molecules
Article
Fast Quantum Approach for Evaluating the Energy of
Non-Covalent Interactions in Molecular Crystals: The Case
Study of Intermolecular H-Bonds in Crystalline Peroxosolvates
Alexander G. Medvedev 1 , Andrei V. Churakov 1 , Mger A. Navasardyan 1 , Petr V. Prikhodchenko 1 ,
Ovadia Lev 2 and Mikhail V. Vener 1, *
1
2
*
Citation: Medvedev, A.G.;
Churakov, A.V.; Navasardyan, M.A.;
Prikhodchenko, P.V.; Lev, O.;
Vener, M.V. Fast Quantum Approach
for Evaluating the Energy of
Non-Covalent Interactions in
Molecular Crystals: The Case Study
of Intermolecular H-Bonds in
Crystalline Peroxosolvates. Molecules
2022, 27, 4082. https://doi.org/
Kurnakov Institute of General and Inorganic Chemistry, Russian Academy of Sciences, Leninskiy Prospekt 31,
119991 Moscow, Russia; mag@igic.ras.ru (A.G.M.); churakov@igic.ras.ru (A.V.C.);
navasardyan@igic.ras.ru (M.A.N.); prikhman@gmail.com (P.V.P.)
The Casali Center of Applied Chemistry, The Institute of Chemistry, The Hebrew University of Jerusalem,
Jerusalem 9190401, Israel; ovadia@mail.huji.ac.il
Correspondence: vener@igic.ras.ru
Abstract: Energy/enthalpy of intermolecular hydrogen bonds (H-bonds) in crystals have been calculated in many papers. Most of the theoretical works used non-periodic models. Their applicability
for describing intermolecular H-bonds in solids is not obvious since the crystal environment can
strongly change H-bond geometry and energy in comparison with non-periodic models. Periodic
DFT computations provide a reasonable description of a number of relevant properties of molecular
crystals. However, these methods are quite cumbersome and time-consuming compared to nonperiodic calculations. Here, we present a fast quantum approach for estimating the energy/enthalpy
of intermolecular H-bonds in crystals. It has been tested on a family of crystalline peroxosolvates in
which the H···O bond set fills evenly (i.e., without significant gaps) the range of H···O distances from
~1.5 to ~2.1 Å typical for strong, moderate, and weak H-bonds. Four of these two-component crystals
(peroxosolvates of macrocyclic ethers and creatine) were obtained and structurally characterized for
the first time. A critical comparison of the approaches for estimating the energy of intermolecular
H-bonds in organic crystals is carried out, and various sources of errors are clarified.
Keywords: peroxosolvates; macrocyclic ether; amino acid; periodic DFT computations; bifurcated
H-bonds; multicomponent crystals; B3LYP vs. PBE-D3
10.3390/molecules27134082
Academic Editor: Yanlu Li
Received: 18 May 2022
1. Introduction
Accepted: 21 June 2022
A quantitative description of intermolecular (non-covalent) interactions in the gas phase
can be achieved using a variety of theoretical approaches: NBO analysis [1], SAPT [2], etc. [3–5].
There is a large amount of experimental data from which “reference” values of the
energy/enthalpy of intermolecular interactions can be obtained [6–10]. Molecular
structure, chemical bond theory, and “standard” quantum chemical approaches focus
on non-periodic systems. Their “generalization” to molecular crystals is not straightforward because of the need to explicitly take into account the periodic boundary
conditions [11,12]. The identification and quantitative description of intermolecular
interactions in crystals is a non-trivial task, both experimentally and theoretically.
An exception are molecular crystals in which the sublimation enthalpy/lattice energy is determined by one type of intermolecular (non-covalent) interactions, such as
CF4 [13,14]. A quantitative characteristic of intermolecular interactions in a crystal
is the lattice energy or sublimation enthalpy, extrapolated to 0 K [15]. A number of
theoretical approaches to the lattice energy assessment are reported in the literature.
They mostly concern single-component crystals and use either quantum chemical modeling [16–18] or semi-empirical schemes [19–21]. In most of the papers cited above, short
Published: 24 June 2022
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Molecules 2022, 27, 4082. https://doi.org/10.3390/molecules27134082
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Molecules 2022, 27, 4082
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(strong) intermolecular H-bonds [22] are not considered since they are practically not
realized in single-component crystals. On the other hand, such H-bonds determine the
structure of multicomponent molecular crystals used in pharmaceuticals [23–25] and
crystal engineering [26]. Intermolecular H-bonds of different types and strengths play
a decisive role in the structure of crystal hydrates [27,28] and peroxosolvates [29,30].
