Noble gas bound beryllium chromate and beryllium hydrogen phosphate: a comparison with noble gas bound beryllium oxide

Abstract

A comparative study is made on the noble gas (Ng) binding ability of beryllium hydrogen phosphate (BeHPO₄), beryllium chromate (BeCrO₄), and beryllium oxide (BeO) via density functional theory and ab initio calculations. BeO serves as a prototype example of a Be based Lewis acid with remarkable Ng binding capability. Although NgBeHPO₄ and NgBeCrO₄ have lower Ng–Be bond dissociation energy by 1.4–4.6 and 2.4–6.3 kcal mol⁻¹, respectively, than NgBeO, the corresponding free energy changes at the standard state show that Ar–Rn analogues may be viable even at an ambient condition. The nature of bonding in all these Ng bound complexes is exactly the same, being exclusively a donor–acceptor type of interaction as indicated by the natural bond orbital, electron density and energy decomposition analyses (EDA) in conjunction with natural orbitals for chemical valence calculations. The negative local energy density values at the bond critical points of Ng–Be bonds involving Kr–Rn imply the covalent nature of the bonding which is further supported by the dominant orbital contribution (80–88%) towards the total stabilization as obtained from the EDA. In fact, the variation in the orbital term is responsible for the observed trend of their Ng binding ability in changing either the Ng atoms or the Be system. Further, Ng → BeY (Y = HPO₄, CrO₄, O) σ-donation is the key contributor (70–82%) of the orbital term, whereas Ng ← BeY π-back donation is responsible only for 15–21% of the total orbital interaction.

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Introduction

Although the noble gas (Ng) chemistry^1–11 was launched in 1962 through the accidental synthesis of a yellowish compound, named xenon hexafluoro platinate (Xe⁺[PtF₆]⁻),¹² the efficacy of the Be center to bind Ng atoms was only noted in 1988 through the prediction of quite highly stable NgBeO complexes¹³ which were later detected by Thompson and Andrews.¹⁴ Among the main group elements, the ability to attain high ionic potential makes Be a very efficient Ng binding center since the bonding between a positively charged center (say Be) and Ng is predominantly of acceptor–donor type interaction. To develop the ideal situation for binding Ng, in a neutral molecule Be should be linked to one or more numbers of electronegative elements so that it yields a highly polarized bond imposing a large positive charge on the Be center. BeO is a perfect example for this. Another way to enhance the efficacy to bind Ng atoms is to choose molecules containing cationic Be centre. For example, BeO⁺ has an enhanced Ng binding ability compared to its neutral analogue^15a and even mono- and di-cationic BeO (BeO^q+; q = 0 ≤ q ≤ 2) can bind multiple Ng atoms.^15b Pyykkö¹⁶ hinted on the strong attraction between Ng and BeS through his study in 1989 in which he found via ab initio computation that Lewis acid of the type, SX (X = Be, B⁺, C²⁺, N³⁺, O⁴⁺) can form strong bond with Ne. Recent detection of NgBeS (Ng = Ne–Xe) at low temperature matrices by Wang et al.¹⁷ further supports their high stability. It was also shown that the Ng binding ability of BeY (Y = O, S, Se, Te) gradually decreases on moving from O to Te, due to the decreased electronegativity of Y along the same.^15a

BeO is often regarded as a prototypical example of Be based compounds having ability to bind Ng.¹³ It is also used in the comparative study of examining the Ng binding ability of other Be systems relative to it. Studies on a series of NgBeY (Y = O, NH, NCN, NBO, CO₃, SO₄) complexes show that BeNCN, BeNBO, BeNH and BeSO₄ are superior to BeO in binding Ng.¹⁸ In fact, BeNCN has the largest Ng binding ability among the neutral Be compounds. Further, one can increase the Ng binding ability by attaching a Lewis acid (say BH₃) at the other end of Be (for examples BeO–BH₃ and BeNCN–BH₃), although such an arrangement does not belong to a global minimum.^10e Grandinetti et al.^10a studied the variation of Ng binding ability of BeNH upon the substitution of –H with different groups. Even, different polynuclear complexes with –NBeHe moieties were computationally predicted.^10g Various Ng bound beryllium nitrides (Be₂N₂, Be₃N₂ and BeSiN₂), monocationic 1-tris(pyrazolyl)borate beryllium (TpBe⁺) and CN₃Be₃⁺ clusters were also found to be viable candidates.^19,15a

