Download Origin and Interpretation of Spin Hamiltonian Parameters, Frank

Origin and Interpretation
of
Spin Hamiltonian
Parameters
Frank Neese
Max Planck Insitut für Bioanorganische Chemie
Mülheim an der Ruhr
Outline
1. Introduction
2. Molecular Electronic Structure
a)
Construction of Many Electron States
b)
The Non-Relativistic Hamiltonian
c)
Relativistic Quantum Theory
3. Ligand Field Theory as a Simple Model
a)
One Electron in a Ligand Field
b)
Many Electrons in a Ligand Field
c)
Tanabe Sugano Diagrams and Optical Spectra
Exchange
Couplings
g- And DValues
4. Perturbation Theory of Spin-Hamiltonian Parameters
a)
Partitioning Theory and Effective Hamiltonians
b)
g-Tensor
c)
D-Tensor
5. Spin-Hamiltonian Parameters in dN Configurations
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
Why a Spin-Hamiltonian ?
Molecular
Hamilton-Operator
Hˆ 0 = Tˆel +VˆEN +VˆEE +VˆNN
Hˆ 1 = Hˆ SO + HZe +K
Theory
Spin
Hamilton-Operator
r r r r
Hˆ eff = βBgS + SDS + K
Reviews:
Dir
ect
C
Molecularstructure
alc
ula
tio
n
Spectra
Fit
Simulation
(1) FN (2003) Curr. Op. Chem. Biol., 7, 125
(2) FN; Solomon, E.I. in: Miller, JS, Drillon, M. (Eds) Magnetism Vol. IV, Wiley-VCH, 2003, pp 345-466
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
Terms in the Spin Hamiltonian
IB2QB2IB2
IA1QA1IA1
βNBgNA1IA1
IIA1
A1
S
S
AA
A2 S
A
A
SSAA
βe B
gB S
B
SAJSB
SB
SSBB
SBDBSB
SADASA
B1
A
SB
S
BA
B2 S
B
S
IA1JNMRIA2
A
g AS
B
βe
IIB1
B1
IB1JNMRIB2
AA
A1
BB
βNBgNB1IA1
IA2QA2IA2
IIA2
A2
J
D
βeg
A
βNgN
Q
JNMR
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
~
~
~
~
~
~
~
0 -103 cm-1
10-1-101 cm-1
10-1-101 cm-1
0 -10-2 cm-1
0 -10-2 cm-1
0 -10-3 cm-1
0 -10-8 cm-1
IB2QB2IB2
IIB2
B2
One Electron Systems
Properties of Electrons:
• Mass
• Electric Charge
• Spin
• Magnetic Dipole Moment
: me
: -e0
: s=1/2 („up“=ms=1/2 or „down“=ms=-1/2)
: |µ|=s(s+1)geβ ~ 1.5 β (ge=2.002319...)
Schrödinger‘s Nonrelativistic Equation for a Single Electron:
Hˆ ψ i (r ) = ε iψ i (r )
Ĥ
ψ i (r )
: „Hamiltonian“. Includes all Energy Terms and Interactions (H=T+V)
: An „Orbital“ (=Square Integrable One-Electron Wavefunction)
ψ i (r )
2
= Probability per Unit Volume of Finding the Electron at r
∫ ψ i (r ) dr = 1 ~ „We are Sure it‘s Somewhere“
2
εi
:„Orbital Energy“. Energy of the Electron in Orbital
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
ψi
ε
.
.
.
ε2 = ε3
ε1
ε0
Nonrelativistic Many Electron Systems
In many Electron Systems Orbitals are – Strictly Speaking – no Longer Well Defined
or Necessary to Describe the Physics!
What Matters is the Many Electron Wavefunction Ψ (r1σ 1 , r2σ 2 ,..., rN σ N )
*
Born-Interpretation: Ψ (r1σ 1 , r2σ 2 ,..., rN σ N )Ψ (r1σ 1 , r2σ 2 ,..., rNσ N )
Conditional Probability for Finding Electron 1 at r1 with Spin σ1, Electron 2 at r2 with Spin σ2, ...
Ψ (r1σ 1 ,..., riσ i ,..., r jσ j ,..., rN σ N ) = − Ψ (r1σ 1 ,..., r jσ j ,..., riσ i ,..., rN σ N )
Pauli-Principle:
Antisymmetry with Respect to Particle Interchanges (Electrons are Fermions)
Born-Oppenheimer Hamiltonian:
e-
Te
VeN
A
e-
Te
Vee
VeN
VeN
VeN
VNN
B
Hˆ BO = Tˆe + VˆEN + VˆEE + VˆNN + TˆN
Contains only the Electronic Kinetic Energy plus the Dominant
Electrostatic Interactions => Looks Easy!
However, the BO Schrödinger Equation:
Hˆ BO ΨI (r1σ 1 ,..., rN σ N R1 ,..., R M ) = E I ΨI (r1σ 1 ,..., rN σ N R1 ,..., R M )
Is Frightening - Unsolvable Equation!
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
Nonrelativistic Many Electron Systems (ctd.)
Since the BO Problem Cannot be Solved Eactly (Except for the Simplest Systems)
a Reasonable Approximation is the Hartree-Fock Meanfield Theory
e-
Hartree-Fock Wavefunction:
Te
VeN
B
VNN
A
1
ψ 1 ...ψ N
N!
Ψ (x 1 ... x N ) =
VeN
Vee
Hartree-Fock Energy:
E HF = T e + E eN + J ee + X ee + E NN
142 4
3
LCAO-Approximation
„Hartree-Fock-Sea“
Slater Determinant
E ee
r
ψ i = ∑ c jiϕ j (r )
j
Successful Approximation (>99% of the Exact NR Energy)
Errors Still Large (>100 kcal/mol)
Need to Describe Correlated Electron Movement: Post-HF
Nice: Reintroduces Orbitals!
