bond is based on the association between p valence orbitals of M and the 3p ... ongoing from phosphorus to bismuth showed an increase of ionic character and it might explain ... structure considered as aligned (Bi4S6)n ribbons bound to each other by the ... The electron lone pair of the M atom is localized in hybrid orbital.
Computational Study of Structural, Vibrational and Electronic Properties of the Highly
Symmetric Molecules M4S6 (M= P, As, Sb, Bi).
E. Semidalas and A. Chrissanthopoulos
Laboratory of Inorganic Chemistry, Department of Chemistry, National and Kapodistrian University of
Athens, University campus, Zografou, GR-15771, Greece.
A systematic computational investigation of the structural, electronic and vibrational properties
of the group 15 sulfides M4S6 at Td symmetry was carried out. The performance of DFT and
MP2 theoretical methods was assessed compared to the high-level CCSD method. The M-S
bond is based on the association between p valence orbitals of M and the 3p of sulfur according
to the natural population analysis. Both polarizability and polarizability volume of the cage
molecules increase as the size of the atoms increases from P to Bi. A structural ‘relaxation’
ongoing from phosphorus to bismuth showed an increase of ionic character and it might explain
the chemical instability of the heavier cage compounds. For the P4S6 molecule, the functionals
wB97XD and CAMB3LYP yielded excellent structural data, while for the heavier molecules
As4S6, Sb4S6 and Bi4S6, the M06 and M06L functionals showed high accuracy. We validated
eight functionals BP86, M06L, B3LYP, M06, Μ06-2Χ, CAMB3LYP, wB97XD, B2PLYP
which span from conventional GGA functionals to long-range corrected hybrid ones, and MP2,
CCSD ab initio methods. Experimentally, these molecules could be useful in the structural
investigation of the isolated gas phase species, besides solving complex structures of liquid,
crystalline or amorphous phases.
26 27 28
Keywords: M4S6 Td molecules, ab-initio, DFT, structural properties, electronic properties, vibrational
The M4S6 highly symmetric sulfides of group 15 elements of the periodic table (M: P, As, Sb,
Bi) are examples of prototype cage like inorganic molecules. These serve as structural building
blocks of network like condensed matter and could be used to describe and predict the
physicochemical properties of various solid- and liquid-state systems . Investigating the
properties of these molecules could reveal reaction mechanisms for the development of less
expensive catalysts as well as the formation of novel atomic-level controlled nanomaterials .
The description of the chemical bonding in the M4S6 molecules is a rather challenging task. The
knowledge of the structural parameters (bond lengths and bond angles) is inadequate to indicate
the strength of the metal-sulfur bond. It is essential to ascertain additional molecular properties
(especially in the case of more ionic systems), such as the population analysis, calculation of
the electrostatic potential surface and bond stretching frequencies [3–7]. From the
computational aspect, our conclusions are important in selecting the most accurate ‘low-cost’
level of theory in predicting the geometrical parameters and frontier orbitals, description of the
metal-sulfur bond, electrostatic potential surfaces and vibrational spectrum of metal sulfides.
The molecules M4S6 were considered as spherical top species with Td symmetry. To our
knowledge neither experimental structural data from electron diffraction studies nor vibrational
spectra of the gas phase Td sulfides have been published yet, so the existence of these species
is currently under consideration. Sulfides of group 15 of lower symmetry than Td have been
reported in the literature. Jason has suggested the possible structures of α-P4S6 (C1 symmetry),
β-P4S6 and γ-P4S6 (Cs symmetry) employing the
structurally characterized by Blachnik et al.  Furthermore, most of the arsenic sulfide
minerals are composed of cage-like molecules interacting through weak van der Waals (vdW)
forces. The uzonite mineral contains the cage molecules As4S5 . The cyclo- anion Sb4S62−
has been obtained as its PPh4+ salt, where two exocyclic Sb-S bonds and an Sb-Sb bond are
P-NMR technique . β-P4S6 has been
present . In individual Sb2S3 nanowires embedded in anodic alumina templates
piezoelectric and ferroelectric properties have been observed . Also, Sb2S3 crystals of the
urchin-like nanostructure, 3−4 μm in length and 30−150 nm in diameter have been obtained at
a mild reaction temperature . Bi2S3 has a very anisotropic one-dimensional orthorhombic
structure considered as aligned (Bi4S6)n ribbons bound to each other by the vdW forces. It has
multiple functions in solar cells, such as a sensitizer, light absorber or electron acceptor material
. In addition, the synthesis and the promising thermoelectric properties of highly oriented
bulk crystalline ingots of n-type bulk Bi2S3 doped with BiCl3 have been reported .
In this work, the formation of the M-S bond of group 15 sulfides M4S6 was characterized by
analyzing the electron density of the metal’s p valence orbitals and the 3p orbitals of sulfur
rearrangement between the two nuclei. In the literature has been reported charge transfer from
the 3p orbitals of sulfur to the p valence orbitals of M for sulfur complexes with M=Au, Ag, Cu
. It is well known that phosphorous like nitrogen forms mostly covalent bonds while
compounds of arsenic, antimony, and bismuth are characterized by more ionic bonding.
