Transcript Lecture 4

Modelling of the Spectroscopic and Magnetic
Properties of Transition Metal Complexes
Using DFT and Ligand Field Theory (LFDFT)
Mihail Atanasov
ACI,Univ.of Heidelberg, Germany
Inst. of General and
Inorg.Chem.Bulgarian Acad.of Sciences
SCM, Vrije Universiteit,Theoretical Chemistry
Amsterdam, 05. 10. 2006
What are coordination compounds?
The LFDFT
Electronic
Structure
Multidet.
DFT
Calculations:
 (CI - DFT)
Ab-initio,
Semi-empirical,
DFT, etc. …


eff
ˆ
Exps.: H
e.g. LFT, AOM,

HVVD, SpinHamiltonian, etc.
Ligand Field Theory:
ligand field orbitals
3d-M1
<3d|h|L> <3d1|h|3d2>
3d1´
ns, np L
exchange:
direct or
indirect
(via
bridging
ligands)
<3d1´|h|3d2´>
<3d|h|L>
3d2´
3d-M2
Ligandfield states of a d2 ion in a cubic field
1
S
1
1
A1
A1
1
E
1
A1
1
G
3
A2
1
1
T2
1
T2
T1
3
1
P
1
T1
3
3
T1
T2
3
1
D
2
e
E
t2e
T1
1
T2
1
A1
1
E
3
A2
3
F
1
T2
3
T2
3
3
T1
T1
Ion
libre
Champ
faible
2
t2
Champ cristallin
Champ
fort
Configurations
The hole function
Fermi hole
Coulomb hole
MultiDeterminant
DFT
Problem:
The present functionals do’nt include
Near-Degeneracy Correlation !
Solution:
Multi-Determinantal DFT is needed
Consider a simple example
H2 Dissociation Energy Curve:
an illustrative Example
Dissociation curve of H2 using DFT
Spin restricted
Spin unrestricted
Coulson-Fisher point
Total Spin of unrestricted DFT wave function
<S2>=S(S+1)
Coulson-Fisher point
Comparison with exact calculations
Spin restricted
Spin unrestricted
Exact (Kolos-Wolniewicz)
Multi-Determinant method for near-degeneragy correlation
u 
1sA

 1sB
2 1S2



1sA

1sB

g 
1sA

 1sB
2 1 S2


Non-Redundant Single Determinants
u

g
 2g
 g u
 g u
 u2





1 
g
 1g  1u 
1 
u
 u2
1 
g
 2g

 3  u
Energies of the single determinants
 u2

 g u
 2g



 g u


Secular equation
 

E  2g


K

  g u

E + 0 
g u 
 

2 
0
E

E  u 
- 

K
 
 
E    E
ˆ  + -  E  J
E  +g -u  E av   +g -u G
g u
av
 g u
+ +
g u

+ + ˆ
+ +



G

av
g u
g u  E av  J g u  K g u
 
K g u  E  +g -u  E  +g u+

Multiplet energies
 
 

E  2g

E  + -  E  + +
g u
g u


   E
E
1 
g
 2g
  
E
3 
u
E
 
 

   E
E 1 g  u2

 g  u


E  +g -u  E  +g u+ 


E  u2



 