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
εi
3d
4s
3s
2s
1s
3p
2p
Aufbau Principle for
Atoms and Molecules!
Exchange Couplings
„Exchange Coupling“
Hˆ = −2 JSˆ A Sˆ B
3
Ψ
1
Ψ
2J
From the Talk of Dr. Glaser Recall:
1.
2.
3.
4.
The „Exchange Hamiltonian“ Does NOT Follow from Magnetic Interactions
(There is No Such Thing as an „Exchange Interaction“ in Nature)
The Born-Oppenheimer Hamiltonian Is Enough to Describe the Splitting of
States of the Same Configuration but Different Multiplicity
The Splitting is Related to the Electron-Electron Interaction and the
Antisymmetry of the N-Particle Wavefunction
The Simplest (Anderson) Model Leads to Two Contributions:
a) The „Direct Exchange“ : K = a(1)b(2) r a(1)b(2) (Ferromagnetic)
b) The „Kinetic Exchange“ : − β / U ≈ − a f b /( E (ionic) − E (neutral ))
(Antiferromagnetic)
a f b ≈∝ S
−1
12
2
2
We can Talk and Argue this Way, but How do We Calculate it ?
A Model Calculation: [Cu2(µ-F)(H2O)6]3+
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
A Model Calculation: [Cu2(µ-F)(H2O)6]3+
The Hartree-Fock SOMOs of the Triplet State („Active“ Orbitals)
ψu
~0.7 eV
ψg
ψg
ψu
The Pseudo-Localized „Magnetic Orbitals“
a = 2 −1/ 2 (ψ g +ψ u )
b = 2 −1/ 2 (ψ g −ψ u )
Notes:
• ‚a‘ and ‚b‘ have Tails on the Bridge (and on the Other Side)
• ‚a‘ and ‚b‘ are Orthogonal and Normalized
• ‚a‘ and ‚b‘ do not Have a Definite Energy
• THE Orbitals of a Compound are not Well Defined! (ROHF, MC-SCF, DFT, Singlet or Triplet
Optimized, ...)
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
Values of Model Parameters:
„Direct“ (Potential) Exchange Term:
a(1)b(2 ) r12−1 a(1)b(2) = 17 cm −1
Hˆ = −2 JSˆ A Sˆ B
2
−1
Exactly Calculated „Kinetic“ Exchange Term: − 2β / U = −57 cm
J = −40 cm −1
Is that Accurate? Look at the Singlet Wavefunction:
1
Ψ0 = 99.94% neutral + 0.06% ionic
BUT:
• The Ionic Parts are too High in Energy and Mix too Little with the
Neutral Configuration (Electronic Relaxation)
•
What About Charge Transfer States?
Need a More Rigorous Electronic Structure Method:
a)
b)
c)
Difference Dedicated CI (DDCI): Malrieu, Caballol et al.
CASPT2: Roos, Pierloot
MC-CEPA: Staemmler
Recommended Literature:
Calzado, C. J.; Cabrero, J.; Malrieu, J. P.; Caballol, R. J. Chem. Phys. 2002, 116, 2728
Calzado, C. J.; Cabrero, J.; Malrieu, J. P.; Caballol, R. J. Chem. Phys 2002, 116, 3985
Fink, K.; Fink, R.; Staemmler, V. Inorg. Chem. 1994, 33, 6219
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr Ceulemans, A.; et al., L. Chem. Rev. 2000, 100, 787
Refined Ab Initio Calculation
Include Relaxation and LMCT/MLCT States via the Difference Dedicated
CI Procedure:
a(1)b(2) r −1 a (1)b(2 ) = + 17 cm −1
12
(~105 Configurations)
− 2β 2 / U = − 57 cm −1
all others = −166 cm −1
J = −206 cm −1
Look at the Singlet Wavefunction
1
Ψ0 = 92.3% neutral + 3.3% ionic + 4.4% LMCT
Reduced
Increased!
NEW+IMPORTANT
The
TheAnderson
AndersonModel
ModelisisNot
NotReally
ReallyRealistic
Realisticand
andShould
Shouldnot
notBe
BeTaken
TakenLiterally
Literally
Even
EvenThough
Thoughits
itsCI
CIIdeas
Ideasare
areReasonable.
Reasonable.
••Relaxation
Relaxationof
ofIonic
IonicConfigurations
Configurationsare
areImportant
Important(„Dressing“
(„Dressing“by
byDynamic
Dynamic
Correlation:
Correlation:(DDCI,
(DDCI,CASPT2,MC-CEPA,...)
CASPT2,MC-CEPA,...)
••LMCT
LMCTStates
Statesare
areImportant
Important
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
Treatment of LMCT States in Model Calculations: VBCI Model:
Tuczek, F.; Solomon, E. I. Coord. Chem. Rev. 2001, 219, 1075
Exchange Coupling by DFT
The Dominant part of the Triplet Wavefunction is:
3
neutral = core aα bα
The Dominant part of the Singlet Wavefunction is:
1
neutral =
1
2
[core a b
α β
]
− core aβ bα
This is a Wavefunction that Can NOT be Represented by a Single Slater Determinant!
• BIG Problem – DFT Can Only Do Single Determinants (Triplet: Fine,Singlet:
Noodleman (J. Chem. Phys., (1981), 74, 5737)
• Use Only Either core aα bβ or core aβ bα as Starting Point BUT Reoptimize
Energy !!!

