Computational investigation of isotopic effect in H 2 @X 12 N 12 and H 2 O@X 12 N 12 (X = B and Al) molecules

1 Introduction

A valuable method for illustration of the reaction mechanisms is compression-induced isotope effects. Mass-dependent isotope effects on the chemical reactions are attributed only from the changes in ZPE of isotopic species [1–5]. It is used as an explicit tool exploring the reaction mechanisms, classical or tunneling; the last is recognized to toughly depend on the molecular motion, which influences the activation barriers, governing their quantum permeability. Deuterium and ¹³C pressure induced isotope effects do not observed on the reaction of oxidation of benzyl alcohol by yeast alcohol dehydrogenase YADH [6, 7]. Enzymatic reactions of hydrogen (deuterium) atom transfer in the oxidation of linoleic acid by soybean lipoxygenase illustrated the huge isotope effect [8, 9]. This irregular isotope effect was a noteworthy happening in chemistry.

Compression of the molecules and the atoms is influences their properties [10]. For examples, electron affinities, ionization potentials, and vibrational frequencies are modified due to this effect. Vibrational frequencies enhance the differences of zero-point energies (ZPEs) of isotopic molecules, which outcome to growing isotope effects. These effects induced by compression are apparently the merely income to amount molecular compression; they are considered to be significance for exploring molecular machines-enzymes, which are recognized to motivate enzymatic reactions by compression of reagents [11]. In a computational study, isotope effects Induced by molecular compression have been reported on the hydroxyl ion and hydrogen and water molecules, H₂@C₆₀, H₂O@C₆₀, and OH ^- @C₆₀ molecules [12]. As a result, similar study in the inorganic nano-cages can be interesting. In the basis of our knowledge, this investigation has been not reported.

X_nY_n(X = Group III, Y = Group V elements) nano-structures have been attracted much attentions in last ago years [13–28]. The XY bond causes X_nY_n nano-cages to indication a reactivity form dissimilar in compared to carbon analogue [29]. Valuable chemical and physical properties of these molecules have been explored [30–42].

In the present work, we interested to computational realization of the induced isotope effect on the compressed hydrogen and water molecules in B₁₂N₁₂ and Al₁₂N₁₂ nano-cages (H₂@X₁₂N₁₂ and H₂O@X₁₂N₁₂; X = B and Al) were calculated.

2 Computational methods

Optimization and vibrational analysis of the considered molecules were provide by the Gaussian 09 package [43]. All calculations were done with the Long range-corrected version of wPBE hybrid function (LC-ωPBE) method [44–47] and standard 6-311 G(d,p) basis set [48–51]. This method has been revealed to be accurate for vibrational and electronic properties in neutral and charged systems [52, 53]. No imaginary frequencies obtained for molecules.

The ZPE of compressed molecule (ɛ_c) was provided as subtract of the caged molecule and the free cage ZPE values:

for H₂:

ɛ c ( H 2 ) = ZPE ( H 2 @ X 12 N 12 ) - ZPE ( X 12 N 12 ) ; X = B or Al

for HOH

ɛ c ( H 2 O ) = ZPE ( H 2 O @ X 12 N 12 ) - ZPE ( X 12 N 12 )

The excess of compression energy Δ ɛ that was the excess of ZPE acquired by compressed molecule under compression is considered as the difference between the ZPE of compressed and free molecules:

Δ ɛ = ɛ c - ɛ f

In this equation, ɛ_f considers as ZPE of the free molecule.

The compression-induced isotope effect on ZPE of hydrogen and deuterium molecules was considered as:

ɛ c * = ɛ c ( H 2 ) - ɛ c ( D 2 )

Likewise, the compression-induced isotope effect on ZPE of HOH and HOD molecules was provided as:

ɛ c * = ɛ c ( HOH ) - ɛ c ( HOD )

Isotope effect on ZPE of free molecules was provided as:

ɛ f * = ɛ f ( H 2 ) - ɛ f ( D 2 )

ɛ f * = ɛ f ( HOH ) - ɛ F ( HOD )

Isotope effects on the dissociation rate constants of free and compressed molecules were the following equations:

( k H k D ) f = exp ( ɛ f * RT )

( k H k D ) c = exp ( ɛ c * RT )

The same equations are usable similarly for tritium molecules and tritium isotope effects.