 
E 1 u  2E  g  u  E  g  u

Energies of the Multiplets
1 
g

1  1 1
 u  g u

1 
g
 u2 

 2g 
3 
u


 1g 1u 
H2 Dissociation Energy Curve:
• Conventional DFT gives a good description near the equilibrium
geometry
• Restricted DFT (same orbital for different spins) does not give
the correct long range behaviour
• The Multi-Determinant Method yields the correct behaviour at
all distances
• Allowing the spin  and  wave functions to have different
spatial components (Unrestricted scheme) yields a Localised (BS)
state and includes near-degeneracy correlation (correct asymptotic
behaviour)
LF-DFT
Ligandfield calculation of MXn based on DFT
Hamiltonian
n
n
n
ZM N Zl  n 1
1
H    i         d  i  si
2 i 1
i 1  ri
l rli  i j 1rij
i 1
n
n
1
H  H0 VLF   d 
i j 1rij
i 1
i
 si
The following parameters are needed
Ligandfield parameters:
d VLF d   ci ci iKS
Electrostatic matrix elements:
i
d d d d  f Racah parameters
B,C
Calculation of ligandfield and Racah’s parameters
Step 1: optimise the geometry of the ground state using LDA
Step 2: get SCF, spin-restricted, KS-orbitals for an averaged
ligandfield dn configuration: e2n/5 t23n/5
10
Step 3: calculate the energies of all   single determinants
n 
using the frozen KS-orbitals obtained in step 2
Step 4: use the energies of all these microstates to fit the
ligandfield and Racah’s parameters
step ii)
SYMMETRY T(D)
SCF
ITERATIONS 200
MIXING 0.20
END
mv tape21 feo4_aoc.t21
RESTRICTED
CHARGE -2
OCCUPATIONS
A1 4
E 4 0.8
T1 6
T2 18 1.2
END
fully occupied
1A1 -11.297 1% 4sFe
1T2 -10.762 4% yz,xz,xy
2T2
1.306 46% yz,xz,xy
1E
1.655 53% z2,x2-y2
2A1 2.400 2% 4s Fe
3T2 3.425
2% 4px,y,z
1T1 4.804 0%
partly occupied:
2E 6.520 47%z2,x2-y2
4T2 7.941 47% yz,xz,xy
empty:
3A1 8.344 99% 4s
mv tape21 feo4_aoc.t21
to be used in step iii)
Step iv)
SYMMETRY NOSYM
DEFINE
xx=0.967
END
ATOMS cartesian
Fe 0 0 0
f=Av
O xx xx xx f=Av
O -xx -xx xx f=Av
O xx -xx -xx f=Av
O -xx xx -xx f=Av
END
FRAGMENTS
Av /home/ata/Documents/mult/feo4_aoc.t21
END
SLATERDETERMINANTS
SD 1
A1 2 // 2
E:1 1 1 // 1 1 E:1 , E:2, T2:1, T2:2, T2:3 denote IRREPS for
E:2 1 0 // 1 0 symmetry species spanned by the MO-orbitrals
T1:1 1 // 1
of dz2, dx2-y2, dyz, dxz, dxy (Td-symmetry)
T1:2 1 // 1
T1:3 1 // 1
Consult ADF manual for other symmetries,
T2:1 3 0 // 3 0 including cases of lower symmetry or cases
T2:2 3 0 // 3 0 without any symmetry.
T2:3 3 0 // 3 0
SUBEND
….
SD 45
….
E:1 1 0 // 1 0
E:2 1 0 // 1 0
T2:1 3 0 // 3 0
T2:2 3 0 // 3 0
T2:3 3 1 // 3 1
SUBEND
END
Using this input ADF generates the energies of all SD, in this case:
1 2
3
4
5 6 7 8 9 10
z2+z2- x2-y2+ x2-y2- yz+ yz- xz+ xz- xy+ xy-1.20944200 z2+ z2- 12B+3C
-2.25626566 z2+ x2-y2+
0// 0
-2.02183988 z2+ x2-y2- 4B+C
-0.52452271 z2+ yz+
9B+10Dq
-0.35645889 z2+ yz- 10B+C+10Dq
-0.52452269 z2+ xz+
9B+10Dq
-0.35645888 z2+ xz- 10B+C+10Dq
-0.83723367 z2+ xy+
10Dq// 1.419
-0.55996008 z2+ xy- 4B+C+10Dq
………………………………
1.56350192 xz+ xz- 12B+3C+20Dq
0.66546225 xz+ xy+ 3B+20Dq
1.56350188 xy+ xy- 12B+3C+20Dq
E(SD) = A.X ; X= (AT.A)-1.AT.E
FeO42-
o LFDFT 10Dq=11369 cm-1
* KS-DFT B=285 cm-1
C=1533 cm-1
d2
1A (1G,1S)
1
t22
e2
10B+5C 6(2B+C)
6(2B+C) 8B+4C
1T (1D,1G)
2
t22
B+2C
23B
t2e
23B
2C
1E (1D,1G)
t22 e2
B+2C -23B
-23B 2C
3T (3F,3P)
1
t22 t2e
-5B
6B
6B
4B
t2e 1T1(1G) 4B+2C
t2e 3T2(3F) -8B
e2 3A2(3F) -8B
1)
