith the lattice Horn.4calculation, 0 G/2hoi which is not infinitesimal,so that all the terms of the exponentialexpansion The expectationvalue of theHamiltonianmust be included, for this state is evaluatedby pairing the phonon

creationand annihilation operatorsgivingexpP(c~e~± de~— ce~ — de~~)0,0>

= expP e~~(c~— a) exppe~(d~— c) 0,0> <H>~1 =

=exp(~P~1~ ~ 1÷~O V=D ~ P9(c÷s)

2~1.~~_312± }÷(c—s)

2{1~, ±J%f~3/3L(c ÷s)2 ‘11- ÷(c — s)2 1~,_

c~t p2V~fl(c±)V~(d~)V 0,0> -

-2kpcos ~V! (v ÷n)!

where the argumentof all themodified BesselThe rotation of the spin statesis given by

R~T3 > = exp ( — iT3) T3> andthus the ~ functions is 2,O

2and c= cos 6/2, S = sin 6/2.integration picks out termswith ri = ~. For This energy hasbeenminimised numerically for

M> 0 the variational stateis = and3/2 anda rangeof valuesof k(= G/hw

8) by choosingvaluesof 6 andp. The

F~> =

4exp( — p2)~cos~ ÷sin 6) + valuesof k were chosenequal to thoseused by2 2 Longuet-Higgins et at. so that the groundstate

energy <H>~andthe excitation of the first levelo2V÷W- A = <H>

372 — <H>! could be comparedwith theirv=Q exactvalues, (given in Table 1).

(c~)~1~(d~ 0,0>

V! (V ÷~i— ~)! Thus the maximum discrepanciesoccuraround k2 = 3, the maximum error in the ground-

— (cos ~ _sin~\ — ~> stateenergy being0.4 per centof the vibrational2 2)

quantum,and 1 per cent in the excitation energyp2V1~M~~i(c+)~M~(d~)1’I0,0> A. Both errors approachzero for strong or weak

V! (V ± ÷ 1)! coupling.

where A is a normalisationconstant. This state A more rigoroustest of the form of this

can also be written in the coordinaterepresentation wavefunction is obtainedby evaluatingthewith 03 = Rcos 0, Q

2 = Rsin 0 reduction factor, p for 112 symmetry perturbationsof the degenerategroundstatesand comparing

> = Nexp — y(R

2 + R~os~ this with the valuescalculatedby Child and

2 Longuet-Higgins. (Table 2).

÷sin ~)1M~i(2yRRo)eM~ ± k> = T3 = (10 — 1~) ÷sin 6(1~÷I~)

(!~+ 1~) ÷sin ô(I~— I)

— (cos4_sin ~)J 2yRR0)e~M+i~ ~> The argumentof the mothEedBessel

functions, 2P2andthe angle 6 are determined

where~, = ~,/(mo)/2h, ~ is a modified Bessel from the energyminimisation.

function and R~ is a variational parameterequalto The agreementis good for weakor moderate

~ p2 coupling but in the limit of strong coupling,

althoughboth valuesof p approachzero it hasbeenshownthat the variational values aretwice

Vol. 11, No. 5 A VARIATIONAL JAHN—TELLER GROUNDSTATE: 603

Table 1.

K2 0.25 0.5 0.75 1.0 2.0 3.0 5.0 7.0 10.0 15.0 20.0 30.0

Exact E~0.774 0.578 0.400 0.233 —0.369 —0.919 —1.961 —2.976 —4.485 —6.991 —9.493 —14.496A 0.834 0.738 0.667 0.611 0.455 0.356 0.235 — 0.115 0.074 0.054 0.035

Variational E~0.774 0.579 0.401 0.236 —0.365 —0.915 —1.959 —2.975 —4.484 —6.990 —9.493 —14.495A 0.835 0.740 0.672 0.616 0.464 0.366 0.244 0.176 0.119 0.076 0.055 0.035

Table 2.

k2 0.25 0.5 0.75 1 2 5

Exact p 0.663 0.486 0.374 0.297 0.137 0.023Variational p 0.664 0.493 0.386 0.312 0.156 0.032

the exactvalues. The reduction factor q, for E CONCLUSION

symmetryperturbationsis relatedto p by the Comparisonof the variational calculationexactequation2q = 1 ±p. The limiting values of energy levels with the exactresults for theof q for strongandweakcoupling are 1 and ~ cluster model shows excellentagreementover the

respectivelyandthesevalues aregiven correctly whole rangeof interactionstrength. The matrixby the variational method. The maximum elementsof E symmetryperturbationscalculateddiscrepancyin q is 0.01 and occurs for moderate from the variational statesagreeto within aboutinteraction strength, k2—~2. Generallythe 1 per centwith exact values. It follows that thereductionfactor q is experimentallymore variational methodgives a good descriptionofsignificant than~ the low energystatesof theJahn—Tellercluster

model, which supportsits use for the Jahn—Tellerinteraction in alattice.

REFERENCES

1. FLETCHER J.R., J. Phys. C. 5, 852 (1972).

2. LONGUET-HIGGINSH.C., OPIK U., PRYCE M.H. andSACK R.A., Proc. R. Soc. A244, 1(1958).

3. CHILD M.S. andLONGUET-HIGGINSH.C., Phil. Trans. R. Soc. A254, 259 (1961).

4. ENGLMAN R. andHORN D., ParainagneticResonance,Vol. I, 329 (1963).

Eine kürzlich für die Jahn—TellerWechselwirkungin einemGittervorgeschlageneVariationsmethodeist an einemGruppenrnodellverwandtworden. Der Vergleich der resultierendenEnergiestufenurid Matrixelementebringt eine guteUbereinstimmungmit den genaubekanntenErgebnissenund befOrwortetdie Anwendungder Methodefür dasWechselwirkungsgitter.