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3. Electrons, states, energy levels

3. Electrons, states, energy levels

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3. Electrons, states, energy levels

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  1. 3. Electrons, states, energy levels

  2. Single electron spatial quantum numbersn: principal (radial) quantum numberl: orbital angular momentum (new azimuthal) quantum numberml: magnetic quantum number For one electron atoms, wavefunctions characterized by even l have even parity: Pi |nlml> = (-1)l |nlml> H&I Ch. 2

  3. Some basic principles Eigenvalue equation wave function operator H Hamiltonian energy matrix elements: 1 if a = b; 0 if a ≠ b orthonormality expectation value diagonal matrix energies are state energies

  4. Perturbation theory describes how state a is perturbed by another state b, giving the corrected state a’. States must be of same symmetry, and mixing is inversely proportional to energy difference. describes how the Hamiltonian describing state a is changed by a perturbation. Example: a Hamiltonian H0 has the perturbation V which changes the energies Ei and wavefunctions ni. H = H0 + V

  5. Single electron spin quantum numbers: electron spin ½ms: magnetic quantum number, ±1/2No two electrons in an atom can have the same values of these four quantum numbers Ce3+ 4f1 - a configuration shows electron structure odd electron systems have Kramers degeneracy: 2 different states with same energy, in absence of magnetic field. E Pr3+ 4f2 Electrons are indistinguishable these 2 are the same state no Kramers degeneracy for even electron systems E

  6. Spin orbit coupling: j = l + s |nls j=l+1/2 mj> |nlsmlms> or |nlsjmj> [(2l+1)/2] ζ E For light atoms, l and s are good quantum numbers = an unique value can specify a state |nls j=l-1/2 mj> e.g. l=3, s=1/2 splits into j = 5/2 and j = 7/2. For lanthanides, only the total angular momentum is (nearly) a good quantum number

  7. Many electron systems Coulomb interaction (e-e repulsion) splits 4fN configuration into LS multiplet terms, whilst spin-orbit coupling (SOC) splits terms into J multiplets. 2S+1LJ : total degeneracy 2J+1 (or J+1/2, Kramers systems) Example: 3P splits into 3P0, 3P1, 3P2 since S and L can vector couple in different ways.

  8. J multiplets To find the multiplet terms for a given 4fN configuration: consider microstates |l ml s ms> easy for 4f1 Ce3+l = 3; s = ½; j = 3/2 or 5/2 gives 2F7/2 and 2F5/2. more complicated for many electron systems, e.g. 4f2 Pr3+ ml1 = -3…+3; ms1 = ±½ ml2 = -3…+3; ms2 = ±½ (no 4 quantum numbers can be the same for 1 and 2)

  9. Microstates of Pr3+ ms written as + or - 4f2 Pr3+ has 91 microstates (14×13)/(1×2) can you fill in 3F 3P 3H

  10. Clebsch-Gordon coefficients When two angular momentum states are coupled, the new states can be expressed as the coupled representation: The Clebsch-Gordon coefficient is related to a 3-j symbol: 3-j symbols have special properties, including: invariant to even permutation of columns m1+m2=M j1+j2+J integer triangle rule: (j1-j2)≤ J ≤ (j1+j2)

  11. Example of Clebsch-Gordon coefficients or abbreviated: Example for j1 =1/2; j2 = 1/2 H&I, page 134

  12. Electronic ground state Hund’s rule tells us the ground state multiplet. • Terms with maximum spin multiplicity (2S+1) has lowest energy. • Then choose the one with largest L. • For less than, or half-filled shells: choose the one with smallest J. For more than half-filled, choose largest J. e.g. Multiplets for Pr3+ and Tm3+ are 1S0, 3P0,1,2, 1I6, 1D2, 1G4, 3F4,3,2, 3H4,5,6's_rules

  13. Crystal field The environment of Ln3+ is different in a crystal than in the free ion since the symmetry is lower. The crystal field (CF) = all surrounding charges, multipoles, etc. The descending symmetry causes splitting of degenerate levels. The SOC interaction is of greater magnitude than the CF interaction for Ln3+ 4fN configurations.

