PHI User Manual

i

Contents

License ...................................................................................................................................... iv

Acknowledgements ................................................................................................................... vi

1. Introduction ............................................................................................................................ 1

2. Theoretical Background ......................................................................................................... 2

2.1 Notation ............................................................................................................................ 2

2.2 Theory .............................................................................................................................. 2

Spin-orbit coupling ............................................................................................................ 3

Exchange coupling ............................................................................................................. 4

Crystal-field potential ........................................................................................................ 5

Zeeman Effect .................................................................................................................... 9

Orbital reduction factor ...................................................................................................... 9

Magnetic properties ........................................................................................................... 9

Accuracy and approximations.......................................................................................... 11

Powder integration ........................................................................................................... 13

Pseudo g-tensors .............................................................................................................. 13

Transition probabilities .................................................................................................... 14

J-mixing ........................................................................................................................... 14

Non-collinearity ............................................................................................................... 14

TIP, intermolecular interactions and magnetic impurities ............................................... 15

Error residuals .................................................................................................................. 16

Electron Paramagnetic Resonance ................................................................................... 16

3. Code Description ................................................................................................................. 18

3.1 PHI ................................................................................................................................. 18

3.2 GUI ................................................................................................................................ 19

4. User Guide ........................................................................................................................... 20

4.1 Binaries and compilation ............................................................................................... 20

How to use the Makefile .................................................................................................. 20

4.2 Program execution ......................................................................................................... 21

4.3 Input files and syntax ..................................................................................................... 21

.input specification ........................................................................................................... 22

Other blocks ..................................................................................................................... 25

sus.exp specification ........................................................................................................ 36 mag.exp specification ...................................................................................................... 36 mce.exp specification ....................................................................................................... 36

heat.exp specification ....................................................................................................... 37 epr.exp specification ........................................................................................................ 37 levels.exp specification .................................................................................................... 37

G.exp specification .......................................................................................................... 37

4.4 Output files and interpretation ....................................................................................... 38 sus.res, mag.res, mce.res, heat.res, epr.res, levels.res and G.res specification ................ 38 zeeman.res specification .................................................................................................. 38 survey.res specification .................................................................................................... 38 states.res specification ..................................................................................................... 38

4.5 Use of the GUI ............................................................................................................... 39

4.6 Example ......................................................................................................................... 40

4.7 Testing ............................................................................................................................ 41

5. Bugs and Feedback .............................................................................................................. 42

ii

6. References ............................................................................................................................ 43

iii

License

PHI

Copyright © 2011 – 2015 Nicholas F. Chilton email: [email protected]

This document is part of PHI.

Any results obtained through the use of PHI that are published in any form must be accompanied by the following reference: N. F. Chilton, R. P. Anderson, L. D. Turner, A.

Soncini and K. S. Murray, J. Comput. Chem., 2013, 34, 1164 – 1175

Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.

Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND

CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES,

INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF

MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE

DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR

CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,

SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT

LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF

USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED

AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT

LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN

ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE

POSSIBILITY OF SUCH DAMAGE.

By downloading and/or using this software you agree to the terms of this license. iv

v

Acknowledgements

The author wishes to acknowledge advice, assistance and contributions from the following people:

Prof. Keith Murray a

Prof. Stuart Batten a

Prof. Richard Winpenny b

Prof. Eric McInnes b

Prof. David Collison b

Prof. Fernande Grandjean c

Dr. Lincoln Turner d

Dr. Russell Anderson d

Dr. Alessandro Soncini

Dr. Angus Gray-Weale e e

Dr. David Paganin d

Dr. Stuart Langley a

Dr. Willem van den Heuvel e

Dr. Marta Estrader f

Dr. James Walsh b

Mr. Chris Billington d

Mr. Philip Chan g a b

School of Chemistry, Monash University, Clayton, Victoria, Australia

School of Chemistry, The University of Manchester, Manchester, United Kingdom c

Institut de Physique, Université de Liège, Belgium; Department of Chemistry, Missouri d

S&T, United States of America

School of Physics, Monash University, Clayton, Victoria, Australia e

School of Chemistry, University of Melbourne, Victoria, Australia f

School of Chemistry, Universitat de Barcelona, Catalonia, Spain g

Monash eResearch Centre, Monash University, Clayton, Victoria, Australia vi

vii

1. Introduction

PHI is a computer package designed for the calculation and interpretation of the magnetic properties of paramagnetic compounds. While the use of phenomenological Hamiltonians is not at all a new concept, the program was conceived as an ‘update’ to older methods while adding new functionality, new approaches and access increased computational power.

The program was designed, primarily, for the treatment of systems containing orbitally degenerate and strongly anisotropic ions, through the inclusion of Spin-Orbit (SO) coupling and Crystal-Field (CF) effects. Thus, PHI was written with the explicit inclusion of orbital angular momentum. The intra-atomic coulomb interaction is treated with the Russell-

Saunders (or LS) formalism, such that only the total spin and the total orbital moments of the ground term are employed. Whilst designed for anisotropic calculations, PHI is also optimized for calculations involving magnetically isotropic or spin-only systems.

Another major design feature was to employ the Zeeman term in the Hamiltonian such that non-perturbative field dependent magnetic properties could be calculated. This also facilitates the calculation of field dependent properties such as Electron Paramagnetic Resonance (EPR) and Zeeman spectra.

One of the main goals is for the program to be approachable by non-experts; a goal that has been facilitated though the use of plain text input files and the provision of pre-compiled binaries for common operating systems. A Graphical User Interface (GUI) is also available to aid running calculations and provide real-time visualization of data, such that the program is even more accessible to beginners. This feature makes the program perfect for use as a teaching aid for magneto-chemical studies.

1

2. Theoretical Background

2.1 Notation

This manual uses the following notation for common mathematical quantities.

Table 2.1.1 – Mathematical notation

Quantity

Scalar

Vector

Vector component

Matrix

Matrix component

Operator

Vector operator

Vector operator component

Symbol

𝐴

𝐴

𝐴⃑ 𝛼

𝐴̿

𝐴 𝛼,𝛽

𝐴̂

𝐴̂

𝐴̂⃑ 𝛼

2.2 Theory

For systems in thermodynamic equilibrium, the underlying postulate is the solution of the time independent Schrödinger equation, Equation 2.2.1. The action of the Hamiltonian operator,

𝐻�, on the wavefunction, Ψ, gives the energy of the state, 𝐸. The wavefunction is usually separated into radial and angular parts and in the domain of spin Hamiltonians, the angular part is solved explicitly while the radial integrals essentially become parameters to be determined. For a given problem, the Hilbert space is constructed from

𝑁 sites with angular momentum basis states of either

�𝑆 𝑖

, 𝑚

𝑆𝑖

⟩, �𝐽 𝑖

, 𝑚

𝐽𝑖

⟩ or �𝐿 only a single term is used to describe each ion, i.e.

𝑆 𝑖 𝑖

,

𝐽 𝑖

, 𝑚

𝐿𝑖

or

𝐿

, 𝑆 𝑖 𝑖

, 𝑚

𝑆𝑖

⟩, where 𝑖 ∈ 𝑁. Note:

and

𝑆 𝑖

are fixed. The total uncoupled basis of the system is the direct product of all the individual basis states, Equation

2.2.2. This system is solved by evaluating the matrix elements of the Hamiltonian over the basis states and diagonalizing the Hamiltonian matrix. The dimension of the Hilbert space and therefore the Hamiltonian matrix is given by Equation 2.2.3.

𝐻�Ψ = 𝐸Ψ (2.2.1)

|𝐿, 𝑆, 𝑚

𝐿

, 𝑚

𝑆

⟩ = �𝐿

1

, 𝑆

1

, 𝑚

𝐿1

, 𝑚

𝑆1

⟩ ⊗ �𝐿

2

, 𝑆

2

, 𝑚

𝐿2

, 𝑚

𝑆2

⟩ ⊗ … �𝐿 𝑖

, 𝑆 𝑖

, 𝑚

𝐿𝑖

, 𝑚

𝑆𝑖

⟩, 𝑖 ∈ 𝑁 (2.2.2)

𝑁 𝑑𝑖𝑚 = �(2𝐿 𝑖 𝑖=1

+ 1)(2𝑆 𝑖

+ 1) (2.2.3)

2

The Hamiltonian is composed of operators which act on the angular momentum basis functions to yield the matrix elements. The Hamiltonian is split into four components: the SO coupling,

𝐻�

𝑍𝐸𝐸

.

𝐻�

𝑆𝑂

, the exchange coupling,

𝐻�

𝐸𝑋

, the CF interaction,

𝐻�

𝐶𝐹

and the Zeeman effect,

𝐻� = 𝐻�

𝑆𝑂

+ 𝐻�

𝐸𝑋

+ 𝐻�

𝐶𝐹

+ 𝐻�

𝑍𝐸𝐸

(2.2.4)

Spin-orbit coupling

The SO coupling operator is usually given as Equation 2.2.5, however this first order model results in the SO multiplets following the Landé interval rule. This is correctly obeyed by ions of low atomic mass, such as the 3d ions, however deviations from the Landé interval rule for heavy ions due to term mixing by SO coupling are significant and must be accounted for.

Thus in PHI, the SO operator is expanded as a power series following the parameterization of

Karayianis,

1

Equation 2.2.6. The sum extends to order

2𝑆 term in question. The coefficients, 𝜆

1𝑖

,

𝜆

2𝑖

and

𝜆

3𝑖 𝑖

, where

𝑆 𝑖

is the total spin of the

were tabulated for the tripositive lanthanides by Karayianis, however we have optimized these and included higher orders where required,

2

Table 2.2.1.

𝑁

𝐻�

𝑆𝑂

= � 𝜆 𝑖 𝑖=1

�𝜎 𝑖

𝐿��⃑ 𝑖

⋅ 𝑆̂⃑ 𝑖

� (2.2.5)

𝐻�

𝑆𝑂

𝑁

2𝑆 𝑖

= � � 𝜆 𝑗𝑖 𝑖=1 𝑗=1

�𝜎 𝑖

𝐿��⃑ 𝑖

⋅ 𝑆̂⃑ 𝑖

� 𝑗

(2.2.6) where 𝜆 𝑗𝑖 𝜎 𝑖

are the SO coupling constants

are the orbital reduction parameters

Ion

Ce

III

Ref. 1

Table 2.2.1 – Optimized spin-orbit parameters for the triply ionized rare-earths 𝝀

𝟏

(cm

-1

)

𝝀

𝟐

(cm

-1

)

𝝀

𝟑

(cm

-1

)

𝝀

𝟒

(cm

-1

)

𝝀

𝟓

(cm

-1

)

𝝀

𝟔

(cm

-1

)

640 - - - - -

Opt.

Pr

III

Ref. 1

Opt.

Nd

III

Ref. 1

Opt.

Pm

III

Ref. 1

691

390

421

299

326

251

-

-4.63

-5.78

-2.48

-2.66

-1.99

-

-

-

0.0475

0.0247

0.0239

-

-

-

-

-

0

-

-

-

-

-

-

-

-

-

-

-

-

Opt.

Sm

III

Ref. 1

Eu

Opt.

III

Ref. 1

Opt.

Tb

III

Ref. 1

Opt.

269

228

241

214

230

-252

-260

-1.85

-2.16

-2.34

-3.82

-3.28

-4.50

0.997

0.00977 -0.000920

0.0368

0.147

0.269

-0.267

0.223

0

0

-0.0402

-

0

0.0315 -0.000743 -0.00000883

0

0.000715

0

-

-

-

0

-0.00164 -0.000144

0

-0.00685

0

-0.000267

3

Dy

III

Ho

III

Er

III

Ref. 1

Opt.

Ref. 1

Opt.

Ref. 1

-357

-362

-497

-515

-629

-4.40

-2.73

-7.06

-7.83

-18.2

-0.121

-0.221

-0.139

-0.121

-0.517

0

-0.00655

0

0.00629

-

0

0.000110

-

-

-

-

-

-

-

-

Opt.

Tm

III

Ref. 1

-572

-875

Opt.

-684

Yb

III

Ref. 1

-2910

Opt.

-2957

-12.6

-123

-177

-

-1.85

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

- - - - -

Exchange coupling

For both spin-only and orbitally degenerate cases, the exchange Hamiltonian (Equation 2.2.7) is parameterized with the complete

𝐽 𝚤𝚥 separated into two components; the (an)isotropic exchange (Equation 2.2.9) and the antisymmetric exchange (Equation 2.2.10), in which case

𝐽 𝚤𝚥

2.2.11. While such an approach is commonplace in spin-only situations, the subject of magnetic exchange between orbitally degenerate ions is non-trivial and a number of attempts have been made to determine an effective operator for such cases.

3–7

Currently in PHI, the exchange interaction for orbitally degenerate ions follows the treatment of Lines,

8

which includes only the spin-spin interaction between the true spins in the

�𝐿 𝑖

, 𝑚

𝐿𝑖

, 𝑆 𝑖

, 𝑚

𝑆𝑖

⟩ basis. In

PHI however, the interaction can also be anisotropic and/or antisymmetric, thus is much more general than the original Lines model. The exchange coupling using the Lines approach may also be calculated in the

�𝐽 𝑖

, 𝑚

𝐽𝑖

⟩ basis when used in conjunction with the lanthanide ions in the simple input method (see section 4.3), utilizing a Clebsch-Gordan decomposition.

By default the reference frame of the exchange matrix is coincident with the global coordinate system, however this can be rotated such that the anisotropic and antisymmetric interactions can be described in simple, local reference frames.

Note that upon swapping the site indices of the exchange Hamiltonian, the exchange tensor becomes its transpose, i.e.

