Analogy Between Particle in a Box and Jahn Teller Effect

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Anlogy Between Prticle in Box nd Jhn Teller Effect MNMurty MNMurtyis Professor in Physics t Ntionl Institute of Science nd Technology, Plur Hills, Berhmpur, Odish.

1 Anlogy Between Prticle in Box nd Jhn Teller Effect MNMurty MNMurtyis Professor in Physics t Ntionl Institute of Science nd Technology, Plur Hills, Berhmpur, Odish. The energy levels of prticle in box re degenerte. The degenercy of n energy level is reduced or removed on slight distortion in the dimensions of the box without chnging its volume. This phenomenonisnlogous to the Jhn{Teller E ect, which sttes tht in n electroniclly degenerte stte (i.e., more thn one degenerte orbitl is vilble for n electron), nonliner molecule undergoes distortion to remove the degenercy by lowering the symmetry nd thus lowering the energy. Prticle in Cubicl Box Aprticle is enclosed inside rectngulr boxhving edges, b nd c in length (Figure 1). The potentil U (x; y; z) of the prticle is de ned by 9 U(x; y; z) = 0 for 0 x ;0 y b; >= 0 z c (1) >; = 1; otherwise: Figure 1. Prticle moving freely in rectngulr box. Solving the SchrÄodinger eqution, the llowed energy levels of the prticle re given by Ã! E nxnyn z = h2 n 2 x 8m + n2 y 2 b + n2 z ; (2) 2 c 2 where h =Plnck's constnt = 6: Js, m = mss of the prticle, nd quntum numbers n x ;n y ;n z = 1; 2; 3; 4;:::. Keywords Degenercy, distrortion, Jhn Teller effect. For cubiclbox, = b = c nd energy cn be expressed s E nxnyn z = h2 ³ n 2 8m 2 x + n 2 y + n2 z : (3) 646 RESONANCE July 2013

2 Eqution (3) shows tht only certin vlues of energy E my occur, i.e., the energy is quntized. It cn be seen tht di erent sets of quntum numbers n x ;n y nd n z hve exctly the sme energy. This sitution is known s degenercy, nd the energy levels re sid to be degenerte. Degenercy occurs in generl whenever system is lbelled by two or more quntum numbers nd di erent combintions of quntum numbers give the sme vlue of the energy. In cse of tomic physics, the degenercy is mjor contributor to the structure nd properties of toms. Figure 2 shows the energy levels for prticle in cubicl box. These energy levels re degenerte to some degree. For convenience, let E 0 = h2. Then the only 8m 2 llowed energies for the prticle re 3E 0 ; 6E 0 ; 9E 0 ;:::; etc., nd ll intermedite vlues re forbidden. The lowest energy stte is known stheground stte nd the sttes with higher energies re known s excited sttes. The ground stte energy E 111 is obtined from (3) by putting n x = n y = n z =1: Degenercy occurs in generl whenever system is lbelled by two or more quntum numbers nd different combintions of quntum numbers give the sme vlue of the energy. E 111 = 3h2 8m 2 =3E 0 : (4) Figure 2. Energy levels nd quntum numbers of prticle in cubicl box. RESONANCE July

3 The wve functions corresponding to three sets of quntum numbers of the triply degenerte second energy level re different. There is only one set of quntum numbers tht gives this energy stte nd hence the ground stte is sid to be non-degenerte. If we now consider the second energy stte, it is seen tht therere three sets (211), (121) nd (112) of the quntum numbers n x ; n y nd n z tht will give the sme energy level. From (3), theenergyof this level is given by E = 6h2 8m =6E 2 0 : (5) Such level is sid to be triply degenerte or three-fold degenerte nd we cn lso sy tht the degenercy of this level is 3. The wve functions corresponding to the three sets of quntum numbers of this energy level re di erent. Distortion of the Box Consider rectngulrboxof sides x; y nd z. Forsmll distortions +dx nd dy long x nd y xes respectively, keeping volume constnt, we hve (x + dx) (y dy) z = xyz or y dx x dy =0(neglecting dx dy), or x y = dx dy : For cubicl box,x = y = z nd hence dx = dy. Now we consider slight distortion of the cubicl box by +d long x-xis nd d long y-xis, keeping its volume constnt. Using (3), theground stte energy is now " E 111 = h2 1 8m ( + d) ( d) + 1 # RESONANCE July 2013

