Microsolvation of the Be-F bond in complexes of BeF2, BeF3 –1, and BeF4 –2 with nH2O, for n = 1–6

Ab initio MP2/aug’-cc-pVTZ calculations have been carried out to determine the structures and binding energies of complexes of BeF2, BeF3 –1, and BeF4 –2 with n H2O molecules, with n = 1–6. For each series of complexes with BeF2, BeF3 –1, and BeF4 –2, the binding energies increase as the number of water molecules increases, but the binding energies per intermolecular bond for complexes formed between BeF2 and BeF4 –2 and nH2O decrease as the number of water molecules increases. The binding energies per bond of the BeF3 –1 complexes show little dependence on the number of H2O molecules present. Intermolecular O-H  …  F hydrogen bonds (HB) stabilise all complexes of H2O with the beryllium bases except for BeF2:1H2O and BeF2:2H2O. Complexes of BeF2 are also stabilised by O  …  Be beryllium bonds (BeB) and O-H  …  O hydrogen bonds. EOM-CCSD calculations have also been performed to determine the spin–spin coupling constants. The one-bond coupling constants 1J(Be-F) increase as the Be-F distance decreases, and exhibit an excellent second-order correlation with that distance. 2hJ(O-F) coupling constants across O-H  …  F hydrogen bonds also exhibit a second-order dependence on distance. Coupling constants 1beJ(O-Be) and 2hJ(O-O) are found in complexes with BeF2 and exhibit a linear dependence on the Be-O and O-O distances, respectively. GRAPHICAL ABSTRACT


Introduction
The solvation of ions by water through the formation of hydrogen bonds is of utmost importance for understanding the structure of aqueous solutions and the role that these interactions play in biological processes [1]. The polar character of water leads to the dissociation of salts in aqueous solution and the dispersion of the resulting ions. The solvation of anions through hydrogen bonding in aqueous solution [2] has been investigated theoretically for a number of anions [3,4], including Fand Cl - [5][6][7][8][9], BF 4 - [10] and PF 6 - [11]. Among the myriad of possible anions, we have turned our attention to BeF 3 -1 and BeF 4 -2 . These anions are isoelectronic with the neutral molecules BF 3 and CF 4 , respectively, and share their planar D 3h and tetrahedral geometries, respectively. BeF 3 -1 has been observed in numerous X-ray structures of proteins, with 141 entries found in the PDB database, October-2020. BeF 4 -2 has only been observed in two systems, namely, PDB (1L5Y [12] and 1XHF [13]). BeF 3 -1 and BeF 4 -2 have been used to mimic transition state structures in phosphoryl transfer processes [14]. Moreover, the similarity in size and electronic characteristics of BeF 4 -2 and SO 4 -2 is responsible for the formation of tetrafluoroberyllates with crystal structures analogous to those of sulfates such as Langbeinites [15] and Tutton's salts [16]. In the Cambridge Structural Database [17], BeF 4 -2 appears in eight structures (KIPPEE [18], MOPGAZ [19], QQQBLD [20], QQQBLG [20], QQQBLJ [20], TGLYBE02 [21], TGLYBE20 [21], and XUGHAI [22]) with different protonated amines as counterions which form N-H···F hydrogen bonds. In addition, they are observed in four systems with metals as counterions (QQQGND [23], TITKEN [24], ZZZALG [25], ZZZALJ [25]). 19 F-NMR spectroscopy has been used to identify BeF 4 -2 , BeF 3 -1 , BeF 2 and BeF + in aqueous solutions [26][27][28][29] and the solid state [30]. Gas-phase theoretical studies have shown that beryllium dianions are metastable [31,32]. In the case of BeF 4 -2 a small barrier prevents its spontaneous dissociation into the more stable ions, BeF 3 -1 + F - [32]. To better understand the bonding in aqueous solution of the neutral molecule BeF 2 and the anions BeF 3 -1 and BeF 4 -2 , we have now undertaken a detailed study of these species in complexes with nH 2 O molecules, for n = 1-6. In this paper we report and discuss the equilibrium geometries of these complexes, the binding energies of these complexes, and their binding energies per intermolecular bond. We also examine the beryllium bonds and hydrogen bonds that bind the monomers together. In addition, we report EOM-CCSD intramolecular Be-F and intermolecular Be-O, O-F, and O-O coupling constants in these complexes. It is the purpose of this paper to report the results of this study.

Methods
The structures of the isolated monomers BeF 2 , BeF 3 -1 , and BeF 4 -2 , and their complexes with nH 2 O molecules for n = 1-6, were optimised at second-order Møller-Plesset perturbation theory (MP2) [33][34][35][36] with the aug'cc-pVTZ basis set [37]. This basis set was derived from the Dunning aug-cc-pVTZ basis set [38,39] by removing diffuse functions from H atoms. Frequencies were computed to establish that the optimised structures correspond to equilibrium structures on their potential surfaces. The binding energies of the complexes are given as -E for the reaction that forms the complex from the corresponding isolated monomers. These calculations were carried out with the Gaussian-16 program [40].
The electron densities of the complexes have been analysed using the Atoms in Molecules (AIM) methodology [41][42][43][44] employing the AIMAll program [45]. The topological analysis of the electron density produces the molecular graph of each complex. This graph identifies the location of electron density features of interest, including the electron density (ρ) maxima associated with the various nuclei, and saddle points which correspond to bond critical points (BCPs) or ring critical points, the latter indicating a minimum electron density within a ring. The zero gradient line which connects a BCP with two nuclei is the bond path.
Spin-spin coupling constants for the complexes were evaluated using the equation-of-motion coupled cluster singles and doubles (EOM-CCSD) method in the CI (configuration interaction)-like approximation [46,47] with all electrons correlated. For these calculations, the Ahlrichs qzp basis set [48] was placed on 17 O and 19 F, the Dunning cc-pVDZ basis set on 1 H atoms, and the hybrid basis set developed previously on 9 Be [49]. All terms which contribute to the total coupling constant, namely, the paramagnetic spin orbit (PSO), diamagnetic spin orbit (DSO), Fermi contact (FC), and spin dipole (SD), have been evaluated. The EOM-CCSD calculations were performed using ACES II [50] on the HPC cluster Owens at the Ohio Supercomputer Center. Table 1 provides the symmetries of the complexes of BeF 2 , BeF 3 -1 , and BeF 4 -2 with nH 2 O molecules, the complex binding energies and binding energies per bond, and the number and type of bonds found in these complexes. Complexes of BeF 2 with one and two H 2 O molecules are stabilised by Be ... O contacts. These interactions are known as beryllium bonds since the beryllium is the atom in the Lewis acid directly involved in the interaction [51,52]. In addition to beryllium bonds,