Multiplets of Transition-Metal Ions in Crystals,
S.Sugano,Y.Tanabe, H.Kamimura, Acad.Press, NY, 1970
2) J.S.Griffith, The Theory of Transition-Metal Ions, Cambridge,
At the University Press, 1971.
LFDFT vs Difference Dedicated CI
Approaches: FeO42LFDFT
SORCI
exp.
T.Brunold
El.trans.
F.Neese
3A →1E
5197
8344
6215
2
3A →1A
2
1
9446
11799
9118
3A
3T
→
2
2
11952
11366
12940
3A
3T
→
2
1
14645
13507
17700
In the general case of a dn metal complex we get: B,C and the 5 „3d“
KS orbital energies: 1KS 2KS 3KS 4KS 5KS
U-eigenvectors
… … … … …
dxy 1 U11 U12 U13 U14 U15 dominated
dyz 2 U21 U22 U23 U24 U25 by d-functions
U
dz2 3 U31 U32 U33 U34 U35
neither
dxz 4 U41 U42 U43 U44 U45
normalized dx2-y25 U51 U52 U53 U54 U55
nor othogonal
… … … … … …
S=UTU U – the (5x5) matrix; C=US-1/2 eigenfunctions of the
Effective one-electron Hamiltonian heff we seek;
5
i= ci d; iKS=(i|heff|i);
=1
5
vLF=C.E.CT=h=  ci. iKS.ci; E=diag iKS
Obtained without any assumptions as is done in crystal field theory
or the AOM; accounts both for electrostatic and covalent contributions
to the LF: h(d|ho+h´ |d)+(L) (d|h´|L)(L|h´|d)/(d-L)
Application to Co(H2O)62+ and Co(H2O)63+
LFDFTa
4T
4T
1g
2g
4A
4T
2E
2g
1g
g
10Dq
B
C
Co(H2O)62+
SORCIb
0
7755
16617
19548
0
6630
14313
19970
6224
13130
expc
8100
16000
19400
21550
11300
8862
9300
860
3013
a this work
b Neese et al. Coord.Chem.Rev.,
in press. cJorgensen,Abs.Sp.Chem.Bonding.
Co(H2O)63+
LFDFTa SORCIb exp.c
1A1g 0.0
0.0
1T1g 15370 15670 16600
1T2g 24537 23600 24900
3T1g 9212
5257
3T2g 13782 10779
5T2g 9660
10Dq 16994
18200
B
775
670
C 2781
this work
b Neese et al. Coord.Chem.Rev.,
in press.
cJorgensen,Abs.Sp.Chem.Bonding.
a
[Co(H2O)63+][H2O]12
LFDFTa
0.0
15669
24726
10271
14798
12649
17102
750
2415
athis
work
Conclusion
It is possible to predict UV-Vis solution spectra with
experimental accuracy using DFT.
Deviations between experiment and theory larger
than 2’000cm-1 (0.25eV) necessitate refined model
chemistry e.g. embedding, structure, etc.
However, calculation of excited states still requires
some expertise and thinking in many cases.
Lanthanides
THE DFT BASED LIGAND FIELD
MODEL AND ITS APPLICABILITY
TO LnCl63- (Ln=Ce to Yb) COMPLEXES
M. Atanasov, C. Daul, H.U.Güdel, T.A.Wesolowski: Inorg. Chem.
2005,44,2954-2963
Introduction
In this work we extend and explore the applicability of the DFT based ligand field (LF) model
(LFDFT, put forward and applied with success to electronic and ESR spectra of 3d-transition
metal (TM) complexes1,2,3) to complexes of rare earths (LnIII). Results from calculations and
experiment are compared to test the method. We describe a computational scheme within DFT
which is able to predict with success LF properties of Ln complexes without recourse to
experimental data.
The LFDFT method
The LFDFT-procedure consists of the following steps:
i) get a geometry of the complex: either from experiment or from a geometry optimization.
ii) calculate the electron density from an average-of-configuration Kohn-Sham DFT SCF
calculation (n/7 occupation of each MO-orbital, identified as being dominated by TM 4ffunctions beforehand).
iii) Using this electron density calculate energies of all Slater determinants resulting from the
14
total of n replacements of n f-electrons over 14 spin-orbitals. The electron density is taken as
 
