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arXiv:0904.2742v1 [physics.chem-ph] 17 Apr 2009

Modeling molecular crystals formed by spin-active metal complexes

by atom-atom potentials

Anton V. Sinitskiy∗

Poncelet Laboratory, Independent University of Moscow,

Bolshoy Vlasyevskiy Pereulok 11, 119002, Moscow, Russia

Andrei L. Tchougr´eeﬀ†

Poncelet Laboratory, Independent University of Moscow,

Bolshoy Vlasyevskiy Pereulok 11, 119002, Moscow, Russia and

JARA, Institut f¨ur Anorganische Chemie, RWTH Aachen, Landoltweg 1, 52056 Aachen, Germany

Andrei M. Tokmachev and Richard Dronskowski

JARA, Institut f¨ur Anorganische Chemie, RWTH Aachen, Landoltweg 1, 52056 Aachen, Germany

(Dated: April 17, 2009)

We apply the atom-atom potentials to molecular crystals of iron (II) complexes with bulky

organic ligands. The crystals under study are formed by low-spin or high-spin molecules

of Fe(phen)2(NCS)2(phen = 1,10-phenanthroline), Fe(btz)2(NCS)2(btz = 5,5′,6,6′-tetrahydro-

4H,4′H-2,2′-bi-1,3-thiazine), and Fe(bpz)2(bipy) (bpz = dihydrobis(1-pyrazolil)borate, and bipy =

2,2′-bipyridine). All molecular geometries are taken from the X-ray experimental data and assumed

to be frozen. The unit cell dimensions and angles, positions of the centers of masses of molecules,

and the orientations of molecules corresponding to the minimum energy at 1 atm and 1 GPa are

calculated. The optimized crystal structures are in a good agreement with the experimental data.

Sources of the residual discrepancies between the calculated and experimental structures are dis-

cussed. The intermolecular contributions to the enthalpy of the spin transitions are found to be

comparable with its total experimental values. It demonstrates that the method of atom-atom

potentials is very useful for modeling organometalic crystals undergoing the spin transitions.

I. INTRODUCTION

The Crystal Field Theory (CFT), proposed in [1] and

known to majority of chemists through [2], suggests that

coordination compounds of d-elements with electronic

conﬁgurations d4,d5,d6or d7can exist either in high-

spin (HS) or low spin (LS) forms (sometimes interme-

diate values of the total spin are also possible). In the

case of strong-ﬁeld ligands the d-level splitting measured

by the average crystal ﬁeld parameter 10Dq exceeds the

average Coulomb interaction energy of d-electrons Pand

the ground state is LS. In the case of weak-ﬁeld ligands

with 10Dq ≪P, the ground state is bound to be HS.

If, however, 10Dq ∼

=P, the LS and HS forms of the

complex may coexist in equilibrium, and the fraction

of either spin form depends on temperature, pressure,

and/or other macroscopic thermodynamic parameters.

The process when the fraction of molecules of diﬀerent

total spin changes due to external conditions is called a

spin crossover (SC) transition. For the ﬁrst time this

phenomenon was reported in 1931 [3]. Nevertheless, ex-

tensive studies of SC started only in 1960s-70s. Nowa-

days, dozens of complexes capable to undergo spin tran-

sitions (spin-active complexes) are known, and most of

them are those of Fe(II). A general review of the ﬁeld

∗Electronic address: sinitsk@mail.ru

†Electronic address: andrei.tchougreeﬀ@ac.rwth-aachen.de

can be found in [4].

A wealth of potential practical applications like dis-

plays and data storage devices (see a detailed review in

[5]) is one of the reasons for research activity in this area.

Industrial applications pose strict demands on the char-

acteristics of the materials to be used. As a consequence,

the problem of predicting SC transition characteristics

(whether it is smooth or abrupt, the transition temper-

ature, the width of the hysteresis loop, the inﬂuence of

additives [6]) is of paramount importance. Theoretical

description of spin transitions is a great challenge by it-

self, and until now a coherent theory allowing to relate

the composition of the materials with the characteristics

of the transition has not been developed. Discussion of

these issues and an overview of the existing theories are

given in [7].

In general, the SC modeling includes two aspects: (i)

that of the interactions within one molecule of a spin-

active complex, and (ii) that of the interactions between

these molecules. The latter is crucially important for un-

derstanding of speciﬁc features of the SC transitions in

solids because the SC manifesting itself as a ﬁrst-order

phase transition is controlled by intermolecular interac-

tions. These ideas are built in the simplest model ca-

pable of describing spin transitions in solids proposed

by Slichter and Drickamer [8]. This model considers the

solid as a regular solution of molecules in the LS and HS

states. The model predicts, in agreement with the exper-

iments, that the spin transition may be either smooth or

abrupt or may exhibit hysteresis, and its character is de-

2

termined by a phenomenological intermolecular parame-

ter Γ, speciﬁc for each material. However, the experimen-

tal data on the heat capacity and the X-ray diﬀraction

contradict to this model.

The thermal dependence of the heat capacity of the

Fe(phen)2(NCS)2crystal is better explained by an alter-

native domain model [9]. Diﬀraction patterns of spin

transition crystals, measured at intermediate temper-

atures, simultaneously contain the Bragg peaks corre-

sponding to the pure LS and HS phases, while no peaks

for intermediate lattice of a solution were observed [10].

Another problem is that the parameter Γ is phenomeno-

logical one, and it cannot be sequentially derived in

terms of microscopic characteristics of the constituent

molecules or their interactions. At the same time, within

the Slichter-Drickamer model, the type of behavior is

tightly related to the sign and magnitude of Γ, so that

a smooth transition requires Γ >0, an abrupt transi-

tion occurs at Γ <0 and hysteresis is possible only if

Γ<0 is less than some critical threshold, which in its

turn depends on the transition temperature [8]. It has

been shown that if the relaxation of the lattice is not al-

lowed, then under very natural assumptions Γ is positive

[11], but the lattice relaxation can lead to Γ of either sign

[12].

Signiﬁcant progress in the understanding of the spin

transitions in crystals is attributed to the Ising-like mod-

els of intermolecular interactions in spin-active materials

[13]. Adaptations of the initial Ising model to the spin

transitions include corrections for intramolecular vibra-

tions, domain formation, parameters distribution, elastic

distortions, presence of two metal atoms in a spin-active

molecule, etc. [7, 14]. These models do not have an-

alytical solutions and they are solved either in a mean

ﬁeld approximation which leads to results analogous to

(or even coinciding with) the Slichter-Drickamer model

[7] or numerically.

In spite of the diversity of the models used in the liter-

ature, the theoretical description of the spin transitions

is not yet satisfactory. First, the existing theories are not

capable to reproduce the whole set of the experimental

data (e.g. asymmetry of the hysteresis loop [7]). Second,

all of them contain phenomenological parameters, like Γ

in the Slichter-Drickamer model, or the energy gap ∆i

or the interspin interaction constants Jij in the Ising-like

models, or the bulk modulus Kand the Poisson ratio σ

in [15] (Kand σcan be measured, but for the purpose

of the theory they must be independently predicted),

etc. Third, even if the models include microscopic level

consideration, they use oversimpliﬁed description of the

molecules (as spheres, ellipsoids), which is not suﬃcient

for constructing a complete theory, especially due to im-

portance of the short intermolecular contacts tentatively

responsible for the cooperativity eﬀects (π-πinteractions,

S···H−C interactions, hydrogen bonds, etc. [16]).

These shortcomings can be overcome by using explicit

potentials for intra- and intermolecular interactions. In

this case one may expect to obtain independent estimates

of the numerous parameters required by the phenomeno-

logical theories. These potentials should also be a help-

ful tool for checking the validity of the initial postulates,

such as the formation of a regular solution or the domain

structure, thus clarifying some obscure points in the the-

ory itself.

An adequate ab initio calculation of the energies of iso-

lated transition metal complexes, and moreover those of

the crystals formed by these complexes, is a very compli-

cated problem. Signiﬁcant electron correlation within the

d-shells breaks the self-consistent ﬁeld approximation, so

that explicit account of nontrivial (static) electron corre-

lation is unavoidable. The existing implementations of ab

initio approaches for solids fail to provide the necessary

quality of the results.