Currently, there is a great demand for approaches that make it possible to quantitatively
describe various intermolecular (non-covalent) interactions in organic crystals. In most
cases, empirical approaches are used that relate the energy of the intermolecular interaction
with one or another parameter of the electron density at the bond critical point [31]. In this
case, the calculated values of the electron density, the values of the parameters retrieved
from the precise X-ray diffraction data, and hybrid approaches are used [32]. The above
approaches are often used to evaluate intermolecular interactions of different natures in
various crystals [33–35], which gives rise to well-founded criticism [36,37]. Possible reasons
for this criticism include the following. (1) The identification of a non-covalent interaction
is based solely on the presence of the bond critical point [38]. (2) The limits of applicability of these approaches are not clear. For example, the approach developed in ref. [31]
was used to estimate energies from ~1.5 [33] to ~600 kJ/mol [39]. (3) The dependence of
the theoretical values of the electron density parameters at the bond critical point on the
approximations used in the calculations by the DFT method has not been studied.
Peroxosolvates of organic molecules are convenient objects for a critical comparison of various DFT functionals and empirical approaches to assessing the energy of
intermolecular interactions, that is, conventional H-bonds. At first, identification of
the conventional H-bonds in crystals may be made without Bader analysis of periodic
electron density [40]. Secondly, the enthalpy of H-bonds can be estimated by methods
that do not use electron density characteristics, namely, changes in IR intensities [41],
frequency shifts [40], and metric parameters of intermolecular H-bonds [42]. Third, organic crystals imply calculations with B3LYP and PBE-D3 functionals, which are widely
used in calculations for periodic [12,14,17,18,34,43] and non-periodic [35] structures.
Finally, H-bonds of various nature and strengths are realized in peroxosolvates [30].
In the present paper, we focus on the applicability of the B3LYP and PBE-D3 functionals
for evaluating the geometric parameters of intermolecular H-bonds of different types and
strengths in peroxosolvates of macrocyclic ethers and amino acids (Scheme 1). Four of
these two-component crystals were obtained and structurally characterized for the first
time. A critical comparison of several methods for estimating the energy of intermolecular
H-bonds is carried out, and various sources of errors are clarified. Finally, a fast quantum
approach for evaluating the energy of non-covalent interactions in molecular crystals is
suggested. This approach is based on non-periodic (cluster) models. Its accuracy and limits
of applicability are demonstrated.
Scheme 1. Structures of the macrocyclic ethers (1–3) and creatine (4), the peroxosolvates I–IV of
which were obtained and studied by X-ray diffraction.
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2. Results and Discussion
2.1. Crystal Structures and Hydrogen Bonding
The structure of 18-crown-6 peroxosolvate C12 H24 O6 • 5H2 O2 (I) comprises one
independent cyclic ether molecule lying on a crystallographic inversion center and
three solvent peroxide molecules (Figure 1). Two of them occupy general positions,
and the third, namely H8-O8-O8A-H8A, is sited on a crystallographic two-fold axis.
The molecule H4-O4-O5-H5 is disordered over two sites with an occupancy ratio
of 0.927(3)/0.083(3) (Figure 1). In I, the ether molecule adopts an ordinary crown
conformation with O1, O3, and O2A atoms lying on one side of the molecule and
O2, O1A, and O3A on the other. The C–O bond lengths vary within the normal
range (1.4209(12)–1.4288(11) Å) for 18-crown-6 derivatives (CSD ver. 5.43 November
2021 [44]). All oxygen atoms of organic coformer are involved in one hydrogen bond
with neighboring H2 O2 molecules. In the solvent molecules, O-O distances lie within the
normal range for peroxosolvates (1.441(10)–1.4692(10) Å) [45]. The torsion H–O–O–H
angles (80.8(18)–139.4(15)◦ ) are also typical for solvent peroxide molecules.
Figure 1. Asymmetric unit of I. Minor components of disorder are shown by open lines. Displacement
ellipsoids are shown at the 50% probability level. The H-bonds are given by the dashed lines.
As in the structures of all known peroxosolvates, H2 O2 molecules in I participate
in two donor hydrogen bonds [30]. Additionally, two of three peroxide molecules are
involved in acceptor H-bonding. In the case of H6-O6-O7-H7 molecules, this hydrogen
bonding results in the unprecedented formation of 2-fold cyclic dimers with R2 2 (6)
topology (Figure S1). Of interest, atom H7 is involved in bifurcated H-bonding, and I
represents the second example of a structure containing an H2 O2 molecule with a bifurcated donor hydrogen bond [28]. Recently, it was supposed that the usage of organic
coformers without donor H-atoms results in the targeted formation of peroxosolvates
with infinite 1D-hydrogen peroxide clusters [46,47]. The structure of I is direct evidence
of this rule (Figure 2). Within the chains, central network-forming H2 O2 molecules
are hydrogen bonded only with adjacent peroxides, while lateral molecules are additionally linked to crown ether coformers by donor HOOH ··· OR2 bonds. To the best of
our knowledge, this untrivial chain topology was not observed previously either for
peroxosolvates or for hydrate structures.