Binding energies
Table S1 of the Supplemental Data provides the structures, molecular graphs, and total energies of the complexes of BeF 2 , BeF 3 -1 , and BeF 4 -2 with nH 2 O molecules. Table S2 provides simplified renderings of these complexes which illustrate the bonding patterns and the numbering system used for each. Examples of these renderings are shown in Figure 1 Figure 1, electron donation by O6 to O4-H occurs across the O4-H ... O6 hydrogen bond. O4 is an electron donor to Be for the formation of the Be ... O4 beryllium bond. Some of this electron density is distributed to F2, which is an electron donor to O6-H for the O6-H ... F2 hydrogen bond. Thus, there is cyclic cooperativity of electron transfer among these bonds. A similar pattern of electron As noted in the Introduction, BeF 4 -2 is a metastable ion in gas phase, [29] since it is 259.8 kJ . mol -1 less stable than the sum of the energies of BeF 3 and Fin agreement with analogous dianion beryllium derivatives. The effect of microsolvation on the stability of BeF 4 -2 can be estimated from the energies of the complexes BeF 4 -2 :2nH 2 O with n = 0-3, with the assumption that in the formation or dissociation of these complexes, BeF 3 and Fare bonded to the same number of water molecules. The reaction for the formation of BeF 4 -2 :2nH 2 O is then given as That solvation significantly stabilises BeF 4 -2 can be seen in Figure 3, which is a plot of the binding energies of these complexes versus the number of water molecules. The correlation coefficient of the second-order trendline is 1.000, or more precisely, 0.999996.

Structures
The   range from 2.82-2.95 Å. The shortest distances are found in the complexes with BeF 2 , which range from 2.62-2.67 Å, while longer distances between 2.66 and 2.81 Å are found in the complexes with BeF 4 -2 . The complexes BeF 2 :1H 2 O and BeF 2 :2H 2 O are stabilised solely by beryllium bonds. When two intermolecular bonds form at the same site, it is expected that these bonds should lengthen, that is, the Be-O distance in BeF 2 :2H 2 O is greater than that distance in BeF 2 :1H 2 O. When more than two H 2 O molecules are present, there are two types of intermolecular interactions, the Be ... O beryllium bond and the O-H ... F hydrogen bond. As noted above, there is a cooperative effect among these bonds such that the beryllium bonds are strengthened by the presence of hydrogen bonds. As a result, the average Be-O distance should decrease as the number of water molecules and therefore the number of hydrogen bonds increases. This is illustrated in Figure 4 by the secondorder trendline which has a correlation coefficient of 0.994. It should be noted that the average Be-O distance is identical to the individual Be-O distance except for complexes with n = 3 and 4 B.  Table S3 of the Supporting Information provides the PSO, DSO, FC, and SD components of the coupling constants 1 J(Be-F), 2h J(O-F), 1be J(Be-O), and 2h J(O-O) for the complexes of BeF 2 , BeF 3 -1 , and BeF 4 -2 with nH 2 O molecules. It is apparent that the one-bond coupling constants 1 J(Be-F) are dominated by the FC term, with the PSO term making contributions ranging from 2 to 5 Hz. Since the FC term is negative and the PSO term positive, the FC term overestimates the absolute value of

1 J(Be-F)
The one-bond coupling constants 1 J(Be-F) in the complexes of BeF 2 , BeF 3 -1 , and BeF 4 -2 with H 2 O are always negative since the magnetogyric ratio of 9 Be is negative.  Figure 5 which is a plot of 1 J(Be-F) versus the Be-F distance. The correlation is excellent, with a second-order trendline that has a correlation coefficient of 0.977. The significant overlap of these coupling constants for the complexes with BeF 2 and BeF 3 -1 is evident in this figure.

2h J(O-F)
Coupling constants 2h J(O-F) across the O-H ... F hydrogen bonds are also given in Table 3. These coupling constants are negative in the complexes with BeF 2 , ranging from -

1be J(Be-O) and 2h J(O-O).
There are two coupling constants that are unique to the BeF 2 :nH 2

Conclusions
Ab initio MP2/aug'-cc-pVTZ calculations have been carried out to determine the structures and binding energies of complexes of BeF 2 , BeF 3 -1 , and BeF 4 -2 with n H 2 O molecules, with n = 1-6. EOM-CCSD calculations have then been performed on these complexes to determine the spin-spin coupling constants 1 J(Be-F), 2h J(O-F), 1be J(Be-O), and 2h J(O-O). The results of these studies support the following statements. (a) a. The one-bond coupling constants 1 J(Be-F) increase as the Be-F distance decreases, and exhibit an excellent second-order correlation with that distance.