Moreover, in molecules with M-S bonds, such as [PbII(S2COEt)n]2-n (n = 1,2,3,4), Ghosh et al.
 concluded that the PbII-S bond is formed by the 6p orbitals of PbII and the 3p of the S
atoms. Both PbII and MIII have the electronic structure ns2p0 but the s orbital of M has limited
contribution to the bond. The electron lone pair of the M atom is localized in hybrid orbital
having a higher percentage of s character. Along the P4S6 to Bi4S6 series, the localization of the
lone pair is significantly increased in an orbital with a higher percentage of s character.
The adamantane structure of P4S6 has been proposed by Gimarc and Ott but without any
experimental evidence . Theoretical studies have been conducted on the Td form of the
phosphorus decasulfide P4S10 at HF/6-31G*, MP2/6-31G* and B3LYP/6-31G* levels of theory
. The first reported calculations for As4S6 at Td symmetry were carried out by Fukui et al.
They performed obsolete ASMO calculations for all valence electrons (INDO type) and
proposed that the As4S6 unit is stable and is a reasonable candidate for a structural unit in As2S3
glass . Babić et al. have made computations for As4S6 (Td) using the outdated Vosko
exchange-correlation parameterization without nonlocal terms correction and an s, p orbital
basis set without d-type functions for arsenic atoms . Their theoretical gas phase results
were compared with the experimental XPS spectrum of amorphous As2S3 solid. The geometry
and vibrations of As4S6 and Sb4S6 of C3ν symmetry have been calculated at HF/SBK level of
theory . Calculations of Bi2S3 ribbon-like nanostructures at B3LYP/def-SV(P) and
PBE/def-SV(P) have been reported .
In the present research work, we systematically investigated the modification of the structural,
vibrational and electronic properties of the group 15 sulfides M4S6 at Td symmetry. Moreover,
we assessed the performance of DFT and MP2 methods by comparing to CCSD results. The
conclusions from this work are beneficial for further experimental and computational studies.
Experimentally, they may be useful in the structural investigation of the isolated gas phase
species, besides solving complex structures of liquid, crystalline or amorphous phases. From
the computational aspect, they are important in selecting the most accurate reasonable-cost level
of theory in predicting the geometrical parameters, vibrations, HOMO-LUMO gaps, and
electrostatic potential surfaces of the studied group 15 sulfides. We validated eight functionals
BP86, M06L, B3LYP, M06, Μ06-2Χ, CAMB3LYP, wB97XD, B2PLYP which span from
conventional GGA functionals to modern dispersion corrected hybrid-meta-GGA functionals,
and MP2 ab initio method. The comparison of geometrical parameters was made based on the
results from the CCSD calculations.
103 104 105
2. MATERIALS AND METHODS
All calculations were performed with the Gaussian 09 (version C.01) program package , in
the Linux Opensuse Leap 42.3 environment. We initially considered the M4O6 (M = P, As, Sb
and Bi) optimized structures of Td symmetry which have been previously reported .
Employing the chemical editor Avogadro (version 1.1.1)  we replaced each oxygen atom
with sulfur in order to obtain the four molecules P4S6, As4S6, Sb4S6 and Bi4S6. Then, we
optimized all structures employing the molecular mechanics UFF force field  implemented
in Avogadro. These input geometries have been fully optimized at each level of theory (MP2,
CCSD and DFT), setting very tight optimization criteria (see supporting information where
listed the final optimized geometries). For all molecules, positive frequencies were obtained
corresponding to stable conformations at energy minima.
Analysis of the electrostatic potential surface has been performed by the program Multiwfn
3.4.1 . All related parameters were calculated according to the equations from Murray et al.
. The natural bond orbital population analysis was carried out with the NBO 3.1 program
 on wavefunctions calculated at the CCSD/LANL08(d) level of theory. The NBO method
considers localized bonds and lone electron pairs as the basic units of the molecular structure.
The program Multiwfn was also used to analyze NBO and NPA results. Harmonic vibrations
were assigned with the aid of Chemcraft and VEDA 4 programs [29,30]. The results were
visualized with Avogadro, VMD (Visual Molecular Dynamics)  and Chemcraft.
The following methods were utilized: (i) CCSD coupled cluster method [32,33] and (ii) MP2
based on the Møller-Plesset second-order perturbation theory . (iii) BP86 is classified as a
generalized gradient approximation (GGA) and consists of the Becke88 exchange functional
 and the correlation functional Perdew86 . (iv) M06L  is classified as a meta-
generalized gradient-approximation (meta-GGA) where the term ‘meta’ denotes dependence
on kinetic energy density. (v) B3LYP is a hybrid functional of generalized gradient
approximation (hybrid GGA). It consists of the Becke88 exchange functional  and the Lee-
Yang-Parr (LYP) correlation functional [38,39]. (vi) M06 and (vii) M06-2X, are hybrid
functionals of meta-generalized gradient approximation (hybrid meta-GGA) . The M06-2X
method has been configured to include a medium-range correction . (viii) CAMB3LYP and
(ix) wB97XD are hybrid exchange-correlation functionals with short and long-range correction.
