Collision Cross Sections for O + Ar+

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Collision Cross Sections for O + Ar Collisions in The Energy Range 0.03 eV to 500 eV Anastasia A. Sycheva, Gabriel G. Balint-Kurti, and Alexander P. Palov J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.5b09151 • Publication Date (Web): 07 Jan 2016 Downloaded from http://pubs.acs.org on January 13, 2016

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The Journal of Physical Chemistry

Collision Cross Sections for O + Ar+ Collisions in The Energy Range 0.03 eV to 500 eV A.A. Sycheva1, G.G. Balint-Kurti2 and A.P. Palov3* 1

Department of Physics, Moscow State University, Leninskie Gory, Moscow 119991 Russian Federation

2

Centre for Computational Chemistry, School of Chemistry, University of Bristol, Bristol BS8 1TS, United Kingdom

3

Skobeltsyn Institute of Nuclear Physics, Moscow State University, Leninskie Gory, Moscow 119991 Russian Federation

*

Corresponding author: E-mail: [email protected], Phone: +7-495-939-46-14

Abstract: The interatomic potentials of the a²Π and b²Π states of the OAr+ molecule are calculated using the relativistic complete-active space Hartree-Fock method followed by a multi-reference configuration interaction calculation with an aug-cc-pwCVNZ-DK basis sets where N is 4 and 5. The calculations were followed by an extrapolation to the complete basis set limit. An avoided crossing between the two potential energy curves is found at an internuclear separation of 5.75 Bohr (3.04 Ǻ). As the transition probability between the curves is negligible in the relative collision energy range 0.03-500 eV of interest here, collisions on the lower adiabatic a²Π potential may be treated without reference to upper state. For low energies and orbital angular momentum quantum numbers the one dimensional radial Schrödinger equation is solved numerically using a Numerov’s algorithm method to determine the phase shift. The semi-classical JWKB approximation was employed for relative energies greater than 5 eV and orbital angular quantum numbers higher than 500. Differential, integral, transport (diffusion) and viscosity cross sections for elastic collisions of oxygen atoms with argon ions are calculated for the first time for the range of relative collision energies studied. The calculated cross sections are expected to be of utility in the fields of nanotechnology and arc welding. The combination of an Ar+(2P) ion and a O(3P) atom gives rise to a total of 12 different molecular electronic states that are all coupled by spin-orbit interactions. Potential energy curves for all 12 states are computed at the complete active space self-consistent field (CASSCF) level and scattering calculations performed. The results are compared with those obtained using just the lowest potential energy curve. Keywords: arc welding, plasma etching, deposition, ab initio, scattering phase shifts, elastic scattering cross section, differential scattering cross section, integral scattering cross section, transport (diffusion) cross section, viscosity cross section, argon, oxygen 1. Introduction The data describing the ion-atom cross sections for argon ion plus oxygen atom collisions are required in several different plasma fields from gas discharge1 and arc welding2 to nanotechnologies3. The collision partners in a gas discharge are normally the positive oxygen ion and the argon atom and calculations of the elastic collision cross section are based on a consideration of the ground X4Σ- electronic state of the ArO+ molecular ion. Investigation of scattering on this potential energy curve started with experiments in 19774,5 and has since continued with recent ab initio calculations6, that also considered spin-orbit interactions. The contribution of the Ar+ ion to the mobility of O+ through an argon-oxygen mixtures is negligible, less than 0.1%6, and this is probably the reason for the fact that the electronically excited states of OAr+ that correlate with the Ar+ + O product channel have not been properly considered until now. The most recent paper we have found7 was published in 1989 and describes an ab initio calculation of a²Π at the MP2 level of theory. 1

