Nova Biotechnol Chim (2020) 19(2): 138-153 

                          DOI: 10.36547/nbc.v19i2.769 
 

 

 Corresponding author: roman.boca@ucm.sk  

Nova Biotechnologica et Chimica 

Thermodynamics and cooperativeness of the spin crossover 

Roman Boča 

Department of Chemistry, Faculty of Natural Sciences, University of SS Cyril and Methodius in Trnava, Trnava, SK- 917 01, 

Slovak Republic 

 
 

Article info 
 

Article history: 

Received: 17th July 2020 

Accepted: 11th September 2020 

 

Keywords: 

Cooperativeness  

Spin crossover 

Statistical analysis 

Thermodynamic parameters 
 

 

Abstract 
 

Spin transition – a passage from the low-spin electronic state to the high-spin one  

of Fe(III) and Fe(II) complexes is assessed from several points of view: theoretical 

modelling, magnetic susceptibility data, and calorimetric measurements. 

The concept of the cooperativeness in the solid state is discussed in detail. 

Thermodynamic parameters are mutually correlated for a set of analogous Fe(III) 

complexes by using modern statistical methods. 
 

 University of SS. Cyril and Methodius in Trnava

 
Introduction 

 

Thermally driven passage from the low-spin 

electronic state to the high-spin one is usually 

termed the spin crossover though also spin 

transition, spin conversion, and spin equilibrium is 

frequently used in this content and confused in their 

meaning. This phenomenon can be considered as  

a kind of unimolecular reaction where  

the conversion from L (low-spin) to H (high-spin) 

states is an process driven by entropy. For such  

a case S > 0 and H ~ kBT > 0 hold true so that 

there exists a critical temperature given by the ratio 

T1/2  =  H/S;  above   T1/2   the  change  in   Gibbs 

T

E
n

e
rg

y

0

G



T
1/2

1/T

ln
K 0

HS

LS

T
1/2

 
Fig. 1. Schematic representation of the spin crossover as 

entropy driven unimolecular reaction. 

energy alters to G < 0 so that the conversion 

progresses spontaneously. Thermal development  

of the high-spin mole fraction xH can be used  

in monitoring conversion degree; the equilibrium 

constant is expressed as K = xH/(1 – xH). In an ideal 

case the lnK vs 1/T dependence is a straight line 

(Fig. 1). 

The spin conversion is often explained using  

the orbital diagram, as presented in Fig. 2 for 

mononuclear Fe(II) and Fe(III) complexes. 

Electrons promoted from the non-bonding orbitals 

t2g into the antibonding orbitals eg cause a softening 

of the adiabatic potential E = f(R) (force constants 

k(H) < k(L) with its minimum lying at higher 
 

High-spin 5T2g

eg

t2g

Low-spin 1A1g

eg

t2g

High-spin 6A1g

eg

t2g

Low-spin 2T2g

eg

t2g

Fe(II) Fe(III)  
Fig. 2. Orbital diagram showing a difference between  

the low-spin and high-spin complexes Fe(II) and Fe(III)  

in an octahedral geometry. 

mailto:roman.boca@ucm.sk


Nova Biotechnol Chim (2020) 19(2): 138-153 

139 

LS     HS 

Distance

E
n

e
rg

y

R
L RH

k
H

k
L

 
Fig. 3. Interrelation of the adiabatic potentials for the low-

spin and high-spin states.  

 

metal-ligand distances, R0(H) > R0(L) (Fig. 3). 

The electronic contribution to the transition entropy 

is given by the spin multiplicities as follows Sel = 

Rln[(2SH+1)/(2SL+1)]; this amounts to 9.1 and 13.4 

J K-1 mol-1 for Fe(III) and Fe(II) complexes, 

respectively. A softer adiabatic potential for the HS 

state implies denser vibrational energy levels which 

enhances the vibrational contribution to  

the transition entropy: Svib(H) > Svib(L).  

 

Generalized crystal field theory 
 

In explaining the spin crossover phenomenon often 

an orbital picture is utilized. However, this simple 

approach abstracts from the mutual repulsion  

of energy and also the spin-orbit coupling. 

Therefore, there is a need of a more sophisticated 

approach to the spin crossover by using quantum-

chemical calculations. Herein the generalized 

crystal field theory (GCF) has been applied  

for such a purpose with numerical outputs (Boča 

2006).  

The interelectron repulsion is involved by 

considering the set of atomic terms labelled by  

the orbital and spin quantum numbers, i.e. 

, , , ,
L S

L S M M . A passage to the complex 

belonging to a point group G requires considering  

a set of crystal-field (CF) terms , , , ,
S

S M    

(like 2T2g, 
6A1g, etc.) where g is the component 

of the multidimensional irreducible representation 

G. The spin-orbit interaction (SOI) splits the CF-

terms into a set of crystal-field multiplets 

, ,    ; they need be classified using the 

irreducible representations of the respective double 

group (1 through 8 for O’ in Bethe notation).  

In terms of the GCF, a theoretical modelling has 

been done for Fe(III) [and Fe(II)] systems  

by involving the interelectron repulsion via Racah 

parameters B = 1,122 [897] cm-1 and C = 4.2 B, 

crystal-field poles for individual ligands F4(L), spin 

orbit interaction with the coupling constant  = 460 
[400] cm-1, orbital-Zeeman and spin-Zeeman 

interactions (Boča and Herchel 2015). The overall 

interaction matrix (Eq. 1): 

 

ee cf soi oZ sZ

4

diagonalization

{ ( , ) ( ) ( ) ( ) ( )}

( )
i

V B C V F V V B V B

E B

   


(1) 

is diagonalized and the calculated Zeeman levels 

Ei(B) form the partition function (Eq. 2): 

B
( , ) exp[ ( ) / ]

i

i

Z B T E B k T           (2) 

Finally, the formulae of the statistical thermo-

dynamics can be utilized in order to calculate 

magnetization and magnetic susceptibility (Eq. 3 

and 4):  

mol

ln
( , )

T

F Z
M B T RT

B B





   
     

   
        (3) 

0

mol
mol 0

( , )
( )

B

M B T
T

B
 

 
  

 
         (4) 

Since the mapping of ( )
i k

E B  proceeds for discrete 

field values Bk, numerical derivatives are required 

that, in fact, are provided after a parabolic fit  

(Eq. 5).  
(0) (1) ( 2) 2

, , ,
( ; ) Δ Δ

k m k m k m k k m k
Z B T c c B c B            (5) 

Effective magnetic moment constructed from  

the magnetic susceptibility displays a thermal 

development that strongly depends upon the crystal 

field strengths – see Fig. 4 for an octahedral Fe(III) 

system. In a narrow interval of the crystal-field 

strengths the spin crossover occurs: for the pole 

strength F4 = 17,700 cm
-1, the ground state is high-

spin 6A1g but for F4 = 18,200 cm
-1 it is low-spin 

2T2g. 