CAMB3LYP consists of 19% Hartree-Fock (HF) and 81% Becke 1988 (B88) for the
interactions of short-range exchange, while for long range exchange, it consists of 65% Hartree-
Fock (HF) and 35% Becke 1988 (B88) . The wB97XD consists of 100% exact long-range
exchange, 22% exact short-range exchange, one modified B97 density exchange functional for
short-range interactions, the B97 correlation density functional and empirical dispersion
corrections [42,43]. (x) B2PLYP is a double hybrid exchange-correlation functional. The first
part consists of Becke (B) exchange terms and Lee-Yang-Parr (LYP) correlation terms [38,39],
while the second one consists of exchange terms from Hartree-Fock (HF) and correlation terms
from the second order perturbation theory (PT2) .
2.3. Basis sets
The used basis sets for the sulfur, phosphorous, arsenic and heavier elements (Sb and Bi) are
the Los Alamos National Laboratory LANL2DZ [45–47] and its completely uncontracted basis
denoted as LANL08 . The LANL08 basis sets for main group elements have been derived
from the Hay-Wadt LANL2DZ sets and correspond to triple-ζ valence orbital quality. The
choice of the basis sets is based on its ability to offer an effective core potential (ECP), which
reduces the number of electrons that are considered explicitly and speeds up the calculations. It
employs a core including for phosphorous and sulfur 10 electrons ([Ne]), for arsenic 28
electrons ([Ar] + 3d), for antimony 46 electrons ([Kr] + 4d) and for bismuth 78 electrons ([Xe]
+ 5d4f). For a more accurate description of the nature of the chemical bond between metal and
sulfur and for a better prediction of the vibrational energies, the addition of diffuse p-
function(s) as well as of polarization d- function(s) is necessary. A set of polarization functions
for the main group atoms has been determined by Gilbert and co-workers and these are denoted
as LANL08(d) and LANL2DZpd . An all-electron, fully optimized contracted Gaussian-
basis set of triple zeta valence quality, named TZVp  for atoms P, S and As has been also
used for calculations on P4S6 and As4S6 molecules. All basis sets were obtained from the EMSL
Basis Set Library and the Basis Set Exchange (BSE) software [51,52].
3. RESULTS AND DISCUSSION
3.1. Molecular structure
The equilibrium structure of the four cage-like molecules M4S6, computed at
CCSD/LANL08(d) level, is depicted in Figure 1. The structural details obtained at various
methods are reported in Table 1. In the M4S6 (M= P, As, Sb, Bi) molecules at Td symmetry,
there are two groups of atoms equivalent by symmetry, the four metal and the six sulfur atoms.
There are one type of M-S bond and two bond angles (∠M-S-M, ∠S-M-S).
An increase in the length of the M-S bond across the 15th group was observed at all levels of
theory. For the CCSD/LANL08(d) optimized geometries, r(P-S) (2.158Å) < r(As-S) (2.274Å)
< r(Sb-S) (2.461Å) < r(Bi-S) (2.532Å). This increase is due to the different degree of covalent
bonding as well as to the fact that the heavier Sb and Bi atoms occupy a larger atomic volume.
The P-S bond length of β-P4S6 has a value equal to 2.145Å  and this is in excellent agreement
with the calculated value equal to 2.158Å at CCSD/LANL08(d) level of theory. The ∠M-S-M
is increased from M = P to M = Sb, whereas in Sb4S6 and Bi4S6 these angles are approximately
equal. The ∠S-M-S for P4S6 and As4S6 are approximately equal while they are decreasing for
Sb4S6 and Bi4S6.
The performance of popular density functionals has been investigated, following the
methodology presented in the DFT evaluation study of Minenkov et al. for the bond lengths of
ruthenium catalysts . For each level of theory, the mean absolute error (𝜀𝑚.𝑎.𝑒. ) of the
structural parameters of the four molecules and the absolute error (𝜀𝑎.𝑒. ) of each molecule were
calculated by the following equations: 1
𝜀𝑚.𝑎.𝑒 = 𝑁 ∑𝑁 𝑖=1|𝑥𝑖 − 𝑐𝑖 |
𝜀𝑎.𝑒 = |𝑥𝑖 − 𝑐𝑖 |
where 𝑥𝑖 denotes the calculated bond length with DFT or MP2 methods and 𝑐𝑖 expresses the
calculated bond length values with the CCSD method.
The mean absolute error of bond lengths for the four M4S6 molecules relative to
CCSD/LANL08(d) is presented in Figure 2.
For all four molecules, it was found that MP2 method underestimates the bond lengths
(𝜀𝑚.𝑎.𝑒. = 0.008Å) relative to the CCSD. Similar bond elongation from MP2 to CCSD has been
reported on the sulfur-hydrogen bond (rSH) of H2S and the sulfur-sulfur bond in H2S dimers, as
well as an excellent agreement of CCSD calculated data with the experimental rSH bond length
within 0.001 Å . Moreover, the CCSD method gives the best results for the geometries of
larger molecules such as SeO, SeCl, and AsO . Helgaker et al. calculated MP2 geometries
with longer bonds than the CCSD ones for 19 molecules consisted of the light-weight elements
H, F, O, N and C  but this is not applicable to the studied molecules which consisted of
heavier atoms P, As, Sb, Bi, and S. Considering the above remarks we selected CCSD as the
reference method for our DFT calculations.
