Quantum Effects in Cold Molecular Collisions from Spatial Polarization

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Spectroscopy and Photochemistry; General Theory

Quantum Effects in Cold Molecular Collisions from Spatial Polarization of Electronic Wave Function Debarati Bhattacharya, Mariusz Pawlak, Anael Ben-Asher, Arie Landau, Idan Haritan, Edvardas Narevicius, and Nimrod Moiseyev J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.8b03807 • Publication Date (Web): 07 Feb 2019 Downloaded from http://pubs.acs.org on February 8, 2019

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The Journal of Physical Chemistry Letters

Quantum Effects in Cold Molecular Collisions from Spatial Polarization of Electronic Wave Function Debarati Bhattacharya,†,⊥ Mariusz Pawlak,‡,⊥ Anael Ben-Asher,† Arie Landau,† Idan Haritan,† Edvardas Narevicius,∗,¶ and Nimrod Moiseyev∗,†,§,k †Schulich Faculty of Chemistry, Technion–Israel Institute of Technology, Haifa, 32000, Israel ‡Faculty of Chemistry, Nicolaus Copernicus University in Toru´n, Gagarina 7, 87-100 Toru´n, Poland ¶Department of Chemical Physics, Weizmann Institute of Science, Rehovot, 76100, Israel §Department of Physics, Technion–Israel Institute of Technology, Haifa, 32000, Israel kRussell-Berrie Nanotechnology Institute, Technion–Israel Institute of Technology, Haifa 32000, Israel ⊥These two authors contributed equally E-mail: [email protected]; [email protected]

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Abstract The quantum phenomena of electronic and nuclear resonances are associated with structures in measured cross-sections. Such structures were recently reported in cold chemistry experiment of ground state hydrogen isotopologues (H2 /HD) colliding with helium atoms in the excited triplet P-state (He(23 P)) [Shagam et al. Nature Chem. 2015, 7, 921], without giving theoretical explanation as to their appearance. This work presents a quantum explanation and simulation of this experiment, which is strictly based on ab initio calculations. We incorporate complex potential energy surfaces into adiabatic variational theory, thereby reducing the multidimensional scattering process to a series of uncoupled 1D scattering “gedanken experiments”. Our theoretical result, which is in remarkable agreement with the experimental data, manifests that the structures in the observed reaction rate coefficient are due to the spatial arrangement of the excited He p-orbitals with respect to the interaction axis, consequently, changing the system from a normal two-rotor model to a three-rotor one. This theoretical scheme can be applied to explain and predict cross-sections or reaction rate coefficients for any resonance related phenomenon.

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The breakdown of a metastable quantum state, into two or more sub-systems, after a finite period of time is known as a resonance phenomenon. 1 Resonances are quite widespread in nature and their occurrence is common in various fields of physics and chemistry. For example, in nuclear physics, resonances are associated with radioactive radiation. In quantum optics, they are associated with leaky modes in wave-guides and optical fibres. Whereas, in chemistry they are associated with autoionization states. Such states can be formed in electron scattering experiments, 2–4 where an electron can be temporarily trapped by an atom or molecule. More specifically, autoionization states of molecular species were observed in cold-chemistry experiments. 5,6 A recent paper of the group of E. Narevicius 5 associates these resonances with structures in the rate of a chemical reaction during an atom-molecule collision at low energies. This pioneering work was followed by several other works from the same group highlighting the occurrence of distinct structures in the reaction rate coefficient and their role in nuclear dynamics in sub-kelvin temperature regime. 6–8 Specifically, these experiments generated a super-molecule in a metastable state upon collision of an excited helium atom (He∗ ) with argon and molecular hydrogen. 5 These works revealed the quantum phenomena of resonance, where the decay channel was dominated by Penning ionization (PI). Within the super-molecule, He∗ +H2 , an electron transfer occurs from the hydrogen molecule to the helium atom. This electron is transferred into the core orbital of helium and completes the occupancy of the inner shell of the helium atom. At the transition state, He∗ has partial negative charge, while H2 has partial positive charge. After a finite period of time, this metastable super-molecule decays by the ejection of the electron from the outer valence orbital of He. This mechanism has been theoretically confirmed for the He(23 S)+H2 system. 9 A recent reaction rate measurement where the helium atom was excited to the triplet P state showed dramatically different result. 6 In that case rigorous ab initio calculations are lacking. Here, we present the quantum explanation and simulation of the collision between He(23 P) and H2 /HD at millikelvin regime. Molecular hydrogen, naturally exists in two nuclear spin states: the spin anti-symmetric para state, which is lower in energy with only even rotational states ( j = 0, 2, ...), and the spin symmetric ortho state, which is higher in energy with only odd rotational states ( j = 1, 3, ...). At normal

