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RIGOUROUS THEORY FOR NEXAFS: MULTIPLETS and MORE
Paul S. Bagus, Chemistry, UNT, Denton, TX
Eugene S. Ilton, PNNL, Richland, WA
History – Many Body Effects For The Mn 3s XPS in MnO Work Spans Over 3 Decades
NEXAFS Selection Rules & Branching Ratios Multiplet Composition of Levels Power & Limitations For NEXAFS Intensities Levels Determine Energies – Multiplets Reflect Intensity
Ligand NEXAFS Edges In Actinide Complexes Pre-Edge Structure Reflects Spin-Orbit Splitting In 5f Shell Leads To Major Broadening
DOE Support Gratefully Acknowledged 1 Utrecht_2013
Mn 3s XPS IN MnO – 7S & 5S MULTIPLETS Fadley, Shirley, Freeman, Bagus, Mallow, PRL 23 (1969)
Mn2+ Models MnO
Consider Only 7S & 5S Couplings
Large Errors
E Too Large By Factor of 2
Irel Also Too Large By 2x
No Satellites
Error Ascribed To Covalency
“As Expected From Covalent Bonding”
Utrecht_2013 2
3d
Fadley Group Experiment – PRL 1988
Mn atoms, MnO, & MnF2 Similar XPS
or 3s
Relative binding energy (eV)
Utrecht_2013 3
Addition of 3p2 3s3d FACs (Frustrated Auger Configurations)
3d
3p
3s
ACTIVE SPACE: NEAR DEGENERATE Cis Bagus, Freeman, Sasaki, PRL 30 (1973)
Calculated splitting with FAC too small by ~ 2eV
Okada & Kotani: “.. we reduce the radial integral … to 75% of its ab initio value …”
Does Not Explain Why Reduction Needed
Three Decades Later, We Understand
Utrecht_2013 4
INCREASE ACTIVE SPACE FOR Mn 3s XPS Bagus, Broer, and Ilton (2004) – Chem. Phys. Lett.
3d
3p
3s
4f
Differential effect - ΔE(7S-5S) = 6.5 eV
Integrals not scaled
Other consequences of correct physics
New satellite at ~25 eV
Absolute value of BE(3s) reduced 3eV
4f FAC
3p3d 3s4f
MULTIPLET SELECTION RULES
Analysis of Closed Shell Cations – Bagus, Freund, et.al. CPL (2008)
V5+ For V2O5 / LII,III (2p) Edge
U6+ For UO3 / NIV,I (4d) & OIV,V (5d) Edges
Selection Rules For J Levels
J=0 To J=1
Selection Rules For Russell-Saunders Multiplets 1S0 To 1P1 – Excitations To 3P1 & 3D1 Dipole Forbidden
Project R-S Multiplets On J Levels
4 Component Spinors Calculated With Dirac Program System
RAS CI Wavefunctions – Full Intermediate Coupling For Levels
Non-Relativistic Limit For Speed of Light Set Very Large
Lowest J=1 Level of V5+ (2p53d1)
90% 3P1, 8.5% 3D1 & 0.5% 1P1 Very Weak XAS Peak
As Expected 3P Is Lowest Multiplet / Level
Utrecht_2013 5
MULTIPLET ANALYSIS OF NEXAFS
V5+ 2p6 To 2p53d1 : J=1 Levels
Remarkable Consistency Between Irel & %RS
Reasonable Agreement With Experiment For V2O5
U6+ 4d10 To 4d95f1
Less Good Agreement Between Irel & %RS
%RS Neglects Changes In Orbital Character: 4d3/2 & 4d5/2
Consistent With Van der Laan Sum Rule Analysis – PRL (2003)
Utrecht_2013 6
Erel Irel %RS(1P) – [Norm]
0 0.02 0.5 [0.02]
4.8 1 28.3 [1]
11.9 2.44 71.2 [2.52]
Erel Irel %RS(1P) – [Norm]
0 0.01 0.5 [0.01]
2.3 1 57.7 [1]
44.4 0.55 41.7 [0.72]
SPIN-ORBIT COUPLING IN ACTINIDE CATIONS
5fn Open Shell – Often Use Hund’s Rules With Max Ms – Neglect S-O Coupling
Compare U4+, UO8 For UO2, and UCl62-
