Download Applying Computational Chemistry to Transition Metal

ESQC 2013
Quantum Chemistry @ Work
Applying Computational Chemistry to
Transition Metal Complexes
Frank Neese
Max Planck Institute for Chemical Energy Conversion
Stiftstr.34-36
Mülheim an der Ruhr
A Primer on Ligand Field Theory
Theory in Chemistry
UNDERSTANDING
NUMBERS
Molecular
Relativistic
Schrödinger
Equation
Nonrelativistic
Quantum
Mechanics
Observable
Facts
(Experiments)
Chemical
Language
Density
Functional
Theory
Hartree-Fock
Theory
Correlated
Ab Initio
Methods
Phenomenological
Models
Semiempirical
Theories
What is Ligand Field Theory ?
★ Ligand Field Theory is:
‣ A semi-empirical theory that applies to a CLASS of substances (transition
metal complexes).
‣ A LANGUAGE in which a vast number of experimental facts can be
rationalized and discussed.
‣ A MODEL that applies only to a restricted part of reality.
★ Ligand Field Theory is NOT:
‣
An ab initio theory that lets one predict the properties of a compound ‚from
scratch‘
‣
A physically rigorous treatment of transition metal electronic structure
Principles of Ligand Field Theory
R-L|
M
δδ+
Strong Attraction
Example:
Electrostatic Potential
Negative
Positive
Complex Geometries
3
Trigonal
Trigonal Pyramidal
4
Quadratic Planar
Tetrahedral
5
Quadratic Pyramidal
6
Octahedral
Trigonal Bipyramidal
T-Shaped
Coordination Geometries
- Approximate Symmetries Observed in Enzyme Active Sites Tetrahedral
Td
Rubredoxin
Trigonal
Bipyramidal
D3h
3,4-PCD
Tetragonal
Pyramidal
C4v
Tyrosine
Hydroxylase
Octahedral
Oh
Lipoxygenase
The Shape of Orbitals
z
y
x
z
z
y
y
x
z
x
z
y
y
x
x
A Single d-Electron in an Octahedral Field
-
e
The Negatively Charged Ligands Produce an Electric Field+Potential
-
The Field Interacts with the d-Electrons on the Metal (Repulsion)
-
The Interaction is NOT Equal for All Five d-Orbitals
dz2
1. The Spherical Symmetry of the Free Ions is Lifted
2. The d-Orbitals Split in Energy
3. The Splitting Pattern Depends on the Arrangement of the Ligands
-
e-
-
-
dyz
2D
dx2-y2 dz2
eg
dxy dxz dyz
t2g
2T
2g
Making Ligand Field Theory Quantitative
Charge Distribution of Ligand Charges:
Ligand field potential:
Expansion of inverse distance:
qi=charges
Hans Bethe
1906-2005
Slm=real spherical harmonics
r<,> Smaller/Larger or r and R
Insertion into the potential:
„Geometry factors“:
Ligand-field matrix elements:
Ligand-field splitting in Oh:
=Gaunt Integral
(tabulated)
Don‘t evaluate these integrals analytically, plug in and compare to experiment! LFT is not an ab initio theory (the
numbers that you will get are ultimately absurd!). What we want is a parameterized model and thus we want to
leave 10Dq as a fit parameter. The ligand field model just tells us how many and which parameters we need what
their relationship is
Optical Measurement of Δ: d-d Transitions
Excited State
eg
[Ti(H2O)6]3+
t2g
2T →2E Transition
2g
g
(„Jahn-Teller split“)
hν=ΔOh
eg
Δ≡ΔOh ≡10Dq
t2g
Ground State
The Spectrochemical Series
A „Chemical“ Spectrochemical Series
I- < S2- < F- < OH- < H2O < NH3 < NO2- < CN- < CO~ NO < NO+
Δ SMALL
Δ LARGE
A „Biochemical“ Spectrochemical Series (A. Thomson)
Asp/Glu < Cys < Tyr < Met < His < Lys < HisΔ SMALL
Δ LARGE
Ligand Field Stabilization Energies
Central Idea:
➡ Occupation of t2g orbitals stabilizes
the complex while occupation of eg
orbitals destabilizes it.