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temperature (room temperature) and pressure, when the molecule is in the lowest state, p-H2 and o-H2 reside in the j = 0 and j = 1 rotational levels, respectively, with the ratio between the para and ortho states being 1:3. However, on controlled cooling to temperatures around 0 K, it is possible to recover pure para-hydrogen (p-H2 ). This makes p-H2 an easier and interesting molecular species to study experimentally. Since the p-H2 ( j = 0) wavefunction is spherically symmetric in space, the intermolecular interaction has no orientation dependency. The anisotropic components of the interaction contribute when the molecule starts rotating (i.e., when the molecule is rotationally excited). This renders some simplicity in the theoretical calculation as well. The quantum effect on the structure of the rate coefficient is obtained only when the hydrogen molecule (H2 /HD) is in its’ ground rotational state (j=0), and not in the higher (j=1) rotational state. Therefore, the present work focuses on the experimentally observed structures on collision of para molecular hydrogen and its’ isotopic form HD, in the lowest rotational state with excited helium atom in the P state. Several theoretical quantum mechanical scattering techniques can be applied to study lowenergy molecular collisions. 8,10,11 We recently developed adiabatic variational theory (AVT) for cold atom–diatom collision experiments. 12,13 We used it together with time-independent nonHermitian scattering theory 14–17 to derive a closed-form expression for the rate coefficient. 18 AVT reduces the multi-dimensional problem, to many uncoupled 1D sub-problems. This approach is universal and can be applied to any kind of ionization process (PI, interatomic (intermolecular) Coulombic decay (ICD), 19,20 or Auger decay). The interaction potential that is used as a basis in adiabatic theory is well described by a complex potential energy surface (CPES). A CPES correctly predicts the position (relative to the ionization threshold) and decay rate (inverse lifetime) of a resonance state with respect to the geometry of the system. 1 Due to the complexities involved in the ab initio treatment of many electron resonance states, generating CPESs is not a trivial task. Complex scaling (CS) 21–30 and complex absorbing potentials (CAPs) 31–39 are regularly used to calculate resonances. However, CS cannot be applied even to any simple molecular species due to singularities that arise in the Hamiltonian of multi-nuclei systems. 40 On the other hand, removal of the artificial complex potential that is introduced in CAP calculations, is cumbersome. 41 A simpler yet effec-

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tive way to generate complex surfaces is by analytic continuation of the Hamiltonian eigenvalues through scaling of the basis functions. 42–47 This approach has been successfully applied by the authors, to generate an accurate CPES for the He(23 S)+H2 molecular system. 9 The methodology, termed as the resonance via Padé (RVP), uses the Padé approximant to analytically dilate from the real into the complex plane. 47,48 Herein, we explain the structures observed in the experimental rate coefficient for the predominant PI decay of He(23 P)+H2 /HD using our in-house RVP and AVT techniques. We show how the presence of the single electron in the excited p-orbital of the Heatom, changes the entire system to a 3-rotor problem rather than the simpler 2-rotor model in the He(23 S)+H2 system. The spatial polarization of this orbital brings in pronounced anisotropic effect in the interaction potential that provides a significant contribution to the calculated rate coefficient. Our theoretical calculations do not include any external scaling or fitting parameters with respect to the complex potentials, and can be considered to be numerically exact for the resonance position and widths, as a function of the configuration of the super-molecule. Therefore, it is expected that the rate coefficients calculated using these CPESs should have the same shape and structure as observed in experiments, provided that the reaction rate coefficients are also calculated numerically exact. In the experimental work, 6 the authors interpreted the asymptotic behaviour of the rate coefficient at high temperatures using the classical Langevin power rules that were derived for cold collisions. We see a clear demarcation in the calculated rate coefficient, where the “classical” prediction meets the complete quantum prediction. This transition zone that appears as a drop in the experimentally observed rate coefficient around 1 K, is slightly shifted towards lower temperatures in the theoretically calculated rate coefficient. We also show that the shape of the rate coefficient over the entire temperature range can be predicted fairly accurately, by using only the complex potential energy surface of the most symmetric configuration. In the present work, we demonstrate how the power law holds true at high temperatures, while explaining the drop in the reaction rate coefficient below 1 K using our quantum calculations, thus proving the general applicability of our theoretical approach. The theoretical PI rate coefficients obtained for the He(23 P)+H2 ( j = 0) and He(23 P)+HD( j =