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Figure 2. Infinite H-bonded peroxide chains in the structure I. Minor parts of disorder are not shown.
Hydrogen bonds are indicated to be dashed lines.
Trialkyl amines easily oxidize by hydrogen peroxide forming N-oxides [48]. We
found that [2.2.2]-cryptand follows this rule and transforms to di-N-oxide in H2 O2 solutions. The asymmetric unit in the structure of [2.2.2]-cryptand di-N-oxide peroxosolvate
C18 H36 N2 O8 •3H2 O2 (II) contains one cryptand coformer and three peroxide molecules
(Figure 3). All molecules lie on general positions. In the organic molecule, all bond lengths
and angles have ordinary values [44]. The coformer adopts “OO” conformation [49] with all
ether oxygen atoms directed toward the center of the molecule. Obviously, this compact conformation prevents ether oxygen atoms from hydrogen bonding and denies the inclusion
of solvents inside the cryptand skeleton. All solvent peroxide molecules possess skew conformation (85(2)-96(2)◦ ) with O-O bond lengths varying within 1.4561(19)-1.4675(18) Å. All
H2 O2 molecules participate only in two approximately linear (170(2)-178(2)◦ ) donor hydrogen bonds HOOH···− O-N+ with short O···O separations (2.6669(17)-2.7732(16) Å). Of interest, other possible H-bonds (HOOH···OR2 or intersolvent HOOH···O2 H2 ) are not found in
the structure. Probably, the latter results from the fact that charge-assisted HOOH···− O-N+
hydrogen bonds are energetically preferable than all other possible HOOH···O interactions
in II. Of interest, both N-oxide oxygen atoms are involved in three hydrogen bonds with
N+ -O···H angles ranging within 97.2–123.3◦ .
Figure 3. Asymmetric unit of II. Displacement ellipsoids are shown at the 50% probability level. The
H-bonds are given by the dashed lines.
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In the crystal, the neighboring molecules are combined in N+ -O− ···(HOOH)n···− O-N+
(n = 1, 2) synthons. These fragments were shown to be structure-forming in peroxosolvates
of organic N-oxides [50]. Finally, H-bonded layers perpendicular to ab-diagonal are formed
in the structure II (Figure 4; Figure 5). To the best of our knowledge, II is the second
example of a crystal structure containing [2.2.2]-cryptand di-N-oxide molecule [51].
Figure 4. Hydrogen bonded layer in II. Hydrogen bonds are shown by dashed lines.
Figure 5. Hydrogen bonded layers in II. View along abc-diagonal.
The structure of dibenzo-18-crown-6 (db18c6) peroxosolvate C20 H24 O6 •4H2 O2 (III)
comprises one independent cyclic ether molecule and four solvent peroxide molecules. All
molecules lie on general positions (Figure 6). The metric parameters of the macrocyclic
ether are ordinary with bond lengths close to those found in I. The organic coformer
adopts an unusual “planar” conformation with approximately parallel aromatic rings
(dihedral angle 1.5(4)◦ ) and with both alkyl oxygen atoms O2 and O5 deviating in opposite
directions from the mean plane of the molecule. In contrast, db18c6 possesses a bent
conformation in the unsolvated structure [52], which coincides with the most stable ground
state conformation in the gaseous phase [53]. According to CSD, a bent conformation
with a dihedral angle between the aromatics of greater than 30◦ is also preferable for
metal and H-bonded complexes of db18c6 (290 of total 298 entries). The remaining eight
entries (BOYVED, DIHZEA, NOKVIU, OYAHAZ02, RAXJAB, RORXOL, VADTEZ, and
XIGPEJ) correspond to planar conformation with a dihedral angle less than 5◦ , most of
them possessing a crystallographically imposed central symmetry. Five ether oxygen atoms
participate in hydrogen bonds with hydrogen peroxide. Only catecholate atom O3 is not
involved in H-bonding. The peroxide molecules possess skew conformation (77(7)-117(7)◦ )
with O-O bond lengths ranging from 1.427(6) to 1.465(6) Å. All of them participate in
two donor hydrogen bonds and one inter-solvent acceptor HOOH···O2 H2 linkage. As
expected, the latter results in formation of hydrogen-bonded peroxide chains with only
lateral peroxide molecules H11-O11-O12-H12 and H41-O41-O42-H42 interacting with the
organic coformer (Figure 7).
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Figure 6. Asymmetric unit of III. Displacement ellipsoids are shown at the 50% probability level. The
H-bonds are given by the dashed lines.
Figure 7. Hydrogen bonded chains along a-axis in III. Hydrogen bonds are shown by dashed lines.
The asymmetric unit in the structure of creatine peroxosolvate C4 H9 N3 O2 •2H2 O2
(IV) contains one independent zwitterionic molecule of natural amino acid and two H2 O2
molecules (Figure 8). All geometric parameters of the organic molecule (including conformation) are close to those previously found for the structures of creatine monohydrate and
three polymorphs of unsolvated creatine [54–56].
