5.80 Small-Molecule Spectroscopy and Dynamics

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1 MIT OpenCourseWare Small-Molecule Spectroscopy and Dynamics Fall 008 For information about citing these materials or our Terms of Use, visit:

2 5.80 Lecture #6 Fall, 008 Page 1 Outline of approach: Configuration which L S J states? H eff [,, F k,g k ] alternative Limiting Case Coupling Schemes L S J vs. j 1 j J off-diagonal vs. F Lecture #6: Many e Atoms energy levels eigenstates transition intensities 1. list of orbital occupancies e.g. 1s s p configuration. Which L S terms result? * list all spin-orbitals p: (overbar means m S = 1/ or ) * list all Slater determinants (anti-symmetrized microstates) consistent with Exclusion Principle e.g. 11 M L = M S =0 standard order of spin-orbitals (needed to get correct signs of matrix elements) * Classify Slaters according to M L and M S * Method of crossing out of microstates: M L,M S ranges for each L S term (p 3 4 S, D, P) * Total degeneracy of a configuration p : = Find the linear combinations of Slaters that are eigenstates in either representation: LM L SM S OR LSJM J These cause H eff to be block diagonalized. 4. Compute matrix elements of H in selected basis set. SO * H = (r i ) i s i ME of 1e operator z n (n) * H ee = e r ij ME of e operator f k F k (n,n) + g k G k (n,n) i> j k,n,n n H eff expressed in terms of fit parameters: orbital energy spin-orbit direct exchange F k G k,, F k,g k orbital radial integrals fit parameters z n,f k,g k are exactly calculable ANGULAR INTEGRALS Would take or 3 lectures to explain how to compute: *1e and e operator matrix elements of Slater determinants

3 5.80 Lecture #6 Fall, 008 Page * how to work out z n,f k,g k factors (see Group Theory and Quantum Mechanics, M. Tinkham, pp ) My goal here is to expose you to the atomic H eff models. Often the relevant 1/r ij and spin-orbit matrices can be found in a book like Condon and Shortley. Read Tinkham pages for methods of evaluating these matrix elements yourselves. Examine the n = 3 and n = levels of H atom explicitly and compare against the observed spectrum. n = can have =0and1 j = 1/ j = 1/ and 3/ n=3hasl=0,1, j = 5/, 3/ n j s E /hc H SO /hc 0 1/ 1/ / 1/ / 1/ / 1/ / 1/ / 1/ / 1/ / 1/ See Figures Energy Level Diagram Spectrum (Theoretical from C & S) Compare P 3/ P 1/ energy level splitting for hydrogenic systems vs. Z H I cm 1 Li III 30 cm 1 Na XI 5400 cm 1

4 5.80 Lecture #6 Fall, 008 Page 3 Energy level Diagram for H I n=3 and n = s p d / / 3/ n=3 1/ 1/ f g d e a b c n= 1/ / 1/ none Transition Label CALC OBS energy order of transition a = = b = = c = = highest d = = " e = = f = = g = = lowest parity selection rule = ±1 vector coupling = j favored (propensity rule, solid lines a, c, d, g)

5 5.80 Lecture #6 Fall, 008 Page 4 Image of The Theory of Atomic Spectra by E. U. Condonand G. H. Shortley removed due to copyright restrictions. Figure removed due to copyright restrictions. Please see: Fig. 35 in Condon, E. E., and G. H. Shortley. The Theory of Atomic Spectra. Cambridge, UK. Cambridge University Press, This illustrates the spin-orbit fine structure of the H atom n = 3 n = spectrum. Transitions are labeled (a) (g) following the table on the previous page. The lengths of the lines represent the calculated relative intensities (assuming equal populations in all m j components of the n = levels). Orbital-Based Periodic Correlations: what do we look at to see the correlations? Lowest L S J state? Center of gravity of entire configuration? Degeneracy d 1 10 d 10 9 =45 D d 9 (L + 1)(S + 1) = 10 1 G, 3 F, 1 D, 3 P, 1 S d 8 d H, G, 4 F, F, D, D, 4 P, P d 7 3 = 10 d incredible d = 10 d = 4 unbelievable d Yet all is given by (nd), F 0,F,F 4 (no G k s for p or any set of identical n orbitals) Massively complicated spectra for 3d series. No corresponding states for 3d N 3d N+1 Magic decoder is,f 0,F,F 4 in effective Hamiltonian. We know how each of these parameters should scale vs. Z for isoelectronic series or across row as Z Z +1, or as N N +1 vs. n for Rydberg series Example (3d) for 3d N 4s of Sc Cu 4 Goes as Z eff (imperfect shielding of one 3d by others) as Z increases. The plot of (LS) is for the lowest L S term (MAX S, MAX L) (Hund s rules)