𝑆̂⃑ 𝑖

⋅ 𝐽

��� ⋅ 𝑆̂⃑ 𝑗

= 𝑆̂⃑ 𝑗

⋅ 𝐽 𝚤𝚥

𝑇

⋅ 𝑆̂⃑ 𝑖

. 𝑖,𝑗∈𝑁

𝐻�

𝐸𝑋

= −2 � 𝑆̂⃑ 𝑖<𝑗 𝑖

⋅ 𝐽

��� ⋅ 𝑆̂⃑ 𝑗

(2.2.7)

𝐽

��� = �

𝐽

𝐽 𝑖𝑗𝑥𝑥

𝐽 𝑖𝑗𝑦𝑥 𝑖𝑗𝑧𝑥

𝐽 𝑖𝑗𝑥𝑦

𝐽 𝑖𝑗𝑦𝑦

𝐽 𝑖𝑗𝑧𝑦

𝐽 𝑖𝑗𝑥𝑧

𝐽 𝑖𝑗𝑦𝑧

𝐽 𝑖𝑗𝑧𝑧

� (2.2.8) 𝑖,𝑗∈𝑁

𝐻�

(𝑎𝑛)𝑖𝑠𝑜

= −2 � 𝐽 𝑖𝑗𝑥 𝑖<𝑗

𝑆̂ 𝑖𝑥

𝑆̂ 𝑗𝑥

+ 𝐽 𝑖𝑗𝑦

𝑆̂ 𝑖𝑦

𝑆̂ 𝑗𝑦

+ 𝐽 𝑖𝑗𝑧

𝑆̂ 𝑖𝑧

𝑆̂ 𝑗𝑧

(2.2.9)

4

𝑖,𝑗∈𝑁

𝐻� 𝑎𝑛𝑡𝑖

= −2 � 𝑑⃑ 𝑖𝑗 𝑖<𝑗

⋅ �𝑆̂⃑ 𝑖

× 𝑆̂⃑ 𝑗

� 𝑖,𝑗∈𝑁

= −2 � 𝑑 𝑖𝑗𝑥 𝑖<𝑗

�𝑆̂ 𝑖𝑦

𝑆̂ 𝑗𝑧

− 𝑆̂ 𝑖𝑧

𝑆̂ 𝑗𝑦

� + 𝑑 𝑖𝑗𝑦

�𝑆̂ 𝑖𝑧

𝑆̂ 𝑗𝑥

− 𝑆̂ 𝑖𝑥

𝑆̂ 𝑗𝑧

� + 𝑑 𝑖𝑗𝑧

�𝑆̂ 𝑖𝑥

𝑆̂ 𝑗𝑦

− 𝑆̂ 𝑖𝑦

𝑆̂ 𝑗𝑥

�

(2.2.10)

𝐽 𝑖𝑗𝑥

𝐽

��� = �

−𝑑 𝑖𝑗𝑧 𝑑 𝑖𝑗𝑦 𝑑 𝑖𝑗𝑧

𝐽 𝑖𝑗𝑦

−𝑑 𝑖𝑗𝑥

−𝑑 𝑖𝑗𝑦 𝑑 𝑖𝑗𝑥

𝐽 𝑖𝑗𝑧

� (2.2.11)

Crystal-field potential

The CF potential is constructed from spherical harmonics to represent the environment in which the spin carrier resides. While twenty-seven terms exist in the full expansion, the number required may be reduced as the CF Hamiltonian must be invariant under the operations of the point group of the molecule (see below for a brief outline of the rules for non-zero parameters). Many approaches have been attempted over the years to determine

Crystal-Field Parameters (CFPs), such as the Point Charge (PCM),

9

Angular Overlap

(AOM)

10,11

and Superposition (SM)

12

models, however all have fallen short of consistently predicting these parameters. This is because, in reality, the electrostatic CF is inadequate due to the overlooked contributions from covalency, non-orthogonality, screening and polarization of the orbitals.

13

In spite of these criticisms, the CF model succeeds in describing experimental results when it is considered a phenomenological Hamiltonian, where the resultant parameters have no direct physical interpretation. Given this interpretation and the close similarity of the operators (see below), the CF Hamiltonian is also used in PHI to model

Zero Field Splittings (ZFS) of effective spins.

There are numerous parameterization schemes for the effective CF Hamiltonian and care must be taken to avoid confusion. For a good grounding, see Mulak and Gajek

13

, Hutchings

9 and Rudowicz.

14

As PHI constructs the Hamiltonian within a total spin-orbit basis, the operator equivalent technique of Stevens et al.

15,16

was chosen as the most efficient method for the evaluation of matrix elements, even though the notation is less transparent than others,

Equation 2.2.12. Here, the definitions of the

Mulak and Gajek

13

and Stevens,

16

𝑂� 𝑞 𝑘

operators are consistent with Hutchings,

9

however for clarity definitions of all the positive and negative operators are given below in Table 2.2.2. The operator equivalents themselves are polynomials of angular momentum operators, derived from the tesseral harmonics and PHI includes all even, odd, positive and negative orders ( 𝑞) for the 2 nd

, 4 th

and 6 th

rank ( 𝑘) operators. The rank, 𝑘, is restricted to 𝑘 = 2, 𝑘 = 4 and 𝑘 = 6 as only the ground terms of the ground configuration are considered. The use of the negative 𝑞 operators is equivalent to the ‘sine’ type operators of Hutchings

9

and the ‘imaginary’ CFPs in Wybourne notation.

13

The method relies on the use of the operator equivalent factors, 𝜃 𝑘

, to relate the total angular momentum matrix elements to the single electron matrix elements. These factors have been tabulated for the ground multiplets for all lanthanides,

17

but not as far as the author is aware for the ground terms of the lanthanides; these are now presented in Table 2.2.4.

5

In PHI, the CF Hamiltonian is applied to either the orbital or the total angular momentum components of a given centre. That is, if the centre possesses a non-zero orbital moment the

CF Hamiltonian directly acts on the orbital component as a true CF. However, if the centre does not possess an orbital moment, the CF Hamiltonian acts on the effective spin or total angular momentum, depending on one’s interpretation of the assigned ‘spin’. Note that the orbital reduction parameter, 𝜎 𝑖

, is only relevant when the CF Hamiltonian is applied to an orbital moment directly.

𝐻�

𝐶𝐹

𝑁 𝑘

= � � � 𝜎 𝑖 𝑘

𝐵 𝑞 𝑘 𝑖 𝑖=1 𝑘=2,4,6 𝑞=−𝑘 𝜃 𝑘

𝑂� 𝑞 𝑘 𝑖

(2.2.12)

𝐵 𝑞 𝑘 where 𝜎 𝑖

are the CFPs (

𝐴 𝜃 𝑘 𝑖

are the orbital reduction parameters 𝑞 𝑘 𝑖

⟨𝑟 𝑘

⟩ 𝑖

in Steven’s notation)

are the operator equivalent factors

𝑂� 𝑞 𝑘 𝑖

are operator equivalents

Table 2.2.2 – Definition of the Stevens operators

𝑂�

𝑂�

𝑂�

𝑂�

𝑂�

𝑂�

4

−4

𝑂�

4

−3

𝑂�

4

−2

=

=

4 �𝐿

= 3𝐿� 𝑧

=

1

=

−𝑖

−𝑖

+ 𝑧

− 𝐿�

2

�𝐿�

2

− 𝐿�

+

−

1 𝑧

�𝐿�

+

2

+ 𝐿�

+ 𝐿�

−

2

−

2

− 𝐿�

�

�

−

� + �𝐿�

+

� + �𝐿�

+

− 𝐿�

+ 𝐿�

−

−

�𝐿�

�𝐿� 𝑧 𝑧

�

�

Operator

=

=

=

−𝑖

+

−𝑖

+

4

− 𝐿�

−𝑖 𝑧

�𝐿�

−𝑖 𝑧

2

+

3

−

4

− 𝐿�

−

− 𝐿�

2

�

3

� + �𝐿�

− 5� �𝐿�

− 𝐿� 𝑧

+

+

2

3

− 𝐿�

− 𝐿�

−

−

2

3

� 𝐿� 𝑧

�

� + �𝐿�

+

2

− 𝐿�

−

2

� �7𝐿�

𝑂�

𝑂�

𝑂�

4

+1

𝑂�

4

−1

4

0

4

+2

=

4 ��7𝐿

= 35𝐿� 𝑧

=

1

=

1 𝑧

3 𝑧

3

− 30𝐿�

− 3𝐿�

2

𝐿�

− 3𝐿� 𝑧

2

2

2

𝐿�

𝐿� 𝑧

+ 25𝐿� 𝑧 𝑧

− 𝐿� 𝑧

− 5� �𝐿�

+

2

� �𝐿�

+ 3𝐿�

� �𝐿�

2

+

+ 𝐿�

+

+ 𝐿�

−

− 𝐿�

22

2

−

−

� + �𝐿�

− 6𝐿�

� + �𝐿�

2

� + �𝐿�

+

2

+

+

− 𝐿�

+ 𝐿�

−

+ 𝐿�

−

2

−

� �7𝐿�

� �7𝐿�

� �7𝐿� 𝑧

3 𝑧 𝑧

2

𝑂�

𝑂�

𝑂�

4

+3

4

+4

=

=

=

2

1 𝑧

�𝐿�

+

1 𝑧

−𝑖

+

4

2 �𝐿

+

6

− 𝐿�

3

+ 𝐿�

−

− 𝐿�

+ 𝐿�

4

−

2

�

6

−

�

3

� + �𝐿�

+

3

+ 𝐿�

−

3

� 𝐿� 𝑧

� 𝑧

3

2

− 𝐿�

2

− 3𝐿�

− 3𝐿�

2

− 𝐿�

2

2

𝐿�

− 5��

𝐿� 𝑧 𝑧

− 𝐿�

− 𝐿�

− 5�� 𝑧 𝑧

��

��

6

𝑂�

𝑂�

𝑂�

𝑂�

𝑂�

𝑂�

𝑂�

𝑂�

𝑂�

𝑂�

𝑂�

𝑂�

=

=

=

=

−𝑖

−𝑖

+

−𝑖

+

4 ��𝐿

+

5

4

3

− 𝐿�

− 𝐿�

− 𝐿�

−

−

−

5

4

3

� 𝐿�

� �11𝐿�

� �11𝐿�

+ �11𝐿� 𝑧

3 𝑧

+ 𝐿�

− 3𝐿� 𝑧 𝑧

2

2

3 𝑧

𝐿� 𝑧

�𝐿�

+

5

− 𝐿�

2

− 3𝐿�

2

− 𝐿�

− 38� + �11𝐿�

𝐿�

− 59𝐿� 𝑧 𝑧

−

5

� �𝐿�

��

− 59𝐿�

+

3 𝑧

�

− 𝐿� 𝑧

2

−

− 𝐿�

3

��

2

−𝑖

4 ��𝐿

+

2

− 𝐿�

−

2

+ �33𝐿� 𝑧

4

� �33𝐿� 𝑧

4

− 18𝐿�

2

− 18𝐿�

𝐿� 𝑧

2

2

𝐿� 𝑧

2

− 123𝐿�

− 123𝐿� 𝑧

2

+ 𝐿�

22 𝑧

2

+ 𝐿�

22

+ 10𝐿�

2

− 38� �𝐿�

+

4

+ 10𝐿�

2

+ 102� �𝐿�

+

− 𝐿�

+ 102�

2

−

− 𝐿�

4

��

−

2

��

=

−𝑖

4 ��𝐿

+

− 𝐿�

−

� �33𝐿�

+ �33𝐿� 𝑧

5 𝑧

5

− 30𝐿�

− 30𝐿�

2

𝐿� 𝑧

3

2

𝐿� 𝑧

3

+ 15𝐿�

+ 15𝐿� 𝑧

3 𝑧

3

+ 5𝐿�

+ 5𝐿�

22

𝐿� 𝑧

22

𝐿� 𝑧

− 10𝐿�

− 10𝐿�

2

𝐿� 𝑧

2

𝐿� 𝑧

+ 12𝐿�

+ 12𝐿� 𝑧

� �𝐿� 𝑧

+

�

− 𝐿�

−

��

= 231𝐿� 𝑧

6

− 315𝐿�

− 60𝐿�

2

2

𝐿� 𝑧

4

+ 735𝐿� 𝑧

4

+ 105𝐿�

22

𝐿� 𝑧

2

− 525𝐿�

2

𝐿� 𝑧

2

+ 294𝐿� 𝑧

2

− 5𝐿�

23

+ 40𝐿�

22

=

1

+

+ 𝐿�

−

� �33𝐿� 𝑧

5

− 30𝐿�

2

𝐿� 𝑧

3

+ 15𝐿� 𝑧

3

+ 5𝐿�

22

𝐿� 𝑧

− 10𝐿�

2

𝐿� 𝑧

+ 12𝐿� 𝑧

�

=

=

=

=

=

+ �33𝐿� 𝑧

5

− 30𝐿�

2

𝐿� 𝑧

3

+ 15𝐿� 𝑧

3

+ 5𝐿�

22

𝐿� 𝑧

− 10𝐿�

2

𝐿� 𝑧

+ 12𝐿� 𝑧

� �𝐿�

+

+ 𝐿�

−

��

1

4 ��𝐿

+

2

+ 𝐿�

−

2

� �33𝐿�

+ �33𝐿� 𝑧

4 𝑧

4

− 18𝐿�

2

− 18𝐿�

𝐿� 𝑧

2

2

𝐿� 𝑧

2

− 123𝐿�

− 123𝐿� 𝑧

2

+ 𝐿� 𝑧

2

22

1

1

+

1

+

1

+

6

2 �𝐿

+

3

4

5

+ 𝐿�

+ 𝐿�

+ 𝐿�

−

−

−

+ 𝐿�

−

6

3

4

5

�

� �11𝐿� 𝑧

3

� �11𝐿� 𝑧

2

� 𝐿� 𝑧

+ 𝐿� 𝑧

− 3𝐿�

− 𝐿�

�𝐿�

+

2

5

2

𝐿� 𝑧

+ 𝐿�

− 59𝐿�

−

5

�� 𝑧

+ 𝐿�

+ 10𝐿�

� + �11𝐿�

− 38� + �11𝐿� 𝑧

2

− 𝐿�

22 𝑧

2

3

2

+ 10𝐿�

2

+ 102� �𝐿�

− 3𝐿�

2

𝐿� 𝑧

− 38� �𝐿�

+ 102�

+

− 59𝐿� 𝑧

� �𝐿�

+

3

4

+

2

+ 𝐿�

+ 𝐿�

−

2

��

−

4

��

+ 𝐿�

−

3

��

Table 2.2.3 – Operator equivalent factors for the lanthanides in the

�𝐽, 𝑚

𝐽

⟩ basis

Ion Multiplet 2 nd

Rank 4 th

Rank 6 th

Rank

Ce

Pr

III

Nd

Pm

Sm

Eu

Gd

Tb

Dy

Ho

III

III

III

III

III

III

III

III

III

2

F

5/2

3

H

4

4

I

9/2

5

I

4

6

H

5/2

7

F

0

8

S

7/2

7

F

6

6

H

15/2

5

I

8

-2/35

-52/2475

-7/1089

14/1815

13/315

0

0

-1/99

-2/315

-1/450

2/315

-4/5445

-136/467181

952/2335905

26/10395

0

0

2/16335

-8/135135

-1/30030

0

272/4459455

-1615/42513471

2584/42513471

0

0

0

-1/891891

4/3864861

-5/3864861

7

Er

III

Tm

III

Yb

III

4

I

15/2

3

H

6

2

F

7/2

4/1575

1/99

2/63

2/45045

8/49005

-2/1155

8/3864861

-5/891891

4/27027

Table 2.2.4 – Operator equivalent factors for the lanthanides in the

|𝐿, 𝑚

𝐿

, 𝑆, 𝑚

𝑆

⟩ basis

Ion Term 2 nd

Rank 4 th

Rank 6 th

Rank

Ce

III

Pr

III

3

2

F

H

-2/45

-2/135

2/495

-4/10395

-4/3861

2/81081

Nd

III

Pm

III

Sm

III

6

4

5

I

I

H

7

F

-2/495

2/495

2/135

2/45

-2/16335

2/16335

4/10395

-2/495

-10/891891

10/891891

-2/81081

4/3861

Eu

III

Gd

III

Tb

III

Dy

III

Ho

III

6

8

7

S

F

H

0

-2/45

-2/135

-2/495

0

2/495

-4/10395

-2/16335

0

-4/3861

2/81081

-10/891891

Er

Tm

Yb

III

III

III

5

I

4

I

3

H

2

F

2/495

2/135

2/45

2/16335

4/10395

-2/495

10/891891

-2/81081

4/3861

Note that only 2 nd whereas the 6 th

and 4 th

rank operators are required to describe CFs for d-block ions,

rank is also, generally, required for f-block ions. Of course, however, higher rank operators may be required to accurately describe ZFS effects.