4 2Ã = h d! 2 Ã + 1 d! m 2 " = h2 1 2d # 8m d +1 = 3h2 8m 2 which is the sme s (4). Therefore the ground stte energy remins the sme due to slight distortion of the cube. Using (3), the new energy of the 2nd energy sttehving wve function Ã (2; 1; 1) is given by " E = h2 4 8m ( + d) ( d) # 2 2 Ã = h d! 2 Ã + 1 d! m 2 = h2 8m 2 " Ã 4 1 2d! Ã d! # + 1 = 6h2 8m 3h2 d: (6) 2 4m3 Similrly the new energies of the 2nd energy stte for the other two sttes Ã (1; 2; 1) nd Ã (1; 1; 2) re given by " E = h2 1 8m ( + d) ( d) # 2 = h2 8m 2 2Ã 4 1+ d! 2 +4Ã 1 d! = 6h2 8m + 3h2 d ; (7) 2 4m3 " E = h2 1 8m ( + d) ( d) # 2 The degenercy of n energy level is reduced or removed on slight distortion of system.this phenomenon is nlogous to the Jhn Teller Effect (Distortion). RESONANCE July

5 The Jhn Teller theorem sttes tht in nonliner molecule, if degenerte orbitls resymmetriclly occupied distortion will occur to remove the degenercy. The Jhn Teller effect is most often encountered in octhedrl complexes of the trnsition metls, whose ions hve incompletely filled d orbitls. = h2 8m 2 2Ã d! 2 Ã + 1 d! = 6h2 8m 2 : (8) Thus, the initil three-fold degenerte 2nd energy level given by (5), issplit on slight distortion of thecubeinto three di erent energy levels given by (6), (7) nd (8). Hence the degenercy is removed. In generl the degenergy of n energy level is reduced or removed on slight distortion of system. This phenomenon is nlogous to the Jhn{Teller e ect (distortion). Jhn{Teller E ect The Jhn{Teller e ect, sometimes lso known s Jhn{ Teller distortion, describes the geometricl distortion of non-liner molecules under certin situtions. Thise ect ws rst predictedin1937by Hermnn Arthur Jhn nd Edwrd Teller, using group theory. The Jhn{ Teller theorem sttes tht in nonliner molecule, if degenerte orbitls re symmetriclly occupied, distortion will occur to remove the degenercy. This theorem essentilly sttes tht ny non-liner molecule with degenerte electronic ground stte will undergo geometricl distortion tht removes the degenercy. An electroniclly degenerte stte represents the vilbility of more thn one degenerte orbitl for n electron. In this condition the degenerte orbitls re symmetriclly occupied nd get more energy. Therefore the system tries to get rid of this extr energy by lowering the overll symmetry of the molecule, i.e., undergoing distortion, which is otherwise known s Jhn{Teller distortion. The Jhn{Teller e ect ismost often encountered in octhedrl complexes of the trnsition metls, whose ions hve incompletely lled d orbitls. In n octhedrl 650 RESONANCE July 2013

