The Halogen-Bond Nature in Noble Gas–Dihalogen Complexes from Scattering Experiments and Ab Initio Calculations

Abstract

In order to clarify the nature of the halogen bond (XB), we considered the prototype noble gas–dihalogen molecule (Ng–X₂) systems, focusing on the nature, range, and strength of the interaction. We exploited data gained from molecular beam scattering experiments with the measure of interference effects to obtain a suitable formulation of the interaction potential, with the support of high-level ab initio calculations, and charge displacement analysis. The essential interaction components involved in the Ng–X₂ adducts were characterized, pointing at their critical balance in the definition of the XB. Particular emphasis is devoted to the energy stability of the orientational Ng–X₂ isomers, the barrier for the X₂ hindered rotation, and the influence of the X₂ electronic state. The present integrated study returns reliable force fields for molecular dynamic simulations in Ng–X₂ complexes that can be extended to systems with increasing complexity and whose properties depend on the selective formation of XB.

1. Introduction

The intermolecular halogen bond (XB), the non-covalent interaction involving an electrophilic region of a halogen atom, is known to control important static and dynamic properties of matter [1]. The role and selectivity of the XB are of high relevance in supramolecular chemistry for the synthesis of materials with a wide variety of technological properties and drug design [2,3]. Moreover, there are important energy transfer processes and predissociation phenomena where the chemical kinetics is guided by interaction forces associated with the XB formation [4,5,6,7,8].

Intermolecular interactions are usually regarded as constituted by two kinds of main contributions: electrostatic (V_el) and non-electrostatic (V_nel) [9,10]. The former contribution originates from the interaction of charges and permanent electric dipoles or multipoles present in the interacting species. V_el is often dominating and it makes the extent and role of V_nel, although often important, difficult to quantify. The nature of V_nel is difficult to assess precisely, as it arises from the delicate balance of various components: Pauli size-repulsion (V_sr), dispersion (V_d), and induction (V_i) attractions, and charge-transfer (V_ct), which all possess different dependence and scaling properties with the distance and relative orientation of the interacting partners. In this respect, the study of simple systems with high orientational symmetry, and where V_el is small or even vanishing, is especially important to understand and model the various V_nel components. Noble-gas–halogen adducts, Ng–X₂ (X = Cl, Br, I), are such ideal prototypes to investigate the role of V_nel components in XB emergence. In a recent study we carried out, the Ng–Cl₂ system showed, for example, that XB formation selectively stabilizes the collinear configuration of the adducts when Cl₂ is in its X ¹Σ_g ground state [11], while this effect disappears if Cl₂ is in its first triplet excited state, B ³Π_u [12,13]. We further studied in detail how the XB formation and the charge transfer (CT) component make the interaction potential deviate from the typical van der Waals model, V_vdW (defined as V_vdW = V_sr + V_d), and how these additional components can be accurately modeled in general [12].

In the present work, we build on our previous findings in order to characterize the entire family of Ng–X₂ complexes. This systematic study will enable us to establish, in a consistent fashion, trends and features that govern the XB formation, and its dependence on the nature of Ng and X₂, in an effort to continue the development of a general predictive model to describe the XB—in particular, components other than the electrostatic one, which is of crucial importance for the design and analysis of more complex systems. As in previous works [13,14,15,16], we use an integrated phenomenological–theoretical approach. The phenomenological procedure is based on the available data resulting from collisional experiments, combined with (semi)empirical methods developed over the years to predict the range and strength of weak interactions and to provide an adequate analytical representation of the potential energy surface (PES). This is flanked by highly accurate theoretical ab initio calculations, which allow us to validate and improve the PES and the models, and to understand and characterize the fundamental components of the interaction. Key to this approach is our ab initio charge–displacement (CD) analysis, which enables us in particular to assess and quantify in a transparent and consistent way the elusive CT component [16].

2. Results and Discussion

2.1. Scattering Results

The total integral cross section Q(v), as a function of the collision velocity v, was measured by means of the molecular beam technique, the noble gases He, Ne, Ar being the projectiles, and the electronic ground state Cl₂/Br₂ halogen molecules being the targets [12,13,17]. The results are reported in Figure 1 and clearly point out that the measured cross-section is made up of the superposition of a smooth average component and an oscillating pattern arising from the quantum “glory” interference. In particular, the cross sections were plotted as Q(v)·v^2/5 to emphasize the well resolved oscillatory patterns due to the quantum interference. As is well known [18,19], the average component is directly related to the absolute scale of the long range mean attractive interaction potential of the colliding system. Frequency, amplitude, and position of the ”glory” oscillations are instead determined by the absolute value, form, and anisotropy of the potential near the equilibrium configuration. Overall, the collected scattering data returned direct information on the basic features of the phenomenological PES describing the intermolecular interaction in Ng–X₂ systems. The data in Figure 1 are evidence that both the smooth average and the oscillating cross section components vary along the Ng series for a selected halogen molecule and with the halogen moiety for a selected Ng, attesting to the sensitivity of the experiment to the details of the interaction.