A delicate situation exists for the intermediate 

crystal field (Fig. 5): with F4 = 18,000 cm
-1  

the ground CF-term is high-spin 6A1g so that  

the spin crossover would not apply. However,  

the close-lying excited CF-term 2T2g(×6) is split 

due to the  spin-orbit interaction by a rather high 



Nova Biotechnol Chim (2020) 19(2): 138-153 

140 

T/K

0 50 100 150 200 250 300 350 400


e
ff
/ 

B

0

1

2

3

4

5

6

7

17500

17700

17800

17900

17950

18000

18100

18200

18500

F
4
/cm



14 000

Fe(III)6A
1g

2
T

2g

 

Fig. 4. Calculated temperature evolution of the effective 

magnetic moment for octahedral Fe(III) systems having 

various crystal-field strengths F4 = 6(Dq).  

amount (SOI = 690 cm
-1) that overcomes the inter-

term gap (0 = 425 cm
-1). Consequently, the ground 

crystal-field multiplet is the component 7(×2) ← 
2T2g that it is doubly degenerate: g(7) = 2. Thus,  

the change of the electronic entropy (Eq. 6): 

1 7
Δ ln[ (A ) / (Γ )] ln[6 / 2] 0

g
S R g g R              (6) 

is positive so that the spin crossover develops 

according to the bold curve drawn in Fig. 4.  

For Fe(II) systems the situation is different in 

several aspects (Fig. 6). With a weaker crystal field 

strength F4 = 12,800 cm
-1 (10Dq = 7,680 cm-1),  

the effective magnetic moment corresponds to  

the high-spin state with a typical course passing 

through a round maximum on heating. With  

a bit higher crystal field of F4 = 12,870 cm
-1,  

the ground state is low-spin and on heating  

the effective magnetic moment increases from zero 

to the value eff ~ 5.0 B at the room temperature.  
A typical spin-crossover behaviour proceeds 

at F4 = 12,900 cm
-1. For F4 = 13,000 cm

-1 

E
n
e
rg

y
 /

c
m




-400

-200

0

200

400

600

800


8


8


7


7

2
T

2g
(x6)


0


SOI

6
A

1g
(x6)

E
n

e
rg

y
 /

c
m




-200

0

200

400

600

800


3


5


4


5

5
T

2g
(x15)


4

1


0


SOI

1
A

1g
(x1)

 
Fig. 5. Left – energies of the lowest crystal-field terms and 

crystal-field multiplets for F4 = 18,000 cm
-1 in Fe(III); right – 

for F4 = 12,900 cm
-1 in Fe(II); calculations involve SOI.

T/K

0 50 100 150 200 250 300 350 400


ef

f/


B

0

1

2

3

4

5

6

12800

12850

12860

12870

12900

12950

13000

13100

14000

F
4
/cm



14 000

Fe(II)

5
T

2g

1
A

1g

 
Fig. 6. Calculated temperature evolution of the effective 

magnetic moment for octahedral Fe(II) complexes having 

various crystal-field strengths F4 = 6(Dq).  

the spin conversion is apparently incomplete  

(it continues to the completeness at the much 

higher temperature). With F4 = 14,000 cm
-1 t 

he spin transition occurs far above the room 

temperature and only a “temperature-independent 

paramagnetism” is visible until T = 400 K.  

The excited crystal-field term 5T2g is split due  

to the spin-orbit interaction into three groups  

of the crystal-field multiplets 5, {3 + 4} and {1 

+ 4 + 5}. This means that four groups  

of energy levels are involved in the spin crossover 

of an octahedral Fe(II) system (Fig. 6).  

In octahedral systems the electron repulsion and the 

crystal field strength are interrelated by the Tanabe-

Sugano (TS) diagrams where the term energy (not 

involving SOI) is plotted versus Dq/B parameter 

( = 10Dq = (10/6)F4) – Fig. 7. These diagrams are 
helpful in identifying the critical ratio when  

the high-spin complex turns to the low-spin one.  

For Fe(II) systems, the crossover of the terms  
5T2g ↔ 

1A1g exists at the ratio Dq/B = 2.38; this 

implies Dq = 2,135 cm-1 and F4 = 12,809 cm
-1.  

The last value matches the “observed” on-set  

of the spin transition. Analogously for Fe(III),  

the crossover 6A1g
 ↔ 2T2g appears at Dq/B = 2.70 

giving rise Dq = 3,029 cm-1 and F4 = 18,176 cm
-1; 

this again matches the region in which the spin 

crossover is observed. 

The octahedral geometry, however, is rather 

hypothetical for real spin crossover systems and 

at least tetragonal and/or trigonal distortions would 

be more close to the reality. A two-dimensional 

map of the lowest crystal field terms is presented  

in Fig. 8 and it can be considered as a generalized 



Nova Biotechnol Chim (2020) 19(2): 138-153 

141 

Dq/B

0.0 0.5 1.0 1.5 2.0 2.5 3.0

E
/B

0

10

20

30

40

50

60

2
T

2g

6
A

1g

6
A

1g
2
T

2g

4
T

1g

4
T

2g

4
A

1g

      
Dq/B

0.0 0.5 1.0 1.5 2.0 2.5 3.0

E
/B

0

10

20

30

40

50

60

1
A

1g

1
T

1g

3
T

1g
3
T

2g

5
T

2g5
T

2g

5
E

g
1
A

1g

 
Fig. 7. Tanabe-Sugano diagrams for octahedral Fe(III) and Fe(II) complexes (experimental B and C parameters were used  

in calculations).  

F
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A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1

A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1

A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1

A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1

A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1

A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1

A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1

A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1

A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1

A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1

A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1

A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1

A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 Eg

A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 Eg

A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 Eg Eg

A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 Eg Eg

A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 Eg Eg

A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 B2 T2 Eg

A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 B2 B2 B2 T2

F
4
(xy)

5000 10000 15000 20000

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T1 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2

Eg T1 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2

Eg Eg T1 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2

Eg Eg Eg T1 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2

Eg Eg Eg Eg T1 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2

Eg Eg Eg Eg Eg T1 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2

Eg Eg Eg Eg Eg Eg T1 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 Eg

Eg Eg Eg Eg Eg Eg Eg T1 A2 A2 A2 A2 A2 A2 A2 A2 Eg Eg Eg

Eg Eg Eg Eg Eg Eg Eg Eg T1 A2 A2 A2 A2 A2 A2 Eg Eg Eg Eg

Eg Eg Eg Eg Eg Eg Eg Eg Eg T1 A2 A2 A2 A2 Eg Eg Eg Eg Eg

Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg T1 A2 Eg Eg Eg Eg Eg Eg Eg

Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg T2 Eg Eg Eg Eg Eg Eg Eg

Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg B2 T2 Eg Eg Eg Eg Eg Eg

Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg B2 B2 B2 T2 Eg Eg Eg Eg Eg

Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg B2 B2 B2 B2 T2 Eg Eg Eg Eg

Eg Eg Eg Eg Eg Eg Eg Eg Eg B2 B2 B2 B2 B2 B2 T2 Eg Eg Eg

Eg Eg Eg Eg Eg Eg Eg Eg Eg B2 B2 B2 B2 B2 B2 B2 T2 Eg Eg

Eg Eg Eg Eg Eg Eg Eg Eg B2 B2 B2 B2 B2 B2 B2 B2 Eg T2 Eg

Eg Eg Eg Eg Eg Eg Eg Eg B2 B2 B2 B2 B2 B2 B2 A1 Eg Eg T2

F
4
(xy)

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T1 Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg

Eg T1 Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg

Eg Eg T1 Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg

Eg Eg Eg T1 Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg

Eg Eg Eg Eg T1 Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg

Eg Eg Eg Eg Eg T1 Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg

Eg Eg Eg Eg Eg Eg T1 Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg

Eg Eg Eg Eg Eg Eg Eg T1 Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg

Eg Eg Eg Eg Eg Eg Eg Eg T1 Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg

Eg Eg Eg Eg Eg Eg Eg Eg Eg T1 Eg Eg Eg Eg Eg Eg Eg Eg Eg

Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg T1 Eg Eg Eg Eg Eg Eg Eg Eg

Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg T2 Eg Eg Eg Eg Eg Eg Eg

Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg T2 Eg Eg Eg Eg Eg A1

Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg T2 Eg Eg Eg Eg A1

Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg T2 Eg Eg A1 B2

Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg T2 Eg B2 B2

Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg T2 B2 B2

Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg T2 B2

Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg T2

 

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4
(xy)

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4
(z

)

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10000

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T2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2

Eg T2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2

Eg Eg T2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 A1

Eg Eg Eg T2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 A1 A1

Eg Eg Eg Eg T2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 A1 A1 A1

Eg Eg Eg Eg Eg T2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 A1 A1 A1

Eg Eg Eg Eg Eg Eg T2 B2 B2 B2 B2 B2 B2 B2 B2 A1 A1 A1 A1

Eg Eg Eg Eg Eg Eg Eg T2 B2 B2 B2 B2 B2 B2 A1 A1 A1 A1 A1

Eg Eg Eg Eg Eg Eg Eg Eg T2 B2 B2 B2 B2 A1 A1 A1 A1 A1 A1

Eg Eg Eg Eg Eg Eg Eg Eg Eg T2 B2 B2 B2 A1 A1 A1 A1 A1 A1

Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg T2 B2 A1 A1 A1 A1 A1 A1 A1

Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg A1 A1 A1 A1 A1 A1 A1 A1

Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg A1 A1 A1 A1 A1 A1 A1 A1

Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg A1 A1 A1 A1 A1 A1 A1 A1 A1

Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg A1 A1 A1 A1 A1 A1 A1 A1 A1

Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg A1 A1 A1 A1 A1 A1 A1 A1 A1

Eg Eg Eg Eg Eg Eg Eg Eg Eg A1 A1 A1 A1 A1 A1 A1 A1 A1 A1

Eg Eg Eg Eg Eg Eg Eg Eg Eg A1 A1 A1 A1 A1 A1 A1 A1 A1 A1

Eg Eg Eg Eg Eg Eg Eg Eg A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1

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T2 Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg A1 A1 A1 A1

Eg T2 Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg A1 A1 A1 A1

Eg Eg T2 Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg A1 A1 A1 B2

Eg Eg Eg T2 Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg A1 A1 A1 B2 B2

Eg Eg Eg Eg T2 Eg Eg Eg Eg Eg Eg Eg Eg Eg A1 A1 B2 B2 B2

Eg Eg Eg Eg Eg T2 Eg Eg Eg Eg Eg Eg Eg Eg A1 A1 B2 B2 B2

Eg Eg Eg Eg Eg Eg T2 Eg Eg Eg Eg Eg Eg A1 A1 B2 B2 B2 B2

Eg Eg Eg Eg Eg Eg Eg T2 Eg Eg Eg Eg Eg A1 B2 B2 B2 B2 B2

Eg Eg Eg Eg Eg Eg Eg Eg T2 Eg Eg Eg A1 B2 B2 B2 B2 B2 B2

Eg Eg Eg Eg Eg Eg Eg Eg Eg T2 Eg Eg A1 B2 B2 B2 B2 B2 B2

Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg T2 Eg B2 B2 B2 B2 B2 B2 Eg

Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg T2 B2 B2 B2 B2 B2 B2 Eg

Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg T2 B2 B2 B2 B2 Eg Eg

Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg T2 B2 B2 B2 Eg Eg

Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg T2 B2 B2 Eg Eg

Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg T2 Eg Eg Eg

Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg A2 T2 Eg Eg

Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg A2 A2 A2 T2 Eg

Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg A2 A2 A2 A2 A2 T2

F
4
(xy)

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T2 Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg

B2 T2 Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg

B2 B2 T2 Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg

B2 B2 B2 T2 Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg

B2 B2 B2 B2 T2 Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg

B2 B2 B2 B2 B2 T2 Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg

B2 B2 B2 B2 B2 B2 T2 Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg

B2 B2 B2 B2 B2 B2 B2 T2 Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg

B2 B2 B2 B2 B2 B2 B2 B2 T2 Eg Eg Eg Eg Eg Eg Eg Eg Eg Eg

B2 B2 B2 B2 B2 B2 B2 B2 B2 T2 Eg Eg Eg Eg Eg Eg Eg Eg Eg

B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 T2 Eg Eg Eg Eg Eg Eg Eg Eg

B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 T2 Eg Eg Eg Eg Eg Eg Eg

B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 A1 Eg T2 Eg Eg Eg Eg Eg Eg

B2 B2 B2 B2 B2 B2 B2 B2 B2 B2 Eg Eg Eg T2 Eg Eg Eg Eg Eg

B2 B2 B2 B2 B2 B2 B2 B2 B2 A1 Eg Eg Eg Eg T2 Eg Eg Eg Eg

B2 B2 B2 B2 B2 B2 B2 B2 A1 A1 Eg Eg Eg Eg Eg T2 Eg Eg Eg

B2 B2 B2 B2 B2 B2 B2 A1 A1 Eg Eg Eg Eg Eg Eg Eg T2 Eg Eg

B2 B2 B2 A2 A2 A2 A2 A1 A1 Eg Eg Eg Eg Eg Eg B2 B2 T2 Eg

A2 A2 A2 A2 A2 A2 A2 A1 Eg Eg Eg Eg Eg Eg Eg B2 B2 B2 T2

 
Fig. 8. Generalized Tanabe-Sugano diagrams showing three lowest energy levels for 

equ ax

4 2
[FeL L ]  bipyramidal complex: top – 

d6, bottom – d5 system; left panel – ground state, centre (right) – first (second) excited state or a component of the degenerate 

state. Solid line passes through the octahedral arrangement and separates the elongated and compressed tetragonal bipyramid.



Nova Biotechnol Chim (2020) 19(2): 138-153 

142 

TS-diagram (SOI not involved) allowing  

a bordering of the spin crossover. When the pole 

strengths F4(z) = F4(xy), the diagram collapses  

to the TS one.  

In the TS diagram the crossover point refers to  

a situation when the LS and the HS state are 

accidentally degenerate. According 

to Fig. 3, the metal-ligand distances obey  

rL < rH. In Fe(II) complexes, these typically are 

rL ~1.96 - 2.00 Å and rHL= rH - rL ~ 0.16 - 0.21 Å. 

This shift manifest itself into the ligand field 

strengths owing to the relationship  

R= (10DqL/10DqH) = (rH/rL)
5. This ratio amounts  

to R ~ (2.2/2.0)5 = 1.6 so that on passing from HS 

to LS, the ligand strength increases approximately 

by a factor of 2 and vice versa. In the TS diagram 

of iron(II) spin crossover compounds 10DqH will 

be situated at the left and 10DqL to the right  

of the crossover point. 

It must be emphasized that the above modelling 

refers to electronic factors only and they 

completely ignore the important contribution of the 

molecular vibrations to the spin crossover.  

Also, vertical excitations in electronic transitions 

are assumed instead of the adiabatic ones.  

The spin crossover can appear also for Mn(III) 

systems (SL = 0 to SH = 2 transition) as well as 

Co(II) complexes (SL = 1/2 to SH = 3/2 crossover). 