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Reaction rate coefficient (cm3/s)


He*+H2

10-9

E1/6

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He*+HD

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Figure 1: Theoretical and experimental 6 PI rate coefficient of He(23 P) colliding with H2 /HD in the ground rotational state ( j = 0). The theoretical results are shown in blue and cyan, while the experimental ones in black and grey with error bars for H2 and HD, respectively, over a temperature range of 5 mK till 100 K. The red dotted line is the theoretical Langevin power law (∝ E 1/6 ) with a coefficient value of 122 a0 , as calculated in Ref. 6. As observed, the power law matches well with the He(23 P)+H2 rate coefficient till about 0.8 K. Below this temperature, the power law fails to explain the drop in the rate coefficient, whereas our ab initio theoretical calculation reproduces the structure. A similar behaviour can be seen for the He(23 P)+HD rate coefficient, given here in arbitrary units as in Ref. 6. By properly taking into account the electronic as well as nuclear resonances, the reaction rate coefficient is successfully reconstructed for almost the entire range of temperature, especially for the He(23 P)+H2 collision pair. 0) systems are shown in Fig. 1. The figure compares the theoretical rate coefficients (in blue and cyan) with the experimental ones (in black and grey) for the temperature range of 5 mK to 100 K. The experimental results for He(23 P)+HD are in arbitrary units, 6 scaled down by 30% relative to the measurement for H2 . Our result is clearly in very good agreement with the experiment over nearly the entire temperature range. The measured reaction rate coefficient by the Narevicius group, found the PI product to weigh 90% at all collision energies. 6 We have thus assumed that the process is dominated by the PI decay channel and have neglected the associative ionization channel. The figure also shows the Langevin power law (in red dotted line) that scales as E 1/6 . In the experimental paper, 6 the authors had related their theoretical rate coefficient to the long range van der Waals interaction, where the potential scales as 1/R6 . Based on those calculations, the 6


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claim was that the entire rate coefficient would be controlled by the classical Langevin power law. We show from our quantum calculations that there is a clear transition from this “classical" regime to the quantum region which is characterised by the drop in the reaction rate coefficient below 1 K for He(23 P)+H2 ( j = 0) and just above 1 K for He(23 P)+HD( j = 0). 3000 2000

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Figure 2: Complex potential energy curves for the reaction of He(23 P) and H2 : a) The estimated real part of the potential energy curves calculated using a high level of theory. More details are provided in the methods section. The most attractive potential with B1 symmetry has a potential depth of about 4800 K at 2 Å. A total of three potentials are attractive in nature while two are repulsive. b) The ionization widths Γ (inverse lifetimes, in logarithmic scale) of ground state molecular hydrogen on collision with He(23 P). The most symmetric potential (B1 ) which is also the most attractive potential surface has the fastest decay rate [in black]. The linear variation of all the decay functions, show how the decay rate increases exponentially as the intermolecular separation decreases. The exponential decay rate is a signature of PI. 50 To determine the theoretical reaction rate coefficient, we first need to calculate the interaction between He(23 P) and H2 as a function of the intermolecular separation. Within the framework of the Born–Oppenheimer approximation, isotopic substitution does not have an impact on the electronic potential. To get an estimate of the strength of the interaction we have calculated the potential in two of the most symmetric configurations. When the super-molecule is in linear geometry, there are two adiabatic potential energy surfaces with Σ and Π symmetries. In any non-linear geometry (say, with point group C s ), the linear Π state splits into A and A electronic states, whereas 0

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Σ transfers to A electronic state. 51 Thus, an ab initio calculation under the Born–Oppenheimer 0

approximation neglecting the spin-orbit coupling will produce two or three potential energy sur7