In this study, we have investigated the Ng binding ability of beryllium hydrogen phosphate (BeHPO₄) and beryllium chromate (BeCrO₄). Further, a comparison regarding their Ng binding ability and nature of bonding is made between the present two species and BeO. The bond dissociation energy (BDE), change in enthalpy (ΔH) and Gibbs' free energy (ΔG) for the dissociation process are evaluated to explore the Ng binding ability of BeHPO₄ and BeCrO₄. The nature of bonding is analyzed through the natural bond orbital (NBO), electron density²⁰ and energy decomposition in conjunction with natural orbital for chemical valence (EDA-NOCV) analyses.^21,22

Computational details

The geometrical structures of BeHPO₄ and BeCrO₄ and their Ng bound analogues are optimized at the MPW1B95/def2-QZVPPD level^23,24 with Gaussian 09 program package.²⁵ A quasi-relativistic pseudo potential is used for the 28 and 60 core electrons of Xe and Rn, respectively.²⁶ For more information about the relativistic pseudo potential and the basis sets for outer electrons, the readers are referred to the original contribution of Peterson and co-workers.²⁶ The frequency calculations are also done at the same level to find out the nature of the stationary state and to make zero point energy (ZPE) corrections. The basis set superposition error (BSSE) is computed by the counterpoise method proposed by Boys and Bernardi.²⁷ Single point energies are evaluated at the CCSD(T)/def2-QZVPPD level²⁸ taking the optimized geometries obtained at the MPW1B95/def2-QZVPPD level to get better energies. MPW1B95 functional is chosen for the current study as MPW1B95/6-311+G(2df,2pd) was found to be the most robust level to reproduce the bond dissociation energy and bond distance obtained at the CCSD(T) level for the Ng bound complexes.²⁹ Due to the non-availability of Pople split-valence basis sets for the heavier Ng atoms and in order to include the relativistic correction, def2-QZVPPD basis set is considered. Nevertheless, we have studied at the MP2/def2-QZVPPD level³⁰ taking the lighter Be system, BeHPO₄ as a case study in order to verify that the stability of Ng bound analogues is not an artifact of density functional theory (DFT). The results assure us that both the levels act similarly, except for the fact that MP2 yields larger BDE values than MPW1B95. On the other hand, as it was already reported in the literature³¹ that MP2 overestimates the BDE values for Ng-compounds, we have provided those results in the ESI (Table S1†) and here we have continued with the MPW1B95 results. NBO analysis is performed at the same level to compute the natural charges on each atom and Wiberg bond index (WBI)³² between two bonded atoms. Electron density analysis²⁰ is done using Multiwfn software³³ at the MPW1B95/def2-QZVPPD level. ADF(2013.01) program package³⁴ is used to carry out the EDA-NOCV^21,22 at the PBE-D3/QZ4P//MPW1B95/def2-QZVPPD level.³⁵ For heavier atoms, scalar relativistic effects are taken into consideration by employing zeroth-order regular approximation (ZORA).³⁶ The interaction energy (ΔE_int) is decomposed into Pauli repulsion (ΔE_Pauli), electrostatic (ΔE_elstat), orbital (ΔE_orb) and dispersion (ΔE_disp) energy terms as:

ΔE_int = ΔE_Pauli + ΔE_elstat + ΔE_orb + ΔE_disp (1)

For further details about the energy terms, the readers are referred to two excellent reviews on EDA.^21d,37

