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.
1
Computational Study of Structural, Vibrational and Electronic Properties of the Highly
2
Symmetric Molecules M4S6 (M= P, As, Sb, Bi).
3 4
E. Semidalas and A. Chrissanthopoulos
5 6
Laboratory of Inorganic Chemistry, Department of Chemistry, National and Kapodistrian University of
7
Athens, University campus, Zografou, GR-15771, Greece.
8 9
ABSTRACT
10 11
A systematic computational investigation of the structural, electronic and vibrational properties
12
of the group 15 sulfides M4S6 at Td symmetry was carried out. The performance of DFT and
13
MP2 theoretical methods was assessed compared to the high-level CCSD method. The M-S
14
bond is based on the association between p valence orbitals of M and the 3p of sulfur according
15
to the natural population analysis. Both polarizability and polarizability volume of the cage
16
molecules increase as the size of the atoms increases from P to Bi. A structural ‘relaxation’
17
ongoing from phosphorus to bismuth showed an increase of ionic character and it might explain
18
the chemical instability of the heavier cage compounds. For the P4S6 molecule, the functionals
19
wB97XD and CAMB3LYP yielded excellent structural data, while for the heavier molecules
20
As4S6, Sb4S6 and Bi4S6, the M06 and M06L functionals showed high accuracy. We validated
21
eight functionals BP86, M06L, B3LYP, M06, Μ06-2Χ, CAMB3LYP, wB97XD, B2PLYP
22
which span from conventional GGA functionals to long-range corrected hybrid ones, and MP2,
23
CCSD ab initio methods. Experimentally, these molecules could be useful in the structural
24
investigation of the isolated gas phase species, besides solving complex structures of liquid,
25
crystalline or amorphous phases.
26 27 28
Keywords: M4S6 Td molecules, ab-initio, DFT, structural properties, electronic properties, vibrational
29
properties.
30 31
32
1. INTRODUCTION
33
The M4S6 highly symmetric sulfides of group 15 elements of the periodic table (M: P, As, Sb,
34
Bi) are examples of prototype cage like inorganic molecules. These serve as structural building
35
blocks of network like condensed matter and could be used to describe and predict the
36
physicochemical properties of various solid- and liquid-state systems [1]. Investigating the
37
properties of these molecules could reveal reaction mechanisms for the development of less
38
expensive catalysts as well as the formation of novel atomic-level controlled nanomaterials [2].
39
The description of the chemical bonding in the M4S6 molecules is a rather challenging task. The
40
knowledge of the structural parameters (bond lengths and bond angles) is inadequate to indicate
41
the strength of the metal-sulfur bond. It is essential to ascertain additional molecular properties
42
(especially in the case of more ionic systems), such as the population analysis, calculation of
43
the electrostatic potential surface and bond stretching frequencies [3–7]. From the
44
computational aspect, our conclusions are important in selecting the most accurate ‘low-cost’
45
level of theory in predicting the geometrical parameters and frontier orbitals, description of the
46
metal-sulfur bond, electrostatic potential surfaces and vibrational spectrum of metal sulfides.
47
The molecules M4S6 were considered as spherical top species with Td symmetry. To our
48
knowledge neither experimental structural data from electron diffraction studies nor vibrational
49
spectra of the gas phase Td sulfides have been published yet, so the existence of these species
50
is currently under consideration. Sulfides of group 15 of lower symmetry than Td have been
51
reported in the literature. Jason has suggested the possible structures of α-P4S6 (C1 symmetry),
52
β-P4S6 and γ-P4S6 (Cs symmetry) employing the
53
structurally characterized by Blachnik et al. [9] Furthermore, most of the arsenic sulfide
54
minerals are composed of cage-like molecules interacting through weak van der Waals (vdW)
55
forces. The uzonite mineral contains the cage molecules As4S5 [10]. The cyclo- anion Sb4S62−
56
has been obtained as its PPh4+ salt, where two exocyclic Sb-S bonds and an Sb-Sb bond are
2
P-NMR technique [8]. β-P4S6 has been
31
57
present [11]. In individual Sb2S3 nanowires embedded in anodic alumina templates
58
piezoelectric and ferroelectric properties have been observed [12]. Also, Sb2S3 crystals of the
59
urchin-like nanostructure, 3−4 μm in length and 30−150 nm in diameter have been obtained at
60
a mild reaction temperature [13]. Bi2S3 has a very anisotropic one-dimensional orthorhombic
61
structure considered as aligned (Bi4S6)n ribbons bound to each other by the vdW forces. It has
62
multiple functions in solar cells, such as a sensitizer, light absorber or electron acceptor material
63
[14]. In addition, the synthesis and the promising thermoelectric properties of highly oriented
64
bulk crystalline ingots of n-type bulk Bi2S3 doped with BiCl3 have been reported [15].
65
In this work, the formation of the M-S bond of group 15 sulfides M4S6 was characterized by
66
analyzing the electron density of the metal’s p valence orbitals and the 3p orbitals of sulfur
67
rearrangement between the two nuclei. In the literature has been reported charge transfer from
68
the 3p orbitals of sulfur to the p valence orbitals of M for sulfur complexes with M=Au, Ag, Cu
69
[4]. It is well known that phosphorous like nitrogen forms mostly covalent bonds while
70
compounds of arsenic, antimony, and bismuth are characterized by more ionic bonding.