→ core τ αAτ βB ≡ BS
Orbitals core aα bβ Minimize
• NOTE: τ αA ,τ βB Are No Longer Orthogonal in Their Space Parts
−E
E
• ASSUME: BS ~ 50% Singlet, 50% Triplet J = − ( SHS + S BS) 2
A
B
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
Better:
J =−
EHS − E BS
S
− S2
(Yamaguchi)
2
HS
BS
)
The Broken Symmetry Wavefunction
Relaxation of Orbitals ‚a‘ and ‚b‘:
a
τA
b
τB
Terrible Mistake:
Negative Spin
Density
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
Energy Gain Through Partial
Delocalization Gives
Nonorthogonality of Space
Parts (S~0.16 here)
In
Inthe
theReal
RealWorld
World
Everywhere
EverywhereZero
Zerofor
for
AASinglet
SingletState!!!
State!!!
Positive Spin
Density
My View:
FN (2003) J. Phys. Chem. Solids, 65, 781
Interpretation of the BS Wavefunction
Diradical Index:
200 x Percentage Neutral
Character on Top of the
Closed Shell
Wavefunction (=50%)
Energy
100%
Situation in
Antiferromagnetic
Coupling
Closed Shell
Dominates
50%
BS-DFT
%d=100x(1+|S|)x(1-|S|)
%d=100x(1+|S|)x(1-|S|)
50%
0.5
1.0
1.5
Ab Initio CI
2.0
2.5
Distance
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
3.0
3.5
2 2
2
2 1/2
%d=200x(c
/(c002+c
+cdd2))))1/2
%d=200x(c002ccdd2/(c
My View:
FN (2003) J. Phys. Chem. Solids, 65, 781
Corresponding Orbitals
How to Calculate the Overlap of the Relaxed Magnetic Orbitals ? (Problem: How
to Find and Define them in the Many MOs composing the BS Determinant)
My Suggestion: Use the Corresponding Orbitals
ψ iα → ∑ U αjiψ iα
j
ψ iβ → ∑ U βjiψ iβ
α
β
such that ψ i ψ j ≠ 0
j
S2
α
BS
N −N
= 
2

β
α
 N − N

2

Overlap
For at most 1 j
β

+ 1 + N β − ∑ niα niβ S iiαβ
i

2
120:
1.00000
121:
1.00000
122:
0.99999
123:
0.99999
...
τ
NOTE
BUT
A
τB
146:
0.99475
147:
0.99267
148:
0.16195...
‚Magnetic Pair‘
: Larger Overlap → Stronger AF Coupling
: Overlap is Result of Variational Calculation NOT of Intuition
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
Ghosh et al. (2003) JACS, 125, 1293;
Herebian et al. (2003) JACS, 125, 10997
FN (2003) J. Phys. Chem. Solids, 65, 781
Numerical Comparison
Singlet-Triplet
Singlet-TripletGap
Gap
Difference Dedicated CI
B3LYP
BP
:
:
:
-411 cm-1
-434 cm-1
-1035 cm-1
Conclusion:
• Ab Initio Methods Accurate, Reliable, (Beautiful ☺) but Expensive
• BS-DFT Can Be Used With Caution and Insight BUT Depends Rather
Strongly on the Functional (GGA: , Hybrid: ☺) and Can Have Substantial
Errors
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
Zero-Field Splittings
And
g-Values
Relativistic Quantum Mechanics: 1 Electron
At the Nonrelativistic Level (HF or not!) Nothing can be Understood about EPR
Spectroscopy but almost Everything about Chemistry!
1.
Need to Introduce the Electromagnetic Fields
2.
Need to Introduce the Electron (and Nuclear) Spin in the Hamiltonian
3.
Need to Consider the Additional Relativistic Interactions
For a One Electron System this is Nicely Accomplished by the Dirac Equation:
External Potential
Electron Like States („Large Component“)
ψ Lα 
=
ψ
Two „Spin Components“ L  β 
ψ L 
Positron Like States („Small Component“)
ψ Sα 
=
ψ
Two „Spin Components“ S  β 
ψ S 
Generalized Momentum: π = p − A
r
Introduces the Fields through their Vector Potential A ( ∇A = 0 :Coulomb gauge)
0
1