faces based on the relative spatial orientation of the p-orbital with respect to the intermolecular axis. We used only two of the most symmetric orientation of the super-molecule: T-shape and linear configurations, with H2 perpendicular and parallel to the collision trajectory, respectively. The T-shape configuration has C2v point group symmetry and generates potentials with A1 , B1 and B2 symmetries. A real (Hermitian) calculation of the potential energy curves for the two configurations thus gave rise to five different potential curves. At first, approximate calculations were carried out assuming that the target electronic state was a bound state. Such bound state calculations revealed that all the five potentials were actually embedded in the continuum, i.e., their energy is above the ionization threshold. Thus, all the states are resonance states. A resonance state, within non-Hermitian formalism, is characterised by complex energy, where the real part of the complex energy gives the resonance position and the imaginary part is related to the decay strength or inverse lifetime. 1 Our previous bound state calculations had given us a rough estimate of the resonance position, however, in order to estimate the decay rate we needed to move into the complex plane. This was achieved by applying the RVP methodology which has been developed by the authors 47 and successfully implemented to predict CPES for the simpler He(23 S)+H2 9 case. The electronic resonance positions (real parts) and corresponding decay rates of the five potential energy surfaces for the He(23 P)+H2 system are shown in Fig. 2. The decay rates for the systems He(23 S) and He(23 P) colliding with H2 have been recently calculated using the Fano–algebraic diagrammatic construction technique 52–55 by Yun et al. in Ref. 56, where they seem to be almost similar in nature. The decay rates presented in Fig. 2, show that they differ in each molecular configuration. The contrasting character of the decay rates between our findings and those presented by Yun et al. 56 is due to the different methods and level of calculations involved. While, the analytic continuation (in the RVP methodology) of the ab initio real potential energy surface to the complex plane provides simultaneously, the numerically exact values of the resonance energy position and the width, the Fano approach used in Ref. 56 is based on the approximation that detach the calculation of the resonance position from the calculation of its width. Within the framework of the Fano approximation, the resonance is described as a bound state in the continuum (neglecting the

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resonance energy shift) which is coupled to the degenerate continuum (that is usually calculated at a lower accuracy level). The most symmetric potential (with the B1 irreducible representation in the T-shape geometry) has the deepest well (it reaches to about 4800 K at an intermolecular distance of 2 Å), and it decays rapidly. The degenerate asymptotes of all the real potentials have been shifted to zero to facilitate comparability. Since, the asymptote denotes the entry channel of the reactants in the collision process, it is expected that all states will contribute. However, as will be discussed in details below, the collision is dominated by the B1 potential, which is the lowest in energy. The decay rates of all the CPESs, are found to be exponential in nature. Such an exponential decay rate is a fingerprint of PI. 50 The calculated interaction potentials were then expanded in a series of renormalized spherical harmonics that were derived keeping in mind the extra dimension introduced by the He p-orbital. Here, the system is treated as a three-rotor problem. On the contrary, the He(23 S)+H2 system, 9 in which the excited electron in the He-atom was occupying the spherically symmetric 2s-orbital, was treated as a two-rotor model. In the event of the hydrogen molecule in the ground rotational state, where the wavefunction is spherically symmetric, the interaction potential simplifies to only the isotropic term V000 (R) and the anisotropic term V020 (R)P2 (cos θa ) arising from the orientation of the p-orbital with respect to the intermolecular axis. This can be contrasted with the rotationally excited hydrogen molecule, where the quadrupole-quadrupole interaction gives rise to a purely anisotropic term as well. 6 Since the isotropic and anisotropic radial interaction potentials are complex, we plot in Fig. 3 both the real and imaginary parts as a function of intermolecular distance, where the asymptotes are shifted to zero. The rapidly increasing real anisotropic radial term Re(V020 ) seems to dominate over the real isotropic radial term Re(V000 ) as the intermolecular separation decreases. However, note that V000 is the leading term in the renormalized spherical harmonics expansion of the interaction potential between reactants (see the methods section for more details), thus the effect of anisotropy on the rate coefficient can be significant but not dramatic. The absolute value of the imaginary part of V020 is smaller than that of V000 only at short intermolecular distance. All in all, the anisotropic

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Figure 3: Isotropic and anisotropic radial interaction potentials of the He(23 P)+H2 ( j = 0) system, V000 = Re(V000 ) − iΓ000 /2 and V020 = Re(V020 ) − iΓ020 /2, respectively. The three label indices are associated with the quantum excitations of the three modes of the activated three body complex which is described here by three rotors. The real part of the anisotropic term has a well and is seen to increase exponentially from an intermolecular separation of around 7.5 Å down to R < 7.5 Å. This shows that as the intermolecular separation decreases the anisotropic effect starts increasing rapidly. The imaginary part of the anisotropic term is weaker than of the isotropic one only at very small values of R (R < 4.3 Å). The inset shows this effect as a zoom-in version of the main plot. term is expected to play an important role in the collision process as the absolute value of V020 is large. In order to get an insight into the investigated collision, we wish to isolate the effect of anisotropy. In addition, in order to pinpoint the most relevant factors in this reaction, we recalculate the reaction rate coefficients using only the T-shape B1 potential. Fig. 4a shows the behaviour of the rate coefficients with contribution from only V000 interaction term. The rate coefficients obtained with only the isotropic term, gives a fairly good representation of the experimental findings. This rate coefficient is marginally lower than the experimental one. However, the effect of anisotropy in the collision process is pronounced. Only by proper inclusion of the anisotropic term we were able to predict the experimental result, see Fig. 1. The anisotropy not only increases the rate coefficient of the reaction, but also changes the relative height and position of the resonance peaks that came