There is a number of attempts to use the DFT-based

methods to take into account the electron correlation in

the SC complexes [17]. These methods yield good results

for many characteristics of isolated spin-active molecules

(optimal molecular geometry, M¨ossbauer parameters, vi-

brational frequencies, nuclear inelastic scattering spec-

tra) [18, 19, 20, 21]. However, the DFT in its traditional

form, as it is demonstrated in [22], is not capable to re-

produce coherently the static correlations, which are ex-

tremely important for the correct description of the spin

transitions even in an isolated molecule. For that rea-

son the results for the energy gap between the LS and

HS states, and hence for the transition temperature, ob-

tained by the DFT techniques are absolutely disastrous.

The common versions of DFT, such as B3LYP, often pre-

dict a wrong ground state multiplicity, let alone the value

of the energy diﬀerence [17]. For example, the tempera-

ture of the spin transition in Fe(phen)2(NCS)2was found

to be an order of magnitude too large (1530 K instead

of 176 K) [23]. In addition, most DFT studies are lim-

ited to isolated molecules in vacuo, and the heat of the

spin transitions in a crystal is identiﬁed with the energy

diﬀerence of isolated molecules. The inﬂuence of the in-

termolecular interactions is thus neglected.

Only a few isolated attempts to explicitly model

a spin-active crystal by the DFT method have been

reported [20, 21, 24, 25]. The application of the

LDA approximation with the periodic boundary con-

ditions to the crystals of Fe(trim)2X2(X = F, Cl,

Br, or I, and trim = 4-(4-Imidazolylmethyl)-2-(2-

imidazolylmethyl)imidazole) formed by either LS or HS

molecules [20] demonstrated that the intermolecular in-

teractions strongly aﬀect the energy splitting between

the LS and HS isomers, thus necessitating their adequate

treatment within coherent SC models. The experimen-

tal X-ray structures for some complexes are available, so

that the optimal geometry of the crystals found by LDA

can be veriﬁed. This comparison showed that the unit

cell volumes were overestimated by 20-24%. At the same

time, the calculated N...X distances were 0.1-0.3 ˚

A lower

than the experimental ones, and the π-stacking distances

were underestimated by 0.3-0.7 ˚

A. Although in general

it is diﬃcult to separate the errors from intra- and inter-

3

molecular interactions, the geometry of individual spin

isomers is usually described much better than the rela-

tive position of the molecules in the crystal. The GGA

approximation has been used to optimize molecular ge-

ometry and lattice parameters of the LS and HS crys-

tals of [Fe(pyim)2(bipy)](ClO4)2·2C2H5OH (pyim = 2-

(2-pyridyl)imidazole) [26]. The bond lengths were found

to be quite reasonable. However, the lattice parameters

were poorly reproduced, so that even the wrong sign of

the unit cell volume change for the LS to HS transition

was obtained: (−6.82 ˚

A3instead of experimental value

of +228.02 ˚

A3). The authors explain it by “the well-

known shortcomings of DFT methods in application to

weak intermolecular interactions” [26]. The DFT+Uap-

proach with the GGA approximation has been applied

to model spin-active crystals of Fe(phen)2(NCS)2and

Fe(btr)2(NCS)2(H2O) (btr = 4,4′-bis-1,2,4-triazole) [25].

The studies of the Fe(phen)2(NCS)2crystal have demon-

strated that DFT is capable of reproducing the lattice

parameters with the precision of 1-5% and the unit cell

volume with the precision of 2 −7% [27]. Unfortunately,

these works do not employ a much better substantiated

approach to the DFT-based treatment of van der Waals

interactions previously proposed by the same authors,

based on explicit treatment of correlations coming from

the long range part of the electron-electron interactions

[28].

Summarizing, DFT models either produce poor re-

sults for spin-active complexes or require parameters (like

DFT+U) adjusted to reproduce the experimental data.

At the same time, the very idea of modeling such com-

plex system as a crystal formed by spin-active transition

metal complexes at a uniform level of theory seems to

be incorrect. The systems under consideration consist

of numerous components, and it is much more natural

to treat these components separately – each at the ade-

quate level of the theory. The most important separation

is that on intra- and intermolecular interactions. On the

level of molecules one can further separate a highly cor-

related d-shell from the rest of the molecule. This idea

has been implemented as a specialized quantum chemical

method – Eﬀective Hamiltonian of Crystal Field (EHCF)

[29] which has been successfully applied to describe the

spin isomers of Fe(phen)2(NCS)2[30]. Furthermore, it

has been demonstrated that the geometry of spin-active

complexes can be adequately described by the EHCF

technique with ligands treated by molecular mechanics

force ﬁelds [31].

On the level of interactions between molecules the

paramount fact is that the molecular crystals formed by

spin-active molecules consist of complexes with bulky or-

ganic ligands. Intermolecular contacts in such crystals

are those between the organogenic atoms like C, H, N,

S, etc. The d-shells of the central ions are eﬀectively

shielded by the ligands. Thus, it is reasonable to assume

[11] that the d-shells do not directly aﬀect the interac-

tions between the molecules of the diﬀerent total spin in

the crystal, but inﬂuence it indirectly: through the vari-

ation of the equilibrium interatomic distances Fe−N in

these complexes, which is further translated into diﬀer-

ent ”sizes” of the LS and HS isomers. In this context,

the standard methods developed for organic molecular

crystals can be successfully applied in this case as well.

The main purpose of the present work is to identify an

adequate way to model intermolecular interactions for

crystals formed by spin-active molecules.

II. ATOM-ATOM POTENTIALS METHOD

In order to avoid unnecessary complications, we limit

our task in the present paper to checking the possibil-

ity of applying the simplest method of modeling inter-

molecular interactions – atom-atom potentials [32] – to

crystals formed by spin-active complexes. The method

assumes that the energy of the molecular crystal (calcu-

lated relative to the system of isolated molecules) can be

represented as:

U=1

2X

αα′mm′rr′

Eαα′(R(αα′mm′rr′)) ,(1)

where each term is the energy of the interaction between

the α-th atom of the m-th molecule in the unit cell num-

ber r= (ra, rb, rc) and the α′-th atom of the m′-th

molecule in the unit cell r′depending on the distance

R. Due to the equivalence of all unit cells, we can get rid

of summation over r′, and the energy per molecule ucan

be written as:

u=1

2MX

αα′mm′r

Eαα′(R(αα′mm′r0)) ,(2)

where Mis the number of molecules per unit cell.

A number of approximations have been suggested for

the atom-atom interaction. The most widespread ones

are the Buckingham potential (6-exp):

Eαα′(R) = −Aαα′

R6+Bαα′e−Cαα′R,(3)

and the Lennard-Jones potential (6-n):

Eαα′(R) = −Aαα′

R6+Bαα′

Rn.(4)

In the above formulae the Aαα′and Bαα′parameters

for the interaction between atoms of diﬀerent types are

often calculated as the geometric mean values of the cor-

responding homogeneous interaction parameters:

Aαα′=pAααAα′α′, Bαα′=pBααBα′α′,(5)

while the Cαα′parameter is approximated in a similar

way as an arithmetic mean value:

Cαα′=1

2(Cαα +Cα′α′).(6)

4

Due to these approximations, the energy can be rep-

resented as a fast computable function depending on the

lattice parameters and relative positions and orientations

of the molecules in the unit cell provided that molecular

geometry of the complex is ﬁxed. Having found the min-

imum of this function, one gets estimates of the inter-

molecular interaction energy (sublimation energy), the

equilibrium unit cell parameters, and the positions and

orientation of the molecules in the unit cell at the abso-

lute zero temperature and absence of external pressure.