In these four structures, the characteristic torsion angle C-N(Me)-CH2 -C between
two planar fragments (guanidinium and amino acid) varies from 68 to 76◦ , while in IV, this
angle is equal to 81.9(1)◦ . In IV, all creatine hydrogen atoms are involved in H-bonding
(Figure 8). Of interest, all four sp2 -hybridized lone electron pairs of the carboxylate group
participate in hydrogen bonds with neighboring peroxide molecules resulting in [SA; SA]
linkage mode [57].
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Figure 8. Hydrogen bonding of creatine in IV. Hydrogen bonds are shown by dashed lines.
H2 O2 molecules adopt skew geometry (96.4(15) and 96.9(16)◦ ) with O-O bond lengths
equal to 1.4602(11) and 1.4643(10) Å. They both are involved in two donor HOOH···− O2 Cand two acceptor NH···O2 H2 hydrogen bonds. In the crystal, creatine and peroxide
molecules are combined in a complex 3D hydrogen-bonded motif (Figure 9).
Figure 9. Three-dimensional packing motif in IV. Hydrogen bonds are shown by dashed lines.
The crystal structure of phenylserine peroxosolvate (V) (Figure S5) was discussed in
detail previously [58].
2.2. H-Bond Enthalpies/Energies in Peroxosolvates of Macrocyclic Ethers and Amino Acids
It was noted that there are several empirical approaches to assessing the enthalpy
∆HHB of the intermolecular H-bond in a solid. The most accurate of these is Iogansen’s
approach to exploring changes in IR intensities [41]. In contrast to the liquid state [59],
the experimental determination of the integral intensities of OH stretching vibrations in
crystals is a non-trivial problem. Periodic DFT calculations of IR intensities are usually
performed in the “double harmonic approximation” [60]. This has limited applicability for
Molecules 2022, 27, 4082
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estimating the frequencies and IR intensities of stretching vibrations of OH/NH groups
involved in the formation of short (strong) H-bonds due to significant mechanical and
electro-optical anharmonicity. A theoretical description of the considered bands beyond the
“double harmonic approximation” is a time-consuming procedure [61–63]. On the other
hand, the Iogansen approach exploring changes in OH stretching frequencies to estimate
∆HHB has limited applicability to ionic H-bonds [30] since some + N-H···O− bonds are not
realized in the gas phase. Moreover, the application of this approach to bifurcated H-bonds
is not obvious. This is why we use the approaches described below.
The ∆HHB values are estimated using the Rozenberg approach [42]:
−∆HHB [kJ/mol] = 0.134·R(H···O)−3.050
(1)
where the R(H···O) is the H···O distance (nm). Equation (1) describes −∆HHB varying from
10 to 80 kJ/mol [42]. The R(H···O) values are obtained as a result of geometry optimization.
The energy of intermolecular H-bonds EHB in the considered crystals is evaluated
according to ref. [64] as:
EHB [kJ/mol] = 1124·Gb [atomic units]
(2)
where Gb is the positively-defined local electronic kinetic energy density at the H···O bond
critical point [38]. Equation (2) will be referred to as the Gb –EHB approach below.
We start with crystal I (crown Figure S1) since it has weak H-bonds (−∆HHB < 25 kJ/mol,
R(H···O) > 1.80 Å and ρb < 0.04 a.u.). Approaches (1) and (2) lead to similar results for these
H-bonds (Table S1a). Moderate H-bonds (25 < −∆HHB < 50 kJ/mol, 1.80 > R(H···O) > 1.60 Å
and 0.04 < ρb < 0.06 a.u.) exist in crystalline II (cryptand), III (benzocrown), and creatine
diperoxosolvate IV, (Figures S2–S4). The differences between the two approaches can reach
10 kJ/mol (Tables S2a–S4a), and the absolute enthalpy values always turn out to be lower
than the calculated values of the energy of the moderate H-bonds. The obtained results
agree with the literature data [65–68].
For short (strong) H-bonds (−∆HHB > 50 kJ/mol, R(H···O) < 1.60 Å and ρb > 0.60 a.u.)
which exist in crystalline 3-phenylserine peroxosolvate V [69] (refcode VILGAB, Figure S5)
the differences between −∆HHB and EHB can be 15 kJ/mol (Table S5a) or more (Table 2
in [65], Table 1 in [68]).
The −∆HHB and EHB values evaluated for the considered molecular crystals versus
the calculated H···O distances are given in Figure 10. With a decrease in the H···O distance
(that is, an increase in the strength of H-bonds), the difference between the energy values
calculated using Equations (1) and (2) increases. Since the accuracy of the Rozenberg
approach for the considered H-bonds is several kJ/mol, we conclude that Equation (2) can
significantly overestimate EHB for short (strong) H-bonds in molecular crystals.