➡ Ligand Field Stabilization Energy (LSFE)
dN
1
2
3
4
5
6
7
8
9
10
LSFE
-2/5Δ
-4/5Δ
-6/5Δ
-3/5Δ
0
-2/5Δ
-4/5Δ
-6/5Δ
-3/5Δ
0
+3/5Δ
-2/5Δ
eg
t2g
3d
Experimental Test:
➡ Hydration energies of hexaquo M2+
High-Spin and Low-Spin Complexes
QUESTION: What determines the electron configuration?
OR
High-Spin
Low-Spin
ANSWER: The balance of ligand field splitting and electron
repulsion (‚Spin-Pairing Energy‘ P=f(B))
Δ/B-SMALL
(Weak Field Ligand)
High Spin
Δ/B ~20-30≡ LARGE
(Strong Field Ligand)
Low-Spin
Inside Ligand Field Theory: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)
Δ/B
Strength of Ligand Field Increases Relative
to the Electron-Electron Repulsion
Figuring out the Possible Multiplets
★ The irrep of the state follows from the direct product of the singly occupied MOs
(closed shells are always totally symmetric!)
★ For non-degenerate point groups (only 1-dimensional irreducible representations),
the result is unique and follows from the direct product table.
★ For degenerate point groups the direct product is itself reducible and several
possibilities arise (Multiplet structure).
★ Example: D2h
Singly occupied MOs: ag, b1g, b2u, au:
State Symmetry: ag ⊗ b1g = b1g
b1g ⊗ b2u = b3u
b3u ⊗ au = ag ➯ B3g
Normal:
a ⊗ a = a, b ⊗ b = a, a ⊗ b = b
D2h:
b1 ⊗ b2 = b3 and cyclic
Always:
g ⊗ g = g, u ⊗ u = g g ⊗ u = u
★ Example: Oh
Singly occupied MOs: (t2g)3(eg)2 and S=5/2
State Symmetry: t2g ⊗ t2g ⊗ t2g = a2g (totally antisymmetric)
eg ⊗ eg
= a2g (totally antisymmetric)
a2g ⊗ a2g
= a1g ➯ A1g
ti ⊗ ti = a1 ⊕ e ⊕ [t1] ⊕ t2 (i=1,2)
t1 ⊗ t2 = a2 ⊕ e ⊕ t1 ⊕ t2
ti ⊗ e = t1 ⊕ t2
(i=1,2)
ei ⊗ ei = e ⊕ a1 [⊕ a2]
(i=1,2)
e1 ⊗ e2 = e ⊕ b1 ⊕ b2
Optical Properties:d-d Spectra of d2 Ions
dx2-y2 dz2
dxy
dxz
dyz
hν
dx2-y2 dz2
dxy
dxz
dyz
[V(H2O)6]3+
Optical Properties:d-d Spectra of d3 Ions
dx2-y2 dz2
dxy dxz dyz
hν
dx2-y2 dz2
dxy dxz dyz
dx2-y2 dz2
dxy dxz dyz
d-d Spectra of
d5
Ions
III
(Fe ,
II
Mn )
dx2-y2 dz2
dxy dxz dyz
ε
hν
All
Spin
Forbidden
Ligand Field Splittings in Different Coordination Geometries
Octahedral
Oh
x2-y2
Tetragonal
Pyramidal
C4v
z2
Square
Planar
D4h
x2-y2
xy
xz
yz
xz
yz
Tetrahedral
D3h
Td
z2
x2-y2
xy xz
z2
xy
Trigonal
Bipyramidal
xz
xy
z2
xz yz
yz
yz
x2-y2 z2
xy x2-y2
Coordination Geometry and d-d Spectra: HS-Fe(II)
10,000 cm-1
6C
eg
t2g
5,000 cm-1
10,000
cm-1
7,000 cm-1
<5,000 cm-1
5C
b2
e
a1
b1
5C
a1
e
e
4C
5,000 cm-1
t2
e
5,000
10,000
Wavenumber (cm-1)
15,000
Applying LFT in Studying Enzymes
Rieske-Dioxygenases
O2,
2e-,
2H+
Active Site Geometry from d-d Spectra
+Substrate
-Substrate
Δε (M-1 cm-1 T-1)
Rieske only
Δε (M-1 cm-1 T-1)
Holoenzyme
Difference
6000
8000 10000 12000 14000 14000
Energy (cm-1)
Solomon et al., (2000) Chem. Rev., 100, 235-349
6000
8000 10000 12000 14000 14000
Energy (cm-1)
Mechanistic Ideas from Ligand Field Studies
-OOC
COO-
-OOC
Fe2+
COO-
Fe2+
2e- from