One can easily extend the method to account for the

external pressure. For this purpose one should optimize

the enthalpy Hinstead of the potential energy U. The

enthalpy is deﬁned as

H=U+P V, (7)

where the volume Vis determined by the lattice parame-

ters. As for the thermal dependence of the lattice param-

eters, the matter is not so simple. One should basically

minimize the Gibbs energy Gto estimate the equilibrium

values of the lattice parameters at a non-zero tempera-

ture (and pressure). This procedure includes calculation

of the entropy of the crystal undergoing the spin transi-

tion, which is a separate non-trivial challenge, as shown

in [35]. To avoid this, one may conﬁne to minimiza-

tion of the internal energy Uor the enthalpy H, but the

resulting lattice parameters will be relevant only for the

absolute zero of temperature. On the other hand, in prac-

tice the parameters of atom-atom interaction are ﬁtted

in such a way that the lattice parameters corresponding

to the minimum of the model internal energy Ubest re-

produce the experimental lattice structures measured at

the room temperature (see e.g. [36]). In this case the

model includes the entropy factor implicitly, and the lat-

tice parameters found by direct minimization of Ushould

actually refer to the room temperature.

The accuracy of the atom-atom approach is corrobo-

rated by extensive statistics obtained for organic molec-

ular crystals [32, 33, 34]. Typically it provides the ac-

curacy level of ca. 0.1÷5 kcal/mol in energy terms for

a wide range of organic crystals. However, in theory we

can expect much better precision for the relative energies

of the crystals undergoing the spin transition, since the

LS and HS crystals are very similar to each other (as is

shown below, the shortest contacts are the same).

III. MODELING METHOD

We performed calculations for the molecular crystals

formed by each of the spin isomers of Fe(phen)2(NCS)2,

Fe(btz)2(NCS)2, and Fe(bpz)2(bipy). The ligands are de-

picted in Fig. 1 and the molecules themselves are shown

in Figs. 2-4. The objects were chosen based on the fol-

lowing considerations. First, all these crystals consist of

neutral molecules only, without ions or solvents. As a re-

sult, the molecules are held together in the crystal by the

FIG. 1: Structure formulae for the ligands of the spin-active

complexes studied.

FIG. 2: Molecular structure of Fe(phen)2(NCS)2.

van der Waals forces (no strong Coulomb forces or ob-

vious hydrogen bonds are involved), which dramatically

simpliﬁes modeling of the energy. Second, these three

substances represent all main types of spin transitions:

abrupt one in the Fe(phen)2(NCS)2crystal, smooth one

in the Fe(btz)2(NCS)2crystal, and the transition with

hysteresis in the Fe(bpz)2(bipy) crystal. Finally, the crys-

tallographic data (including the molecular geometries)

for both HS and LS forms of these three substances are

available in the literature.

The energy of van der Waals interactions was described

by the Lennard-Jones (6-12) potential with the parame-

ters of the ”Universal Force Field” (UFF) parameteriza-

tion [37] and by the Buckingham (6-exp) potential with

the parameters provided in [36] (see Tables I, II). In the

latter case the parameters for the C···H, N···H, S···H,

C···N, and S···C interactions are given in [36] explicitly

and there is no need to use eqs. (5) and (6). Unfortu-

5

FIG. 3: Molecular structure of Fe(btz)2(NCS)2.

FIG. 4: Molecular structure of Fe(bpz)2(bipy).

nately, the system of parameters [36] for the (6-exp) po-

tential has not been extended to boron. Hence we took

the minimum depth and the interatomic separation at

the minimum for the B···B pair from [38], estimated the

corresponding A,Band Cparameters and found the pa-

rameters for the B···H, B···C, B···N, and B···S pairs

following eqs. (5) and (6). The parameters of the (6-exp)

potential for pairs involving Fe atom(s) are not deter-

mined, but they are immaterial in the present context,

and we set them to be equal to zero.

The MOLCRYST program suite [39] capable of calcu-

TABLE I: Parameters of the Lennard-Jones (6-12) potential

[37] used in the calculations.

H B C N S Fe

A, kcal·˚

A6/mol 50.9 1668 685 332 2365 15.9

B, 107·kcal·˚

A12/mol 0.147 38.6 11.2 3.99 51 0.048

lation and minimization of molecular crystals energy and

enthalpy with use of the Lennard-Jones and Bucking-

ham atom-atom potentials was employed. This program

has been thoroughly tested on the examples of molecu-

lar crystals of aromatic hydrocarbons. The geometries

of HS and LS forms of the complexes were taken from

experiments [40, 41, 42] and were assumed to be ﬁxed

(frozen) throughout the modeling. The validity of the

rigid-body approximation can be tested [43] and the anal-

ysis of the diﬀerence vibrational parameters for an SC

crystal demonstrated [44] that the non-rigidity is rela-

tively small for both HS and LS forms.

When calculating the energy according to eq. (2), we

restricted ourselves to summation over three layers of

unit cells around the central ”0-th” unit cell. In other

words, only those r= (ra, rb, rc) were included into the

sum, for which |ra| ≤ 3, |rb| ≤ 3, and |rc| ≤ 3. It was

found that extending this limit to 4 or more layers does

not aﬀect the ﬁnal result for the energy or enthalpy (the

diﬀerences are less than 0.01 kcal/mol). As for the equi-

librium values of the lattice parameters, their values are

stable (within 0.1%) already with one layer of the sur-

rounding unit cells (those adjacent to the ”0-th” cell)

included into the summation.

To ﬁnd the equilibrium values of the lattice parame-

ters, positions of the centers of masses (CM), and the

rotation angles of molecules in the unit cell, minimiza-

tion of the enthalpies of six pure crystals (three HS

and three LS) was performed. Pressure was set to be

1 atm. In all the cases the experimental crystallographic

data were taken as initial approximations. At the ﬁrst

stage we minimized the enthalpy as a function of ﬁve or

six parameters (a,b,c, one non-trivial rotation angle,

and one non-trivial CM coordinate in the cases of the

Fe(phen)2(NCS)2and Fe(btz)2(NCS)2crystals; the same

plus the unit cell angle βin the case of the Fe(bpz)2(bipy)

crystals), preserving the symmetry of the crystal (Pbcn,

Pbcn and C2/c correspondingly); after that we checked

that the ﬁnal point of the previous step is the global min-

imum, allowing for variation of all 27 parameters (a,b,

c, three unit cell angles, three rotation angles for each of

four molecules in the unit cell, three CM position coordi-

nates for three out of four molecules in the unit cell; the

fourth molecule position is not independent due to the

crystal translational symmetry). The optimized struc-

tures are shown on Figs. 5-7.

The enthalpy was calculated as the sum of the internal

energy and the product of the pressure and the volume of

the crystal with 1 mole of molecules. In all the cases the

internal energies were found to be about −50 kcal/mol

6

TABLE II: Parameters of the Buckingham (6-exp) potential [36] used in the calculations.

H· · · H C···H N···H S···H C· · · C C· · · N S···C N···N S···S B· · · B [38]

A, kcal·˚

A6/mol 26.1 113 120 279 578 667 1504 691 2571 3.688

B, 103·kcal/mol 5.774 28.87 54.56 64.19 54.05 117.47 126.46 87.3 259.96 19.84

C, ˚

A−14.01 4.10 4.52 4.03 3.47 3.86 3.41 3.65 3.52 6.82

Note: parameters for the S· · · N interaction and interactions of

B with other atoms were calculated according to the superposition

approximation (5) and (6).

FIG. 5: Crystal structure of Fe(phen)2(NCS)2.

FIG. 6: Crystal structure of Fe(btz)2(NCS)2.

FIG. 7: Crystal structure of Fe(bpz)2(bipy).

relative to the isolated molecules. The experimental data

to verify this result are not available. However, the en-

ergy magnitude is quite reasonable in comparison with

the available data on organic molecular crystals [33, 45],

taking into consideration that the numbers of interatomic

contacts per molecule in the crystals under study are a

few times higher than those in ordinary organic crystals.

The diﬀerences between the internal energies and the en-

thalpies in all the cases at 1 atm are rather small, less

than 0.01 kcal/mol, which is not surprising, since we deal

with solid substances.

As mentioned above, the (6-exp) potential parameter-

ization from [36] implicitly includes the entropy contri-

bution since it was ﬁtted to reproduce the room temper-

ature geometries of crystals by minimization of the inter-

nal energy rather than the Gibbs energy. So the results of

our calculation with the Buckingham potential should be

compared with the room temperature experimental data.