3 Results and discussion

Optimized geometries of the H₂@X₁₂N₁₂ and H₂O@X₁₂N₁₂ (X = B, Al) molecules are presented in Fig. 1 at the LC-ωPBE/6-311 G(d,p) level of theory.

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Fig. 1

Optimized geometries of structure of H₂@X₁₂N₁₂ and H₂O@X₁₂N₁₂ (X = B and Al) molecules.

3.3 Energetic aspects

3.3.1 Zero-point energies

Sum of electronic and zero-point energies (E_elect+ZPE) and zero-point energy (ZPE) values of H₂@B₁₂N₁₂ and H₂@Al₁₂N₁₂, hydrogen, water and various isotopes of their (D₂, T₂, HD, HT, D₂O, T₂O, HOD and HOT) are listed in Table. Evidently, the compression of H₂, D₂, T₂, HD, HT, H₂O, D₂O, T₂O, HOD and HOT molecules in X₁₂N₁₂ increases their ZPE (Fig. 2). Larger ZPE values are observed in the presence of X = B than X = Al.

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Fig. 2

Diagrams of ZPE values of compression H₂, D₂, T₂, HD, HT, H₂O, D₂O, T₂O, HOD, HOT in X₁₂N₁₂ (X = B and Al) molecules.

3.3.2 Compression energy

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Table 1 (a)

Sum of electronic and zero-point energies (E_elect+ZPE) and zero-point energy (ZPE) values of (a) hydrogen, water and (b) H₂@B₁₂N₁₂ and H₂@Al₁₂N₁₂ and various isotopes of their (D₂, T₂, HD, HT, D₂O, T₂O, HOD and HOT) at the LC-ωPBE/6-311 G(d,p) level of theory

Molecule	Eelc+ZPE	ZPE, kcal/mol
H2	–1.1792	6.35
HD	–1.1691	5.50
HT	–1.1696	5.19
D2	–1.1707	4.49
T2	–1.1720	3.67
H2O	–76.3831	13.61
D2O	–76.3890	9.90
T2O	–76.3916	8.31
HOD	–76.3861	11.78
HOT	–76.3873	11.01

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(b)

	X = B		X = Al
Molecule	Eelc+ZPE	ZPE, kcal/mol	Eelc+ZPE	ZPE, kcal/mol
X12N12	–955.5124	83.17	–3565.8846	51.90
H2@X12N12	–956.5857	94.51	–3567.0272	60.88
D2@X12N12	–956.5914	90.96	–3567.0315	58.22
T2@X12N12	–956.5939	89.39	–3567.0333	57.05
HD@X12N12	–956.5885	92.80	–3567.0292	59.61
HT@X12N12	–956.5896	92.11	–3567.0301	59.10
HOH@X12N12	–1031.6042	99.03	–3642.2150	68.00
HOD@X12N12	–1031.6081	96.61	–3642.2184	65.84
HOT@X12N12	–1031.6098	95.56	–3642.2199	64.91
D2O@X12N12	–1031.6119	94.19	–3642.2219	63.67
T2O@X12N12	–1031.6153	92.08	–3642.2249	61.79

Compression energy (Δɛ) obtained by the molecule under compression is defined as:

Δ ɛ = ɛ c - ɛ f

Where, for H₂:

ɛ c ( H 2 ) = ZPE ( H 2 @ X 12 N 12 ) - ZPE ( X 12 N 12 )

for HOH

ɛ c ( HOH ) = ZPE ( HOH @ X 12 N 12 ) - ZPE ( X 12 N 12 )

and, ɛ_f is ZPE of the free molecule.

The excess of compression energy Δɛ obtained by the molecule under compression is moderately noteworthy, about 5.00–2.60 (X = B) and 1.48–2.63 (X = Al) kcal/mol. It is notable that ɛ_c nearly linearly rises with increasing of Δɛ:

X = B:

For: H₂, D₂, T₂, HD, HT; Δɛ= 0.4729 ɛ_c –0.4086; R² = 0.998

For: H₂O, D₂O, T₂O, HOD, HOT; Δɛ= 0.2366 ɛ_c –1.5112; R² = 0.9984

X = Al:

For: H₂, D₂, T₂, HD, HT; Δɛ= 0.2963 ɛ_c –0.0649; R² = 0.9927

For: H₂O, D₂O, T₂O, HOD, HOT; Δɛ= 0.1447 ɛ_c + 0.1462; R² = 0.9955

It quantitatively confirms an obvious declaration: the more compression makes the more change in ZPE of isotopic molecules and greater isotope effect.