reductase
-OOC
Fe4+
COO-
H
H
-O
O-
Fe4+
-OOC
H
HO
COOH
OH
products
Solomon et al., (2000) Chem. Rev., 100, 235-349
2H+
-OOC
COO-
O
+O2
O
Fe2+
-OOC
COO-
(O
-OOC
-O
O)2-
Fe4+
COO-
or
Fe3+
O (H+)
Beyond Ligand Field Theory
Beyond Ligand Field Theory
„Personally, I do not believe much of the electrostatic
romantics, many of my collegues talked about“
(C.K. Jörgensen, 1966 Recent Progress in Ligand Field Theory)
Experimental Proof of the Inadequacy of LFT
[Cu(imidazole)4]2+
Ligand Field Picture
N
N
N
N
Wavefunction of the Unpaired
Electron Exclusively Localized on
the Metal
No Coupling Expected
Clearly Observed Coupling Between
The Unpaired Electron and the Nuclear Spin
of Four 14N Nitrogens (I=1)
Need a Refined Theory that
Includes the Ligands Explicitly
MO Theory of ML6 Complexes
t1u
Metal-p
(empty)
Key Points:
a1g
Metal-s
(empty)
eg
t2g
Metal-d
(N-Electrons)
Ligand-p
(filled)
t1g,t2g,3xt1u,t2u
2xeg,2xa1g
Ligand-s
(filled)
‣ Filled ligand orbitals are lower in
energy than metal d-orbitals
‣The orbitals that are treated in
LFT correspond to the antibonding metal-based orbitals
in MO Theory
‣ Through bonding some electron
density is transferred from the
ligand to the metal
‣ The extent to which this takes
place defines the covalency of
the M-L bond
MO Theory of ML6 Complexes
t1u
Metal-p
(empty)
Key Points:
a1g
Metal-s
(empty)
eg
t2g
Metal-d
(N-Electrons)
Ligand-p
(filled)
t1g,t2g,3xt1u,t2u
2xeg,2xa1g
Ligand-s
(filled)
‣ Filled ligand orbitals are lower in
energy than metal d-orbitals
‣The orbitals that are treated in
LFT correspond to the antibonding metal-based orbitals
in MO Theory
‣ Through bonding some electron
density is transferred from the
ligand to the metal
‣ The extent to which this takes
place defines the covalency of
the M-L bond
MO Theory and Covalency
[Cu(imidazole)4]2+
The Unpaired Electron is Partly
Delocalized Onto the Ligands
Magnetic Coupling Expected
1. Theory Accounts for Experimental Facts
2. Can Make Semi-Quantitative Estimate
of the Ligand Character in the SOMO
π-Bonding and π-Backbonding
[FeCl6]3dx2-y2
dxy
[Cr(CO)6]
dx2-y2
dz2
dxz
dyz
dxy
dz2
dxz
dyz
Interpretation of the Spectrochemical Series
I- < S2- < F- < OH- < H2O < NH3 < NO2- < CN- < CO~ NO < NO+
Δ SMALL
Δ LARGE
π-DONOR
π-‘NEUTRAL‘
eg
π-ACCEPTOR
eg
eg
L-π∗
t2g
t2g
M-d
M-d
L-π,σ
M-d
L-σ
t2g
L-σ
Computational Treatment of Transition Metals
Is Every Transition Metal Complex a Multireference Problem?
The „conventional wisdom“ dictates that transition metal
complexes are multireference cases
..... Really ?
Is Every Transition Metal Complex a Multireference Problem?
The „conventional wisdom“ dictates that transition metal
complexes are multireference cases
..... Really ?
Let‘s look at the ground states of octahedral complexes:
2T
3T
5E
g
4T
2g
1g
1g
4A
2g
5E
g
6A
1g
3A
2g
2E
g
1A
1g
So, What is the Case?