The specific feature of Equations (1) and (2) is the possibility of using the Gb and
R(H···O) values obtained from experiments (high precision X-ray diffraction, neutron
diffraction) or the periodic DFT computations followed by the Bader analysis of the periodic
electronic density. From a theoretical point of view, Equation (1) presents the simplest case
because the R(H···O) values are obtained as a result of geometry optimization. From the
experimental point of view, the applicability of this approach is not straightforward due to
the usual lack of accuracy that X-ray structures have on the H atoms. It should be noted that
neutron diffraction data on H-bonded crystals are very limited [69]. Initially, Equation (2)
was developed for gas-phase systems with intermolecular H-bonds with energies ranging
from 3 to 90 kJ/mol [31,64]. Currently, approaches that use the characteristics of the electron
density at the point (3, −1) are used to estimate the energies of non-covalent interactions
in various systems in gas and condensed media, and the EHB values vary within wide
limits [33,39]. From a theoretical point of view, errors in the energy values obtained using
Equation (2) arise due to inaccuracies in the calculated values of the metric parameters of Hbonds and the use of non-periodic models in the computations. From an experimental point
of view, errors in the EHB values are due to the use of the Kirzhnitz approximation [70].
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Figure 10. The H-bond enthalpies, −∆HHB (Equation (1), blue circles) and the energies, EHB (Equation (2), red circles), evaluated for the considered molecular crystals versus the H···O distances
computed at the B3LYP/6-31G** level.
To estimate the error in determining the −∆HHB value caused by the use of B3LYP,
the R(H···O) values were computed at the PBE-D3/6-31G** level (Tables S6–S8). The
−∆HHB values calculated using PBE-D3 agree nicely with those obtained using B3LYP. In
accordance with the literature ([28] and Table S1a in [71]), the PBE-D3 calculations slightly
overestimate the H-bonded energy compared to the B3LYP calculations.
2.3. Fast Quantum Approach for Evaluating the Energy of Intermolecular H-Bonds in
Molecular Crystals
The cluster approximation is widely used to describe the structure and properties
of organic crystals: the energy of the intermolecular interactions [72,73], the mobility of
charges in organic semiconductors [74,75], and the chemical shift of the nucleus [76]. This
requires knowledge of the structure of the considered crystal, that is, the cif file. Usually,
a cluster consisting of several molecules is cut out of a crystal, and the geometry is partially
optimized, i.e., the position of only hydrogen atoms is varied [77]. In contrast to C-H
bonds whose length can be uniformly normalized [78], the length of OH and NH depends
on the strength of the H-bonds formed by these groups (Table 2 in [22]). Note that the
complete optimization of the cluster under non-periodic conditions can significantly change
the mutual orientation of the molecules, that is, the number and strength of the studied
H-bonds [79].
The cluster size is determined by the condition that all “unique” H-bonds that occur in
the crystal under study are taken into account. In some cases, it can be quite large. For all
the crystals considered, except for crystal III, the cluster includes no more than 100 atoms.
Calculations of such clusters are an order of magnitude faster than calculations of crystals
(geometry optimization). The total enthalpies/energies of all unique H-bonds in each of the
studied crystals, calculated using periodic DFT, are given in Tables SXa in Supplementary
Materials, where X = 1, 2, . . . 5. The corresponding values obtained in the cluster approximation are given in Tables SXb in Supplementary Materials, where X = 1, 2, . . . 5. It follows
that the total enthalpies/energies calculated using the cluster approximation are in good
agreement with the values obtained from periodic DFT calculations. This justifies the use
of the cluster approximation for estimating the energies of H-bonds in molecular crystals.
The Gb -EHB approach gives reasonable results for energies from 2.0 [79] to 50 kJ/mol and is
applicable to describe bifurcated H-bonds formed by the OH group. This has been called
the fast quantum approach.
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It should be emphasized that the analysis of normal vibrations of the cluster structure,
obtained as a result of partial optimization usually leads to imaginary frequencies. Therefore, this approximation is hardly applicable for calculating the experimentally observed
characteristics of molecular crystals, namely: sublimation enthalpies [17,18,43], IR [80,81]
and Raman spectra [68,71].
Unlike Equation (1), the Gb -EHB approach can be applied to intramolecular H-bonds [82,83],
which play a structure-directed role in pharmaceutical multicomponent organic crystals [84,85]. It should be noted that empirical approaches use different geometric parameters to estimate the energies of inter- and intramolecular H-bonds in organic crystals,
compare [42] with [86].
3. Materials and Methods
3.1. Compounds and Solvents
Macrocyclic ethers (dibenzo-18-crown-6, 18-crown-6, [2.2.2]cryptand), creatine, and
50% hydrogen peroxide were purchased from Sigma-Aldrich (St. Louis, MO, USA), Protein Company (Moscow, Russia), and Fisher Scientific (Waltham, MA, USA), respectively.