0 −i
1 0 
 σ y = 
Pauli Matrices: σ x = 
 σ z = 

c = Speed of Light (~137 in atomic units)

1 0
i 0 
 0 − 1
cσ π ψ L 
ψ L 
 V


 


=
E
2 

ψ
−
c
V
c
σ
π
2

 S 
ψ S 
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
Relativistic Quantum Mechanics: Many Electrons
Pragmatic Approach: Add the Nonrelativistic Electron-Electron Repulsion to the
One Electron Dirac Equation (Dirac-Coulomb Operator)
Approximate Schemes:
1.
Full Four Component Relativistic Born-Oppenheimer Calculations (Even Harder than
the Nonrelativistic BO Theory).
2.
Four Component Hartree-Fock Like Approximation (Dirac-Fock Method)
3.
Decoupling of The Large and Small Component (Two Components Methods)
4.
Decoupling of the Spin Components and Introduction of Spin- and Field Dependent
Terms via Perturbation Theory
Types of Additional Terms to be Considered:
1.
Scalar Relativistic (Kinematic) Term (do not Depend on Spin)
2.
Spin-Orbit Coupling
3.
Terms Involving the External Fields and/or the Electron and Nuclear Spins
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
Additional Terms due to Magnetic Fields and Relativity
(Noncomprehensive List)
Details: Neese, F.; Solomon, E.I. in: Miller, JS, Drillon, M. (Eds) Magnetism Vol. IV, Wiley-VCH, 2003, pp 345-466
1.
Scalar Relativistic Terms
a)
b)
π
Darwin Term
2
α 2 ∑∑ Z Aδ (riA )
− 18 α
Mass-Velocity Term
A
2
•„Pauli Expansion“. Alternatives: ZORA,
Douglas-Kroll-Hess,...
• Important in the Core Region
i
∑∇
4
i
i
2.
Spin-Orbit Coupling
a)
b)
One-Electron Part
Two Electron Part
1
2
α 2 ∑∑ Z A riA−3l iAs i
− 12 α
A
2
i
i
3.
l iA = (ri − R A )× p i
i
∑ s ∑ r {l
j ≠i
−3
ij
j
i
• Breit-Pauli Form (Others Possible)
• !! Mixes Different Multiplicities !!
• !! Reintroduces Angular Momentum
in Nondegenerate States !!
• Contributes to almost ALL EPR
Properties (g,D,A)
λLˆ Sˆ don‘t
+ 2l ij
}
l ij = (ri − r j )× p i
Spin-Field Interaction Terms
a)
Electron-Spin-Zeeman Term
b)
Electron-Orbit-Zeeman Term
c)
Nuclear Zeeman Term
1
2
αg e B∑ sˆ i
i
1
• g-Tensor
• Nuclear Shielding Tensor
ˆ
2 αB ∑ l i
i
β N B∑ g N( A )Iˆ A
A
4.
Spin-Spin Terms
a)
Electron-Spin-Spin Term
1
2
α 2 ∑∑ rij−5 {rij2s i s j − 3(s i rij )(s j rij )}
j ≠i
i
b)
Electron Spin-Nucleus Dipole/Dipole
1
c)
Electron Orbital-Nucleus Dipole
1
d)
Fermi Contact Term
4π
2
{
A
2
}
αg e β N ∑ g N( A ) ∑ riA−5 riA2 s i Iˆ ( A ) − (s i riA )(Iˆ ( A )riA )
i
αβ N ∑ g N( A ) ∑ riA−3l iAIˆ ( A )
A
3
i
αg e β N ∑ g N
( A)
A
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
• Zero-Field Splitting
• Hyperfine Interaction
∑ s Iˆ ( )δ (r )
A
i
Ai
i
α = c −1 =Fine Structure Constant
Summary
Dirac Equation
(1-Electron)
Dirac Coulomb Operator
(Relativistic; N-Electrons)
Nonrelativistic
Limit
Additional (Small) Terms
1.
2.
3.
Scalar Relativistic Corrections
Relativistic Spin-Orbit Coupling
Magnetic Field Interactions
Perturbation Theory
Born-Oppenheimer Operator
(Nonrelativistic; N-Electrons)
Schrödinger Equation
(Nonrelativistic; 1-Electron)
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
EPR-Parameters
EPR-Parameters
1.1.
2.2.
3.3.
4.4.
5.5.
6.6.
7.7.
g-Tensor
