Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy...
Transcript of Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy...
Institute for
Nanoscale Physics and Chemistry
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Liviu F. Chibotaru and Liviu Ungur
Magnetic anisotropy in complexes
and its ab initio description
Nanoscale superconductivity, fluxonics and photonics
K.U.Leuven Methusalem Group
Theory of Nanomaterials Group, Department of Chemistry,
Katholieke Universiteit Leuven
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Outline:
2
• Basic notions of atomic magnetism
• What is the magnetic anisotropy?
• When does it appear?
• How does it manifest in magnetic properties?
• Zeeman and ZFS interaction for weak s-o coupling: spin magnetic Hamiltonians
• ---- # ---- for intermediate/strong s-o coupling: pseudospin magnetic Hamiltonians
• Exchange interaction for weak s-o coupling: spin exchange Hamiltonians
• ---- # ---- for intermediate/strong s-o coupling: pseudospin exchange Hamiltonians
• The Lines’ model
• The sign of the g tensor
• Ab initio calculations of electronic coupling of complexes
• Calculation methodology of anisotropic magnetic properties
• SINGLE_ANISO module for mononuclear complexes and fragments.
Examples of applications.
• POLY_ANISO package for polynuclear complexes.
Examples of applications.
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 3
Classical magnetic and angular momenta
3
2
5
Zeeman ext
1) Currents induce magnetic field
Biot and Savart low:
2) In finite systems of currents:
3,
m
2
= a
z
dd I
loops
I Id S
c c
E
l xB
x
x x μ x μB x l
xμ e
μ H
B
B B
3) In atom one electron in the Bohr's orbit:
2 2
gnetic momen
Bohr mag
orbital a
2
ngular moment neton
t
z z
v eI e r mv L
r mc
e
mcμ r pL L
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 4
In quantum mechanics (one electron):
B
B
B
B
ˆ , in units of 2
For each electron: angular momentum (spi
ˆ
ˆ
ˆ
n):
2.0023
magnetic moment:
ˆ ˆˆ ˆ 2
( ) of
ˆ
ˆ
ˆ
o
s s s
o s
e
mc
internal
Total
Eigenstates quantiza
g
t
g
ion
μ l
μ μ sμ l
μ
l
s
s
2 2 22 2
22 2 2 2
angular momenta:
ˆ ˆ ˆˆ ( 1) , l l l
0,1,2, , 1, , 2 1 definite projections
ˆ ˆ ˆ ˆ ( 1) , s s s
1
ˆ ˆl
ˆ ˆs
2
x y z
s s yz s s s z
z
x
lm m lm lm l l lm
l m l l l l
sm m sm sm s s sm
s m
l
s
l
s
1 1,
2 2 s
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 5
Conservation of angular momenta in atoms
2 2 2
2 2
2 2
A physical quantity is conserved if , 0
atom (H)
ˆ1ˆ ˆ ˆ , , , 2
ˆ ˆ ˆˆ , , 0 , are conser
One-electron
ˆ ˆ
ved
ˆ nlm
Ze
A
H r l fm r r r
A
r
H
H
H
r
H
ll
l s l s
2 2 2
1 1 1( )
, , ,
atom:
1ˆ 2 2
Many-electron
s s
el el el
nl nlm nlm nl lm nlmm nlm m
N N N
i
i i ji i jj i
R r Y r
Ze eH
m r
r r
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
1) Independent electrons in the central field: 2 2
0
1
0 0
ˆ ˆ ˆ, ( ),2
( ) - average electrostatic field from all electrons
ˆ ˆ ˆˆ, , 0 , are conserved
n
i i i i
i i
i
i i i i
ZeH H H V r
m r
V r
H Hl s l s
2) Permutational symmetry of electrons:
1 2
Antisymmetrized w. f. are characterized by a total spin
only the total spin is conservedn
definite S
S s s s
3) Interacting electrons:
2
int
1 2
1 1 1ˆ ˆ ˆ( ) , 0, , 02
only the total orbital moment is conserved
n n
i i i j
i j ii j i j i j
n
eV V r
l l lr r r r r r
L l l l
6
Solutions of SE are definite and te Lrms S
1) Independent electrons in the central field:
2 1l nl
configurationnl
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
4) Spin-orbit interaction
The interaction of electron spins with magnetic fields of their orbital motions:
2 2
2 2
d ( )ˆ ˆ ˆ( ) , ( ) , ( ) ( )2 d
isl i i i i i i
i i i i
U r ZeV r r U r V r
m c r r r l s
Within a given electronic configuration nl:
2 2 4ˆ ˆ ˆ , ( ) ( ) d ,sl i i nl nl
i
V R r r r r Z l s
Hund’s rule for the GS atomic term::
1) maximal within a given configuration
2) maximal possible for this
S
L S
Within a given atomic term LS:
' 'ˆOne should calculate
L S L SLM SM sl LM SMV
For half-filled shell 2
are with respect to spin coordinates off all electronsL SLM SM
nn S
symmetric n
7
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
1 2
ˆThe matrix elements of are the same for all 1,..
Since m.e. m.e.