 

Master equation 
 

For description of the spin crossover, a number  

of different theoretical models have been developed 

so far (Wajnflasz 1970; Bari and Sivardiére 1972; 

Slitcher and Drickamer 1972; Sorai and Seki 1974; 

Zimmermann and König 1977; Rao et al. 1981; 

Spiering et al. 1982; Adler et al. 1987; Bousseksou 

et al. 1995; Cantin et al. 1999; Boča et al. 2003). 

Their attempt is to simulate a development  

of the high-spin mole fraction xH under the thermal 

propagation: xH = f(T). One of these models is  

the regular solution & domain model based upon 

general principles of thermodynamics.  

The first step in the regular solution model  

is consideration of mixing entropy Smix; this results 

from the distribution of the LS and HS molecules 

within the system of N molecules that is simplified 

by exploiting the Stirling formula for factorials  

(Eq. 7): 

mix B

B

!
ln

( )![(1 ) ]!

{ ln (1 ) ln(1 )}

N
S k

xN x N

k N x x x x




    

                    (7) 

Here, we utilized that xN molecules are in the HS 

state and (1 – x)N in the LS one. 

Let us consider domains of like spin and of uniform 

size. Then the number of molecules per domain  

is /n N D  and the mixing entropy alters to  

(Eq. 8): 

mix B

B

!
ln

( )![(1 ) ]!

{ ln (1 ) ln(1 )}

D
S k

xD x D

k D x x x x




    

                   (8) 

The molar mixing entropy becomes expressed as 

follows (Eq. 9): 

mix
( / ){ ln (1 ) ln(1 )}S R n x x x x            (9) 

where 
A B

R N k  is the ideal gas constant.  

The second contribution in the play is 

intermolecular interactions with energy 
int

E  

constituted as follows (Eq. 10): 
2 2

int HH LH LL

2

0 1 2

2 (1 ) (1 )E E x E x x E x

J J x J x

    

  
           (10) 

where 
HH LH LL

, ,E E E  are the interaction energies 

between HS-HS, LS-HS and LS-LS pairs, 

respectively. To this end, a rearrangement offers 

(Eq. 11): 

0 LL

1 LH LL

2 LL HH LH

2( )

2

J E

J E E

J E E E

 


  
   

         (11) 

The molar Gibbs energy adopts the form (Eq. 12):  

H L mix int
(1 )

x
G xG x G TS E           (12) 

where 
L

G  and 
H

G  refer to the molar Gibbs 

energies for the LS and HS units, respectively.  

The equilibrium condition requires (Eq. 13): 

H L

,

1 2

( / ) ln
1

2 0

x

T p

G x
G G R n T

x x

J J x





   
     

  

  

      (13) 

which yields the equation for the mole fraction 

of the high-spin species (Eq. 14): 

  
1

1 2
1 exp (Δ Δ 2 ) /x n H T S J J x RT



       (14) 

An alternative expression is (Eq. 15): 



Nova Biotechnol Chim (2020) 19(2): 138-153 

143 

1/[1 ( )]x f x                      (15) 

with the factor (Eq. 16): 

 1 2( ) exp [(Δ ) Δ 2 ] /f x H J T S J x n RT        (16) 

Such an implicit equation requires solution  

by an iterative procedure.  

The entropic term contains two contributions: 

electronic and vibrational (Eq. 17). The partition 

function of a set of m = 3N – 6 harmonic oscillators 

(m = 15 for a hexacoordinate complex) is 

L, B

vib,L

1 L, B

L, B

1 1 L, B

exp( / 2 )

1 exp( / )

1
exp ( / 2 )

1 exp( / )

m
i

i i

mm

i

i i i

h k T
z

h k T

h k T
h k T










 




 

 
  

  



 

 

and analogously for the HS state. Then,  

in the approximation of an averaged (Einstein) 

modes 
L

h  and 
H

h  the entropic term becomes 

(Eq. 18): 

el,H vib,H

el vib

el,L vib,L

H L B

L H B

Δ Δ Δ ln

2 1 1 exp( / )
 ln

2 1 1 exp( / )

m

Z z
S S S R

Z z

S h k T
R

S h k T





 
     

 

    
   

    

      (18) 

and the enthalpic one is (Eq. 19):  

0 HS LS
Δ Δ ( )/ 2H E m h h           (19) 

Now the Eqs. 15 – 16 need be solved  

by an iterative process starting with a trial set  

of parameters n, H + J1, Lh , Hh   

and J2 for each temperature point. Finally,  

the equilibrium constant is expressed as  

(Eq. 20): 

H

H

1 2 H

ln ln
1

[( ) 2 ] /

x
K

x

H J T S J x n RT




      

         (20) 

The effect of the individual parameters to  

the conversion curve and/or equilibrium constant is 

presented in Fig. 9.  

The condition for the equilibrium (Eq. 13), defines 

the transition temperature T1/2 at which the high-

spin and low-spin mole fractions are equal,  

or 
H

0.5x   (Eq. 21): 

1 / 2
H L 1 2

0 0
T T

G G J J


            (21) 

T/K

0 100 200 300

x
H

S

0.0

0.5

1.0

(1/T)/K


0.00 0.01 0.02

ln
K

-5

0

5

T/K

0 100 200 300

x
H

S

0.0

0.5

1.0

(1/T)/K


0.00 0.01 0.02

ln
K

0

5

a) b) 

c) d)

 
Fig. 9. Conversion curves modelled by the master equation: 

a) change of the enthalpy H/R = 100 K (short dashed),  

500 K (long dashed) and 1000 K (solid) for fixed T1/2 = 

H/S = 150 K and J = 0; b) effect of the domain size n = 1 

(solid), 5 (long dashed), 50 (short dashed) for fixed H/R = 

500 K and T1/2 = 150 K – steepness of the transition; c) effect 

of the solid-state cooperativeness J/R = 0 (solid), 100 K (long 

dashed), 300 K (short dashed) for fixed H/R = 500 K  

and S/R = 5 – deviations from the linearity in the van’t Hoff 

plot; d) effect of molecular vibrations 
HS B

/h k   140 K 

(solid), 160 K (dashed), 180 K (dot-dashed) for fixed 

LS HS
1.5  , 

e e

HS LS
/ 5g g  , and 0/kB = 600 K. 

The transition temperature (Eq. 22): 

1/2 1 2
( ) /T H J J S             (22) 

includes also two cooperativity factors; 

a cancellation of 
1 2

W J J   can be assumed in the 

theoretical model. One can overcome the two-body 

interactions (Koudriavtsev 1999) by considering 

there-body ones producing a third-order term 
3

3 H
J x  

in Eq. 10; this yields an additional term 

1/2 3
(3 / 4) /T J S    in Eq. 22.  