6 crystl, the t 2g orbitls occur t lower energy thn the e g orbitls. Considerble distortions re observed when n oddnumberof electrons occupy the e g orbitls in d 9 [Cu(11), Ag(11)], low spin d 7 [Co(11), Ni(111)] nd high spin d 4 [Cr(11), Mn(111)] con gurtions in the octhedrl environment (Figure 3). Jhn{Teller distortion is signi cnt in these con gurtions due to symmetriclly occupied e g orbitls. The ground sttes of ll these complexes re doubly degenerte. The degenercy of the orbitls is removed by lowering the symmetry of the molecule either by elongtion of bonds long the z-xis (z-out distortion) or by shortening of the bonds long the z-xis (z-in distortion). Thus nocthedrlly symmetricl molecule is distorted to tetrgonl geometry. In cse of z-out distortion, the energies of d-orbitls with z fctor re lowered. The energies of orbitls with z fctor re incresed incse of z-in distortion (Figure 4). Usully the octhedrl d 2 ; d 4 high spin, d 7 low spin, d 8 low spin nd d 9 con gurtions show the z-out distortion. The octhedrl d 1 con gurtion like Ti (111) in [Ti (H 2 O) 6 ] 3+ shows z-in distortion. Figure 3 (left). Configurtion with considerble Jhn Teller distortion. Figure 4 (right). z-out nd z-in distortions in octhedrl complexes. RESONANCE July

7 Jhn{Teller Distortion in [Cu(OH 2 ) 6 ] 2+ Ion The Jhn{Teller e ect is responsible for the tetrgonl distortion of the hexcqucopper(11) complex ion, [Cu(OH 2 ) 6 ] 2+. TheCu(11)ioninthecqueous medium is surrounded by six wter molecules in tetrgonl geometry { four of these re t thecorners of squre plne nd t shorter distnces with stronger interctions, wheres the remining two re wekly intercting with the metl ion t distnt xil positions s shown in Figure 5. The two xil Cu-O distnces re 238pm, wheresthe four equtorilcu-odistnces re» 195pm. The d 9 electronic con gurtion of this ion gives three electrons in e g orbitls nd the two wys of lling the e g level gives doubly-degenerte electronic ground stte (Figure 6). The distortion occurs long one of the moleculr four-fold xes (lwys lbelled s z-xis). This distortion normlly tkes the form of elongting the bonds to the lignds lying long the z-xis, but occsionlly OH 2 H 2 O OH 2 Cu H 2 O OH 2 OH 2 Figure 5 (top). Structure of [Cu(OH 2 ) 6 ] 2+ ion. Figure 6 (bottom). Two wys of filling e g level of [Cu(OH 2 ) 6 ] 2+ ion. 652 RESONANCE July 2013

8 Figure 7. Removl of degenercy in Cu(11) ion. occurs s shortening of these bonds insted. The removl of degenercy or J{T distortion is shown in Figure 7. Dynmic Jhn{Teller Distortion In some molecules, the distortion is negligible. However the distortion cn beseenbyfreezing the molecule t lowertempertures. This condition is known s dynmic Jhn{Teller distortion. The complexes of the type M 2 PbCu(NO 2 ) 6 show dynmic J{T distortion. Here, M =K,Rb,Cs,Tl. Suggested Reding [1] A K Chndr, Introductory Quntum Chemistry, 4th Ed., Tt McGrw- Hill Publishing Compny Limited, New Delhi, [2] Kenneth S Krne, Modern Physics, 2nd Ed., Weley Indi Pvt. Ltd., New Delhi, [3] K Veer Reddy, Symmetry nd Spectroscopy of Molecules, 2nd Ed., New Age Interntionl Publishers Pvt. Ltd., Kolkt, Address for Correspondence M N Murty Gundlvri Street Ner Hnumn Bzr Berhmpur Odish, Indi. Emil: mnrynmurty@rediffmil.com RESONANCE July

Energy Bands Energy Bands and Band Gap. Phys463.nb Phenomenon

Energy Bands Energy Bands and Band Gap. Phys463.nb Phenomenon Phys463.nb 49 7 Energy Bnds Ref: textbook, Chpter 7 Q: Why re there insultors nd conductors? Q: Wht will hppen when n electron moves in crystl? In the previous chpter, we discussed free electron gses,