2.2. Interaction Potential Components

An analytical form of the interaction PES, which in this case is expressed as a function of the vector R length, with R joining Ng and X₂ midpoint, and of the Θ angle between R and X₂ axis (see Figure S1 in the Supplementary Materials, SM), was used to fit the observable experiment by tuning a few physically meaningful parameters defining strength and range of the fundamental interaction components. Specifically, each PES was defined as a combination of the ubiquitous van der Waals (V_vdW) component, given in a suitable anisotropic form, with a three-body contribution V_3B, arising from the strongly asymmetric electronic charge distribution around the X₂ molecule, to which was added the stereo-selective role of a CT stabilization effect (V_CT). An important test of the analytical form of the interaction potential and of its parametrization was obtained by calculating the scattering cross sections for each system from the spherical average and from the full PESs, to reproduce, at low and high collision velocity, respectively, values and velocity dependence of the measured data, including also the frequency and amplitude of the oscillating patterns. In particular, results calculated under the two limiting conditions were combined according to the different dynamical regimes, emerging in different collision velocity ranges (see refs [12,13,17] and below), in order to provide a better comparison with the experimental data. As it is shown in Figure 1, the employed potential model adequately reproduced cross section values and both amplitude and frequency of the glory patterns observed in the investigated Ng–X₂ systems. The potential analytical form was further validated by comparing selected cuts of the phenomenological PESs with the ab initio ones (see the next sections).

When experimental cross section data are missing, the analytical expression for the PES can be gained by merely adopting potential parameters, obtained by proper scaling laws defined in terms of fundamental physical properties of the interacting partners [10,11,12,13,14], and then applying the fine-tuned phenomenological model potential—see the Ng–X₂ complexes involving the heavier (Kr, Xe) Ngs or I₂ and/or when X₂ is in the lowest excited B state.

Further details on the PES parametrization are given in the next section. The main features of the obtained PESs, in terms of equilibrium distances R_m and binding energies E_m are given in

Table 1. Equilibrium distances (R_m, Å) and binding energy values (E_m, meV; 1 meV = 0.02306 kcal/mol) of the phenomenological interaction potential considered as spherical average (V_sph) or in the selected linear (V_lin) and perpendicular (V_perp) orientation for the noble gas–dihalogen molecule (Ng–X₂) complexes in the X ¹Σ_g ground state.

	Vsph		Vlin		Vperp
	Rm	Em	Rm	Em	Rm	Em
He–Cl2	4.16	3.23	4.27	5.93	3.39	6.15
Ne–Cl2	4.16	6.98	4.29	12.21	3.44	12.10
Ar–Cl2	4.33	17.99	4.53	27.01	3.70	27.12
Kr–Cl2	4.43	23.01	4.61	34.67	3.83	32.78
Xe–Cl2	4.56	27.77	4.73	41.45	4.00	37.71
He–Br2	4.41	3.24	4.51	6.51	3.50	6.43
Ne–Br2	4.40	7.29	4.52	14.09	3.54	13.22
Ar–Br2	4.57	18.42	4.75	30.03	3.81	29.28
Kr–Br2	4.61	25.69	4.77	43.92	3.88	38.91
Xe–Br2	4.73	32.24	4.86	55.02	4.04	46.18
He–I2	4.85	2.76	4.95	5.54	3.79	5.54
Ne–I2	4.80	6.46	4.90	13.21	3.78	12.21
Ar–I2	4.88	19.20	5.04	34.66	3.94	32.98
Kr–I2	4.93	26.34	5.05	49.82	4.02	43.19
Xe–I2	5.02	34.64	5.11	65.77	4.16	54.26

and

Table 2. Equilibrium distances (R_m, Å) and binding energy values (E_m, meV; 1 meV = 0.02306 kcal/mol) of the phenomenological interaction potential considered as spherical average (V_sph) or in the selected linear (V_lin) and perpendicular (V_perp) orientation for the Ng–X₂ complexes in the B ³Π_u excited state.