 

Cooperativeness 
 

Within the regular solution model, the interaction 

term involves the solid-state cooperativity factor – 

cooperativeness  through the expression (Eq. 23): 

int
(1 )E x x           (23)

(17) 



Nova Biotechnol Chim (2020) 19(2): 138-153 

144 

This formula originates in the intercentre 

interaction (Eq. 24): 

2

int 0 1 2

2

LL LH LL LL HH LH
2( ) ( 2 )

E J J x J x

E E E x E E E x

  

     
   (24) 

which yields the relationship valid for the regular 

solution&domain model (Eq. 25):  

int
1 2 1 2 2

2 ( ) (1 2 )

(1 2 )

E
J J x J J J x

x

W x


     



  

      (25) 

The remainder (Eq. 26): 

1 2 HH LL
W J J E E            (26) 

can be absorbed into the effective parameter  

of the site formation, or it is omitted (Eq. 27): 

eff 1 2
ΔH J J             (27) 

Then the factor entering the iteration process  

(Eq. 14) relaxes to (Eq. 28): 

 eff( ) exp [ Δ (1 2 )] /f x T S x n RT             (28) 

Finally, the cooperativeness becomes expressed  

in the form (Eq. 29):  

2 LH LL HH
2J E E E              (29) 

This expresses a tendency for molecules  

of one type to interact effectively with molecules  

of the same spin.  

Parameter distribution model was outlined because 

behaviour of the solid-state samples is non-ideal:  

a reduction of the cooperativeness can be described 

using a statistical distribution. High 

cooperativeness leads to the thermal hysteresis  

a rectangular shape of the hysteresis loop. 

Observed profile of the conversion curves, 

however, is often distorted with marked deviations 

from the rectangular towards angled shape (Boča et 

al. 2001). The parameter distribution model 

considers the optimum (maximum) cooperativeness 

J2 that drops as follows (Eq. 30): 

2, 2i i
J n J          (30)       (29), 

where i is the grid point (e.g. 1/100 of the optimum 

value opt 1n  ). To this end the equation (Eq. 31):  

1/[1 ( )]
i i

x f x             (31) 

contains the factor (Eq. 32): 

 1 2,( ) exp [(Δ ) Δ 2 ] /i if x H J T S J x n RT      (32)                (31) 

This equation need be solved by an iterative 

procedure for the trial set of parameters,  

for selected temperature, and for each mesh point. 

The iteration procedure starts with 
(0)

0
i

x

   

in the heating direction, and 
(0)

1
i

x

  on the 

cooling path. The averaged value is given by  

the formula (Eq. 33): 

Mesh Mesh

H

1 1

/
i i i

i i

x w x w
 

   
    
   
                                   (33)   (32), 

using the weights obeying the Gaussian distribution 

(Eq. 34): 

2

opt
exp[ ( ) / ]

i i
w n n            (34) 

T/K

350 400 450

x
H

S

0.0

0.5

1.0

T/K

350 400 450

0.0

0.5

1.0
a) b) 

T/K

350 400 450


e

ff
/

B

0

1

2

3

4

5

T/K

350 400 450
0

1

2

3

4

5
a) b) 

 
Fig. 10. Parameter distribution model of the spin crossover. 

Distribution model width: d = 0.00001 (abrupt step) and 0.1 

(gradual step). Used parameters: D0/kB = 2144 K, J/kB =  

452 K, reff = 205 and gH = 2.0. Modelled using the program 

MIF&FIT (Boča 2016).  

 

The width of the distribution leads to  
the following effects (Fig. 10):



Nova Biotechnol Chim (2020) 19(2): 138-153 

145 

1) 0   causes a sharp distribution so that  
the model collapses to the abrupt step on heating 

and cooling, respectively. The conversion curve 

displays a hysteresis loop possessing  

the rectangular walls.  

2) Increased d causes that the hysteresis loop has 
angled walls and a decreased width.  

3) With increasing d the conversion curve is less 
complete and visibly smoother.  

Thermal hysteresis originates in existence of two 

minima of the Gibbs energy at different 

temperatures so that on the heating/cooling  

the system falls into one of them.  

 

Cooperativeness as the chemical/physical 

hardness 
 

Let us reconsider pair-wise interactions among 

solid-state particles (Eq. 35): 

2 2

int H LL H LH H H HH H
( ) (1 ) 2 (1 )E x E x E x x E x      (35) 

which can be rearranged into the form of a Taylor 

expansion (Eq. 36):  

2

int H 0 1 H 2 H
( )E x J J x J x           (36) 

The coefficients of the Taylor series are Eq. 37 – 

39 (see Fig. 11): 

0 LL
J E           (37) 

int
1 LH LL

H

2 2 0
E

J E E
x


 

     
 

       (38) 

2

int
2 LL HH LH 2

H

1
2 0

2

E
J E E E

x


 
      

 
      (39) 

where m – the chemical potential that equals to 

minus absolute electronegativity (Sen and 

Jorgensen 1987);  – the Pearson’s chemical 
hardness (Sen 1993; Pearson 2005).  

The cooperativeness J is then expressed through  

an excess of the interaction energy (Eq. 40): 

2 LH HH LL
/ 2 ( ) / 2 0J J E E E                (40) 

so that it interrelates to the chemical hardness. 

Another cooperative contribution is (Eq. 41): 

1 2 HH LL
0W J J E E                       (41) 

and it eventually can be neglected.  

Recall some additional definitions according  

to Pearson (Pearson 2005).  

ELL = J0

(EHH + ELL)/2

U = J
1
/2

EHH

ELH

W = J
1
 + J

2

J = J
2
/2

 
Fig. 11. Interrelations of the interaction parameters.  

 

a) Electronic chemical potential (Eq. 42) – 

derivative of the energy with the number  

of electrons at the constant potential generated  

by a system of nuclei: 

E

N


 
  

 
          (42) 

b) Absolute electronegativity expressed (Eq. 43) as 

an average of the ionization energy Ei and electron 

affinity Eeg: 

i eg

2

E E
 


              (43) 

c) Chemical hardness (Eq. 44): 

2

2

1

2

E

N


 
  

 
,   

i eg

2

E E



         (44) 

d) Electronegativity shift for two reactants owing to 

a transfer of N electrons from 2 to 1 (Eq. 45 and 

46): 

o

1 1 1
2 N              (45) 

o

2 2 2
2 N              (46) 

d) Electronegativity (chemical potential) 

equalization 
1 2
   yields (Eq. 47): 

o o

1 2

1 2

1

2 ( )
N

 

 


 


                    (47) 

Electrons move from the site of lower 

electronegativity to the site of higher 

electronegativity (Eq. 48). This causes an energy 

lowering: 



Nova Biotechnol Chim (2020) 19(2): 138-153 

146 

o o

1 2

1 2

1

4 ( )
E

 

 


  


         (48) 

e) Bulk modulus B and compressibility  of a solid 
is (Eq. 49): 

1

T

p
B V

V

 
    

 
, [Pa]        (49) 

f) Physical hardness becomes (Eq. 50): 

02

,T V

V
BV H

N N





 
   

 
, [J mol-1]     (50) 

where V0 – molar volume, N – number  

of particles.  

g) Fluctuations in the number of particles  

for a grand canonical ensemble (Eq. 51): 

2
2

0,

1 1
( )

T V

N N
N N

V kT V H

 



 
     

 
     (51) 

An ensemble can be crystals of identical volume 

but with varying numbers of component atoms,  

i.e. crystals which are physically soft (inverse  

of hard) possess large fluctuations in N.  This set  

of equations represents a good starting point  

for investigation of the physical and chemical 

nature of the solid-state cooperativeness.  