The most accurate functionals are the M06L, B2PLYP, CAMB3LYP and M06 with similar
performance according to the 𝜀𝑚.𝑎.𝑒. (Table S2). The ranking of the methods relative to CCSD
and based on the 𝜀𝑚.𝑎.𝑒. for all molecules is the following:
BP86< B3LYP< wB97XD< M06-2X< MP2< M06L≤ B2PLYP≤ CAMB3LYP≤ M06< CCSD.
In terms of 𝜀𝑎.𝑒 (see Table S2) the most accurate methods are wB97XD and CAMB3LYP for
P4S6, M06L and M06 for As4S6, B2PLYP and M06 for Sb4S6, and M06 and M06L for Bi4S6
(Figures S1-S4). The M06 and M06L methods show high accuracy for the molecules with the
higher-weight elements such as As, Sb and Bi while the wB97XD shows an excellent
performance for the lowest-weight P4S6.
3.2. Atomic charges
The natural atomic charges as calculated with natural population analysis (NPA) are presented
in Table 2. The M and S atoms are positively and negatively charged for all M4S6 molecules,
respectively. The charge increases from P to Bi, indicating an increase in the ionic character of
the bonds M-S and that the electrons’ density moves from metal to sulfur, as one could predict
on the basis of electronegativities. In particular, NPA shows the ability of sulfur atoms to host
part of the negative charge provided by the M atoms resulting in the stabilization of the
adamantane structure. Both MP2 and CCSD methods provide almost equal values for the
natural atomic charges as recorded in Table 2. The metal’s charge values are in the range 0.42
– 1.17 for M4S6 species showing a less ionic character of M-S than M-O bond for heavier
metals, compared with the values 0.35 - 2.05 from phosphorous to bismuth, for M4O6 species
3.3. Electrostatic Potential Analysis
The electrostatic potential (ESP) on the vdW molecular surface provides information about the
strength and the orientation of intermolecular interactions such as hydrogen or halogen bonding
[57,58] as well as the electrophilic and nucleophilic positions of the molecule where chemical
reactions are expected. The following parameters have been described in published work by
Murray et al. . 𝑉̅𝑆+ and 𝑉̅𝑆− indicate the mean positive and negative ESP values on the vdW
surface respectively, Π is the mean deviation on the surface, which is an index of the internal
2 charge separation, and 𝜎𝑡𝑜𝑡 is the total ESP variance which is the sum of the positive 𝜎+2 and
the negative 𝜎−2 parts. The greater the 𝜎+2 and 𝜎−2 , the greater the molecule's tendency to interact
with other molecules through the positive and negative ESP domains. The degree of charge
balance equals ν and when the 𝜎+2 and 𝜎−2 are equal, then ν is maximized and equals 0.250. As
long as ν is closer to 0.250, the more likely it is that the molecule interacts with other ones
2 through the positive and negative regions to a similar extent. The product 𝜎𝑡𝑜𝑡 𝜈 is also a very
useful quantity; a large value indicates a relatively strong tendency to interact with other
molecules of the same kind electrostatically.
According to the data in Table 3, there is an increase in the values of all parameters V, d, Π,
2 2 𝜎𝑡𝑜𝑡 , 𝜎+2 , 𝜎−2 , 𝜈 and 𝜈𝜎𝑡𝑜𝑡 along the group 15, from M = P to M = Bi. The internal charge
separation is increased from P4S6 (𝛱 = 6.12 𝑘𝑐𝑎𝑙 𝑚𝑜𝑙 −1 ) to the heavier Bi4S6 (𝛱 =
16.14 𝑘𝑐𝑎𝑙 𝑚𝑜𝑙 −1 ). This separation is also confirmed by the charge values (Table 2) where
the charge difference between Bi and S atoms in Bi4S6 is greater than that between P and S
atoms in P4S6. The 𝜈 parameter shows the degree of charge balance and if it receives the
maximum value 𝜈 = 0.25 then the molecule interacts with other molecules to the same extent
with its positive and its negatively charged region. Also, the 𝜈 parameter increases from P4S6
to Bi4S6, so more intermolecular interactions with different molecules for the heavier Bi4S6 are
2 expected. Moreover, a high value of 𝜈𝜎𝑡𝑜𝑡 indicates that a molecule has strong electrostatic
2 interactions among other molecules of the same species. The 𝜈𝜎𝑡𝑜𝑡 increases from P4S6
2 2 (𝜈𝜎𝑡𝑜𝑡 = 3.55 (𝑘𝑐𝑎𝑙 𝑚𝑜𝑙 −1 )2) to Bi4S6 (𝜈𝜎𝑡𝑜𝑡 = 19.04 (𝑘𝑐𝑎𝑙 𝑚𝑜𝑙 −1 )2) whereby Bi4S6
molecules interact strongly with each other. For example, it has been found that in saturated
2 hydrocarbons 𝜈𝜎𝑡𝑜𝑡
is about 1 (𝑘𝑐𝑎𝑙 𝑚𝑜𝑙 −1 )2
2 while for formamide is 𝜈𝜎𝑡𝑜𝑡 =
62.5 (𝑘𝑐𝑎𝑙 𝑚𝑜𝑙 −1 )2  so that it will have strong electrostatic forces through its positive and
negative regions and formamide molecules should interact strongly with each other. That was
confirmed experimentally after the investigation of the N-H stretching domains at the infrared
and Raman spectra in the solid and liquid state of formamide . The Figures S5-S8 illustrate
the electrostatic potential surfaces of M4S6 molecules at CCSD/LANL08(d) level of theory. The
colors are related to the electrostatic potential values. Red color indicates electronic deficient
areas (𝑉(𝑟) > 0) and blue indicates electron rich areas (𝑉(𝑟) < 0).