2T
2g
3T
4A
2g
5E
g
6A
1g
5T
2g
4T
3A
2g
2E
g
1A
1g
1g
1g
Nondegenerate term, fine
STRONG Jahn-Teller
distortion will lift
degeneracy
Weak Jahn-Teller
distortion will lift the
degeneracy
Either naturally or by means of the Jahn Teller effect or by means of a low
symmetry ligand environment, the ground states of mononuclear complexes are
practically always conceptually well described by one determinant
A single determinant is not automatically appropriate for „magnetically interacting“ ions
Excited multiplets may formally require higher excitation levels from a single det.
An example of a missed multiplet
[Ni(H2O)6]2+
FN, J. Biol. Inorg. Chem. 11 (2006) 702; FN Coord. Chem. Rev. 253 (2009) 526.
Many Missed Multiplets: L-edge X-ray Absorption
continuum
Laporte
Rule
3d
3p
3s
L-edge
2p
700
710
720
730
u
g
g
g
Energy (eV)
2s
K-edge
1s
7105
7115
7125
Energy (eV)
Dipole
forbidden!
Multiplets and Spin Couplings
L-edge excitations lead to final states with a (n-1)p core hole, e.g.:
nd
X-Ray
(n-1)p
Electronic State:
6 x 25 6Γ,4Γ, 2Γ = 550 CSFs (1512 STATES)
only 15 particle/hole pairs
The p/h space does only spans a small part of the final state
manifold!
Its eigenspectrum cannot possibly coincide with the true
(non-relativistic) eigenspectrum
An Example: The Mn2+ Ion (6S, high-spin d5)
Many particle spectrum
(RAS-CI)
p/h spectrum
(BP86 TD-DFT +30 eV)
Transition Energy (eV)
This looks and actually is
unrealistic
Focusing only on the
sextet states the p/h
pattern is qualitatively
correct but all multiplets
arising from spin-flips are
missing.
6P
6D
6F
6P
4P
6D
6F
However, once SOC is
turned on the grave
deficiencies of the
p/h spectrum will show up
... Hartree-Fock (based) theory may still be a
graveyard for transition metal chemistry
38
Choosing the reference determinant: (FeO)2+
Fe 3dz2
Fe 3dx2-y2
Fe 3dxz
Fe 3dyz
Much studied
intermediate, ready to
attack Taurine
Fe 3dxy
O px
O pz
O py
Relative Energy (kcal/mol)
30
20
10
0
-10
-20
-30
-40
-50
-60
A. Ghosh, E. Tangen, H. Ryeng, P.R. Taylor, Eur. J. Inorg. Chem., 2004, 23, 4555; Bassan, A.; Borowski, T.; Siegbahn, P.E.M. Dalton, 2004, 3153; F. Neese, J. Inorg. Biochem., 2006, 100, 716;
DeVisser, S. J. Am. Chem. Soc., 2006,128, 9813; Klinker, E.J.; Jackson, T.A.; Jensen, M.P.; Stubna, A.; Juhasz, G.; Bominaar, E.L.; Münck , E.; Que Jr., L. Angew. Chem.-Int. Edit., 2006,
39 45,
7394; L. Bernasconi, M. J. Louwerse and E. J. Baerends, E. J. Inorg. Chem, 2007, 3023; Hirao, H.; Que, Jr., L.; Shaik, S. Chem. Eur. J., 2008, 14, 1740;Decker, J. U. Rohde, E. J. Klinker, S. D.
Wong, L. Que Jr. and E. I. Solomon, J. Am. Chem. Soc., 2007, 129, 15983;
Problems with finding a reference determinant
S=2/S=3 Splitting of the iron-oxo intermediate:~20 kcal mol-1 with B3LYP
Discussion: FN, Liakos, DG; Ye, S. J. Biol. Inorg. Chem., 2011,16, 821
40
Problems with finding a reference determinant
S=2/S=3 Splitting of the iron-oxo intermediate:~20 kcal mol-1 with B3LYP
.... must be too low based on ligand field considerations
Discussion: FN, Liakos, DG; Ye, S. J. Biol. Inorg. Chem., 2011,16, 821
40
Problems with finding a reference determinant
S=2/S=3 Splitting of the iron-oxo intermediate:~20 kcal mol-1 with B3LYP
.... must be too low based on ligand field considerations
.... try CCSD(T)
Discussion: FN, Liakos, DG; Ye, S. J. Biol. Inorg. Chem., 2011,16, 821
40
Problems with finding a reference determinant
S=2/S=3 Splitting of the iron-oxo intermediate:~20 kcal mol-1 with B3LYP
.... must be too low based on ligand field considerations
.... try CCSD(T)
.... get 7 kcal mol-1 ????