The matter is not so clear in the case of the UFF parame-

ter system [37]. The authors introduce their parameters

of the van der Waals interaction explicitly referring to

ionization potentials, polarizabilities, and Hartree-Fock

calculations, so these parameters seem to be providing

unadjusted estimates of the internal energy. Neverthe-

7

less, direct comparisons of the numbers produced with

their empirical parameters and experimental geometries

are widely used. Strictly speaking, we do not have suﬃ-

cient information to judge whether our results obtained

with this (6-12) potential describe physical properties for

the absolute zero temperature or for the room tempera-

ture. However, comparing the experimental data on the

lattice parameters of Fe(phen)2(NCS)2at 15 K, 32 K, 130

K and 298 K [46] with the results of our calculations, we

can see that the latter are somewhat closer to the high-

temperature values of the lattice parameters, rather than

to the low-temperature ones.

The room temperature crystallographic data are avail-

able only for the HS crystals. As for the LS crystals,

we need to extrapolate their experimental lattice param-

eters to the room temperatures to make the comparison

with the results of our calculations possible. This is es-

pecially important for the analysis of the changes of the

lattice parameters ∆V, ∆a, ∆b, ∆c,etc. in the course

of the spin transition, otherwise the calculated experi-

mental values would include not only the contribution

of the spin transition itself, but also of thermal expan-

sion of the crystal. In the cases of the Fe(phen)2(NCS)2

and Fe(btz)2(NCS)2compounds, the dependences of the

V,a,b,cparameters and the HS molecules fraction x

(from the magnetic susceptibility data) on temperature

are known in the range from ca. 130 K to 293 K [41]

(each series consists of 22-25 observations). In a linear

approximation,

V(T, x(T)) = (1 −x) (Vo,LS +κV,LS (T−To))

+x(Vo,HS +κV ,HS (T−To)) ,(8)

and similarly for the a,band cparameters. We deter-

mined the coeﬃcients Vo,LS ,κV,LS,Vo,HS ,κV ,HS ,etc. by

the method of least squares (R2of such models are typi-

cally 0.995÷0.999), and made extrapolation of the lattice

parameters to the room temperature and unchanged frac-

tion of the HS molecules. These extrapolated values were

used for comparison with the results of the method of

atom-atom potentials. As for the Fe(bpz)2(bipy) crystal,

the lattice parameters, published in the literature, were

measured only at few temperatures [42, 47]. Thermal

coeﬃcients of expansion, calculated for the LS form on

diﬀerent temperature intervals, diﬀer signiﬁcantly, which

does not allow for a reliable extrapolation of the lattice

parameters to the room temperature. On the other hand,

high-temperature coeﬃcients of expansion are more sta-

ble. Because of these reasons, we extrapolated the HS

crystal lattice parameters to 139 K to estimate ∆V, ∆a,

∆band ∆cfree of thermal distortions, though only at

139 K. The results of the extrapolations made are used

in the next Section for comparison with calculated opti-

mal lattice parameters.

IV. RESULTS AND DISCUSSION

A. Crystal geometries

The most important for thermodynamical description

of the spin transitions characteristic of the lattice is the

unit cell volume. The estimates of this quantity obtained

by the atom-atom potentials model are given in Tables

III-V. The average error in the computed volume is 1.8%,

ranging from 0.5% to 4.0%. The Lennard-Jones and the

Buckingham potentials provide comparable levels of ac-

curacy. These numbers should be compared with the

discrepancy of 20 −24% in [20] and 1 −8% in [26] (both

calculated with the DFT method), the only analogues

published so far. At the same time one should remember

that these data include relatively small errors in the ge-

ometries of separate molecules while our calculations are

free of them because we used the experimental structures

for the molecules.

The changes of the unit cell volumes in the course of

the spin transition are relatively small diﬀerences of two

large numbers, and their correct estimation is diﬃcult.

For example, ∆Vof [Fe(pyim)2(bpy)](ClO4)2·2C2H5OH

was found to be negative [26], though all complexes stud-

ied experimentally have positive ∆V, in agreement with

the fact that the Fe−N bonds are longer in the HS com-

plexes, and thus the HS molecules should have a larger

”size”. The calculated value of ∆Vfor [Fe(trim)2]Cl2,

published in [20], has the correct sign, but the experi-

mental volumes of the LS and HS crystals are available

only for diﬀerent temperatures, which makes it impossi-

ble to compare the experimental and calculated values.

The values of ∆Vof the Fe(phen)2(NCS)2compound,

calculated by us with both Lennard-Jones and Bucking-

ham atom-atom potentials, are fairly close to the ex-

perimental values (extrapolated to the room tempera-

ture), being probably overestimated by 10 −17% (while

the uncertainty in the extrapolated experimental value

is ca. 10%). In the case of Fe(btz)2(NCS)2, the er-

rors are correspondingly about +15% and −3% for the

two potentials, while the uncertainty in the extrapolated

experimental value is ca. 7%. Finally, in the case of

Fe(bpz)2(bipy) the calculated values diﬀer from the ex-

perimental one, extrapolated to 139 K, by 4 −14%. We

would like to stress that the temperature dependence of

∆Vis much stronger, than that of the unit cell vol-

ume V. For example, the low-temperature (at 15 K)

∆V(Fe(phen)2(NCS)2) equals to 61.2˚

A3, the ∆Vvalue

extrapolated to 293 K is about 70÷79˚

A3, and the diﬀer-

ence between the experimental unit cell volume of the HS

form at 293 K and that of the LS form at 130 K is 119.1

˚

A3. The presumable errors of the atom-atom potentials

method in calculations of ∆V(ca. 5−15 ˚

A3) are compa-

rable with the uncertainties in the extrapolated estimates

for experimental values (ca. 3−9˚

A3) and much less than

the changes in the volumes of the crystals caused by tem-

perature expansion of the crystals (dozens of ˚

A3).

As for the unit cells themselves, in all the cases the

8

TABLE III: Comparison of experimental and calculated unit cell parameters for Fe(phen)2(NCS)2(at 1 atm).

a,˚

Ab,˚

Ac,˚

Aβ,◦V,˚

A3angle,◦CM y/b H, kcal/mol

The LS isomer

calc. (6-12) 13.185 9.922 17.347 90 2269.2 142.85 0.1112 -54.46

calc. (6-exp) 12.992 9.861 17.281 90 2214.0 144.48 0.1138 -54.38

calc. (6-exp) modif. 13.017 9.991 17.469 90 2271.7 144.59 0.1065 -54.04

exp. 15 K [46] 12.762 10.024 17.090 90 2186.3 143.84 0.0943 -

exp. 130 K [40] 12.770 10.090 17.222 90 2219.1 140.51 0.0925 -

exp. extrap. to 293 K 12.77 10.18 17.40 90 2259 - - -

The HS isomer

calc. (6-12) 13.525 9.910 17.583 90 2356.7 147.36 0.1071 -52.65

calc. (6-exp) 13.264 9.869 17.542 90 2296.2 149.35 0.1088 -53.80

calc. (6-exp) modif. 13.227 10.017 17.815 90 2360.4 149.19 0.0993 -52.50

exp. 15 K [46] 13.185 9.948 17.135 90 2247.5 153.84 0.0989 -

exp. 293 K [40] 13.161 10.163 17.481 90 2338.2 147.09 0.0938 -

The diﬀerence between the HS and LS isomers

calc. (6-12) 0.340 -0.012 0.236 0 87.5 4.51 -0.0040 1.81

calc. (6-exp) 0.272 0.008 0.260 0 82.2 4.87 -0.0050 0.57

calc. (6-exp) modif. 0.210 0.026 0.347 0 88.7 4.60 -0.0072 1.54

exp. extrap. to 293 K 0.39 -(0.02÷0.04) 0.05÷0.08 0 70÷79 - - -

exp. (15 K) [46] 0.423 -0.076 0.045 0 61.2 10.00 0.0045 -

Note: see detailed explanation of ”(6-exp) modif.” parameteri-

zation in Subsection (IV B).