71
Moreover, in molecules with M-S bonds, such as [PbII(S2COEt)n]2-n (n = 1,2,3,4), Ghosh et al.
72
[16] concluded that the PbII-S bond is formed by the 6p orbitals of PbII and the 3p of the S
73
atoms. Both PbII and MIII have the electronic structure ns2p0 but the s orbital of M has limited
74
contribution to the bond. The electron lone pair of the M atom is localized in hybrid orbital
75
having a higher percentage of s character. Along the P4S6 to Bi4S6 series, the localization of the
76
lone pair is significantly increased in an orbital with a higher percentage of s character.
77
The adamantane structure of P4S6 has been proposed by Gimarc and Ott but without any
78
experimental evidence [17]. Theoretical studies have been conducted on the Td form of the
79
phosphorus decasulfide P4S10 at HF/6-31G*, MP2/6-31G* and B3LYP/6-31G* levels of theory
80
[18]. The first reported calculations for As4S6 at Td symmetry were carried out by Fukui et al.
81
They performed obsolete ASMO calculations for all valence electrons (INDO type) and
3
82
proposed that the As4S6 unit is stable and is a reasonable candidate for a structural unit in As2S3
83
glass [19]. Babić et al. have made computations for As4S6 (Td) using the outdated Vosko
84
exchange-correlation parameterization without nonlocal terms correction and an s, p orbital
85
basis set without d-type functions for arsenic atoms [20]. Their theoretical gas phase results
86
were compared with the experimental XPS spectrum of amorphous As2S3 solid. The geometry
87
and vibrations of As4S6 and Sb4S6 of C3ν symmetry have been calculated at HF/SBK level of
88
theory [21]. Calculations of Bi2S3 ribbon-like nanostructures at B3LYP/def-SV(P) and
89
PBE/def-SV(P) have been reported [22].
90
In the present research work, we systematically investigated the modification of the structural,
91
vibrational and electronic properties of the group 15 sulfides M4S6 at Td symmetry. Moreover,
92
we assessed the performance of DFT and MP2 methods by comparing to CCSD results. The
93
conclusions from this work are beneficial for further experimental and computational studies.
94
Experimentally, they may be useful in the structural investigation of the isolated gas phase
95
species, besides solving complex structures of liquid, crystalline or amorphous phases. From
96
the computational aspect, they are important in selecting the most accurate reasonable-cost level
97
of theory in predicting the geometrical parameters, vibrations, HOMO-LUMO gaps, and
98
electrostatic potential surfaces of the studied group 15 sulfides. We validated eight functionals
99
BP86, M06L, B3LYP, M06, Μ06-2Χ, CAMB3LYP, wB97XD, B2PLYP which span from
100
conventional GGA functionals to modern dispersion corrected hybrid-meta-GGA functionals,
101
and MP2 ab initio method. The comparison of geometrical parameters was made based on the
102
results from the CCSD calculations.
103 104 105
4
106
2. MATERIALS AND METHODS
107 108
2.1. Programs
109
All calculations were performed with the Gaussian 09 (version C.01) program package [23], in
110
the Linux Opensuse Leap 42.3 environment. We initially considered the M4O6 (M = P, As, Sb
111
and Bi) optimized structures of Td symmetry which have been previously reported [3].
112
Employing the chemical editor Avogadro (version 1.1.1) [24] we replaced each oxygen atom
113
with sulfur in order to obtain the four molecules P4S6, As4S6, Sb4S6 and Bi4S6. Then, we
114
optimized all structures employing the molecular mechanics UFF force field [25] implemented
115
in Avogadro. These input geometries have been fully optimized at each level of theory (MP2,
116
CCSD and DFT), setting very tight optimization criteria (see supporting information where
117
listed the final optimized geometries). For all molecules, positive frequencies were obtained
118
corresponding to stable conformations at energy minima.
119
Analysis of the electrostatic potential surface has been performed by the program Multiwfn
120
3.4.1 [26]. All related parameters were calculated according to the equations from Murray et al.
121
[27]. The natural bond orbital population analysis was carried out with the NBO 3.1 program
122
[28] on wavefunctions calculated at the CCSD/LANL08(d) level of theory. The NBO method
123
considers localized bonds and lone electron pairs as the basic units of the molecular structure.
124
The program Multiwfn was also used to analyze NBO and NPA results. Harmonic vibrations
125
were assigned with the aid of Chemcraft and VEDA 4 programs [29,30]. The results were
126
visualized with Avogadro, VMD (Visual Molecular Dynamics) [31] and Chemcraft.
127 128
2.2. Methods
129
The following methods were utilized: (i) CCSD coupled cluster method [32,33] and (ii) MP2
130
based on the Møller-Plesset second-order perturbation theory [34]. (iii) BP86 is classified as a
131
generalized gradient approximation (GGA) and consists of the Becke88 exchange functional
5
132
[35] and the correlation functional Perdew86 [36]. (iv) M06L [37] is classified as a meta-
133
generalized gradient-approximation (meta-GGA) where the term ‘meta’ denotes dependence
134
on kinetic energy density. (v) B3LYP is a hybrid functional of generalized gradient
135
approximation (hybrid GGA). It consists of the Becke88 exchange functional [35] and the Lee-
136
Yang-Parr (LYP) correlation functional [38,39]. (vi) M06 and (vii) M06-2X, are hybrid
137
functionals of meta-generalized gradient approximation (hybrid meta-GGA) [40]. The M06-2X
138
method has been configured to include a medium-range correction [37]. (viii) CAMB3LYP and
139
(ix) wB97XD are hybrid exchange-correlation functionals with short and long-range correction.