The second order CF operators are intimately related to those of the standard ZFS Spin

Hamiltonian

17

and using the definitions of the CF operators as in Table 2.2.2, the relationships between the ZFS parameters and the CFPs are therefore expressed in Equations

2.2.13 and 2.2.14.

𝐷 = 3𝐵 𝜃

2

(2.2.13)

𝐸 = 𝐵 𝜃

2

(2.2.14)

The non-zero CFPs are determined solely by the point group of the ion in question. Often the assumed point group symmetry does not include the entire molecule, but only the first coordination sphere of the paramagnetic ion, as this is the largest contribution to the perturbation. Often, idealized symmetry may be used initially, followed by small corrections to allow for distortions of lower symmetry. For a full C

1

representation, all 27 CFPs are required. If the group is not C

1

then only CFPs with even 𝑞 are required. If a C n

axis is present, only CFPs with 𝑞 = 𝑗𝑛, where 𝑗 is an integer, are required. Only the following groups need negative 𝑞 CFPs: C

1

, C i

(S

2

), C

3

, C

3i

(S

6

), C

4

, S

4

and C

6

. A comprehensive list of non-zero CFPs for all point group symmetries can be found in Gorller-Walrand and

Binnemans' chapter.

18

8

Zeeman Effect

The Zeeman Effect is the stabilization and destabil1ization of angular momentum projections parallel and anti-parallel to a magnetic field, Equation 2.2.15. It is this response to the magnetic field which is responsible for the observable magnetic properties, such as magnetization and magnetic susceptibility.

𝑁

𝐻�

𝑍𝐸𝐸

= 𝜇

𝐵

� �𝜎 𝑖 𝑖=1

𝐿��⃑ 𝑖

⋅ 𝐼̿ + 𝑆̂⃑ 𝑖 where

𝐼̿ is the identity matrix 𝑔

(2.2.15)

Orbital reduction factor

Note that in all Hamiltonians above, the 𝜎 parameter has been included with all orbital operators. This is the combined 𝜎 = −𝐴 ⋅ 𝜅 factor, required when using the T≡P equivalence for orbital triplet terms.

19

A is required when making the T → P substitution and takes the value of 1.0 when representing a T

2

term and 1.5 when representing a T

1 term. κ (or k in some texts) is the orbital reduction factor which is an empirical constant,

0 < 𝜅 < 1, and accounts for a reduction in the effective orbital angular momentum due to covalency or low symmetry effects. It can be effectively removed setting 𝜎 to unity (default). Note that for the SO and CF

Hamiltonians the orbital reduction factor is included as 𝜎 𝑖

, 𝜎 𝑖

2

, 𝜎 𝑖

3

, 𝜎 second, third, fourth, fifth and sixth rank, respectively, where required. 𝑖

4

,

𝜎 𝑖

5

or 𝜎 𝑖

6

for first,

Magnetic properties

The inclusion of the Zeeman Hamiltonian allows the magnetic properties to be calculated from first principles

20

without resorting to perturbation theory. Thus, full mixing of all states by the magnetic field is implicitly included. The fundamental definitions for the magnetic properties are expressed in Equations 2.2.16 and 2.2.17.

𝑀 ∝ −

𝜕𝐸

𝜕𝐵 (2.2.16) 𝜒 ∝

𝜕𝑀

𝜕𝐵 (2.2.17)

The molar magnetization is the sum of the magnetization of each state weighted by its

Boltzmann population, Equation 2.2.18, where Z is the partition function, Equation 2.2.20, giving the magnetization for a single Cartesian direction, 𝛼 ∈ x, y, z, in Bohr Magnetons per mole (µ

B

mol

-1

). Equivalently, the Magnetization can be calculated using Equation 2.2.19.

𝑀 𝛼

=

1

𝑍𝜇

𝐵 𝑑𝑖𝑚

� − 𝑖=1

𝜕𝐸 𝑖

𝜕𝐵 𝛼 𝑒

−𝐸 𝑖 𝑘

𝐵

𝑇 (2.2.18)

𝑀 𝛼

= 𝑘

𝐵 𝜇

𝐵

𝑇 𝜕 ln 𝑍

𝜕𝐵 𝛼

(2.2.19)

9

𝑑𝑖𝑚

𝑍 = � 𝑒

−𝐸 𝑖 𝑘

𝐵

𝑇 (2.2.20) 𝑖=1

Following Equation 2.2.17, the molar magnetic susceptibility is the first derivative of

Equation 2.2.18, resulting in Equation 2.2.21, which contains terms that depend on the first and second derivatives of the eigenvalues with respect to the magnetic field. As there are two derivative steps there are nine possible combinations of the Cartesian directions, 𝛼, 𝛽 ∈ 𝑥, 𝑦, 𝑧, leading to the definition of the 3 × 3 magnetic susceptibility tensor. Equation 2.2.21 reduces to the traditional vanVleck formula in the limit of zero magnetic field, however the numerical method employed here is capable of accurately determining the susceptibility in the presence of non-zero fields as used in experiment. Following Equation 2.2.19, Equation

2.2.22 is entirely equivalent to Equation 2.2.21. 𝜒 𝛼,𝛽

=

𝜕𝑀

𝜕𝐵 𝛼 𝛽

=

10𝑘

𝑁

𝐴

𝐵

𝑇𝑍 𝑑𝑖𝑚

− �� 𝑖=1

2

�𝑍 ��

𝜕𝐸 𝑖

𝜕𝐵 𝛼 𝑑𝑖𝑚

𝜕𝐸 𝑖

𝜕𝐵 𝛼 𝑖=1 𝑒

−𝐸 𝑖 𝑘

𝐵

𝑇 𝑑𝑖𝑚

� �� 𝑖=1

𝜕𝐸 𝑖

𝜕𝐵 𝛽

𝜕𝐸

𝜕𝐵 𝑖 𝛽 𝑒

−𝐸 𝑖 𝑘

𝐵

𝑇 𝑒

−𝐸 𝑖 𝑘

𝐵

𝑇

− 𝑘

��

𝐵 𝑑𝑖𝑚

𝑇 � 𝑖=1

𝜕

𝜕𝐵 𝛼

2

𝐸 𝑖

𝜕𝐵 𝛽 𝑒

−𝐸 𝑖 𝑘

𝐵

𝑇 �

(2.2.21) 𝜒 𝛼,𝛽

=

𝑁

𝐴 𝑘

10

𝐵

𝑇 𝜕

2 ln 𝑍

𝜕𝐵 𝛼

𝜕𝐵 𝛽

(2.2.22)

The entropy change associated with the application and removal of a magnetic field is the quantity associated with the Magnetocaloric Effect (MCE). The magnetic entropy change is easily calculated for isotropic or anisotropic systems through Equation 2.2.23.

is the molecular mass of the complex and the entropy change is in units of J kg

21

-1

Note that

𝑀

K

-1

. 𝑟

−∆𝑆 𝛼

=

−1000𝑁

𝑀 𝑟

𝐴

𝐵 𝛼

=∆𝐵

�

𝜕𝑀 𝛼

𝜕𝑇 𝑑𝐵 𝛼

(2.2.23)

𝐵 𝛼

=0

The low temperature heat capacity of a paramagnetic system can be very sensitive to the magnetic interactions. PHI is equipped to calculate the magnetic heat capacity through

Equation 2.2.24, which includes a phenomenological term to capture the effect of the lattice heat capacity.

22

The heat capacity is given in units of R (N

A k

B

), where

𝑇 temperature and 𝛼 is the lattice exponent.

𝐷

is the Debye

𝐶 =

𝑍 �∑ 𝑑𝑖𝑚 𝑖=1

𝐸 𝑖

2 𝑒

−𝐸 𝑖 𝑘

𝐵

𝑇 𝑘

𝐵

� − �∑

2

𝑇 2 𝑍 2 𝑑𝑖𝑚 𝑖=1

𝐸 𝑖 𝑒

−𝐸 𝑖 𝑘

𝐵

𝑇 �

2

+ 234 �

𝑇

𝑇

𝐷

� 𝛼

(2.2.24)

10

Accuracy and approximations

Originally in PHI, the magnetic properties were determined using Equations 2.2.18 and

2.2.21 directly, where the derivatives were calculated using the finite difference method.

23,24

This method was prone to some instability due to level crossings where the eigenvalues changed index, as well as hypersensitivity to Hamiltonian parameters when numerical limitations of double precision arithmetic became significant. From version 1.8, PHI now uses Equations 2.2.19 and 2.2.22 which prove to be more numerically stable and have the added benefit of being 1.5 and 1.67 times faster overall, respectively.

Whilst the general method for the calculation of the magnetic properties of arbitrary systems has been given above, a useful simplification of the method is possible when considering magnetically isotropic ‘spin-only’ compounds. Taking advantage of the spherical symmetry of the Hamiltonian in conjunction with first order approximation methods can lead to a substantial reduction in the computational demands of the problem. While the uncoupled basis is most useful for anisotropic systems easily allowing formulation of the SO and CF

Hamiltonians, isotropic systems requiring only the isotropic exchange Hamiltonian are block diagonal in a total spin basis. In this case the problem can be solved by considering each block independently, greatly reducing the dimension of the problem and speeding up the calculation. The matrix elements can be calculated using Irreducible Tensor Operators (ITOs) and the Wigner-Ekhart theorem and while the literature is well established, the necessary equations and procedures are presented to clarify frequent typographical errors and to present a consistent notation.

In this example, the coupled basis is formed by first coupling

𝑆 followed by coupling

𝑆̃

1

to

𝑆

3

to make

𝑆

123

or

𝑆̃

2

1

and

𝑆

2

to make

𝑆

12

or

𝑆̃

1

,

etc., to the final total spin

𝑆, expressed in bra-ket notation in Equation 2.2.25. Recall that these are vector sums such that Equation

2.2.26 must be satisfied for all coupling steps.

�𝑆

1

, 𝑆

2

, 𝑆̃

1

, 𝑆

3

, 𝑆̃

2

, … , 𝑆, 𝑚

𝑆

⟩ ≡ ��𝑆̃�, 𝑆, 𝑚

𝑆

⟩ (2.2.25)

�𝑆 𝑖

− 𝑆 𝑗

� ≤ 𝑆̃ 𝑘

≤ 𝑆 𝑖

+ 𝑆 𝑗

(2.2.26)

The isotropic exchange Hamiltonian can be represented by use of a 0 th

rank tensor operator,

25

Equation 2.2.27. The matrix elements of spherical tensor operators are evaluated by applying the Wigner-Ekhart theorem

26

followed by a decoupling procedure to calculate the reduced matrix elements,

27,28

as expressed in Equation 2.2.28, where the numerator of the fraction is a

Clebsch-Gordan coefficient and the quantities in braces are Wigner 9j symbols

29

(see below for simplifications). The remaining reduced matrix element can be easily calculated (see below). Note that the 𝑘 𝑖

and 𝑘� 𝑖

values are the ranks of the component and intermediate spins, respectively, and can be easily determined using simple rules.

28 𝑖,𝑗∈𝑁 𝑖,𝑗∈𝑁

𝐻�

𝐸𝑋

= −2 � 𝐽 𝑖𝑗 𝑖<𝑗

�𝑆̂⃑ 𝑖

⋅ 𝑆̂⃑ 𝑗

� = 2√3 � 𝐽 𝑖𝑗

𝑇�

(0) 𝑖<𝑗

(2.2.27)

11

𝑁

× ��⟨𝑆 𝑖 𝑖=1

=

|| 𝑆̂

⟨𝑆, 𝑚

(𝑘 𝑖

)

𝑆

; 0,0|𝑆

√2𝑆

||𝑆 𝑖

⟩

′

′

+ 1

��𝑆̃

, 𝑚

𝑁−1 𝑖=2

⟩

′

�, 𝑆

� �� ��2𝑘� 𝑖

′

, 𝑚

��2𝑘�

1

�𝑇�

(0)

��𝑆̃�, 𝑆, 𝑚

𝑆

+ 1��2𝑆̃

+ 1��2𝑆̃ 𝑖

1

+ 1��2𝑆̃ 𝑖

′

�

+ 1��2𝑆̃

1

′

+ 1� �

𝑆

𝑆

1

𝑆̃

′ 𝑖−1

2

1

′

+ 1� �

𝑆̃

𝑆 𝑖+1

𝑆̃ 𝑖

′

𝑆

1

𝑆

2

𝑆̃

1

𝑆̃ 𝑖−1

𝑆 𝑖+1

𝑆̃ 𝑖 𝑘

1 𝑘

2 𝑘�

1 𝑘� 𝑖−1 𝑘

� 𝑖+1 𝑘� 𝑖

� �

(2.2.28)

As the tensor is rank zero, the Clebsch-Gordan coefficient is equivalent to two Kronecker delta functions, Equation 2.2.29. While this simplifies the calculations, it also implies something much more meaningful – there is no dependence on the magnetic quantum number at all, such that it may be excluded from the basis and the dimensionality of the Hamiltonian matrix further reduced. Coupled with block diagonalization of the matrix, this leads to a tremendous reduction in the computational effort required for the problem.

⟨𝑆, 𝑚

𝑆

; 0,0|𝑆

′

, 𝑚

⟩ = 𝛿

𝑆,𝑆

′ 𝛿 𝑚

𝑆

,𝑚

(2.2.29)

The occurrence of Wigner 9j symbols in every matrix element is unfortunate due to their computational complexity, however in this case there are only four possible 9j symbols which are easily simplified.

28

They are presented below in Equations 2.2.30 ‒ 2.2.3 1, where the quantities in the braces on the right hand side of the equations are Wigner 6j symbols.

� 𝑎 𝑏 0 𝑐 𝑑 0 𝑒 𝑓 0

� = 𝛿 𝑎,𝑏 𝛿 𝑐,𝑑 𝛿 𝑒,𝑓

�(2𝑎 + 1)(2𝑐 + 1)(2𝑒 + 1)

(2.2.30)

� 𝑎 𝑏 1 𝑐 𝑑 0 𝑒 𝑓 0

� = 𝛿 𝑐,𝑑

(−1) 𝑎+𝑓+𝑐+1

�3(2𝑐 + 1) 𝑒 𝑓 𝑐� (2.2.31)

� 𝑎 𝑏 1 𝑐 𝑑 1 𝑒 𝑓 0

� = 𝛿 𝑒,𝑓

(−1) 𝑏+𝑐+𝑒+1

�3(2𝑒 + 1) 𝑑 𝑐 𝑒� (2.2.32)

� 𝑎 𝑏 0 𝑐 𝑑 1 𝑒 𝑓 1

� = 𝛿 𝑎,𝑏

(−1) 𝑎+𝑑+𝑒+1

�3(2𝑎 + 1) 𝑐 𝑑 𝑎

� (2.2.33)

The reduced matrix elements remaining in Equation 2.2.28 are easily calculated, depending on the rank of the particular spin operator:

27

⟨𝑆 𝑖

|| 𝑆̂

(0)

||𝑆 𝑖

⟩ = �2𝑆 𝑖

+ 1 (2.2.34)

⟨𝑆 𝑖

|| 𝑆̂

(1)

||𝑆 𝑖

⟩ = �𝑆(𝑆 + 1)(2𝑆 + 1) (2.2.35)

12

Once the matrix elements have been calculated, the matrix is diagonalized to determine the eigenvalues and eigenvectors of the coupled states. To evaluate the magnetic properties it is necessary to determine the effective g-factors for the coupled spin multiplets. In general, the spin multiplets, 𝜓 𝑎

, originate from a mixture of the different

��𝑆̃�, 𝑆 𝑎 necessitating further ITO algebra.