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He*+H2

10-9


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He*+HD

10-10

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He*+H2

10-9

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10-1

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Figure 4: Comparison of theoretical (curves) and experimental 6 (black and grey dots with error bars) rate coefficients of He(23 P) colliding with H2 ( j = 0) and HD( j = 0): a) The calculations were performed using only the isotropic radial interaction potential (V000 ) [blue and cyan solid curves]. Although the rate coefficients given only by the isotropic term lie slightly beneath the experimental rate coefficient for both the reactions, it does give a reasonable description of the entire reaction rate coefficient. The sudden drop in the rate coefficient is clearly depicted by the isotropic term as well. Inclusion of the single anisotropic term that arises from the spatial distribution of the partially filled p-orbital of helium, raises the theoretical rate coefficient to match with the experimental one. A peak that was pronounced in the isotropic rate coefficient (at 0.7 K) for the reaction He(23 P)+H2 is subdued by the anisotropic term (compare with Fig. 1). A similar peak that is seen at 0.3 K in the reaction He(23 P)+HD, is also subdued by including the anisotropic term. b) The calculations were performed with only the most symmetric potential, B1 in the T-shape configuration [green solid curves]. The rate coefficient has been shifted up in this case for better comparison with the experimental results. The red dotted line represent the Langevin power law that scales as E 1/6 . The calculated rate coefficient is in good accord with a classical prediction down to 0.8 K for the He(23 P)+H2 and He(23 P)+HD reactions. The extra peaks that are seen at the high temperature region arise from the adiabats that support extra resonances but their effect gets nullified when taking into account the contribution from all other potential surfaces. up in the isotropic rate coefficient. Fig. 4b shows the contribution from B1 configuration alone. The calculated rate coefficients reconstruct most of the structures observed in the experiment. This shows that the majority of the reactants will populate and collide along this particular adiabatic surface. The actual rate coefficient obtained from this most symmetric configuration alone was lower than the measured rate coefficients of both the reactions by an order of magnitude. In the Fig. 4b we have shifted the rate coefficients obtained using complex potential energy surface with B1 symmetry (shown in solid green lines) for a clear comparison with the experimental data. Moreover,

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in this calculation, we get convergence of the classical behavior and the quantum one. According to Langevin rules, which are classical, the behavior of the reaction rate should scale as E 1/6 . Here, we obtain this behaviour down to 0.8 K, but as the temperature decreases, the theoretical results show a sharp drop which is very similar to the observed one. This clearly show that the measured rate coefficients are dominated by quantum effects below 1 K. The fact that the calculated rate coefficients had to be moved up vertically without changing the shape to match the experimental finding, proves that even though this is the most dominant collision channel, the contribution from all other potentials cannot be neglected for an accurate prediction of rate coefficients. To generate the final reaction rate coefficient as shown in Fig. 1, we have extended the adiabatic variational theory 12,13 that was initially derived for diatomic molecules colliding with atoms in the S-state (two-rotor model). In general, the considered system He(23 P)+H2 /HD is a three-rotor system, hence we included in our approach all proper rotations. Within the framework of AVT, the many degrees of freedom problem (where the distance between nuclei in H2 /HD is fixed), is replaced with a set of uncoupled one-dimensional Schrödinger equations with adiabats (effective potentials). These adiabats are determined on the basis of isotropic and anisotropic radial interaction potentials. We applied a simple closed form formula for the rate coefficient presented in Ref. 18. This expression does not require any knowledge of the products in the collision process. Rather, it includes knowledge of only the eigenenergies of the radial Schrödinger equation of all the different adiabatic potentials that are complex in nature. The explicit expression for the collision rate coefficient requires the well defined rotational state of the molecule and the energy density of states. Only by the use of the 2D density of states we were able to correctly map the experimental data for entire temperature range. Thus, in our calculations we have used 2D density of states, which is energy independent (based on quantum mechanical definition) to get good agreement as such. In the calculation of the rate coefficient using only the B1 potential, we have used the 1D density of states. Nevertheless, it must be stressed that AVT is suitable for smooth complex effective potentials, in which avoided crossings do not occur. Strong anisotropic effects may introduce major avoided crossings and in such cases AVT might not be appropriate. In the present case, they