Structures and stability

The optimized geometries of BeHPO₄ and BeCrO₄ possess point groups of C_s and C_2v, respectively, in which Be is coordinated with two O atoms and central atom, as shown in Fig. 1. As an outcome of the highly polarized Be–O bond, the Be centers carry high positive charge, being 1.40 |e| in BeHPO₄ and 1.36 |e| in BeCrO₄. Therefore, the Be in former system is slightly more electropositive in nature than that in the latter and importantly both the Be centers carry larger positive charge than that in BeO (1.29 |e|). Since the interaction between Ng and Be is mainly as of donor–acceptor type, it will be interesting to examine whether the Ng binding ability varies linearly with the positive charge on Be center, i.e., larger the positive charge on Be, larger would be the Ng binding ability. Provided that not only the positive charge on Be would decide the extent of such interaction, but also the energy of the two participating orbitals and the corresponding symmetries would play important role in deciding their Ng binding ability. Hence, the ligand attached to Be is also an important factor.

Fig. 1 The optimized geometries of BeHPO₄ and BeCrO₄ at the MPW1B95/def2-QZVPPD level. Their corresponding point groups are given in parentheses. The natural charges on Be centers (in atomic unit) are provided.

Nevertheless, since the positive charge on Be in the present cases is quite large, they would be able to polarize the so-called rigid electron density of Ng atoms facilitating chemical bond formation. Further, as the electron cloud becomes more diffused (hence more polarizable) along He to Rn, it is also expected to get larger Ng–Be BDE for the heavier Ng atoms. The Ng bound analogues along with the important geometrical parameters, WBIs and natural charges are shown in Fig. 2. The symmetries of the structures remain unaltered in the Ng bound analogues. The potency of these two compounds to bind Ng is evident from the ZPE and BSSE corrected BDE (D^BSSE₀), ΔH and ΔG values for the dissociation of NgBeHPO₄ and NgBeCrO₄ into Ng and BeHPO₄ or BeCrO₄ (see Table 1).

Fig. 2 The minimum energy structures of Ng bound BeHPO₄ and BeCrO₄ obtained at the MPW1B95/def2-QZVPPD level. The natural charges on Be and Ng centers (in atomic unit) are provided above the respective atoms. The Ng–Be bond distances are in Å unit and are given below the Ng–Be bonds without parentheses. The values in square bracket represent the Wiberg bond index.

Table 1 ZPE and BSSE corrected dissociation energy (D^BSSE₀, kcal mol⁻¹), dissociation enthalpy (ΔH, kcal mol⁻¹) and free energy change (ΔG, kcal mol⁻¹) at 298 K for the dissociation processes: NgBeY (Y = HPO₄, CrO₄, O) → Ng + BeY computed at the MPW1B95/def2-QZVPPD level. D_CCSD(T) is the dissociation energy computed at the CCSD(T)/def2-QZVPPD//MPW1B95/def2-QZVPPD level, and ZPE and BSSE are taken from the MPW1B95/def2-QZVPPD level

Ng	BeHPO4				BeCrO4				BeO
	D^BSSE0	DCCSD(T)	ΔH	ΔG	D^BSSE0	DCCSD(T)	ΔH	ΔG	D^BSSE0	ΔH	ΔG
He	2.4	2.4	2.8	−3.7	1.3	0.9	1.8	−5.1	4.5	5.2	−1.4
Ne	3.5	4.2	3.8	−2.8	2.5	2.6	2.7	−4.1	4.9	5.3	−1.2
Ar	8.4	9.5	8.6	1.9	6.8	7.4	7.0	0.0	11.5	12.0	5.4
Kr	9.9	11.3	10.1	3.4	8.2	9.2	8.4	1.4	13.5	14.0	7.5
Xe	11.7	13.6	11.9	5.3	10.0	11.4	10.1	3.2	16.2	16.6	10.2
Rn	12.4	15.1	12.6	6.1	10.7	13.0	10.8	4.0	17.0	17.4	11.1