30

⟩ coupled basis states,

The g-factors for each multiplet can be calculated according to Equations 2.2.36 and 2.2.37, where the 𝑐 eigenvector describing state 𝜓 𝑎,𝑆̃

′

factors are the components of the 𝑎

. With the g-factors known, the magnetic properties are calculated by considering the first order Zeeman perturbation to the 𝑚

𝑆

states. 𝑔 𝑎

=

�𝑆 𝑎

(𝑆 𝑎

1

+ 1)(2𝑆 𝑎

+ 1)

𝑁

� 𝑏 𝑖=1 𝑎,𝑖 𝑔 𝑖

(2.2.36)

𝑁

× ��⟨𝑆 𝑖 𝑖=1 𝑏 𝑎,𝑖

= � � 𝑐 𝑎,𝑆̃

′

𝑆̃

′

𝑆̃

= � � 𝑐 𝑎,𝑆̃

′

𝑆̃

′

𝑆̃ 𝑐 𝑎,𝑆̃

��𝑆̃

′

�, 𝑆��𝑆̂

(𝑘 𝑖

)

���𝑆̃�, 𝑆� 𝑐 𝑎,𝑆̃

𝑁−1

|| 𝑆̂

(𝑘 𝑖

)

||𝑆 𝑖

⟩

� �� ��2𝑘� 𝑖 𝑖=2

��2𝑘�

1

+ 1��2𝑆̃

+ 1��2𝑆̃

1 𝑖

+ 1��2𝑆̃

+ 1��2𝑆̃ 𝑖

′

1

′

𝑆

1

+ 1� �

𝑆

2

1

′

′ 𝑖−1

+ 1� �

𝑆̃

𝑆̃

𝑆 𝑖+1

𝑆̃ 𝑖

′

𝑆

1

𝑆

2 𝑘

1 𝑘

2

𝑆̃

1 𝑘�

1

𝑆̃ 𝑖−1

𝑆 𝑖+1

𝑆̃ 𝑖

� 𝑘� 𝑖−1 𝑘 𝑖+1 𝑘� 𝑖

� �

(2.2.37)

Powder integration

For powder measurements on anisotropic systems, the calculations must be integrated over many orientations to accurately reflect the experiment. While poly-crystalline samples contain a finite number of crystallites with discrete orientations, it is usually assumed that the size of the crystals is small enough that it is closely representative of a powder sample with an infinite number of orientations evenly distributed on the sphere. For the magnetic susceptibility, it is sufficient to use the ‘xyz’ integration scheme, as this is exact for a second rank tensor property. For the magnetization however, the ‘xyz’ scheme is inadequate and should not be used. A number of orientation integration schemes are possible; PHI uses the

Zaremba-Conroy-Wolfsberg (ZCW) scheme as presented by Levitt.

31

The implementation in

PHI samples the magnetic properties over a hemisphere, as magnetic properties are invariant under inversion of the magnetic field.

Pseudo g-tensors

For calculations involving anisotropic ions which give rise to doublet states, pseudo g-tensors may be calculated within the basis of each doublet. This is equivalent to treating each doublet as a pseudo-spin

𝑆̃ = 1/2 state whose magnetic anisotropy is given by the g-tensor. For

Kramers systems these doublets are related by time inversion symmetry and the treatment is rigorous, however for non-Kramers systems the g-tensors for pseudo doublets are only approximate and only 𝑔 conjugate states.

17 𝑧

is non-zero due to vanishing off-diagonal elements between the

The theory is well established,

32,33

but a brief overview of the method will be given. Note that PHI does not currently support the g-tensor calculation in bases of other values of pseudo-spin.

13

For a given system, the expectation values of the three Cartesian magnetic moment operators are evaluated in the basis of the zero field wavefunction, Equation 2.2.38. The g-tensor is then constructed for each doublet through Equation 2.2.39, where 𝜓 and 𝜓

′

are the wavefunctions of the doublet and 𝛼, 𝛽 ∈ 𝑥, 𝑦, 𝑧.

†

𝐻�

𝑍𝐸𝐸 𝜇

𝐵

𝐵 𝛼

=1

Ψ� (2.2.38)

𝐺 𝛼,𝛽

= 2 � � 𝜇 𝛼𝑢,𝑣 𝑢=𝜓,𝜓′ 𝑣=𝜓,𝜓′ 𝜇 𝛽𝑣,𝑢

(2.2.39)

This g-tensor is then diagonalized to yield three principle values and their corresponding directions, leading to the definition of the anisotropic g-tensor for each pseudo-spin doublet.

By convention, the directions are defined such that 𝑔 𝑥

< 𝑔 𝑦

< 𝑔 𝑧

.

Transition probabilities

For anisotropic systems, the zero-field average transition probability between states 𝑢 and 𝑣 is calculated through Equation 2.2.40, using the expectation values of the three Cartesian magnetic moment operators, Equation 2.2.38. The transition probabilities are in units of squared Bohr magnetons ( 𝜇

𝐵

2

).

2

𝑇 𝑢,𝑣

=

1 𝛼=𝑥,𝑦,𝑧

𝑁

3 � ��𝑣� � �𝜎 𝑖

𝐿� 𝑖𝛼

+ 𝑆̂ 𝑖𝑥 𝑔 𝑖𝑥𝛼

+ 𝑆̂ 𝑖𝑦 𝑔 𝑖𝑦𝛼

+ 𝑆̂ 𝑖𝑧 𝑔 𝑖𝑧𝛼

� �𝑢�� (2.2.40)

J-mixing

For calculations on single lanthanide ions in the expressed also in the

�𝐽 𝑖

, 𝑚

𝐽𝑖

�𝐿 𝑖

, 𝑚

𝐿𝑖 means of investigating the extent of J-mixing by the CF.

, 𝑆 𝑖

, 𝑚

𝑆𝑖

⟩ basis, the wavefunction is

⟩ basis, though a Clebsch-Gordan decomposition. This provides a

Non-collinearity

For single magnetic centres the orientation of the reference frame is always an arbitrary choice and any symmetry elements that may be identified, by crystallography or other means, can be related to this axis. When considering multiple magnetic sites in a single compound, while the global reference frame is still arbitrary, the individual reference frames, which may possess defined symmetry elements, may not be coincident and in which case it would not be ideal to enforce the global frame upon all sites. Therefore PHI allows users to rotate individual reference frames of the magnetic centres to allow for a description of each centre in its own, most convenient, reference frame. The two sources of magnetic anisotropy in PHI are the anisotropic g-tensor and the CF Hamiltonian. The diagonal g-tensor is easily rotated into the local frame, using the Z-Y’-Z’’ convention according to Equations 2.2.41 – 2.2.44.

The rotation of the CFPs is performed according to Mulak and Mulak’s convention,

34

with a slight modification. The rotation of a set of CFPs of a given rank, in Wybourne notation, is given by Equation 2.2.45, where the elements of the unitary rotation matrix,

𝐷�, are given by

Equation 2.2.46. The symbols in brackets in Equation 2.2.46 are binomial coefficients. The rotation convention in PHI is different to that of EasySpin; the rotation matrices are the transpose of each other, therefore

𝑅

𝑃𝐻𝐼

������������(−𝛾, −𝛽, −𝛼).

14

cos 𝜃 − sin 𝜃 0 sin 𝜃 cos 𝜃 0

0 0 1

(2.2.41) cos 𝜃 0 sin 𝜃

0 1 0

− sin 𝜃 0 cos 𝜃

(2.2.42)

𝑅 𝑃𝐻𝐼 𝑧

(𝛼) ∙ 𝑅 𝑦

(𝛽) ∙ 𝑅 𝑧

(𝛾) (2.2.43)

𝐺

′

������(𝛼, 𝛽, 𝛾) ∙ 𝐺̿ ∙ 𝑅

𝑃𝐻𝐼

𝑇

(𝛼, 𝛽, 𝛾) (2.2.44)

′ 𝑘

(𝑘)

(𝛼, 𝛽, 𝛾) ⋅ 𝐵

𝐷

(𝑘) 𝑚,𝑛

(𝛼, 𝛽, 𝛾) = 𝑒 𝑖(𝑚𝛼+𝑛𝛾) �

(𝑘 + 𝑚)! (𝑘 − 𝑚)!

(𝑘 + 𝑛)! (𝑘 − 𝑛)!

2𝑘 𝑘 − 𝑚 − 𝑝� � 𝑘 − 𝑛 𝑝 � (−1) 𝑘−𝑚−𝑝 𝑐𝑜𝑠 �

−𝛽

2 �

2𝑝+𝑚+𝑛 𝑠𝑖𝑛 �

−𝛽

2 �

2𝑘−2𝑝−𝑚−𝑛 𝑝=0

(2.2.46)

TIP, intermolecular interactions and magnetic impurities

A Temperature Independent Paramagnetic (TIP) component can be added to the calculated magnetic susceptibility, directly in units of cm

3

mol

-1

, Equation 2.2.47.

Intermolecular interactions between spin systems can be modelled using the mean-field approximation, Equation 2.2.48; this expression changed as of version 2.0 to allow its use in anisotropic systems.

Magnetic impurities are included employing analytical expressions for the field and temperature dependent magnetization and magnetic susceptibility for pure spin centres with g

= 2.0. As of version 2.0, the impurity value represents the fraction of the system, Equations

2.2.49 and 2.2.50.

These effects are included in the order of TIP, zJ, magnetic impurity, giving the final expression for the magnetic susceptibility, Equation 2.2.51. 𝜒

𝑇𝐼𝑃

= 𝜒 𝑐𝑎𝑙𝑐

+ 𝑇𝐼𝑃 (2.2.47) where

𝑇𝐼𝑃 is the temperature independent paramagnetism 𝜒 𝑧𝐽

= 𝜒

𝑇𝐼𝑃

� 𝜒

𝑇𝐼𝑃

(2.2.48)

𝑁

𝐴 𝜇

𝐵

2 where 𝑧𝐽 is the intermolecular interaction parameter

15

𝜒 = (1 − 𝐼𝑀𝑃)𝜒 𝑧𝐽

+ (𝐼𝑀𝑃)𝜒

𝐼𝑀𝑃

(2.2.49)

𝑀 = (1 − 𝐼𝑀𝑃)𝑀 𝑐𝑎𝑙𝑐

+ (𝐼𝑀𝑃)𝑀

𝐼𝑀𝑃

(2.2.50) where 𝜒

𝐼𝑀𝑃

is the field and temperature dependent magnetic susceptibility of the impurity

𝑀

𝐼𝑀𝑃

is the field and temperature dependent magnetization of the impurity

𝐼𝑀𝑃 is the fraction of magnetic impurity 𝜒 = (1 − 𝐼𝑀𝑃)

𝑁

𝐴 𝜒 𝑐𝑎𝑙𝑐 𝜇

𝐵

2

+ 𝑇𝐼𝑃

� (𝜒 𝑐𝑎𝑙𝑐

+ 𝑇𝐼𝑃)

+ (𝐼𝑀𝑃)𝜒

𝐼𝑀𝑃

(2.2.51)

Error residuals

In all cases, the error for a particular data set is calculated following the sum of squares approach, Equation 2.2.52 as an example for magnetization. When calculating the total error for a simultaneous comparison to multiple data sets, the total residual is calculated as the product of the individual sum of squares errors for each data set. In this way, different error scales of the individual data sets will not obscure any features. 𝑝𝑜𝑖𝑛𝑡𝑠

𝑅𝑒𝑠𝑖𝑑𝑢𝑎𝑙 = � �𝑀 𝑖=1 𝑒𝑥𝑝

− 𝑀 𝑐𝑎𝑙𝑐

�

2

(2.2.52)

Electron Paramagnetic Resonance

The simulation of EPR spectra is not a simple task. Due to the field swept nature of the experiment, the action of the magnetic field on the sample must be accounted for and generally cannot be treated as a perturbation. Therefore, evaluation of the field dependent wavefunctions is required. Many approaches for this task have been employed, using various approximations, most of which involve searching the energy manifolds for transitions.

35,36

In

PHI, the EPR spectrum is calculated via a ‘brute-force’ approach which considers the transition probability for every pair of states at each field point explicitly. While this approach is very computationally intensive, it does not rely on any approximations and includes all transitions, whether deemed to be ‘allowed’ or ‘forbidden’ as well as looping transitions. The EPR absorption as a function of field is calculated through Equation 2.2.53 or

2.2.54 for perpendicular or parallel mode, respectively.

There is also the possibility in PHI to calculate the EPR spectrum using an infinite order perturbation theory approach. The structure of the routine is almost identical to that for the full method, however in place of diagonalization of the full Hamiltonian, the exchange and

Zeeman components are treated as perturbations to the zeroth order wavefunctions, which are the eigenfunctions of the SO and CF Hamiltonians. This method therefore assumes knowledge of the single-site properties for each ion, which can then be perturbed by the exchange interaction and magnetic field. Of course this is inappropriate in large magnetic fields or if the exchange interactions are stronger than the SO or CF terms.

In both cases the linewidth function assumes a pseudo-voigt profile (Equation 2.2.55),

37 which has shown to be required in certain applications.

38

The linewidth is treated in frequency space and therefore no frequency-field conversion factor (commonly referred to as

16

the

1 𝑔

factor) is required.