ˆ ˆ ˆˆ ˆ ˆ ˆ ˆˆ
ˆ ˆ ˆ , , 0 Total angular mome is cˆˆ ˆn m onu : st
i
n i
sl i i i
i i
sl
i n
n
Vn
V
n
s
SS s s s s
l s S l S L L
J JS JL
S
2
22 2 2 2 2
, 1
GS
erved
ˆ ˆ eigenstates of ,J : , , 1, ,
ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ2 2
( 1) ( 1) ( 1) , - Lande rule2
h.f. :
Multiplets -
n >0 or - mm l ua l
Jz JM J
J J J
M J J J
E J J L L S S E J
n J L S
J
L S L S L S J L S
GS
tiplet structure
ˆ ˆ ˆ h. invertedf. : 0 - m.s.
sl i i
i
n V J L Sl s
8
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Hierarchy of interactions in atoms
Configurations Terms Multiplets
(shells)
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
3Nd: Example
Configurations Terms Multiplets
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Zeeman interaction
In applied homogeneous field :H
B B
B
Spherical symmetry of atoms rotational symmetry around
Each multiplet splits into (2 1) -components :
ˆ ˆˆ ˆ2
ˆ
Since J is the only conservin
J
J
J
JM J J J J J z z J
J J zM
z
J J M
E E JM JM E JM J S JM H
E M S H
H
H
L S H
2
22 2 2 2
B
g momentum = const
One can prove: const const = / ( 1)
1 1 1But ( 1) ( 1) ( 1
Zeeman
)2 2 2
splitt, , 1, , -
)1
in
1
g
(
J
J
JM
JM J JJ
M
J J
J J S S L L
E E M J J J
J J
gM H
g
2
S J S
J S J J S
J S J S J S J S L
( 1) ( 1) - Lande (gyromagnetic) factor
2 ( 1)
S S L L
J J
e BZeˆ = ˆˆ 2 V μ L SH H
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
, 1, ,
= of
Isotropy o
f sp
degene
racy aftace r
e
M J J J
MJM
'
'
'
all quantization axes:
of eigenfunctions of angular momentum :
they have w.r.t.
ˆEigenfunctions of J :
ˆEigenfunctions of
the same form
J : '
' '
where
z
z
J
MMM J
JM
JM
JM
JM D JM
Basic property
- Euler angles of relative to
The same property also for and
Magnetic properties of consverved angular momen isotrota pic
' '
e
'
ar
x y z xyz
S L
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Origin of magnetic anisotropy: removal of
degeneracy after M
Atoms in a given multiplet state J are magnetically isotropic: the Zeeman splitting is the same for all directions of the applied H
This is the consequence of the (2J+1) degeneracy of multiplets
Example: a d2 configuration (l=2, L=3, S=1, J=2) in electric field (E || z))
Zeeman at B
B
Zeeman
B
splits into nondegenerate 2,0 and two degenerate 2, 1 and 2, 2 levels.
ˆ ˆˆIn applied magnetic field:
2 0ˆFor the ground doublet 2, 2 :
0 2
GS orbital magnet
J
J z
J z
V g
g HV
g H
μ H J H
B
at B B
B
Zee B BZeeman splittin
2 , 0
ˆ ˆˆic moment 0, 0
2 , 0
: 4 4g cos
J z
J J z z
J z
J z J
g H
g g J H
g H
E g anisotrop cH g H i
μ J
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Magnetic anisotropy in mononuclear complexes
1. All spin levels corresponding to S>1/2 become split into:
• Kramers doublets for half-integer spin
• non-degenerate levels for integer spin
Spin-orbit coupling on metal ions leads to the following effects:
g g
2. The Zeeman interaction becomes anisotropic,
the gyromagnetic factor becomes a tensor:
3. The magnetization becomes anisotropic,
the susceptibility function becomes a tensor:
( , ) ( , ,) MH HT H T
ZFS
,
- tensorH of zero fieldS D S splitting
D
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Weak spin-orbit coupling effects
LS << ECF
No orbital states close to the
ground one within the spin-
orbit energy range
Magnetic centers have a spin-only ground state
• Most of transition-metal ions - Cr3+, Mn2+, Fe3+, Ni2+
• Magnetic moment has a spin origin
• The spin of the magnetic centers is a good quantum number
ZFS tensor is small
D ~ 2 /ECF
/ECF ~ 0.01 - 0.1
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Perturbational treatment of spin-orbit coupling for
Zeeman
0 0
0
ˆEffective Zeeman Hamiltonian: H
Gyromagnetic tensor:
Zero-field-splitting tensor:
where
ˆ ˆL L
ˆ ˆL , L - Cartesian components of the total orbital momentum
e
i i
i i
S g H
g
E E
2
g 1
D
2 1
0
S
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Strong spin-orbit coupling effects
There are orbital states close to
the ground one within the spin-
orbit energy range
• f ions - lanthanides and actinides - Ce3+, Nd3+, Dy3+, Yb3+, U4+ et al.
• Some d ions and complexes – Co2+, [MoIII(CN)7]4-, [Fe(CN)6]
3- .
(quasi-) degenerate magnetic centers
= unquenched orbital moments
• The total spin on magnetic centers is no longer a good
quantum number
D ~
ZFS tensor is large
ECF ~
Pseudospin descrip tion
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Non-perturbational treatment of spin-orbit coupling
effects
1) Spin-orbit interactions are included in quantum chemistry calculations
exact treatment of zero field splitting of all molecular terms
2) All paramagnetic properties are expressed through matrix
Zeeman B
1
of
magnetic moments operators:
ˆ ˆˆˆ ˆH ,
on calculated (exact) wave functions of spin-orbit-coupling multiplets
elN
e i i
i
elements
g
μ H μ s l
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Exact derivation of pseudospin Hamiltonians and ZeeH ZFSH
1 2
1) Choice of the relevant manifold:
wave functions 2 , , 1Nab init Sio N
1
*
ZFS
1
2) Definition of pseudospin (in a given coordinate system):
' S
ˆ3) Decomposition of ' ' and H
in pseudospin matrices S , S S ,
N
i in n z
n
N
i j in jn n
n
c SM SM M SM
c c E
, =x,y,z
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Zeeman and ZFS tensors for various pseudospins
ZeeH
.