140
CAMB3LYP consists of 19% Hartree-Fock (HF) and 81% Becke 1988 (B88) for the
141
interactions of short-range exchange, while for long range exchange, it consists of 65% Hartree-
142
Fock (HF) and 35% Becke 1988 (B88) [41]. The wB97XD consists of 100% exact long-range
143
exchange, 22% exact short-range exchange, one modified B97 density exchange functional for
144
short-range interactions, the B97 correlation density functional and empirical dispersion
145
corrections [42,43]. (x) B2PLYP is a double hybrid exchange-correlation functional. The first
146
part consists of Becke (B) exchange terms and Lee-Yang-Parr (LYP) correlation terms [38,39],
147
while the second one consists of exchange terms from Hartree-Fock (HF) and correlation terms
148
from the second order perturbation theory (PT2) [44].
149 150
2.3. Basis sets
151
The used basis sets for the sulfur, phosphorous, arsenic and heavier elements (Sb and Bi) are
152
the Los Alamos National Laboratory LANL2DZ [45–47] and its completely uncontracted basis
153
denoted as LANL08 [48]. The LANL08 basis sets for main group elements have been derived
154
from the Hay-Wadt LANL2DZ sets and correspond to triple-ζ valence orbital quality. The
155
choice of the basis sets is based on its ability to offer an effective core potential (ECP), which
156
reduces the number of electrons that are considered explicitly and speeds up the calculations. It
6
157
employs a core including for phosphorous and sulfur 10 electrons ([Ne]), for arsenic 28
158
electrons ([Ar] + 3d), for antimony 46 electrons ([Kr] + 4d) and for bismuth 78 electrons ([Xe]
159
+ 5d4f). For a more accurate description of the nature of the chemical bond between metal and
160
sulfur and for a better prediction of the vibrational energies, the addition of diffuse p-
161
function(s) as well as of polarization d- function(s) is necessary. A set of polarization functions
162
for the main group atoms has been determined by Gilbert and co-workers and these are denoted
163
as LANL08(d) and LANL2DZpd [49]. An all-electron, fully optimized contracted Gaussian-
164
basis set of triple zeta valence quality, named TZVp [50] for atoms P, S and As has been also
165
used for calculations on P4S6 and As4S6 molecules. All basis sets were obtained from the EMSL
166
Basis Set Library and the Basis Set Exchange (BSE) software [51,52].
167 168
3. RESULTS AND DISCUSSION
169
3.1. Molecular structure
170
The equilibrium structure of the four cage-like molecules M4S6, computed at
171
CCSD/LANL08(d) level, is depicted in Figure 1. The structural details obtained at various
172
methods are reported in Table 1. In the M4S6 (M= P, As, Sb, Bi) molecules at Td symmetry,
173
there are two groups of atoms equivalent by symmetry, the four metal and the six sulfur atoms.
174
There are one type of M-S bond and two bond angles (∠M-S-M, ∠S-M-S).
175
An increase in the length of the M-S bond across the 15th group was observed at all levels of
176
theory. For the CCSD/LANL08(d) optimized geometries, r(P-S) (2.158Å) < r(As-S) (2.274Å)
177
< r(Sb-S) (2.461Å) < r(Bi-S) (2.532Å). This increase is due to the different degree of covalent
178
bonding as well as to the fact that the heavier Sb and Bi atoms occupy a larger atomic volume.
179
The P-S bond length of β-P4S6 has a value equal to 2.145Å [9] and this is in excellent agreement
180
with the calculated value equal to 2.158Å at CCSD/LANL08(d) level of theory. The ∠M-S-M
181
is increased from M = P to M = Sb, whereas in Sb4S6 and Bi4S6 these angles are approximately
7
182
equal. The ∠S-M-S for P4S6 and As4S6 are approximately equal while they are decreasing for
183
Sb4S6 and Bi4S6.
184
The performance of popular density functionals has been investigated, following the
185
methodology presented in the DFT evaluation study of Minenkov et al. for the bond lengths of
186
ruthenium catalysts [53]. For each level of theory, the mean absolute error (𝜀𝑚.𝑎.𝑒. ) of the
187
structural parameters of the four molecules and the absolute error (𝜀𝑎.𝑒. ) of each molecule were
188
calculated by the following equations: 1
189
𝜀𝑚.𝑎.𝑒 = 𝑁 ∑𝑁 𝑖=1|𝑥𝑖 − 𝑐𝑖 |
(1)
190
𝜀𝑎.𝑒 = |𝑥𝑖 − 𝑐𝑖 |
(2)
191
where 𝑥𝑖 denotes the calculated bond length with DFT or MP2 methods and 𝑐𝑖 expresses the
192
calculated bond length values with the CCSD method.