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The D^BSSE₀ value ranges from 2.4 to 12.4 kcal mol⁻¹ in NgBeHPO₄ with a gradual increase in going from He to Rn at the MPW1B95/def2-QZVPPD level. NgBeCrO₄ is noted to have 1.0–1.7 kcal mol⁻¹ smaller D^BSSE₀ value than that in NgBeHPO₄. In comparison to NgBeO, while the latter system possesses 1.4–4.6 kcal mol⁻¹ smaller D^BSSE₀ value, the former system has 2.4–6.3 kcal mol⁻¹ lower D^BSSE₀ value. On the other hand, the computed BDE values (D_CCSD(T)) at the CCSD(T)/def2-QZVPPD//MPW1B95/def2-QZVPPD level, corrected from the ZPE and BSSE as obtained at the MPW1B95/def2-QZVPPD level show almost similar results, except that the values are slightly towards the upper side (by 0.6–2.7 kcal mol⁻¹ for Ar–Rn cases) compared to those at the MPW1B95/def2-QZVPPD level. For a given Ng the differences in the D_CCSD(T) values (1.5–2.3 kcal mol⁻¹) between NgBeHPO₄ and NgBeCrO₄ are also quite similar to D^BSSE₀ values.

Therefore, despite the higher positive charge on Be, BeHPO₄ and BeCrO₄ show less efficacy in binding Ng than BeO, reflecting the impact of attached ligands on their Ng binding ability. On the other hand, since the attached ligands in BeHPO₄ and BeCrO₄ are almost same in shape and size, here the increased positive charge results in an improved Ng binding ability. The associated ΔH value suggests the endothermic nature of dissociation process and shows more or less similar behavior as that of D^BSSE₀. Inclusion of entropic factor and thermal correction makes only Ar–Rn bound BeHPO₄ and Kr–Rn bound BeCrO₄ viable at standard temperature and pressure, whereas the He and Ne analogues would spontaneously dissociate into individual fragments. The corresponding ΔG value in case of ArBeCrO₄ implies that at slightly lower temperature, it would remain in bound form. On the other hand, for He and Ne analogues such alteration of temperature should be in larger amount in order to make them stable. Hence, they would be viable only at very cryogenic situation. Note that in case of BeO also, the dissociation process is noted to be endergonic in nature only for Ar–Rn bound analogues. Now, we have checked whether the relative stability in terms of D^BSSE₀ value alters significantly than that based on ΔG value. It is found that for a given Ng, although the overall trend remains same, the relative stability between two systems slightly increases when it is expressed in terms of ΔG. For example, NgBeHPO₄ and NgBeCrO₄ have 1.6–5.0 kcal mol⁻¹ and 2.9–7.1 kcal mol⁻¹ smaller ΔG value than that in NgBeO, respectively. Hence, the corresponding differences are slightly larger than those based on D^BSSE₀ value. Comparison between NgBeHPO₄ and NgBeCrO₄ based on ΔG value also shows the similar trend.

We have also studied the effect of diffuse functions on energy and geometry of the Ng bound complexes taking Ar and Kr bound BeHPO₄ and BeCrO₄ as case studies. They are reoptimized and characterized at the MPW1B95/def2-QZVPP level. In all cases, although the symmetry of the overall structures remains unchanged, the Ng–Be bonds are noted to be longer (by 0.023–0.032 Å) at the MPW1B95/def2-QZVPP level than that at the MPW1B95/def2-QZVPPD level and consequently, the corresponding Ng–Be BDE values are also smaller (by 0.1–0.6 kcal mol⁻¹) at the former level than that at the latter. Therefore, inclusion of a diffuse function has an impact on energetic and geometrical parameters.

Natural bond orbital analysis

As the positive charge on Be attracts the electron cloud from Ng, it results in an Ng → Be electron flow which is reflected in the positive natural charges on Ng and a diminished positive charge on Be after Ng binding. Such electron transfer is small for He and Ne, Ne being the least; however it is considerable in magnitude (0.20–0.30 |e|) for Ar–Rn cases with a gradual increase from Ar to Rn. Similar to the D^BSSE₀ value, the degree of electron transfer is less in NgBeCrO₄ than that in NgBeHPO₄ and such transfer is the largest in NgBeO. It further confirms that the strength of the interaction of such complexes is mainly dictated by the electron accepting property of Be center. Low WBI of Ng–Be bond for He and Ne (0.14–0.19) indicates the non-covalent nature of bonding, whereas it is considerably high (0.34–0.52) for Ar–Rn cases with an increased value for the heavier homologues (see Fig. 2). In fact, almost half WBI is noted in Ng–Be bonds involving Kr, Xe and Rn atoms. Therefore, significant covalent interaction exists in the Ng–Be bonds for the heavier Ng atoms which gradually increases along the bottom of the group. Furthermore, for a given Ng corresponding WBI value in NgBeHPO₄ is slightly larger than that in NgBeCrO₄ implying a somewhat greater covalent character in the former case. The NBO analysis further shows the formation of Ng–Be σ-orbitals in Ar–Rn cases which are highly polarized towards Ng centers (see Table 2).