TABLE IV: Comparison of experimental and calculated unit cell parameters for Fe(btz)2(NCS)2(at 1 atm).

a,˚

Ab,˚

Ac,˚

Aβ,◦V,˚

A3angle,◦CM y/b H, kcal/mol

The LS isomer

calc. (6-12) 13.266 10.518 16.889 90 2356.4 125.60 0.0385 -54.15

calc. (6-exp) 13.099 10.498 16.741 90 2302.1 126.45 0.0445 -57.10

calc. (6-exp) modif. 13.160 10.652 16.963 90 2377.9 127.77 0.0493 -52.55

exp. (130 K) [41] 13.055 10.650 16.672 90 2318.1 127.48 0.0421 -

exp. extrap. to 293 K 13.17 10.80 16.88 90 2397 - - -

The HS isomer

calc. (6-12) 13.242 10.724 16.947 90 2406.6 129.45 0.0451 -54.32

calc. (6-exp) 13.077 10.669 16.803 90 2344.3 130.14 0.0498 -58.20

calc. (6-exp) modif. 13.190 10.786 16.973 90 2414.5 130.77 0.0527 -53.61

exp. (293 K) [41] 13.288 10.861 16.920 90 2441.9 129.79 0.04150 -

The diﬀerence between the HS and LS isomers

calc. (6-12) -0.023 0.206 0.059 0 50.2 3.85 0.0066 -0.17

calc. (6-exp) -0.022 0.172 0.062 0 42.2 3.68 0.00535 -1.10

calc. (6-exp) modif. 0.030 0.134 0.010 0 36.6 3.00 0.0034 -1.06

exp. extrap. to 293 K 0.12 0.06 0.03÷0.04 0 42÷45 - - -

Note: see detailed explanation of ”(6-exp) modif.” parameteri-

zation in Subsection (IV B).

symmetry for the energy minimum points, according to

our calculations, coincides with the experimental one.

Orientation of a molecule in the unit cell can be charac-

terized by three angles, corresponding to the transforma-

tion of coordinates from the molecular coordinate system

(e.g. that of the principal axes of inertia tensor) to the

laboratory (or crystal) coordinate system. In all the con-

sidered cases, two of these angles have trivial values (0,

90 or 180◦); the values of the third angle, corresponding

to rotation around the C2axis of the molecule, are given

in Tables III-V. The same is true for the CM positions

of molecules within a unit cell. Two parameters out of

three for each molecule are trivial (0, 1/4, 1/2, or 3/4

of the corresponding translation period). The remaining

parameter (corresponding to the ycoordinate in the units

of b) is given in Tables III-V as well. One can see that

the calculated values are fairly close to the experimental

ones both for the rotation angles and the CM positions.

The discrepancy between the calculated and experi-

mental values of the lattice parameters a,b,cis in the

0.1% to 3.2% range, on average being equal to 1.3% for

the (6-12) potential and 1.4% for the (6-exp) potential.

9

TABLE V: Comparison of experimental and calculated unit cell parameters for Fe(bpz)2(bipy) (at 1 atm).

a,˚

Ab,˚

Ac,˚

Aβ,◦V,˚

A3angle,◦CM y/b H, kcal/mol

The LS isomer

calc. (6-12) 16.319 14.840 10.685 113.97 2364 92.55 0.2699 -49.78

calc. (6-exp) 16.136 14.661 10.697 114.25 2307 91.93 0.2724 -40.22

exp. (139 K) [42] 16.086 14.855 10.812 114.18 2357 90.91 0.2754 -

The HS isomer

calc. (6-12) 16.242 15.178 10.823 113.60 2445 85.06 0.2703 -48.59

calc. (6-exp) 16.032 14.995 10.834 113.92 2381 84.48 0.2728 -39.75

exp. (293 K) [42] 16.307 15.075 11.024 114.95 2457 85.02 0.2782 -

exp. extrap. to 139 K 16.16 14.99 11.04 114.9 2426÷2429 - - -

The diﬀerence between the HS and LS isomers

calc. (6-12) -0.077 0.338 0.138 -0.37 81 -7.49 0.0004 1.19

calc. (6-exp) -0.104 0.334 0.137 -0.33 74 -7.45 0.0004 0.47

exp. extrap. to 139 K 0.07 0.14 0.23 0.7 69÷72 - - -

exp. (30 K) [47] -0.076 0.347 0.219 1.09 71.2 - - -

As for the changes of these parameters in the course of the

spin transition, in most cases the results predicted by the

method of atom-atom potentials are in good agreement

with the experimental data (the errors are ca. 0.05 −0.1

˚

A). What is especially impressing is that the method is

capable of reproducing decrease of some lattice periods

in the course of the spin transition, which may happen in

spite of the overall increase of the unit cell volume (the

parameter bof the Fe(phen)2(NCS)2crystal, the param-

eter aof the Fe(bpz)2(bipy) crystal). However, we have

three problematic cases: the variation of the parameter c

of the Fe(phen)2(NCS)2crystal (underestimated by the

factor of 3÷5 times), and the variation of the parame-

ters aand bof the Fe(btz)2(NCS)2crystal (wrong sign

of the result for aand underestimation by the factor of

3÷4 times for b). It is especially important that both

Lennard-Jones and Buckingham potentials yield close re-

sults. Trying to ﬁnd an explanation for these errors, we

noted that these three parameters are most sensitive to

temperature changes. For example, the poorly predicted

∆cof the Fe(phen)2(NCS)2crystal (calculated from the c

values extrapolated to the same temperature) changes in

relative terms by 0.4% per 100 K, while both ∆aand ∆b

– only by 0.2% per 100 K. Similarly, ∆a, ∆band ∆cof

the Fe(btz)2(NCS)2crystal decrease by 0.7%, 1.1% and

0.6% per 100 K. This allows us to suggest that omission

of the explicit treatment of the entropy contribution to

the Gibbs energy, and thus the uncertainty in renormal-

ization of the empirical parameters of the potentials, is

one of the main sources of errors in the method in its

current form, even if it is partially compensated by data

correction for the thermal expansion.

Another possible explanation (which does not exclude

the previous one) is that some speciﬁc interactions take

place in these crystals, diﬀerent from those in ordinary

organic crystals used for ﬁtting the presumably pure van

der Waals interaction parameters. In this case, the per-

formance of the method can be improved by correcting

the parameters of atom-atom interactions.

FIG. 8: Contacts in the crystal of Fe(phen)2(NCS)2.

B. Contacts analysis and parameters adjustment

To study this problem and yet further improve the

performance of the method, we analyzed intermolecular

contacts in the crystals, comparing atom-atom distances

found in the experimental studies with those optimized

with the parameters from [36, 37]. The lists of the short-

est atom-atom contacts (we selected those separated by

less than the sum of the corresponding van der Waals

radii) are given in Tables VI-VIII and they are also de-

picted on Figs. 8-10. It is important to note that in all

three materials the spin transition does not much aﬀect

the picture of intermolecular contacts. In other words,

the shortest contacts in a LS crystal are also short (typi-

cally, though not always, shorter than the sum of the van

der Waals radii) contacts in its HS form, and vice versa.

First of all, one can see that in most cases the shortest

10

TABLE VI: Shortest intermolecular contacts in the LS and

HS crystals of Fe(phen)2(NCS)2: interatomic distances (˚

A)

determined experimentally (exp.) or calculated with (6-12)

and (6-exp) potentials and the sum of the van der Waals radii

(vdW) of the atoms [36].

Pair R(exp.) R(vdW) R(6-12) R(6-exp)

The LS isomer

H···H 2.093 2.34 2.374 2.289

S···C 3.314 3.55 3.370 3.341

C· · · H 2.589, 2.784 2.92 2.620, 2.801 2.520, 2.866

S···H 2.832, 2.891, 2.97 3.162, 3.311, 3.185, 3.419,

2.911, 2.951 3.052, 2.907 2.983, 2.915

The HS isomer

H···H 2.211 2.34 2.411 2.307

S···C 3.357 3.55 3.345 3.339

C· · · H 2.570, 2.750 2.92 2.635, 2.872 2.509, 2.792

S···H 2.941 2.97 3.121 3.156

TABLE VII: Shortest intermolecular contacts in the LS and

HS crystals of Fe(btz)2(NCS)2: interatomic distances (˚

A) de-

termined experimentally (exp.) or calculated with (6-12) and

(6-exp) potentials and the sum of the van der Waals radii

(vdW) of the atoms [36].