In total, 98% hydrogen peroxide was prepared by an extraction method [87]. Handling
procedures for concentrated hydrogen peroxide are described in detail (danger of explosion!) [88,89].
3.2. Cocrystal Preparation
Colorless crystals of I–IV were obtained by cooling to −21 ◦ C saturated solutions (rt)
of respective compounds in 98% hydrogen peroxide. Compounds I and III are stable just
for a few minutes without mother liquor, while crystal IV is stable for several hours. In
contrast, the peroxosolvate of cryptand II demonstrates high stability and does not exhibit
noticeable decay for several months.
3.3. Single Crystal X-ray Diffraction Experiments
The crystals were extracted from the mother liquor and covered immediately with
inert oil to prevent contact with atmospheric moisture. Experimental data sets were
collected on a Bruker SMART APEX II diffractometer (for I, III, and IV) and a Bruker
SMART Photon II diffractometer (for II). Absorption corrections based on measurements
of equivalent reflections were applied [90]. The structures were solved by direct methods
and refined by full-matrix least-squares on F2 with anisotropic thermal parameters for all
non-hydrogen atoms [91]. The studied thin plate crystal of II was found to be twinned
by the law (1–10) [5–31] with an approximate domain ratio of 0.79/0.21. Both domains
were indexed separately, and full data were integrated and scaled for absorption with
two matrices (APEX III software). The data for the main domain were used for calculations.
In I, one of the peroxide molecules was disordered over two sites with an occupancy ratio
of 0.927(3)/0.083(3). All hydrogen atoms in I and IV were found from difference Fourier
synthesis and refined with isotropic thermal parameters (except for a minor part of the
disordered H2 O2 molecule). In II and III, all carbon H atoms were placed in calculated
positions and refined using a riding model, while “active” hydrogen atoms were found from
difference Fourier synthesis, and their positional parameters were refined. In all structures,
the partial substitutional disorder of hydrogen peroxide by water molecules [92–95] was
not observed since no residual peaks with an intensity more than 0.17 e/A3 were seen in the
hydrogen peroxide molecule regions. Crystal data and details of X-ray analysis are given
in Table S5a. CCDC 2158207-2158210 contains the supplementary crystallographic data for
I–IV. These data can be obtained free of charge via http://www.ccdc.cam.ac.uk/conts/
retrieving.html (accessed on 18 May 2022) (or from the CCDC, 12 Union Road, Cambridge
CB2 1EZ, UK; Fax: +44-1223-336033; E-mail: deposit@ccdc.cam.ac.uk).
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3.4. Periodic (Solid-State) DFT Computations
B3LYP and PBE-D3 are the most popular functionals in periodic (solid-state) DFT
calculations of organic crystals [12,17,18,81]. These functionals provide a grounded tradeoff between the accuracy and the rate of calculations of experimentally observed properties
of multicomponent organic crystals [28,43,68,71]. Computations with all-electron Gaussiantype localized orbital basis 6-31G** were conducted using the CRYSTAL17 package [96]. We
employed B3LYP [97,98] and PBE [99]. The London dispersion interactions were taken into
account by introducing the D3 correction with Becke–Jones damping (PBE-D3) developed
by Grimme et al. [100]. The space groups and the unit cell parameters of the considered twocomponent crystals (I, II, III, IV, and V) obtained from the X-ray diffraction experiment
were fixed, and the structural relaxations were limited to the positional parameters of
the atoms (AtomOnly). The model structure of I was obtained from the crystal structure,
which has one crystallographically independent hydrogen peroxide molecule disordered
over two sites with an occupancy ratio of 0.927(3)/0.073(3). Only the higher occupancy
was considered in our computations. Tolerance on energy controlling the self-consistent
field convergence for geometry optimizations and frequencies computations is set to 10−10
and 10−11 hartree, respectively. The shrinking factor of the reciprocal space net is set to
3. All structures optimized at the B3LYP/6-31G** level are found to correspond to the
minimum point on the potential energy surface. Further details of calculations are given in
the Supplementary Materials.
4. Conclusions
Four new crystalline peroxosolvates of 18-crown-6, [2.2.2]-cryptand, dibenzo-18crown-6, and creatine were crystallized from the solution of the corresponding coformer in
98% hydrogen peroxide. H2 O2 participate in the formation of moderate hydrogen bonds
with organic coformer and adjacent molecules of hydrogen peroxide.
Hydrogen bond enthalpies/energies in peroxosolvates of macrocyclic ethers and
amino acids are evaluated using the approaches based on the H···O distance (the R(H···O)∆HHB approach) and the local electronic kinetic Gb densities at the hydrogen bond critical
point (the Gb -EHB approach). In the case of weak and moderate hydrogen bonds, both
approaches give similar results. For short (strong) hydrogen bonds (−∆HHB > 50 kJ/mol,
R(H···O) < 1.60 Å and ρb > 0.60 a.u.), the differences −∆HHB and EHB can be 15 kJ/mol or
more. We conclude that the Gb -EHB approach should be used with caution for describing
short (strong) hydrogen bonds in organic crystals.