	Vsph		Vlin		Vperp
	Rm	Em	Rm	Em	Rm	Em
He–Cl2	4.37	2.69	4.82	3.24	3.37	5.79
Ne–Cl2	4.40	5.52	4.84	6.77	3.41	11.59
Ar–Cl2	4.58	14.45	5.00	17.60	3.67	27.37
Kr–Cl2	4.68	18.66	5.10	22.68	3.78	34.18
Xe–Cl2	4.82	23.18	5.25	28.11	3.95	40.61
He–Br2	4.57	2.67	5.03	3.15	3.61	5.37
Ne–Br2	4.58	5.66	5.03	6.77	3.63	11.21
Ar–Br2	4.73	16.01	5.16	19.24	3.82	29.41
Kr–Br2	4.81	21.25	5.24	25.43	3.92	38.03
Xe–Br2	4.93	27.14	5.35	32.51	4.07	46.35
He–I2	5.12	2.13	5.62	2.85	3.83	4.88
Ne–I2	5.10	4.66	5.59	6.32	3.82	10.62
Ar–I2	5.19	14.69	5.66	20.05	3.94	31.73
Kr–I2	5.26	20.15	5.73	27.30	4.02	42.49
Xe–I2	5.35	26.94	5.83	36.42	4.14	55.01

for a useful comparison of all systems. Since general and common trends were observed among the Ng–X₂ complexes, selected values related to Ng–Br₂ and Ar–X₂ (X = Cl, Br, I) complexes were considered, as seen in Figure 2 and Figure 3, showing the angular minimum energy path (MEP) associated with the PESs (see Figures S2–S5 in SM for Ng–Cl₂ and Ng–I₂ systems). For the ground state Ng–Br₂(X) complexes, the Figure 2 (left side) emphasizes the increase of the interaction along the Ng series and shows the occurrence of a saddle point at intermediate orientation angle (Θ is about 50–60°). For each system the ground state linear configuration (Θ = 0°) results were more stable than those of the excited state (right side of Figure 2) and for heavier Ng–Br₂(X) systems, such configuration became in absolute the most stable one. In all cases a relevant selectivity arose from the different angular dependence of the interaction components involved. The various components concurring with the overall interaction potential, evaluated along the MEPs, are reported in Figure 3 for the selected Ar–X₂(X) adducts (X = Cl, Br, I). Interestingly, it points out that the V_vdW component tended to stabilize the perpendicular configuration (Θ = 90°), while the V_CT contribution emerged more efficiently in proximity of the collinear configuration, and V_3B provided a less important, although repulsive, contribution at intermediate configurations. We verified that for the excited PESs the contributions of V_3B and V_CT were found to be vanishing, suggesting that the involved interactions showed a pure van der Waals behavior. Therefore, the full topography of each PES depended on the critical balance of the effectiveness of the V_vdW, V_CT, and V_3B components. All these peculiarities of the intermolecular interaction in Ng–X₂ complexes were confirmed by the ab initio calculations (see below).

2.3. Ab Initio Potential Energy Surfaces

As discussed above, ab initio calculations were performed on the Ng–X₂ adducts aimed at assessing the leading interaction components, with a particular focus on their origin in the electron density displacement, and to further test the validity of the adopted intermolecular potential formulation. We focused on the cuts of the PESs most relevant for the anisotropic behavior of the Ng–X₂ interaction, which are the linear and the T-shaped configurations. For the saddle isomer both the ab initio and the phenomenological PESs return a Θ value of ca. 50–60° for all the Ng–X₂ systems. The ab initio computed cuts of the PES for the three considered configurations were compared with those derived from the semi-empirical potential model parametrization, eventually fitted through the analysis of cross section data. In Figure 4 we report the ab initio and model cuts of the PES for the Ar–X₂ complexes (X = Br, I) in the X ground state, while those for the Cl₂ adducts are reported in Reference [13]. A satisfactory agreement (within the combined own uncertainty) was clearly indicated between the ab initio and the model potentials, thus further confirming the validity of the parametrized semi-empirical model for the accurate representation of the Ng–X₂ interaction (X = Cl, Br, I) over the whole range of interaction distances, orientation angles, and strengths. The optimized intermolecular distance (R_m) of the equilibrium structures and the corresponding binding energies (E_m) of the considered cuts are reported in the SM section, Table S1.

For the selected Ar–Br₂ adduct, we also report in Figure 5 the computed cuts of the electronically excited PES, compared with those provided by the phenomenological potential model. Here too, the agreement was entirely satisfactory. The interpretation afforded by the model revealed that, at variance with the ground state, the interaction in the excited state was entirely of van der Waals nature, thus explaining the higher energy stability of the T-shaped vs linear configuration, with the V_CT and V_3B components playing a minor role.