193
The mean absolute error of bond lengths for the four M4S6 molecules relative to
194
CCSD/LANL08(d) is presented in Figure 2.
195
For all four molecules, it was found that MP2 method underestimates the bond lengths
196
(𝜀𝑚.𝑎.𝑒. = 0.008Å) relative to the CCSD. Similar bond elongation from MP2 to CCSD has been
197
reported on the sulfur-hydrogen bond (rSH) of H2S and the sulfur-sulfur bond in H2S dimers, as
198
well as an excellent agreement of CCSD calculated data with the experimental rSH bond length
199
within 0.001 Å [54]. Moreover, the CCSD method gives the best results for the geometries of
200
larger molecules such as SeO, SeCl, and AsO [55]. Helgaker et al. calculated MP2 geometries
201
with longer bonds than the CCSD ones for 19 molecules consisted of the light-weight elements
202
H, F, O, N and C [56] but this is not applicable to the studied molecules which consisted of
203
heavier atoms P, As, Sb, Bi, and S. Considering the above remarks we selected CCSD as the
204
reference method for our DFT calculations.
8
205
The most accurate functionals are the M06L, B2PLYP, CAMB3LYP and M06 with similar
206
performance according to the 𝜀𝑚.𝑎.𝑒. (Table S2). The ranking of the methods relative to CCSD
207
and based on the 𝜀𝑚.𝑎.𝑒. for all molecules is the following:
208
BP86< B3LYP< wB97XD< M06-2X< MP2< M06L≤ B2PLYP≤ CAMB3LYP≤ M06< CCSD.
209
In terms of 𝜀𝑎.𝑒 (see Table S2) the most accurate methods are wB97XD and CAMB3LYP for
210
P4S6, M06L and M06 for As4S6, B2PLYP and M06 for Sb4S6, and M06 and M06L for Bi4S6
211
(Figures S1-S4). The M06 and M06L methods show high accuracy for the molecules with the
212
higher-weight elements such as As, Sb and Bi while the wB97XD shows an excellent
213
performance for the lowest-weight P4S6.
214 215
3.2. Atomic charges
216
The natural atomic charges as calculated with natural population analysis (NPA) are presented
217
in Table 2. The M and S atoms are positively and negatively charged for all M4S6 molecules,
218
respectively. The charge increases from P to Bi, indicating an increase in the ionic character of
219
the bonds M-S and that the electrons’ density moves from metal to sulfur, as one could predict
220
on the basis of electronegativities. In particular, NPA shows the ability of sulfur atoms to host
221
part of the negative charge provided by the M atoms resulting in the stabilization of the
222
adamantane structure. Both MP2 and CCSD methods provide almost equal values for the
223
natural atomic charges as recorded in Table 2. The metal’s charge values are in the range 0.42
224
– 1.17 for M4S6 species showing a less ionic character of M-S than M-O bond for heavier
225
metals, compared with the values 0.35 - 2.05 from phosphorous to bismuth, for M4O6 species
226
[3].
227 228
3.3. Electrostatic Potential Analysis
229
The electrostatic potential (ESP) on the vdW molecular surface provides information about the
230
strength and the orientation of intermolecular interactions such as hydrogen or halogen bonding
9
231
[57,58] as well as the electrophilic and nucleophilic positions of the molecule where chemical
232
reactions are expected. The following parameters have been described in published work by
233
Murray et al. [27]. 𝑉̅𝑆+ and 𝑉̅𝑆− indicate the mean positive and negative ESP values on the vdW
234
surface respectively, Π is the mean deviation on the surface, which is an index of the internal
235
2 charge separation, and 𝜎𝑡𝑜𝑡 is the total ESP variance which is the sum of the positive 𝜎+2 and
236
the negative 𝜎−2 parts. The greater the 𝜎+2 and 𝜎−2 , the greater the molecule's tendency to interact
237
with other molecules through the positive and negative ESP domains. The degree of charge
238
balance equals ν and when the 𝜎+2 and 𝜎−2 are equal, then ν is maximized and equals 0.250. As
239
long as ν is closer to 0.250, the more likely it is that the molecule interacts with other ones
240
2 through the positive and negative regions to a similar extent. The product 𝜎𝑡𝑜𝑡 𝜈 is also a very
241
useful quantity; a large value indicates a relatively strong tendency to interact with other
242
molecules of the same kind electrostatically.