  14. Crystal field levels The split J levels are described by irreducible representations (irreps) of the site symmetry point group of Ln3+. We can determine the symmetry irreps of a given J if we know the site symmetry, but we do not know the energy level ordering without detailed calculation. Ch. 3 HI

  15. Character table and irreps

  16. Crystal field irreps from J-values χ(φ) = [sin(J+1/2)φ]/[sin(φ/2]) e.g. for C2, φ = π; For E, χ(φ) = lim φ→0[sin(J+1/2)φ]/[sin(φ/2]) =(J+1/2)φ]/[(φ/2])=(2J+1)

  17. Theoret. Chim. Acta 74 (1988) 219

  18. Bases for irreps in Oh Γ1 J=0: |0> J=4: (24)-1/2 [(14)1/2 |0> + (5)1/2 (|4>+|-4>)] J=6: (1/4)[(2)1/2 |0>- (7)1/2(|4>+|-4>)] Γ2 J=3: (2)-1/2(|2>-|-2>) J=6: (32)-1/2[(5)1/2(|6>+|-6>)- (11)1/2(|2>+|-2>)] etc.

  19. Calculation of energy levels of Ln3+ 1. ab initio methods accurate to hundreds thousands of cm-1 2. Semi-empirical methods accurate to 5-50 cm-1. Calc A Calc B A. Reid, Eur. J. Inorg. Chem. doi:10.1002/ejic.20100182 B.

  20. Calculation of energy levels of Ln3+ Need only to consider 4fN electrons since electrons in closed shells form spherical charge distribution which do not result in energy splitting - only a shift. (H – Ei)Ψi= 0 (1) 3. Diagonalize matrix using trial parameter values to get agreement between calculated and observed energy levels. Wavefunctions generated can be tested by calculations of spectral intensities, etc. 1. Choose all possible arrangements of 4fN as bases. 2. Calculate the Hamiltonian matrix. Wybourne book

  21. Basics of energy level calculations For a Ln3+ system: abbreviate (just consider two terms) Eigenvalue equation is or simply: then solve for the 2 energies E1, E2 Eigenvectors a1, a2 must be orthonormal

  22. Electronic Hamiltonian and states of 4fN systems The electronic states of rare earth ions in crystals are N-body localized states, since the N electrons of 4fN are coupled strongly, and move around the corresponding ion core without extending far away. The semiempirical calculations for the 4fN energy level systems employ a parametrized hamiltonian Hunder the appropriate site symmetry for Ln3+: H = HAT+ HCF + HADD(2) where HAT comprises the atomic hamiltonian, which includes all interactions which are spherically symmetric; HCFis the operator comprising the nonspherically symmetric CF; HADD contains other interactions. Crosswhite J. Opt. Soc. Am. B 1 (1984) 246

  23. Pr3+ energy levels major interactions for Ln3+ are Coulomb and SOC Further smaller splittings due to crystal field Free ion: Coulomb repulsion Spin-orbit coupling H&I p. 398

  24. SOC and CF Transition metal d-electron systems: CF > SOC Lanthanide 4fN: SOC > CF H = HCoul(ff) + Hso(f) + Hcf(f)+.... Most important interactions are e-e repulsion > SOC > CF i.e. weak CF Do not judge by parameter values: But by energy splittings, e.g. Ce3+ 4f1 levels in Oh symmetry n.b. for Ce3+, 4f1, no Coulomb interaction Label states 2S+1LJ Energy (cm-1) 3048 2661 2160 (7/2) ×ζ4f 570 0

  25. The atomic Hamiltonian:

  26. Other interactions The two-body configuration interaction parameters α, β, γ parametrize the second-order Coulomb interactions with higher configurations of the same parity. For fN and f14-N, N>2 the three body parameters Ts (s = 2,3,4,6,7,8) are employed to represent Coulomb interactions with configurations that differ by only one electron from fN. With the inclusion of these parameters, the free ion energy levels can usually be fitted to within 100 cm-1. The magnetic parameters Mj (j = 0,2,4) describe the spin-spin and spin-other orbit interactions between electrons, and the electrostatically correlated spin-orbit interaction Pk (k = 2,4,6) allows for the effect of additional configurations upon the spin-orbit interaction. Usually the ratios M0:M2:M4 and those of P2:P4:P6 are constrained to minimize the number of parameters, which otherwise already total 20.

  27. Crystal field parameters CFP Garcia, Handbook on Phys. Chem. Rare Earths 21 (1995) 263 Gőrller-Walrand, Handbook on Phys. Chem. Rare Earths 23 (1996) 121

  28. Crystal field parameters CFP • Since the hamiltonian is a totally symmetric operator, only those values of k whose angular momentum irreps transform as the totally symmetric representation of the molecular point group are included. • In C1 symmetry: 27 CFP • In higher symmetries, the angular interactions cancel. • For atoms, spherical symmetry, no CFP. • For LnCl63- systems, Oh symmetry, 2 CFP.