2.4. Halogen Polar Flattening in the Ng–X₂ Complexes

A fundamental enabling feature of the XB is the so-called σ-hole on the halogen atoms, the axial electron deficiency that permits an approaching donor to bind. Associated with the σ-hole is a flattening of the halogen electron density on the outer side of the X₂ molecule along the bond direction. Remarkably, the strong anisotropy of the halogen electron density in the X₂ molecule can be related to a shift at lower distances of the repulsive wall, thus favoring a closer approach of the incoming Ng interacting partner in the linear configuration and rationalizing the higher energy stability of the linear vs T-shaped isomer. A rough estimation of the polar flattening in the X₂ series, therefore, is highly desirable, also enabling a fine calibration of the semi-empirical parameters in the analytical potential form. One possibility consists in the comparison of the electron density profile of the X₂ molecules along two selected directions, namely the X₂ bond axis and the plane perpendicular to it, at the fixed X nucleus position. The results are reported in Figure 6, focusing on the low electron density regions, in the outermost part of halogens, which are of interest of the polar flattening. For each dihalogen molecule, the electron density profile along the bond axis direction was systematically lower than that along the direction perpendicular to it, thus confirming the polar flatting of the electron density for each halogen atom in the X₂ molecule. It is reasonable that the shift in the position of the repulsive wall in the linear isomers had to be strictly related to the shift at lower distance of the electron density profile along the X₂ bond vs the plane perpendicular to it. A measure of the shift may be provided by the Δz value, defined as the distance between two points of X₂ density profiles at a selected density value. At the isodensity value of 1 me/bohr³, which is usually employed to map the Coulomb potential for the analysis of the σ-hole [20,21], the Δz values were computed respectively as 0.23/0.25/0.25 Å for Cl₂/Br₂/I₂, thus suggesting a comparable flattening of the electron density on the outer region of the covalent bond in the halogen series. The estimated polar flattening enabled a decrease at shorter distances of the repulsive wall position in the linear vs T-shaped isomer of about 4%–5%.

2.5. Charge Displacement Analysis

The next subsection is addressed to characterization of the dependence of CT on the basic physical properties of the involved partners. This effort could be crucial to obtain a formulation of its stabilization contribution in terms of general validity. As previously done in the framework of our fine-tuned integrated experimental and theoretical approach [13,14,15], the presence of a CT component in the Ng–X₂ bond, which is required by the PES model to quantitatively reproduce the experimental cross sections, was ascertained by resorting to the so-called CD function [22], which has proved especially useful for the identification and estimation of the CT in weakly bound systems [22,23,24,25]. The CD curves for selected Ng–X₂ (X = Br, I) systems in the linear configuration are reported in Figure 7, together with the 3D contour plots of the electron density difference between the complex and the non-interacting fragments. The X₂ substrate pronouncedly polarized the spherical cloud of the Ng center, which underwent a depletion/accumulation in the region opposite/towards X₂. A visible amount of charge rearrangement was present even on the X₂ moiety. Moreover, the CD curves were distinctly positive everywhere, indicating a corresponding electron charge flow in the direction from Ng to X₂. This effect was a clear fingerprint of a donor role of Ng and was related to the presence of an XB interaction in the Ng–X₂ complexes. Remarkably both the σ-hole and the electron density polar flattening on the halogen concurred to strengthen the CT in the linear isomer. For a plausible quantitative estimate of the amount of CT, we could take, as we have consistently done in previous works, the Δq value at the so-called “isodensity boundary”, i.e., the point on the z-axis where the electron densities of the non-interacting fragments become equal. The results are detailed in the top left inset of Figure 7, and yielded comparable CT values for the Br₂ and I₂ complexes, ranging between 0.5 and 5.6–7.6 millielectron (me), going from the lighter to the heavier Ng. A similar trend was encountered for the Ng–Cl₂ complexes (Ng = He, Ne, Ar) [13].

The CD analysis was also carried out on the selected Ar–Br₂ complexes involving the Br₂ molecule in the excited B ³Π_u state, showing that the CT was essentially negligible, as already discussed for the Ng–Cl₂ complexes [13].

2.6. CT Dependence on the First Ionization Potential

A proper rationalization of the increasing CT values along the Ng and X₂ series in terms of the properties of the involved fragments is highly desirable, since the interaction CT component plays a key role in the determination of differences in the energy stability among the various conformers. We expect that the CT (and, consequently, the related energy amount, V_CT) must be depending on the extent of the overlap between involved atomic/molecular orbitals. The associated contribution is ruled also by the energy separation between the states of the system coupled by CT. These effects can be modeled in terms of donor ionization energy and acceptor electron affinity, as they concur to determine both the dependence on the distance of the overlap integral, and of the separation energy between states of the system coupled by CT [26,27]. Taking into account that present systems, as many other weakly bounded adducts, are affected by non-resonant CT couplings, considered in the perturbation limit [24], a meaningful general relation, defining strength and range of the V_CT component may be expressed as follows:

$$V_{CT}\left(\mathrm{R}\right)=\frac{B_X}{I_{Ng}-A_X-\frac{q^2}{R}}\cdot e^{-0.512\cdot \left(\sqrt{I_{Ng}}+\sqrt{A_X}\right)\cdot R}$$

(1)

For the present systems, I_Ng is the donor first ionization energy and $A_X$ is the electron affinity of the electrophile X₂. The numerator is correlated to the value of overlap integral between Ng and X₂ orbitals, and, in particular, the pre-exponential term, $B_X$, returns information on the size of the X₂ electron charge distribution. The denominator provides an estimate of the radial dependence of the energy difference between states coupled by CT, with the ($I_{Ng}-A_X$) term being the energy separation between the Ng–X₂ and the Ng⁺–X₂⁻ asymptotic energy, and the $\frac{q^2}{R}$ accounting for the Coulomb attraction between Ng⁺ and X₂⁻ at the distance R.