243
According to the data in Table 3, there is an increase in the values of all parameters V, d, Π,
244
2 2 𝜎𝑡𝑜𝑡 , 𝜎+2 , 𝜎−2 , 𝜈 and 𝜈𝜎𝑡𝑜𝑡 along the group 15, from M = P to M = Bi. The internal charge
245
separation is increased from P4S6 (𝛱 = 6.12 𝑘𝑐𝑎𝑙 𝑚𝑜𝑙 −1 ) to the heavier Bi4S6 (𝛱 =
246
16.14 𝑘𝑐𝑎𝑙 𝑚𝑜𝑙 −1 ). This separation is also confirmed by the charge values (Table 2) where
247
the charge difference between Bi and S atoms in Bi4S6 is greater than that between P and S
248
atoms in P4S6. The 𝜈 parameter shows the degree of charge balance and if it receives the
249
maximum value 𝜈 = 0.25 then the molecule interacts with other molecules to the same extent
250
with its positive and its negatively charged region. Also, the 𝜈 parameter increases from P4S6
251
to Bi4S6, so more intermolecular interactions with different molecules for the heavier Bi4S6 are
252
2 expected. Moreover, a high value of 𝜈𝜎𝑡𝑜𝑡 indicates that a molecule has strong electrostatic
253
2 interactions among other molecules of the same species. The 𝜈𝜎𝑡𝑜𝑡 increases from P4S6
254
2 2 (𝜈𝜎𝑡𝑜𝑡 = 3.55 (𝑘𝑐𝑎𝑙 𝑚𝑜𝑙 −1 )2) to Bi4S6 (𝜈𝜎𝑡𝑜𝑡 = 19.04 (𝑘𝑐𝑎𝑙 𝑚𝑜𝑙 −1 )2) whereby Bi4S6
255
molecules interact strongly with each other. For example, it has been found that in saturated
10
256
2 hydrocarbons 𝜈𝜎𝑡𝑜𝑡
is about 1 (𝑘𝑐𝑎𝑙 𝑚𝑜𝑙 −1 )2
2 while for formamide is 𝜈𝜎𝑡𝑜𝑡 =
257
62.5 (𝑘𝑐𝑎𝑙 𝑚𝑜𝑙 −1 )2 [27] so that it will have strong electrostatic forces through its positive and
258
negative regions and formamide molecules should interact strongly with each other. That was
259
confirmed experimentally after the investigation of the N-H stretching domains at the infrared
260
and Raman spectra in the solid and liquid state of formamide [59]. The Figures S5-S8 illustrate
261
the electrostatic potential surfaces of M4S6 molecules at CCSD/LANL08(d) level of theory. The
262
colors are related to the electrostatic potential values. Red color indicates electronic deficient
263
areas (𝑉(𝑟) > 0) and blue indicates electron rich areas (𝑉(𝑟) < 0).
264 265
3.4. Bonding analysis
266
The analysis of the chemical bonding and of the HOMO-LUMO orbitals was based on results
267
from the CCSD/LANL08(d) method. For the compounds M4S6, the simple σ bond between M
268
and S can be written as: 𝜎𝑀𝑆 = 𝑐𝑀 ℎ𝑀 + 𝑐𝑆 ℎ𝑆
269
(3)
270
2 where 𝑐𝑀 and 𝑐𝑆 are the polarity coefficients of the hybrid orbitals ℎ𝑀 and ℎ𝑆 , while 𝑐𝑀 + 𝑐𝑆2 =
271
1. The bond ionicity parameter provided by the Eq. (4) and quantifies the polarity of the bond
272
[60].
273
𝑐 2 −𝑐 2
𝑆 𝑖𝑀𝑆 = 𝑐𝑀 2 +𝑐 2 𝑀
(4)
𝑆
274
For the P4S6 molecule, all 𝜎𝑃𝑆 bonds are equivalent and each one is expressed according to Eq.
275
(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
276
electrons in an orbital with hybridization 𝑠𝑝0.59 (62.83% 3s of P) while each S atom has two
277
lone pairs of electrons, the first one in 𝑠𝑝0.39 (71.65% 3s of S) and the second one in the 3p of
278
S (99.86%). The |iMS | parameter was found equal to 0.193. In each of the other molecules
279
As4S6, Sb4S6 and Bi4S6, the σ bonds are equivalent, and the lone electron pairs are in a similar
280
configuration to P4S6, i.e. one in M and two in S. The results are summarized in Table 4.
11
281
Based on the NBO theory, it follows from the values of the bond ionicity |iMS | that the most
282
covalent bond M-S exists in P4S6 with |iMS | = 0.193. As M varies from P to Bi, the bond
283
becomes more polar, since for M = Bi, |iMS | = 0.464. For all the studied species it was found
284
that 𝑐𝑀 < 𝑐𝑆 and the bond is more polarized to the S atoms. In addition, the M-S bond is formed
285
from the p valence orbitals of M and the 3p orbital of S. According to Table 4, the participation
286
of the p orbitals of M and of S to the 𝜎𝑀𝑆 bond is increased from P4S6 (𝜎𝑀𝑆 =
287
0.635(𝑠𝑝6.93 𝑑 0.14 )𝑃 + 0.772(𝑠𝑝6.00 𝑑0.07 )𝑆 )
288
0.856(𝑠𝑝7.58 𝑑0.04 )𝑆 ).
289
In molecules with M-S bonds such as [PbII(S2COEt)n]2-n (n=1,2,3,4), Ghosh et al. concluded
290
that the PbII-S bond is formed from the 6p orbitals of PbII and the 3p ones of S atoms [16]. A
291
similar conclusion about the M-S bond was reached for the studied M4S6 molecules in this
292
work. The NBO analysis of the 𝜎𝑀𝑆 bond showed that the donor NBOs are composed mainly
293
of the 3p orbital of sulfur while the acceptor NBOs are mostly comprised of the valence p
294
orbitals of M. Therefore, a general tendency is observed in the M-S bond of association between
295
p valence orbital of the metal and the 3p of sulfur.
296
The lone electron pair of M denoted as nM is found in hybrid orbitals with higher s character.