frozen (no SCF iterations are done).

iv) Use these energies and a least-squares fit to get the LF-model parameters (MATLAB scripts
are used to facilitate the work).
v) Introduce these parameters into a conventional CI - LF-program to calculate electronic
energies and properties of all electronic states.
The model is parametric but its model parameters are obtained from first principles, i.e.
without recourse (fit) to experiment, as different from usual applications of ligand field
theory.
Results
Figure: Left : Model cluster for DFT based ligand fi eld calculations on LnCl63complexes. Right : Symmetry species, orbital shapes and para meters defi ning the ligand
fi eld splitting of the f-orbitals in an octahedra l coordination
•
•
•
•
•
•
•
•
The parameters 1 and 2 calculated from the LFDFT and obtained from a fit to experimental
high-resolute spectra are compared in Figure next. Calculated 1 and 2 values show
deviations from experimental ones, which increase from left to right of the Ln series.
Artificial splitting is due to overestimate of TM-ligand covalency by the DFT method. To
eliminate this drawback of the DFT method, we propose the following procedure.
From the KS matrices of eigenvectors C (in columns) and eigenvalues E(diagonal) one
reconstructs the KS-Hamiltonian (S is the overlap matrix) :
H.C =S.C.E
(2)
Taking a transformation to an orthogonal basis:
C’=S1/2 .C
(3)
Explicitly:
(S-1/2 . H . S-1/2) . (S1/2 . C)=(S1/2 . C) . E
(4)
•
H’ . C’=C’ . E
H'ff
 C'E C'   '
HLf
(5)
H'fL 
(6)
' 
HLL
•
H' S
•
We neglect H’fL and focus on H’ff . Diagonalization of H’ff yields eigenvalues which account
both for Coulombic (crystal field like) and Pauli (exchange) repulsions.
We call this model BLDFT. Application to LnCl63- (Fig.) yields values of 1 and 2 which
agree with experimental for nearly all members of the Ln-series.
Only for Ce, Pr and Nd the LFDFT method, which includes in addition ligand-metal charge
transfer yields results which agree better with experiment.
•
•
1 2
1 2
 H S
T
Figure: Values of 1 and 2 (in cm-1) from LFDFT calculations and from the interpretation of the
f-f spectra (experimental values are taken from Ref.4) for LnCl63- complexes. Computational
details: DFT Program: ADF, release ADF2003.01, basis set: triple zeta plus polarization (ZORA)
; core: no-core, all-electron calculation; relativistic: scalar ZORA, Functional: PW91
Figure: Values of 1 and 2 (in cm-1) from BFDFT calculations and from the interpretation of the
f-f spectra (experiment) for LnCl63- complexes; computational details.
Conclusion
1. Mixing of 4f with 3p-Cl orbitals in LnCl63- does not exceed 1%, as an analysis of
4f4f and ligand-to-metal CT spectra shows.
2. DFT using current functionals and ADF- data base (basis sets) overestimates