It is quite instructive to compare the behavior of V_CT, provided by Equation (1), with that gained from the semiempirical model (see Equation (9) in Section 3), for three series of Ng–X₂ systems in the collinear configuration. In all cases V_CT was evaluated at a selected intermediate distance value of R = 3 Å, here representing the separation distance between the Ng center and the X atom closer to Ng, being the only effectively involved in the CT from Ng to X₂. The use of such condition is suitable to emphasize the V_CT dependence on basic properties of the partners. Adopting Equation (1), the $B_X$ parameter was optimized in order to obtain the best comparison with the reference V_CT contribution (also including the model uncertainties) gained from the semi-empirical parametrization (see below).

In the comparison we also included the Ng–Cl adduct, which was previously investigated through state selected Cl atom beam experiments, carried out with the same molecular beam apparatus used here [26]. We report, in

Table 3. Ng first ionization potential energies (I_Ng), X/X₂ electron affinities $(A_X)$ and $B_X$ optimized parameters.

	INg (eV) (a)		${\mathit{A}}_{\mathit{X}}\left(\mathbf{eV}\right){}^{\left(\mathbf{a}\right)}$	${\mathit{B}}_{\mathit{X}}$$[\left(\mathbf{eV}{)}^2\right] {}^{\left(\mathbf{b}\right)}$
He	24.587	Cl	3.621	3430
Ne	21.564	Cl2	2.38 ± 0.1	1300
Ar	15.759	Br2	2.55 ± 0.1	2500
Kr	13.999	I2	2.55 ± 0.05	5000
Xe	12.130

^(a) Values from reference [28]; ^(b) Absolute error of ±10%.

, the optimized parameters for the V_CT formula (Equation (1)), together with the I_Ng and $A_X$ values, as taken from the literature.

From Figure 8, we notice that the semi-empirical function (Equation (1)) reproduces, within the error range, the calculated V_CT values, suggesting the following trend: Ng–Cl₂ < Ng–Cl ≃ Ng–Br₂ < Ng–I₂. This behavior can be explained introducing the V_CT dependence on the dihalogens’ dimensions and electronic affinities: the more electrophilic they are, the greater their V_CT component is. For the Cl atom, we have a different electronic deficit, because it has, properly, an electron missing on the 3p orbital.

2.7. CT Proportional Constant

As a final important aspect of the CT component, contributing to the formation of the weak intermolecular XB, we briefly discuss the relation between the amount of electron charge transferred and the energy stabilization of the adduct arising from this interaction component. It is clear that, when CT is sufficiently small, as it was in the present cases, we should expect the energy stabilization V_CT to be roughly proportional to CT, that is: V_CT = k CT. Wang et al. [29] indeed found an excellent linear correlation. Obtaining a reliable estimate of the energy stabilization per transferred charge unit (k) is clearly a result of general interest, and the employed phenomenological–theoretical approach enables us to do just that.

We evaluated the k value for each Ng–X₂ system in the collinear isomer as the ratio between the V_CT strength, as predicted by the semiempirical model at the ab initio optimized equilibrium distance, and the ab initio CT quantity. The results, with their appropriate error bars, are shown in Figure 9 for the whole series of Ng elements, displayed on the scale of their ionization energy. The average proportional constant results were $\left(4.6\pm 1.0\right)$ eV/e, with the associated global error being estimated considering the combined uncertainty in the V_CT (pre-exponential factor A_CT has a 15% error) and in the separation between the polar flattening and the CT stabilization contributions.

3. Materials and Methods

3.1. Phenomenological Approach

The phenomenological approach is solidly founded on the experiments carried out with the molecular beam technique, using the He, Ne, and Ar noble gases as projectiles and the electronic ground state Cl₂/Br₂ halogen molecules as targets. At given angle and energy resolution conditions, we measured the total integral cross-section Q(v) as a function of the collision velocity v, so that quantum “glory” interference effects were resolved. Unfortunately, on the present experimental apparatus, limit-angle problems prevented the measurement of the true quantum cross-section for heavier Kr or Xe beams. Similarly, substantial modifications of the apparatus and different operative conditions would be required to enable experiments with I₂ targets.