Discussion: FN, Liakos, DG; Ye, S. J. Biol. Inorg. Chem., 2011,16, 821
40
Problems with finding a reference determinant
S=2/S=3 Splitting of the iron-oxo intermediate:~20 kcal mol-1 with B3LYP
.... must be too low based on ligand field considerations
.... try CCSD(T)
.... get 7 kcal mol-1 ????
.... what is happening here ?
Discussion: FN, Liakos, DG; Ye, S. J. Biol. Inorg. Chem., 2011,16, 821
40
Results based on a UHF determinant
The UHF wavefunction:
5A
<S2>
6.94 (6.0)
„5A“
7A
12.01 (12.0)
ρFe
4.54
4.60
ρO
-0.78
1.11
Linearized CCSD density:
ρFe
4.23
4.37
ρO
-0.70
1.05
HS-FeIIIO●-
41
Results based on a UHF determinant
The UHF wavefunction:
5A
<S2>
6.94 (6.0)
7
„5A
A“
7A
12.01 (12.0)
ρFe
4.54
4.60
ρO
-0.78
1.11
Linearized CCSD density:
ρFe
4.23
4.37
ρO
-0.70
1.05
HS-FeIIIO●-
41
Results based on a restricted determinant
The CASSCF(10,8) wavefunction:
5A
7A
6.0
12.0
ρFe
3.76
4.71
ρO
0.11
1.08
<S2>
5A
CCSD with the leading natural det.
ρFe
3.41
4.37
ρO
0.32
1.05
IV
2HS-Fe O
42
Results based on a restricted determinant
The CASSCF(10,8) wavefunction:
5A
7A
6.0
12.0
ρFe
3.76
4.71
ρO
0.11
1.08
<S2>
5A
7
CCSD with the leading natural det.
ρFe
3.41
4.37
ρO
0.32
1.05
IV
III
2●HS-Fe
HS-Fe O
O
42
Improved energetics
Energetics:
5A
Eh
7A
Eh
ΔE
kcal mol-1
-1678.098...
-1678.086...
7.6
CASDET/CCSD(T): -1678.148...
-1678.086...
40.2
...
50.5
UHF/CCSD(T):
SORCI(10,8):
...
43
What is the problem with UHF?
The problem with open-shell transition metals and UHF is the fundamental imbalance
that is built into Hartree-Fock theory
✓ Electrons of like spin experience an enormously strong exchange
stabilization (... leading to „inverted bonding“ where the metal d-orbitals
fall below the ligand orbitals)
✓ Electrons of opposite spin experience the full blown, unscreened
electron-electron interaction that is strongly repulsive
‣
‣
‣
Hartree-Fock theory is OVERWHELMINGLY, HUGELY biased in favor of unpaired
electrons and high spin states
Hartree-Fock will frequently break symmetry and converge to „crazy“ solutions (there
are many of those)
No reasonable level of correlation can „repair“ this inappropriate electronic structure
YOU have to LOOK AT what you are calculating!
Spin State Energies: Hartree-Fock vs DFT
E(S=2)-E(S=0) in kcal/mol (ZPE is not subtracted but is easy to calculate).
[Co(NH3)6]3+
[Fe(NH3)6]2+
BP
47,4
1,4
PBE
47,3
1,6
TPSS
49,9
4,4
PBE0 (25% HFX)
28,5
-19,5
TPSSh (10% HFX)
40,4
-4,6
B3LYP (20% HFX)
32,3
-14,4
B2PYLP
42,3
-12,5
UHF
-43,4
-81,1
SORCI
-
-32,2
★ Results vary a lot with functional.