Table 2 The NBO, occupancy (|e|), contribution from the atoms constructing that NBO and atomic orbital contributions towards that NBO for NgBeHPO₄ and NgBeCrO₄ (Ng = Ar–Rn) complexes are presented at the MPW1B95/def2-QZVPPD level

Complex	NBO	Occupancy	Contribution from atoms to NBO	Atomic orbitals contribution to NBO
				Ng	Be
ArBeHPO4	Ar–Be	1.994	Be (8.53%)–Ar (91.47%)	s (28.13%), p (71.77%)	s (16.11%), p (83.27%)
KrBeHPO4	Kr–Be	1.996	Be (10.48%)–Kr (89.52%)	s (22.67%), p (77.22%)	s (26.77%), p (72.76%)
XeBeHPO4	Xe–Be	1.994	Be (12.63%)–Xe (87.37%)	s (18.12%), p (81.75%)	s (28.10%), p (71.44%)
RnBeHPO4	Rn–Be	1.993	Be (13.06%)–Rn (86.94%)	s (13.62%), p (86.27%)	s (28.73%), p (70.82%)
ArBeCrO4	Ar–Be	1.997	Be (9.11%)–Ar (90.89%)	s (29.87%), p (70.01%)	s (22.41%), p (77.18%)
KrBeCrO4	Kr–Be	1.996	Be (10.97%)–Kr (89.03%)	s (24.07%), p (75.79%)	s (22.89%), p (76.71%)
XeBeCrO4	Xe–Be	1.995	Be (13.14%)–Xe (86.86%)	s (19.17%), p (80.67%)	s (23.71%), p (75.85%)
RnBeCrO4	Rn–Be	1.994	Be (13.57%)–Rn (86.43%)	s (14.51%), p (85.35%)	s (23.96%), p (75.61%)

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The degree of polarization slightly diminishes on moving from Ar to Rn. From both Ng and Be sides, p-orbitals take major part in the bonding (70–86%). Note that although Ng binding ability follows a regular trend along the group, an inconsistency is noted in the degree of electron transfer and WBI values between He and Ne. Despite the fact that Ne is more polarizable than He, the former shows the least electron transfer and WBI value. Such perplexing behavior of Ne was already reported in the literature. Grandinetti nicely elaborated it in his article ‘Neon behind the signs’.³⁸

Electron density analysis

One can categorize a bond as covalent or as non-covalent based on different electron density based descriptors evaluated at the bond critical point (BCP) and their fulfillment of certain criteria. In general, the positive and negative values of Laplacian of electron density (∇²ρ(r_c)) indicate depletion and concentration of electron density (ρ(r_c)) at BCP, respectively, which further represent a non-covalent and covalent situation, respectively. However, in many instances the above criterion fails to describe a bond.³⁹ For examples, it cannot explain the nature of bonding in very simple molecules like CO and F₂.^20,39d Cremer and Kraka^39d argued for the use of the criterion, local energy density, H(r_c) < 0 to describe the covalency of a bond, particularly in the cases where ρ(r_c) is small. H(r_c) is the sum of local kinetic energy density (G(r_c)) and local potential energy density (V(r_c)). In the present cases, ρ(r_c) is small and ∇²ρ(r_c) is positive for all Ng–Be bonds (Table 3). On the other hand, H(r_c) is negative for Kr–Rn bonds in both NgBeHPO₄ and NgBeCrO₄ complexes, implying their covalent nature. The contour plots of ∇²ρ(r) for NgBeHPO₄ complexes are provided in Fig. 3 in which green solid lines show the region of ∇²ρ(r) > 0 and blue dotted lines highlight the area of ∇²ρ(r) < 0 (for the contour plots of ∇²ρ(r) of NgBeCrO₄ and NgBeO see Fig. S1†).