Using the results from the measured integral cross-section Q(v) as a function of the collision velocity v (see Figure 1), integrated with other experimental data on various types of weakly bounded interacting systems, we proposed a well-defined and simple formulation of the intermolecular potential in the Ng–X₂ systems. By denoting with R the distance of the Ng atom from the midpoint of the X₂ bond (the latter assumed of fixed length) and with Θ the angle between R and the X₂ bond axis (see Figure S1 in the SM), this formulation is as follows:

V(R, Θ) = V_vdW(R, Θ) + V_3B(R, Θ) + V_CT(R, Θ)

(2)

As previously ascertained, V_vdW is best represented as the sum of individual interactions between Ng and each of the two X atoms (X_a and X_b), V_vdW = V_Ng–Xa + V_Ng–Xb, each of them expressed as an improved Lennard–Jones (ILJ) function of the distance r_i between Ng and X_i and of the angle φ_i between r_i and the X₂ axis (see Figure S1 in SM):

$$V_{Ng-X_i}\left(r_i,{\phi }_i\right)=\epsilon \left({\phi }_i\right)\left[\frac{6}{n\left(r_i,{\phi }_i\right)}\cdot {\left(\frac{r_m\left({\phi }_i\right)}{r_i}\right)}^{n\left(r_i,{\phi }_i\right)}- \frac{n\left(r_i,{\phi }_i\right)}{n\left(r_i,{\phi }_i\right)-6}\cdot {\left(\frac{r_m\left({\phi }_i\right)}{r_i}\right)}^6\right].$$

(3)

Here, the $\epsilon \left({\phi }_i\right)$ and $r_m\left({\phi }_i\right)$ parameters (potential energy minimum and equilibrium distance, respectively) are calculated as follows:

$$\epsilon \left({\phi }_i\right)= {\epsilon }_{\left|\right|}\cdot co{s}^2\left({\phi }_i\right)+{\epsilon }_{\cdot }\cdot si{n}^2\left({\phi }_i\right)$$

(4)

$$r_m\left({\phi }_i\right)= r_{m\left|\right|}\cdot co{s}^2\left({\phi }_i\right)+r_{m\cdot }\cdot si{n}^2\left({\phi }_i\right)$$

(5)

where the || and ⊥ symbols refer to the collinear (${\phi }_i=0$) and perpendicular (${\phi }_i=\pi /2$) configurations, respectively. The factor $n\left(r_i,{\phi }_i\right)$, which modulates the fall-off of the repulsion and the radial dependence of the intermediate and long-range attraction, depends on β, an additional parameter related to the “hardness” of the interacting partners, as follows:

$$n\left(r_i,{\phi }_i\right)=\beta +4\cdot {\left(\frac{r_i}{r_m\left({\phi }_i\right)}\right)}^2$$

(6)

Here, β is optimally found to lie in the interval 7.0–7.5 for all atom pairs. The partial long-range dispersion attraction coefficients are described only by the asymptotic V_vdW component of the potential and, according to the ILJ formulation, their expression is:

$$C_{6i}=\left[\epsilon \left({\phi }_i\right)\cdot r_m^6\left({\phi }_i\right)\right]$$

(7)

with the global attraction coefficient resulting as the sum of the two angularly averaged C_6i terms.

The values of the ε and the $r_m$ parameters were predicted in an internally consistent way for all Ng–X₂ systems from the polarizability components (see refs 10–14 and references therein), and they were validated by comparing the calculated cross-sections with the experimental results. During the analysis, an additional constraint of providing global average asymptotic attractions was imposed in order to obtain a satisfactory agreement (within about 10%) with the results reported in reference [30].

As previously done for other systems involving halogen atoms [13,31,32], the zero order values of $r_{m\left|\right|}$ were decreased by about 4% to account for the polar flattening effect in the ground state of X₂. Such effect is related to the peculiarity of the electronic charge distribution of the X atom in the direction pointing at the approaching Ng atom in the collinear isomer, the origin and details of which are rationalized in the theoretical sections of this paper. The reduction of $r_{m\left|\right|}$ must be accompanied by corresponding increase of ${\epsilon }_{\left|\right|}$ in order to maintain the $C_6$ coefficient constant.

The second term, V_3B, in the expression for $V\left(R, \Theta \right)$, can be conveniently formulated as:

$$V_{3B}\left(R, \Theta \right)=A_{3B}\cdot si{n}^2\left(2\Theta \right)e^{-3.0\cdot R}$$

(8)

This term is crucial to properly represent the angular dependence of the full PES, especially in the proximity of the saddle point between the collinear and perpendicular configurations, where the molecular repulsion by occupied π* orbitals is more prominent. With respect to a previous formulation [11], the expression for this term was slightly improved to guarantee a smooth second derivative in vicinity of Θ = π/2.

Finally, the charge-transfer term of the potential formulation was defined as:

$$V_{CT}\left(R,\Theta \right)=-{\displaystyle \sum }_{i=a,b}{A}_{CT}\cdot co{s}^4\left({\phi }_i\right)\cdot e^{-3.0\cdot r_i}$$

(9)

The dynamical treatment used for the scattering data analysis was extensively described in recent papers. It affords a satisfactory reproduction of the measured Q(v) (full lines in Figure 1) by essentially only varying the parameters A_3B and A_CT, while keeping the other parameters fixed at their predicted value to within very small adjustments.

