Pitfalls on evaluating pair exchange interactions for modelling molecule-based magnetism

This jou nal is © The Royal Socie y o Chemis y 2021 J. Ma e . Chem. C, 2021, 9, 10647–10660 | 10647
Ci e his: J. Ma e . Chem. C, 2021,
9, 10647
Pi alls on e alua ing pai exchange in e ac ions
o modelling molecule-based magne ism
Ma ia Fumanal,
a
Joaquim Jo ne -Somoza,
abc
Se gi Vela,
a
Juan J. No oa,
a
Jo di Ribas-A ino *
a
and Me ce
`Deumal *
a
Molecule-based magne ism is a solid-s a e p ope y ha esul s om he mic oscopic in e ac ion
be ween magne ic cen es o adicals. The obse ed magne ic esponse is due o unpai ed elec ons
whose coupling leads o a pa icula magne ic opology. The e o e, o unde s and he magne ic
esponse o a gi en molecule-based magne and ep oduce he a ailable expe imen al magne ic
p ope ies by means o s a is ical mechanics, one has o be able o de e mine he alue o he J
AB
magne ic exchange coupling be ween adicals. The calcula ion o J
AB
is hus a key poin o modelling
molecule-based magne ism. In his Pe spec i es a icle, we will build upon ou expe ience in modelling
molecula magne ism o poin ou some pi alls on e alua ing J
AB
couplings. Special a en ion mus be paid
o he clus e models used o e alua e J
AB
, which should accoun o coope a i e effec s among J
AB
in e ac ions and also conside he en i onmen (coun e ions, hyd ogen bonding) o he wo adicals whose
in e ac ion has o be e alua ed. I will be also necessa y o assess whe he a DFT-based o a wa e unc ion-
based me hod is bes o s udy a gi en adical. Finally, in addi ion o model and me hod, he J
AB
couplings
ha e o be able o adap o changes in he magne ic opology due o he mal luc ua ions. The e o e, i is
mos impo an o app aise in which sys ems molecula dynamics simula ions would be equi ed. Gi en he
la ge numbe o issues one mus ackle when choosing he co ec model and me hod o e alua e J
AB
in e ac ions o modelling magne ic p ope ies in molecule-based ma e ials, he ‘‘human ac o ’’ is a mus o
c oss-examine and challenge compu a ions be o e us ing any esul .
a
Dep Cie
`ncia de Ma e ials i Quı
´mica Fı
´sica and IQTCUB, Facul a de Quı
´mica, Uni e si a de Ba celona, Ma ı
´i F anque
`s 1, E-08028 Ba celona, Spain.
E-mail: j. [email protected], [email p o ec ed]
b
IZO-SGI SGike , Euskal He iko Unibe si a ea (UPV/EHU), Joxe Ma i Ko a Cen e , A . Tolosa 72, 20018 Donos ia, Euskadi, Spain
c
The Max Planck Ins i u e o he S uc u e and Dynamics o Ma e (MPSD), Bldg. 99 (CFEL) Lu upe Chaussee 149, 22761 Hambu g, Ge many
Ma ia Fumanal
D Ma ia Fumanal ob ained he
PhD in 2015 a he Uni e si y o
Ba celona unde he supe ision
o P o . Juan No oa and D Jo di
Ribas-A ino. He PhD ocused on
he modeling o o ganic p-s acked
a chi ec u es o he de elopmen
o magne ic swi ches. She joined
a e he Uni e si y o S asbou g
o wo k wi h D Chan al Daniel
in exci ed s a es dynamics o
unc ional Re(I) complexes. Since
2018, she wo ks a he E
´cole
Poly echnique Fe
´de
´ ale de
Lausanne whe e she was awa ded a Ma ie Skłodowska-Cu ie
Indi idual Fellowship o pe o m he esea ch in he design o
copolyme s o single ission in he g oup o P o . Cle
´mence
Co minboeu .
Joaquim Jo ne -Somoza
D Joaquim Jo ne -Somoza cu -
en ly wo ks as Scien i ic Com-
pu ing Expe a he Ad anced
Resea ch Facili ies (UPV/EHU,
Spain) and is a gues esea che
a The Max Planck Ins i u e o
he S uc u e and Dynamics o
Ma e , Hambu g (Ge many).
D Jo ne -Somoza did his PhD
on he Theo e ical S udy o
Molecula Magne s unde he
supe ision o D Deumal and
P o . No oa. He was awa ded
wi h se e al na ional and
in e na ional pos doc o al g an s, like he Ma ie Cu ie Slwadoska
Indi idual Fellowship in 2019, and became an expe on he ab
ini io s udy o exci ed s a es p ope ies o condensed ma e and i s
in e ac ion wi h elec omagne ic ields.
Recei ed 8 h Ma ch 2021,
Accep ed 17 h June 2021
DOI: 10.1039/d1 c01083b
sc.li/ma e ials-c
Jou nal o
Ma e ials Chemis y C
PERSPECTIVE
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1. In oduc ion
In molecule-based magne ic ma e ials, he mic oscopic in e -
ac ions be ween magne ic cen es o adicals, whose unpai ed
elec ons couple leading o a speci ic magne ic opology, play a
key ole in de ining hei mac oscopic magne ic p ope ies.
1
The e a e wo main s a egies o ep oduce expe imen al mag-
ne ic p ope ies (e.g. magne ic suscep ibili y, hea capaci y,
magne iza ion) by means o s a is ical mechanics
2
depending
on whe he he elec onic s uc u e is explici ly conside ed o
no . The la e app oach aims a p o iding a nume ical mag-
ne ic esponse o a gi en molecula ma e ial. A e y success ul
app oach,
3a
which does no explici ly accoun o he c ys al
geome y (excep in a pa ame ic o m
3
), is based on he use o
a gene alized spin Hamil onian. Al e na i ely, he elec onic
s uc u e-based s a egy a ge s on cap u ing he mic oscopic
complexi y o he molecula ma e ial o unde s and, as well as
ep oduce, he expe imen al magne ic da a.