1 3 5
μ H
μ = μ +μ +μ + .. S HZFS
1/2
1
3/2
2
5/2
3
7/2
4
1
1
1 3
1 3
1 3 5
1 3 5
1 3 57
1 3 57
H0
H2
H2
H2+H4
H2+H4
H2+H4+H6
H2+H4+H6
H2+H4+H6+H8
Individual tensor components:
1
0
2
ZFS
ZFS
0
ˆˆ , ,
ˆ ˆˆ
ˆ ˆH , ,
ˆˆ ˆH
1 1ˆ ˆ ˆ ˆ ˆˆ, - Stewens operators2 2
ˆ - spherical func
nn m m
nm n nm n
m
nn m m
nm n nm n
m
m m m m m m
n n n n n n
m
n n m
g S g g
b O c
S D S D D
e O f
O O O O Oi
O Y
S S tions of , ,x y zS S S
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Calculation of the parameters of pseudospin Hamiltonians
2 2
ˆ ˆˆ ˆ2 1 Sp 2 1 Spm m
n n
nm nm
n n
n O nb c
S O S S O S
ZFS ZFS
2 2
ˆˆ ˆ ˆ2 1 Sp H 2 1 Sp Hm m
n n
nm nm
n n
n O ne f
S O S S O S
0
'
(2 )!(2 1)!ˆ '(2 )!2 1
nm SM
n SM n m
O S S n S nSM O SM C
SS
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Definition of pseudospin for arbitrary : method of A tensor
1main main
diag
B
1ˆSp , - matrix of in arbitrary basis
2
Diagonalization:
, ,
Main values of the tensor 2 1 :
6, , ,
1 2 1
Pseudospin eigenfunc
r r XX YY ZZ
i ii
A N N
A A A
g N S
g A i X Y ZS S S
R A R A
1 3 5
S
Application: ,
t
,
ions:
Z ZSM M SM diagonalization of
μ μ μ
S
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Exchange interaction in polynuclear complexes
1) No spin-orbit coupling
2) Weak spin-orbit coupling effects
3) Strong spin-orbit coupling effects
1 2ˆ ˆ Hamiltonian: HexchIsotropic J spin S S
1 2 1 2 1 2ˆ ˆ ˆ ˆ ˆ ˆˆ Hamiltonian: Hexch symmAnisotropic J spin S S d S S S J S
2
, CF CF
J JE E
d J
1 2ˆ Hamiltonian: H
exchAnisotropic higher order
terms
pseudospin S J S
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Pseudospin exchange Hamiltonians: 4 4U U
5
3 3
5 4 4
3 3
5 4 4
3 3
5 4 4
: 1
7 1, 1 , 3 , 1
8 8
1 1,0 ,
Ground state triple
2 , 22 2
7 1, 1 , , 1
8
t
38
S
H H
H H
H H
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
2 2 2 2 2 2
ZFS
1 1
(1)
exch
1
(1) (2) (3)ZFS exch exch exch
H
4.6 cm 5.2 cm
H
S S
14.6 cm ,
S S S S
S S S S
2
S S
H H H H H
z z x y x y
A B A A B B
x x y y z z
A B A B Ax y z
x
B
x
D E
D E
J J J
J J
1 1
(2)
exch 1 2 3 4
1 1 1 1
1 2 3
e
2 2 2 2
4
S S S S S S S S S S S
2.1 cm , 18.8 cm
H
0.3 cm , 0.4 cm , 0.8 cm , 0.6
S S S S S
cm
H
x x z z y y z z x x z z y y z z
A B A B A B A B A B A B A
z
B A B
J
j j j j
j j j j
(1) (1) (2) (2) (3) (3) (1) (2) (1) (2)
2(1)
(3)
xch 1 2 3 4
1 1
(2)
1 1
1 2 3 4 1.8 cm , 2.6 cm , 0.6 cm , 0.2 cm
O O O O O O O O O O
O S 1 / 3
O S
A B A B A B A B A B
z
k k
x
k k
q q q q
q q
S
q
S
q
2 2
(3)
S
O S S S S
y
k
x y y x
k k k k k
V.S. Mironov, L.F. Chibotaru, A. Ceulemans
Adv. Quant. Chem. 44, 600-616 (2003)
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Practical approach: combined exact-
phenomenological treatment of anisotropic
exchange in polynuclear complexes
exch
exc
ˆ1) introduce an effective H
ˆ 2) diagonalize H
exact quantum chemical treatment of individual metal fragments
treatment of exchange interactions:
Lines model
ij
ij
Ji j
S S
h in the basis of lowest SO multiplets on sites:
(J - fitting parameters)
- contains one single Lines parameter for each exchange-coupled pair
- becomes exact
Advan
in t
tages
he lim
:
it of
ij
ji
isotropic and strongly anizotropic (Ising) interactions
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Polynuclear Ln complexes - weak exchange limit
Spin-orbit coupling effects are strong:
3
CF splitting of multiplets on Ln ions exchange splitting
Non-collinear magnetic structure
: D
>>
a y compl exExample
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Anisotropic exchange interactions for axial ions
11zs22 zs
1 21 2ˆ
ex z zH Js s
11zs 12 zS
1 11 2ˆ
ex z zH Js S
Ln - Ln: Ln - M,R:
axial
isotropic
s
S
1 2 1 2 1 2 1 2ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆFor two isotropic ions: ex x x y y z zH J J S S S S S S S S
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
g>0 for
conserved
angular
momenta
B
B
B
ˆˆ , 1
ˆˆ , 2.0023
1 11ˆˆ , 1 02 2 1
L L
s s
sJ J s
g g
g g
S S L Lgg g g
J J
μ L
μ S
μ J
Sign[g] not evident for pseudospins
B
e.g. , of a KD (crystal-field splitting of an atomic )
1/2Without symmetry no way
-1/2
1S ,
1/ 2
S , , , ,2
J
Suniqu
S
eS
g x y z
Negative g factors in complexes with strong spin-orbit
coupling effects
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Dynamics of the magnetic moment in external H
ZeeH μ H
Diagonalization of g: , , , ,
, ,X Y Z
x y z X Y Z
g g g g
Zee B
, ,
H S i i i
i X Y Z
g H
Equation of motion (M.H.L. Pryce, 1959) :
Zee B 2 2
ˆ 1 1ˆˆ ˆ ˆ,HXX X Y Z Z Y Y Z
Z Y
di i g g g H H
dt g g
The direction of precession depends Sigon n X Y Zg g g
6
6
was found in:
with circularly polarized lNpF
U
ight (1967)
with circularly polarized EPR (1967)F
0X Y Zg g g
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Ab initio calculation of X Y ZSign g g g
B
are matrices in RASSI-SO basis
X Z X Z Z X
Y Y
i
g g i
g
L. Chibotaru, A. Ceulemans, H. Bolvin, Phys. Rev. Lett. 2008
(L.Ungur, LC, PRL, 2012)
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
X Y ZSign g g g for weak s-o coupling
2 2 2
ZFS BH S 1 S S , SZ X Y eD S S E g
For individual
Kramers doublets
1/ 2S
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Signs of individual gi – unique definition of g tensors
L.Chibotaru, A. Ceulemans, H. Bolvin, Phys. Rev. Lett. 2008
• use of symmetry:
• principle of adiabatic connection
Example: Tetragonally distorted 2
6PaCl
6
7
g
g
7
(GS)
7
(ES)
6
(GS)
SM
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Ab initio calculation of magnetic properties of mononuclear complexes
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
What is an ab initio calculation?
