Transport Cross Sections and Collision Integrals for O($^{3}$P)–O($^{3}$P) Interaction

Abstract

New collision integrals and transport cross sections for O($^{3}$P)–O($^{3}$P) interaction are reported in the 300–30000 K range. Those values are based on a new set of potential energy curves (PECs) calculated with the multireference configuration interaction method. The results of the classical and semiclassical WKB (Wentzel–Kramers–Brillouin) methods are compared, excellent performance of the classical approach is shown (discrepancy much lower than 1% even at room temperature). In particular, the classical and WKB methods agree very well for the repulsive potentials effectively reducing overall uncertainty.

Introduction

In the study of low-temperature plasmas, the region of 300–1000 K is crucial and possibly the most prone to inaccuracies [1, 2]. For modeling of transport properties of gases and plasmas, the collision integral approach is typical [3]. For an accurate description of plasmas, heavy species (atoms, ions, molecules) and electrons have to be considered and appropriate data are stored in the databases such as LXCat [4] and IAEA databases (CollisionDB and ALADDIN).

The interaction potentials themselves are also needed for the calculation of transport properties, however, those are often not available. The LXCat and IAEA databases store some atom/ion interactions [4], the Quantemol database [5] and plasma studies of nitrogen-oxygen mixtures [6] make extensive use of the Lennard–Jones model which in case of interactions of atoms and atomic ions is almost always not appropriate.

It was already verified that the quality of PECs of interacting atoms/ions is of prime importance, especially at lower temperatures [1, 2, 7], so that reporting of accurate PECs of the ground and exited states is crucial. Knowing that sensitivity, only the high-quality PECs (preferably ab initio) should be used and simplified models such as Morse, Lennard–Jones, hardsphere, variable hard sphere, and variable soft sphere [8, 9] should be avoided for atom/ion interactions if possible. Some latest studies tend to use ab initio based collision integrals when it is possible [10, 11].

The PECs for the interactions of the ground state oxygen atoms are calculated by the ab initio method to compare the classical and semi-classical WKB methods for the system of heavier atoms (without light hydrogen atoms involved as previously compared for hydrogen-nitrogen systems in Refs. [12, 13]). The oxygen atoms interaction PECs and collision integrals are also needed, apart from the modeling of oxygen-containing plasmas, in rarefied gas dynamics (hypersonic flows, shock waves) but the use of the exact potentials is difficult [14]. In particular, in air heated in hypersonic flow, oxygen dissociates more easily than nitrogen and oxygen atoms assist the decomposition of other molecules [15, 16]. It is also worth noting that in the situation of fast gas heating, the production of atomic oxygen species is particularly significant, even at temperatures below 1000K [17, 18]. Similar high production of oxygen atoms is known in plasma-assisted ignition in turbulent flows in which the reactive oxygen atoms enhance combustion and stabilize flame [19].

The excited states PECs of the oxygen molecule are needed [20, 21]. This study reports the PECs dissociating to the ground state atoms but the interaction of excited atoms poses a challenge which only recenltly have been addressed by one study [22].

Transport cross sections and collision integrals will be given in a wide range of temperatures 300–30000 K. The lowest temperature of 300K was used as in the well-known compilation by Wright et. al [23]. The WKB approximation for the scattering phase shifts is employed in this study to compare the collision integrals to the classically calculated counterparts. The other results used for comparison with the present ones use the WKB approach (but different code and details) [24] and the classical approach [22] (also different code).

Section 2 presents the methods used and the ab initio calculations of PECs, in Section 3 the results are discussed (with subsection devoted to comparisons with the recent results by Zhao et al. [22]), and in Section 4 the conclusions of the study are presented. This study uses atomic units, except for collision integrals which are traditionally given in Å$^2$.

Methodology

Ab initio Calculations of Potential Energy Curves

The MOLPRO-2022 package [25] was employed for the quantum chemical calculations. All of the PECs of the oxygen atoms interactions are obtained with the internally contracted explicit correlated multireference configuration interaction method (icMRCI-f12) with Davidson correction using the aug-cc-pVQZ basis set. All calculations were carried out in the $C_{2v}$ symmetry group. Furthermore, we also used the projection of the $L_z$ operator to specific molecular states in order to ensure angular momentum quantum of the desired states. The molecular orbitals were constructed using the state averaged complete active space self-consistent field (CASSCF) method with a full valance active space consisting of 12 electrons on 9 orbitals (5$a_{1}$,2$b_{1}$,2$b_{2}$,0$a_{2}$). The previous similar research of PECs for oxygen were conducted in [26] and [22]. The calculated PECs dissociating to the ground state O($^{3}P$)+O($^{3}P$) atoms are shown in Fig. 1. The $^1{\Pi _g}$ and $^3{\Pi _g}$ PECs (correlating to ground state oxygen atoms) show a barrier around 3.0 bohr which originates from the avoided crossing from the interaction with higher energy $^1{\Pi _g}$ and $^3{\Pi _g}$ PECs correlating to excited states atoms. These two higher energy $^1{\Pi _g}$ and $^3{\Pi _g}$ states are also displayed on Fig. 1.