3.4. Bonding analysis
The analysis of the chemical bonding and of the HOMO-LUMO orbitals was based on results
from the CCSD/LANL08(d) method. For the compounds M4S6, the simple σ bond between M
and S can be written as: 𝜎𝑀𝑆 = 𝑐𝑀 ℎ𝑀 + 𝑐𝑆 ℎ𝑆
2 where 𝑐𝑀 and 𝑐𝑆 are the polarity coefficients of the hybrid orbitals ℎ𝑀 and ℎ𝑆 , while 𝑐𝑀 + 𝑐𝑆2 =
1. The bond ionicity parameter provided by the Eq. (4) and quantifies the polarity of the bond
𝑐 2 −𝑐 2
𝑆 𝑖𝑀𝑆 = 𝑐𝑀 2 +𝑐 2 𝑀
For the P4S6 molecule, all 𝜎𝑃𝑆 bonds are equivalent and each one is expressed according to Eq.
(3) as 𝜎𝑀𝑆 = 0.635(𝑠𝑝6.93 𝑑0.14 )𝑃 + 0.772(𝑠𝑝6.00 𝑑 0.07 )𝑆 . Also, each P atom has a lone pair of
electrons in an orbital with hybridization 𝑠𝑝0.59 (62.83% 3s of P) while each S atom has two
lone pairs of electrons, the first one in 𝑠𝑝0.39 (71.65% 3s of S) and the second one in the 3p of
S (99.86%). The |iMS | parameter was found equal to 0.193. In each of the other molecules
As4S6, Sb4S6 and Bi4S6, the σ bonds are equivalent, and the lone electron pairs are in a similar
configuration to P4S6, i.e. one in M and two in S. The results are summarized in Table 4.
Based on the NBO theory, it follows from the values of the bond ionicity |iMS | that the most
covalent bond M-S exists in P4S6 with |iMS | = 0.193. As M varies from P to Bi, the bond
becomes more polar, since for M = Bi, |iMS | = 0.464. For all the studied species it was found
that 𝑐𝑀 < 𝑐𝑆 and the bond is more polarized to the S atoms. In addition, the M-S bond is formed
from the p valence orbitals of M and the 3p orbital of S. According to Table 4, the participation
of the p orbitals of M and of S to the 𝜎𝑀𝑆 bond is increased from P4S6 (𝜎𝑀𝑆 =
0.635(𝑠𝑝6.93 𝑑 0.14 )𝑃 + 0.772(𝑠𝑝6.00 𝑑0.07 )𝑆 )
0.856(𝑠𝑝7.58 𝑑0.04 )𝑆 ).
In molecules with M-S bonds such as [PbII(S2COEt)n]2-n (n=1,2,3,4), Ghosh et al. concluded
that the PbII-S bond is formed from the 6p orbitals of PbII and the 3p ones of S atoms . A
similar conclusion about the M-S bond was reached for the studied M4S6 molecules in this
work. The NBO analysis of the 𝜎𝑀𝑆 bond showed that the donor NBOs are composed mainly
of the 3p orbital of sulfur while the acceptor NBOs are mostly comprised of the valence p
orbitals of M. Therefore, a general tendency is observed in the M-S bond of association between
p valence orbital of the metal and the 3p of sulfur.
The lone electron pair of M denoted as nM is found in hybrid orbitals with higher s character.
Along the group 15 of the periodic table, the localization of the lone pair increases significantly
in the s orbital of M, since in the P4S6 it is 62.83% s while for the Bi4S6 it is 86.59% s. Similarly,
the first lone electron pair of S denoted as nS , is in a hybrid orbital with more s character.