297
Along the group 15 of the periodic table, the localization of the lone pair increases significantly
298
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,
299
the first lone electron pair of S denoted as nS , is in a hybrid orbital with more s character.
300
From M = P to M = Bi there is small increase in s character: for P4S6 is 71.65% while for Bi4S6
301
it is 76.74%. The second lone electron pair of S, nS
302
molecules.
to
Bi4S6
(𝜎𝐵𝑖𝑆 = 0.518(𝑠𝑝21.00 𝑑0.07 )𝐵𝑖 +
(𝜎)
(𝜎)
(𝜋)
303
12
is located on the 3p orbital in all M4S6
304
3.5. Frontier orbitals analysis
305
The analysis of the frontier molecular orbitals calculated at the CCSD/LANL08(d) level of
306
theory for all M4S6 molecules follows. In all studied molecules, there are three energy-
307
degenerated HOMO orbitals having t1 symmetry (indicated as MO-26, MO-27 and MO-28) and
308
two degenerated LUMO orbitals of e symmetry (indicated as MO-29 and MO-30). These five
309
MO orbitals are presented in Figures 3 and S9-S11.
310
The participation of the atomic orbitals to the aforementioned five frontier orbitals has been
311
analyzed.
312
P4S6: The 3py, 3px and 3pz atomic orbitals of P atoms have the largest contribution (by 34.32%)
313
to the three HOMO orbitals of P4S6 MO-26, MO-27 and MO-28 respectively. Also, these three
314
HOMO orbitals consisted of the 3s orbital of the P atoms, with a contribution of 10.71% and of
315
the 3s of the S atoms with 1.46% participation. The two LUMO orbitals do not consist of 3s of
316
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
317
whereas MO-30 is 30.80% 3px and 28.94% 3pz of P.
318
As4S6: For the As4S6, the 4s of the As atoms contributed a total of 11.97% and the 3s of the S
319
atoms a total of 1.08%. The 4px, 4pz and 4py of As were significantly involved in the three
320
HOMO orbitals MO-26, MO-27 and MO-28 by 30.80% respectively. The two LUMO orbitals
321
did not consist of 4s of As but 11.61% of 3s of S. Also, MO-29 consisted of 37.53% of 4pz and
322
26.26% of 4py of As while MO-30 from 42.20% 4px and 16.94% 4py of As.
323
Sb4S6: In Sb4S6 the three HOMO orbitals consisted mainly of 5py, 5pz and 5px of Sb which
324
contribute 27.69% to MO-26, MO-27 and MO-28 respectively. Also, the three HOMOs
325
consisted of the 5s of the Sb by 16.58%. The two LUMOs were not composed of 5s of Sb but
326
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
327
while MO- 30 from 47.22% 5px and 18.30% 5pz of Sb.
13
328
Bi4S6: For the Bi4S6 the three HOMO orbitals consisted of the 6s orbitals of Bi by 12.04% and
329
of the 3s of S atoms by 0.52%. The 6py, 6pz and 6px participated in MO-26, MO-27 and MO-
330
28 by 20.38% respectively. The two LUMOs were not composed of 6s of Bi but of 9.10% of
331
3s of S. Furthermore, MO-29 consisted of 48.00% of 6px and 14.61% of 6pz of Bi. The MO-30
332
was composed of 38.54% 6py and 33.57% 6pz of Bi atoms.
333 334
All contributions of the atomic orbitals to the aforementioned HOMOs and LUMOs for M4S6
335
molecules are provided in the Table 5 of this paper. In all studied molecules, it was found that
336
their HOMO consisted mainly of the p valence orbitals of the M atoms and of the 3p orbitals of
337
the S atoms. Along the group 15, the contribution of 3p of S atoms is significantly increased,
338
the 3py of the P4S6 participates 37.64% in MO-26 while in Bi4S6 3py participates by 55.41%. In
339
addition, the contribution of the p valence orbitals of M to the LUMOs is significant in all M4S6
340
molecules.
341 342
For the energy gaps between HOMO and LUMO orbitals at the CCSD/LANL08(d) level of
343
theory, it was found that from M = P to M = As there is a decrease in these values whereas there
344
is no significant difference between Sb4S6 and Bi4S6 molecules. That is, the following
345
relationship applies to the HOMO-LUMO energy gaps:
346 347
𝛥𝐸𝑃4 𝑆6 (10.5 𝑒𝑉) > 𝛥𝐸𝐴𝑠4 𝑆6 (10.0 𝑒𝑉) > 𝛥𝐸𝑆𝑏4 𝑆6 , 𝛥𝐸𝐵𝑖4 𝑆6 (9.3 𝑒𝑉)
348 349
Since all 𝛥𝐸𝑀4 𝑆6 values are high, M4S6 molecules are expected to be chemically stable.
350
14
351
3.6. Molecular Polarizability
352
The perturbation of electron density when a molecule interacts with the electric field of
353
radiation, ions, polar molecules, etc. is fundamental to understand the behavior of molecules in
354
chemical reactions, their solvation properties, the recognition processes and spectroscopic
355
properties. Electric polarizability is currently of importance as it is extensively used to model
356
intermolecular interactions [61], basic molecular characteristics as the acidity and basicity [62],
357
hardness and softness [63–65] and chemical reactivity.