Fig. 1

PECs that correlates to the ground state oxygen atoms: O($^{3}$P)–O($^{3}$P) interactions

Classical and Quantum Approaches for Cross Sections

The quantum-mechanical cross sections related to transport properties (transport cross sections) are [27] - momentum-transfer cross section

$$\begin{aligned} Q^{(1)}(E)=\frac{4\pi }{k^2}\sum _{l=0}^{\infty }(l+1)\sin ^{2}(\delta _{l}-\delta _{l+1}), \end{aligned}$$

(1)

and viscosity cross section

$$\begin{aligned} Q^{(2)}(E)=\frac{4\pi }{k^2}\sum _{l=0}^{\infty }\frac{(l+1)(l+2)}{2l+3}\sin ^{2}(\delta _{l}-\delta _{l+2}), \end{aligned}$$

(2)

where $\delta _{l}$ is the scattering phase shift related to the l-th partial wave and $k=\sqrt{2\mu E}$ ($\mu$ - reduced mass of the interacting atoms system). The scattering phase shifts can be obtained from the WKB approximation [28] as

$$\begin{aligned} \delta _{l}(E)=k\lim _{r'\rightarrow \infty }\left( \int _{r_{c}}^{r'}\sqrt{1-\frac{V(r)}{E}-\frac{(l+1/2)^2}{k^2r^2}}{\textrm{d}}r-\int _{(l+1/2)/k}^{r'}\sqrt{1-\frac{(l+1/2)^2}{k^2r^2}}\textrm{d}r \right) , \end{aligned}$$

(3)

where V(r) is the potential energy function describing inter-atomic interaction and $r_{c}$ is the classical turning point (distance of the closest approach).

In classical mechanics the cross sections are expressed by integral over the impact parameter b:

$$\begin{aligned} Q^{(n)}(E)=2\pi \int _{0}^{\infty }b\,(1-\textrm{cos}^{n}[\chi (b,E)])\,db, \end{aligned}$$

(4)

with the deflection angle [29]

$$\begin{aligned} \chi (b,E)=\pi -2b\int _{r_{c}}^{\infty }\frac{{\textrm{d}}r}{r^{2}\sqrt{1-\frac{b^2}{r^2}-\frac{V(r)}{E}}}. \end{aligned}$$

(5)

From the transport cross sections (classical or quantum), the reduced collision integrals can be calculated [30]

$$\begin{aligned} \sigma ^2\Omega ^{(l,s)*}=\frac{1}{(k_{B}T)^{s+2}\pi (s+1)!\left[ 1-\frac{1}{2}\frac{1+(-1)^l}{1+l}\right] }\int _{0}^{\infty }e^{-E/(k_{B}T)}E^{s+1}Q^{l}(E)dE. \end{aligned}$$

(6)

The above expressions were implemented in Python programming language with the help of numerical methods of the SciPy library.

Collision Integrals

The momentum-transfer cross section for the ground state oxygen atoms interacting on the ground molecular state $(X^{3}\Sigma ^{-}_g)$ is presented in Fig 2. A good agreement between the WKB and classical values can be observed. This agreement is also confirmed by the collision integrals based on that PEC shown in Table 1, the percent discrepancies are almost always less than $1\%$. In Table 2 the collision integral on the repulsive $^{3}\Pi _{g}$ PEC are shown, and discrepancies between the WKB and classical values are always below $0.1\%$; the excellent agreement of the momentum-transfer cross sections is shown in Fig. 3.

Fig. 2

Momentum-transfer cross section $Q^{(1)}$ for O($^3$P)–O($^3$P) interaction on the ground state PEC $X^{3}\Sigma ^{-}_{g}$ - semi-classical WKB (WKB) and classical mechanics (CM) approaches

Table 1 Collision integrals for O($^3$P)–O($^3$P) interaction on the ground state PEC $X^{3}\Sigma ^{-}_{g}$

Table 2 Collision integrals for O($^3$P)–O($^3$P) interaction on the PEC $^{3}\Pi _{g}$

Fig. 3

Momentum-transfer cross section $Q^{(1)}$ for O($^3$P)–O($^3$P) interaction on the repulsive $^{3}\Pi _{g}$ - semi-classical WKB (WKB) and classical mechanics (CM) approaches

For the colliding ground state oxygen atoms there are 17 PECs, only 6 of them are attractive (and only 3 are strongly attractive) as shown in Fig. 1. The final conclusion for oxygen atoms is that the uncertainty of collision integrals due to the classical description of scattering (even for 300K) is very low (of the order of $0.1\%$).