6 5.80 Lecture #6 Fall, 008 Page 5 Look at Figure 6- from Tinkham: Figure removed due to copyright restrictions. Please see: Figure 6- in Moore, C. E. "Atomic Energy Levels." Natl Bur Standards, Circ 467. [Vols. I (1949) ; Vol II. (1951)]. property of 3d orbital eff ( Z ) 4 scaling. Z eff 3d increases in steps of 0.8. property of lowest L S term of (3d) (4s) configuration. No scaling. Hund s 3 rd Rule. p Example LS jj Coupling p configuration 1 D, 3 P,1,0, 1 S 0 See Condon and Shortley, pages 198, 68, 74-5, 94

7 5.80 Lecture #6 Fall, 008 Page 6 The Russell-Saunders Case: Energy Levels removed due to copyright restrictions.

8 5.80 Lecture #6 Fall, 008 Page 7 1 S 0 F 0 +10F D 0 F 0 + F H ee ( p ) = 3 P 0 0 F 0 5F is fully diagonal 3 P F 0 5F 0 3 P F 0 5F H SO J=1 1 S 0 0 1/ J=0 P 1/ , in different order than above H ee, factors into, 0 0 1/ 0 0 np 1 1, and 1 D 0 1/ J= P 1/ / ( p ) = 3 P 1 ( ) Add H ee + H SO to get secular equations for J =, 1, 0. These matrices were evaluated in L S J basis set. CouldhaveusedLM L SM S. More work, same results. To get secular equations into most convenient form * subtract out center of gravity (C of G) E V * put into form = 0 V E Eigenvalues E ± =±[ /4 + V ] 1/ 1 S 0 F o +10F 1/ F 1 + 1/ J=0 = F o + F + 3 P 0 1/ F o 5F 15 sym F So E ± F + + ± (J = 0) = F 0 + F + F 4 4 J=1 E(J=1)=F 0 5F - / 1 1/ 1 F 1/ 0 + F 3F D 1 J= =F F P 1/ F 0 5F / 3F + 4 1/ E ± 9F ± (J = ) = F 0 F + F /

9 5.80 Lecture #6 Fall, 008 Page 8 Note that these matrices have off-diagonal and E differences that are dominated by F. There are two convenient limits for intraconfigurational energy level patterns. For p : = 0 F =0 L S coupling j j coupling J = L + S J = j 1 + j 10F 1 S (3/, 3/) 0 (3/, 3/) fine 9F 3/ F 1 D (3/, 1/) coarse 1/ structure (3/, 1/) 1 fine 6F 3/ fine 5F 3 P 0,1, (1/, 1/) 0 structure Be sure that total degeneracy of all states is the same in both limits. possible values for j 1 and j for p = 1, s = 1/ are 3/ and 1/ total (j 1,j ) possible values of J degeneracy (3/, 3/) 3 exclusion principle exclusion NOT 0 OBVIOUS 4 (3/, 1/) 1 1 (1/, 1/) 1 exclusion

