Mihai V. Putz: Undergraduate Structural Physical Chemistry Course, Lecture 6 1

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1 Mihai V. Putz: Udergraduate Structural Physical Chemistry Course, Lecture 6 Lecture 6: Quatum-Classical Correspodece I. Bohr s Correspodece Priciple Turig back to Bohr atomic descriptio it provides the requecy o emited waves (photos) whe the trasitio betwee two states i a atom takes place: E E me Z h 8ε h ( )( + ) >, Now, the mai questio is whe this trasitio requecy is becomig evetually equal with the requecies associated with the orbital circular motio o states ad, idividually. For respodig i tha oe may otice that the Bohr quatiicatio supports the classical couterpart picture o electroic circular movemet at optimum distace aroud the ucleus, with the agular velocity ϕ& ad the revolutio requecy, liked by the equatio: rom where we have: T m v opt, opt, m & opt, opt, ( r ϕ ) m r ( π ) e πh πε m Z opt, ( ) h h T Now, the relatio betwee the quatum trasitio requecy ad the classical oes associate to the quatum states ad, ad, ca be clariied i two limitig cases. Whe is about irst eighbor high levels, i.e. Δ &, >>, we have i asymptotical sese that leadig with classical-quatum equivalecy Whe is about o-irst eighbor but still high levels, i.e. Δ > &, >>, so that >> Δ we get the classical-quatum coectio as: Δ Δ Δ

2 Mihai V. Putz: Udergraduate Structural Physical Chemistry Course, Lecture 6 Thereore, the rule is that as much the quatum levels are higher as the quatum ad classical requecies approaches each other, establishig the so called Bohr corespodece priciple betwee the quatum ad classical worlds. A eve more strikig ad practical orm o Bohr correspodece priciple may be uold sice we itroduce the couter o the quatum trasitio states as: Δ I hδ that combied with Bohr quatum trasitio priciple h Δ provides the quatum requecy uder the orm Δ Δ ΔI to be compared with the classical requecy o the state with E, recogized to ca be writte as de h d de di Δ Δ Leadig with the idea tha both requecies aproach each other whe the slopes o the secat o spectrum lies equals the slopes o the tagets o the iitail ad ial poits o the graph E E I ), (,ϕ beig here I, ϕ recogized as the phase itegral I p dϕ h, ϕ ϕ sice providig the agular mometum quatiicatio, p h m & ϕ, ϕ r opt, i accord with above kietic eergy quatiicatio where the classical requecy was rooted. Note that the accompaig radial itegral I, r prdr h gives othig less tha the startig de Broglie-Heiseberg-Bohr quatiicatio. Also ote that while Bohr model cosiders agular ad radial itegrals beig quatiied by the same quatum umber, urther discrimiatio betwee them opes the way to treat D-eliptic Sommereld orbital descriptio. However, while

3 Mihai V. Putz: Udergraduate Structural Physical Chemistry Course, Lecture 6 the last approach is still ot the geeral oe we preer to directly treat the D-case, however i detail i the ext volume o the series. Table.I. Check o the correspodece priciple or asymptotic Bohr s hydroge atom levels. Quatum states Orbit requecy [s - ] Trasitio Iitial Fial Iitial Fial Frequecy [s - ] The Table.I urther ilustrates the Bohr quatum-classical correspodece priciple. II. Moseley Law ad Spectral Atomic Periodicity Atomic Bohr s spectra are based rom the trasitio eergies give beore; however, we may idetiy the so called spectral term Z allowig to rewritee the spectral trasitio i terms o the wave-umber (the so called ayleight-itz priciple): ~ λ hc Still, or a may-electroic atom, the hydrogeic Bohr treatmet ca be still preserved with the price o itroducig the so called shieldig costat σ * ( Z σ ) that evetually depeds o the shell s quatum umber (ad o its urther sub-shell s reiemet). The last relatioship may be urther trasormed to get the liear Z-depedecy: See H. E. White, Itroductio to Atomic Spectra, McGraw-Hill, Ic., New York, 9, pp.

4 Mihai V. Putz: Udergraduate Structural Physical Chemistry Course, Lecture 6 * ( Z σ ) tellig that the spectral terms or a give shell are proportioal with the iverse o the quatum umber o that shell. This has udemetal pheomeological iterpretatio or periods ad groups o periodic Table: paractically, or the the dieret periods dow groups it displays the icreasig so the dimiishig agle o the itted lies amog the K, L, M, etc. trasitios; wherereas withi periods, as the Z icreases the squared o the spectral term to the ydberg costat cotributio icreases. Yet, or practical use the requecy is to be employed, rather tha idividual spectral terms; i this case oe irstly has i geerally that: c c λ ( ) ( Z σ ) ( ) Z σ c while i assumig the same iitial ad ial shieldig costat, trasitio requecy is obtaied as the geeralized Moseley law: σ σ σ, the simpliied squared root ( ) Z σ c. As a specializatio, or istace, or K lies (, ) the shieldig costats ca be urther assumed as σ so that the Moseley workig ormula is obtaied: Kα ( Z ) 5 c Z ( ).8 (Hz) Note that the Moseley rule is less exact comparig with the spectral terms iterpretatio o periodic spectra o elemets through assumig the same shieldig costats betwee the paired levels cosidered. Still, or Kα lies it behaves i air agreemet with experimet see Table.II though comparig with theoretical yield:.6[ ev ] ( Z ).( Z ) [ev]. h Kα Fially, ote that the dierece betwee the spectral term ad requecy pictures o the Moseley law is the same as that betwee the orbital motio requecies ad the trasitio requecies betwee two (Bohr) levels. Moreover, the Moseley law may be regarded also as provided the atomic umber Z i terms o structural quatum iormatio icludig spectral terms, eergies, shieldig costats; i other words, atomic Z may be see as a measure o such ier quatum structure merely as the give costat or a atom.

5 Mihai V. Putz: Udergraduate Structural Physical Chemistry Course, Lecture 6 5 Table.II Experimetal Kα (or KL: trasitio rom the level L with, l ad j/ to the level K with ad l) X-ray eergies, i ev, with experimetal ucertaity i parethesis, as compared with computed oes rom the Moseley law, or the third ad ourth periodic groups o elemets. Z Elemet Kα [ev] Experimetal(uc.) Computed Na.98() Mg 5.7(). Al 86.95() 68.8 Si 79.9() P.7(8) * S 6.7(8) 95 7 Cl 6.86(9) 6. 8 Ar (6) K.956(6).8 Ca 688.8(9) 68. Sc (85) 8. Ti 5.9(9) 98. V 9.67(59) 96.8 Cr 55.58(7) M (8) Fe 69.6(99) Co (9) Ni 76.(5) Cu 87.86(6) Z 865.8(7) Ga 9.85(7) 98. Ge 9855.() 98. As 57.5(5).8 Se 8.5() Br () Kr 595.(56) 95. * Iterpolated rom earby elemets. Experimetal data rom