The modulation of the involved potential parameters was gained through empirical/semi-empirical formulas developed in our laboratory [10,11,12,13,14,27,28], which represent the strength of the basic intermolecular interaction components in terms of fundamental physical properties of the involved partners, such as the electronic polarizabilities, which define strength, range, and anisotropy of the V_vdW component. Such formulas also account for the dependence of specific parameters on the ionization potential of Ng and on the π*⟶σ* electronic excitation of X₂, which is accompanied by a substantial electronic charge rearrangement, causing an increase of the molecular polarizability and of its anisotropy.

By then comparing the model predictions with the results of ab initio calculations, reported in the previous section, the potential parameters can be further refined and thereby the overall potential formulation exhaustively validated for the entire family of Ng–X₂ adducts, with X₂ either in its ground or first excited electronic state. The parameters values derived for the analytical form of the Ng–X₂ PESs are reported in

Table 4. Potential parameters (r_m in Å; ε, Α_CΤ, A_3B in meV) employed for the formulation of the Ng–Cl atom–atom pairwise interaction for Ng–X₂ systems in the (X ¹Σ_g⁺) ground state. || and ⊥ symbols refer to the collinear and perpendicular configurations, respectively. β is equal to 7.0 except for in Ng–Cl₂ systems (β = 7.5). The maximum estimated uncertainty is about 10% for ε, 3% for r_m, and 10%–15% for A_CT and A_3B.

	rm||	ε ||	rm⊥	ε ⊥	ACT	A3B
He–Cl2	3.39	4.47	3.55	2.93	310,000	22,000
Ne–Cl2	3.44	8.76	3.60	5.76	450,000	61,000
Ar–Cl2	3.68	19.70	3.85	13.00	700,000	240,000
Kr–Cl2	3.79	24.40	3.98	15.70	830,000	460,000
Xe–Cl2	3.94	28.20	4.14	18.10	970,000	890,000
He–Br2	3.53	4.71	3.71	3.02	620,000	44,000
Ne–Br2	3.57	9.66	3.75	6.21	900,000	120,000
Ar–Br2	3.80	21.30	4.00	13.90	1,400,000	440,000
Kr–Br2	3.87	28.90	4.07	18.40	1,660,000	900,000
Xe–Br2	4.01	34.50	4.23	21.90	1,900,000	1,750,000
He–I2	3.81	4.00	4.06	2.59	1,550,000	88,000
Ne–I2	3.81	8.76	4.05	5.70	2,250,000	244,000
Ar–I2	3.95	23.60	4.20	15.50	3,500,000	900,000
Kr–I2	4.03	30.90	4.28	20.30	4,150,000	1,800,000
Xe–I2	4.16	38.60	4.41	25.60	4,750,000	3,500,000

and

Table 5. Potential parameters (r_m in Å; ε, Α_CΤ, A_3B in meV) employed for the formulation of the Ng–Cl atom–atom pairwise interaction for Ng–X₂ systems in the (B³Π_u⁺) excited state. || and ⊥ symbols refer to the collinear and perpendicular configurations, respectively. β is equal to 7.0 except for in Ng–Cl₂ systems (β = 7.5). The maximum estimated uncertainty is about 10% for ε and 3% for r_m.

	rm||	ε||	rm⊥	ε⊥	ACT	A3B
He–Cl2	3.62	3.03	3.58	2.88	0	0
Ne–Cl2	3.64	6.32	3.62	5.73	0	0
Ar–Cl2	3.81	16.30	3.87	13.40	0	0
Kr–Cl2	3.91	20.90	3.98	16.70	0	0
Xe–Cl2	4.06	25.70	4.14	19.80	0	0
He–Br2	3.83	2.92	3.81	2.66	0	0
Ne–Br2	3.83	6.27	3.83	5.53	0	0
Ar–Br2	3.96	17.70	4.02	14.40	0	0
Kr–Br2	4.04	23.30	4.11	18.60	0	0
Xe–Br2	4.16	29.60	4.26	22.60	0	0
He–I2	4.12	2.70	4.12	2.40	0	0
Ne–I2	4.09	6.00	4.11	5.20	0	0
Ar–I2	4.16	19.00	4.23	15.40	0	0
Kr–I2	4.23	25.80	4.31	20.60	0	0
Xe–I2	4.33	34.30	4.42	26.60	0	0

.