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indeed introduce some avoided crossings in the adiabats. Therefore, the adiabatic approximation needs to be handled carefully, however as we have shown, it can be used for the 3 P state fairly accurately judging from the final results for almost the entire temperature range, especially for the lower temperatures. In addition, this is the simplest tool that can be universally applied to calculate the cross sections or reaction rate coefficients for any type of decay channels. Overall, the theory can be applied to other systems as well but much depends on the shape of the complex potentials, specially the shape of the anisotropic term. So far, theoretical rate coefficient calculations have had to introduce some scaling or shifting parameter with respect to the complex potentials to match experimental observables. 5,7,8,12,13 In most such calculations, the real and imaginary parts of the interaction potential was seldom treated at the same level of theory. This may have caused some discrepancies in the final rate coefficient results. However, to treat any such low temperature decay process is extremely challenging, partly due to the high level of theory needed to generate accurate CPESs and partly due to the exact working equations needed to generate correct rate coefficient using those CPESs. Thus, rate coefficient calculations involve a two level complexity. We have shown a way to treat the real and imaginary parts of the interaction potential at the same level of theory, thus reducing discrepancies on this front. In conclusion, we put forward a compact and easily applicable theoretical approach for calculating cross sections or reaction rate coefficients absolutely in an ab initio fashion. Although the method has some limitations, it was able to provide very good agreement with recent coldchemistry observations. In particular, we have shown herein that the He(23 P)+p-H2 rate coefficient does indeed follow the power law at the asymptote for relatively higher temperatures, however, below 1 K quantum effects become dominant. There is a clear transition zone in the reaction rate coefficient that is characterised by a sharp drop in the calculated rate coefficients. Just above this drop zone, i.e., above 1 K the classical power law can be applied to predict the rate coefficient. Below 1 K, the classical explanation fails completely and the rate coefficients are governed by quantum laws. The calculation based on the most attractive electronic potential alone, supports the

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E 1/6 scaling law at higher temperature. The reaction rate coefficient is dominated by this T-shape B1 potential, although in order to obtain quantitative agreement with experiment, all the CPESs are required. The behaviour of the rate coefficient is nearly identical for the He(23 P)+HD( j = 0) system. Moreover, we have demonstrated how important a role the non-isotropic interactions play in low-energy collision dynamics. Only by inclusion of the interaction anisotropy, that results from the spatial distribution of the partially filled p-orbital of metastable helium atom, the resonant structure was properly reproduced. By using the AVT approach for the calculation of the rate coefficient (which approximates the multi-dimensional problem to a series of uncoupled subproblems), we are able to pinpoint the exact location of the deviation from the classical behavior. This work illustrates the validity and accuracy of the CPESs presented herein. We note, that using CPESs, which couples electronic and nuclear coordinate, opens the gate for predicting and studying chemical reactions in which autoionization and dissociation take place simultaneously. It is pertinent to reinstate here, that our methodology is system independent, the bottleneck being generation of the real potential energy curve. However, with the rise of efficient Hermitian electronic structure codes our approach can tackle even large multi-electron systems (e.g. DNA/RNA nucleobases) to predict the outcome of any collision/scattering processes.

Methods Real potential energy curves. The complex potentials were calculated in two steps. First, the real potential energy curves were generated under the Born-Oppenheimer approximation. Under this approximation, the nuclei are assumed to be held at fixed geometry as compared to the fast changing geometry of the electrons. We employed the non-relativistic Hamiltonian, as at relatively large intermolecular separation of He and H2 , relativistic effects are indeed negligible. For this particular system of excited He-atom with partially filled p-orbital it was suggested, by Dubernet and Hutson, 51 that even the largest relativistic contribution, i.e., the spin-orbit coupling, is negligible. Thus, we end-up with only three adiabatic potential energy curves. These curves were