From M = P to M = Bi there is small increase in s character: for P4S6 is 71.65% while for Bi4S6
it is 76.74%. The second lone electron pair of S, nS
(𝜎𝐵𝑖𝑆 = 0.518(𝑠𝑝21.00 𝑑0.07 )𝐵𝑖 +
is located on the 3p orbital in all M4S6
3.5. Frontier orbitals analysis
The analysis of the frontier molecular orbitals calculated at the CCSD/LANL08(d) level of
theory for all M4S6 molecules follows. In all studied molecules, there are three energy-
degenerated HOMO orbitals having t1 symmetry (indicated as MO-26, MO-27 and MO-28) and
two degenerated LUMO orbitals of e symmetry (indicated as MO-29 and MO-30). These five
MO orbitals are presented in Figures 3 and S9-S11.
The participation of the atomic orbitals to the aforementioned five frontier orbitals has been
P4S6: The 3py, 3px and 3pz atomic orbitals of P atoms have the largest contribution (by 34.32%)
to the three HOMO orbitals of P4S6 MO-26, MO-27 and MO-28 respectively. Also, these three
HOMO orbitals consisted of the 3s orbital of the P atoms, with a contribution of 10.71% and of
the 3s of the S atoms with 1.46% participation. The two LUMO orbitals do not consist of 3s of
P but consisted by 12.72% of 3s of S. Also, MO-29 is 39.82% of 3py and 10.90% of 3pz of P
whereas MO-30 is 30.80% 3px and 28.94% 3pz of P.
As4S6: For the As4S6, the 4s of the As atoms contributed a total of 11.97% and the 3s of the S
atoms a total of 1.08%. The 4px, 4pz and 4py of As were significantly involved in the three
HOMO orbitals MO-26, MO-27 and MO-28 by 30.80% respectively. The two LUMO orbitals
did not consist of 4s of As but 11.61% of 3s of S. Also, MO-29 consisted of 37.53% of 4pz and
26.26% of 4py of As while MO-30 from 42.20% 4px and 16.94% 4py of As.
Sb4S6: In Sb4S6 the three HOMO orbitals consisted mainly of 5py, 5pz and 5px of Sb which
contribute 27.69% to MO-26, MO-27 and MO-28 respectively. Also, the three HOMOs
consisted of the 5s of the Sb by 16.58%. The two LUMOs were not composed of 5s of Sb but
of 9.83% of the 3s of S. The MO-29 consisted of 41.45% of 5py of Sb and 29.86% of 5pz of Sb
while MO- 30 from 47.22% 5px and 18.30% 5pz of Sb.
Bi4S6: For the Bi4S6 the three HOMO orbitals consisted of the 6s orbitals of Bi by 12.04% and
of the 3s of S atoms by 0.52%. The 6py, 6pz and 6px participated in MO-26, MO-27 and MO-
28 by 20.38% respectively. The two LUMOs were not composed of 6s of Bi but of 9.10% of
3s of S. Furthermore, MO-29 consisted of 48.00% of 6px and 14.61% of 6pz of Bi. The MO-30
was composed of 38.54% 6py and 33.57% 6pz of Bi atoms.
All contributions of the atomic orbitals to the aforementioned HOMOs and LUMOs for M4S6
molecules are provided in the Table 5 of this paper. In all studied molecules, it was found that
their HOMO consisted mainly of the p valence orbitals of the M atoms and of the 3p orbitals of
the S atoms. Along the group 15, the contribution of 3p of S atoms is significantly increased,
the 3py of the P4S6 participates 37.64% in MO-26 while in Bi4S6 3py participates by 55.41%. In
addition, the contribution of the p valence orbitals of M to the LUMOs is significant in all M4S6
For the energy gaps between HOMO and LUMO orbitals at the CCSD/LANL08(d) level of
theory, it was found that from M = P to M = As there is a decrease in these values whereas there
is no significant difference between Sb4S6 and Bi4S6 molecules. That is, the following
relationship applies to the HOMO-LUMO energy gaps:
𝛥𝐸𝑃4 𝑆6 (10.5 𝑒𝑉) > 𝛥𝐸𝐴𝑠4 𝑆6 (10.0 𝑒𝑉) > 𝛥𝐸𝑆𝑏4 𝑆6 , 𝛥𝐸𝐵𝑖4 𝑆6 (9.3 𝑒𝑉)
Since all 𝛥𝐸𝑀4 𝑆6 values are high, M4S6 molecules are expected to be chemically stable.
3.6. Molecular Polarizability
The perturbation of electron density when a molecule interacts with the electric field of
radiation, ions, polar molecules, etc. is fundamental to understand the behavior of molecules in
chemical reactions, their solvation properties, the recognition processes and spectroscopic
properties. Electric polarizability is currently of importance as it is extensively used to model
intermolecular interactions , basic molecular characteristics as the acidity and basicity ,
hardness and softness [63–65] and chemical reactivity.
Our polarizability calculations were performed at the CCSD(full)/LANL08(d) optimized
geometry, using the MP2(full)/LANL08(d) theoretical level. Due to high symmetry there is
only one independent component of the polarizability tensor: αxx = αyy = αzz = 𝛼̅.
In atomic units the calculated values of the mean polarizability are:
𝛼̅(𝑃4 𝑆6 ) = 193.91, 𝛼̅(𝐴𝑠4 𝑆6 ) = 217.87, 𝛼̅(𝑆𝑏4 𝑆6 ) = 267.46, 𝛼̅(𝐵𝑖4 𝑆6 ) = 285.82 e2α02Eh-1.