39,40

Note that the 𝑥

′

and 𝑦 orthogonal to the main magnetic field,

𝐵�⃑, while the 𝑧

′

′

directions are determined as mutually

direction is parallel to it. 𝑖,𝑗∈𝑑𝑖𝑚

𝐴�𝐵�⃗� = � ���𝑗�𝐻�

𝑍𝐸𝐸 𝑖<𝑗

�

𝐵 𝑥′

=1

�𝑖��

2

+ ��𝑗�𝐻�

𝑍𝐸𝐸

�

𝐵 𝑦′

=1 𝑖,𝑗∈𝑑𝑖𝑚

𝐴�𝐵�⃗� = � ��𝑗�𝐻�

𝑍𝐸𝐸 𝑖<𝑗

�

𝐵 𝑧′

(2.2.53)

�𝑒

−𝐸 𝑖 𝑘

𝐵

𝑇

=1

�𝑖��

2

�𝑖��

− 𝑒

𝑍

2

�

�𝑒

−𝐸 𝑗 𝑘

𝐵

𝑇 �

−𝐸 𝑖 𝑘

𝐵

𝑇 − 𝑒

𝑍

𝑉�∆𝐸, 𝜂

−𝐸 𝑗 𝑘

𝐵

𝑇 𝑖𝑗

�

𝑉�∆𝐸, 𝜂 𝑖𝑗

, 𝑣�

, 𝑣� (2.2.54) where

|𝑖⟩ and 𝑗⟩ are two eigenstates evaluated at 𝐵�⃑

𝑉(∆𝐸, 𝜂, 𝑣) = 𝑣

∆𝐸 = ��𝐸 𝑖

− 𝐸 𝑗

� − 𝐸

𝑀𝑊 𝜂 𝑖𝑗

is the linewidth 𝑣 is the voigt parameter

�

2

2

�

+ (1 − 𝑣)

2√ln 2

√𝜋𝜂𝑒

2

(2.2.55)

As with the calculation of powder thermodynamic magnetic properties of anisotropic systems, the EPR absorption signal must be integrated over all possible orientations of the magnetic field; the ZCW scheme as discussed above is used for this purpose. After the absorption spectrum is calculated, it is normalized and if requested the first or second derivative or the integration is taken via finite differences.

The EPR linewidth can be augmented to include the effects of crystal mosacity.

41

The linewidth is augmented for each orientation independently as in Equation 2.2.56. 𝜂 𝑖𝑗

= �𝜂

0

2

+ 𝜔

2

��

𝜕𝐸

𝜕𝐵 𝑖 𝜃

−

𝜕𝐸 𝑗

𝜕𝐵 𝜃

�

2

+ �

𝜕𝐸

𝜕𝐵 𝑖 𝜑

−

𝜕𝐸 𝑗

𝜕𝐵 𝜑

�

2

� (2.2.56) where 𝜔 is the mosacity parameter

17

3. Code Description

3.1 PHI

PHI is written entirely in Fortran95 and split into six modules for easy maintenance. These modules are data.f90, ang_mom.f90, powder.f90, props.f90, fitting.f90 and phi.f90.

data.f90 contains all the explicit variable declarations for global variables and arrays. It also contains a number of subroutines which initialize constants, read input files, write output files and perform diagnostics.

ang_mom.f90 contains all the Hamiltonian operators and tools required for matrix operations.

powder.f90 contains the routines required for powder integration procedures.

props.f90 contains the subroutines for the calculation of the magnetic properties.

fitting.f90 contains the subroutines necessary to perform surveys and fits, containing residual calculation routines and fitting algorithms.

phi.f90 is the main program which controls what calculations are to be performed.

The program can be well understood by means of an operational schematic, Figure 3.1.1.

Note: the boxes are only representative of the flow of the program and do not necessarily correspond to individual subroutines or functions.

Figure 3.1.1 – Operational schematic of PHI

18

The heart of the program is the construction and diagonalization of the Hamiltonian matrix, which is required for any calculation. PHI relies on the use of external linear algebra routines

(BLAS and LAPACK routines) for matrix diagonalization and multiplication.

PHI has been written to take advantage of multiple processor cores, now common in consumer and specialized machines. There are two models of parallelism currently supported by PHI – Symmetric Multi-Processing (SMP) and Single Process Multiple Data (SPMD) – which use fundamentally different ideas to perform tasks more efficiently compared to a sequential model. The simplest approach to increase computational efficiency is to employ multiple cores on a shared memory machine (SMP model) to perform multiple diagonalizations simultaneously, which is achieved in PHI using OpenMP threads to distribute work. However, the SMP model is clearly limited by the size of the machine, both the number of cores and available memory. For this reason, the SPMD model is one of the most common parallel strategies due to the cost effectiveness of multiple smaller machines.

PHI uses the MPI standard to distribute work amongst an arbitrary number of processes connected by a network.

Multi-dimensional non-linear optimization is a difficult problem, often requiring in-depth parameter space analysis to determine the global minimum for a given problem. For this reason, PHI contains two internal fitting algorithms, Powell’s method

42

and the Simplex method,

43

which have been implemented based on those described in Numerical Recipes for

Fortran.

44

The Simplex method is well suited to optimizing nearby minima while Powell’s method is often useful in situations where a good initial guess is not available.

PHI contains several functions from the Fortran version of Stevenson’s anglib library

45

– modified versions of the functions ‘cleb’, ‘sixj’, ‘binom’ and ‘angdelta’ are contained within the source.

3.2 GUI

The GUI is written in Python using the PyGTK, Matplotlib and ReportLab libraries and has been designed in Glade. The GUI is provided as a standalone executable for Windows, while

MacOS and Linux users must run the GUI as a python script.

19

4. User Guide

4.1 Binaries and compilation

PHI is available as both pre-compiled, statically linked binaries and as source code.

There are eight available binaries, as listed below.

Windows, 32-bit: phi_vx.x_windows32.exe

Windows, 32-bit, OpenMP:

Windows, 64-bit:

Windows, 64-bit, OpenMP:

MacOS, 64-bit:

MacOS, 64-bit, OpenMP:

Linux, 64-bit:

Linux, 64-bit, OpenMP: phi_vx.x_windows32omp.exe phi_vx.x_windows64.exe phi_vx.x_windows64omp.exe phi_vx.x_mac64.x phi_vx.x_mac64omp.x phi_vx.x_linux64.x phi_vx.x_linux64omp.x

These binaries have been compiled using the Intel compiler suite and the Intel MKL.

To compile PHI from source, the supplied Makefile must be tailored to the specifications of the system at hand, a Fortran 95 compiler must be available and the appropriate libraries need to be compiled and in known locations. The source files must be listed in the order shown in the Makefile or errors will be encountered. PHI must be compiled using a C pre-processor, which provides a means for compiling different versions of the code from the same source files. Table 4.1.1 shows the compile time options. Even without SMP or SPMD activated, the

C pre-processor must still be invoked. If compiling for SPMD, it is recommended to use your

MPI library’s wrapper compiler, eg “mpifort” with the additional libraries and options required by PHI. It is recommended that PHI be compiled with the highest level of compiler optimization and inter-procedural optimization. Note that the source code is written in freeform Fortran95 and therefore compilers such as gfortran may need the ‘-ffree-form’ flag (or similar).

Table 4.1.1 – Compile time options

Option C preprocessor flag

Other required flags

Additional libraries

Forbidden flags

SMP (OpenMP)

SPMD (MPI)

-Domp

-Dmpi

How to use the Makefile

-openmp

MPI

-Dmpi

-Domp

The supplied Makefile provides a skeleton to set up a custom compilation of PHI. The variables at the top of the file must be set in order to compile the program. COMPILER is your Fortran 95 compiler, e.g. ifort or gfortran. MPI_COMPILER is your wrapper MPI compiler, e.g. mpifort. FLAGS and MPI_FLAGS can be adjusted as the user pleases.

LAPACK must be set to the appropriate library destination and contain links to lapack and blas. SOURCES is the list of source files for PHI and must be in the default order.

The flags in the Makefile may be specific to the ifort compiler and therefore must be

20

substituted for their equivalent flag for your compiler (e.g. -openmp becomes -fopenmp for gfortran)

4.2 Program execution

To run PHI, it is as simple as launching the executable on the command line from the working directory containing the input file, e.g. “phi_vx.x_linux64.x test-job”, where test-job is the name of the input file.

The standalone GUI is launched by simply running the gui executable, whereas the GUI script is run with Python as “python phi.py”. The PHI executable(s) should also be in the same directory as the GUI files, allowing the GUI to automatically select the PHI executable.

This selection can be changed by altering the executable name in the GUI or by placing only the desired PHI executable in the program directory. The default job directory is the same as that containing the GUI files, however this can be changed by selecting “File > New” or “File

> Open”.

4.3 Input files and syntax

Input to the program is via plain text input and data files. The job name used to launch the program (Section 4.2) defines the name of the associated input and data files PHI will look for. It will look for files in the directory that the program was launched from, the current working directory. For the above example, PHI will look for “test-job.input” in the current directory. This input file contains all the instructions that PHI needs to perform calculations.

The other data files required vary based on the type of calculation specified by the input file.

A total list of input and data files is given below, using the example job name. test-job.input test-job_mag.exp

Contains all input specifications and parameters

Contains experimental magnetization data test-job_sus.exp test-job_levels.exp test-job_G.exp test-job_mce.exp test-job_epr.exp test-job_heat.exp

Contains experimental susceptibility data

Contains experimental energy levels

Contains experimental g-tensors

Contains experimental MCE data

Contains experimental EPR data

Contains experimental heat capacity data

Please note that when running PHI on a Windows or Unix machine, the end-of-line characters for the data files must be in DOS or Unix format and that data files prepared on

Macintosh computers may have to be converted before they will work. This can be accomplished with the free utility flip, https://ccrma.stanford.edu/~craig/utility/flip/ .

21

.input specification

This file is delimited into blocks by headers, signified by four asterisks, “****”. The first block must be the ****Spin or ****Ion block (see below) and the input file is terminated by

“****End”. The input file is not case-sensitive, despite the notation given in this manual for clarity. After the “****End” termination line, the input file is not read by PHI and so may contain descriptions, other input specifications or comments. Also, any line that begins with

“#” is interpreted as a comment and not read by PHI.

The first block, which must be the ****Spin or ****Ion block, specifies the number and type of magnetic centres in the problem and this can be done in one of two ways.

Method 1:

****Spin block

In the first method, the ‘full’ input method, the first line must be “****Spin” and the subsequent lines denote the spin angular momentum of the centres. The ‘spins’ are entered as two times the spin (

2𝑆 𝑖

), or the number of unpaired electrons. Note that these spins may be real spins or pseudo-spins. In the following example, three paramagnetic centres are declared with spins

𝑆

1

= 2, 𝑆

2

= 𝑆

3

= 𝑆

4

= 5/2.

****Spin

4

5

5

5

****Orbit block

In the ‘full’ input method, the “****Orbit” block is also used, which details the corresponding orbital angular momentum of each site declared in the ****Spin block. Like

( the ****Spin block, the orbital moments must be entered as two times the orbital moment

2𝐿 𝑖

). If this block is omitted, the orbital moments are all assumed to be zero. This example assigns

𝐿

1

= 𝐿

2

= 𝐿

3

= 0 and 𝐿

4

= 5, corresponding to the spins above.

****Orbit

0

0

0

10

Method 2:

****Ion block

The above two blocks can be efficiently replaced in the case of common situations, by using the ‘simple’ input method. To use the simple input method, the first line must be “****Ion” and the subsequent lines define the centres in a standard notation. The example below would make exactly the same assignments as specified in the examples above, under certain assumptions.

22

****Ion

Mn(III)Oh(w)

Fe(III)Oh(w)

Fe(III)Oh(w)

Dy(LS)

The possible keywords for the ****Ions block are given in Table 4.3.1 with the specifications that they designate.

Table 4.3.1 – Ion types

Keyword

Ee

Ti(III)Oh

Ti(III)Td

Ti(III)FI

Ti(II)Oh

Ti(II)Td

Ti(II)FI

V(IV)Oh

V(IV)Td

V(IV)FI

V(III)Oh

V(III)Td

V(III)FI

V(II)Oh

V(II)Td(w)

V(II)Td(s)

V(II)FI

Cr(III)Oh

Cr(III)Td(w)

Cr(III)Td(s)

Cr(III)FI

Cr(II)Oh(w)

Cr(II)Td(w)

Cr(II)FI

Mn(VI)Oh

Mn(VI)Td

Mn(VI)FI

Mn(IV)Oh

Mn(IV)Td(w)

Mn(IV)Td(s)

Mn(IV)FI

Mn(III)Oh(w)

Mn(III)Td(w)

Mn(III)FI

Mn(II)Oh(w)

Mn(II)Oh(s)

Mn(II)Td(w)

Mn(II)FI

Term S L

λ (cm

-1

)

2

S 1/2 0 -

2

2

T

2g

1/2 1

E 1/2 0

2

D 1/2 2

3

3

T

1g

3

A

2

F

1

1

1

1

0

3

155.0

-

155.0

61.5

-

61.5

2

2

T

2g

1/2 1

E 1/2 0

2

D

3

3

T

1g

3

A

2

F

1/2

1

1

2

1

0

1 3

4

4

A

2g

3/2 0

2

T

1

3/2 1

E 1/2 0

250.0

-

250.0

105.0

-

105.0

-

56.5

-

4

F 3/2 3

4

A

2g

3/2 0

4

T

1

3/2 1

2

E 1/2 0

4

F

5

E g

5

T

2

5

D

3/2

2

2

3

0

1

2

D

2 2

2

2

T

2g

E

1/2 1

1/2 0

1/2 2

4

4

A

2g

3/2 0

2

T

1

E

3/2

1/2

1

0

4

F

5

5

E g

T

2

5

D

3/2

2

2

3

0

1

2 2

6

2

A

1g

5/2 0

6

T

2g

1/2 1

6

A

1

5/2 0

S 5/2 0

56.5

-

91.5

-

91.5

-

57.5

57.5

540.0

-

540.0

-

138.5

-

138.5

-

89.0

89.0

-

-300.0

-

-

σ

-

-1.0

-

1.0

-1.5

-

1.0

-1.0

-

1.0

-1.5

-

1.0

O

Comment

T d

O h h

Radical

O h

symmetry

T d

symmetry

Spherical symmetry

O h

symmetry

T d

symmetry

Spherical symmetry

symmetry

symmetry

Spherical symmetry

symmetry

T d

symmetry

Spherical symmetry

- O h

symmetry

-1.5 T d

symmetry, weak CF

- T d

symmetry, strong CF

1.0 Spherical symmetry

- O h

symmetry

-1.5 T d

symmetry, weak CF

- T d

symmetry, strong CF

1.0 Spherical symmetry

- O h

symmetry, weak CF

-1.0 T d

symmetry, weak CF

1.0 Spherical symmetry

-1.0

-

1.0

-

O h

symmetry

T d

symmetry

Spherical symmetry

-1.5 T d

symmetry, weak CF

- T d

symmetry, strong CF

1.0

-

-

-

-

O h

symmetry

Spherical symmetry

O h

symmetry, weak CF

-1.0 T d

symmetry, weak CF

1.0 Spherical symmetry

O h

symmetry, weak CF

-1.0 O h

symmetry, strong CF

T d

symmetry, weak CF

Spherical symmetry

23

Fe(VI)Oh

Fe(VI)Td

Fe(VI)FI

Fe(III)Oh(w)

Fe(III)Oh(s)

Fe(III)Td(w)

Fe(III)FI

Fe(II)Oh(w)

Fe(II)Td(w)

Fe(II)FI

Co(III)Oh(w)

Co(III)Td(w)

Co(III)FI

Co(II)Oh(w)

Co(II)Oh(s)

Co(II)Td

Co(II)FI

Ni(III)Oh(w)

Ni(III)Oh(s)

Ni(III)Td

Ni(III)FI

Ni(II)Oh

Ni(II)Td

Ni(II)FI

Cu(II)Oh

Cu(II)Td

Cu(II)FI

Ce(J)

Ce(LS)

Pr(J)

Pr(LS)

Nd(J)

Nd(LS)

Pm(J)