Solves the Schrödinger Eq.: 𝑯 𝚿 = 𝑬𝚿
𝑯 -- includes all possible interactions between the particles (electrons, nuclei) 𝚿 -- contains all information about the electronic state of the system
• Uses ONLY the atomic coordinates as input
• Does NOT employ any experimental data / parameters / information
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
What is an ab initio calculation? Exact solution impossible. Approximations are needed…
𝚿 -- single-determinant:
o Hartree-Fock (HF or SCF) -- 𝚿 is represented by one configuration;
o improvements: SCF + PT; SCF + CI; SCF + Coupled Clusters … etc;
Suitable for closed-shell systems! NOT applicable for open-shell systems!
𝚿 -- many-determinants:
o Multiconfigurational Hartree-Fock (MCSCF)
o CASSCF is one of the existing multiconfigurational approaches;
o improvements: CASPT2; NEVPT2, DDCI
Suitable for open-shell systems! Computationally more difficult than HF!
Spin-orbit coupling is described by the DKH Hamiltonian in the AMFI approach, using
CASSCF/CASPT2 states as input.
DFT ≠ is NOT ab initio, because the Exchange-Correlation Functional employed by
DFT is fitted to a list of experimental data;
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Relativistic ab initio computational methodology in MOLCAS
MOLCAS (www.molcas.org) - basic ab initio program package
Get the atomic coordinates from the crystal structure file (CIF)
Define the basis sets for all atoms: which already include scalar relativistic
effects (ANO-RCC; ANO-DK3)
Perform state – average CASSCF calculation of all spin states
Use the RASSI-SO program to mix a large number (preferably all) of spin
states by spin-orbit coupling
Exact energy spectrum and wave functions + matrix elements of L
Use the SINGLE_ANISO program to compute the g tensor of the lowest
groups of states, crystal field parameters of the lowest atomic multiplet,
magnetic susceptibility and molar magnetization
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Expression for magnetic susceptibility
2
1 1
1
Van-Vleck susceptibility expressed in terms of matrix elements of magnetic moment:
ˆ ˆ ˆ ˆˆ ˆ
and
1
3
i
i
ENSS NSSA B kT
ENSSi j i jkT
i
powder X
kT i j j i j i i jNT i j j i e
E Ek e
T T
.
For the calculation of magnetization we need the eigenvalues as function of magnetic field.
X YY ZZT T
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Calculation of molar magnetization
Exact
diagonalization of
the Zeeman matrix
Account of the effect
of excited states via
second - order
perturbation theory
Parameter to
define size of the
Zeeman matrix
The larger the Zeeman matrix
the more accurate Zeeman
spectrum and the resulting
molar magnetization
Energy
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Expression for molar magnetization
3
( )
1
1 1
( )
1 1
3
31
1
ˆ ˆ ˆ ˆ
ˆ( )
ˆ ˆ ˆ ˆ
ˆ ˆ
H H
H
H H
H
E mH H H HNM NSS
kTH H B
m n NM H H
E nH E mNM NSS
kT kT
m n
B
m n n m n m m n H
m m eE m E n
M
e e
n j j n j n n j H
n j j n HkT E n
1 1 1
( )
1 1
:
states obtained by diagonalization of the Zeeman matrix
( ) the Zeeman energy of the state
, states unpertur
H H
H
E nNSS NSS NSS
kT
n NM j j
E nE mNM NSS
kT kT
m n
H
H H H
eE j
e e
where
m
E m m
n j
bed by magnetic field, i.e. the states which are higher in energy than the cut-off energy .
, the spin-orbit energy of the states and .
EM
E n E j n j
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Effect of intermolecular interaction on the magnetization vector and magnetic susceptibility
ˆˆ ˆ ; E
;
Solve self-consistently the equation for and tMolar magnetiza hen substitute intotion he t:
n n
n n
Zee n
E E
kT kT
n n
E EH HkT kT
n n
nH
H zJ S S f S
n n e n S n e
M S
e e
S E
10 0
calculation of
;
T
where susceptibility tensor of the magnetic
he susceptibility tens
center alone
ˆˆ1
o
ˆ
:
ˆ
r
A
n
H
Ei j j i j
kTi j i j
M
zJ zJ
n m m S n me n n n S n
kT
A 1 Β A
ˆˆ
ˆ ˆ ˆ ˆ1 ˆ ˆ
n
n
n
i i j
n ij m n m
E
kT
n
Ei j j i j i i j
kTi j i j
n ij m n m
E
kT
n
n n S m
E E
e
n S m m S n m S n n S me n S n n S n
kT E E
B
e
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Ab initio treatment of magnetism of mononuclear complexes and fragments
- susceptibility tensor
- powder susceptibility
- magnetization vector
- powder magnetization
( , )
- Zeeman split
( , )
( , )
(
ting
, )
T H
T H
M T H
M T H
SINGLE_ANISO
program
Magnetic properties Pseudospin Hamiltonians
- second order ZFS tensor
- first order gyromagnetic tensor
- Higher-order ZFS and
Zeeman Hamiltonians
- the sign of the product
is eva
New:
luated.