g-Tensor
D-Tensor
D-Tensor
A-Tensor
A-Tensor
Q-Tensor
Q-Tensor
Chemical
ChemicalShielding
Shielding
Spin-Spin
Coupling
Spin-Spin Coupling
......
Ligand Field Theory
Ligand Field Theory may Be Viewed as a Simple but Logically Consistent and
Qualitatively Correct Way to Construct the most Important Low-Lying
Nonrelativistic Many Electron Wavefunctions
Shape
Shapeof
ofd-Orbitals
d-Orbitals
One-Electron
One-ElectronininaaLigand
LigandField
Field
z
y
d z 2 −r 2
-
x
-
z
z
d xz
-
y
y
x
x
- d z 2 −r 2
Very Sensitive Function of the
Ligand Nature+Arrangement
-
y
y
-
x
z
z
d x2 − y2
d yz
e-
Nonspherical Electric Field Induces
a Splitting of the Five d-Orbitals
e-
-
d -xy d yz
-
dxy
x
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
dx2-y2
22
DD
dz2
dxz
22
TT2g
2g
dyz
eeg
g
t2g
t
2g
Many Electron Ligand Field States
1.
2.
3.
Distribute the N Electrons Among the Available Orbitals in all Possible Ways
(Configuration).
Couple the Spins of the Unpaired Electrons to get a State of Well Defined Total Spin
(Multiplicity).
Determine the Overall State Symmetry by Taking the Direct Product of the Irreps of
all Partially Filled Shells (Symmetry).
Configuration State Functions: n; ΓM Γ ; SMµ
Term Symbol:
2S+1
dd-2S+1ΓΓ
d = 1,2,...
4.
Use the Rules of Ligand Field Theory to Calculate the Energy of the Configuration
State Function as a Function of the Ligand Field Parameters:
5.
1.
Ligand Field (10Dq, Ds, Dt,...) or AOM (eσ,eπ,eδ,...) Parameters
2.
Interelectronic Repulsion Racah (A,B,C) or Slater-Condon (F0,F2,F4) Parameters
Possibly Allow the Interaction of Configuration State Functions of the Same
Symmetry and Multiplicity (Configuration Interaction)
6.
Possibly Allow the Interaction of the CI States Through the Spin-Orbit-Coupling
Operator
Technical Details: Griffith, J.S. (1964) The Theory of Transition Metal Ions., Camebridge University press, Camebridge
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
Tanabe-Sugano Diagrams
Critical Ligand Field Strength
where the High-Spin to LowSpin Transition Occurs
E/B
Energy
RELATIVE to the
Ground State in
Units of the
Electron-Electron
Repulsion
High-Spin
Ground State
(Weak Field)
Low-Spin
Ground State
(Strong Field)
Energy of a
Given Term
Relative to the
Ground State
Zero-Field
(Free Ion Limit)
D/B
Strength of Ligand Field Increases Relative
to the Electron-Electron Repulsion
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
The d5 Case
dx2-y2 dz2
dxy dxz dyz
ε
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
2+
[Mn(H
[Mn(H22O)
O)66] ]2+
hν
All
Spin
Forbidden
Perturbation Theory of SH Parameters
Divide the Complete Set of Many Electron States into Two Sets
1.
a; S 0 M
„Model Space“:
M=S0,S0-1,...,-S0
The 2S+1 Component of the Orbitally Nondegenerate Ground State
2.
„Outer Space“:
b; S b M
All Other States of Any Multiplicity and Symmetry
Goal: Build an „Effective Hamiltonian“ Which is Only Defined in the „a“ Space
but Gives the Same Eigenvalues and Eigenvectors as the Complete
Hamiltonian Including All Relativistic and Magnetic Field Interactions!
Perturbative Analysis to 2nd Order :