Pair R(exp.) R(vdW) R(6-12) R(6-exp)

The LS isomer

S···C 3.275 3.55 3.352 3.333

S···H 2.706, 2.743, 2.97 2.886, 2.840, 2.755, 2.741,

2.942 2.849 2.907

C· · · H 2.811 2.92 2.931 2.805

C· · · C 3.453 3.50 3.668 3.570

The HS isomer

S···C 3.351 3.55 3.401 3.384

S···H 2.893, 2.925 2.97 2.832, 2.904 2.770, 2.805

C· · · H 2.848, 2.888 2.92 2.802, 2.851 2.693, 2.755

TABLE VIII: Shortest intermolecular contacts in the LS and

HS crystals of Fe(bpz)2(bipy): interatomic distances (˚

A) de-

termined experimentally (exp.) or calculated with (6-12) and

(6-exp) potentials and the sum of the van der Waals radii

(vdW) of the atoms [36].

Pair R(exp.) R(vdW) R(6-12) R(6-exp)

The LS isomer

C· · · H 2.655, 2.658, 2.92 2.732, 2.716, 2.651, 2.606,

2.689, 2.817, 2.815, 2.877, 2.720, 2.763,

2.879, 2.912 2.895, 2.811 2.835, 2.818

H···H 2.283, 2.332 2.34 2.387, 2.390 2.301, 2.333

C· · · C 3.368, 3.374 3.50 3.327, 2.720 3.387, 3.412

The HS isomer

C· · · H 2.579, 2.719, 2.92 2.728, 2.739, 2.622, 2.657,

2.782, 2.813, 3.002, 2.809, 2.892, 2.707

2.878, 2.900 2.838, 3.189 2.769, 3.040

H···H 2.318 2.34 2.477 2.336

C· · · C 3.360, 3.420 3.50 3.250, 3.304 3.194, 3.236

N···H 2.606 2.67 2.786 2.642

FIG. 9: Contacts in the crystal of Fe(btz)2(NCS)2.

FIG. 10: Contacts in the crystal of Fe(bpz)2(bipy).

contacts involve hydrogen atoms (S···H, C···H, N···H,

or H···H). It is well known that coordinates of the hydro-

gen atoms determined from X-ray diﬀraction may be sub-

ject to signiﬁcant errors unless tricks of crystallographic

computing are used. While the X–H bond length is no-

toriously underestimated due to the shift of the bonding

electron pair towards the nonmetal X atom, the position

of the X–H vector in three-dimensional space is correctly

found. Thus, the H atom should ”ride” on the nonmetal

atom with a ﬁxed bond length (e.g., C–H = 1.09 ˚

A, N–H

= 1.01 ˚

A, O–H = 0.96 ˚

A). Because the crystal structures

under study seemingly did not proﬁt from such ”riding”

H atoms approach, we may suggest that one of the main

sources of mistakes in our results is the uncertainty in

11

the H positions. This also indicates that in the future

research, when taking into consideration intramolecular

degrees of freedom, one should take possible deformations

of the C−H bonds into account.

It is reasonable to suggest that the poorly described

atom-atom interactions will be at the top of the list of the

highest atom-atom repulsion energies. Indeed, if some in-

teratomic distance increases when the system goes from

the experimental conﬁguration to the optimized one,

the repulsion between the corresponding atoms weakens.

Thus one can expect that the intensity of that interac-

tion is overestimated, since such relaxation does not oc-

cur in experiment. A similar reasoning applies to the

strongest attractions as well. In practice, the picture is

not so clear because molecules in organic crystals typi-

cally have numerous contacts between various atoms. By

analyzing the crystals formed by the Fe(phen)2(NCS)2

or Fe(btz)2(NCS)2molecules we found that the sulphur

atoms play very important role in the intermolecular in-

teractions (Fe(bpz)2(bipy) does not contain sulphur). As

our calculations demonstrated, the S atoms participate in

many close contacts with other atoms, thus providing a

signiﬁcant contribution to the repulsion within crystals;

at the same time, their contributions to the attraction are

also dominant (attraction energies of various S···S pairs

are the largest by absolute value in these crystals; as for

the S···C contacts, in some of them attraction is also

very strong, while some other S···C contacts are among

extreme cases of repulsion).

The Fe(phen)2(NCS)2molecule has S atoms only in

the NCS groups while in the case of Fe(btz)2(NCS)2the

chelating ligand also contains the S atoms. We found that

the S atoms of both types participate in the contacts with

extremal values of the energy. Taking into consideration

that the parameterization of the van der Waals energy of

the S···X contacts (X = S, C, H) is not so well studied as

compared to the C···C, C···H, and H···H interactions,

and that some involvement of the lone pairs and vacant

d-orbitals of the S atoms can complicate the approxima-

tion of the S···X interactions by the center-symmetric

atom-atom contributions, we suggest that improving the

treatment of S···X (X = S, C, H) interaction energies

may be another way of developing a better model of the

atom-atom potentials for molecular crystals undergoing

spin transitions. For example, the shortest C···S dis-

tances are found to be ca. 0.2˚

A shorter than the sum of

the van der Waals radii of the atoms. In the case of the

S···H contacts, this contraction may reach even 0.27 ˚

A.

Thus it is reasonable to suggest that due to some speciﬁc

interactions, the optimal interatomic distances involving

S atoms may be lower than determined by the standard

parameterization.

For that reason we adjusted the parameters for the

S···C, S···S, S···H interactions in order to improve

agreement between the experimental and modeled crys-

tal lattice parameters. However, improvement of some

of the calculated lattice parameters often increases the

discrepancies for others. The situation is especially diﬃ-

cult for the diﬀerences between the spin isomers ∆a, ∆b,

∆c. Variation of the parameters for atom-atom contacts

similarly aﬀects the lattice parameters in the LS and HS

crystals, hence the resulting change in those parameters

is small and it can be only calculated rather than pre-

dicted from any physical or geometrical reasoning.

We performed a systematic quantitative study of the

inﬂuence of the interaction parameters on the equilib-

rium conﬁgurations of the crystals. To get the general

understanding of this issue, we optimized the crystals of

Fe(btz)2(NCS)2with the interaction parameters slightly

modiﬁed. We increased, one by one, parameters for each

pair of atoms (the well depth and the equilibrium sep-

aration) by 5% to estimate numerically the sensitivity

of the energy contributions to the potential parameters.

The choice of the Fe(btz)2(NCS)2crystals was suggested

by the fact that it is poorly described with the original

parameterization: there are qualitative discrepancies for

the changes in two out of three lattice parameters (∆a

and ∆b). Also we limited the consideration to the (6-

exp) potential only, since the parameters of interaction

between atoms of diﬀerent elements were determined ex-

plicitly [36] without any reference to the superposition

approximation (except for the N···S pairs making little

diﬀerence for the systems studied), and thus they can be

varied separately.

Changes in the optimal lattice parameters δa,δb,δc,

caused by 5% variations of each parameter of the atom-

atom interaction energy, are given in Table IX. The table

also speciﬁes the diﬀerences between the experimental

values of the lattice parameters and those calculated with

the initial parameters of [36] (exp.). As one can see from

the numbers, most of the interaction parameters very

slightly aﬀect the optimal conﬁguration of the crystals.

Corrections caused by the well depth changes by 5% are

a hundred times smaller than the diﬀerence between the

experimental and calculated lattice parameters, leaving

no hope to reduce the discrepancy by ﬁtting the well

depths within reasonable frames. The same applies to

most of the atom-atom equilibrium separations on the

corresponding interaction energy curves (C···C, H···H,

N···N, N···H, etc.), though in this case the changes in

the lattice parameters caused by a 5% increase of the

distances are only tens times smaller than the required

scale of correction.