Pm(LS)

Sm(J)

Sm(LS)

Eu(LS)

Gd(III)

Tb(J)

Tb(LS)

Dy(J)

Dy(LS)

Ho(J)

Ho(LS)

Er(J)

Er(LS)

Tm(J)

3

3

T

1g

3

A

2

F

1

1

1

0

1 3

6

2

A

1g

5/2 0

6

T

2g

1/2 1

6

A

1

5/2 0

S 5/2 0

332.5

-

332.5

-

-460.0

-

-1.5

-

1.0

-

-1.0

-

O

O

T h

Spherical symmetry h d

O

T h d

symmetry

symmetry

symmetry, weak CF

symmetry, strong CF

symmetry, weak CF

- - Spherical symmetry

5

5

T

2g

E

2

2

1

0

5

D

5

5

T

2g

E

5

D

2

2

2

2

1

0

2 2

4

T

1g

3/2 1

4

2

E g

1/2 0

A

2

3/2 0

4

F 3/2 3

4

T

1g

3/2 1

4

2

E g

1/2 0

A

2

3/2 0

4

F 3/2 3

-100.0

-

-100.0

-145.0

-

-145.0

-171.5

-

-

-171.5

-235.0

-

-

-1.0

-

1.0

-1.0

-

1.0

-1.5

-

-

1.0

-1.5

-

-

-235.0 1.0

3

3

A

2g

3

T

1

F

1

1

0

1

1 3

2

E g

1/2 0

2

T

2

1/2 1

2

D 1/2 2

-

-315.0

-315.0

-

-830.0

-

-1.5

1.0

-

-1.0

-830.0 1.0

2

F

5/2

5/2 0

2

F 1/2 3

-

Table 2.2.1

-

1.0

O

O

T

O

O

O

O h h h h d h h

symmetry, weak CF

T d

symmetry, weak CF

Spherical symmetry

symmetry, weak CF

symmetry, weak CF

Spherical symmetry

symmetry, weak CF

symmetry, strong CF

T

Spherical symmetry

symmetry, weak CF

symmetry, strong CF

T

Spherical symmetry

O

T h d

symmetry

symmetry

Spherical symmetry

O

T d d h d

symmetry

symmetry

symmetry

symmetry

Spherical symmetry

Spherical symmetry

Spherical symmetry

4

3

3

H

4

H

4

1

0

5

-

Table 2.2.1

-

1.0

I

9/2

9/2 0

4

I 3/2 6

-

Table 2.2.1

-

1.0

5

I

4

5

I

6

H

5/2

6

H

7

F

8

S

4

2

5/2

5/2

3

7/2

0

6

0

5

3

0

-

Table 2.2.1

-

Table 2.2.1

Table 2.2.1

-

-

1.0

-

1.0

1.0

-

7

F

6

7

F

6

H

15/2

6

H

5

I

8

5

I

6

3

15/2

5/2

8

2

0

3

0

5

0

6

-

Table 2.2.1

-

Table 2.2.1

-

Table 2.2.1

-

1.0

-

1.0

-

1.0

Spherical symmetry

Spherical symmetry

Spherical symmetry

Spherical symmetry

Spherical symmetry

Spherical symmetry

Spherical symmetry

Spherical symmetry

Spherical symmetry

Spherical symmetry

Spherical symmetry

Spherical symmetry

Spherical symmetry

Spherical symmetry

Spherical symmetry

Spherical symmetry

4

I

15/2

15/2 0

4

I 3/2 6

-

Table 2.2.1

-

1.0

3

H

6

6 0 - -

Spherical symmetry

Spherical symmetry

Spherical symmetry

24

Tm(LS)

Yb(J)

Yb(LS)

3

H 1 5 Table 2.2.1 1.0

2

F

7/2

7/2 0

2

F 1/2 3

-

Table 2.2.1

-

1.0

Spherical symmetry

Spherical symmetry

Spherical symmetry

Note that for all d block ions and LS type f-block ions, the isotropic electronic spin g-factor is set to 2.0. For J type lanthanides, the isotropic g-factor is set to the appropriate g

J value.

17

For the d-block free-ions (FI) and LS type f-block ions, the appropriate operator equivalent factors are automatically included.

Other blocks

****Gfactors block

To specify the spin g-factors for the centres, the “****Gfactors” block is used. Unless specified, all g-factors are taken to be 2.0 by default. The syntax requires the site index followed by one or three values, indicating either an isotropic or anisotropic spin g-factor

( 𝑔 𝑥

, 𝑔 𝑦 of 𝑔 𝑥 𝑔 𝑥

, 𝑔 𝑧

= 𝑔 𝑦

). In the following example, the second centre is given an anisotropic spin g-factor

= 0.1, 𝑔 𝑦

= 1.9 and 𝑔 𝑧

= 2.5 and 𝑔 𝑧

= 2.2, the third centre is also given an anisotropic spin g-factor of

= 11.9 and the fourth centre is given an isotropic spin g-factor of

1.98.

****Gfactors

2 1.9 1.9 2.2

3 0.1 2.5 11.9

4 1.98

****Exchange block

To define (an)isotropic exchange coupling interactions between the centres, the

“****Exchange” block is used. The interactions are all zero by default, so only the required interactions should be listed. This is done on one line by specifying the index of the first site, followed by the index of the second site, followed by the isotropic or anisotropic exchange values. Only one or three values should be given, indicating either an isotropic exchange or the three diagonal components of the exchange matrix J xx

, J yy

and J zz

. The following example specifies three coupling pathways between sites 1 and 2, 2 and 3 and 1 and 3, where the exchange involving site 1 is axially anisotropic.

****Exchange

1 2 2.0 2.0 -6.0

2

1

3

3

5.5

1.0 1.0 5.0

****Antisymmetric block

To define the antisymmetric components of exchange interactions between centres, the

“****Antisymmetric” block is used. The interactions are all zero by default, so only the required interactions should be listed. This is done on one line by specifying the index of the first site, followed by the index of the second site, followed by the three components of the antisymmetric exchange. The following example specifies antisymmetric exchange between sites 2 and 3. Note that handedness is preserved with PHI, so that switching the order of the interacting pair is equivalent to negating the antisymmetric exchange vector.

****Antisymmetric

2 3 0.1 -0.1 1.5

25

****Interaction block

To define completely asymmetric exchange interactions between centres, the

“****Interaction” block is used. The interactions are all zero by default, so only the required interactions should be listed. The syntax is similar to that for the “****Fit” and

“****Survey” blocks, see below. The first line for each exchange pair gives the site indices, followed by three lines for each row of the interaction tensor. The final line must be “----” which signifies the end of the interaction tensor. Note that handedness is preserved with PHI, so that switching the order of the interacting pair will imply usage of the transpose of the given exchange tensor.

****Interaction

2 3

Jxx Jxy Jxz

Jyx Jyy Jyz

Jzx Jzy Jzz

----

****SOCoupling block

To define or modify the SO Coupling parameters, the “****SOCoupling” block is used. The syntax requires the site index followed by up to six values representing the first to sixth order

SO Coupling parameters, in wavenumbers. This example sets the parameters for the sites 1 and 5 only, where site 5 has only a first order component, while site 1 has both first and second order.

****SOCoupling

1

5

421.0 -5.78

-165.0

****OReduction block

The combined orbital reduction parameters can be set, through the use of this block. These are specified by the index of the site followed by the value of the total orbital reduction parameter. Note that the value for the parameter should be negative. A value of 1.0 removes the feature (default), i.e. no orbital reduction. In this example, site 5 is given a total orbital reduction parameter of 1.35 (be sure not to confuse the terminology and sign of the parameter).

****OReduction

5 -1.35

****CrystalField block

A CF may be specified by using the “****CrystalField” block. The syntax requires the site index, followed by the rank, order and then value of the parameter. The site index, rank and order must be integers, while the CFPs must be real numbers. If the ‘full’ input method was used, operator equivalent factors are not included by PHI automatically and they must be included in the input parameters explicitly, if required. However if the ‘simple’ input method was used, the operator equivalent factors for the lanthanides and d-block free-ions are automatically included by PHI. The following example specifies the

𝐵

-0.001 for site 1, the

𝐵 equal to 0.230. 𝜃

6 𝜃

2

parameter equal to

parameter equal to

0.0006 for site 2 and the 𝐵

4

−1 𝜃

4

parameter

26

To describe a weak cubic field for the d-block free-ions, it is recommended to use the ‘Cubic

N’ parameter (see Table 4.3.3). As the operator equivalent factors are taken into account using the simple input method, octahedral fields are described in all cases with a positive

𝐵

4

0 parameter and tetrahedral fields with a negative

𝐵

4

0

parameter.

****Crystal Field

1 2 0 -0.001

2

3

6

4

3

-1

0.0006

0.230

****Sus block

This block provides all the options for the calculation of magnetic susceptibility. Table 4.3.2 gives all the available options for this block.

Table 4.3.2 – ****Sus block options

Parameter Options/Syntax Comments

Magnetic field direction / integration

Field STR

Field Powder N

Field Vector X Y Z

Field Angles θ φ

Selection of magnetic field: STR is either x, y or z for single directions or xyz for principal axes integration. If STR is ‘Powder’ then

ZCW integration is used where N ≥ 0. If STR is ‘Vector’ an arbitrary single direction is given. If STR is ‘Angles’ an arbitrary single direction is given in polar coordinates. Default

= “Field z” (isotropic), “Field xyz”

(anisotropic).

Magnetic field

Temperature sweep

BSus A B C D …

Sweep Low High N

Selects the magnetic field(s) in Tesla for the calculation. Any number of fields can be listed on the same line. Default = “BSus 0.01”.

Sets the temperature range (in Kelvin) and number of points. Default = “Sweep 1.8 300

250”.

Sets a TIP (in cm

3

mol

-1

). Default = “TIP 0”. Temperature Independent

Paramagnetism

Intermolecular interaction

TIP X zJ X

Options/Syntax

Field STR

Field Powder N

Field Vector X Y Z

Sets the mean-field zJ parameter (in cm

-1

).

Default = “zJ 0”.

****Mag block

This block provides all the options for the calculation of magnetization. Table 4.3.3 gives all the available options for this block.

Table 4.3.3 – ****Mag block options

Parameter

Magnetic field direction / integration

Comments

Selection of magnetic field: STR is either x, y or z for single directions or xyz for principal axes integration. If STR is ‘Powder’ then

ZCW integration is used where N ≥ 0. If STR is ‘Vector’ an arbitrary single direction is

27

Temperature

Magnetic field sweep

Field Angles θ φ

TMag A B C D …

Sweep Low High N given. If STR is ‘Angles’ an arbitrary single direction is given in polar coordinates. Default

= “Field z” (isotropic), “Field Powder 3”

(anisotropic).

Selects the temperature(s) in Kelvin for the calculation. Any number of temperatures can be listed on the same line. Default = “TMag 2

4 10 20”.

Sets the magnetic field range (in Tesla) and number of points. Default = “Sweep 0 7 10”.

****MCE block

This block provides all the options for the calculation of the magnetocaloric effect. Table

4.3.4 gives all the available options for this block.

Table 4.3.4 – ****MCE block options

Parameter

Magnetic field direction / integration

Options/Syntax

Field STR

Field Powder N

Field Vector X Y Z

Field Angles θ φ

Comments

Selection of magnetic field: STR is either x, y or z for single directions or xyz for principal axes integration. If STR is ‘Powder’ then

ZCW integration is used where N ≥ 0. If STR is ‘Vector’ an arbitrary single direction is given. If STR is ‘Angles’ an arbitrary single direction is given in polar coordinates. Default

= “Field z” (isotropic), “Field Powder 3”

(anisotropic).

Magnetic field

Temperature sweep

Molecular mass

Integration

BMCE A B C D …

Sweep Low High N

Mass X

Integrate N

Selects the magnetic field(s) in Tesla for the calculation. Any number of fields can be listed on the same line. Default = “BMCE 7”.

Sets the temperature range (in Kelvin) and number of points. Default = “Sweep 1.8 50

250”.

Sets the molecular mass for the sample in g mol

-1

. Default = “Mass 2000”.

Sets the number of magnetic field integration points. Default = “Integrate 50”.

****Heat block

This block provides all the options for the calculation of the heat capacity. Table 4.3.5 gives all the available options for this block.

Table 4.3.5 – ****Heat block options

Parameter

Magnetic field direction / integration

Options/Syntax

Field STR

Field Powder N

Field Vector X Y Z

Comments

Selection of magnetic field: STR is either x, y or z for single directions or xyz for principal axes integration. If STR is ‘Powder’ then

ZCW integration is used where N ≥ 0. If STR is ‘Vector’ an arbitrary single direction is

28

Magnetic field

Temperature sweep

Field Angles θ φ

BHeat A B C D …

Sweep Low High N given. If STR is ‘Angles’ an arbitrary single direction is given in polar coordinates. Default

= “Field z” (isotropic), “Field Powder 3”

(anisotropic).

Selects the magnetic field(s) in Tesla for the calculation. Any number of fields can be listed on the same line. Default = “BHeat 0.1”.

Sets the temperature range (in Kelvin) and number of points. Temperatures are on a base-

10 logarithmic scale. Default = “Sweep 0.5 20

250”.

Debye lattice contribution Debye T

D

α

Sets the Debye temperature (in Kelvin) and exponent. Default = “Debye 0 0”

****EPR block

The “****EPR” block is used to specify the options for the EPR calculation. The possible keywords are given in Table 4.3.6.

Table 4.3.6 – ****EPR block options

Parameter

Magnetic field direction / integration

Temperature

Options/Syntax

Field STR

Field Powder N

Field Vector X Y Z

Field Angles θ φ

TEPR A B C D …

Comments

Selection of magnetic field: STR is either x, y or z for single directions or xyz for principal axes integration. If STR is ‘Powder’ then

ZCW integration is used where N ≥ 0. If STR is ‘Vector’ an arbitrary single direction is given. If STR is ‘Angles’ an arbitrary single direction is given in polar coordinates. Default

= “Field Powder 6”.

Sets the temperature(s) for the simulation, in

Kelvin. Default = “TEPR 5”.

Frequency

Spectrum type

Parallel mode

Magnetic field sweep

Linewidth

Pseudo-voigt parameter

FEPR A B C D …

Type N

Parallel

Sweep Low High N

Sets the frequency(ies) for the simulation, in

GHz. Default = “FEPR 9.5 35 94”.

Selects whether the absorption (N = 0), first derivative (N = 1), second derivative (N = 2) or integrated absorption (N = -1) spectrum is to be calculated. Default = “Type 1”.

Selects parallel mode. Default = Off.

Sets the magnetic field range (in Tesla) and number of spectrum points. Default = “Sweep

0 1.6 250”.

Linewidth A B C D … Sets the base FWHM isotropic pseudo-voigt linewidth, in cm

-1

, for each frequency. If only one linewidth is given and multiple frequencies are to be simulated, then all frequencies have the same linewidth. Default =

“Linewidth 0.27”.

Voigt A B C D … Sets the pseudo-voigt parameter for each frequency. If only one parameter is given and multiple frequencies are to be simulated, then

29

Mosaicity

Subspace perturbation

Mosaic A B C D …

Subspace N all frequencies have the same parameter.

