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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 discussed. 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 molecular crystals undergoing the spin transitions.
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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,4H-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···HC 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 FeN 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
ααmmrr
Eαα(R(ααmmrr)) ,(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 rdepending 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
ααmmr
Eαα(R(ααmmr0)) ,(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+BααeCααR,(3)
and the Lennard-Jones potential (6-n):
Eαα(R) = Aαα
R6+Bαα
Rn.(4)
In the above formulae the Aααand Bααparameters
for the interaction between atoms of different types are
often calculated as the geometric mean values of the cor-
responding homogeneous interaction parameters:
Aαα=pAααAαα, Bαα=pBααBαα,(5)
while the Cααparameter is approximated in a similar
way as an arithmetic mean value:
Cαα=1
2(Cαα +Cαα).(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, ˚
A14.01 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 (TTo))
+x(Vo,HS +κV ,HS (TTo)) ,(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 FeN 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. 515 ˚
A3) are compa-
rable with the uncertainties in the extrapolated estimates
for experimental values (ca. 39˚
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 CH 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(104˚
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 δa δb δc
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 101GPa1) 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
0.54%. 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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Ab initio computations within the density functional theory are reported for the spin cross-over complex [Fe(btz)2(NCS)2] (btz = 2.2′-bis-4.5-dihydrothiazine), where 3d6 FeII is characterized by high-spin (HS t2g4, eg2) and low-spin (LS t2g6, eg0) states. Results of infrared and Raman spectra for the isolated molecule are complemented for the crystalline solid with a full account of the electronic band structure properties: the density of states assessing the crystal field effects and the chemical bonding, assigning a specific role to the Fe-N interactions within the coordination sphere of FeII.
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Some transition-metal containing molecules can undergo a spin transition (ST) between a low-spin (LS) and a high-spin (HS) state. The main goal of this paper has been to explore the theoretical status of the equation proposed by Slichter and Drickamer, giving the free energy of an assembly of ST molecules. The parameters of this equation are the enthalpy variation, ΔH, the entropy variation, ΔS, accompanying the LS⇆HS reaction, and an enthalpy term accounting for intermolecular interactions, T. First, a ST molecule has been described by a two-level system, and the ST problem at the microscopic scale has been formulated in the form of an Ising Hamiltonian. Then, the free energy of an assembly of such molecules has been calculated, using the mean-field approximation, which has led to the equation of Slichter and Drickamer. This approach has allowed us to relate the ΔH, ΔS, and T parameters of the thermodynamical equation to molecular (microscopic) data. In particular, T has been found to be equal to 2NzJ; N is Avogadro's number, z is the number of nearest neighbors of a molecule, and J is given by [(ELH+EHL2−(ELL+EHH2]/2; ELL, EHH, ELH, andEHL are the possible values for the energy of a pair of nearest neighbor ST molecules; ELL stands when the two molecules are in the LS state, EHH when they are in the HS state, ELH and EHL when one of the molecules is in the LS state and the other in the HS state. In the mean-field approach the LS and HS molecules are randomly distributed within the crystal lattice, which is a questionable approximation when T is large. Therefor, we have explored what happens beyond the mean-field approximation, when the ST molecules are no longer randomly distributed, using Monte Carlo simulations. Two main results have emerged, namely: the occurrence of a thermal hysteresis is less probable in the non-random-distribution model, and the like-spin molecules tend to assemble in like-spin clusters. These clusters must be viewed in terms of high probability and not of static distribution. Our findings have been discussed at the light of the previous theoretical approaches and of the experimental data.
Article
In this chapter we shall give a general outline of the atom-atom potential method and discuss the ways most frequently used to derive the atom-atom-potential parameters. A large number of parameter sets, which covers the most important atomic species encountered in organic compounds, is presented. In all cases we have indicated the particular compounds and physical properties used in fitting the parameters of a potential. We hope that this material will assist the reader in choosing the set best suited to the particular problem he is interested in.
Article
Molecular mechanics force field for boron-containing compounds based on CHARMM parameters and Gillespie–Kepert (GK) model is developed. GK potential functions are applied to the coordination sphere of three-coordinated (uncharged) or four-coordinated (anionic) boron atom. The force field provides an accurate description of experimental X-ray stereochemistries in a wide range of organoboranes, organoborate complexes with polyhydroxy compounds, mixed oligomers of boric acid and borates.
Article
A new model for the first-order transitions between the low-spin and high-spin phases of molecular crystals of some transition metal complexes is proposed. Tbe abrupt spin transitions (or those exhibiting thermal hysteresis) are attributed to first-order transitions between the different crystal phases with the sublattice order. The necessary conditions for these transitions to take place are formulated in terms of intramolecular parameters of the spin transition and intermolecular interactions.
Article
A series of the d6 iron(II) complexes with bulky organic ligands (like [Fe(bipy)2(NCS)2]) can exist in two spin forms: in the low-spin (S = 0) form at low temperature and in the high-spin (S = 2) form at high temperature. In the crystal phase, the transition between these two forms may be either smooth or abrupt. Recently, the abrupt spin transitions were identified with the first-order transitions between different ordered phases occurring in the binary mixtures of the two spin forms of the complex. Here, we apply the method widely used in the field of binary metal alloys to the analysis of the spin transitions. The molecules undergoing the spin transition are modeled by octahedra of variable size which interact when they are immediate neighbors in the crystal lattice. We show that some simple assumptions concerning the intermolecular interaction and crystal geometry relaxation allows one to get the desired first-order phase transitions together with a satisfactory description for the crystal compressibility as a function of temperature. © 1996 John Wiley & Sons, Inc.