covalent mixing in nearly all LnCl63- except in Ce, Pr and Nd.
3.This leads to artificially large LFDFT splittings for Ln= SmIII to YbIII.
4. A computational scheme proposed in this work is developed (BLDFT) which helps to
exclude artificial charge transfer effects, but to account for Coulombic and Paulirepulsion of the 4f- from the ligand-closed-shell-electrons. It leads to good agreement
between calculated LF energies and experimental values.This lends support of an early
concept (Ballhausen and Dahl [5]) describing ligand field in terms of a pseudopotential.
5. Our results are consistent with a study using first principles embedding potential for
LnCl63- complexes [6].
References
[1] M.Atanasov, C.A.Daul and C. Rauzy, C. Chem.Phys.Lett. 2003, 367, 737.
[2] M.Atanasov, C.A.Daul and C. Rauzy, Structure and Bonding, 2004, 106, 97.
[3] C.Daul, C.Rauzy, M.Zbiri, P.Baettig, R.Bruyndonckx, E.J.Baerends and
M.Atanasov, Chem.Phys.Lett. in press.
[4] M.F.Reid and F.S.Richardson, J.Chem.Phys. 1985, 83, 3831.
[5] C.J.Ballhausen and J.P.Dahl, Theoret.Chim.Acta, 1974, 34, 169.
[6] M.Zbiri, M.Atanasov, C.Daul, J.Garcia-Lastra and T.A.Wesolowski,
Chem.Phys.Lett., CPL 2004.
Fine Structure
Ligandfield calculation of the ZFS in Ni(H2O)6++
Hamiltonian
n
n
n
ZM N Zl  n 1
1
H    i         d  i  si
2 i 1
i 1  ri
l rli  i j 1rij
i 1
n
n
1
H  H0 VLF   d 
i j 1rij
i 1
i
 si
The following parameters are needed
Ligandfield parameters:
d VLF d   ci ci iKS
Electrostatic matrix elements:
d d d d  f Racah parameters
Sin-orbit coupling matrix elements:
d r  d  k orb_ red R 3d r3 R 3d
i
KS-orbitals with dominant d-character in a ZORA DFT calculation
7
8
8
(2)
(4)
(4)
Int.J.Quant.Chem., 102,119-131(2005)
ESR g- and A-tensors, Chem.Phys.Lett. 399,2004,433
Spin-orbit coupling:
<sms,a|Hso|sms´,b>  ζnl<sms,lml(a,)|ls|sms´,lml´(b,)>
ζnl=<Rnl| 1 dV |Rnl> |l,ml>-real spherical harmonics xy,yz,z2,xz,x2-y2
r dr
s, s,, c, c
Co2+, ζ=-598 cm-1
k-orbital reduction
factor;
k=(i,) (ci)2/(2l+1)=0.77
i,=1…5
17b2
(xy)
7a2
19a1
(yz)
(x2-y2)
7b1
(xz)
18a1
(z2)
B
C
(x´y´|hlf|x´y´)
(y´z´|hlf|y´z´)
(z2´|hlf|z2´)
(x´2-y´2|hlf|x´2-y´2)
(z2´|hlf|x´2-y´2)
(x´z´|hlf|x´z´)
ζ
k
P

512±53 cm-1
3118±225 cm-1
-1071±407 cm-1
6308 ±407 cm-1
5052 ±407 cm-1
3731 ±407 cm-1
2771 ±407 cm-1
-24003 ±407 cm-1
460 cm-1
0.77
188.10-4 cm-1
0.147
1. Ground state Kramers doublet |0±> obtained by diagonalization of
the full 120x120 CI matrix:  SDd | h  g  h | SDd 

lf
ER
SO

hlf-ligand field,gER-1/r12-interel.repulsion,hSO-spin-orbit
coupling
2.The g-tensor is obtained by equating the Zeeman matrix elements
<O±|kL+geS|O±> with those of the spin-Hamiltonian
<±|g.Seff|±>:
 g.Seff   O  kL  ge S O 
......... ......................  ...........................O  ..................O 
1
1


g
(
g

ig
)

z
x
y 
 O  kL  g e S O 
2
2
  .