aS 0 M H eff aS 0 M ′ = aS 0 M Hˆ 1 aS 0 M ′ −
∑ (E
b ≠ 0 , M ′′
b
− E0 )
−1
{ aS M Hˆ bS M ′′
0
1
b
aS b M ′′ Hˆ 1 aS 0 M ′
+ aS 0 M Hˆ 1 bSb M ′′ bS b M ′′ Hˆ 1 aS 0 M ′
}
Ĥ1 = Sum of ALL Relativistic and Magnetic Field Interactions
Insertion of the Explicit Forms and Term-by-Term Comparison yields Explicit Expressions
for the Spin-Hamiltonian Parameters in Terms of Many Electron States and Their Energies
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
Straightforward but Sometimes Tedious
The g-Tensor
First Order Contributions:
Isotropic Free Electron
( SZ )
g µν
= δ µν g e
( RMC )
g µν
= δ µν
(GC )
=
g µν
α 2 1 ge
2 S 2
aS 0 S 0 ∑ ∇ i2 s zi aS 0 S 0
Relativistic Mass Correction (Small)
i
1
aS 0 S 0 ∑ ξ (riA ){riAri − riA, µ ri ,ν }siz aS 0 S 0
S
i, A
Gauge Correction (Small)
Second Order Contribution:
(OZ / SOC )
=−
g µν
{
1
(Eb − E0 )−1 aS0 S0 ∑i liµ bS0 S0 bS0 S0 ∑i , A ξ (riA )liAν siz aS0 S0
∑
S b ( Sb = S 0 )
+ aS 0 S 0 ∑i , A ξ (riA )liAµ siz bS 0 S 0 bS 0 S 0 ∑i liν aS 0 S 0
}
Orbital Zeeman/SOC
Usually Dominant
NOTES: • Only Excited States of the Same Total Spin as the Ground State!
• Only Need to Know the „Principal Components“ with M = S!
Lµ b b Lν 0
• The Often Quoted Equation g µν = g − 2λ ∑ (E − E ) 10 4
42443
−1
e
b
b
0
b)
Λ(µν
Is not Correct and Makes Only Limited Sense!
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
FN (2001) Int. J. Quant. Chem., 83, 104
The D-Tensor
First Order Contribution: Direct Electron-Electron Magnetic Dipole Interaction
( SS )
Dµν
1
α2
=
0 S 0 S 0 ∑i ∑ j ≠i rij−5 rij2δ µν − 3(rij ) µ (rij )ν {2 s zi s zj − s xi s xj − s yi s yj }aS 0 S 0
2 S 0 (2 S 0 − 1)
{
}
Complicated!
Assumed Small for Transition Metals
Second Order Contribution: Spin-Orbit Coupling
Neese, F.; Solomon, E.I. (1998) Inorg. Chem., 37, 6568
SOC − (0 )
Dµν
=−
1
S 02
∆
∑
(
)
b Sb = S
−1
b
aS 0 S 0 ∑i , A ξ (riA )liA, µ si , z bS 0 S 0 bS 0 S 0 ∑i , A ξ (riA )liA,ν si , z aS 0 S 0
SOC − ( −1)
Dµν
=−
1
∆−b1 aS 0 S 0 ∑i , A ξ (riA )liA, µ si , +1 bS 0 − 1S 0 − 1 bS 0 − 1S 0 − 1 ∑i , A ξ (riA )liA,ν si , −1 aS 0 S 0
∑
S (2S − 1) b ( Sb = S −1)
SOC − ( +1)
Dµν
=−
1
∆−b1 aS 0 S 0 ∑i , A ξ (riA )liA, µ si , −1 bS 0 + 1S 0 + 1 bS 0 + 1S 0 + 1 ∑i , A ξ (riA )liA,ν si , +1 aS 0 S 0
∑
(S0 + 1)(2S0 + 1) b(Sb = S +1)
NOTES: • Excited States S0 and S0 ± 1 Contribute!
• No Proportionality to the g-Tensor!
• Not Traceless!
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
(Review:
Review: FN (2004
(2004)) ZeroZero-Field Splitting. In: Kaupp,
Kaupp, M.; Bühl,
hl, M.; Malkin,
Malkin, V.
(Eds)
Eds) The Quantum Chemical Calculation of NMR and EPR Properties.
Properties.
WileyWiley-VCH, in press
The g-Tensor in Ligand Field Theory
2
Only MOs of Dominantly Metal d-Character Considered: ψ i ≈ αd i + 1 − α Li
Critical!
Metal
Ligand
„Covalently Diluted“ Metal Orbitals with „Anisotropic Covalency“ (αi≠ αj)
Neglect Ligand Part in All Matrix Elements
ψ i oˆ ψ j ≈ α iα j d i oˆ d j
Critical! See Review Article
g µν ≈ δ µν g e +
ζ ion 
−1
2 2
M
M
∑ ∆ i → sα i α s d i lµ d s d s lν d i
S 0 i → s
DOMO→SOMO
−

