All calculations were performed using the Gaussian09 package (Frisch et al., 2009). The structures were optimized without constraints at MP2(full) method in conjunction with aug-cc-pVTZ-PP for Xe and Rn (Peterson et al., 2003) and aug-cc-pVTZ for the remaining atoms (Dunning, 1989; Woon and Dunning Jr, 1993; Wilson et al., 1999). Vibrational frequencies at the same level of theory were also performed in order to ensure that each minimum has only positive frequencies and that each transition state has only a single imaginary frequency, as well as to obtain the zero point vibrational energy (ZPE). The counterpoise method of Boys and Bernardi (1970) was used to correct the BSSE for binding energy.

AIM analysis (Matta and Boyd, 2007) and graphic representations were performed with the AIMALL program (Keith, 2017) using the MP2(full) density (wavefunction) as input as described in the AIM theory (Dobado et al., 1998; Cortés-Guzmán and Bader, 2005).

The transition state theory (TST)(Truhlar et al., 1996) was developed primarily by Henry Eyring (Eyring, 1935) and Michael Polanyi (Polanyi and Wigner, 1928) between 1928 and 1935. The TST is an improvement over the so-called theory of collisions (Lewis, 1967), and it is widely used to calculate the rate constants of chemical reactions.

The start point of TST is the existence of a transition state (TS) between the reagents and products. Located at the top of the potential energy barrier and it assumes a quasi-equilibrium between reactants and activated transition state complexes. For a bimolecular reaction given by (1)R1+R2⇋TS→P1+P2.

The TS is characterized by a single imaginary frequency along the reaction coordinate of the molecular system which is represented here by ν̄1. In its turn, the reaction coordinate can be represented by angular changes in bond distances during the chemical reaction (Henkelman et al., 2002).

The equation that determines the reaction rate is known as the Eyring equation, given by (2)krate(T)=κ(T)kBThq¯m,TS °qm,R1°qm,R2°NAe-Eb0/RT, where 0 < κ(T) ≤ 1 is the so-called transmission coefficient, k_B is the Boltzmann constant, h is the Planck constant, qm ° is the standard molar partition function, N_A is the Avogadro constant, R is the gas constant and Eb0 is the barrier energy with zero-point energy correction. In addition, the TS, R₁ and R₂ subscripts stand for the transition state and reagents, respectively. Thus, the rate constant is determined by the parameters that characterize both reagents and the TS.

The general partition function is formed by the product of translational q^trans., rotational q^rot., vibrational q^vib. and electronic q^ele. partition functions. The translational partition function for a free particle with mass m moving along the length dimension l_x can be evaluated by considering that the separation of energy levels is small and that a large number of states are accessible at room temperatures. Therefore, the energy levels should be continuous and the sum contribution of the translational partition function becomes an integral. Which the solution for the three-dimensional case is (Atkins et al., 2013) (3)qtrans.=(2πmkBT)3/2h3lxlylz.

Although the system can be excited at normal modes, the energy levels are discrete for the rotational mode. The three degrees of freedom of spatial rotation and the three moments of inertia I_A, I_B and I_C must be taken into account for a non-linear molecule (Atkins et al., 2013), thus (4)qrot.=(π)1/2σ(8π2IAIBICkBTh2)3/2, where σ is the so-called number of symmetry. The vibrational mode has reasonably spaced energy which must be taken into account since they are partially occupied. As a consequence, the vibrational partition function is strictly calculated as a sum over the occupied states. In the case of n vibrational degrees of freedom, the vibrational partition function is given by the product of n partition functions, (5)qvib.=∏in11-e-hνi/kBT, where ν_i is each of the fundamental vibrational frequencies. In most cases, only the lowest energy state is occupied and the electronic energies should not contribute considerably to the total partition function (Atkins et al., 2013). A good approximation is to disregard the contributions of the nuclear and electronic spins and to vanish the fundamental energy level for the electronic partition function. Under these considerations the electronic partition function should be equal to unity (Atkins et al., 2013) (6)qele.=1.