 

Calorimetry vs. magnetometry 
 

The classical thermodynamics deals with the 

volume work d dw p V   and defines two heat 

capacities (Eq. 52 and 53): 

2( , ) ln
V

V V V

U S V Z
C R T

T T T

  

  

    
     
    

     (52)            (51) 

2

( , )

ln ln

ln

p

p

V T p

E S p
C

T

Z Z
R T T

T T V





  

  

 
  
 

    
     

    

         (53)           (52), 

where Z is the partition function. In the case  

of the magnetic work 
0

d dw H M  again two 

kinds of the heat capacities are distinguished  

(Eq. 54, 55): 

2( , ) ln
M

M M M

U S M Z
C R T

T T T

  

  

    
     
    

(54) 

2

( , )

ln ln

ln

H

H

M T H

E S H
C

T

Z Z
R T T

T T M





  

  

 
  
 

    
     

    

     (55) 

(Here, enthalpy is denoted as E, not to be confused 

with the magnetic field strength H.) For the solid 

state, the appropriate formula for the molar excess-

heat capacity measured in the zero field is (Eq. 56): 

ex 2 ln
p M

M M

Z
C C R T

T T

 

 

  
    

  
          (56) 

This allows a comparison of the experimental heat 

capacity (measured using adiabatic or differential 

scanning calorimeters) with that reconstructed by 

the theoretical model of the spin crossover  

(e.g. regular solution&domain model).  

By substituting the partition function (Eq. 57):  

L L

L

H H L

L L H

0 L H

1 2 2

( 1)

[1 exp( / )]

( 1) 1 exp( / )
1

( 1) 1 exp( / )

exp{ [Δ ( ) / 2

( ) (2 1)] /

m

m

S S
Z

h kT

S S h kT

S S h kT

h h m

J J J x kT







  






    
   

    

   
 

    

         (57)            (56) 

into Eq. 56, explicit expression for the heat 

capacity along with its FORTRAN code is obtained 

by exploiting capabilities of the MATHEMATICA 

package (Wass 1999). 

The heat capacity and/or its weighted function then 

become a combination of the underlying lattice 

vibration functions (taken as polynomials)  

and the excess-heat capacity (Eq. 58 and 59): 

2 3

H L L L L

2 3 ex

H H H H H

(1 )( )

( )

p

p

C x a b T c T d T

x a b T c T d T C

    

    
      (58) 

2 3

H L L L L

2 3 ex

H H H H H

( / ) (1 )( )

( ) ( / )

p

p

C T x a b T c T d T

x a b T c T d T C T

    

    
  (59) 

where only some polynomial terms need be 

considered.  

Three complexes under the investigation are 

characterized as follows (Fig. 12). Complex 1 – 

[Fe(2-pic)3]Cl2·MeOH is a non-cooperative system; 

its thermodynamic data were scanned by  

the adiabatic calorimetry (Nakamoto et al. 2001).  



Nova Biotechnol Chim (2020) 19(2): 138-153 

147 

T/K

0 50 100 150 200 250 300 350

C
p
/(

J 
K

-1
 m

o
l-

1
)

0

100

200

300

400

500

600

700

T/K

50 100 150 200 250 300 350

x
H

S

0.0

0.5

1.0

(1/T)/K
-1

0.004 0.006 0.008
ln

K
-6

-3

0

3

6

1

T/K

120 150 180 210 240

(C
p
/T

)/
(J

 K


2
 m

o
l

1
)

4

6

8

T/K

50 100 150 200 250 300 350

x
H

S

0.0

0.5

1.0

(1/T)/K
-1

0.004 0.006 0.008

ln
K

-6

-3

0

3

6

100 150 200

C
p

/(
J 

K
-1

 m
o

l-
1

)

500

1000

1500

2

T/K

0 50 100 150 200 250 300

C
p
/(

J 
K

-1
 m

o
l-

1
)

0

100

200

300

400

500

600

T/K

50 100 150 200 250 300 350

x
H

S

0.0

0.5

1.0

(1/T)/K
-1

0.004 0.006 0.008

ln
K

-6

-3

0

3

6

0 50 100 150 200 250 300
0

2000

4000

6000

3

 
Fig. 12. Experimental (open symbols) and fitted (full lines) 

thermodynamic functions for 1, 2, and 3.  

Complex 2 – [Fe(pybzim)3](ClO4)2 belongs  

to a medium-cooperative system. There is  

no structural change during the spin crossover as 

documented by a continuous increase of the lattice 

parameters. DSC technique has been applied for it 

(Boča et al. 2003).  

Complex 3 – [Fe(phen)2(NCS)2] is a strongly 

cooperative system. The effective magnetic 

moment increases abruptly near the transition 

temperature T1/2 and the structural changes 

accompany the spin transition.  

The thermodynamic data were collected by the 

adiabatic calorimetry (Sorai and Seki 1974).  

The data fitting by the Eq. 56 – 57  gave  the set  of  

the spin crossover parameters which are presented 

in Table 1. 

 
Table 1. Parameters of the spin crossover for complexes 1 

through 3 from fitting the heat capacity.a  

Parameter b Cooperativeness  

1 – small 2 – medium 3 – high 

Site format. energy 

(eff/kB) /K 

1370 351 1039 

Entropic parameters  reff = 

7545 
L

/h hc = 

454 cm-1 

H

L
/ 1.5

h

h






 

reff = 357 

Cooperativeness 

(J/kB) /K 

20 136 182 

H /kJ mol-1 11.39 

[8.88] 

2.92  

[3.04] 

8.64 

[8.60] 

S /J K-1 mol-1 74.2 

[59.5] 

19.0  

[21.0] 

48.9 

[48.8] 

1/2
/T H S    /K 153 

[149] 

153  

[145] 

177 

[176] 
a Values in square brackets are the direct calorimetric 

determination.  
b Simplifications: 

eff
lnS R r  ,

2
/ 2J J  , 

eff 0 L H 1 2
( ) / 2m h h J J        . 

 
Having the experimental heat capacity curve  

and its temperature-weighted function at the 

disposal, the numerical integration offers the 

enthalpy (Eq. 60) and entropy (Eq. 61) of the spin 

transition: 

p max

min p

d d

T T

p p

T T

H C T C T                   (60) 

p max

min p

( / ) d ( / ) d

T T

p p

T T

S C T T C T T           (61) 

The measured Cp and Cp/T data need be corrected 

for the underlying lattice vibrations, for instance, 

by subtracting polynomial functions applied below 

Tmin and/or above Tmax and the integration limit 

contains the peak value of Tp (Fig. 13). 

The integration can be improved by utilizing  

the conversion curve xH vs T known from  

the measurements of the magnetic susceptibility. 

This enables a construction of the smooth baseline 

between Cmin for the LS and Cmax for the HS; the 

excess enthalpy Δ ( )H T  (Eq. 62) associated



Nova Biotechnol Chim (2020) 19(2): 138-153 

148 

T
3
/K

3

3e+6 6e+6 9e+6

0.5

1.0

1.5

T/K

120 140 160 180 200

C
p
/(

k
J
 K


1
 m

o
l

1
)

0.5

1.0

1.5

T/K

120 140 160 180 200

( C
p
)'
/(

k
J
 K

-1
 m

o
l-

1
)

0.0

0.1

0.2

T/K

120 140 160 180 200

(C
p
/ T

)/
(J

 K


 m

o
l

1
)

5

6

7

T
2
/K

2

20000 40000

5

6

7

120 140 160 180 200

( C
p
/ T

)'
/(

J
 K


2
 m

o
l

1
)

0.0

0.4

0.8

 

with the spin crossover at the given temperature is: 

min

min

3

LS LS

HS max min

Δ ( ) [ ( )]d

( )d

T

p

T

T

T

H T C a b T T

x C C T

  

 





     (62) 

The excess entropy at the given temperature Δ ( )S T  

can be evaluated in an analogous way.  