2 2
M
M
−1
α
α
d
l
d
d
l
d
∆
∑ s→a s a s µ a a ν s 
s →a
SOMO→EMPTY
a
Ψ0 Hˆ SOC Ψoa
-
s
Ψ0 Hˆ SOC Ψio
i
Intrinsic
Moment
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
+
„α-Spin
Currents“
„β-Spin
Currents“
The D-Tensor in Ligand Field Theory
Same Assumption as for the g-Tensor; Also Neglect Spin-Spin First Order Term
SOC − (0 )
Dµν
=−
2
ζ ion


−1
2 2
2 2
M
M
−1
M
M
α
α
d
l
d
d
l
d
α
α
d
l
d
d
l
d
∆
+
∆
∑ i → s i j i µ s s ν i
∑ s→a i j s µ a a ν s 
4 S 02 i → s
s →a
No Sign
DOMO→SOMO
SOMO→EMPTY
Change
Only IF: a) No Contributions from ∆S0 ± 1
b) Either only DOMO→SOMO OR SOMO→EMPTY
SOC − (0 )
Dµν
≈±
ζ average
4S0
∆g µν
Contributions from Excited States with S0 ± 1 not as Simple → Detailed Attention
Convention: Principal Values of D are Ordered Such That
D = Dzz − 1 2 ( DXX + DYY )
E=
( DXX − DYY )
and 0 ≤ E / D ≤ 1 / 3
1
2
ζl1s1
ζl 2 s 2
vs
ζ l 3s 3
ζl 2 s 2
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
ζl1s1
ζ l 3s 3
Angular Momentum Matrix Elements
lx
dz2
dxz
dyz
dx2-y2
i 31/2
dz2
dxz
dyz
i
-i 31/2
-i
dx2-y2
i
dxy
ly
-i
dz2
dxz
dyz
dx2-y2
i 31/2
-i
dx2-y2
i
dxy
i
dz2
dxz
dyz
Use of These Tables
Let‘s You Calculate the Ligand
Field Expressions for g, D and A
as a Function of:
• The Excited State Energies
(Experimental Input or Ligand Field and
Interelectronic Repulsion Parameters)
• The Covalency Parameters
-i
dyz
lz
dxy
-i 31/2
dz2
dxz
dxy
(Result of the Analysis or MO Calculations
but be careful – The Connection is not Clean and
the Accuracy of HF and DFT is Often not High)
• The Free Ion SOC Constants
dx2-y2
dxy
(Usually taken from Tables)
dz2
dxz
dyz
-i
i
dx2-y2
dxy
-2 i
2i
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
Two Additional Parameters Arise for Hyperfine
Interactions
• gegNβeβN<r-3>d from Tables
• Aiso from Fit to Experiment
The g-Tensor of d9 Complexes (S=1/2)
dx2-y2
dz2
dz2
dxz , dyz
dxy
g zz = g e +
dxz , dyz
dx2-y2 , dxy
8ζ Cuα x22 − y 2 α xy2
g zz = g e
∆ xy → x 2 − y 2
g xx = g yy = g e +
2ζ Cuα x22 − y 2 α xz2 , yz
g xx = g yy = g e +
∆ xz , yz → x 2 − y 2
2700
2800
2900
3000
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
3100
3200
Magnetic Field (Gauss)
3300
3400
3500
3600
6ζ Cuα z22 α xz2 , yz
∆ xz , yz → x 2 − y 2
Key Message:
g-Values React
Sensitively to
Geometry
ZFS of Tetrahedral d5 Complexes (S=5/2)
4
T2c
4
T1c
}
t2
e
Td
4
T1b
4
T2b
Γ-CT
manifold
6
4
E
4
A1
}
t2
e
t2
e
D2d
xy
xy
xz
yz
decreases
ζ 2α2
= Fe
5
( )
a
2
a
1
t2
e
A1
t2
e
T
T
4
6
xz,yz ψ xz , yz ≈ γd xz , yz + 1 − γ 2 Lxz , yz
z2,x2-y2
x2-y2 z2
D 4 T1a
4
− ∆t2
ψ xy ≈ β1 d xy + 1 − β12 Lxy
decreases
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr


β12
γ2
−


 E0 + 1 2 ∆ t2 E0 − 1 2 ∆ t2 
ψ x 2 − y 2 ≈ αd xy + 1 − α 2 Lx 2 − y 2
ψ z 2 ≈ αd z 2 + 1 − α 2 Lz 2
increases
increases
Key Messages:
1) Anisotropic Covalency can be a
Key Determinant of the ZFS
2) Spin-Flip Contributions can be
Dominant
Neese, F.; Solomon, E.I. (1998) Inorg. Chem., 37, 6568
d-d versus CT States in FeCl4total
6
Γ-CT-states
4
Γ-LF-states
4
Γ LF states
6
Fe-SOC+Cl-SOC
Cl-SOC only
Fe-SOC only
D (cm-1)
Γ CT states
-1
Dexp= - 0.042 cm
∆Θ (degrees)
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
∆Θ (degrees)
Neese, F.; Solomon, E.I. (1998) Inorg. Chem., 37, 6568
g- and D-Values in Octahedral d8 (S=1)
Oh
Lower Symmetry
3T
1g
3T
2g
3E
3A
3A
2g
3A
Contributions from Triplets
2  4α 2 α 2
xy x − y
ζ ion
2
2
1 α xz , yz (3α z 2 + α x 2 − y 2 ) 
D≈
−