On the other hand, the coefficient κ(T) represents the tunneling effect of the reaction coordinate of the chemical system and it is usually important for light atoms or molecules at low temperatures. Thus, tunneling estimates were made using both Wigner (Polanyi and Wigner, 1928) and Eckart (Eckart, 1930) methods.

The Wigner tunneling correction proposes a parabolic potential, (7)VWigner(s)=Eb-12m(2πν̄1)2s2, where E_b corresponds the energy potential barrier of MEP, ν̄1 is the imaginary frequency of transition state and s is the coordinate reaction. This implies in a transmission coefficient given by Bell (1959) (8)κWigner(T)=1-124(hν̄1kBT)2,

For very low temperatures, the Wigner tunneling effect is not very effective, and for this reason, it was also employed Eckart tunneling correction (Truhlar et al., 1985).

The Eckart tunneling correction uses a potential of the type (9)V(x)=Aeαx1 + eαx+Beαx(1 + eαx)2, where α is a parameter described by (10)α2=-μ(ν̄1)2B2Eb0(Eb0 - A) and μ is the reduced mass of the system. These parameters determine the barrier width. Here it is important to note that the A and B can be positive, negative or zero. The A parameter corresponds to the energy difference V(x → −∞) and V(x → +∞), and B is a parameter that measures the height of the barrier given by (11)B=2Eb0-A+2Eb0(Eb0-A).

So the most usual form for the Eckart's potential in the study of reaction rates is (Truhlar et al., 1985) (12)VEckart(s)=Aeα(s-s0)1 + eα(s-s0)+Beα(s-s0)[1 + eα(s-s0)]2, where s is the coordinate of the reaction and s₀ is the reaction coordinate corresponding to the maximum of the barrier, which is given by (13)s0=-1αln (A + BA - B). Finally, the transmission probability (Bell, 1980), obtained through the solution of the Schrödinger equation with Eckart's potential, is expressed by the following equation (14)PEckart(E)=1-cosh[2π(k - β)] + cosh(2πδ)cosh[2π(k + β)] + cosh(2πδ), where k, β and δ depend on ν̄1, A, B and energy (E).

The quantum tunneling correction κ(T) can thus be calculated from the ratio between the quantum rate k_quan.(T) and the classical rate k_class.(T) in which the particles cross the barrier. Thus, the Eckart tunneling correction with transmission coefficient is given by (15)κ(T)=kquan.(T)kclass.(T)=eEb/kBTkBT∫0∞dE PEckart(E)e-E/kBT, where integration is performed over all possible energies.

The details about the generation of the potential energy surface are described in another work of our group (Roncaratti et al., 2014), so it will be commented briefly here. First, all HP geometry parameters were kept frozen at their equilibrium values of D_OO = 1.45Å, D_OH = 0.966Å and the angle HOO = 100.8°. The Ng's position is expressed in terms of the polar coordinates as represented in Figure 1, where R is the distance of Ng relative to the middle of O-O bond and α is the polar angle with respect to an axis perpendicular to the O-O bond (z axis). The two planes defined by O-O-H atoms are then rotated around the O-O axis, with steps of 1°. In addition, α was equal to 0°, 45°, 90° and R distance was varied from 2 to 5Å with steps 0.1Å.

Topological studies performed on this adjusted potential energy surface (PES) showed that the HP and HP-Ng complexes have two overall minimum configurations, termed cis (labeled as θ₋) and trans (labeled as θ₊), separated by two potential barriers, denoted here as cis-barrier and trans-barrier. The potential energy curves (PEC) obtained from the PES are then presented in Figure 2.