(def2-QZVPP basis set, PBE/def2-TZVP geometries)
★ GGA functionals overstabilize the low-spin state
★ Hartree-Fock exchange stabilizes the high-spin states
The Spin State Problem:
Exchange Coupling
What is Exchange ?
The interaction of two paramagnetic ions (or more generally fragments) leads to a
„ladder“ of total spin states which are described phenomenologically by the
Heisenberg-Dirac-VanVleck Hamiltonian
St=4
Ion A
8J
Ion B
St=3
N
N
S
Partially Filled
d-Shell
Net
Magnetic
Moment
SA=5/2
S
„Magnetic“
Interaction
Net
Magnetic
Moment
Partially Filled
d-Shell
6J
St=2
St=1
4J
What is the origin of this „magnetic“ interaction and how do we calculate it?
With no other magnetic interactions, the energy of a given spin-state is simply:
SB=3/2
„Exchange Coupling“: Anderson Model
2e- in 2 orbitals problem:
ψ1,2 = 12 (a ±b )
Neglect overlap for a moment
1
3
Ψ 2 = ψ2 ψ2
1
Ψ3 =
1
2
(ψ ψ
1
2
− ψ1 ψ2
3-Singlets
1-Triplet
4Fab2
J aa −J ab
≡ 2J HDvV
Heisenberg-Dirac-van Vleck „effective“ Exchange (H=-2JSASB)
States
Ionic
)
Ψ1 = ψ1 ψ2
After-CI: E ( 1 Ψ ) − E ( 3 Ψ ) ≈ 2Kab −
Orbitals
1
Ψ1 = ψ1 ψ1
Integrals
(
)
J aa = Jbb = a(1)a(1) | r12−1 | a(2)a(2)
On-Site Coulomb Integral
(
)
J ab = a(1)a(1) | r12−1 |b(2)b(2) ∝ Rab−1
Inter-Site Coulomb Integral
(
)
Kab = a(1)b(1) | r12−1 | a(2)b(2) ∝ e
−αRab
Inter-Site Exchange Integral
Neutral
(
+ (b(1)b(1) | r
)
| a(2)b(2))
Fab = hab + a(1)a(1) | r12−1 | a(2)b(2)
−1
12
Fock Like „Transfer“ Integral
„Exchange Coupling“
2J
1. The „Exchange Hamiltonian“ does NOT follow from magnetic interactions (there is
no such thing as an „exchange interaction“ in nature)
2. The Born-Oppenheimer Hamiltonian is enough to describe the splitting of states
of the same configuration (unlike spin-crossover!) but different multiplicity
3. The splitting is related to the electron-electron interaction and the antisymmetry of
the N-particle wavefunction
4. The simplest (Anderson) model leads to two contributions:
Kab = (a(1)b(1) | r12−1 | a(2)b(2)) > 0 (Ferromagnetic)
a) The „Direct Exchange“ :
2
F
β
ab
b) The „Kinetic Exchange“ :
−
≡− <0
(Antiferromagnetic)
J aa −J ab
U
We can talk and argue this way, but how do we calculate it ?
A model calculation: [Cu2(μ-F)(H2O)6]3+
A Model Calculation: [Cu2(µ-F)(H2O)6]3+
The Hartree-Fock SOMOs of the triplet state („active“ orbitals)
~0.7 eV
The pseudo-localized „magnetic orbitals“
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, ...)
Values of Model Parameters:
(
)
Kab = a(1)b(1) | r12−1 | a(2)b(2) = 17 cm −1
„Direct“ (Potential) exchange term:
Exactly calculated „kinetic“ exchange term:
Is that accurate? Look at the singlet wavefunction:
BUT:
• The ionic parts are too high in energy and mix too little with the neutral configuration
(electronic relaxation)
• What about charge transfer states?
Ø Need a more rigorous electronic structure method:
a) Difference Dedicated CI (DDCI): Malrieu, Caballol et al.
b) CASPT2: Roos, Pierloot
c) 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
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:
(~105 Configurations)
Kab = 17 cm −1
Look at the singlet wavefunction
Reduced
Increased!
NEW+IMPORTANT
The Anderson model is not really realistic and should not be taken literally even
though its CI ideas are reasonable.