Table 3 Electron density descriptors (au) at the bond critical points (BCP) of Ng–Be bonds in NgBeHPO₄, NgBeCrO₄ and NgBeO obtained from the wave functions generated at the MPW1B95/def2-QZVPPD level

Complex	ρ(rc)	∇^2ρ(rc)	G(rc)	V(rc)	H(rc)
HeBeHPO4	0.020	0.192	0.039	−0.031	0.009
NeBeHPO4	0.022	0.218	0.045	−0.036	0.009
ArBeHPO4	0.029	0.183	0.043	−0.041	0.002
KrBeHPO4	0.028	0.145	0.037	−0.038	−0.001
XeBeHPO4	0.029	0.118	0.034	−0.038	−0.004
RnBeHPO4	0.030	0.107	0.033	−0.039	−0.006
HeBeCrO4	0.017	0.146	0.031	−0.025	0.006
NeBeCrO4	0.017	0.151	0.032	−0.027	0.005
ArBeCrO4	0.026	0.157	0.038	−0.037	0.001
KrBeCrO4	0.027	0.126	0.033	−0.034	−0.001
XeBeCrO4	0.027	0.097	0.028	−0.032	−0.004
RnBeCrO4	0.027	0.080	0.025	−0.031	−0.005
HeBeO	0.030	0.300	0.063	−0.051	0.012
NeBeO	0.026	0.278	0.058	−0.047	0.011
ArBeO	0.034	0.222	0.053	−0.051	0.002
KrBeO	0.033	0.178	0.046	−0.047	−0.001
XeBeO	0.033	0.135	0.038	−0.043	−0.005
RnBeO	0.032	0.112	0.034	−0.040	−0.006

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Fig. 3 The plots of Laplacian of electron density (∇²ρ(r)) of NgBeHPO₄ (Ng = He–Rn) complexes at the MPW1B95/def2-QZVPPD level. (Green solid lines show the region with ∇²ρ(r) > 0 and blue dashed lines show the region with ∇²ρ(r) < 0.)

The valence electrons of He and Ne do not affect much due to the presence of Be, whereas they start to deform by polarizing towards Be center from Ar and it gradually increases along Ar–Rn. In fact, in cases of Xe and Rn analogues, a tiny region is generated in between Be and Ng centers where the electron density gets accumulated. It confirms that the degree of covalent interaction in Ng–Be bonds gradually increases as one moves towards the heavier Ng analogues. Note that the bonding situation in Ng–Be bonds of NgBeO is very similar to the present cases where the H(r_c) values are negative for Kr–Rn bonds, although the corresponding ∇²ρ(r_c) values are positive.

Energy decomposition analysis

The results of the EDA computations on NgBeY (Y = HPO₄, CrO₄, O) complexes considering Ng as one fragment and BeY as another are provided in Table 4. The results show that the Ng–Be bonds are stabilized mostly via orbital contacts having contribution of 72–88% of total attraction. On the other hand, while the ΔE_elstat term only contributes 8–18% of total stabilization, the contribution from the ΔE_disp term is negligible. Note that the associated energy term for orbital contact gradually enhances along He–Rn, which is same in the line of increased covalent character. Further, for a given Ng both the magnitude and percentage of orbital term is smaller in NgBeCrO₄ than those in NgBeHPO₄ implying smaller covalent character in former than that in the latter and NgBeO has the highest value for this energy term among the three complexes. Here, it would be worthy to mention that there exist significant differences in the origin of attractive interaction in Ng bond involving Be and a transition metal (say Cu, Ag, Au).⁴ While in the latter bonds the contribution from ΔE_elstat and ΔE_orb is almost equal,⁴⁰ in the former bonds the attractive interaction is almost exclusively originated from the ΔE_orb term.