10 5.80 Lecture #6 Fall, 008 Page 9 LS (j, j) J Coupling Patterns jj states LS states Ge(4p) Pb(6p) 1 S 0 ~ 3 P 0 1 S 0 1 S 0 1 S 0 ( 3 P 0 ) 1 D ~ 3 P (3/, 3/) 0 and electrostatic F = 1017 cm 1 F = 9 cm 1 (3/, 3/) spin-orbit = 880 cm 1 = 794 cm 1 1 D ( 3 P ) 3 P 1 ~ 1 D 1 D 3 P 1 (3/, 1/) and 1 D (3/, 1/)1 3 P ( 1 D ) 3 P 1 3 P 3 P 0,1, 3 P 1 P 0 3 P 0 ( 1 S 0 ) 3 P 0 ~ 1 S 0 (1/, 1/) 0 See Condon and Shortley, pages 74-5, plotted on scale to keep E MAX E MIN = constant. p Matrix in L S J BASIS SET 1 S 0 F 0 +10F 1/ 3 P 0 1/ F 0 5F 3 P 1 F 0 5F / 1 D F 0 + F 1/ 3 P 1/ F 0 5F + /

11 5.80 Lecture #6 Fall, 008 Page 10 Perturbation Theory in /F 1 limit 1 o S 0 = E + 1 S 15F + 3 o P 0 = E 3 P 15F + 3 o P 1 = E 3 / P 3 P = E o3 P + / + 6F Landé Interval Rule negligible H ij Perturbation Theory works when o. Thus, our L S J basis set matrix for H is suitable for E i E o j perturbation theory if 1/ 1/ and are 1 15F + 6F (3/,3/ ) = P / 1/ (3/,1/ ) 3 P 1 = D 3 3 1/ 1 1/ (3/,3/ ) 1 S 3 0 = 0 P / 1/ (1/,1/ ) 1 0 = S J=0 J= (i.e. F ) Alternatively, we can transform to the jj basis set using the transformation given on page 94 of Condon and Shortley. (3/,1/ ) 1 = 3 P 1 1/ 1/ 1 D 3 P 0

12 5.80 Lecture #6 Fall, 008 Page 11 And the matrices are given by 1 1/ H (3/,3/),( 3/,3/) H 3 H P 1 H D 3 = P 3 1 D P 3 1 1/ H (3/,1/),( 3/,1/) H 3 H P 1 H D 3 = P 3 1 D P 3 1 H (3/,3/),( 3/,1/) H 1 = H 3 3 P 3 P 3 D 1 D 3 H 3 P 1 D 1 D 1 D MATRICES TRANSFORMED TO j 1 j J BASIS SET 3F + 3/ F (3/, 3/) H (J = ) = 3/ F F 1 (3/, 1/) H (J = 1) = 5F / (3/, 1/) 1 5F + 5 F (3/, 3/) 0 H (J = 0) = 5 F (3/, 1/) 0 Goes to limiting j j pattern when F 0. Note: is off-diagonal in L S J basis F is off-diagonal in j j J basis This is an example of the battle between two terms in H eff. Suggested homework problem: repeat all of the steps in pages 5-11 for the d configuration. Use tables in Condon and Shortley.

13 5.80 Lecture #6 Fall, 008 Page 1 Transitions p sp (j 1,j ) J also other weak 3 P J 3 P J lines S 0 p (upper states) (3/, 3/) 0 propensity rules (3/, 3/) = ±1 j 1 =J J = L strong or S = 0 j =J 1 D J = 0, ±1 (3/, 1/) but (3/, 1/) 1 J=0 J=0 3 P 3 P 1 same as for sp 3 P 0 (1/, 1/) 0 (3/, 1/) 1 P 1 (3/, 1/) 1 sp (lower states) 1 3 P 3 P 1 (1/, 1/) 1 3 P 0 (1/, 1/) 0 other forbidden transition strengths predictable from mixing coefficients =±1 J = L strong S = 0 p p s active e j 1 = 0 (inactive e ) j =0, ±1 J = 0, ±1 J=0 / J = 0 J = j strong

14 5.80 Lecture #6 Fall, 008 Page 13 Figure removed due to copyright restrictions. Please see: Pages in Condon, E. E., and G. H. Shortley. The Theory of Atomic Spectra. Cambridge, UK: Cambridge University Press, 1951.

15 5.80 Lecture #6 Fall, 008 Page 14 Table of matrices of spin-orbit interaction removed due to copyright restrictions.

16 5.80 Lecture #6 Fall, 008 Page 15 Transformations in the theory of complex spectra text removed due to copyright restrictions.