4  E ( 4 A)
4
E(4 E)


≈ (ζ ion / 4) ∆g zz − 1 2 (∆g xx + ∆g yy )
2
[
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
2
]
g- and D-Values in d4 (S=2)
Weak
Axial
Strong
Axial
Octahedral
Usually the Complete D is Attributed to
the Lowest Spin Quintet:
z2
2 2
x-y
2 2
x-y
z2
2 2
x-y
However, the Triplets are at Least Equally
Important:
2
z
yz
xz
5B
1g
5
xy
xz
xy
yz
5E
g
Eg(xz,yz→x2-y2) >0
5
B2g(xy→x2-y2)
<0
5A
1g
=0
5
T2g
>0
<0
3
3
yz
xz
xy
A2g (z2→xy) =0
Eg (z2→xz,yz)
3
<0
T2g
Dr. Thorsten Glaser, Münster
5
B2g(xy→z2)
5
Eg(xz,yz →z2)
3
Eg(x2-y2→xz,yz)
3
2 2
>0 A2g(x -y →xy)
5
5
A
2
2 2
1g(z →x -y )
5
1g
B
D<0
gz< gx,y< ge
5
5
E
B1g(x2-y2→z2)
A1g
D>0
gx,y<gz=ge
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
Mn
Cr
g1
1.994
1.984
g2
2.001
1.988
g3
2.001
1.992
-2.0
-0.4
E/D
0.03
0.33
Contr. to D (S=2; 3/2)
-0.3
-0.2
Contr. to D (S=1; 1/2)
-1.7
-0.2
D
(cm-1)
Quantum Chemistry
Problem:
• Theory is Formulated in Terms of an Infinite Number of Many Electron
States
General but Intellectually Very Inefficient Way to Deal with Small Perturbations
• Most Succesful Quantum Chemical Methods (MPn, CC, DFT, ...) Only
Apply to the Ground State
(Generally: Lowest State of a Given Symmetry)
Alternative to Sum-Over-States is Needed
Response Theory
Study the Total Energy of the Ground State in the Presence of Multiple Small
Perturbations λ1hλ1, λ2hλ2, λ3hλ3,... (Any of the Operators Listed Before)
EE0 →
E (λ ,λ ,λ ,...)
0 → E00(λ11,λ22,λ33,...)
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
Response Theory (ctd.)
Series Expansion:
∂E
E0 (λ1 , λ2 ,...) = E0 (0,0,...) + ∑ λ n 0
∂λn
n
Define Properties:
∂E0
∂λn
λ1 = λ2 =
+ ...
λ =0
First Order
λ =0
∂ 2 E0
∂λn ∂λm
Examples: λ1 =
∂ 2 E0
1
+ ∑ λ n λm
2 n
∂λn ∂λm
λ =0
Second Order
λ1 = Electric Field
λ =0
Electric Field
Dipole Moment
Electric Field
Polarizability
λ1 =
Magnetic Field
λ2 =
Electron Spin
EPR g-Tensor
λ1 =
Magnetic Field
λ2 =
Nuclear Spin
NMR Chemical Shielding Tensor
λ1 =
Electron Spin
λ2 =
Nuclear Spin
Hyperfine Coupling
Elegant
Elegantand
andGeneral
Generalbut
butDerivatives
DerivativesMay
Maybe
beVery
VeryHard
Hardto
toCalculate!
Calculate!
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
g-Tensor: DFT and HF
∂2E
• g-Values can be Defined by: g rs = ∂B ∂S
r
s
• Four Contributions Arise:
g rs = g eδ rs + ∆g RMCδ rs + ∆g rsGC + ∆g rsOZ / SOC
• In the CPKS Approach the Kohn-Sham Molecular Orbitals are Calculated
in the Presence of a Magnetic Field B
σ
j
ψ =ψ
σ (0 )
j
r σ (1)
r
σ (0 )
+ iBψ j + ... = ψ j + iB ∑ U σjbψ bσ (0 ) + ...
b∈σ
• The UKS-CPKS Equations Become:
 Aαα

 0

• Gives:
∆g
OZ / SOC
rs
 Vα 
0  Uα 
 = − β 

ββ  β 
V 
A  U 


1
(− 1)δ σβ
=−
∑
β S σ =α , β
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
with: Viaσ = ψ iσ l ψ aσ
∑σ Im (U )Im ( ψ
i , a∈
σ ;r
ia
σ (0 )
i
r
∑ A ξ (rA )l A,s ψ aσ (0 )
)
FN. (2001) J. Chem. Phys., 115, 11080
Hyperfine Treatment see:
FN (2003) J. Chem. Phys. 3939
FN (2001) J. Phys. Chem. A, 105, 4290
DFT Results: g-Values
Small
SmallRadicals
Radicals
Transition
Transition Metals
Metals
40000
200
O2H-∆gmax
20000
ClO2-∆gmid
10000
+
H2O -∆gmax
0
Calculated g-shift (ppt)
Calculated g-shift (ppm)
30000
BP Functional
B3LYP Functional
250
BP-Functional
B3LYP-Functional
150
100
∆gmax-Cu(II)
50
Observed correlation:
slope=0.71 R=0.968 σ=21 ppt
0
-50
-10000
-100
-10000
0
10000
20000
30000
40000
Experimental g-shift (ppm)
-100
-50
0
50
100
150
200
250
Experimental g-shift (ppt)
• Reasonable but Still Some Problems Especially for Metals
• Zero-Field Splittings Maybe Approached Similarly but there are
Even Some Conceptual Problems
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
FN (2001) J. Chem. Phys., 115, 11080
Spectroscopy ORiented Configuration Interaction
General ab Initio Method For Calculating Excited States and SOS
Based Properties. Designed for Energy Differences
hν
dx2-y2 d
z2
dxy
dxz
dx2-y2 d
z2
dxy
dyz
8
-1
-1
dyz
[Ni(H2O)6]2+
12
ε (M cm )
dxz
SOR-CI
4
0
30000
B3LYP
25000
20000
15000
10000
-1
Wavenumber (cm )
Accurate
AccurateDescription
Descriptionofofd-d
d-dMultiplets
MultipletsShould
Shouldgive
giveGood
GoodEPR
EPRParameter
ParameterPredictions
Predictions
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
FN (2003) J. Chem. Phys., 119, 9428
FN (2003) Chem. Phys. Lett., 380/5-6, 721
SORCI: [Cu(NH3)4]2+
Exp
g-Shifts
SORCI B3LYP
0.241
0.047
0.248
0.058
0.142
0.040
-596
-71?
-591
1
-611
-34
39
31
37
26
49
34
(eV)
2.03
2.08
2.18
2.09
2.37
2.55
2.69
2.55
2.72
∆E(CT)
5.58
5.38
5.22
63Cu-HFC‘s
(MHz)
14N-HFC‘s
(MHz)
∆E(d-d)
(eV)
Max Planck Institute for Bioinorganic Chemistry, Mülheim an der Ruhr
FN (2004) submitted