Reasonable agreement between the hydrogen bonds enthalpies/energies obtained
using the H···O distances and the local electronic kinetic Gb densities at the hydrogen bond
critical point in crystalline peroxosolvates indicates that the use of the Bader approach for
localization and characterization of intermolecular hydrogen bonds in organic crystals is
fully justified.
It is shown that the cluster approximation can be used to evaluate intermolecular
interactions using the R(H···O)-∆HHB and Gb -EHB approaches. The latter approach gives
reasonable results for energies from 2.0 to 50 kJ/mol and is applicable to describe bifurcated
hydrogen bonds formed by the OH group. It has been called the fast quantum approach.
Due to the use of partial optimization, the cluster approximation is hardly applicable for
calculating sublimation enthalpies, IR and Raman spectra, and thermodynamic and elastic
properties of molecular crystals.
Supplementary Materials: The following supporting information can be downloaded at: https:
//www.mdpi.com/article/10.3390/molecules27134082/s1, References [38,75,101–105] are cited in
Supplementary Materials; Section S1, Details of periodic (solid-state) DFT computations; Section S2,
Non-periodic DFT computations; Figure S1: H-bonding network of H2 O2 molecules in the crystal
structure of I. Peroxide molecule O(4B)-O(5B) represents minor part of disorder. Displacement
ellipsoids are shown at the 50% probability level. The H-bonds are given by the dashed lines.
Symmetry operation: (i) 1 − x, 1 − y, 1 − z; (ii) 2 − x, y, 1.5 − z; (iii) 1 − x, y, 1.5 − z; (iv) x, 1 − y,
Molecules 2022, 27, 4082
12 of 17
0.5 + z.; Figure S2: Part of the H-bond network in II. H atoms omitted for clarity. Displacement
ellipsoids are shown at the 50% probability level. The H-bonds are given by the dashed lines.
Symmetry operation: (i) 1 − x, 2 − y, 1 − z; (ii) −x, 1 − y, 2 − z; (iii) x, y, 1 + z; Figure S3: Part of the
H-bond network in III. Displacement ellipsoids are shown at the 50% probability level. H atoms
of macrocyclic ether are omitted for clarity. The H-bonds are given by the dashed lines. Symmetry
operation: (i) −0.5 + x, 1.5 − y, z; (ii) 0.5 − x, 0.5 + y, −0.5 + z.; Figure S4; Part of the H-bond network
in creatine peroxosolvate IV. Displacement ellipsoids are shown at the 50% probability level. H
atoms of macrocyclic ether omitted for clarity. The H-bonds are given by the dashed lines. Symmetry
operation: (i) x, 1 − y, −0.5 + z; (ii) x, −1 + y, z; (iii) 0.5 − x, 1.5 − y, 1 − z; (iv) 1 − x, 1 − y, 1 − z;
Figure S5: Part of the H-bond network in 3-phenylserineperoxosolvate V. Displacement ellipsoids
are shown at the 50% probability level. Only «active» hydrogen atoms are shown. The H-bonds are
given by the dashed lines. Symmetry operation: (i) −x, y, −z; (ii) 0.5 − x, 0.5 − y, 0.5 + z; (iii) x, y,
−1 + z; Table S1a: B3LYP/6-31G** values of the (H···O) distance, R(H···O), the electron density, ρb ,
and the local electronic kinetic energy density, Gb , at the H···O bond critical point in I. The H-bond
energy EHB and enthalpy ∆HHB evaluated using Equations (2) and (1), respectively, are given in
the last two columns; Table S1b: The values of (H···O) distance, R(H···O), the electron density, ρb ,
and the local electronic kinetic energy density, Gb , at the O···O bond critical point obtained using
partial B3LYP/6-31G** optimization of the I cluster. The H-bond energy EHB and enthalpy ∆HHB
evaluated using Equations (2) and (1), respectively, are given in the last two columns.; Table S2a:
B3LYP/6-31G** values of the electron density, ρb , and the local electronic kinetic energy density, Gb ,
at the H···O bond critical point in II. The H-bond energy EHB and enthalpy ∆HHB evaluated using
Equations (2) and (1), respectively, are given in the last two columns.; Table S2b: The values of (H···O)
distance, R(H···O), the electron density, ρb , and the local electronic kinetic energy density, Gb , at
the O···O bond critical point obtained using partial B3LYP/6-31G** optimization of the II cluster.
