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”

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3d

Fadley Group Experiment – PRL 1988

Mn atoms, MnO, & MnF2 Similar XPS

or 3s

Relative binding energy (eV)

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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

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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

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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)

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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)

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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

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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

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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

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OUR NEW COLLABORATOR

Marty Devours Basic Concepts

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Background Slides

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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)

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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