A direct fitting of the magnetic susceptibility data 

(T) is possible by considering three contributions 
in the form of a Curie-Weiss law, namely for the 

low-spin species, high-spin system, and eventual 

paramagnetic impurity (Eq. 63, 64, and 65): 

2

L 0 L L L L L
( 1) / 3( )C g S S T           (63) 

2

H 0 H H H H H
( 1) / 3( )C g S S T           (64) 

2

PI 0 PI PI PI PI PI
( 1) / 3( )C g S S T           (65) 

where the reduced Curie constant consists of the 

physical constants, 
2

0 A 0 B B
/C N k  . For Fe(II) 

centres (SL = 0, SH = 2, SPI = 5/2) the appropriate 

set of magnetic parameters consists of L, gH, H, 

gPI = 2, PI, and PI. For Fe(III) centres (SL = 1/2, 

SH = 5/2) the active set is gL, L, aL, gH = 2.0 and 
H. Some of these parameters can be fixed  
or omitted in order to avoid an overparametrization. 

In addition, there are four parameters of the spin 

crossover that enter evaluation of the conversion 

curve xH(T), i.e. eff, J, Lh  and Hh  (the last 

again can be fixed). Then the susceptibility is 

balanced as follows (Eq. 66):  

H PI L H H PI PI
( ) (1 )T x x x x              (66) 

and the equilibrium constant (Eq. 67) is: 

H H PI
/ (1 )K x x x                       (67) 

The enthalpy of the spin transition (Eq. 68)  

is calculated as a temperature–independent  

quantity: 

0 A eff eff B
Δ ( / )H N R k           (68) 

which absorbs the site formation energy, zero-point 

vibration correction, and eventually the cooperati-

veness parameters (Eq. 69): 

eff 0 H L 1 2
Δ ( ) / 2 ( )E h h m J J             (69) 

The entropy of the transition (Eq. 70) is  

a temperature-dependent quantity which at the 

transition temperature is: 

1/2

L B 1/2H

L H B 1/2

1 exp( / )2 1
(Δ ) ln

2 1 1 exp( / )

m

T

h k TS
S R

S h k T





   
   

    

     (70) 

This can be approximated through an effective 

degeneracy ratio (Eq. 71): 

Fig. 13. Heat capacity analysis 

for [Fe(pybzim)3](ClO4)2:  

a) raw data; b) data for 

subtraction of underlying lattice 

vibrations; c) data for numerical 

integration yielding H and S.  

a)                                      b)                                              c)                   



Nova Biotechnol Chim (2020) 19(2): 138-153 

149 

(1/T)/K
-1

0.004 0.006 0.008 0.010

ln
K

-5

0

5

T/K

50 100 150 200 250 300

x
H

S

0.0

0.5

1.0

Susceptibility

Susceptibility, fitted

DSC, fitted

Fig. 14. Comparison of magnetic and calorimetric data  

for [Fe(pybzim)3](ClO4)2. Empty points – experimental data, 

lines – fitted.  

eff
Δ lnS R r                     (71) 

where reff is subjected to the fitting procedure.  

The fitting of the magnetic data and calorimetric 

data is compared in Fig. 14 on the common basis – 

temperature evolution of the high-spin mole 

fraction (left) and the van’t Hoff plot (right).  

 

Statistical analysis 
 

A number of organic species H2L, acting  

as pentadentate ligands L2-, has been prepared 

(Fig. 15) by a Schiff condensation between  

the substituted salicylaldehyde (R-sal) and the 

asymmetric or symmetric triamine (pet or dpt). 

T/K

0 50 100 150 200 250 300

x
H

0.0

0.5

1.0

T/K

0 50 100 150 200 250 300


e
ff
/

B

0

2

4

6
B = 0.1 T

T
1/2

 
Fig. 16. Magnetic data for [Fe(napet)NCS]: left – temperature 

dependence of the effective magnetic moment, right –  

a temperature evolution of the calculated high-spin fraction 

(right); grey circles – experimental data, solid line – fitted.  

Analogously, a set of Schiff-base ligands was 

obtained using the naphthyl skeleton. They were 

complexed with Fe(III) salts yielding 

hexacoordinate [FeIIILX] complexes.  

Thermal evolution of the effective magnetic 

moment showing spin crossover is exemplified  

in Fig. 16 along with the fitted curve based upon 

the regular solution model. 

In a series of hexacoordinate [FeIIIL5X] complexes, 

the transition temperature T1/2 of the spin crossover 

can be modified by appropriate coligands X  

(Fig. 17) (Šalitroš  et al. 2009; Nemec  et al. 2011; 

Krüger et al. 2013, 2015; Masárová et al. 2015). 

This is affected by the enthalpic and entropic terms 

since T1/2 = H/S holds true. It is expected that 

N

OH

N

OH

NH

N

OH

N

OH

NH
N

OH

N

OH

NH

N

OH

N

OH

NH

OEt EtO

N

OH

Cl

N

OH

NH

Cl

N

OH

MeO

N

OH

NH

OMe

N

OH

N

OH

NH

N

OH

N

OH

NH

Cl Cl

N

OH

N

OH

NH

napet 2anapet

1anapet

salpet 5Cl-salpet

3EtO-salpet

saldpt
5Cl-saldpt 3MeO-saldpt  

Fig. 15. Sketch of the related pentadentate ligands H2L
5. 



Nova Biotechnol Chim (2020) 19(2): 138-153 

150 

Fe
(III) N

O

N

O

R1 R1

NH

X

R2
R2  

Fig. 17. General form of the [Fe(R-salpet)X] type complexes; 

coligands X = Cl, N3
, NCO, NCS, NCSe, and CN. 

the value of H can be altered by varying  

the crystal field strength of involved ligands. 

However, the factors influencing the value of S 

are more complex as they include electronic (net 

spin), vibrational, and other contributions.  

Temperature evolution of the effective magnetic 

moment for related complexes is shown in Fig. 18 

and 19. It can be concluded that an increase  

of the crystal field strength 10Dq of the coligand 

X causes a switch  of the spin states  of complexes  

T/K

0 50 100 150 200 250 300


e

ff
/

B

0

1

2

3

4

5

6

[Fe(saldpt)py](BPh
4
)
2
, SC

[Fe(5Cl-saldpt)py](BPh
4
)
2
, SC

[Fe(3MeO-saldpt)py](BPh
4
)
2
, SC

SC

SC

 
Fig. 19. Spin crossover in complexes of [FeIII(R-saldpt)py] 

(BPh4)2 type. 

from the high-spin state (X = Cl–, NCO–), through 

the spin crossover (X = NCS–, NCSe–), to the low-

spin state (X = CN–).  

In order to unhide latent correlations among 

thermodynamic parameters influencing the spin 

crossover, modern multivariate methods were 

utilized for processing data listed in Table 2: 

Pearson correlation (PC), cluster analysis (CA)  

and the principal component analysis (PCA) 

(Augustín and Boča 2015). 