  29. CFP in higher symmetry

  30. Theoretical analysis More convincing for LnX63- systems because fewer independent crystal field parameters are involved.

  31. 4f1 and 4f13 systems These are simple because only one electron (or one hole) so no Coulomb repulsion.

  32. Crystal field analysis of 4f13 Inorg. Chem. 16 (1976) 1694 J. Chem. Phys. 94 (1991) 942 2F5/2 Different notations employed we use: 2F7/2

  33. (Equation 1) (Equation 2) Crystal field analysis of 4f13 Griffith, The Theory of Transition –Metal Ions; Cambridge University Press, 1961

  34. Γ6: Γ8: Γ7: Crystal field analysis of 4f13 Energy matrices of 4f13 Take advantage of symmetry factorization

  35. The published parameters of , and were used Comparison of experimental and calculated energy levels of Cs2NaYbCl6. J Alloys Compds 215 (1994) 349

  36. Γ8: Γ7: Crystal field analysis of 4f13 Anticipate the effect of crystal field parameters on the energy gaps. The effect of on Γ7 should be more prominent than that on Γ8 . The effect of on Γ8 might be more prominent than that on Γ7.

  37. Crystal field analysis of 4f13 Energy against crystal parameters of Cs2NaYbCl6, calculated by f-shell program.

  38. Fitting programs • Prof. M. Reid: f shell programs, from Prof. F.S. Richardson’s group. • Profs. Edvardsson, Åberg: • Dr. Michèle Faucher: ATOME

  39. Determination of site symmetry molecular point group of Ln3+Need published crystal structure. • Use International Tables of Crystallography: Vol 4A, Space Group symmetry: Hermann-Mauguin notation.

  40. 2. Use Appl. Spectrosc. 25 (1971) 155: Schoenflies notation. number of equivalent atoms in Bravais cell number of different kinds of site with this symmetry

  41. What use are the parameters from energy level calculations? If useful: 1A. Energy level fitting accurately reproduce the experimental data set. 1B Predict missing or unexplored energy levels. 1C. Energy level dataset representative (i.e. extending over a wide range) and fairly complete. 1D. Wave functions resulting from the parametrization should be capable of accurately predicting other properties such as g-factors and spectral intensities. Duan, J. Phys. Chem. A 114 (2010) 6055

  42. What use? 2. Parameters expected to show some type of systematic variation for materials comprising a series of closely-related elements. 3. Parameters should be related to other physical quantities in a systematic manner. Also the parameters should show explicable trends over various crystal hosts for a particular ion.

  43. Critical test: Cs2NaLnCl6 1. Ln3+ in octahedral symmetry: only 2 CFP 2. Representivity (100 × Nexp/Ntotal) of the dataset does vary considerably for different Ln3+, being 100% for Ce3+, Yb3+; over 90% for Pr3+, Tm3+, but much less for the more extensive 4fN configurations, such as 3% for Gd3+. 3. Standard deviations of most fits are around 20 cm-1.

  44. Results: Prediction of energy levels 421 Energy levels of Pm3+ in Cs2NaPmCl6 Using interpolated parameters

  45. Prediction of luminescent levels

  46. Prediction of spectral intensities Discussed later for Gd3+ TPA

  47. Systematic variation F2 = (61573±610) +(3223.3±79.4)N (9) F4 = (46213±830) + (2054±108)N (10) • F6 = (25631±1620)+(3594.3±18)N -(121.6±36.4) N2 (11) Central Field approximation: F4/F2 and F6/F2 being stable for Ln3+, at 0.70 and 0.54 F4/F2 = (0.7437±0.0138) - (0.00316±0.00180)N (12) F6/F2 = 0.4413 + (0.02261 ± 0.0069)N - (0.0014 ± 0.00048)N2 (13) Comparison with the free-ion values: for Pr3+Fk are 7.5±1.2% smaller in the Cs2NaPrCl6 crystal. This Nephelauxetic Effect has been ascribed to various causes, including the reduced repulsion between 4f electrons due to interpenetration of ligand electrons.

  48. Variation of spin-orbit coupling ζ4f = (539.4±10.2) + (87.82±3.33)N + (7.095±0.233)N2 (12)

  49. Variation of crystal field parameters B4 = (2176.2±29.2) - (56.7±3.6)N (13) B6 = (285.5±24.6) - (12.6±3.0)N (14)