The H-bond energy EHB and enthalpy ∆HHB evaluated using Equations (2) and (1), respectively, are
given in the last two columns; Table S3a; B3LYP/6-31G** values of the (H···O) distance, R(H···O),
the electron density, ρb , and the local electronic kinetic energy density, Gb , at the O···O bond critical
point in III. The H-bond energy EHB and enthalpy ∆HHB evaluated using Equations (2) and (1),
respectively, are given in the last two columns; Table S3b: The values of (H···O) distance, R(H···O),
the electron density, ρb , and the local electronic kinetic energy density, Gb , at the O···O bond critical
point obtained using partial B3LYP/6-31G** optimization of the III cluster. The H-bond energy EHB
and enthalpy ∆HHB evaluated using Equations (2) and (1), respectively, are given in the last two
columns; Table S4a; B3LYP/6-31G** values of the (H···O) distance, R(H···O), the electron density,
ρb , and the local electronic kinetic energy density, Gb , at the O···O bond critical point in creatine
peroxosolvate IV. The H-bond energy EHB and enthalpy ∆HHB evaluated using Equations (2) and (1),
respectively, are given in the last two columns; Table S4b: The values of (H···O) distance, R(H···O),
the electron density, ρb , and the local electronic kinetic energy density, Gb , at the O···O bond critical
point obtained using partial B3LYP/6-31G** optimization of the IV cluster. The H-bond energy EHB
and enthalpy ∆HHB evaluated using Equations (2) and (1), respectively, are given in the last two
columns; Table S5a: B3LYP/6-31G** values of the (H···O) distance, R(H···O), the electron density, ρb ,
and the local electronic kinetic energy density, Gb , at the O···O bond critical point in 3-phenylserine
peroxosolvate V (refcode VILGAB). The H-bond energy EHB and enthalpy ∆HHB evaluated using
Equations (2) and (1), respectively, are given in the last two columns; Table S5b: The values of (H···O)
distance, R(H···O), the electron density, ρb , and the local electronic kinetic energy density, Gb , at
the O···O bond critical point obtained using partial B3LYP/6-31G** optimization of the V cluster.
The H-bond energy EHB and enthalpy ∆HHB evaluated using Equations (2) and (1), respectively, are
given in the last two columns; Table S6: Distances between the atoms involved in the formation of
intermolecular H-bonds R(O···O) and R(H···O) in Ia) , obtained using periodic DFT computations
(PBE-D3/6-31G** and B3LYP/6-31G**). Theoretical values of the enthalpy, ∆HHB of intermolecular
H-bonds evaluated using Equation (1) are given in the last two columns; Table S7: Distances between
the atoms involved in the formation of intermolecular H-bonds R(O···O) and R(H···O) in II, obtained
using periodic DFT computations (PBE-D3/6-31G** and B3LYP/6-31G**). Theoretical values of the
enthalpy, ∆HHB of intermolecular H-bonds evaluated using Equation (1) are given in the last two
columns; Table S8: Distances between the atoms involved in the formation of intermolecular H-bonds
R(X···O) and R(H···O) in creatine diperoxosolvate IV, where X = N or O, obtained using periodic DFT
computations (PBE-D3/6-31G** and B3LYP/6-31G**). Theoretical values of the enthalpy, ∆HHB of
Molecules 2022, 27, 4082
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intermolecular H-bonds evaluated using Equation (1) are given in the last two columns; Table S9:
X-ray structure determination summary.
Author Contributions: Conceptualization, M.V.V. and A.V.C.; experimental methodology, M.A.N.,
A.V.C. and A.G.M.; theoretical methodology, M.V.V.; investigation, A.G.M., A.V.C., M.A.N., P.V.P.,
O.L. and M.V.V.; single-crystal XRD experiment, M.A.N., A.V.C. and A.G.M.; validation, A.V.C. and
P.V.P. writing—original draft preparation, A.G.M., A.V.C. and M.V.V.; writing—review and editing,
M.V.V., P.V.P., A.V.C. and O.L.; supervision, A.V.C. and P.V.P.; project administration, A.V.C. and
P.V.P.; funding acquisition, A.V.C. All authors have read and agreed to the published version of
the manuscript.
Funding: The authors thank Russian Foundation for Basic Research for financial support (grant
20-03-00449).
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Supplementary crystallographic data for I–IV can be obtained free of
charge via http://www.ccdc.cam.ac.uk/conts/retrieving.html (accessed on 18 May 2022). The data
presented in this study are available on request from the corresponding author.
Acknowledgments: The single-crystal X-ray diffraction studies were performed at the Centre of
Shared Equipment of IGIC RAS within the State Assignment on Fundamental Research of the
Kurnakov Institute of General and Inorganic Chemistry. O.L. acknowledges the financial support of
the Israel Science Foundation (grant no. 1215/19). M.V.V. thanks G. L. Perlovich for the possibility of
conducting computations using Crystal17.
Conflicts of Interest: The authors declare no conflict of interest.
Sample Availability: No possibility.
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