4
As compu a ional
chemis s, we seek modelling molecule-based ma e ials as a ool
o a ionalize hei physical p ope ies. Acco dingly, wi hin he
amewo k o molecula magne ism, one has o be able o
e alua e he magni ude o he J
AB
magne ic exchange coupling
be ween adicals. The calcula ion o he J
AB
adical adical
in e ac ion is hus one o he key poin s o modelling
Se gi Vela
D Se gi Vela g adua ed wi h a
chemis y deg ee om he Uni-
e si y o Ba celona in 2009. He
ob ained his PhD in 2014 unde
he supe ision o P o . Juan
J. No oa and P o . Me ce
`Deumal.
Then,hejoined heLabo a oi ede
Chimie Quan ique a he Uni-
e si y o S asbou g, whe e he
wo ked in he compu a ional
modelling o Spin C osso e
ma e ials wi h P o . Vincen
Robe . In 2018, he joined he
g oup o P o . Cle
´mence
Co minboeu a EPFL unde a Ma ie Sklodowska-Cu ie Fellowship,
o wo k in he s udy o molecula pho o-swi ches, and on he design
and disco e y o no el ma e ials wi hin he Ma el p og am.
Juan J. No oa
D Juan J. No oa ob ained his
PhD in 1981 a he Uni e si y o
Ba celona (UB). He hen did
pos doc o al s ays wi h P o . R.
Ca bo
´and P o . M. A. Robb.
Since 1997, he has been Full
P o esso a he Depa men o
Ma e ials Science and Physical
Chemis y o UB. Du ing his
ca ee he has been a isi ing
scien is in se e al ad anced
esea ch labo a o ies (CRAY
Resea ch, Minneso a; IBM
Resea ch Labo a o y in Zu
¨ ich),
and Visi ing P o esso a se e al uni e si ies (No h Ca olina S a e
Uni e si y, Uni . Bologna, Cla k Uni e si y, Uni . o U ah, Uni . o
Sao Ca los, and Osaka P e ec u e Uni e si y). His cu en esea ch
ocuses on compu a ional s udies o molecula ma e ials.
Jo di Ribas-A ino
D Jo di Ribas-A ino ecei ed his
PhD in 2006 wi h P o . Juan
J. No oa a he Uni e si y o
Ba celona (UB). Du ing his PhD,
he pe o med wo esea ch s ays
a he U.S. Na al Resea ch
Labo a o y, wi h D Ma k R.
Pede son. The ea e , he wo ked
as a Pos doc o al Resea che a
he Ruh -Uni e si a
¨ -Bochum
wi h P o . Dominik Ma x. A
Humbold Fellowship suppo ed
pa o his pos doc. In 2010, he
e u ned o UB wi h a ‘‘Ramo
´ny
Cajal’’ con ac and since 2015, he has been Associa e P o esso a
he same uni e si y. His cu en esea ch in e es s conce n he
de elopmen and applica ion o compu a ional ools o s udy
mul i unc ional molecule-based ma e ials.
Me ce
`Deumal
D Me ce
`Deumal is ull P o esso
a Seccio
´de Quı
´mica Fı
´sica
(Uni e si a de Ba celona), whe e
she de elops he asks bo h as a
esea che and lec u e a BSc,
MSc and PhD le els. She is he
local coo dina o a UB o he
in e uni e si y doc o al p og am
on ‘Theo e ical Chemis y and
Compu a ional Modeling’, and
belongs o ‘Ins i u de Quı
´mica
Teo
` ica i Compu acional’
(awa ded as Uni o Excellence
Ma ı
´a de Maez u), and ‘Xa xa
de Re e e
`ncia en Quı
´mica Teo
` ica i Compu acional’. He esea ch
is amed in he ield o Ma e ial Science and Compu a ional
Chemis y, and ocuses on he a ional design o ad anced
mul i unc ional molecula ma e ials using mul iscale simula ion
me hods, which encompass he use o ab ini io quan um
mechanics, molecula dynamics, Mon e Ca lo, s a is ical
mechanics, machine lea ning and solid s a e calcula ions.
Pe spec i e Jou nal o Ma e ials Chemis y C
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magne ism, and can be a demanding ask i he numbe o J
AB
couplings is la ge. A his poin , anyone wo king in he ield o
molecula magne ism has wished he e was a ully au oma ed
p ocedu e o de e mine he magne ic esponse o a gi en
molecule-based compound in a black-box ashion. Such au o-
ma ed code would mos p obably eso o he simples model
(dime ) and he leas expensi e me hod (DFT b oken symme y
le el)
5
o e alua e all J
AB
coupling in e ac ions.
5b,c,6
Also he
magne ic uni cell would pe o ce ha e o be ound in an
au oma ed way. Model, me hod and magne ic uni cell would
ha e a difficul alida ion since, by de ini ion, he e is no
diagnosis o check whe he he esul s ob ained om a
‘‘black-box p ocedu e’’ a e co ec o no . We mus acknow-
ledge ha he selec ion o he igh model and me hod o
compu e he J
AB
magne ic coupling is some imes e y challen-
ging. The e o e, due o he la ge numbe o issues one mus
ackle, he au oma ized selec ion o he model and me hod can
be an ex emely difficul ask. The ake-home message o his
pape is no ha i s design is no easible, bu ha he e mus
be some ca e ul hough behind i . This Pe spec i e pape aims
a e lec ing on how molecule-based magne ism is modelled,
which pi alls could be encoun e ed, and how o de ec and
p e en hem.
Usually, he wo king s a egy ha we ollow o calcula e
mac oscopic magne ic p ope ies om mic oscopic magne ic
in e ac ions among adicals uses only he knowledge o X- ay
c ys allog aphic da a. A ou -s ep p ocedu e
7
can be ollowed.
Fi s , he c ys al packing is analysed o selec pai s o A, B
adicals ha migh be magne ically impo an , in e ms o
adical adical dis ances. Secondly, once all hose AB
dime s ha a e uni ocally de ined wi hin he c ys al ha e been
iden i ied, he mic oscopic J
AB
adical adical magne ic in e -
ac ions a e compu ed using quan um chemis y me hods.