‣ Relaxation of ionic configurations are important („dressing“ by dynamic
correlation: (DDCI, CASPT2,MC-CEPA,...)
‣ LMCT states are important
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 a αbα
The dominant part of the singlet wavefunction is:
1
neutral =
1
core a b
(
2
α β
− core a βbα
)
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 aαbβ or core a βbα as starting point BUT reoptimize
Minimize Energy !!!
orbitals core aαbβ ⎯ ⎯
⎯ ⎯ ⎯ ⎯→ core τ αAτ βB ≡ BS
‣
NOTE:
‣
ASSUME:
are no longer orthogonal in their space parts
~ 50% Singlet, 50% Triplet
Better:
(Yamaguchi)
Broken Symmetry Limits
Weak-Limit:
Strong-Limit:
Intermediate-Region:
BS → core a αbβ →
1
S
(
2
BS → core ψ1α ψ1β → CS
BS → mixture
0
+ T0
) ⎯ ⎯→ E (S ) − E (T ) = 2(E (BS ) − E (T ))
0
( )
0
( )
0
(
( ))
⎯ ⎯→ E S 0 − E T0 = 2 E (CS ) − E T0
( )
( )
⎯ ⎯→ E S 0 − E T0 = −2
Yamaguchi
( )
E (BS ) − E T0
S2
BS
− S2
T
Soda, T.; et al. Chem. Phys. Lett. 2000, 319, 223
The Broken Symmetry Wavefunction
Relaxation of Orbitals ‚a‘ and ‚b‘:
Energy gain through partial
delocalization gives
nonorthogonality of space parts
(S~0.16 here)
Terrible mistake:
Negative spin
density
In the real world
everywhere zero for a
singlet state!!!
Positive spin
density
My View:
FN (2003) J. Phys. Chem. Solids, 65, 781
Interpretation of the BS Wavefunction
Diradical Index:
100%
Closed Shell
Dominates
50%
Situation in
Antiferromagnetic
Coupling
200 x Percentage Neutral
Character on Top of the
Closed Shell Wavefunction
(=50%)
BS-DFT
%d=100x(1+|S|)x(1-|S|)
50%
Ab Initio CI
%d=200x(c02cd2/(c02+cd2))1/2
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?)
Our suggestion: Use the corresponding orbitals
ψiα →
β
i
ψ →
α α
U
∑ ji ψj
j
∑U
j
β
ji
α
i
such that
β
j
ψ
ψ |ψ
⎛N − N
⎜
= ⎜⎜
⎜⎝
2
α
S
2
BS
β
j
β
≠ 0 For at most 1 j
⎞⎟⎛ N − N
⎞⎟
⎟⎟⎜⎜
⎟⎟ + N − ∑ n αn β S αβ
+
1
⎜
β
i i
ii
⎟⎟⎜
⎟⎟
2
i
⎠⎝
⎠
α
Overlap
β
2
120:
1.00000
121:
1.00000
122:
123:
0.99999
0.99999
...
146:
0.99475
147:
0.99267
148:
0.16195...
NOTE : Larger overlap → Stronger AF coupling
BUT : Overlap is result of variational calculation NOT of intuition
‚Magnetic Pair‘
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 Gap
Difference Dedicated CI :
B3LYP
:
BP
:
-411 cm-1
-434 cm-1
-1035 cm-1
Conclusion:
➡ Ab Initio Methods accurate, reliable, (beautiful J) but expensive
➡ BS-DFT can be used with caution and insight BUT depends rather
strongly on the functional (GGA: L, hybrid: J) and can have substantial
errors
Summary
★ Transition metals are difficult for quantum
chemistry, but that there is magic surrounding
them
★ Ligand field theory is the language of choice in
transition metal chemistry and having some
working knowledge of LFT, much better quantum
chemical calculations will likely be performed.
★ Hartree Fock theory is a terrible starting point for
transition metals because the Coulomb/Fermi
imbalance leads to severely poor reference
determinants
★ CASSCF is better, but not that much better. Both
severely overestimate the ionicity of metal-ligand
bonds.
★ Single reference treatments are usually
applicable (with some caution). for
„magnetically“ interacting ions, broken symmetry
can be thought of as „poor man‘s CASSCF“
http://www.cec.mpg.de/downloads.html