Molecular Spectra and Molecular Structure - Lec 20

Page 1 of 8 pages Lecture #20: Second-Order Effects Last time: perturbations = accidental degeneracy Today: effects of “Remote Perturbers”. What terms must we add to the effective H so that we can represent all usual behaviors with minimum number of parameters. Use the Van Vleck transformation. Two effects to be discussed * centrifugal distortion of all zero- and first-order parameters. e.g. B→D A → AD [interaction with all v’s of same Λ-S state] * Λ-doubling and other 2nd-order parameters [interaction with all v’s of all other states] We will work with 2∏, 2∑s example Recipe * Heff in terms of E, B, A, (λ, γ), α, β * Van Vleck transformation: diagrammatically in the form of “railroads” for each location in Heff * each term in Van Vleck transformation is explicit function f (v, J ) * ∑ e′, v′ H ev, e′v′ H e′v′ E oev − E oe′v′ new 2nd order parameter e/f 2 ∏3/2 2 2 parameter ∏3/2 E v∏ + A∏ 2 + Bv∏ (y 2 − 2 ) ∏1/2 −Bv∏ (y 2 −1) 1/2 E v∏ − A∏ 2 + Bv∏ (y 2 ) ∏1/2 2 2 ∑s 2 ∑s −βsv∏ v∑ (y 2 −1) 1/2 α s + βs ⎡⎣1 m(−1)s y⎤⎦ E v∑ + Bv∑ ⎡⎣y 2 m(−1)s y⎤⎦ y ≡ J + 1/2 For simplicity we do not include γ terms (λ terms are not possible for S < 1 states). Page 2 of 8 pages What do we do with these ⎞ ⎛ m interesting H′mn ⎟ ⎜ m′ ⎟ ⎜ H= ⎟ ⎜n remote ⎟ ⎜ ⎠ ⎝ n′ follows rules for matrix multiplication H VV m, m ′ ≡ E om δ mm ′ λ2 + λ H′mm ′ + 2 1 ∑ n ⎡ H′mn H′nm ′ H′mn H′nm ′ ⎤ + o ⎢ o o o ⎥ E − E E − E ⎣ m n n ⎦ m′ ~ λ2 ∑ n H′mn H′nm′ + E om ′ − E on 2 E om We are going to write Heff in terms of zero-order parameters E, B, A perturbation parameters α, β second-order parameters D, AD, o, p, q H=H ROT +H SO ( H )2 = ( H ROT ) ( SO ( ROT + H + H ) 2 2 ×H SO ) 2 e/f dependent e/f independent q (Λ-doubling) D (centrifugal distortion of B) e/f dependent e/f independent o (Λ-doubling) λ (2nd-order spin-spin) e/f dependent e/f independent p (Λ-doubling) γ (2nd-order spin-rotation) AD (centrifugal distortion of A) Generate many 2nd-order parameters — not all are linearly independent. Page 3 of 8 pages 2 2 Let’s first work through all paths from ∏1/2, v∏ to remote state and back to ∏1/2, v∏. RAILROAD matrix element +Bv∏ v∏′ y 2 − A v∏ v∏′ y 2 2 −Bv∏ v′∏ (y 2 −1) same 2 ∏1/2, v′∏ 2 ∏3/2, v′∏ same ∑s, v′∑ same 1/2 2 α ∏1/2, v∏ s v∏ v′∑ +β s v∏ v′∑ ⎡⎣1 m(–1) y⎤⎦ s 2 2 same other states [2∆3/2, 4∆3/2, 4∏3/2, 4 ∑s3/2 ] collect terms and sum ⎛e⎞ B2vv′ (y 4 + y 2 −1)+ A 2vv′ 4 − Bvv′ A vv′y 2 = ⎜ ⎟ ∑ 2 1/2 , ∏1/2 E ov∏ − E ov∏′ ⎝ f ⎠ v∏′ H(2) 2 ∏ α + (β ) ⎡⎣1 m(−1) 2y + y ⎤⎦ + (α ( ) +∑ E −E 2 s v∏ v∑′ 2 s v∏ v∑′ s 2 o v∏ v∑′ o v∑′ Now define some 2nd-order parameters. D≡− B2v∏ v∏′ ∑ ′ v∏ AD ≡ 2 A v∏ v∏′ Bv∏ v∏′ ∑ ′ v∏ ′ v∏ E ov∏ − E ov∏′ ≠v∏ 2 A v∏ v∏′ ∑ A0 ≡ (defined so that D > 0 for v∏ = 0) o o ≠v∏ E v∏ − E v∏′ o o ≠v∏ E v∏ − E v∏′ (α ) o( ∑ )≡ ∑ E −E 2 2 s v∏ v∑′ s v∑′ (≠ v∑ ) p( ∑ )≡ 4 2 s ∑ v∑′ (≠ v∑ ) o v∏ o v∑′ α sv∏ v∑′ βsv∏ v∑′ E ov∏ − E ov∑′ (β ) q( ∑ )≡ 2 ∑ E −E 2 2 s v∏ v∑′ s v∑′ (≠ v∑ ) o v∏ o v∑′ s v∏ v∑′ ) s βsv∏ v∑′ 2 ⎡⎣1 m(−1) y⎤⎦ ∏1/2, v∏ Page 4 of 8 pages Thus H (2) 2 ∏1/2 ,2 ∏1/2 ⎛e⎞ 1 4 2 = −D y + y −1 − A D y 2 + A0 4 + o (2 ∑s ) ( ) ⎜ ⎟ 2 ⎝f ⎠ 1 1 + p (2 ∑s )⎡⎣1 m(−1)s y⎤⎦ + q (2 ∑s )⎡⎣1 m(−1)s 2y + y 2 ⎤⎦ 2 2 These same parameters appear in other locations in 2∏ Heff. Non-Lecture −Bv∏ v∏′ (y 2 −1) 1/2 2 A v∏ v′∏ 2 + Bv∏ v′∏ (y − 2 ) ∏1/2, v′∏ 2 2 ∏3/2, v∏ Thus H (2) 2 ∏ 3/2 ,2 ∏ 3/2 −β s v∏ v′∑ / (y – 2 etc. ∏3/2, v′∏ same ∑s, v′∑ same 2 2 same other states 2 ∏3/2, v∏ same ⎛e⎞ 1 1 2 s 2 4 2 2 ⎜ ⎟ = −D ⎡⎣y − 3y + 3⎤⎦ + AD (y − 2 )+ A0 4 + q ( ∑ )⎡⎣y −1⎤⎦ 2 2 ⎝f ⎠ −Bv∏ v∏′ (y 2 −1) 1/2 2 A v∏ v∏′ 2 + Bv∏ v′∏ (y − 2 ) ∏1/2, v′∏ 2 2 ∏3/2, v∏ Thus H(2) 2 ∏ 3/2 , 2 −β s v∏ v′∑ / (y – etc. ∏3/2, v′∏ same ∑s, v′∑ same 2 2 2 same other states 2 ∏3/2, v∏ same ⎛e⎞ 1/2 1/2 ⎤ 1/2 1/2 ⎤ ⎡1 2 1 1 2 ⎡ 2 2 2 2 + y −1 + y − 2 y −1 A y −1 − y −1 = +D y ( ) ( ) ( ) ( ) ( ) ⎜ ⎟ ⎣ ⎦ D⎢ ⎥⎦ ∏1/2 ⎣2 2 2 ⎝f ⎠ 2 2 1/2 2(y – 1)(y – 1) 0 1/2 1/2 ⎤ 1 1 2 s ⎡ ⎤ ⎡ + p ( ∑ )⎣− (y 2 −1) ⎦ + q (2 ∑s )⎣−⎡⎣1 m(−1)s y⎤⎦(y 2 − 1) ⎦ 2 4 3/2 1/2 1/2 1 1 = +D2 (y 2 − 1) − p (y 2 −1) − q (2 ∑s )(1 m(−1)s y )(y 2 −1) 4 2 Page 5 of 8 pages Non-Lecture Bv∑ v′∑ (y 2 m(−1)s y ) −βv∑v∏′ (y 2 −1) 2 1/2 2 ∑sv∑ α s v∑ v′∏ +β s v∑ v′∏ (1 m(–1) y) s ∑sv′∑ same ∏3/2, v′∏ same 2 ∏1/2, v′∏ same other states same 2 –s 2 ∑ , ∏1/2, 4∑–s, 4∏1/2 2 2 ∑s, v∑ H (22 ∑) s ,2 ∑s = −D∑ ⎡⎣y 4 m(−1)s 2y 3 + y 2 ⎤⎦ 1 1 + q ∑ (2 ∏)⎡⎣y 2 − 1+ (1 m(−1)s 2y + y 2 )⎤⎦ = q ∑ (2 ∏)⎡⎣2y 2 m(−1)s 2y⎤⎦ 2 2 1 + p ∑ (2 ∏)⎡⎣2 (1 m(−1)s y )⎤⎦ 4 +o ∑ (2 ∏) q∑(2∏) is exactly correlated with B∑ because it has same J-dependence. o∑(2∑) is exactly correlated with E∑. 1 2 2p∑( ∏) is exactly correlated with γ∑. These second-order parameters cannot be determined by a fit to the observed energy levels. They also cause the mechanical meaning of the E, B, γ parameters to be contaminated. Page 6 of 8 pages 2 2 s eff Now that I have worked out all of the correction terms for the ∏, ∑ H , we can examine the structure of this matrix. For simplicity, specialize to 2∑+ (s = 0). () 2 –D∏(y4 – 3y2 + 3) 1 + AD(y2 – 2) 2 1 + q∏(y2 – 1) 2 + A0/4 +D∏2(y2 – 1)3/2 1 – p∏(y2 – 1)1/2 4 1 – q∏(1 ∓ y)(y2 – 1)1/2 2 2 sym –D∏(y4 + y2 – 1) + A0/4 1 + ADy2 + o∏ 2 1 1 + p∏(1 ∓ y) + q∏[1 ∓ 2 2 2y + y2] ∏1/2 2 ∏1/2 2 ∏3/2 ∑+ 2 ∏3/2 H(2) ef 2 ∑+ –D∑(y4 ∓ 2y3 + y2) +q∑(y2 ∓ y) + NOTE: 1 p∑(1 ∓ y) + o∑ 2 ** Centrifugal Distortion matrix elements are not trivial replacement of B by [B – DJ(J + 1)] ** e/f degeneracy in 2∏ is lifted in H(2) ** all Λ-doubling in 2∏ states comes from 2∑±, none from 2∏, 2∆, 4∏, 4∆, etc. Now apply perturbation theory to H(0) + H(1) + H(2) matrices to analyze where specific effect (e.g. Λ-doubling) originates. Often want to do this in order to: * identify parameter responsible for an observed splitting with a certain J-dependence; * prove that two fit parameters are correlated and therefore not independently determinable; * build in correction for expected not quite remote perturber; * determine whether a certain fit parameter can actually be determined for your specific data set. Page 7 of 8 pages EXAMPLE - Λ-Doubling EXPLICIT IMPLICIT E2∏ 1/2 E2∏ 3/2 e − E2∏ e − E2∏ 1/2 3/2 e/f dependence on-diagonal in Heff e/f dependence off-diagonal in Heff f = −yp ∏ − 2yq ∏ + “second order” f =0 + “second order” largest parity largest parity H 23/2,1/2 “second order” = o ≈ + o E 3/2 − E1/2 dependent term independent term largest term H 3/2,1/2 = −Bv∏ (y 2 −1) + D∏ 2 (y 2 −1) 1/2 3/2 − 1/2 1/2 1 1 p∏ (y 2 −1) − q ∏ (1 my)(y 2 −1) 4 2 parity dependent part 1/ 2 1/ 2 1 H 23 / 2,1/ 2 = ∓2 Bv∏ q∏ y ( y 2 − 1) ( y 2 − 1) = ∓ Bv∏ q∏ y ( y 2 − 1) 2 o o E3/ 2 − E1/ 2 ≈ A∏ So E 3/2e − E 3/2f B B 3 2 ≈ −2 qy (y − 1) ≈ 2 qJ A A Similar algebra for 2 Π1/2: J E1/2e − E1/2f Usually p ∏ >> q ∏ because B 3 ≈ − (p ∏ + 2q ∏ )y + 2 qJ A p ∝ αβ q ∝ β2 p/q ≈ α A = β B At low-J, leading contribution to Λ-doubling in 2∏1/2 is –Jp∏ linear in J in 2∏3/2 is –(2Bq/A)J3 cubic in J Page 8 of 8 pages Structure of 2∑+ state lumped into E v∑ ⎛ e⎞ 1 E ⎜ 2 ∑+ ⎟ = (E v∑ + o ∑ )+ (Bv∑ + q ∑ )(y 2 my)+ p ∑ (1 my) − D∑ (y 4 m2y 3 + y 2 ) f⎠ 2 ⎝ same as γR·S lumped into Bv∑ A mixture of mechanical and magnetic significance is what we determine by fitting a spectrum! Finally, replace y by N as follows: for 2∑+ y2 ∓ y 1∓y y4 ∓ 2y3 + y2 e J = N + 1/2 y=N+1 N(N + 1) –N N2(N + 1)2 f J = N – 1/2 y=N N(N + 1) 1+N N2(N + 1)2 ⎛ 2 + e⎞ ⎤ 1 ⎡1 2 E ⎜ ∑ ⎟ = E v∑ + Bv∑ N(N + 1) − D∑ [N(N +1)] + p ∑ ⎢ m(N + 1 / 2)⎥ ⎦ f⎠ 2 ⎣2 ⎝ N is pattern-forming quantum number! E 2 ∑+ e − E 2 ∑+ f = −yp ∑ = − (N +1 / 2 ) p ∑