Default = “Voigt 0” (Gaussian).

Sets the mosacity parameter for each frequency. If only one parameter is given and multiple frequencies are to be simulated, then all frequencies have the same parameter.

Default = “Mosaic 0”.

Turns on the subspace perturbation method and selects how many states to include.

Default = Off.

****Zeeman block

The “****Zeeman” block is used to specify the options for the Zeeman calculation. The possible keywords are given in Table 4.3.7.

Table 4.3.7 – Zeeman options

Parameter

Magnetic field direction

Options/Syntax

Field STR

Magnetic field sweep

Field Vector X Y Z

Field Angles θ φ

Sweep Low High N

Comments

Selection of magnetic field: STR is either x, y or z for single directions or xyz for principal axes integration. If STR is ‘Vector’ an arbitrary single direction is given. If STR is

‘Angles’ an arbitrary single direction is given in polar coordinates. Default = “Field z”.

Sets the magnetic field range (in Tesla) and number of points. Default = “Sweep 0 7 250”.

Either the full calculation or the approximation scheme (in cases where applicable) may be used to perform such a sweep. If using the full calculation method, please note that due to the convention of matrix diagonalization routines, the eigenvalues are returned in ascending order, thus presenting artefacts at level crossings which appear like avoided crossings. This is demonstrated in Figure 4.3.1 with a simple isotropic case using the full calculation (top) and the approximation method (bottom), both with 10 steps. It is clearly seen that the level crossings are not correctly displayed in the top figure due to the width of the steps and the eigenvalue re-ordering. This can be corrected visually by increasing the number of steps.

30

Figure 4.3.1 – Zeeman plots using a full calculation (top) and the approximation method (bottom)

****Survey block

To perform a parameter sweep, the “****Survey” block is utilized. The first line of this block specifies what the user wishes to survey. For example, if the first line is ‘Residual’ the output will be the residual error between the calculation and experiment against the parameters in the survey. Other options include ‘M(i,j)’, ‘S(i,j)’, ‘C(i,j)’ or ‘H(i,j)’, which represent the value of the magnetization, susceptibility, MCE or heat capcity respectively, for the i th

field and the j th

temperature. Following the first line, this block is internally delimited into sections which belong to the same variable, by “----”. The start and end values for the parameter and the number of steps required are first specified, followed by the properties that they control.

In the following example, the exchange coupling parameter is varied between -10 and 10 cm

-

1

in 20 steps and the isotropic g-factors of sites 1 and 4 are varied from 1.7 to 2.3 in 10 steps.

31

****Survey

Residual

-10.0 10.0 20

EX 1 2

----

1.7 2.3 10

GF

GF

----

1

4

4

4

4

0

0

Table 4.3.8 lists the syntax for different properties; note that the dummy integers (zeros) must be present.

Table 4.3.8 – ****Fit and ****Survey block syntax

Syntax

EX SiteA SiteB 1/2/3/4/5/6/7

Comment

Exchange coupling, third integer represents x, y, z, isotropic, antisymmetric x, antisymmetric y or antisymmetric z.

IN SiteA SiteB 1/2/3/4/5/6/7/8/9

SO Site 1/2/3/4/5/6 0

Interaction tensor, third integer represents Jxx,

Jxy, Jxz, Jyx, Jyy, Jyz, Jzx, Jzy or Jzz.

Spin-orbit coupling, second integer is the order.

GF Site 1/2/3/4 0

CF

RC

RE

OR

LW

VO

Site

Site

Rank Order

1/2/3 0

SiteA SiteB 1/2/3

Site 0

Freq. 0

Freq. 0

0

0

0

G-factor, second integer represents x, y, z or isotropic.

Crystal field parameter.

Reference frame rotation, second integer represents α, β or γ.

Exchange frame rotation, third integer represents α, β or γ.

Orbital reduction parameter.

EPR linewidth, second integer selects corresponding frequency, where 0 implies all frequencies.

EPR pseudo-voigt parameter, second integer selects corresponding frequency, where 0 implies all frequencies.

MO Freq. 0

TI 0

DT 0

DA

ZJ

IM

0

0

0

0

0

0

0

0

0

0

0

0

0

0

EPR mosacity, second integer selects corresponding frequency, where 0 implies all frequencies.

Temperature Independent Paramagnetism.

Debye temperature.

Debye exponent.

Mean-field intermolecular interaction.

Monomeric impurity.

****Fit block

To fit experimental data, the “****Fit” block must be detailed. This block is very similar in syntax to the ****Survey block, however in place of the start, finish and number of steps,

either the starting value for the parameter or the lower limit, starting value and upper limit is required. Also, before the beginning of the variable sub-blocks, the first line is either

32

“Powell” or “Simplex”, specifying the fitting algorithm to be used. The example below would fit the isotropic exchange coupling between sites 1 and 2 and the isotropic g-factor for site 1, limited between 1.9 and 2.1, using the Simplex method. The user should be reminded that the residual in fitting modes is not an absolute reference and will vary dramatically from problem to problem. Also, it is advisable to always visually check the results of the minima obtained from fits to aid in the determination of the global minimum. Note that while some fits may be numerically better than others, it does not necessarily mean that they are actually better – this must be visually confirmed. Note: the ordering of parameters will not affect a

Simplex minimization, however it will affect a Powell minimization. The Simplex method is often more useful than the Powell method when approximate values for the parameters are already known.

****Fit

Simplex

10.0

EX 1

----

1.9 2.0 2.1

GF

----

1

2

4

4

0

****Params block

Finally, the “****Params” block is used to choose the operation mode and other calculation options. Table 4.3.9 gives the options available in this section.

Table 4.3.9 – ****Params block options

Parameter Options/Syntax Comments

Operation mode

Magnetism approximation

OpMode STR1 STR2 Selection of operation mode, STR1 and STR2 are strings, see Table 4.3.10. Must be present.

Approx Turns on the block diagonal approximation for isotropic systems. Default = Off.

Monomeric impurity

Zero field splitting

Cubic crystal field

Static magnetic field

Rotate reference frame

Rotate exchange frame

IMP N x

ZFS I J K …

Cubic I J K …

Adds a monomer impurity of spin S = N/2, with fraction x, i.e. x = 1 for one uncoupled spin. Default = Off.

Alters the convention of

𝐵

such that it equals

𝐷. Any number of sites can be listed on the

Forces cubic CFP ratios for on

𝐵

4

0 same line. Default = Off.

and

𝐵

𝐵

4

4

and

𝐵

6

4

based

. Any number of sites can be listed on the same line. Default = Off.

StaticB |B| X Y Z Includes the presence of a static magnetic field

StaticB |B| θ φ

Rotate N 𝛼 𝛽 𝛾 of magnitude |B| Tesla, with vector (X,Y,Z) or polar coordinates (θ,φ). Default = Off.

Rotates the reference frame (CFPs and or g) for site N, through the Euler angles 𝛼, 𝛽 and 𝛾, given in degrees. Default = Off.

EXRotate N M 𝛼 𝛽 𝛾

Rotates the exchange frame for the exchange defined between sites N and M, through the

Euler angles 𝛼, 𝛽 and 𝛾, given in degrees.

33

Number of CPU cores

Fitting algorithm display

Full wavefunction printing

Save survey calculations

Disable operator equivalent factors

G-tensor multiplets

Single crystal experiment

Survey percentage completion

Residual type

G-tensor direction residual

Fitting algorithm vigour

Fitting algorithm limiting

MaxCPU N

NoPrint

FullWF

Save

NoOEF

Mults N A B C D …

Single

Percent

Residual STR

Default = Off.

Sets the upper limit of CPU cores available, N is an integer. Default = “MaxCPU 1”.

Turns off the printing of fit progress to the terminal and intermediate results to disk.

Default = Off.

Prints the full wavefunction in the states.res file. Default = Off.

Saves a file for each step of the survey calculation. Default = Off.

Disables the Operator Equivalent Factors such that CF input values are assumed to contain θ k

.

Default = Off.

Gives the multiplicities of the multiplets for the calculations of pseudo-spin

𝑆̃ =

1

2

states. N gives the number of multiplets, followed by N integers giving the multiplicity. Default = Off.

Circumvents checking for the need to integrate magnetic properties – i.e. requested single field direction is allowed.

Default = Off.

Prints survey percentage completion. Default

= Off.

Selects residual calculation method. STR is a string, see Table 4.3.13. Default = Off.

GDir STR

FitVigour X

Selects which directions to include in the residual calculation for g-tensors (and directions). STR is a string of x, y, z or a combination thereof. Default = “GDir xyz”.

Sets how vigourous the fitting algorithm starts, as a parameter percentage. Default =

“FitVigour 10”.

FitLimit X Sets how strongly the fitting algorithm enforces parameter limits. Limiting function is 𝑒

𝑋|∆|

, where

∆ is difference between the fitting parameter and its limit. Default = “FitLimit

12”.

Table 4.3.10 – Operation modes

OpMode

Sim STR2

Fit STR2

Comments

Simulation; STR2 is a string, see below and Table 4.3.11.

Fit; STR2 is a string, see below and Table 4.3.11.

Sur STR2 Survey; STR2 is a string, see below and Table 4.3.11.

Coupling Report Reports the block diagonal structure of the matrix

Matrix Elements Prints Hamiltonian matrix

34

The second required string is composed of letters representing the calculations to be performed. For example, “MS” would represent Magnetization and Susceptibility, whilst

“LSG” would represent energy Levels, Susceptibility and G-tensors. The possible letters are given in Table 4.3.12. Note that a Simulation involving L (energy levels) will result in the printing of the wavefunction in states.res, that G and D are mutually exclusive and that the letter codes may be in any order

(i.e. LMSG ≡ MGLS etc.).

Table 4.3.11 – Operation mode STR2

STR2

L

M

S

Comments

Energy levels

Magnetization

Susceptibility

G

D

C

H

G-tensors

G-tensors with directions

MCE

Heat capacity

E

Z

EPR

Zeeman

The number of directions used with the ZCW integration scheme is not a linear trend with

ZCW level. Table 4.3.12 shows the number of directions for each ZCW level up to ZCW 20.

Table 4.3.12 – Number of directions in ZCW integration

ZCW Number

0 21

1

2

3

4

34

55

89

144

9

10

11

12

5

6

7

8

233

377

610

987

1597

2584

4181

6765

13

14

15

16

17

18

19

10946

17711

28657

46368

75025

121393

196418

35

20 317811

The options for the residual types can be used to favour the better fitting of particular regions of data.

Table 4.3.13 – Residual types

Residual type string

LowT

HighT

LowB

HighB

Comments

Low temperature bias

High temperature bias

Low field bias

High field bias

LowT/LowB

LowT/HighB

HighT/LowB

HighT/HighB

LowE

Low temperature and low field bias

Low temperature and high field bias

High temperature and low field bias

High temperature and high field bias

Low energy bias

HighE

sus.exp specification

High Energy bias

This file is used to define the experimental data for fitting purposes. The file is plain text composed of floating point numbers, where the first column represents the temperature in K and the subsequent columns represent the experimental data (

χ

M

T) in cm

3

mol

-1

K for the different fields as defined in the .input file, for example 0.01, 0.1 and 1 T respectively. Note that there should be no blank lines at the end of the file.

T1

T2

T3 etc.

B1

B1

B1

B2

B2

B2

B3

B3

B3

mag.exp specification

This file is very similar to the sus.exp file, however the first column represents the magnetic field in T and the subsequent columns represent the experimental data (

𝑀) in Bohr

Magnetons per mole (

𝑁

𝐴 𝜇

𝐵

) for the different temperatures as defined in the .input file.

B1

B2

B3 etc.

T1

T1

T1

T2

T2

T2

T3

T3

T3

mce.exp specification

This file is very similar to the sus.exp file, however the first column represents the temperature points in K and the subsequent columns represent the experimental data (

−𝛥𝑆) in

J kg

-1

K

-1

for the different magnetic fields. Note that the appropriate molecular mass must be defined in the ****Params block.

36

T1 B1 B2 B3

T2

T3 etc.

B1

B1

B2

B2

B3

B3

heat.exp specification

This file is very similar to the sus.exp file, however the first column represents the temperature points in K and the subsequent columns represent the experimental data (

𝐶) in units of R (J mol

-1

K

-1

) for the different magnetic fields.

T1

T2

T3 etc.

B1

B1

B1

B2

B2

B2

B3

B3

B3

epr.exp specification

This file is very similar to the sus.exp file, however the first column represents the magnetic field points in T and the subsequent columns represent the experimental data (either integrated absorbance, absorbance, first derivative or second derivative) for the different frequencies and temperatures. The data should be normalized to the magnitude of the largest peak (positive or negative), at the first temperature in the input file. Note that the temperature is the inner loop and varies first, followed by the frequency.

B1

B2

B3 etc.

F1,T1 F1,T2 F1,T3 F2,T1 F2,T2 F2,T3

F1,T1 F1,T2 F1,T3 F2,T1 F2,T2 F2,T3

F1,T1 F1,T2 F1,T3 F2,T1 F2,T2 F2,T3

levels.exp specification

This file defines the experimental energy levels which are only required if a fit or survey with respect to the energy levels is required. The format for this file is one floating point value per line, for each energy level, given in wavenumbers. It is possible to specify unknown energy levels in the file, using a question mark (see example below), which will not be used when calculating error residuals. Note that there should be no blank lines at the end of the file.

E1

E2

?

?

E5

E6 etc.

G.exp specification

This file is used to define the experimental g-tensors used for the fitting and survey modes.

The file consists of lines with three or twelve values, defining either 𝑔 𝑔

𝐷 𝑦 𝑖 𝛼

, 𝑔

,

𝐷 𝑧 𝑗 𝑔 𝛼

,

𝐷

,

𝐷 𝑖 𝑥 𝑘 𝛼

,

𝐷 𝑗 𝑔 𝑥

,

𝐷 𝑔 𝑘 𝑥

,

𝐷 𝑖 𝑦

,

𝐷 𝑗 𝑔 𝑦

,

𝐷 𝑘 𝑦

,

𝐷 𝑖 𝑧

,

𝐷 𝑗 𝑔 𝑧

,

𝐷 𝑘 𝑧 𝑥

, 𝑔 𝑦

and 𝑔 𝑧

or 𝑔 𝑥

for each Kramers doublet, where

is the unit vector denoting the direction of 𝑔 𝛼 diagonalized g-tensor for a pseudo-spin 1/2 Kramers doublet. Therefore, the number of g-

,

. Each line represents the

37

tensors must be less than or equal to half the total dimension of the problem. Note that there should be no blank lines at the end of the file. g x

1 g y

1 g z

1 g x

2 g y

2 g z

2 g x

3 g y

3 g z

3 etc.

or

g x

1 g y

1 g z

1 D i g x

1 D j g x

1 D k g x

1 D i g y

1 D j g y

1 D k g y

1 D i g z

1 D j g z

1 D k g z

1 g x

2 g y

2 g z

2 D i g x

2 D j g x

2 D k g x

2 D i g y

2 D j g y

2 D k g y

2 D i g z

2 D j g z

2 D k g z

2 g x

3 g y

3 g z

3 D i g x

3 D j g x

3 D k g x

3 D i g y

3 D j g y

3 D k g y

3 D i g z

3 D j g z

3 D k g z

3 etc.