Only three parameters signiﬁcantly aﬀect the optimal

structure of the crystal: the equilibrium separations for

the S···C, S···H, and S···S pairs (listed in the order of

decreasing eﬀect). This conﬁrms our assumption made

above on the basis of the interatomic contacts analysis

that the contacts involving the S atoms need an improved

treatment ﬁrst of all.

To do this, we performed a numerical minimization

of the sum of squares of residuals fas a function of the

equilibrium separations rfor the S···C, S···H, and S···S

12

TABLE IX: Changes in the optimal lattice parameters a,band c(10−4˚

A) of Fe(btz)2(NCS)2caused by 5% increase of the

(6-exp) potential parameters.

The LS isomer The HS isomer The HS/LS diﬀerence

Pair δa δb δc δa δb δc δ∆a δ∆b δ∆c

well depth

H···H 12 14 -3 28 -1 -5 16 -15 -2

C· · · H -14 -11 1 6 -9 11 20 2 10

N···H -13 -5 -13 -15 -6 -13 -2 -1 0

S···H 126 24 6 85 41 -10 -41 17 -16

C· · · C -28 -20 -19 -27 -20 -14 1 0 5

N···C -25 -10 -31 -29 -12 -27 -4 -2 4

S···C -25 27 51 -11 18 31 14 -9 -20

N···N -11 -4 -1 -13 -4 3 -2 0 4

S···S -23 -13 4 -26 -8 19 -3 5 15

equilibrium distance

H···H 251 216 5 483 24 -33 232 -192 -38

C· · · H 396 98 365 755 74 421 359 -24 56

N···H -60 -29 -48 -41 -23 -21 19 6 27

S···H 2084 828 700 1741 991 533 -343 163 -167

C· · · C 168 -55 269 277 -61 263 109 -6 -6

N···C -148 -83 -70 -137 -90 -2 11 -7 68

S···C 714 1131 2075 894 958 1821 180 -173 -254

N···N -80 -27 271 -109 -18 305 -29 9 34

S···S 11 -58 1116 -22 28 1226 -33 86 110

exp. 705 2930 1354 2098 1843 1131 1393 -1087 -223

atoms:

f= (aLS,calc −aLS,exp)2+

(bLS,calc −bLS,exp)2+ (cLS,calc −cLS,exp)2+

(aHS,calc −aH S,exp)2+ (bH S,calc −bHS,exp)2+

(cHS,calc −cH S,exp)2+ (∆acalc −∆aexp )2+

(∆bcalc −∆bexp)2+ (∆ccalc −∆cexp )2,

(9)

where acalc,bcalc ,ccalc stand for the optimal lattice pa-

rameters calculated with the Buckingham potential pa-

rameterization diﬀerent from one in [36] by rSC ,rS H and

rSS separations, and aexp ,bexp ,cexp are the experimen-

tal lattice parameters (extrapolated to 293 K, if neces-

sary). The result is that the equilibrium separation of

the S···C contact should be increased by 6.6% (+0.26

˚

A), that of the S···H one – decreased by 2.3% (−0.08

˚

A), and that of the S···S one – decreased by 15% (−0.57

˚

A). Optimal values of the crystal lattice parameters of

the Fe(btz)2(NCS)2crystal, calculated with the adjusted

parameters, are given in Table IV (calc. (6-exp) modif.).

After ﬁtting the S···C, S···H and S···S equilibrium

distances all six unit cell dimensions became closer to the

experimental values: the error in b(LS) decreased from

−0.32 ˚

A to −0.15 ˚

A, the error in a(HS) – from −0.21

˚

A to −0.10 ˚

A, in b(HS) – from −0.18 ˚

A to −0.08 ˚

A,

and so on. The values of the CM position and the ro-

tation angle change, by contrast, insigniﬁcantly, in spite

of the fact that they were not included in the treatment

by the least squares method. The performance of the

model in predicting the quantities ∆a, ∆band ∆calso

signiﬁcantly improved. The value of ∆ashifted towards

the experimental one and changed the sign to the correct

one (positive instead of negative). ∆bmoved towards the

experimental value, though this correction was only 1/3

of the initial discrepancy. Finally, the ∆cvalue shifted in

the correct direction, but this time the change was even

larger than the required one. An attempt to improve

yet further the relation between the predicted and actual

values of ∆a, ∆b, ∆cand ∆Vby another modiﬁcation

of the atom-atom interaction parameters (for example,

by increasing the weights ascribed to the corresponding

squares in the treatment by the least squares method)

leads to catastrophic results for a,band cof the pure

LS and HS crystals: a tiny improvement by 0.01 ˚

A in

∆a, ∆b, ∆csimultaneously leads to the growth in the

discrepancies in a,band cby ca. 0.1˚

A.

We applied the same modiﬁed parameterization to the

LS and HS crystals of Fe(phen)2(NCS)2. The results

are given in Table III (calc. (6-exp) modif.). In regard

to the unit cell dimensions of the LS and HS crystals,

the modiﬁcation improved 4 out of 6 periods, especially

those poorly described by the original parameterization:

the error in b(LS) decreased from −0.32 ˚

A to −0.19 ˚

A,

in b(HS) – from −0.29 ˚

A to −0.15 ˚

A. Error in unit cell

volumes Vdecreased 2-3 times. At the same time, a sig-

niﬁcant error in the cvalue for the HS form appeared

(0.33 ˚

A instead of 0.06 ˚

A). As for the changes in the lat-

tice parameters, their values became more distant from

the experimental values by 0.01 −0.09 ˚

A. To sum up, the

suggested modiﬁcation generally improves the results of

the model for both S-containing materials, though it fails

to eliminate the errors completely.

13

TABLE X: Comparison of experimental and calculated unit

cell parameters for Fe(phen)2(NCS)2(at 1 GPa).

System a,˚

Ab,˚

Ac,˚

AV,˚

A3

LS(1 GPa) calc. (6-12) 13.060 9.773 17.183 2193.2

calc. (6-exp) 12.838 9.700 17.089 2128.0

exp. (298 K) [48] 12.656 9.848 16.597 2068.6

Diﬀerence, calc. (6-12) -0.465 -0.137 -0.399 -163.5

LS(1 GPa)/ calc. (6-exp) -0.426 -0.169 -0.453 -168.2

HS(1 atm) exp. -0.505 -0.315 -0.884 -269.6

Diﬀerence, calc. (6-12) -0.125 -0.149 -0.163 -76.0

LS(1 GPa)/ calc. (6-exp) -0.155 -0.161 -0.193 -86.0

LS(1 atm) exp. -0.114 -0.242 -0.625 -150.5

The adjustment of interaction parameters, described

in this Subsection, does not claim to produce a new sys-

tem of atom-atom parameters. We undertook it just to

estimate how much improvement in the performance of

the method at the expense of minor changes within the

same theoretical paradigm may be done, and to illustrate

that accurate treatment of the intermolecular contacts

involving sulphur atoms are of primary importance for

modeling the spin transition in S-containing materials.

C. Contributions of intermolecular interactions to

enthalpy

The results described in the previous Sections demon-

strate that the method of atom-atom potentials is capa-

ble of modeling intermolecular interactions and reproduc-

ing experimental data on the geometry of the unit cells.

This allows us to go on to estimate the contributions of

the van der Waals intermolecular forces to the energy

(enthalpy) of the spin transitions, which cannot be ex-

tracted from experimental data. The results are given in

the last columns of Tables III-V. First of all, one can see

that this contribution may be either positive or negative,

which corroborates the theoretical conclusion of [12]. An-

other important point is that the lattice contribution to

the enthalpy of the spin transition is comparable with

its total value. Though the estimates obtained with the

Lennard-Jones and Buckingham potentials are somewhat

diﬀerent, the general picture is the same. For example,

in the case of the Fe(phen)2(NCS)2crystal we found this

component to be equal to +1.81 kcal/mol (6-12) or +0.57

kcal/mol (6-exp) or +1.54 kcal/mol (6-exp modiﬁed),

while the total experimental enthalpy (from the calori-

metrical data) is +2.05 kcal/mol [9]. It means that one

cannot neglect intermolecular interactions in calculating

thermodynamical characteristics of the spin transitions

in molecular crystals. (This conclusion was also made

in [20] on the basis of DFT calculations; however, the

contribution of intermolecular interactions, which can be

extracted from their results and ranging from 2 to 23

kcal/mol, seems to be strongly overestimated).