1
1
  O  kL  g e S O 
 ( g  ig )

g
x
y
z
2
2

=x,y,z; L=(i=1,n)li; S=(i=1,n)si;
O  kL  g e S O 
O  kL  g e S O 

.


or:
gz  O  kL  g e S O   O  kL  g e S O 
gx  O  kL  g e S O   O  kL  ge S O 
gy  i( O  kL  g e S O   O  kL  g e S O  )
A-tensor
Interaction between nuclear and electron angular momenta:
H HF  ΔHF .I
ΔHF
n
 HF - the hyperfine coupling operator
1
 P (l i  ai   .si )
7
i 1
1
2
3
1
orbital angular momentum of the
electron
aF
electron spin
 
the Fermi contact term.
P
2
ai  4.si  (l i .si ).l i  l i .(l i .si ) 3
3
P

g


r
P – electron-nuclear dipolar coupling constant:
e
N
3d
8
 - related to the Fermi hyperf.const.
aF  ge  N  i  (0)  i  (0)
3
i
The A-tensor is calculated similarly to the g-tensor:
Az  O  hf O   O  hf O 
Ax  O   O   O   O 
hf
hf
Ay  i( O  hf O   O  hf O  )
Results:g-tensor
gxx
ZORA
LDA GGA
2.85
2.76
LFDFT-GGA
A
B
3.21 2.80
Exp.
gyy
1.89
1.93
1.87
1.94
1.90±0.03
gzz
1.91
1.92
1.87
2.11
2.00±0.02
giso
2.22
2.20
2.28
2.32
2.92/3.26
A: two states model 97%|dyz1dxy2,2A2>+3%|dz21dxy2,2A1>
B: full calculation, giso=(gxx+gyy+gzz)/3
Exp.values: range of values, because of strong dependence on the
host lattice.
Multiplet splittings: LFDFT(GGA)
LFDFT
exp.
2A
2
0.0
-
2A
1
4665
-
2B
1
7036
4000
2A
1
10885
8000
4B
1
13021
-
4A
1
12835
-
4B
1
14694
-
A-tensor values of Co(acacen) determined by spin-orbit restricted
ZORA calculation and the LFDFT approach:
ZORA
LDA GGA
LFDFT GGA
A
B
Exp.
Axx
151
108
95
55
100/128
Ayy
25
28
12
14
32/40
Azz
66
71
39
19
29/34
A: two states model 97%|dyz1dxy2,2A2>+3%|dz21dxy2,2A1>
B: full calculation, giso=(gxx+gyy+gzz)/3
Exp.values: range of values, because of strong dependence on the
host lattice.
ZORA references;
g-tensor, A-tensor: spin-restricted, spin-orbit ZORA calculations
(variational relativistic), homogeneous magnetic field treated
as a first order perturbation:
g-tensor: van Lenthe et al, J.Chem.Phys. 107, 1997, 2488-2498;
A-tensor: van Lenthe et al, J.Chem.Phys. 108, 1998, 4783-4796.
LFDFT references:
1. Chem.Phys.Lett. 367(2003) 737-746. LFDFT
2. Struct.and Bonding, 106(2004) 97-125. LFDFT
3. Chem.Phys.Lett. 399(2004) 433-439. g, A-tensors
4. Int.J.Q.Chem. 102 (2005), 119-131, spin-orbit coupling
5. Inorg.Chem. 44(2005), 2954-2963. rare earths.
6. Chem.Phys.Lett. 427(2006) 449-454 CoII,CuII porphyrins gtensors.
Review articles: C.R.Chimie, 8(2005) 1421-1433;
Chimia, 59(2005) 504-510.