X Y Z
D
g
g g g
Now implemented in MOLCAS - 7.8
Crystal-Field for Ln
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
SINGLE_ANISO as a module in MOLCAS
Molcas flowchart
required programs to run
before SINGLE_ANISO
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Ab initio Crystal Field for Lanthanides
Use the lowest CASSCF/RASSI wave functions {Yn} of the complex
corresponding to the ground atomic multiplet on Ln
Make the correspondence {Yn} → {|J,M>}:
Define the coordinate system and the main quantization axis;
Make the correspondence {Yn} → {|J,M>}
Rewrite the CASSCF/RASSI matrix
Use the irreducible tensor technique to extract parameters of the Crystal-Field
Y ,i iM
M
c J M
Y Y 2 1 2 1
*
''J J
i i i i iM iMi i
E M M E c cCF CFH H
L. Ungur, L. F. Chibotaru, J. Am. Chem. Soc. , 2013, submitted.
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Ab initio Crystal Field for Lanthanides
Irreducible tensor operators:
The Crystal Field Matrix can be recovered as follows:
This is a one-to-one mapping of the ab initio calculated energy matrix!
The obtained set of parameters is unique!
Implemented in SINGLE_ANISO in MOLCAS 7.8
2,4,6 0 1
n nc s
n nm nmn m m
B Bm m
CF n nH O
1 1
O 1 Ο O 1 O O2 2
m mm m m m m mn n n n n ni
L. Ungur, L. F. Chibotaru, J. Am. Chem. Soc. , 2013, submitted.
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Ab initio Crystal Field for Lanthanides
Extended Stevens Operators (ESO) as defined in:
• Rudowicz, C.; J. Phys. C: Solid State Phys.,18 (1985) 1415-1430.
• in the "EasySpin" function in MATLAB, www.easyspin.org.
The Crystal Field Matrix:
• k - the rank of the ITO, = 2, 4, 6, 8,10,12.
• q - the component (projection) of the ITO, = -k, -k+1, ... 0, 1, ... k;
Implemented in SINGLE_ANISO in MOLCAS 7.8
,
q qk k
k q
BCFH O
L. Ungur, L. F. Chibotaru, J. Am. Chem. Soc. , 2013, submitted.
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Ab initio extracted Crystal-Field: [Tb(pc)2]-
0 50 100 150 200 250 300
10.2
10.5
10.8
11.1
11.4
11.7
12.0
Experiment
calculation
T
/cm
3K
mo
l-1
T / K
zJ'=0.005 cm-1
• CASSCF/RASSI calculations of the entire molecule with MOLCAS
7.8 package
• SINGLE_ANISO calculation of magnetic properties
Tb
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Real vs symmetrized [Tb(pc)2]-
N M 𝑩𝒏𝒎𝒄 𝑩𝒏𝒎
𝒔 𝑩𝒏𝒎𝒄 𝑩𝒏𝒎
𝒔 𝑩𝒏𝒎𝒄 𝑩𝒏𝒎
𝒔
2 0
1
2
414.0
0.0
0.0
------
0.0
0.0
614.6
-7.9
5.0
------
15.8
227.0
531.3
0.0
0.0
------
0.0
0.0
4
0
1
2
3
4
-228.0
0.0
0.0
0.0
10.0
------
0.0
0.0
0.0
0.0
-98.2
-5.0
2.3
1.1
-18.6
------
6.3
54.7
-2.7
4.7
-111.9
0.0
0.0
0.0
0.0
------
0.0
0.0
0.0
0.0
6
0
1
2
3
4
5
6
33.0
0.0
0.0
0.0
0.0
0.0
0.0
------
0.0
0.0
0.0
0.0
0.0
0.0
-13.5
0.5
-0.3
2.0
-4.3
3.7
14.9
------
-0.8
25.4
-2.0
1.2
-3.8
24.1
-7.3
0.0
0.0
0.0
0.0
0.0
0.0
------
0.0
0.0
0.0
0.0
0.0
0.0
Real Symm- D4d Ishikawa
Dexp=260 cm-1
D
D
6 6
5
5
Ab initio
6
5
4
3
2
1 0
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Ab initio extracted Crystal-Field: Dy3+ from DyZn3
0 50 100 150 200 250 300
10
11
12
13
14
0 1 2 3 4 5 6 7
0
1
2
3
4
5
Exp. T = 2 K
Exp. T = 3 K
Exp. T = 5 K
Theory T = 2 K
Theory T = 3 K
Theory T = 5 K
M /
B
H / T
T
/ c
m3K
mol-1
T / K
N M 𝑩𝒏𝒎𝒄 𝑩𝒏𝒎
𝒔
2 0
1
2
-136.1
-11.7
352.9
------
-72.4
35.1
4
0
1
2
3
4
-48.5
-1.5
85.9
-14.4
-97.6
------
0.6
-14.3
-34.0
30.2
6
0
1
2
3
4
5
6
11.9
5.2
-4.6
13.8
8.3
3.4
22.5
------
61.7
5.2
50.5
-1.2
2.0
-5.6
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Ab initio extracted Crystal-Field: Ho3+ from Ho2
0 50 100 150 200 250 3000
5
10
15