Depending on he sys em, one can eso o a a ie y o
ei he Densi y Func ional Theo y DFT
5
o wa e unc ion-based
(e.g. CASSCF, RASSCF, DDCI)
8
me hods. Also depending on he
sys em, one can eso o a a ie y o diffe en schemes o
ob ain he nume ical alue o J
AB
magne ic coupling: localized
s. delocalized, p ojec ed s. unp ojec ed, e c.
5,6
Due o he
molecula na u e o he magne ic ma e ials we a e in e es ed
o s udy, he J
AB
magne ic coupling be ween adicals is o sho -
ange, which allows e alua ing he J
AB
in e ac ions using un-
ca ed clus e models. The simples dime model wo ks in mos
o he cases. Fo a pai o S= 1/2 adicals, he J
AB
mic oscopic
exchange in e ac ion is calcula ed as he ene gy diffe ence
be ween open-shell single and iple s a es.
5
Howe e , a dime
model is no always he bes op ion o compu e J
AB
. The
enla gemen o he size o he clus e model (e.g. using e a-
me , decame models) is some imes a mus o accoun o
coope a i e effec s among J
AB
in e ac ions no included in a
adical– adical pai calcula ion (i.e. dime model).
9
The selec ion
o he co ec model mus also conside he en i onmen o he
wo adicals whose in e ac ion has o be e alua ed, namely
coun e ions,
10
hyd ogen bonding,
11
e c. To sum up, i espec i e
o he model, wi hin his app oach, he J
AB
in e ac ions a e
e alua ed a he ixed ela i e posi ions (an X- ay esol ed
s uc u e o , al e na i ely, an op imized s uc u e
12
) o he
selec ed pai s o adicals wi hin he c ys al. Since he he mal
oscilla ions o he spin ca ying uni s a ound hei equilib ium
posi ions a e igno ed in his ype o analysis, i can be s a ed
ha he s anda d app oach is based on a s a ic pe spec i e.
Howe e , o ma e ials wi h dominan exchange in e ac ions
p opaga ing h ough p–plabile ne wo ks,
13
one migh ha e o
eso o a dynamic pe spec i e, in which he mal ib a ions a e
explici ly conside ed.
14
The e o e, in o de o ge a physically
co ec in e p e a ion o he magne ic esponse o a gi en
molecula ma e ial one has o be able o de e mine whe he
he s a ic pe spec i e will su ice o , on he con a y, molecula
dynamics simula ions would be equi ed.
Le us now dig ess o commen on he possibili y o using a
pe iodic app oach a he han a clus e model o e alua e he
J
AB
magne ic coupling be ween adicals. This al e na i e migh
appeal as being an easie op ion since uni cells wi h diffe en
spin se ings could be used o calcula e he J
AB
magne ic
coupling in e ac ions be ween diffe en spin cen es wi hou
ha ing o wo y abou he ep esen a ion o he en i onmen .
Ye pe o ming pe iodic compu a ions on he c ys al s uc u e
is compa a i ely mo e difficul o wo main easons. Fi s , he
numbe o adicals in he uni cell (Z) can be la ge, which would
equi e he e alua ion o a la ge numbe o spin s a es ha , in
mos cases, will become in ac able (e.g. a ai ly small Z=4in
e . 9bgene a es o e 14 po en ial J
AB
magne ic in e ac ions
ha need o be assessed). Mo eo e , in mos cases hose spin
s a es a e no he g ound s a e o a gi en spin mul iplici y and,
hus, he de ini ion and – especially – he p ese a ion o he
desi ed spin con igu a ion ep esen s a challenge o solid s a e
codes. Second, e alua ion o J
AB
in e ac ions wi h accu a e
me hods in solid s a e has an ex emely la ge compu a ional
cos and, in many cases, bo h non-hyb id and hyb id DFT
unc ionals accessible o pe iodic compu a ions may ha e
di icul ies in cap u ing he ue elec onic na u e o he
adicals wi hin he c ys al. Fo hese easons, clus e models
o e g ea ad an ages o calcula e J
AB
magne ic in e ac ions in
molecula ma e ials a high le el accu a e me hods. S ill, he
choice o he ac ual clus e model (dime , e ame , e c.) needs
o be alida ed o ensu e he eliabili y o he J
AB
alues.
Once all J
AB
’s ha e been e alua ed, he magne ic opology
can be nex de ined in e ms o all compu ed non-negligible J
AB
magne ic coupling in e ac ions. F om he magne ic opology,
we selec a ep esen a i e magne ic model, which will enable us
o conside he unpai ed elec ons being comple ely coupled
an i e omagne ically (AFM), comple ely coupled e omagne i-
cally (FM), o in any o he o he many s a es ob ained as a
esul o sol ing he secula equa ion p oblem oge he wi h he
ene gy spec a and co esponding spin quan um numbe s. I is
impo an o s ess ha he de ini ion o he magne ic model is
c ucial since he ull diagonalisa ion
15
will be done in he space
expanded by his model. One could hink ha he selec ion
o he magne ic model is a ques ion o compu e powe and
should no be difficul o au oma ically build a model
Hamil onian based on he hie a chy o he J
AB
magne ic
couplings. Howe e , he choice o he mos app op ia e magne ic
Jou nal o Ma e ials Chemis y C Pe spec i e
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model is inhe en ly difficul in molecule-based magne s as hey
a e usually cha ac e ised by many J
AB
magne ic couplings. As a
consequence, con e gence wi h espec o he model Hamil o-
nians size is equi ed, which complica es i s au oma ic selec ion.
Finally, he esul ing ene gy spec a and spin quan um
numbe s a e used in he app op ia e s a is ical mechanics exp es-
sions o simula e he mac oscopic magne ic p ope ies.