The PECs obtained for HP-Ng complexes are similar in shape and depth. The cis-barriers for the HP-Ng complexes are all smaller than the respective value for the free HP. The free HP has a cis-barrier of 7.5594 kcal/mol whereas the values for the complexes increase monotonically from 6.9828 kcal/mol for HP-Rn up to 7.5107 kcal/mol for HP-He. In addition the trans-barrier values are also lower than the respective value for the free HP, which is 1.0427 kcal/mol, and their values are 1.0928 kcal/mol for HP-He, 1.0817 kcal/mol for HP-Ne, 1.0651 kcal/mol for HP-Ar, 1.0676 kcal/mol for HP-Kr, 1.0736 kcal/mol for HP-Rn and 1.0749 kcal/mol for HP-Xe complexes. For the free HP, the cis-barrier and trans-barrier experimental energies (Hunt et al., 1965) are 7.0334 ± 0.0715 kcal/mol and 1.1036 ± 0.0114 kcal/mol, respectively. These values are in a good agreement with our results. However, for the HP-Ng complexes we did not find experimental data for comparison.

The results concerning geometric parameters, interactions and their characterization are summarized in Tables 1–3. The geometrical parameters obtained at MP2(full)/aug-cc-pVTZ, optimized without any constraints for two minimum structures and transition states (presented as cis and trans barriers) are given in Table 1 together with the graphical representation in Figure 3. Table 2 lists the binding energies corrected and uncorrected for the BSSE. Table 3 shows the numerical results for AIM analysis and Figure 4 depicts the ∇²ρ(r) contour plots for cis and trans barrier configurations.

The PESs yield θ₋ and θ₊ as true minima, i.e. without any imaginary frequencies in accordance with results from literature (Maciel et al., 2006; Roncaratti et al., 2014). In addition, all transition state structures displayed a well characterized imaginary frequency around 600 cm⁻¹ for cis and 400 cm⁻¹ for trans barriers (see Supplementary Information for further details). Figure 5 describes a schematic representation of vibrational modes of the isolated HP in the transition state with the actual frequencies and an imaginary frequency, which represents the frequency along the reaction coordinate.

For the HP-Ng complexes, the geometrical parameters are almost the same when compared with isolated HP in agreement with the weak interaction of these systems. The HP-Ng distances increase from He up Rn. On average, they are close to 2.55Å(He), 2.65Å(Ne), 2.75Å(Ar), 2.85Å(Kr), 3.00Å(Xe), and 3.06Å(Rn).

Regarding the binding energies, HP-He and HP-Ne are all repulsive, being less repulsive for the cis barrier configuration. This can be understood as a consequence of the fact that the noble gases turn out to be the hardest elements (Furtado et al., 2015) and this hardness decreases when the Ng atomic number is increased (the hardness in this context is a resistance to changes in its electronic population Furtado et al., 2015 coupled to Ng's high electronegativity Allen and Huheey, 1980). Although the BSSE increases monotonically from He to Rn, the binding energies also become more attractive as the atomic number increases.

For the four structures of each HP-Ng, the higher binding values are always observed for the cis-barrier configuration. As it will pointed latter, the decrease of the rate through the two barriers are not correlated with the binding energy, suggesting the hyperconjugation effects on HP may be important for the decrease of the interaction rate.

Regarding the AIM analysis, the existence of (3,−1) bond critical point (BCP) and its associated atomic interaction line indicates that electronic charge density is accumulated between the linked nuclei (Bader, 1991). In its turn, the values of the charge density ρ(r) in BCP are small while their corresponding ∇²ρ(r) are positive in accordance with a closed shell type of interaction. As a consequence, all configurations of all complexes show an interaction of a van der Waals type. Since higher ellipticity suggests conjugation and hyperconjugation effects of electron delocalization, these effects seem more pronounced in the HP-He and HP-Ne complexes. Another interesting feature is that all cis-barrier configurations of all complexes show a (3,+1) BCP indicating a cyclic nature.