20
25
30
0 1 2 3 4 5 6 70
2
4
6
8
10
M /
B
H / T
T
/cm
3K
mol-1
T / K
J = -0.44 cm-1
N M 𝑩𝒏𝒎𝒄 𝑩𝒏𝒎
𝒔
2 0
1
2
302.0
32.3
130.3
------
-40.5
-257.0
4
0
1
2
3
4
93.0
5.4
17.4
8.2
-46.9
------
4.8
-83.5
25.6
-53.0
6
0
1
2
3
4
5
6
8.6
-27.9
5.2
-11.6
-12.8
-12.8
-17.8
------
25.0
0.5
-30.6
5.8
-5.6
-17.9
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Ab initio extracted Crystal-Field:
Yb3+ in an almost Oh environment
S=1/2 KD
(J=7/2)
g
2F
0.0 49.7 87.7
346.0 477.4 524.8 572.0
0.0
gX
gY
gZ
3.2098
2.6944
1.7634
187.9
gX
gY
gZ
1.9936
2.2654
3.6065
300.0
gX
gY
gZ
0.1556
0.7852
3.1483
485.0
gX
gY
gZ
1.8265
3.2849
4.8065 0 50 100 150 200 250 300
0.0
0.4
0.8
1.2
1.6
2.0
2.4
0 10 20 30 40 50 60 70
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
2 K
3 K
5 K
M /
N
H / kOe
M
T /
cm
3 m
ol-1
K
T / K
H = 500 Oe
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Ab initio extracted Crystal-Field:
Yb3+ in an almost Oh environment
N M 𝑩𝒏𝒎𝒄 𝑩𝒏𝒎
𝒔
2 0
1
2
-42.3
-164.9
107.8
-------
-11.1
-50.7
4
0
1
2
3
4
-542.0
943.7
-33.6
472.1
1231.9
-------
-881.5
858.4
335.2
44.0
6
0
1
2
3
4
5
6
-11.9
7.9
1.6
-32.3
-4.6
14.4
-2.6
-------
-7.7
-5.8
-27.8
3.0
-6.8
24.0
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Ab initio extracted Crystal-Field: [Cp*-Er-COT]-
Energy N M 𝑩𝒏𝒎𝒄 𝑩𝒏𝒎
𝒔 0.0
0.0
177.5
177.5
192.3
192.3
206.8
206.8
225.9
225.9
242.6
242.6
268.5
268.5
319.5
319.5
2 0
1
2
307.3
157.3
-33.6
-----
2.7
0.4
4
0
1
2
3
4
327.6
101.3
-1.1
2.6
3.3
-----
5.7
-1.6
0.2
1.9
6
0
1
2
3
4
5
6
22.8
-17.9
-1.2
0.8
-0.2
-9.5
-1.7
-----
0.2
0.0
0.0
0.1
-0.1
0.1
Gao Song et. al, Inorg. Chem., 2012, 51 (5), pp 3079–3087
0 50 100 150 200 250 3000
1
2
3
4
5
6
7
8
9
10
11
12
0 1 2 3 4 5 6 70.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
T=1.8 K
/
B
H / T
Experiment
model A
T
/ c
m3K
mo
l-1
T / K
gX = 0.0000849
gY = 0.0009137
gZ = 17.9366602
Er
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Ab initio extracted Crystal-Field: [Cp*-Tb-COT]-
0 50 100 150 200 250 3000
1
2
3
4
5
6
7
8
9
10
11
12
0 1 2 3 4 5 6 70.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
T=1.8 K
/
B
H / T
Experiment
Calculation
T
/ c
m3K
mol-1
T / K
Energy N M 𝑩𝒏𝒎𝒄 𝑩𝒏𝒎
𝒔
0.0
17.0
31.6
86.9
88.9
149.5
149.9
167.1
167.1
234.5
234.7
341.4
341.4
2 0
1
2
-211.8
15.3
-18.0
-------
-36.7
2.1
4
0
1
2
3
4
-66.6
-6.3
60.4
1.8
-0.8
-------
147.9
5.1
-13.6
-0.1
6
0
1
2
3
4
5
6
27.0
-2.4
32.8
6.3
-7.2
-2.0
3.5
-------
3.7
4.9
-27.6
-1.0
4.5
2.3
Tb
Song Gao et. al, Inorg. Chem., 2012, 51 (5), pp 3079–3087
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Ab initio extracted Crystal-Field: [Cp*-Tm-COT]-
Tm
Gao Song et. al, Inorg. Chem., 2012, 51 (5), pp 3079–3087
Energy N M 𝑩𝒏𝒎𝒄 𝑩𝒏𝒎
𝒔
0.0
0.0
215.2
215.8
295.9
306.2
348.4
358.6
360.6
404.7
411.4
436.4
437.7
2 0
1
2
-317.0
1.9
38.8
-----
-3.2
64.8
4
0
1
2
3
4
-90.6
-20.4
1.9
1.8
0.0
-----
35.7
3.3
0.0
0.0
6
0
1
2
3
4
5
6
-8.3
-14.1
5.6
5.1
0.1
-1.9
-2.7
-----
25.4
9.1
0.0
0.0
-3.2
0.1
0 50 100 150 200 250 3000
1
2
3
4
5
6
7
0 1 2 3 4 5 6 70.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
T=1.8 K
/
B
H / T
Experiment
Calculation
T
/ c
m3K
mo
l-1
T / K
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Semi ab initio calculation of magnetic properties of polynuclear complexes
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
• Employing a model fragmentation of the polynuclear complex.
• Ab initio treatment of individual fragments:
CASSCF (CASPT2) RASSI-SO SINGLE_ANISO
SINGLE_ANISO produces an input file for POLY_ANISO.