2
He e we
mus highligh ha all simula ions a e always compa ed o he
a ailable expe imen al da a o alida ion pu poses. The e o e,
one can calcula e magne ic suscep ibili y, hea capaci y, o
magne iza ion, depending on he a ailable expe imen al da a.
In summa y, he wo king s a egy abo e ou lined is i s -
p inciples (FP) because he J
AB
magne ic in e ac ions a e com-
pu ed using DFT o o he wa e unc ion-based echniques om
c ys allog aphic da a, and is bo om-up (BU) because we use
mic oscopic da a o calcula e mac oscopic p ope ies. This
Fi s -P inciples Bo om-Up FPBU
7
wo king s a egy has been
applied success ully o compu e he magne ic esponse o a
a ie y o p o o ypical molecule-based magne s o diffe en
magne ic dimensionali y. Howe e , i is no jus he magne ic
esponse ha is a ge ed, since we ha e been able o unde -
s and and a ionalize he magne ic beha iou depending
on a ange o diffe en si ua ions such as in luence o
empe a u e,
16
effec o he Madelung ield,
10
effec o he
c ys al de ec s,
17
effec o he ligands,
11,18
magne ic dimen-
sionali y,
19
bis abili y
20
.... A his poin , one has o ealize ha
a oo simple au oma ed p ocedu e would imply applying oo
many unsu eyed es ic ions: dime model and un es ic ed
DFT o compu e ene gies and, in u n, J
AB
; es ima ion o he
magne ic uni cell upon quan i a i e c i e ia; e c. In ac , he
only a emp we ha e come ac oss which au oma ically handle
all hese issues is applied o neu al o ganic-based 1D
magne s.
4d
The choice o hese ma e ials g ea ly simpli ies
calcula ions. Ye i would mos p obably ace se ious challenges
when applied o mo e complex ma e ials (as he examples we
discuss below). The e o e, howe e app aising, black-box and
modelling magne ism do no necessa ily hold hands. Basically,
a black-box p ocedu e based on he s a ic e alua ion o J
AB
’s
using a dime model wi h un es ic ed DFT limi s p edic abili y
and migh only be use ul o desc ip ion pu poses. Desc ip ion
pu poses a e ob iously impo an . Howe e , as compu a ional
chemis s, we usually ha e mo e ques ions han answe s, and
he ques ions usually come ou unexpec edly. The e o e using a
oo simple au oma ed code is hough less. In ac , we will nex
p oceed o discuss issues we mus conside in e ms o models
and in e ms o me hods in o de o e alua e he adical adical
magne ic in e ac ions and, in he long un, simula e magne ism.
Fi s , we will show ha he dime model canno be aken o
g an ed o e alua e he magne ic J
AB
in e ac ion be ween wo
adicals. In ac , he size o he clus e model o calcula e J
AB
mus be assessed in o de o be ce ain abou i . No e ha a
w ong alue o J
AB
coupling will lead o he w ong magne ic
opology which, in u n will gi e ise o he w ong minimal
magne ic model whose eigen alues ha e o be used o simula e
he ele an magne ic da a. This is p ecisely he case o Cu
2
(1,4-
diazacyclohep ane)
2
Cl
4
(CuHpCl in Fig. 1a):
9b,21
he use o he
w ong model o e alua e J
AB
esul s in a spin-ladde magne ic
opology ins ead o a 3D ne wo k o in e connec ed squa ed-
plaque es. Secondly, we will add ess he choice o he igh
me hod o calcula e J
AB
. The calcula ion o a gi en J
AB
in ol es
he e alua ion o he ene gy o , a leas , wo s a es. This ene gy
e alua ion can be done by means o ei he DFT-based o
wa e unc ion-based me hods. Since DFT calcula ions a e less
demanding, in e ms o esou ces and compu a ional ime, i is
conside ed as he de aul me hod o be used o calcula e he
ene gy o he s a es in ol ed. Al hough DFT is known o p o ide
accep able alues o J
AB
coupling in many sys ems,
9–11,14,16–20
we will exempli y which a e he e ec s o choosing he w ong
me hod wi h he phenylsemiquinone-b idged bisdi hiazolyl
(PhBBO in Fig. 1b) compound.
22,23
Finally, we will add ess
he use o molecula dynamics simula ions when s udying
bis able compounds whose adicals pack o ming p-s acks
along a gi en c ys allog aphic di ec ion and, hus, he mal
luc ua ions migh ha e an impac on he magne ic opology.
We ha e encoun e ed ha J
AB
couplings ha e o be able o
adap o changes due o in e molecula ib a ions as in he case
o 1,3,5- i hia-2,4,6- iazapen alenyl (TTTA in Fig. 1c),
14a,24
which unde goes phase ansi ion be ween a low empe a u e
LT and a high empe a u e HT phases.
2. Me hodology
The s anda d s a ic Fi s -P inciples Bo om-Up (FPBU) p ocedu e
is applied o s udy he alleged magne ically isola ed cup a e
spin-ladde CuHpCl sys em, and o a ionalize he magne ism o
he pu ely o ganic PhBBO semiquinone-b idged bisdi hiazolyl
compound. Ye i is ound ha he s udy o he LT and HT
phases o TTTA equi es a dynamic pe spec i e. He ea e bo h
wo king s a egies will be desc ibed.
The i s -p inciples bo om-up FPBU
7
p ocedu e implies
ou s eps, as summa ized he ein. Fi s , a e inspec ion o
he c ys al s uc u e, he symme y-unique adical pai s ha
a e likely o be magne ically ele an a e iden i ied (using a
adical adical dis ance cu off alue be ween spin-ca ying
moie ies). Second, hei magne ic exchange in e ac ions, J
AB
,
a e compu ed and he magne ic opology o he c ys al (i.e., he
ne wo k o connec i i y de ined by all ele an J
AB
alues) is
de ined. When necessa y, wa e unc ion-based mul i e e ence
CASSCF/PT2 and RASSCF/PT2 ene gies (E
LS
,E
HS
)
8,25
using a DZV
basis se
26
a e ob ained om Molcas 7.6.