Table 4 EDA results of the NgBeY (Y = HPO₄, CrO₄, O) complexes considering Ng as one fragment and BeY as another fragment at the PBE-D3/QZ4P//MPW1B95/def2-QZVPPD level. All energy terms are in kcal mol⁻¹

Complex	ΔEPauli	ΔEelstat^a	ΔEorb^a	ΔEdisp^a	ΔEint	ΔEorb (σ)^b	ΔEorb (π‖)^b	ΔEorb (π⊥)^b
a The percentage values within the parentheses show the contribution towards the total attractive interaction ΔEelstat + ΔEorb + ΔEdisp.b The percentage values within the parentheses show the contribution towards the total orbital term.
HeBeHPO4	5.7	−1.0 (10.3)	−8.2 (86.9)	−0.3 (3.7)	−3.7	−7.3 (89.0)	—	—
NeBeHPO4	7.7	−2.3 (18.3)	−9.7 (78.7)	−0.4 (3.0)	−4.7	−7.0 (72.2)	−1.1 (11.3)	−1.1 (11.3)
ArBeHPO4	12.8	−2.8 (12.3)	−19.3 (84.9)	−0.6 (2.8)	−9.9	−14.4 (74.6)	−2.1 (10.9)	−2.0 (10.4)
KrBeHPO4	13.3	−2.8 (11.1)	−21.4 (85.8)	−0.8 (3.0)	−11.6	−16.6 (77.6)	−2.2 (10.3)	−1.9 (8.9)
XeBeHPO4	13.8	−2.6 (9.4)	−24.2 (87.5)	−0.9 (3.1)	−13.8	−19.4 (80.2)	−2.2 (9.1)	−1.9 (7.9)
RnBeHPO4	13.9	−2.5 (8.6)	−25.4 (88.0)	−1.0 (3.5)	−15.0	−20.8 (81.9)	−2.1 (8.3)	−1.7 (6.7)
HeBeCrO4	6.0	−1.3 (15.0)	−7.0 (81.9)	−0.3 (3.0)	−2.6	−6.1 (87.1)	—	—
NeBeCrO4	7.1	−2.7 (24.7)	−7.7 (71.8)	−0.4 (3.5)	−3.7	−5.7 (74.0)	—	—
ArBeCrO4	13.8	−4.0 (18.0)	−17.4 (78.7)	−0.7 (3.3)	−8.3	−13.0 (74.7)	−1.9 (10.9)	−1.7 (9.8)
KrBeCrO4	14.5	−4.1 (16.6)	−19.6 (79.8)	−0.9 (3.6)	−10.0	−15.2 (77.6)	−2.0 (10.2)	−1.7 (8.7)
XeBeCrO4	15.1	−3.9 (14.2)	−22.5 (82.1)	−1.0 (3.8)	−12.3	−18.0 (80.0)	−2.0 (8.9)	−1.6 (7.1)
RnBeCrO4	15.2	−3.8 (13.2)	−23.7 (82.6)	−1.2 (4.2)	−13.6	−19.4 (81.9)	−2.0 (8.4)	−1.5 (6.3)
HeBeO	5.9	−0.8 (6.5)	−11.4 (92.7)	−0.1 (0.8)	−6.4	−9.2 (80.7)	—	—
NeBeO	7.9	−2.6 (18.4)	−11.4 (80.9)	−0.1 (0.7)	−6.2	−7.8 (68.4)	−1.4 (12.3)	−1.4 (12.3)
ArBeO	13.4	−3.5 (13.3)	−22.6 (85.9)	−0.2 (0.8)	−12.9	−15.9 (70.4)	−2.8 (12.4)	−2.8 (12.4)
KrBeO	14.5	−3.9 (13.2)	−25.4 (86.1)	−0.2 (0.7)	−15.0	−18.5 (72.8)	−2.9 (11.4)	−2.9 (11.4)
XeBeO	15.2	−3.9 (11.2)	−29.0 (87.3)	−0.3 (0.9)	−18.0	−21.9 (75.5)	−2.9 (10.0)	−2.9 (10.0)
RnBeO	15.1	−3.9 (11.4)	−30.0 (87.7)	−0.3 (0.9)	−19.0	−23.3 (77.7)	−2.8 (9.3)	−2.8 (9.3)