• Lines model: Treatment of exchange interactions
1. Introduce an effective
2. The dipolar magnetic coupling can be considered exact:
3. Diagonalize in the basis of lowest spin–orbit states on sites
(Jij - fitting parameters)
4. Calculate all magnetic properties and parameters of magnetic Hamiltonians
ji
exchH ij
ij
Ji j
S S
exch
2
1 2 1 1 2 2
3
3B
r
μ μ μ r μ r
Semi - ab initio treatment of magnetism in polynuclear complexes and fragments
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
( , ); ( , )
( , ); (
Zeeman splitting
, )
T H T H
M T H M T H
POLY_ANISO
program
Magnetic properties Magnetic Hamiltonians
- Higher-order ZFS
- tensor;
and
gyromag
-
n
tens
etic tens r
r
o s
oD g
Fragment 1 Fragment 2 Fragment 3 Fragment N … J12 J23 J J
Semi - ab initio treatment of magnetism in polynuclear complexes and fragments
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Dy3 isosceles triangle:
KD Dy1 Dy2 Dy3
1 2 3 4 5 6 7 8
0.000 108.778 202.571 286.735 322.050 385.913 458.087 524.465
0.000 239.577 459.170 546.174 587.706 642.290 709.691 979.880
0.000 225.784 485.721 634.670 672.382 711.704 753.551 834.336
Main values of the g-tensor for the lowest doublets
gX gY gZ
0.0528 0.0793
19.7092
0.0020 0.0024
19.8247
0.0024 0.0028
19.8525
Angle between the main magnetic axis gZ and the Dy3 plane (degrees)
6.657 0.568 10.398
Interaction Jexch Jdip Jexch + Jdip 1-2 2-3 3-1
1.608 1.608 1.311
5.428 5.293 2.959
7.036 6.901 4.270
Y.-X. Wang, W. Shi, H. Li, Y. Song, L. Fang, Y. Lan, A. K. Powell, W. Wernsdorfer, L. Ungur, L. F. Chibotaru, M. Shenc, P. Cheng, Chem. Sci., 2012, 3, 3366-3370.
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Dy3 isosceles triangle: Y.-X. Wang, W. Shi, H. Li, Y. Song, L. Fang, Y. Lan, A. K. Powell, W. Wernsdorfer, L. Ungur, L. F. Chibotaru, M. Shenc, P. Cheng, Chem. Sci., 2012, 3, 3366-3370.
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 KUL 2013, Leuven, 22 May
2013
Isostructural series of mixed 4f-3d SMM’s :
Co-Ln-Co, Ln = Gd, Tb, Dy
Co-Gd-Co Co-Tb-Co Co-Dy-Co
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
0.04 K
0.3 K
0.4 K
0.5 K
0.6 K
0.7 K
0.8 K
0.9 K
1 K
M/M
s
0H (T)
0.002 T/s
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
0.04 K0.2 K0.3 K0.4 K0.5 K0.6 K0.7 K0.8 K0.9 K1.0 K1.1 K
M/M
s
0H (T)
0.14 T/s
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
0.04 K
0.2 K
0.3 K
0.4 K
0.5 K
0.6 K
0.7 K
0.8 K
0.9 K
1.0 K
1.1 K
M/M
s
0H (T)
0.14 T/s
Yamaguchi,T.; Costes JP, et. al., Inorg. Chem. 2010, 49, 9125–9135.
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Isostructural series of mixed 4f-3d SMM’s :
Co-Ln-Co, Ln = Gd, Tb, Dy
Co-Gd-Co > Co-Tb-Co > Co-Dy-Co
The SMM behavior is mainly due to the strong
anisotropy on Co2+ centers
The approximate
symmetry of the first
coordination sphere of
Co2+ centers is a
distorted trigonal prism
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Ab initio calculation of Co fragments
(CASSCF-calculations)
2S+1 CoGdCo CoTbCo CoDyCo Co1 Co2 Co1 Co2 Co1 Co2
4
0.0
49.6
3410.2
7868.3
8167.9
8959.1
16091.0
23555.8
23655.9
23980.6
0.0
120.0
3551.6
8384.0
8417.1
9364.8
16784.3
23965.2
24108.9
24166.4
0.0
170.6
3477.2
8153.9
8175.9
9082.5
16287.7
23691.6
23814.0
24083.1
0.0
89.0
3412.4
7985.3
8224.8
8994.5
16157.6
23732.3
23784.3
23893.2
0.0
95.2
4142.0
7195.5
7381.0
8262.2
13963.9
23018.0
23083.2
23314.7
0.0
56.6
4033.8
7510.5
7567.4
8512.8
14595.8
23165.0
23382.7
23497.5
2
14594.1
14783.7
19775.6
19854.3
20396.0
21012.2
21134.3
21643.7
25940.3
26112.0
…
14367.8
14488.8
19798.0
19873.0
20473.7
21125.9
21161.3
21710.8
26268.2
26282.4
…
14594.9
14751.3
19790.9
19956.5
20475.6
21105.9
21159.1
21699.4
26080.7
26201.6
…
14592.0
14723.2
19828.5
19864.4
20409.4
21036.6
21152.1
21648.4
26015.9
26129.2
…
15865.5
16017.1
19925.2
19946.6
20382.5
21020.6
21063.3
22124.4
25266.8
26035.9
…
15530.8
15640.9
19861.0
19933.8
20395.8
21039.2
21074.8
22031.2
25498.3
26105.2
…
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Ab initio calculation of Co fragments
(RASSI –calculations )
CoGdCo CoTbCo CoDyCo Co1 Co2 Co1 Co2 Co1 Co2
0.0
248.6
536.5
852.0
3771.5
3847.3
8240.1
8303.5
8612.5
8676.8
9398.1
9426.7
0.0
242.2
549.0
856.1
3880.7
3954.7
8646.4
8749.4
8856.8
8967.1
9767.6
9794.4
0.0
235.8
565.8
868.0
3786.4
3862.0
8393.3
8498.0
8602.7
8713.2
9469.3
9495.7
0.0
246.5
544.1
857.3
3756.8
3832.8
8328.2
8400.1
8654.1
8726.5
9415.4
9445.9
0.0
257.8
570.6
896.0
4467.1
4549.2
7528.7
7626.4
7842.7
7932.6
8712.6
8732.9
0.0
258.6
558.8
883.8
4381.7
4460.8
7815.0
7931.2
8059.1
8171.5
8976.7
8988.8
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Ab initio calculation of Co fragments
(SINGLE_ANISO )
KD CoGdCo
Co1 Co2
E g E g
1
gX
gY
gZ
0.000
0.4197
0.4233
9.2928
0.000
0.4066
0.4155
9.2570
2
gX
gY
gZ
248.686
0.5064
0.9614
5.2042
242.221
1.4578
1.9117
4.9886
3
gX
gY
gZ
536.549
0.6653
0.8109
1.2123
549.084
0.9821
1.6751
1.7170
4
gX
gY
gZ
852.041
0.0425
0.0639
2.7922
856.123
0.0063
0.0258
2.8266
The Co2+ centers have an
unquenched orbital moment in
the ground state:
<LZ>≈ 1.6B
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Unquenched orbital moment of Co2+ in a
perfect prismatic environment
CoO3N3 core has original geometry from
X-Ray experiment. The H atoms saturate
the broken O-C or N-C bonds.