27
O he wise ene gies a
Fig. 1 Chemical o mula o (a) Cu
2
(1,4-diazacyclohep ane)
2
Cl
4
, CuHpCl,
(b) phenylsemiquinone-b idged bisdi hiazolyl, PhBBO, and (c) 1,3,5- i hia-
2,4,6- iazapen alenyl, TTTA.
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UB3LYP
28
o ange-co ec ed CAM-B3LYP
29
le el using 6-31+G(d)
o 6-311++G(d,p) basis se s
30
a e compu ed using Gaussian09
31
o he dime model and O ca 3.0
32
package o he e ame
model. F om he gene al Heisenbe g Hamil onian o a pai o
S= 1/2 spin cen es,
H
ˆ=2J
AB
S
ˆ
A
S
ˆ
B
, (1)
he J
AB
alue is de ined as 2J
AB
=E
S
E
T
, whe e E
S
and E
T
a e
he ene gies o he open-shell single and iple s a es, espec-
i ely, o a wo- adical clus e . In DFT calcula ions, he ene gy o
he single s a e can be app oxima ed using ha o he single-
de e minan b oken-symme y (BS) solu ion.
5b,c
Wi hin his
app oxima ion, he exp ession chosen o compu e he ene gy
diffe ence is
6d
ESET¼
2ES
BS ET

1þSab2(2)
whe e E
S
BS
is he ene gy o he BS solu ion
5b,c
and S
ab
is he
o e lap in eg al be ween he magne ic a and b o bi als o he BS
solu ion. In ou case, hose o bi als a e localized on each o he
wo adicals. This leads o S
ab
= 0 and o he inal exp ession
ha was used o compu e J
AB
alues:
J
AB
=E
S
BS
E
T
. (3)
I should be no ed ha in he case o he e ame model, he e
a e (a leas ) ou diffe en J(di)’s among pai s o hese ou
adicals (see Fig. 2). The e o e, he calcula ion o ou diffe en
J(di)’s equi es he e alua ion o he ene gy o i e diffe en spin
s a es. Wi hin he e ame app oach, he spin s a es ha ha e
been e alua ed a e he high spin quin uple (HS), wo low spin
iple (LS1, LS2) and wo low spin single (LS1/4, LS1/3) s a es
(he e numbe s deno e adicals wi h spin down in Fig. 2).
Speci ically Fig. 2 shows he se o equa ions used o calcula e
he non-negligible J
AB
in e ac ion be ween CuHpCl adicals in
dime s d1, d3, d5, d6, and d7 (namely, J(d1), J(d3), J(d5), J(d6),
and J(d7)).
Thi d, he Heisenbe g Hamil onian is applied o a model
space (i.e. a subse o he magne ic opology), which is designed
in such a way ha , ideally, he esul ing se o eigen alues
ep oduces hose ha esul om he applica ion o he
Heisenbe g Hamil onian o he ull in ini e c ys al. Finally,
he esul ing ene gies and o al spin numbe s a e in oduced
in o he p ope s a is ical mechanics exp essions o calcula e
he mac oscopic p ope ies o he sys em, such as he magne ic
suscep ibili y wT(T), hea capaci y C
p
(T) and magne iza ion
M(H).
2
The compu a ional scheme adop ed o he s udy o he
in e play be ween he mal luc ua ions and magne ism in TTTA
consis s o h ee s eps: (i) ab ini io molecula dynamics (AIMD)
simula ions
33
o bo h LT and HT phases o TTTA; (ii) compu-
a ion o J
AB
alues be ween pai s o adicals o a la ge numbe
o ames along he AIMD ajec o ies; and las ly, (iii) calcula-
ion o he ib a ionally-a e aged magne ic suscep ibili y 
w
ib
on he basis o ull diagonaliza ions o he Heisenbe g
Hamil onian buil om he p e iously e alua ed J
AB
alues.
Acco dingly, i s o all, AIMD simula ions
33
a e pe o med a
300 K o bo h he LT and HT phases o TTTA (ca. 10 ps, ime
s ep 4 a.u.) as implemen ed in he CPMD package.
34
No e he
empe a u e is chosen o be 300 K because TTTA is bis able a
ha empe a u e and he e is c ys allog aphic da a a ailable o
bo h LT and HT phases. Supe cells include 32 TTTA molecules
(8 s acks o 4 adicals each). DFT calcula ions a e ca ied ou a
PBE le el
35
wi hin he spin un es ic ed o malism (b oken
symme y single M
S
= 0 s a e) using plane wa e pseudo-
po en ials
36
expanded a a kine ic ene gy cu off o 25 Ry,
oge he wi h Vande bil ul aso pseudopo en ials,
37
a ic i-
ious mass o he o bi als o 400 a.u., and G-poin sampling o
he B illouin zone. The semiempi ical dispe sion po en ial
in oduced by G imme,
38
in i s DFT-D2 pa ame e iza ion is
also conside ed. In addi ion, he AIMD simula ions a e pe -
o med in he canonical (o NVT) ensemble using Nose
´–Hoo e
chain he mos a s.
39
Pe iodic bounda y condi ions in all h ee
di ec ions a e imposed in all AIMD simula ions. Pai s o
adicals a e hen excised om he supe cell o 32 adicals a e
AIMD simula ions e e y 0.97 s. Calcula ions o J
AB
a UB3LYP/
6-31+G(d)
28,30
le el a e conduc ed using dime models: app oxi-
ma ely 20 000 and 60 000 J
AB
e alua ions a e ca ied ou o he
LT and HT polymo phs, espec i ely. Finally, he ib a ionally-
a e aged magne ic suscep ibili y 
w
ib
o he HT phase a 300 K
is compu ed by a e aging he w alue o e he whole se o
con igu a ions ha a e used o de e mine he ime-e olu ion o
he J
AB
alues be ween adjacen adicals wi hin a s ack. Tha is
o say, he w alue is compu ed o an o e all o ca. 10 000
di e en molecula con igu a ions (each con igu a ion was col-
lec ed e e y 0.97 s h oughou he AIMD simula ions) and, hen,
a e aged. The esul ing 
w
ib
a 300 K
14a
is las compa ed o he w
alue ob ained om a s a ic analysis
20
and om expe imen .