Spinor Energies, E in eV
Covalent Character From Projection – NP = <| A(nl)A†(nl)|>
Totals For Average Occupation Within 5f2
XAS Is For Cl K-Edge NEXAFS Cl(1s) To U(5f)
Utrecht_2013 7
5/2 7/2
U4+ 0 0.75
UO2 0 0.17 0.62 0.67 1.31
UCl62- 0 0.13 0.66 0.78 0.86
N(5f) N(6d) - Large Uncertainty For N(6d)
UO2 2.35 3.024
UCl62- GS 2.29 2.84
XAS 3.10 2.36 - Note Reductions
Cl K-EDGE XAS FOR UCl6
Covalent Mixing of Cl(3p) With U(5f) & U(6d)
Provides Intensity For Cl(1s) To Nominally U(5f) & U(6d)
Mixing of Cl(3p) In UCl62- Spinors < ~5% For U(5f) & 25% For U(6d)
Wavefunctions
Symmetry – GS Oh / XAS D4h – Hole Localized On Pair of Cl Atoms
Optimize Spinors For Average of Configuration With Dirac Program
GS - (5f)2 & XAS (1sg)1(5f)3
RAS CI For Intermediate Coupling For Final States
GS - 1
XAS -1s to 5f In Range 0 to 8 eV
Intensity From Dipole Transition Matrix Element – Computed Exactly
Utrecht_2013 8
Cl(1s) TO U(5f) XAS
Voigt Broaden w/ 0.25 eV Gaussian & 1.0 eV Lorentzian
Representative States With I |<GS|z|Ex>|2 and E
Utrecht_2013 9
E(eV) Irel N[5f(5/2)] N[5f(7/2)]
0.0 1 2.76 0.24
0.12 1.44 2.55 0.45
0.58 0.34 1.86 1.14
2.23 0.35 1.79 1.21
2.75 0.29 1.26 1.74
Sum of I(x) + I(y) + I(z)
Intensity Broadened ~ 4 eV
Major Origin S-O Coupling
SUMMARY
Intermediate Coupling Is An Important Aspect of Heavy Metals
Analysis With 2s+1LJ Russell-Saunder Multiplets Is Incomplete
Broad Range of Consequences
NEXAFS Intensities & Branching Ratios
Multiplet Composition of Levels Provide A Guide
Changes In Orbital Character May Modify Predictions
Ligand NEXAFS Edges
A First Study That Includes S-O Splitting & Intermediate Coupling
Intensity Known To Reflect Covalent Character
But Treated With RS High Spin Alignment
Multiplet Splitting & Spin-Orbit Coupling of Actinide 5f Gives Major Broadening
Further Work Required To Compare With Experiment
Utrecht_2013 10
Lausanne_WOXS_2008 13
MULTIPLET SPLITTING AND U OXIDATION STATE
Compare 4f XPS For Different 5fn
Multiplet Splitting Largest for U5+
Opposite To Trend For Transition Metals
Exchange Splitting E nK
For n Unpaired Electrons
Narrowing of 4f XPS From 5f SO Splitting
Ms Is Not Good Quantum Number
Consequences For Magnetic Properties
Test With High Resolution XPS
U5+(5f1)
U4+(5f2)
U3+(5f3)
Lausanne_WOXS_2008 14
TRANSITION INTENSITIES
Cannot Model Spectra Without Accurate Transition Intensities
Many Final States – Some Have Small I & Some Have Large I
For XPS Use Sudden Approximation, SA, Relative Intensities
Remove Electron From ith core orbital Leaving All Orbitals Unchanged
Irel Is Square of Overlap Matrix Element Between Two (N1) Electron WF’s
For XANES Use Dipole Matrix Elements: I(k)<irf>2
i & f Are N Electron WF’s
Matrix Elements are Trivial When One Orbital Set Is Used
General Formula Involves Sum of Products of One Electron Integrals
Theory Well Known – Löwdin, 1955- But Calculation Is Complicated
Exact Transition Matrix Elements Used In This Work
Forms Major Part of Computational Time