Similar calculations were performed for the M4O6 molecules:
𝛼̅(𝑃4 𝑂6 ) = 89.106, 𝛼̅(𝐴𝑠4 𝑂6 ) = 107.17, 𝛼̅(𝑆𝑏4 𝑂6 ) = 142.24, 𝛼̅(𝐵𝑖4 𝑂6 ) = 155.27 e2α02Eh-1
A quantity related to polarizability is the polarizability volume, α', defined as α' = α/(4πε0)
For M4S6 molecules these volumes have been calculated:
α'(P4S6)= 28.705, α'(As4S6)= 32.252, α'(Sb4S6)= 39.593, α'(Bi4S6)= 42.312 Å3
As it is expected polarizability and polarizability volume increase as the size of atoms (P to Bi
and O to S) increases.
3.7. Vibrational properties
The highly symmetric M4S6 Td molecules have the following irreducible representation of the
Γvib. = 2Α1(R) + 2Ε(R) + 2T1(i.a.) + 4T2(IR,R)
The vibrational modes having A1, E and T2 symmetry are Raman (R) active and only the ones
having T2 symmetry are IR active (IR). The vibrational modes having T1 symmetry are both IR
and Raman forbidden (i.a.).
In Table 6 the harmonic wavenumbers calculated at the optimized geometry and their
symmetries, are presented.
A non-linear molecule with N atoms exhibits 3N - 6 normal modes of vibration. There are 24
fundamental vibrations - symmetric and antisymmetric – for all M4S6 cages. Each fundamental
mode can be expressed by a normal coordinate. Then, those coordinates are transformed to
internal ones, which are a superposition of multiple local modes including stretching, bending
and torsion [29,30]. The Greek letters ν, β, δ, τ denote stretching, in plane bending, out of plane
bending and torsion modes respectively (Table 6). By using the symmetry considerations and
theoretical results for IR and Raman intensities, it is possible to assign the predicted energies
of vibrations to specific vibrational modes.
For all systems investigated here and for all calculated modes, the corresponding vibrational
wavenumbers decrease when the atomic number of the central atom increases. For the M-S
stretching vibrations this decrease is expected since the quadratic M-S force constants decrease
in the series from P to Bi whereas the reduced mass increases reinforcing the behavior expected
from the quadratic force constant.
There is also a correlation between bond angles and hybridization. The percentage of s character
of the M to the 𝜎𝑀𝑆 bond decreases from P to Bi. However, the contribution of the p orbitals of
M increases from P to Bi. In addition, we found that the angle ∠S-M-S decreases from P4S6 to
Bi4S6. Therefore, P4S6 with the highest %s contribution of phosphorus to the 𝜎𝑀𝑆 bond, has the
largest value of ∠S-P-S equal to 106.97°. This trend is in agreement with the prediction of Bent
rule, which was originally derived from sp-hybridized main group elements .
In the present work, the structural, electronic and vibrational properties of the M4S6, M = P, As,
Sb, Bi inorganic clusters were investigated by means of computer simulation methods. The
calculations were performed with the MP2, CCSD and DFT methods employing the basis set
LanL08(d) with ECP for the P, As, Sb, Bi, S atoms. Comparing with the available CCSD
theoretical data, we concluded that M06 and M06L functionals show high accuracy for
molecules with the higher-weight elements such as As, Sb, and Bi while the wB97XD and
CAMB3LYP show an excellent performance for the lowest-weight P4S6 concerning the
prediction of the structural data. We also present for the first time data for the hypothetical
Sb4S6 and Bi4S6 molecules. The vibrational spectral analysis is presented for the whole series
of group 15 metal sulfides. A structural ‘relaxation’ ongoing from phosphorous to bismuth
indicates an increase of ionic character and in some extent can explain the instability of these
compounds when the atomic number of the metal increases.
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Figure 1. The adamantane structure of the M4S6 molecules at Td symmetry, as computed at CCSD/LANL08(d).
Figure 2. Mean absolute error for the four M4S6 molecules relative to CCSD/LANL08(d).
Figure 3. Molecular orbital diagrams of HOMO and LUMO orbitals of P4S6.