24
3. Resul s and discussion
In o de o s ess he impo ance o he models and me hods
used o e alua e he J
AB
magne ic coupling be ween adicals,
h ee examples will be epo ed in which a oo simple black-box
s udy (based on s a ic DFT calcula ions pe o med on dime s)
would ha e p e en ed us om cap u ing he ue na u e o he
magne ism o hose h ee diffe en molecule-based magne s.
3.1. On he size o he clus e model o e alua e J
AB
magne ic
coupling be ween adicals
The i s example is he Cu
2
(1,4-diazacyclohep ane)
2
Cl
4
molecula
ansi ion me al AFM complex (in sho CuHpCl, see Fig. 1a).
Fig. 2 Te ame model and se o equa ions used o calcula e J(di)
magne ic in e ac ions be ween CuHpCl adicals (numbe ed 1 o 4). HS/
LS s ands o high spin and low spin, espec i ely.
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Since i was syn he ized in 1997,
21
CuHpCl has been he p o o-
ypical compound o quo e when a magne ically isola ed spin-
ladde sys em was equi ed due o i s c ys al packing consis ing
in well-sepa a ed ladde a angemen s o coppe a oms (see
Fig. 3a). The e o e, om di ec c ys al obse a ion, he magne ic
opology consis s o (appa en ly) non-connec ed spin-ladde s
wi h uni o m ails (see Fig. 3b). In e es ingly he e is a la ge a ie y
o expe imen al da a, and he e o e i s s udy is e y appealing.
The o e all AFM beha iou o CuHpCl has been cha ac e ized
by measu es o magne ic suscep ibili y, hea capaci y, magne-
isa ion, spin gap, inelas ic neu on sca e ing, e c. Al hough
he magne ic opology appea s o be clea , acco ding o li e a-
u e, he i ings o expe imen al magne ic suscep ibili y da a o
diffe en ladde models we e no conclusi e.
40
I hus ollows
ha his coppe de i a i e is mo e challenging han an ici-
pa ed, as i mus ha e many compe ing mic oscopic magne ic
Fig. 3 (a) [101] iew o c ys al packing o CuHpCl consis ing o well-sepa a ed ladde a angemen s o Cu a oms (highligh ed as ed a ows). (b) View o
an isola ed ladde showing in a-ladde i s nea es neighbou s nn pai s o adicals (d1–d5). Colou ed lines ha e been added be ween Cu a oms o
dis inguish di e en adical adical pai s. E alua ion o J(d3) (in g een) using as a clus e model: (c) a ba e d3 dime , (d) a e ame -based model ha
explici ly accoun s o he adical d3 pai unde s udy and he poin cha ges o i s d1 coun e pa s [dime /d3-d1PC], (e) an eigh adical model consis ing
on a wo adical d3 pai embedded in six adicals ep esen ed by poin cha ges [dime /d3-d1PC]-4PC, and ( ) a e ame . Colou code: C (black), H
(pale pink), N (ligh blue), Cu (blue) and Cl (g een). PCs ep esen ed as shadowed a oms in blue.
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J
AB
in e ac ions. The alleged spin ladde magne ic opology o
CuHpCl has been a long las ing issue.
Fo CuHpCl, all possible in a-ladde nea es -neighbou pai s
o adicals we e selec ed, namely d1, d2, d4 as ungs and d3, d5 as
ails (see Fig. 3b). Once selec ed, all i e co esponding J
AB
magne ic couplings we e e alua ed using a dime model. The
only magne ically signi ican adical pai s we e d1 ( ung), and d3,
d5 ( ails) (see Table 1). Acco dingly, he magne ic opology could
beasc ibed oconsis o spin-ladde s. S ikingly, he J
AB
magne ic
in e ac ion o d3 and d5 ( he ail in e ac ions) has opposi e
sign. This magne ic beha iou is appa en ly a odds wi h hei
geome y, which would sugges ha he spin-ladde should ha e
uni o m ails. The e o e, o be ce ain abou he nume ical alue
o J
AB
o d3 and d5 pai s o adicals, he impo ance o he
su ounding adicals was assessed by using diffe en size clus e
models wi h o wi hou poin cha ges (PC)
41
ep esen ing he
en i onmen o he adicals whose magne ic s eng h was e al-
ua ed. Fo ins ance, ega ding d3, we e alua ed J(d3) using a
dime wi h i s i s nea es -neighbou eplaced by poin cha ges
(namely dime /d3-d1PC), using he p e ious model embedded in
he poin cha ges o i s second nea es neighbou adicals
(namely, [dime /d3-d1PC]-4PC), and using a e ame model (see
Fig. 3c– ). The esul s show ha wi h a e ame model bo h ail
in e ac ions a e AFM bu no uni o m a all (see Table 1). I is hus
exceedingly impo an o s ess ha i he en i onmen is no well
desc ibed he coupling in e ac ions a e meaningless. Fo CuHpCl,
i is also undamen al o ealize ha o a good desc ip ion o each
Cu monome one has o accoun explici ly o he d1 pai , which
becomes he magne ic building block o his compound and
co esponds o a e omagne ic FM in e ac ion (see Fig. 4a in
ed, and Table 2). Each wo o hese dinuclea FM building blocks
a e hen connec ed an i e omagne ically by d5 pai s o adicals
(see Fig. 4a in g een). I hus ollows ha he inal 3D magne ic
opology esul s om weake AFM in e ac ions (see Fig. 4a in
o ange;Fig.4bshowsaglobal iew).No ice ha he emaining
non-negligible J(di) in e ac ions a e one o de o magni ude
smalle han d1, d5, d12 and d14 (see Table 2). The e o e, hese
Table 1 Magne ic J(di) in e ac ions (in cm
1
) o d1, d3 and d5 pai s o
adicals (see Fig. 3b) using diffe en clus e models, namely dime , dime
wi h poin cha ges ([dime /di-d1PC] and [dime /di-d1PC] – 4PC) and
e ame (see Fig. 3c– o models o e alua e J(d3)). No e ha X- ay
c ys allog aphic da a a 4 K is used
Model J(d1) J(d3) J(d5)
Dime +3.13 +2.11 3.12
[dime /di-d1PC] +3.13 +0.08 4.31
[dime /di-d1PC] – 4PC +2.02 +0.28 3.68
Te ame +2.30 0.37 3.88
Fig. 4 (a) Schema ic ep esen a ion o wo in e ac ing squa ed-plaque e magne ic mo i s. (b) Magne ic opology o CuHpCl, whe e only coppe a oms
a e ep esen ed. Colou code: in a-plaque e AFM J(d5) g een, and FM J(d1) ed; in e -plaque es AFM J(d12) o ange and AFM J(d14) yellow. (c) Magne ic
suscep ibili y cu es o (i) J
AB
alues ob ained wi h he dime app oach using one isola ed spin-ladde magne ic model ( ull blue ci cles) and a 3D
magne ic model consis ing o h ee in e ac ing ladde models ( ull ed ci cles), and (ii) J
AB
alues ob ained by he e ame app oach using a 3D magne ic
model ha includes ou in e ac ing plaque e-based magne ic building blocks (emp y ed ci cles). No e ha expe imen al da a is gi en o compa ison
pu poses ( ull black ci cles).