T/K

0 50 100 150 200 250 300


e
ff
/

B

0

1

2

3

4

5

6

[Fe(Cl-salpet)Cl], HS

[Fe(Cl-salpet)NCO], HS

[Fe(Cl-salpet)NCS], SC

[Fe(Cl-salpet)NCSe], SC

[Fe(Cl-salpet)CN], LS

T/K

0 50 100 150 200 250 300
0

1

2

3

4

5

6

[Fe(Br-salpet)NCS], HS

[Fe(Br-salpet)N
3
], SC

[Fe(Br-salpet)NCSe], SC

HS

SC

LS

HS

SC


e
ff
/

B

SC

T/K

0 50 100 150 200 250 300


e

ff
/

B

0

1

2

3

4

5

6

[Fe(5Br-salpet)NCS], HS

[Fe(3EtO-salpet)NCS], SC
*

[Fe(5Cl-salpet)NCS], SC

T/K

0 50 100 150 200 250 300
0

1

2

3

4

5

6

-[Fe(1anapet)NCS], HS

-[Fe(1anapet)NCS], SC
*

[Fe(2anapet)NCS], SC

[Fe(napet)NCS].MeCN, SC


e

ff
/

B

HS HS

SC
SC

SC

SC

SC

 
Fig. 18. Comparison of the spin states in complexes of [FeIIIL5X] type. Top – fixing the Schiff-base ligand and varying 

coligand; bottom – varying the Schiff-base ligand and fixing the coligand NCS-. HS – high spin, SC – spin crossover, 

LS – low spin. For details consult Table 2.  



Nova Biotechnol Chim (2020) 19(2): 138-153 

151 

Table 2. Retrieved thermodynamic parameters for spin 

crossover systems.a  

 Compound T1/2 H S J 

1h [Fe(3EtO-salpet)NCS] 84 

82 

0.9 12 98 

2 [Fe(5Br-

salpet)N3]·MeOH 

142 1.6 11 76  

3 [Fe(5Cl-salpet)NCS] 280 5.8 21 180 

4 [Fe(5Cl-salpet)NCSe] 293 6.5 22 197 

5 [Fe(5Br-salpet)NCSe] 326 6.0 18 215 

6 [Fe(salpet)atz] 416 15 37 284 

7i,h -[Fe(1anapet)NCS] 44 

40 

1.6 4 82 

8 [Fe(2anapet)NCS] 114 1.6 14 52 

9h [Fe(napet)N3]MeOH  122 

117 

1.5 11 99 

10 [Fe(napet)NCS]MeCN 151 1.9 12 87 

11 [Fe(napet)NCO] 155 2.5 16  102 

12 [Fe(napet)NCSe]MeC

N 

170 2.3 13 99 

13 [Fe(napet)NCS] 186 3.3 18  150 

14i [Fe(5Cl-saldpt)py]BPh4 78 0.4 6 63 

15 [Fe(3MeO-saldpt)py] 

BPh4 

273 4.5 17 90 

16 [Fe(saldpt)py]BPh4 310 5.4 17 150 
a Spin crossover between S = 1/2 and S = 5/2; h – spin 

crossover with hysteresis; i – spin crossover between 

intermediate spin S = 3/2 and S = 5/2. Data of 6 and 7  

was omitted in the statistical analysis. Units: T1/2/K;  

H/kJ mol-1;S/J K-1 mol-1; (J/kB)/K. Taken from (Boča  

et al. 2000; Šalitroš et al. 2009; Nemec et al. 2011; Krüger  

et al. 2013, 2015; Masárová  et al. 2015).  

The results obtained by the cluster analysis are 

shown in Fig. 20. It can be seen that the complexes 

under study span two clusters: the cluster 1 is 

formed by complexes for which below-room 

temperature spin crossover was observed (No. 1, 2, 

Dendrogram

Ward's Method,Squared Euclidean

0

3

6

9

12

15

18

D
is

ta
n
c
e

1 2 3 4 589

1
0

1
1

1
2

1
3

1
4

1
5

1
6

 
Fig. 20. Dendogram of the cluster analysis (using Ward’s 

method and squared Euclidean distance) for complexes 

numbered according to Table 2. 

8, 9, 10, 12 and 14). The remaining complexes span 

the cluster No 2 and they show near- or above-

room temperature spin transition.  

The PCA analysis yields the biplots of principal 

components shown in Fig. 21. It is seen that  

the transition temperature T1/2 closely relates to  

the enthalpy and entropy of the spin transition 

whereas a correlation with cooperativeness J does 

not exist. The biplot on the right shows the objects 

spanning into individual clusters which are well 

separated between two principal components.  

Finally, the Pearson correlation evaluates  

the correlation coefficients for each pair of 

variables. It is found that T1/2 – H pair possesses  

a high correlation coefficient r = 0.97. As already 

indicated by CA and PCA analysis, the correlation 

of the cooperativeness with the remaining 

parameters is weak. The transition temperature  

and the transition entropy are plotted vs the 

transition enthalpy along with the regression curve. 

H

J

Biplot

-4 -2 0 2 4

Component 1

-1.5

-1

-0.5

0

0.5

1

1.5

C
o
m

p
o
n
e
n
t 

2

Tc

S

Component 1

C
o
m

p
o
n
e
n
t 

2

Plot of PCOMP_2 vs PCOMP_1

-4 -2 0 2 4

-1.5

-1

-0.5

0

0.5

1

1.5 CLUSTNUMS

1

2

 

Fig. 21. Results of Principal component analysis. Top –  

a biplot with a ray diagram, bottom – individual complexes 

classified by the clusters.  



Nova Biotechnol Chim (2020) 19(2): 138-153 

152 

H /kJ mol
-1

0 1 2 3 4 5 6 7


S

 /
J 

K
-1

 m
o

l-
1

0

5

10

15

20

25

H /kJ mol
-1

0 1 2 3 4 5 6 7

T
1

/2
/K

0

50

100

150

200

250

300

350

400

 
Fig. 22. Linear correlations T1/2 vs H and S vs H.  

The confidence intervals (95 %) are drawn as dashed curves.  

According to Fig. 22 the linear relationship is now 

confirmed.  

The aspects discussed in the present 

communication are only a part of the huge story, as 

documented by a number of alterative models  

and plethora of experiments about the spin 

crossover phenomenon. Alternative views were 

comprehensively presented in recent articles 

(Enachescu et al. 2018; Nicolazzi et al. 2018; 

Pavlik et al. 2018). 

 

Conclusions 
 

The spin crossover behaviour is based upon a fine 

tuning of the crystal field strengths balancing  

the interelectron repulsion. The spin orbit coupling, 

however, also plays an important role. Ideal 

thermodynamic behaviour, as described by  

the regular solution&domain model, is further 

modified by the interactions in the solid state that 

are considered as the solid-state cooperativeness. 

The role of the cooperativeness is analogous to  

the role of the activity in solution models: it allows 

using the equation for the ideal system but instead 

of the mole fraction xH, an effective mole fraction 

J·xH is in the play. The nature of the 

cooperativeness lies in the electronic (chemical) 

and nuclear (physical) hardness. Thermo-dynamic 

data about H and S can be obtained not only 

from the direct calorimetric measurements but also 

from an appropriate model of the spin crossover 

adapted to the magnetic data analysis. These 

parameters are mutually interrelated through  

the transition temperature T1/2 as proven by  

the multivariate statistical analysis.  

 

Acknowledgments 
 
Grant agencies of Slovakia are acknowledged for the financial 

support (projects VEGA 1/0013/18 and APVV 16-0039). 

 

Conflict of Interest 
 
The author declares that he has no conflict of interest.  

 

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