Table 1. Optimized geometrical parameters of M4S6 molecules calculated at various levels of theory, using the LANL08(d) or TZVp [in brackets] basis sets. Structural Molecule
Parameter r(P-S) (Å)
r(Sb-S) (Å) Sb4S6
r(As-S) (Å) As4S6
Table 2. Natural atomic charges of M4S6 (Μ = P, As, Sb, Bi) MP2/LANL08(d) CCSD/LANL08(d) M S M S 0.426 -0.284 0.424 -0.283 P4S6 As4S6
Table 3. Electrostatic potential surface analysis results at CCSD/LANL08(d) level of theory. VvdW
Table 4. NBO chemical bonding and bond ionicity parameter for all M4S6 molecules. (𝝈)
𝜎𝑀𝑆 = 0.635(𝑠𝑝6.93 𝑑 0.14 )𝑃 + 0.772(𝑠𝑝6.00 𝑑 0.07 )𝑆
𝜎𝐴𝑠𝑆 = 0.599(𝑠𝑝9.78 𝑑 0.13 )𝐴𝑠 + 0.800(𝑠𝑝6.53 𝑑 0.06 )𝑆
𝜎𝑆𝑏𝑆 = 0.534(𝑠𝑝12.59 𝑑 0.11 )𝑆𝑏 + 0.846(𝑠𝑝6.58 𝑑 0.04 )𝑆
𝜎𝐵𝑖𝑆 = 0.518(𝑠𝑝21.00 𝑑 0.07 )𝐵𝑖 + 0.856(𝑠𝑝7.58 𝑑0.04 )𝑆
Table 5. Orbital composition of frontier orbitals (%) in M4S6 (M = P, As, Sb, Bi) molecules at CCSD/LANL08(d) Atom P P P P S S S S
Orbital 3pz 3py 3px 3s 3pz 3py 3px 3s
MO-26 6.78 34.32 6.78 10.71 0.04 37.64 0.04 1.46
MO-27 6.78 6.78 34.32 10.71 0.04 0.04 37.64 1.46
MO-28 34.32 6.78 6.78 10.71 37.64 0.04 0.04 1.46
MO-29 10.90 39.82 9.05 0.00 2.26 8.27 1.88 12.72
MO-30 28.94 0.03 30.80 0.00 6.01 0.01 6.39 12.72
As As As As S S S S
4s 4px 4py 4pz 3s 3px 3py 3pz
Sb Sb Sb Sb S S S S
5s 5px 5py 5pz 3s 3px 3py 3pz
11.97 30.80 6.19 6.19 1.08 41.42 0.30 0.30 16.58 6.66 27.69 6.66 0.76 1.41 37.73 1.41
11.97 6.19 6.19 30.80 1.08 0.30 0.30 41.42 16.58 6.66 6.66 27.69 0.76 1.41 1.41 37.73
11.97 6.19 30.80 6.19 1.08 0.30 41.42 0.30 16.58 27.69 6.66 6.66 0.76 37.73 1.41 1.41
0.00 1.01 26.26 37.53 11.61 0.17 4.45 6.37 0.00 0.95 41.45 29.86 9.83 0.10 4.16 3.00
0.00 42.20 16.94 5.65 11.62 7.16 2.88 0.96 0.00 47.22 6.73 18.30 9.83 4.74 0.68 1.84
Bi Bi Bi Bi S S S S
6s 6px 6py 6pz 3s 3px 3py 3pz
12.04 4.00 20.38 4.00 0.52 1.40 55.41 1.40
12.04 4.00 4.00 20.38 0.52 1.40 1.40 55.41
12.04 20.38 4.00 4.00 0.52 55.41 1.40 1.40
0.00 48.00 9.65 14.61 9.10 4.54 0.91 1.38
0.00 0.17 38.54 33.57 9.10 0.02 3.64 3.17
Table 6. Calculated harmonic wavenumbers (cm-1) at MP2(full) level of theory using the LANL08(d) or TZVp basis sets, of the M4S6 cage-like molecules, at Td symmetry. In brackets the calculated intensities are presented [Raman activity in Å4·(amu)-1, IR intensity in km·mol-1]. Intensities and activities less than 0.5 have been set equal to zero. Calculations have been performed at the optimized geometry at the specific level of theory (CCSD(full)/LANL08(d) or MP2(full)/TZVp). Νο
1 2 3 4 5 6 7 8 9 10
T1 (i.a.) E (R) T2 (R,IR) T2 (R,IR) A1 (R) T2 (R,IR) T1 (i.a.) E (R) A1 (R) T2 (R,IR)
139 158  205 [11,2] 294 [2,4] 350  390 [1,0] 391 456  431  480 [2,81]
143 163  212 [15,2] 302 [1,5] 362  404 [1,0] 420 474  441  502 [1,84]
106 116  148 [11,2] 212 [0,6] 244  314 [0,5] 312 326  340  397 [2,75]
107 120  152 [13,2] 217 [0,7] 250  331 [0,4] 344 343  356  417 [1,66]
79 85  108 [11,3] 158 [0,6] 181  271 [1,12] 286 273  287  372 [2,110]
70 63  82 [2,8] 133 [1,9] 133  245 [1,19] 268 240  266  359 [3,102]
assignment β(SMS) + τ(MSMS) + δ(SSSM) β(SMS) + β(MSM) + τ(MSMS) β(SMS) + β(MSM) + δ(SSSM) + τ(MSMS) β(SMS) + β(MSM) + δ(SSSM) + τ(MSMS) ν(M-S) ν(M-S) + τ(MSMS) + δ(SSSM) ν(M-S) + β(SMS) ν(M-S) + β(MSM) β(MSM) + δ(SSSM) ν(MS) + τ(MSMS) + δ(SSSM)
The Greek letters ν, β, δ, τ denote stretching, in plane bending, out of plane bending and torsion modes respectively.