The temperature dependence of the rate constant for cis to trans (i.e., through trans-barrier) and trans to cis (i.e., through cis-barrier) conformations for HP and HP-Ng complexes are presented in Figure 6. These results, in addition to conventional rate, are also exhibited with Eckart's or Wigner's tunneling corrections.

It was found that for the entire 100 K up to 4,000 K range the HP-Ne has the lowest rates for both barriers among all noble gas complexes, followed by HP-He. This result suggests that Ne and He are the noble gases more suitable for study the oriented collision dynamics with HP. In fact, the decrease of CRT shows an inverse correlation with respect the average valence electron energy (Allen, 1989), which follows the sequence (from higher to lower values): Ne, He, Ar, Kr; with Xe and Rn having very close values.

Nevertheless, there is a trend of rate increase as are move from Ar up to Rn. It is interesting to note that although this behavior is very similar regarding the cis-barrier for all rates (conventional, Wigner e Eckart), it seems that the tunneling is more important to describe the trans barrier's rate, where there is a significant difference for Eckart's values specially in the 100–200 K range when compared to respective Wigner and conventional results.

The final thermal rate constant can be expressed in the two familiar Arrhenius forms. In this work, the first is the Arrhenius modified form given by (16)k(T)=ATne-Ea/RT, where A is the pre-exponential factor, T a temperature, n is a real number, R is the universal gas constant and E_a is the activation energy. The second is the d-Arrhenius form (Aquilanti et al., 2010; Silva et al., 2013; Carvalho-Silva et al., 2017) expressed by (17)k(T)=A(1-dEaRT)1/d where d is a parameter that yield the degree of deformation of the exponential function.

The curve obtained by the reaction rate constant vs. the temperature can be fitted (Ramalho et al., 2011) to obtain the parameters A, n and E_a for the Arrhenius modified form, as presented in Table 4, and the parameters A, d, and E_a for d-Arrhenius form, as presented in Table 5. This feature confirms the trend of lower k(T) observed for HP-Ng complexes for both barriers when compared to isolated HP.

It can be also observed in Figure 6 that the trans to cis conformation rate of HP is lower (in the range 100–200 K) than the corresponding ones for HP-Ar, HP-Kr, HP-Xe, and HP-Rn. In the case of the chiral transition from cis to trans, the rates of all HP-Ng complexes are lower than that of the isolated HP. These results showed that the transition rate from cis to trans is greater than the corresponding trans to cis for both the isolated HP molecule and for all HP-Ng complexes. This suggests that the most important barrier that separates the chiral configurations of the isolated HP and the HP-Ng complexes is the trans-barrier, since it is the smallest. The energy of the HP's trans-barrier is relatively small (1.0427 kcal/mol) compared to its cis-barrier (7.5595 kcal/mol) as already seen in Figure 2.

An interesting result is presented in the Table 6. Although the increase in the trans-barrier of the HP-Ng complexes relative to HP is considerably small (see Figure 7), the change in the transition rate from cis to trans is relatively high. This is verified for high (4,000 K), room (298.15 K) and also for low temperatures (100 K). The most pronounced decrease in the rate corresponds to the HP-Ne complex, in which the decrease of the trans-barrier of just 0.0389 kcal/mol (see Figure 6) corresponds to a decrease of over 60% for the cis-trans transition rate, followed by HP-He. It is also interesting to note that this small change in energy barrier but with a substantial change in rate was also observed for other HP-Ng complexes. For example, HP-Ar complex showed a decrease of just 0.0224 kcal/mol but a 28.68% decrease of rate at 100 K.

Finally, at a temperature close to 300K, the Boltzmann distribution shows that about 16% of HP's population has higher energy than the trans-barrier with thermal fluctuations of approximately 1.7686 kcal/mol (Ball and Brindley, 2016). It has also been found that at low temperatures the chiral interconversion quantum encapsulation time of HP is very small. At a temperature of 100 K this time is <1 pico-second (Bitencourt et al., 2008), and at temperatures close to 0 K which can reach 3 pico-seconds.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.