21
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magne ic couplings will no be impo an o simula ion pu -
poses. The esul ing magne ic opology can be pic u ed as a 3D
ne wo k o in e ac ing squa ed plaque e magne ic building
blocks. The simula ion o he w(T) magne ic suscep ibili y using
J
AB
magne ic in e ac ions ex ac ed om dime and e ame
models u he suppo s he ailu e o he dime model since
he nume ical e o in ol ed in w(T) when using he J
AB
calcula ed
a dime le el is la ge han 200% compa ed o expe imen ,
i espec i e o he magne ic model used being an isola ed spin-
ladde o a3Dmodelo in e ac ingspin-ladde s (see ull symbols
in Fig. 4c). Ye he simula ion using a 3D magne ic model o
in e ac ing plaque es pa ame e ized wi h J
AB
’s ob ained om a
4- adical e ame clus e model ce ainly ag ees wi h he expe i-
men al da a (see emp y symbols in Fig. 4c). We can hus sa ely
conclude ha o CuHpCl he dime model ails o e alua e J
AB
magne ic couplings. Fu he we can also come o he conclusion
ha he esul ing 3D magne ic opology is by a mo e complex
han he alleged spin-ladde hin ed om di ec obse a ion o he
c ys al packing (compa e Fig. 3b and 4b).
9b
3.2. Assessing he adequacy o he me hod o e alua e J
AB
magne ic coupling be ween adicals
The main objec ion in he CuHpCl case laid on he size o he
dime model, i.e. he dime model was oo small o e alua e
co ec ly he J
AB
mic oscopic in e ac ion be ween wo adicals.
Changing gea , we will ocus on he me hods a he han on he
models o e alua e J
AB
magne ic coupling be ween adicals. The
choice o he igh me hod is also c ucial and has o be assessed
while s udying he molecule-based ma e ial o in e es .
We would like o men ion ha CuHpCl was s udied a
UB3LYP/6-31+G(d) le el,
28,30
which ou expe ience g an s as
app op ia e o a la ge a ie y o sys ems.
9–11,14,16–20
We will
now eso o he PhBBO magne
22
ha belongs o he pu ely
o ganic semiquinone-b idged bisdi hiazolyl amily o com-
pounds (see Fig. 1b). In sho , i has a 1D elec onic s uc u e
due o he p esence o p-s ack mo i s and pauci y o close
in e -column adical adical con ac s (see Fig. 5a). The expe i-
men al magne ic suscep ibili y wT(T) da a sugges s pa amag-
ne ic beha iou , wi h s ong local FM in e ac ions (see black
line in Fig. 5c). This da a was i ed o he Bake model
o Heisenbe g 1D FM chain o S= 1/2 cen e s esul ing in
J= +29.5 cm
1
and a mean ield pa ame e zJ0=2.5 cm
1
.
PhBBO was also ound o exhibi a phase ansi ion a 4.5 K.
In his case, DFT-based me hods (such as un es ic ed
B3LYP
28
o ange-co ec ed CAM-B3LYP
29
)p o ideagooddesc ip-
ion o he spin densi y o an isola ed adical (see monome in
Table 3). Howe e , using a dime model o e alua e he magne ic
in e ac ion be ween wo adicals, DFT-based me hods a e able o
co ec ly desc ibe he spin densi y o he iple s a e bu ail o
desc ibe he open-shell single s a e (see Dime 1 in Fig. 5b and
Table 3 o spin densi y). As a esul , he FM in e ac ion ha we
wan o e alua e is o e es ima ed a DFT le el (+343.5 cm
1
a
UB3LYP, and +193.3 cm
1
a CAM-B3LYP, see Table 4). We mus
hen eso o wa e unc ion-based me hods (such as CASSCF and
RASSCF) in o de o co ec ly desc ibe he adicals spin densi y
and, in u n, compu e adequa ely he J
AB
coupling in e ac ion
be ween hem (see Tables 3 and 4). The J
AB
magne ic in e ac ion
esul s o be +34.6 cm
1
a CASSCF(14,14) and 49.7 cm
1
a
RASSCF(38,2,2;16,6,4) le els. Calcula ions a CASPT2 le el a e
help ul o assess ha dynamic co ela ion does no necessa ily
need o be conside ed o adequa ely e alua e J
AB
magne ic in e -
ac ions (see Table 4). The magne ic suscep ibili y is hen calcula ed
a all le els o heo y. Clea ly, un es ic ed bo h B3LYP and
CAM-B3LYP o e es ima e he alue o wTa any gi en empe a u e
( ed and g een lines in Fig. 5c). Con a ily, a CASSCF(14,14) le el,
wTis unde es ima ed (blue line in Fig. 5c). One has o eso o
RASSCF(38,2,2;16,6,4) o ep oduce co ec ly he expe imen al da a
(o ange line in Fig. 5c). Fo PhBBO, i can hus be concluded ha
DFT me hodology d ama ically ails.