CoO3N3H12 fragment was symmetrized
towards C3v symmetry.
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Unquenched orbital moment of Co2+ in a
perfect prismatic environment
Original geometry
SFS KD g
4
0.0
78.3
4637.8
5814.7
6009.0
7592.0
…
0.0
gX
gY
gZ
0.1858
0.1865
9.5528
260.2
gX
gY
gZ
1.0181
1.2129
5.4266
2
16956.6
17122.0
19763.0
19785.4
20304.1
20775.8
…
573.5
gX
gY
gZ
1.0486
1.1674
1.4079
908.7
gX
gY
gZ
0.0265
0.0537
2.5510
Symmetrized C3v geometry
SFS KD g
4
0.000
0.010
5508.5
5508.5
7512.6
7810.3
9433.9
…
0.0
gX
gY
gZ
0.0066
0.0066
9.8032
286.3
gX
gY
gZ
0.0032
0.0035
5.8128
2
19124.9
19124.9
19658.3
19658.3
20423.0
20423.0
…
613.3
gX
gY
gZ
0.0001
0.0001
1.8155
976.7
gX
gY
gZ
0.0108
0.0108
2.1910
<LZ>≈ 1.9B <LZ>≈ 1.8B
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Unquenched orbital moment of Co2+ in a
perfect trigonal prismatic environment
For a d7 configuration
in a trigonal prismatic environment
4E
Spin-orbit
coupling
Energ
y
1 2C c
2 1C c
0
Energ
y
3d orbitals:
ZL < 2
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Unquenched orbital moment of Co2+ in a
perfect higher-order prismatic environment
4E
Spin-orbit
coupling E
nerg
y 1
2
0
Energ
y
3d orbitals:
For a d7 configuration
in a higher-order prismatic environment ZL = 2
ˆˆ ˆso zi zi
i
H l s
LZ SZ E -2 3/2
2 -3/2
-2 1/2 /3
2 -1/2 /3
-2 -1/2 -/3
2 -1/2 -/3
-2 -3/2 -
2 3/2 -
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 KUL 2013, Leuven, 22 May
2013
Unquenched orbital moment of [Mo(CN)7]4-
V.S. Mironov, L.F. Chibotaru, A. Ceulemans, J. Am. Chem. Soc, 2003, 125, 9750-9760
Figure 1. Structure of the [Mo(CN)7]4-
heptacyanometalate: (a) the idealized
pentagonal bipyramid (D5h symmetry
group) and (b) real distorted bipyramid;
the polar (θ) and azimuthal (φ)
Figure 3. Electronic structure of the [Mo(CN)7]4- heptacyanometalate: (a) Mo 4d orbital
energies in a regular D5h pentagonal bipyramid, (b) the energy spectrum of MoIII(4d3) in
the D5h bipyramid without spin-orbit coupling, and (c) the energy spectrum of MoIII(4d3)
in the D5h bipyramid with the spin-orbit coupling. The orbital composition of the ground
φ(1/2) and excited χ( 1/2) Kramers doublets is shown; (d) the splitting of 4d orbital
energies in distorted [Mo(CN)7]4- complexes.
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Ab initio based calculation of magnetism in
Co-Gd-Co, Co-Tb-Co and Co-Dy-Co
( Poly_Aniso routine )
0 1 2 3 4 50
2
4
6
8
10
12
0 50 100 150 200 250 30010
15
20
25
30
35
/
cm
3K
mol-1
T / K
M /
B
H / T0 1 2 3 4 5
0
2
4
6
8
10
0 50 100 150 200 250 3000
5
10
15
20
25
30
35
40
/
cm3K
mol-1
T / K
M /
B
H / T0 1 2 3 4 5
0
2
4
6
8
10
0 50 100 150 200 250 30010
15
20
25
30
35
/
cm
3K
mo
l-1
T / K
M /
B
H / T
Co-Gd-Co Co-Tb-Co Co-Dy-Co
1 1 2 2 1 2exch Co Ln Co Ln Co CoH J S S S S J S S
J1 = +0.90 cm-1 J1 = +1.10 cm-1 J1 = +1.25 cm-1
J2 = +0.60 cm-1 J2 = +0.35 cm-1 J2 = +0.50 cm-1
zJ = -0.0005 cm-1 zJ = -0.0005 cm-1 zJ = -0.0005 cm-1
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Exchange energy spectrum and orientation of local magnetic
moments in Co-Gd-Co
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Structure of the blocking barrier in Co-Gd-Co
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Exchange energy spectrum and orientation of local magnetic
moments in Co-Tb-Co
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Structure of the blocking barrier in Co-Tb-Co
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Exchange energy spectrum and orientation of local magnetic
moments in Co-Dy-Co
Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013
Structure of the blocking barrier in Co-Dy-Co