4.4 Output files and interpretation

PHI outputs information regarding the operation of the program and the type of calculations it is performing to stdout (shell, command prompt or terminal). This is redirected to the GUI output panel, however when running PHI without a GUI, this can be directed to a specified output file by appending, for example, “> test-job.out” to the execution command, so that it would read on Linux “./phi_vx.x_linux64.x test-job > test-job.out”. PHI writes all calculated data to files in the working directory of the job, in the files described below. Note that the naming of the files is identical to that of the .exp input data files – the job name is appended with an underscore to the following output files, for example “test-job_mag.res”.

sus.res, mag.res, mce.res, heat.res, epr.res, levels.res and G.res specification

Data is written to this file in exactly the same format as the input .exp files.

zeeman.res specification

This file contains the results from a calculation of a Zeeman plot. The file consists of 𝑑𝑖𝑚 + 1 columns, where dim is the dimension of the total Hilbert space of the system. The first column contains the magnetic field strength (

𝐵) in Tesla, followed by the corresponding energy for each state in the system in wavenumbers.

survey.res specification

This file contains a number of columns, one for each operator defined in the ‘****Survey’ block and one more, the final column, which represents the residual for the parameter set defined by the row, noting that the columns are ordered as the variables in the ‘****Survey’ block.

states.res specification

This file is produced when a simulation of the energy levels is requested and contains information regarding the wavefunction and energy levels and transition probabilities, Jmixing and g-tensors, if applicable. In the full calculation case, the wavefunction is printed in a matrix type manner, with row and column headers. The first column contains the row headers which are the basis elements in which the Hamiltonian was constructed, i.e. the single ion states. The subsequent columns are the different eigenstates of the system, with the column headers displaying the energies in wavenumbers. The columns show the expansion coefficients for the basis states that comprise the given eigenstate. Unless “FullWF” is

38

selected, coefficients below 1×10

-10

are not printed. In the case of wavefunctions with imaginary components, two matrices are printed; the top one being the real part of the coefficients and the bottom one being the imaginary components. Below this are shown the percentage contribution of each basis state to the wavefunction. For anisotropic systems, the transition probabilities between the states are printed – note that this matrix is symmetric. For single lanthanide ions calculated under the simple input method, the wavefunction is also transformed into the

�𝐽 𝑖

, 𝑚

𝐽𝑖

⟩ basis and is printed in the same manner as that for the regular wavefunction. If the g-tensor calculation is appropriate, the diagonal g-tensors are printed along with their directions in the internal coordinate system.

In the approximation mode, a list of the intermediate and final spin states along with their energies are provided.

4.5 Use of the GUI

The GUI runs completely independently from PHI and is provided as a visualization aid as well as a tool to abstract the command line from users. It is therefore the perfect platform for teaching the fundamentals of magneto-chemistry in an interactive environment.

Figure 4.5.1 shows a screenshot of the GUI in operation under Debian (Linux). The interface is divided into three sections. The main left pane is for PHI input and output, the right pane is for displaying the results and the bottom left section is the control panel. To begin, an input file can be typed into the “Input” tab in the main left pane or a file opened using the “File >

Open” menu. If a new input is typed into the “Input” tab, then a name must be given to the new job, in the “Job Name/Input File:” box. The current working directory can be changed from that where the GUI was launched through the “File > New” menu, which also clears the

“Input” box. The box to the left of the “Pause” button allows the user to specify the version of PHI to use for the calculation.

Figure 4.5.1 – Screenshot of the PHI GUI in operation on a Debian (Linux) platform

39

When the execute button is clicked (green arrow on top toolbar), the text in the “Input” tab is written to a file with the name given in the “Job Name/Input File:” box and PHI is executed.

The PHI output is re-routed to the “Output” tab in the left pane, while simultaneously the plotting routine initiates, polling the data files for new data. This updating feature is designed to reflect the progress of a fitting routine, however also plots static data. The routine ends when the calculation is complete. The “Reset zoom on update” checkbox controls whether the plot auto-scales with each new refresh and can be unchecked to allow the observation of a particular region during the fitting process. The “Plot Update Interval” field specifies the time in seconds between each update and can be changed to suit the computational demand of the calculations. The “Output Line Buffer” field specifies how many lines are buffered from PHI before they are printed into the “Output” tab and should be increased if the calculations are very quick and producing a copious amount of output.

If performing a fit where the plotting routine is constantly plotting new data, the “Pause” button can be used to freeze the plot to inspect the plot without it changing. Note that this does not pause the calculation by PHI, it only pauses the plotting loop which resumes upon disengaging the “Pause” button.

The calculation can be aborted by clicking the “Abort” button (red cross on top toolbar), killing the PHI instance.

The plot legend can be dragged to the desired location when a fitting is not taking place. Plots can be saved from the plotting section, however the results of the calculation remain in the project directory and can be plotted or examined outside the GUI if desired. A report of the calculation can be generated by clicking the report button (page icon on top toolbar), which produces a .pdf file containing the input, output and any plots that are available.

The most common causes of errors encountered in the GUI are:

− incorrect user input

− output files in use by other programs

Keep in mind when using the GUI that any errors arising from PHI or the GUI will be printed to the terminal (shell, command prompt, etc.) and this information should be supplied if reporting a bug.

4.6 Example

A number of examples utilizing PHI can be found in the literature,

46–55

however a brief example is provided to show how to write a simple input file.

The classic Cu(II)

2

(OAc)

4

dimer, originally investigated by Bleaney and Bowers

56

and subsequently by Gerloch et al.,

57

shows a strong decrease in the

χ

M

T vs. T data with a reduction in temperature. Such behaviour originates from anti-ferromagnetic superexchange between the Cu(II) ions, leading to an S = 0 ground state. To investigate the magnitude of the superexchange interaction, a fit of the

χ

M

T vs. T data to a single-J isotropic HDVV spin

Hamiltonian with a variable g-factor in the Zeeman Hamiltonian, was performed. The entire input file required to perform this calculation with PHI is presented to highlight the simplicity of such operations, Figure 4.6.1 (inset). This analysis found a very good fit to the experimental data, Figure 4.6.1, with a coupling constant of J = -144.6 cm

-1

and g = 2.12.

40

****Ions

Cu(II)Oh

Cu(II)Oh

****Fit

Simplex

-50

EX 1 2 4

----

2.00

GF 1 4 0

GF 2 4 0

----

****Sus

Bsus 1

****Params

OpMode Fit S

****End

Figure 4.6.1 – Magnetic susceptibility of the Cu(II)

2

(OAc)

4

dimer in a field of 1 T, the solid line is a fit to the data using the parameters in the text. Inset: Entire PHI input file required to perform the calculation.

4.7 Testing

The “Coupling Report” Operation Mode is provided to inform the user of the block diagonal structure of the HDVV Hamiltonian matrix in a coupled total spin basis, without performing any demanding calculations. It is useful to check to see the requirements of large problems and determine whether it can be solved on the available hardware.

41

5. Bugs and Feedback

The development of PHI is an ongoing process, which the author hopes will continue with the advent of new technologies and/or interfaces, which may enhance the computational power available to the user. The author welcomes any bug reports, feature requests, comments, suggestions or queries about the code. Please address all correspondence to [email protected].

Please keep in mind that the code is continually under development and bugs may still be present. Updated source code and binaries are uploaded to nfchilton.com/phi regularly.

42

6. References

1. N. Karayianis, J. Chem. Phys., 1970, 53, 2460–2469.

2. N. F. Chilton, R. P. Anderson, L. D. Turner, A. Soncini, and K. S. Murray, J. Comput.

Chem., 2013, 34, 1164–1175.

3. R. J. Elliott and M. F. Thorpe, J. Appl. Phys., 1968, 39, 802–807.

4. O. Kahn, Mol. Phys., 1975, 29, 1039–1051.

5. J. J. Borrás-Almenar, J. M. Clemente-Juan, E. Coronado, A. V. Palii, and B. S. Tsukerblat,

Chem. Phys., 2001, 274, 131–144.

6. W. P. Wolf, J. Phys. Colloq., 1971, 32, C1–26–C1–33.

7. A. Palii, B. Tsukerblat, J. M. Clemente-Juan, and E. Coronado, Int. Rev. Phys. Chem., 2010,

29, 135–230.

8. M. E. Lines, J. Chem. Phys., 1971, 55, 2977–2984.

9. M. Hutchings, in Solid State Physics, Elsevier, Amsterdam, 1964, vol. 16, pp. 227–273.

10. C. E. Schäffer and C. K. Jørgensen, Mol. Phys., 1965, 9, 401–412.

11. W. Urland, Chem. Phys., 1976, 14, 393–401.

12. D. J. Newman and B. Ng, Rep. Prog. Phys., 1989, 52, 699–762.

13. J. Mulak and Z. Gajek, The Effective Crystal Field Potential, Elsevier, 2000.

14. C. Rudowicz, J. Phys. C Solid State Phys., 1985, 18, 1415–1430.

15. B. Bleaney and K. W. H. Stevens, Rep. Prog. Phys., 1953, 16, 108–159.

16. K. W. H. Stevens, Proc. Phys. Soc. Sect. A, 1952, 65, 209–215.

17. A. Abragam and B. Bleaney, Electron Paramagnetic Resonance of Transition Ions, Oxford

University Press, 1970.

18. C. Gorller-Walrand and K. Binnemans, in Handbook on the Physics and Chemistry of Rare

Earths, Elsevier, 1996, vol. 23.

19. F. Lloret, M. Julve, J. Cano, R. Ruiz-García, and E. Pardo, Inorganica Chim. Acta, 2008,

361, 3432–3445.

20.

R. Boča, Theoretical Foundations of Molecular Magnetism, Elsevier, 1999.

21. M. Evangelisti and E. K. Brechin, Dalton Trans., 2010, 39, 4672.

22. M. Evangelisti, F. Luis, L. J. de Jongh, and M. Affronte, J. Mater. Chem., 2006, 16, 2534.

23. C. Voglis, P. E. Hadjidoukas, I. E. Lagaris, and D. G. Papageorgiou, Comput. Phys.

Commun., 2009, 180, 1404–1415.

24. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas,

Graphs, and Mathematical Tables, U.S. Department of Commerce, Washington D.C., 10th edn., 1972.

25. J. J. Borrás-Almenar, J. M. Clemente-Juan, E. Coronado, and B. S. Tsukerblat, J. Comput.

Chem., 2001, 22, 985–991.

26. J. J. Sakurai and S. F. Tuan, Modern quantum mechanics, Addison-Wesley Longman, 2010.

27. J. J. Borrás-Almenar, J. M. Clemente-Juan, E. Coronado, and B. S. Tsukerblat, Inorg. Chem.,

1999, 38, 6081–6088.

28. D. Gatteschi and L. Pardi, Gazzetta Chim. Ital., 1993, 123, 231–240.

29. B. W. Shore and D. H. Menzel, Principles of atomic spectra, Wiley, 1967.

30. A. Bencini and D. Gatteschi, EPR of Exchange Coupled Systems, Springer-Verlag.

31. M. Eden and M. H. Levitt, J. Magn. Reson., 1998, 132, 220–239.

32. M. Gerloch and R. F. McMeeking, J. Chem. Soc. Dalton Trans., 1975, 2443–2451.

33. H. Bolvin, ChemPhysChem, 2006, 7, 1575–1589.

34. J. Mulak and M. Mulak, Phys. Status Solidi B, 2006, 243, 2796–2810.

43

35. G. R. Hanson, K. E. Gates, C. J. Noble, M. Griffin, A. Mitchell, and S. Benson, J. Inorg.

Biochem., 2004, 98, 903–916.

36. S. Stoll and A. Schweiger, J. Magn. Reson., 2006, 178, 42–55.

37. S. D. Bruce, J. Higinbotham, I. Marshall, and P. H. Beswick, J. Magn. Reson., 2000, 142, 57–

63.

38. H. Husein Mor, H. Weihe, and J. Bendix, J. Magn. Reson., 2010, 207, 283–286.

39. G. van Veen, J. Magn. Reson. 1969, 1978, 30, 91–109.

40. R. Aasa and tore Vänngård, J. Magn. Reson. 1969, 1975, 19, 308–315.

41. S. K. Klitgaard, F. Galsbøl, and H. Weihe, Spectrochim. Acta. A. Mol. Biomol. Spectrosc.,

2006, 63, 836–839.

42. M. J. D. Powell, Comput. J., 1964, 7, 155 –162.

43. J. A. Nelder and R. Mead, Comput. J., 1965, 7, 308 –313.

44. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in

Fortran 90: The Art of Parallel Scientific Computing, Cambridge University Press, 2nd edn.,

1996.

45. P. D. Stevenson, Comput. Phys. Commun., 2002, 147, 853–858.

46. S. K. Langley, L. Ungur, N. F. Chilton, B. Moubaraki, L. F. Chibotaru, and K. S. Murray,

Chem. - Eur. J., 2011, 17, 9209–9218.

47. S. K. Langley, N. F. Chilton, B. Moubaraki, and K. S. Murray, Dalton Trans., 2011, 41,

1033–1046.

48. M. R. Razali, N. F. Chilton, A. Urbatsch, B. Moubaraki, S. K. Langley, K. S. Murray, G. B.

Deacon, and S. R. Batten, Polyhedron, 2013, 52, 797–803.

49. S. K. Langley, N. F. Chilton, B. Moubaraki, and K. S. Murray, Dalton Trans., 2012, 41,

9789–9796.

50. S. K. Langley, N. F. Chilton, L. Ungur, B. Moubaraki, L. F. Chibotaru, and K. S. Murray,

Inorg. Chem., 2012, 51, 11873–11881.

51. S. K. Langley, N. F. Chilton, I. A. Gass, B. Moubaraki, and K. S. Murray, Dalton Trans.,

2011, 40, 12656–12659.

52. S. K. Langley, N. F. Chilton, B. Moubaraki, and K. S. Murray, Dalton Trans., 2011, 40,

12201–12209.

53. A. S. R. Chesman, D. R. Turner, K. J. Berry, N. F. Chilton, B. Moubaraki, K. S. Murray, G.

B. Deacon, and S. R. Batten, Dalton Trans., 2012, 41, 11402–11412.

54. M. Nematirad, W. J. Gee, S. K. Langley, N. F. Chilton, B. Moubaraki, K. S. Murray, and S.

R. Batten, Dalton Trans., 2012, 41, 13711.

55. M. R. Razali, A. S. R. Chesman, N. F. Chilton, S. K. Langley, B. Moubaraki, K. S. Murray,

G. B. Deacon, and S. R. Batten, Dalton Trans., 2012, 42, 1400–1405.

56. B. Bleaney and K. D. Bowers, Proc. R. Soc. Math. Phys. Eng. Sci., 1952, 214, 451–465.

57. M. Gerloch and J. H. Harding, Proc. R. Soc. Math. Phys. Eng. Sci., 1978, 360, 211–227.

44