Furthermore, the greater vibration rouses the more repulsive potential of the inner wall of the cage. Therefore, the averaged place of hydrogen atom is nearer to the internal wall of the cage and involvements stronger repulsion than D and T atoms. Explicitly, these changes in amounts and locations of isotopic atoms are accountable for the compression-induced isotope effects.

3.4 Rate constants of dissociation

The isotope effects on the rate constants of dissociation of compressed molecules rise in compared to those of free molecules (Tables 2). Consequently, for molecular hydrogen and water molecule, k_H/k_D rise under compression (Fig. 3). For tritium molecule, the effect k_H/k_T is anticipated to be greater. Obviously, the isotope effects are certain means for examination of molecular compression. Larger k_H/k_D and k_H/k_T values are observed in the presence of X = B than X = Al.

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Fig. 3

Diagrams of k_H/k_D values of compression D₂, T₂, HD, HT, D₂O, T₂O, HOD, HOT in X₁₂N₁₂ (X = B and Al) molecules.

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Table 2

ZPE of compressed molecule (ɛ_c), Compression energy (Δɛ, kcal/mol), compression-induced isotope effect on ZPE of hydrogen, deuterium, HOH and HOD molecules (ɛ_c ^*, kcal/mol) molecules Isotope effect on ZPE of free molecules (ɛ_f ^*, kcal/mol) rate constants of dissociation of the dissociation rate constants of free and compressed molecules (in s ^- 1) at the LC-ωPBE/6-311 G(d,p) level of theory

Molecule	ɛ c	Δ ɛ	ɛ f	ɛ c  *	ɛ f  *	kc	kf	kc/kf
H2@B12N12	11.34	4.99	6.35
D2@B12N12	7.79	3.30	4.49	3.55	1.86	402.94	23.05	17.48
T2@B12N12	6.23	2.56	3.67	5.12	2.68	5650.33	92.28	61.23
HD@B12N12	9.64	4.14	5.50	1.71	0.85	17.84	4.20	4.25
HT@B12N12	8.94	3.76	5.19	2.40	1.16	57.71	7.13	8.09
HOH@B12N12	15.87	2.26	13.61
HOD@B12N12	13.45	1.67	11.78	2.42	1.83	59.57	21.91	2.72
HOT@B12N12	12.39	1.38	11.01	3.47	2.59	352.20	79.98	4.40
D2O@B12N12	11.02	1.12	9.90	4.84	3.70	3567.39	521.28	6.84
T2O@B12N12	8.91	0.60	8.31	6.95	5.30	125900.30	7691.29	16.37
H2@Al 12N12	8.98	2.63	6.35
D2@Al12N12	6.32	1.83	4.49	2.66	1.86	89.11	23.05	3.87
T2@Al12N12	5.15	1.48	3.67	3.83	2.68	644.33	92.28	6.98
HD@Al12N12	7.71	2.21	5.50	1.27	0.85	8.55	4.20	2.04
HT@Al12N12	7.20	2.01	5.19	1.78	1.16	20.32	7.13	2.85
HOH@Al12N12	16.10	2.49	13.61
HOD@Al12N12	13.94	2.16	11.78	2.16	1.83	38.29	21.91	1.75
HOT@Al12N12	13.01	1.99	11.01	3.09	2.59	185.90	79.98	2.32
D2O@Al12N12	11.77	1.87	9.90	4.33	3.70	1486.67	521.28	2.85
T2O@Al12N12	9.89	1.58	8.31	6.21	5.30	35600.74	7691.29	4.63

Computational realization of the induced isotope effect on the compressed hydrogen and water molecules in B₁₂N₁₂ and Al₁₂N₁₂ nano-cages (H₂@X₁₂N₁₂ and H₂O@X₁₂N₁₂; X = B and Al) at LC-ωPBE/6-311 G(d,p) level of theory indicated:

H-H and OH bonds were shorter in H₂@B₁₂N₁₂ and H₂O@B₁₂N₁₂ molecules than free H₂ and H₂O molecules. In contrast, H-H and OH bonds were longer in H₂@Al₁₂N₁₂ and H₂O@Al₁₂N₁₂ molecules than free H₂ and H₂O molecules. Longer H-H and OH bonds were found in compressed Al₁₂N₁₂ than similar compressed B₁₂N₁₂ molecules.