358
Our polarizability calculations were performed at the CCSD(full)/LANL08(d) optimized
359
geometry, using the MP2(full)/LANL08(d) theoretical level. Due to high symmetry there is
360
only one independent component of the polarizability tensor: αxx = αyy = αzz = 𝛼̅.
361
In atomic units the calculated values of the mean polarizability are:
362
𝛼̅(𝑃4 𝑆6 ) = 193.91, 𝛼̅(𝐴𝑠4 𝑆6 ) = 217.87, 𝛼̅(𝑆𝑏4 𝑆6 ) = 267.46, 𝛼̅(𝐵𝑖4 𝑆6 ) = 285.82 e2α02Eh-1.
363
Similar calculations were performed for the M4O6 molecules:
364 365
𝛼̅(𝑃4 𝑂6 ) = 89.106, 𝛼̅(𝐴𝑠4 𝑂6 ) = 107.17, 𝛼̅(𝑆𝑏4 𝑂6 ) = 142.24, 𝛼̅(𝐵𝑖4 𝑂6 ) = 155.27 e2α02Eh-1
366
A quantity related to polarizability is the polarizability volume, α', defined as α' = α/(4πε0)
367
For M4S6 molecules these volumes have been calculated:
368
α'(P4S6)= 28.705, α'(As4S6)= 32.252, α'(Sb4S6)= 39.593, α'(Bi4S6)= 42.312 Å3
369
As it is expected polarizability and polarizability volume increase as the size of atoms (P to Bi
370
and O to S) increases.
371 372
3.7. Vibrational properties
373
The highly symmetric M4S6 Td molecules have the following irreducible representation of the
374
fundamental vibrations:
375
Γvib. = 2Α1(R) + 2Ε(R) + 2T1(i.a.) + 4T2(IR,R)
15
376
The vibrational modes having A1, E and T2 symmetry are Raman (R) active and only the ones
377
having T2 symmetry are IR active (IR). The vibrational modes having T1 symmetry are both IR
378
and Raman forbidden (i.a.).
379
In Table 6 the harmonic wavenumbers calculated at the optimized geometry and their
380
symmetries, are presented.
381
A non-linear molecule with N atoms exhibits 3N - 6 normal modes of vibration. There are 24
382
fundamental vibrations - symmetric and antisymmetric – for all M4S6 cages. Each fundamental
383
mode can be expressed by a normal coordinate. Then, those coordinates are transformed to
384
internal ones, which are a superposition of multiple local modes including stretching, bending
385
and torsion [29,30]. The Greek letters ν, β, δ, τ denote stretching, in plane bending, out of plane
386
bending and torsion modes respectively (Table 6). By using the symmetry considerations and
387
theoretical results for IR and Raman intensities, it is possible to assign the predicted energies
388
of vibrations to specific vibrational modes.
389
For all systems investigated here and for all calculated modes, the corresponding vibrational
390
wavenumbers decrease when the atomic number of the central atom increases. For the M-S
391
stretching vibrations this decrease is expected since the quadratic M-S force constants decrease
392
in the series from P to Bi whereas the reduced mass increases reinforcing the behavior expected
393
from the quadratic force constant.
394
There is also a correlation between bond angles and hybridization. The percentage of s character
395
of the M to the 𝜎𝑀𝑆 bond decreases from P to Bi. However, the contribution of the p orbitals of
396
M increases from P to Bi. In addition, we found that the angle ∠S-M-S decreases from P4S6 to
397
Bi4S6. Therefore, P4S6 with the highest %s contribution of phosphorus to the 𝜎𝑀𝑆 bond, has the
398
largest value of ∠S-P-S equal to 106.97°. This trend is in agreement with the prediction of Bent
399
rule, which was originally derived from sp-hybridized main group elements [66].
16
400
4. CONCLUSIONS
401 402
In the present work, the structural, electronic and vibrational properties of the M4S6, M = P, As,
403
Sb, Bi inorganic clusters were investigated by means of computer simulation methods. The
404
calculations were performed with the MP2, CCSD and DFT methods employing the basis set
405
LanL08(d) with ECP for the P, As, Sb, Bi, S atoms. Comparing with the available CCSD
406
theoretical data, we concluded that M06 and M06L functionals show high accuracy for
407
molecules with the higher-weight elements such as As, Sb, and Bi while the wB97XD and
408
CAMB3LYP show an excellent performance for the lowest-weight P4S6 concerning the
409
prediction of the structural data. We also present for the first time data for the hypothetical
410
Sb4S6 and Bi4S6 molecules. The vibrational spectral analysis is presented for the whole series
411
of group 15 metal sulfides. A structural ‘relaxation’ ongoing from phosphorous to bismuth
412
indicates an increase of ionic character and in some extent can explain the instability of these
413
compounds when the atomic number of the metal increases.
17
414
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22
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).
24
e
t1
Figure 3. Molecular orbital diagrams of HOMO and LUMO orbitals of P4S6.