3.2. Computational Details

The ab initio calculations were performed with the MOLPRO program [33]. Since the investigated non-covalent interactions have a strong dispersion component, it is fundamental that the computational method used to treat a HX system is of highly accurate level, adequately accounting for the consistent electron correlation. Accordingly, the calculations were carried out at the coupled-cluster level of theory [34], with single, double, and perturbatively included triple excitations, CCSD(T), using augmented correlation consistent polarized valence triple- (only in the He case), quadruple-, quintuple-ζ basis sets (aug-cc-PVXZ, hereafter labelled as AVXZ, with X = T, Q, 5). Selected cuts of the ground state PESs for the Ng–X₂ (X = Br, I) complexes were investigated by considering the Ng atom in the ¹S₀ ground state and the Cl₂ molecule in the (X¹ Σ_g⁺) ground state. We also considered the Ar–Br₂ complex with Br₂ being in its first excited state (B³Π_u), characterized by a (π_g*)³(σ_u*)¹ valence shell configuration. Remarkably, the double degeneracy of the Π state was lifted upon approach of the Ng atom. The overall symmetry was lowered to the C_2v (C_s) point groups for the T-shaped (saddle) isomer, and led to two electronic states with A₁/B₂ (A’/A’’) symmetries, which became degenerate for the collinear geometry.

According to the formulation of the phenomenological PES, we used the Jacobi coordinates (R, r, Θ) to describe the triatomic Ng–X₂ complexes. The Br–Br (I–I) bond distance in the adducts was fixed at the equilibrium value 2.28 Å (2.666 Å) in the isolated Br₂ (I₂) molecule in the ground state [35], since the relatively weak interaction with Ng was expected to leave the geometry of X₂ essentially unaffected. For the Br₂ molecule, the first excited state, an equilibrium distance of 2.699 Å was considered [36]. For each Ng–X₂ complex, we varied the intermolecular distance, R, in the range 2.0–7.0 Å, for both Θ = 0° and 90° values. For the X ground state, we also examined peculiar cuts of the Ng–X₂ PESs by varying the Θ angle in the range 0–90° (step by 5°) and optimizing the intermolecular distance R. In this case, the Θ value corresponding to the energy maximum along the path returned the saddle configuration between the two limiting ones.

The basis set superposition errors (BSSEs) on the interaction energy values computed with the AVXZ basis sets were evaluated by applying the counterpoise method [37,38].

3.3. Charge Displacement Function

As the above discussion implies, specific configurations of Ng–X₂ ($\mathrm{X}=\mathrm{Br},\mathrm{I}$) may be selectively stabilized when there is a CT component. A well-established method that permits in a simple yet powerful way to access and quantify the electron displacement occurring upon formation of the bond is the CD function [22]. This function gives, at each point z along an axis joining two interacting fragments, the electron charge ($\Delta q$) that, upon formation of the complex, is displaced from right to left across the plane perpendicular to the axis through z. Its expression is:

$$\Delta q\left(z\right)= {{\displaystyle \int }}_{-\infty }^{+\infty }dx {{\displaystyle \int }}_{-\infty }^{+\infty }dy {{\displaystyle \int }}_{-\infty }^z\Delta \rho \left(x,y,z^{\prime }\right)d{z}^{\prime }$$

(10)

where the integrand $\Delta \rho$ is the electron density difference between the complex under study and its non-interacting fragments, i.e., the change in electron density brought about by the interaction.

The CD curve provides in most cases a straightforward and unambiguous tool to assess the presence and extent of CT in the formation of the adduct, especially in few-atom systems such as the ones at hand. If the function is appreciably different from zero and does not change in sign in the region between the fragments, we can with confidence assert that CT is taking place. Conversely, if the curve crosses zero in this region, CT may be uncertain (both in magnitude and direction). When CT is ascertained, it is useful, for comparative purposes, to obtain a definite numerical estimate of it, by considering the CD function value at a specific point between the fragments along the z axis. In this case, as in previous ones [13,14,15], we chose as fragment separator the point along z at which the electron densities of the non-interacting fragments become equal (isodensity boundary).

4. Conclusions

In the present manuscript we have characterized by a thoroughly integrated experimental and theoretical approach the XB interaction in the series of prototype noble gas–dihalogen adducts. We gain an internally consistent description for the interaction in all the Ng–X₂ families (Ng = He, Ne, Ar, Kr, Xe; X₂ = Cl₂, Br₂, I₂), together with an analytical formulation of all the PESs, which is fine tuned by a few well-defined physical parameters. The nature of the interaction is strongly anisotropic not only in the X₂ ground state, but also for the excited one. In the Ng–X₂ ground state an interaction with XB characteristic clearly emerges limited to the linear configuration, as suggested by the polar flattening of the X₂ electron density along the X–X bond direction and by the presence of a sizable CT accompanying the adduct formation. These peculiar features are missing in the T-shape orientation adducts or in the linear complexes involving the X₂ excited state, so that the interaction has substantially a vdW nature. The analytical potential formulation based on the investigated prototype Ng–X₂ adducts can be easily extended to more complex systems, such as those involving the H₂O or NH₃ polyatomic molecules as interacting partners. Remarkably, in the latter cases a more pronounced anisotropy character is expected in the XB interaction, being affected by the interacting partner orientation.