It is typical also for most other systems that higher statistical weights of repulsive states lowers difference of the WKB and classical collision integrals. It should be put in the further context of the sensitivity of collision integrals to PECs, their inaccuracy may exceed $20\%$ as recent studies of various system have shown [1, 2, 22].

Note also the differences between WKB and classical values given in Table 1—there is a certain variation of discrepancies (the values are not getting closer monotonically with temperature) which can be attributed to the oscillations of WKB cross sections. This confirms the suggestion that values calculated classically may be more accurate than the estimates shown as percent discrepancies by comparison with the WKB method applied here.

The feature of cross sections is instability below $10^{-4}$ hartree, however, it was verified that even if the integration over energy for the collision integral starts from this energy (completely neglecting that low energy part of cross section), the error introduced is only $0.1-0.2\%$ (at 300 K) or smaller (at higher temperatures). It can be concluded that uncertainty introduced by possible problems with both WKB and classical approach even at the lowest temperatures is around $0.1\%$.

Finally, Table 3 gives the collision integrals for the scattering of the ground state oxygen atoms compared with the Stallcop et al. data [24]. The discrepancies between both data sets are rapidly decreasing with temperature and over 1000K are smaller than $4\%$. This confirms the good quality of older data at higher temperatures, especially when compared with other data not calculated directly from ab initio PECs [31]. The verification of usually reasonable quality of Stallcop et al. collision integrals with novel ab initio PECs are known for: the N–H interaction were difference less than $3\%$ was found above 1000K [1] and the N–N interaction where the difference was up to $5\%$ (even at 500 K) [32], higher discrepancies are found at some temperatures for the N–N$^{+}$ system [33].

Table 3 The collision integrals for the O($^3$P)–O($^3$P) interaction compared with the values reported in Stallcop [24]

Comparison of the Known Ab initio Based Collision Integrals

The latest study of the oxygen atoms collision integrals presents results based on the ab initio points fitted by the neural network to the analytical functions [22]. The ab initio points were calculated to the similar highest energies (around 0.6 hartree). That new set of data allows comparison of three independent sets of collision integrals based on ab initio energy points; that comparison is presented in Table 4.

The maximal differences between data are only below $4\%$ in the broad range of 2000–20000 K. Those differences are explained by the high sensitivity of collision integrals to the small variation of PECs and also by different methodologies. At temperatures above 20000K the differences may exceed $4\%$ because of additional extrapolation of the PECs and increased inaccuracy in calculated energy points (in general, the oxygen atoms interactions is a difficult problem for the quantum chemistry software); also use of diabatic PECs in Ref. [22] may influence high temperature collision integrals.

All the values highly differ at temperature of 1000 K (and below) but it is not clear if the sensitivity of collision integrals is enough to explain those differences (the highest sensitivity may be present at the lowest temperatures). Probably the methodology may also play a significant role. Publication [22] also report differences over $10\%$ for nitrogen atoms collision integrals based on ab initio points below 1000 K, this study stress the influence of PECs uncertainties at high interatomic distance.

In summary, the collision integrals at low temperatures have the most significant uncertainty and also the highest temperatures where PECs extrapolation is needed is prone to differences exceeding $1-2\%$ (which are typical discrepancies in the 2000–20000 K region).

Table 4 Comparison of known ab initio based collision integrals for the O($^3$P)–O($^3$P) interaction with maximum percentage difference (max. diff.) between the presented three values

Conclusions

The classical mechanics collision integrals and the momentum-transfer cross sections for the scattering of the ground state oxygen atoms are reported.

The estimation of the error in the collision integrals due to the classical treatment is of the order of $0.1\%$ even at room temperature (that can be compared to $1-3\%$ or better for NH$^{+}$ cases [12]). That uncertainty can be regarded as negligible, especially comparing to the possible uncertainty of PECs. The crucial ingredient of the good performance of the classical approach is its excellent quality for the potentials of repulsive character.

In light of the high sensitivity of collision integrals with respect to PECs as well as the often employed Lennard–Jones or Morse potentials and exponential approximation for the repulsive curves, the use of the classical approach seems to be justified even more.

Taking into account all factors (possible inaccuracy of the PECs, the theoretical and computational methods, and computational complexity) the classical approach to the elastic scattering for the transport properties seems to be the most effective approach. Only for the lightest atoms and temperatures around or below room temperature (depending on the system) the full quantum description is the most appropriate option.