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calculated only in the two most symmetric configurations, i.e., T-shape and linear geometries, under fixed hydrogen bond length equating to the expectation value of the interatomic distance in the vibrational ground state of H2 (r = 1.4487 a0 ). When the orientation of the H2 molecule is perpendicular to the main super-molecular axis, we assigned it to be in the T-shape geometry. The real potential energy curves were calculated using the equation-of-motion coupled cluster theory (EOM-CC) 49 as implemented in the latest version of Q-Chem – a quantum chemistry package. 57 The singles and doubles with perturbative triples truncation, EOM-CCSD(dT), 49 was used for the electronic structure calculations. EOM-CC treats both the static and dynamic electron correlation effects, which allow the description of multi-configuration wavefunctions in complex molecular systems, in addition, inclusion of triples corrections provides very accurate eigenenergies. The 5ZP basis set 58 was augmented by four extra diffuse s- and p-functions on the hydrogen atom. The helium atom was augmented by six extra diffuse s- and p-functions. All the diffuse functions were added in an even tempered basis with a constant ratio of 2.0. This basis set was optimized for this super-molecule in Ref. 9. The particular triplet state was identified on careful inspection of the coefficients that corresponds to the He(23 P)+H2 asymptote. In the T-shape geometry (C2v point group), the three potential energy curves have A1 , B1 and B2 symmetries. On moving to the linear configuration (C∞v point group), the B1 and B2 electronic states merge to form 3 Π state and the 3 A1 electronic state becomes the 3 Σ state. The total of five potential energy curves were obtained assuming that the electronic state of interest was a bound state. Thus, the real surfaces provided only an approximate position of the final resonance state. In order to correctly predict the electronic resonance position and lifetime, we needed to move into the complex plane. The considered state of He(23 P)+H2 is embedded in the continuum of the He(11 S)+H+2 species and hence is indeed a resonance state. Complex potential energy curves. Analytic continuation method was used to move from the real plane to the complex plane. We used the previously generated real potential energy curves to analytically dilate into the complex plane. The method that we applied here, involved the generation of stabilization graphs, at each geometry using standard (Hermitian) quantum chemistry

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codes, followed by analytic dilation using the Padé approximant, it is named – resonance via Padé (RVP). This method has already been used successfully to calculate the CPESs for the He(23 S)+H2 system. 9 A plot of the energy of the state in question, versus the scaling parameter is known as a stabilization graph. 59 We obtained stabilization plots by scaling the basis set exponents by a real scaling parameter (α) that can vary over a chosen range. In the present case, the basis set scaling was restricted to the extra diffuse functions only, i.e., those s- and p-functions whose exponents were appended in the real calculations, while the tight basis functions were left unscaled. The scaling parameter was varied over a wide range (0.6 6 α 6 2.5) with a difference of 0.01 between each α value. The same calculations (as we did for the real potential energy curves) were then re-run with the new scaling parameter for each intermolecular separation. Thus, a large number of grid points were obtained after the scaling of the basis set exponent. The characteristic features of a stabilization graph are: a stable line (energy) that remains fairly unchanged upon small variations of the scaling parameter, and additional lines that strongly depend on α and try to cross the stable line, thus generating avoided crossings. The stable line corresponds to the resonance state, whereas the unstable lines correspond to the continuum states. This behaviour can easily be understood from a physical viewpoint of the resonance and continuum wavefunctions. The resonance wavefunction is localized in the interaction region. Thus, it will not be affected by small changes of the basis functions. 1 On the other hand, continuum wavefunctions are not localized in space and since basis-set scaling changes the basis-set spatial distribution, it leads to massive change of the eigenvalue spectrum of the Hamiltonian with respect to even small variation in the scaling parameter. 1 Such stabilization plots were obtained for each molecular geometry spanning all five potential energy surfaces. The stable regions in a stabilization plot gave a qualitative estimate of the resonance position. However, in order to quantitatively estimate the correct resonance width and position, we needed to dilate into the complex plane. After the generation of stabilization graphs, analytical dilation into the complex plane was achieved by the Padé approximant. The stable regions are also analytic functions of the scaling parameter, unlike the avoided crossings. 48 Thus, we used the energy

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eigenvalues from those stable regions and constructed an energy function as a ratio of two polynomials. That is, a finite set of scaling points and their corresponding eigenvalues were chosen from the stable region as the RVP input data. The resonance energies were then identified as stationary points in the complex plane. The exact working equation for this analytic continuation scheme is presented in the supporting information. On a practical note, the analytical fitting of the energy functional was generated using the Schlessinger point method. 60 The set of chosen points were varied systematically until convergence was obtained in terms of the final complex energy values. A statistical analysis was carried out in order to locate the most stable solution, having the least mean squared deviation. Thus, we obtained a single complex energy for each grid point on the intermolecular axis, where the real part of the complex energy corresponded to the resonance position while the imaginary part corresponded to the decay rate of the resonance state. In this way, we treat both the real and imaginary part of the complex energies at an equal footing. Adiabatic variational theory. After the CPESs were generated (we have five such complex potentials) we had to plug them into adiabatic variational theory and finally calculate the collisional rate coefficient. In order to include the effect of all the five CPESs, they need to be reformulated such that the potential energy surfaces could be expressed in terms of isotropic and anisotropic components. The complete intermolecular interaction potential expanded in a series of renormalized spherical harmonics and radial potentials is presented in the supporting information. For He(23 P) colliding with p-H2 /HD in the ground rotational state ( j = 0), the interaction potential reduces to 51 V CPES (R, θa ) =