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Now, let us see which energy terms are responsible for the obtained trends in Ng binding ability. ΔE_elstat and ΔE_disp terms do not show any notable changes in alteration of either the Ng or BeY. The variation in contribution of ΔE_orb term is key in deciding the Ng–Be bond strength which is further reflected in high linear correlation coefficient (R² = 0.95) in the plot between ΔΔE_orb vs. ΔΔE_int where ΔΔE_orb or ΔΔE_int are the differences between the respective energy terms of an Ng bound complex and its next heavier Ng bound analogue (see Fig. 4). Further decomposition of ΔE_orb term into its σ- and π-contributions reveals that this is actually Ng → BeY σ-donation which is the main contributor of ΔE_orb term (70–82%) and is responsible for the obtained variation in Ng–Be bond strength in changing either the Ngs or BeY.

Fig. 4 The plot of ΔΔE_orb vs. ΔΔE_int at the PBE-D3/QZ4P//MPW1B95/def2-QZVPPD level.

The plots of deformation density for NgBeHPO₄ (Ng = Ar–Rn) are provided in Fig. 5 where the red and blue colors represent the region having Δρ(r) < 0 and Δρ(r) > 0, respectively, which means that the electron density shifts from red to blue region.

Fig. 5 The plots of deformation densities, Δρ(r), of the pair-wise orbital interactions in NgBeHPO₄ (Ng = Ar–Rn) complexes at the PBE-D3/QZ4P//MPW1B95/def2-QZVPPD level. The associated orbital interaction energies are given in kcal mol⁻¹. The color code of the charge flow is red → blue. An isovalue of 0.0005 au is used to better represent the π-back donation.

The plot of the Δρ(σ) shows that the electron density gets depleted from Ng and accumulated in between Be and Ng centers with slight shift of electron density towards adjacent oxygen centers. On the other hand, Δρ(π_‖) and Δρ(π_⊥) represent Ng ← BeY in-plane and out-of-plane π-back donation, respectively. However, the associated orbital energy terms (ΔE_orb(π_‖) and ΔE_orb(π_⊥)) have the contribution of only 15–21% of total ΔE_orb value.

Conclusion

The noble gas (Ng) binding ability of beryllium hydrogen phosphate (BeHPO₄) and beryllium chromate (BeCrO₄) is assessed by an in silico study and is compared with that of BeO, a prototype example of Be based system having notable Ng binding ability. The Ng–Be bond dissociation energy (BDE) lies within the range of 2.4 and 12.4 kcal mol⁻¹ in NgBeHPO₄ with a gradual increase along He–Rn, whereas it is slightly smaller by 1.0–1.7 kcal mol⁻¹ in NgBeCrO₄. However, the Ng–Be bond in NgBeO is stronger by 1.4–4.6 kcal mol⁻¹ than NgBeHPO₄ and by 2.4–6.3 kcal mol⁻¹ than NgBeCrO₄. The Ar–Rn bound analogues may be viable even at standard temperature and pressure as dictated by the corresponding free energy changes. The natural charges on Ng and Be centers indicate the donor–acceptor type interaction in Ng–Be bonds, whereas quite high Wiberg bond indices for the Ng–Be bonds involving heavier Ngs reflect considerable covalent contribution in them which is further confirmed by the results of electron density analysis where the Kr–Rn bonds are categorized as covalent bonds based on the negative local energy density values at the bond critical points of Ng–Be bonds. The stabilization in Ng–Be bonds, which is solely dependent on the orbital contacts as obtained from the energy decomposition analysis, is also in the line of covalent character of the Ng–Be bonds. Further decomposition of the orbital term into its σ- and π-components shows that Ng → BeY σ-donation is the key contributor for the orbital interaction and is responsible for the obtained variation in Ng–Be bond strength in changing either the Ngs or BeY.

Acknowledgements

PKC would like to thank DST, New Delhi for the J. C. Bose National Fellowship. MG thanks CSIR, New Delhi for his Senior research fellowship.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6ra20232b

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