25
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)
P4S6
Bi4S6
2.149
CCSD
wB97XD
M06-2X
M06L
M06
BP86
CAMB3LYP
B2PLYP
B3LYP
2.158
2.156
2.154
2.169
2.168
2.187
2.156
2.164
2.177
∠P-S-P (°)
114.271 [113.999]
114.282
114.445
114.683
114.610
114.716
114.430
114.432
114.407
114.482
∠S-P-S (°)
106.969 [107.115]
106.963
106.874
106.744
106.784
106.726
106.882
106.881
106.894
106.854
[2.248]
2.274
2.265
2.265
2.279
2.278
2.301
2.268
2.279
2.291
∠As-S-As (°)
115.748 [115.301]
115.860
116.118
116.254
116.112
116.298
115.611
116.123
115.840
115.876
∠S-As-S (°)
106.159 [106.406]
106.097
105.953
105.877
105.956
105.853
106.234
105.950
106.108
106.088
2.454
2.461
2.444
2.450
2.468
2.463
2.489
2.454
2.467
2.477
∠Sb-S-Sb (°)
117.594
117.952
118.365
118.428
117.962
118.342
117.095
118.321
117.685
117.689
∠S-Sb-S (°)
105.122
104.918
104.681
104.645
104.912
104.694
105.405
104.707
105.070
105.068
r(Bi-S) (Å)
2.525
2.532
2.515
2.519
2.534
2.528
2.557
2.524
2.538
2.548
∠Bi-S-Bi (°)
117.385
117.961
118.369
118.349
118.009
118.275
116.783
118.325
117.497
117.473
∠S-Bi-S (°)
105.241
104.913
104.679
104.690
104.885
104.733
105.581
104.704
105.177
105.191
r(Sb-S) (Å) Sb4S6
MP2
DFT methods
[2.136]
r(As-S) (Å) As4S6
ab-initio methods
2.266
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
0.665
-0.444
0.663
-0.442
Sb4S6
1.073
-0.715
1.077
-0.718
Bi4S6
1.154
-0.769
1.170
-0.780
27
Table 3. Electrostatic potential surface analysis results at CCSD/LANL08(d) level of theory. VvdW
d
Π
2 𝜎𝑡𝑜𝑡
𝜎+2
𝜎−2
(Å3)
(g cm-3)
(kcal mol-1)
(kcal mol-1)2
(kcal mol-1)2
(kcal mol-1)2
𝜈
2 𝜈𝜎𝑡𝑜𝑡
(kcal mol-1)2
P4S6
285.09
1.84
6.12
28.43
24.26
4.17
0.125
3.55
As4S6
306.78
2.67
7.85
38.91
32.38
6.53
0.139
5.44
Sb4S6
350.17
3.22
11.17
59.84
47.81
12.03
0.161
9.61
Bi4S6
360.56
4.74
16.55
111.42
87.05
24.37
0.171
19.04
28
Table 4. NBO chemical bonding and bond ionicity parameter for all M4S6 molecules. (𝝈)
(𝝈)
(𝝅)
|𝐢𝐌𝐒 |
𝝈𝑴𝑺
𝐧𝐌
P4S6
𝜎𝑀𝑆 = 0.635(𝑠𝑝6.93 𝑑 0.14 )𝑃 + 0.772(𝑠𝑝6.00 𝑑 0.07 )𝑆
𝑠𝑝0.59
𝑠𝑝0.39
3𝑝
0.193
As4S6
𝜎𝐴𝑠𝑆 = 0.599(𝑠𝑝9.78 𝑑 0.13 )𝐴𝑠 + 0.800(𝑠𝑝6.53 𝑑 0.06 )𝑆
𝑠𝑝0.38
𝑠𝑝0.38
3𝑝
0.282
Sb4S6
𝜎𝑆𝑏𝑆 = 0.534(𝑠𝑝12.59 𝑑 0.11 )𝑆𝑏 + 0.846(𝑠𝑝6.58 𝑑 0.04 )𝑆
𝑠𝑝0.28
𝑠𝑝0.36
3𝑝
0.430
Bi4S6
𝜎𝐵𝑖𝑆 = 0.518(𝑠𝑝21.00 𝑑 0.07 )𝐵𝑖 + 0.856(𝑠𝑝7.58 𝑑0.04 )𝑆
𝑠𝑝0.15
𝑠𝑝0.30
3𝑝
0.464
M4S6
29
𝐧𝐒
𝐧𝐒
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
Symmetry
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)
P4S6
P4S6 (TZVp)
As4S6
As4S6 (TZVp)
Sb4S6
Bi4S6
139 158 [15] 205 [11,2] 294 [2,4] 350 [76] 390 [1,0] 391 456 [2] 431 [0] 480 [2,81]
143 163 [16] 212 [15,2] 302 [1,5] 362 [69] 404 [1,0] 420 474 [2] 441 [1] 502 [1,84]
106 116 [11] 148 [11,2] 212 [0,6] 244 [29] 314 [0,5] 312 326 [7] 340 [43] 397 [2,75]
107 120 [13] 152 [13,2] 217 [0,7] 250 [21] 331 [0,4] 344 343 [7] 356 [48] 417 [1,66]
79 85 [10] 108 [11,3] 158 [0,6] 181 [16] 271 [1,12] 286 273 [8] 287 [69] 372 [2,110]
70 63 [8] 82 [2,8] 133 [1,9] 133 [9] 245 [1,19] 268 240 [13] 266 [83] 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.