X

V0q0 (R)Pq (cos θa )

q=0,2

1 = V000 (R) + V020 (R) (3 cos2 θa − 1), 2 where θa is a planar angle between the p-orbital and the intermolecular axis. The only one anisotropic radial interaction term that survives is due to the direction of the p-orbital of the Heatom. This is the V020 term. All other anisotropic radial terms vanish due to the symmetric nature 17



of the p-H2 /HD wavefunction, wherein the p-H2 /HD molecule in the j = 0 rotational quantum state can be assumed to be spherical in shape and can be treated as an atom. Thus, the interaction potential contains only two radial terms, the isotropic V000 (R) and the anisotropic V020 (R). We should emphasize that both of them are complex. Note that the problem here is much more complicated with respect to collisions between p-H2 ( j = 0) and He(23 S) because the interaction when the helium atom is in the S-state is totally isotropic. In our case, the spatial orientation of the helium p-orbital introduces the effect of anisotropy. We considered the Hamiltonian assuming the stiffness of the molecule. This approximation arises from the fact that the frequency of the diatomic vibration is much higher than the intermolecular frequency. 61 We extended and adjusted AVT 12,13 to our problem to break down the multi-dimensional problem to simpler sub-problems. Thus, according to AVT, we constructed the potential matrix in a basis set consisting of the products of three spherical harmonics, then we diagonalized it. We repeated these two steps for different values of R, which was treated as a parameter and not as a variable. In this way we obtained effective one-dimensional potentials (also known as adiabats). Under AVT, the Hamiltonian is dependent only on the distance between sub-systems in the super-molecule but the Schrödinger equation has to be solved for each effective potential. The time-independent Schrödinger equation was solved by means of the sin-DVR method with 3500 basis functions and a box length of 2500 a0 . To calculate the total PI rate coefficient as a function of collision energy we used a simple closed-form expression depending on density of states and imaginary part of eigenvalues. 18 In our present calculations involving described above potential energy surfaces, we have used the 2D density of states. It does not depend on energy (by definition), nonetheless depends on the box length. We have confirmed that the variation of the box length unaffected the whole structure but only moved the results vertically. The theoretical result reported here, shows the He(23 P)+H2 ( j = 0) rate coefficient computed for partial waves up to lmax = 5, i.e., six partial waves from l = 0 to l = 5 were included in the entire calculations. For the heavier system He(23 P)+HD( j = 0), we used lmax = 6. We did observe an increase in the rate coefficient in the higher temperature region on further increasing the value of lmax , as shown in Fig. S2 in the

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supporting information. This behaviour is due to the fact that at higher temperatures, the adiabatic approximation is in principle not applicable. This is supported by the calculations using only the potential of B1 symmetry, in which we do not use this approximation and convergence is achieved throughout the entire temperature range (see Figs. S2c and S2d). The reaction rate coefficients for temperatures lower than 6 K were converged for lmax = 5 and lmax = 6 respectively, for H2 and HD. Therefore, we used this value for the description of the rate coefficient for the higher temperature that shows the classical behavior of E 1/6 dependence. The transition from the classical to the quantum region in the rate coefficient is associated with an increase in the contribution of low partial waves (l ≤ 3) in the collision process. Higher values of l shifts the calculation towards the classical regime. Since, the measured reaction rate was energy-averaged, therefore the theoretical result was convoluted with the experimental energy resolution, using a Gaussian distribution with 10 m/s width.

Acknowledgement This research was financially supported by the I-Core: the Israeli Excellence Center “Circle of Light", by the Israel Science Foundation (Grants No. 298/11 and No. 1530/15), and by the National Science Centre, Poland (Grant No. 2016/23/D/ST4/00341). A.B.A. gratefully acknowledges the support of the Adams Fellowship Program of the Israel Academy of Sciences and Humanities.

Supporting Information Available The calculation details of the Padé approximant and the relevant working equations of adiabatic variational theory are mentioned in the given pdf file.

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Quantum Effects in Cold Molecular Collisions from Spatial Polarization

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