TABLE XI: Comparison of experimental and calculated unit

cell parameters for Fe(btz)2(NCS)2(at 1 GPa).

System a,˚

Ab,˚

Ac,˚

AV,˚

A3

LS(1 GPa) calc. (6-12) 13.072 10.410 16.640 2264.4

calc. (6-exp) 12.877 10.380 16.502 2205.7

exp. (298 K) [48] 12.839 10.454 16.362 2196.1

Diﬀerence, calc. (6-12) -0.171 -0.313 -0.307 -142.2

LS(1 GPa)/ calc. (6-exp) -0.366 -0.343 -0.445 -200.9

HS(1 atm) exp. -0.449 -0.407 -0.558 -245.8

Diﬀerence, calc. (6-12) -0.194 -0.107 -0.249 -92.0

LS(1 GPa)/ calc. (6-exp) -0.389 -0.138 -0.386 -150.7

LS(1 atm) exp. -0.216 -0.196 -0.310 -122.0

D. Pressure eﬀects

Finally, we studied behavior of the crystal lattice

parameters under the external hydrostatic pressure.

Calculations were made for the Fe(phen)2(NCS)2and

Fe(btz)2(NCS)2compounds, since the experimental data

on the pressure eﬀects on the spin transition are available

only for these crystals [48]. We performed minimization

of the enthalpies as a function of the lattice parameters,

CM positions of the molecules, and their rotation an-

gles, at two values of the pressure. The external pressure

was accounted for by the P V term in the function to be

minimized. The starting points of optimization were the

experimental geometries. As previously, at the ﬁrst step

we minimized enthalpy as a function of ﬁve parameters,

preserving the symmetry of the crystal, and after that

checked that we get the global minima by allowing vari-

ation of all 27 parameters mentioned above. The results

for the lattice parameters of the LS forms at 1 GPa and

298 K are given in Tables X, XI, and the compressibil-

ity coeﬃcients at 1 atm and 1 GPa – in Table XII. As

one can see from the tables, the high-pressure lattice pa-

rameters are very well reproduced (errors are below 4%),

though less accurately than those for the low pressure.

As for the compressibility coeﬃcients, in all the cases the

correspondence between the calculated and experimental

values is qualitative (the compressibility coeﬃcients are

underestimated by a factor of 1.5÷3 as compared to the

experimental values). One can see that the Buckingham

potential produces better values than the Lennard-Jones

one. Taking into consideration that the (6-exp) parame-

terization used in the present study is based only on the

crystal structures measured at 1 atm, and very few con-

tacts in those structures have distances corresponding to

the repulsive branch of the potential (see Figs. 5-7 of Ref.

[36]), we conclude that our results for the high-pressure

structures are better than one could expect.

V. CONCLUSION

Numerical modeling of the spin transitions in molec-

ular crystals is important from practical and theoretical

14

TABLE XII: Comparison of experimental and calculated com-

pressibility coeﬃcients (in 10−1GPa−1) for Fe(phen)2(NCS)2

and Fe(btz)2(NCS)2.

System kakbkckV

Fe(phen)2(NCS)2calc. (6-12) 0.14 0.22 0.14 0.50

HS, 1 atm calc. (6-exp) 0.16 0.22 0.17 0.56

exp. (298 K) [48] 0.21 0.33 0.53 1.07

Fe(phen)2(NCS)2calc. (6-12) 0.07 0.12 0.07 0.26

LS, 1 GPa calc. (6-exp) 0.09 0.13 0.09 0.30

exp. (298 K) [48] 0.16 0.28 0.38 0.82

Fe(btz)2(NCS)2calc. (6-12) 0.22 0.14 0.20 0.56

HS, 1 atm calc. (6-exp) 0.25 0.14 0.18 0.57

exp. (298 K) [48] 0.41 0.43 0.37 1.21

Fe(btz)2(NCS)2calc. (6-12) 0.11 0.08 0.11 0.30

LS, 1 GPa calc. (6-exp) 0.13 0.09 0.11 0.34

exp. (298 K) [48] 0.28 0.33 0.28 0.89

viewpoints. There is no alternative to calculations explic-

itly taking into account the composition and structure

of interacting molecules (instead of representing them

by spheres, or ellipsoids, or octahedra etc., immersed in

an elastic media), both for the purposes of theoretical

study of the transition mechanisms and for prediction

of phenomenological parameters for macroscopic models.

Meanwhile, the modern quantum chemical methods are

hardly applicable to such objects, because their accuracy

level is not suﬃcient to calculate the required values (for

example, enthalpies of the spin transitions).

We demonstrate that the atom-atom potentials can be

used for analysis of intermolecular contributions to the

structure and energy of spin-active crystals. Indeed, in-

termolecular contacts in these crystals are those between

the C, H, N, S, etc. organogenic atoms, while the metal

atom and its bonds with the donor atoms of the ligands

are hidden inside the complex. As a consequence of that,

the van der Waals interactions in the spin-active crystals

can be approximated similarly to those in ordinary or-

ganic molecular crystals. In the present paper we checked

for the ﬁrst time the possibility to use the atom-atom po-

tentials method for this class of objects.

In all the cases the symmetry groups of optimized

crystals coincided with those found in experiment; the

unit cell volumes were calculated with the precision of

0.5−4%. Errors in the predicted lattice parameters did

not exceed 3% at the ambient pressure and 4% at 1 GPa.

Direction (sign) and magnitude of the changes of the lat-

tice parameters and molecules positions in the unit cell in

the course of the temperature- and pressure-driven spin

transitions were reproduced correctly. The compress-

ibility coeﬃcients are in a qualitative agreement with

their experimental values, although 1.5÷3 times underes-

timated. Thus the accuracy of the method of atom-atom

potentials is quite suﬃcient at the present level of the

theory. We attempted to improve the parameterization,

which is based on the (6-exp) parameters from [36] and

diﬀers from it in the S···H, S···C and S···S equilibrium

separations. The results of this ﬁtting of the parameters

demonstrate that the performance of the method can be

signiﬁcantly improved by adjustment to the speciﬁc cases

under study, and that the energy of interactions involving

sulphur atoms is the crucial term for adequate treatment

of spin transitions in the crystals studied.

Our study shows that any reliable calculation of spin

transition parameters (such as transition enthalpy) must

take into account intermolecular interactions. According

to our estimates for the Fe(phen)2(NCS)2crystal, the

van der Waals contribution to the transition enthalpy is

about +0.6÷1.8 kcal/mol (as compared with the total

transition enthalpy of +2.05 kcal/mol).

We believe that the accuracy of the method used in this

paper is limited by (i) implicit treatment of the entropy

eﬀects (through ﬁtting the interaction parameters, rather

than explicit calculation of frequencies of intermolecular

oscillations); (ii) uncertainty of the H atoms positions

in the experimental X-ray structures; (iii) description of

the energy of interactions involving the S atoms (due to

possible involvement of lone pairs and vacant d-orbitals

of the sulphur atoms). Nevertheless, even the current

level of precision is enough for using the method of atom-

atom potentials to study the spin transitions in molecular

crystals.

Acknowledgments

This work has been supported by the RFBR through

the grant No 07-03-01128. The ﬁnancial support of this

work through the JARA-SIM research project ”Local

Electron States in Molecules and Solids” is gratefully

acknowledged. The authors are thankful to Prof. J.

´

Angy´an of Universit´e Henri Poincar´e, Nancy for valu-

able discussion and sending his results prior to publica-

tion. Valuable discussions with Prof. A.V. Yatsenko and

Dr. N.V. Goulioukina of the Chemistry Department of

Moscow State University are gratefully acknowledged.

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