23
Ins ead wa e unc ion-based
me hods a e he e equi ed. The eason o he DFT ailu e u ns
ou o be simple. I is because semiquinone-b idged bisdi hiazolyl
compounds a e mul i e e ence sys ems due o he p esence
o low-lying open-shell s a es (wha is called mul i-o bi al effec
by expe imen alis s). This is hus he second example in which a
Table 2 Magne ic J(di) in e ac ions (in cm
1
) be ween adicals using a
e ame model using X- ay c ys allog aphic da a a 4 K. J(di) classi ied
acco ding o c ys al packing mo i
J(di)/‘‘in a’’ J(d1) J(d3) J(d5)
+2.30 0.37 3.88
J(di)/‘‘in e ’’ J(d6) J(d7) J(d10) J(d11) J(d12) J(d14)
0.29 0.12 +0.22 0.31 1.05 1.38
Fig. 5 (a) ab-View o he al e na ing ABAB p-s acks o PhBBO wi h in a-
s acks S3S1 con ac s d
1
= 3.68 Å (in black) and d
2
= 3.81 Å (in ed) along
he a-axis (p-s acking axis). (b) ac-View o Dime 1 along he a-axis
(p-s acking axis). (c) Magne ic suscep ibili y as a unc ion o empe a u e
using J
AB
calcula ed a un es ic ed B3LYP (in ed), CAM-B3LYP (in g een),
CASSCF(14,14) (in blue) and RASSCF(38,2,2;16,6,4) (in o ange). No e ha
he line in black co esponds o expe imen al da a.
22
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black-box s udy based on s anda d DFT me hods would ha e led
o an inco ec in e p e a ion, and o quan i a i ely-w ong alues
o he J
AB
pai in e ac ions o PhBBO.
3.3. Add essing he mal luc ua ions o e alua e J
AB
magne ic
coupling be ween adicals
Finally, in addi ion o model and me hod, he J
AB
couplings
ha e o be able o adap o changes in he magne ic opology
due o he mal luc ua ions. We ha e encoun e ed ha
molecula dynamics simula ions a e necessa y when s udying
bis able compounds whose adicals pack o ming p-s acks
along a gi en c ys allog aphic di ec ion, as in he case o
Table 3 Mulliken spin densi ies o N a oms calcula ed o he double g ound s a e o he PhBBO monome , and o he iple and b oken symme y
(BS) open-shell single s a es o Dime 1 (see Fig. 5b o ac- iew o geome y). The alue o he spin densi y o he N a oms o he second adical wi hin
Dime 1 is indica ed in pa en hesis. The calcula ions we e pe o med a un es ic ed B3LYP and CAM-B3LYP le els using he 6-311++G(d,p) basis se
Me hod Sys em Spin s a e
Spin densi y
N1 N2
B3LYP Monome Double +0.27 +0.14
Dime 1 T iple +0.27 (+0.27) +0.13 (+0.13)
J(Dime 1) = +343.5 cm
1
Single BS +0.27 (0.26) +0.06 (0.05)
CAM-B3LYP Monome Double +0.32 +0.21
Dime 1 T iple +0.30 (+0.30) +0.21 (+0.21)
J(Dime 1) = +193.3 cm
1
Single BS +0.25 (0.24) +0.16 (0.16)
Table 4 Magne ic exchange couplings (in cm
1
)calcula ed o Dime 1using
he oom empe a u ec ys als uc u e.Thecalcula ionswe epe o medwi h
un es ic ed B3LYP and CAM-B3LYP using 6-311++G(d,p) basis se , and wi h
CASSCF and RASSCF using he DZV con ac ion o ANO-RCC basis se (no e
ha ac i e spaces a e de ailed). Also Mulliken spin densi ies o he wo non-
equi alen N a oms o Dime 1 a hei iple s a e a e gi en
Basis Me hod J
AB
/cm
1
Spin densi y
N1 N2
6-311++g(d,p) B3LYP 343.5 +0.27 +0.13
CAM-B3LYP 193.3 +0.30 +0.21
DZV CASSCF(14,14) 34.6 +0.30 +0.15
RASSCF(38,2,2;16,6,4) 49.7 +0.29 +0.15
RASPT2 48.7
Fig. 6 Packing o he monoclinic HT (SAXPOW05) and iclinic LT (SAXPOW06) c ys al s uc u es eco ded a oom empe a u e (RT): ps acks in he (a)
LT ( iewed along c) and (b) HT ( iewed along b) phases. See inse alues o J
AB
magne ic couplings o (a) eclipsed and slipped pai s and (b) egula
p-s acking pai . (c) Magne ic suscep ibili y w(T) o LT and HT phases using s uc u es de e mined a RT SAXPOW06 ( ) and 250 K SAXPOW03 ( )/RT
SAXPOW05 ( ) employing a 16- adical magne ic model. The expe imen al cu e showing he he mal hys e esis p esen in he ange 210–320 K, whe e
HT and LT s uc u es can bo h exis , is also gi en in black. All calcula ions we e done a UB3LYP/Aug-cc-pVTZ le el. (d) w(T) plo wi h he app oxima e
posi ions on hys e esis loop o each o he six published X- ay da a se s o TTTA c ys al (LT: eco ded a 150 K and RT, SAXPOW01 and 06; HT: eco ded a
225 K, 250 K, RT, and 310 K, SAXPOW04, -03, -05 and -02).
24
Re codes SAXPOW01-06 gi en by he Camb idge C ys allog aphic Da a Base CCDC.
42
Jou nal o Ma e ials Chemis y C Pe spec i e
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