Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy...

77
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

Transcript of Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy...

Page 1: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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

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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.

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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

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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

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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

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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

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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

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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

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Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013

Hierarchy of interactions in atoms

Configurations Terms Multiplets

(shells)

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3Nd: Example

Configurations Terms Multiplets

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

Page 23: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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

Page 24: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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

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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)

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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

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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

Page 28: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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

Page 29: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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

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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

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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)

Page 32: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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

Page 33: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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

Page 34: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013

Ab initio calculation of magnetic properties of mononuclear complexes

Page 35: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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

Page 36: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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;

Page 37: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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

Page 38: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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

Page 39: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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

Page 40: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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

Page 41: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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

Page 42: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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

Page 43: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013

SINGLE_ANISO as a module in MOLCAS

Molcas flowchart

required programs to run

before SINGLE_ANISO

Page 44: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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.

Page 45: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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.

Page 46: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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.

Page 47: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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

Page 48: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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

Page 49: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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

Page 50: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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

Page 51: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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

Page 52: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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

Page 53: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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

Page 54: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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

Page 55: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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

Page 56: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013

Semi ab initio calculation of magnetic properties of polynuclear complexes

Page 57: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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

Page 58: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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

Page 59: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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.

Page 60: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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.

Page 61: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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.

Page 62: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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

Page 63: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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

Page 64: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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

Page 65: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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

Page 66: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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.

Page 67: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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

Page 68: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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

Page 69: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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 -

Page 70: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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.

Page 71: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

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

Page 72: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013

Exchange energy spectrum and orientation of local magnetic

moments in Co-Gd-Co

Page 73: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013

Structure of the blocking barrier in Co-Gd-Co

Page 74: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013

Exchange energy spectrum and orientation of local magnetic

moments in Co-Tb-Co

Page 75: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013

Structure of the blocking barrier in Co-Tb-Co

Page 76: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013

Exchange energy spectrum and orientation of local magnetic

moments in Co-Dy-Co

Page 77: Magnetic anisotropy in complexes and its ab initio descriptionschnack/...Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013 Origin of magnetic anisotropy: removal of degeneracy

Magnetic Anisotropy workshop, Karlsruhe 11-12 October 2013

Structure of the blocking barrier in Co-Dy-Co