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There are nevertheless two areas of plasma research where a knowledge of the potential energy curves of the excited states of the OAr+ molecule are important, namely, arc welding and nanotechnology. The elastic cross sections of the Ar+ ion scattering with an O atom are required for modeling the motion of Ar+ ions through a neutral gas in gas shielded arc welding2. In this technique inert gases such as helium and argon are used to protect the non-consumable electrode while oxygen is applied to decontaminate the metal8. The modelling of these processes requires a knowledge of Ar+-O viscosity and diffusion transport coefficients. These coefficients are not currently available from any experimental measurements and there are at present large uncertainties in the theoretically computed values. The uncertainties arise from the approximate nature of the underlying inaccuracies in the interaction potentials which have been used to compute the coefficients. Thus the calculation of accurate potential energy curves (PECs) and corresponding cross sections should lead to a considerable improvement in the modelling of the welding process. The interaction of argon ions with oxygen-containing low-k films is intensively studied in the field of modern nanoelectronics where the sputtering process is simulated through the use of molecular dynamics methods. In most cases a repulsive Moliere potential function is used for the two-body interaction term occurring in Newton’s equations of motion9. Use of such an approximate potential can only provide a very approximate description for the scattering process. The accurate Ar+-O interatomic potential we compute in this paper, can be used as an improved approximation for the Ar+ ion -- the bonded oxygen potential in such molecular dynamics calculations. Below we present an ab initio calculation of the potential energy curves for both the a²Π and b²Π electronic states of the OAr+ molecular ion. We then compute the elastic cross section and also the transport (diffusion) and viscosity cross sections10 for the lower of the two potential energy curves. The paper is organized as follows: in section 2 we give details of our ab initio calculations and of the scattering theory used to compute the elastic scattering phase shifts and the integral, transport (diffusion) and viscosity cross sections, in section 3 we discuss the ab initio potentials, analyze the computed phase shifts and the differential, integral, transport (diffusion) and viscosity cross sections. A conclusion is presented in the final section.

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2. Theory 2.1. Ab initio calculations The MOLPRO computer package (version 2012.1)11,12 for the ab initio calculation of molecular wavefunctions and energies was used in the present work to compute the PECs for the two lowest ²Π electronic states (a²Π and b²Π) of the OAr+ molecule. These two states correlate asymptotically with an Ar+(²P) ion and an O(³P) atom, both in their respective ground states13. The OAr+ molecule belongs to the C∞v symmetry point group. As the MOLPRO package includes only abelian point groups the diatomic molecule must be represented using the a1, a2, b1, b2 irreducible representations of the C2v point group. In order to achieve the required accuracy three methods were used sequentially. At each internuclear separation we first performed a restricted Hartree–Fock calculation. This was followed by a complete active space self-consistent field (CASSCF) method and then a multireference configuration interaction (MRCI) method calculation. So as to include the effect of relativistic corrections we applied the fourth-order Douglas–Kroll Hamiltonian approximation14,15 as implemented in the MOLPRO package. The spin-orbit interaction was not included in the calculations. We chose two relativistic basis sets16,17, aug-cc-pwCVQZ-DK and aug-cc-pwCV5Z-DK, for our PEC calculations as the largest available and used the calculated energies in a two-points fit18 to extrapolate the energy to the complete basis set (CBS) limit:

E  X  1  E X X 3 E  X 1  X  13  X 3 3

(1)

where X=Q=4 and X+1=5 signify the identification numbers of the two largest basis sets used. In the CASSCF calculations 10 molecular orbitals were used in the active space while the rest of the electrons were placed in closed shell orbitals. 2.2. Scattering theory In this subsection we outline the theory needed to compute the differential, integral (ICS), diffusion and viscosity cross sections for Ar  ion - O atom collisions. The central quantity required is the scattering phase shift19. In order to compute the scattering phase shift, δl(E), for the l-th orbital angular momentum quantum number and relative collision energy E the radial Schrödinger equation was solved using a numerical scheme of sixth order accuracy, namely, the Numerov's method

20,21,22

. The resulting wavefunction is

analyzed in the asymptotically large internuclear separation region and, by comparison with the corresponding wavefunction for scattering in the absence of an interaction potential, the phase shift is extracted. The phase shifts are computed in this manner for relative collision energies 0.03 eV - 5 eV. For the energy range of 5 eV - 500 eV we used the numerical Numerov method for l less than 500 and the JWKB (Jeffreys-Wentzel-Kramers-Brillouin) 23,24 approximation for the higher l values. 3



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The JWKB phase shifts were calculated with use of the Langer correction25, to the centrifugal part of the effective potential:



JWKB l

E   r

rm ax 2

 l  1 / 22  dr  rm ax 2 E  l  1 / 22  dr   2 μ E  2 μ V r     OAr   r1  OAr OAr r2 r2    1/ 2

1/ 2

(2) where  OAr  is the reduced mass of OAr+ molecule, rmax is the maximum internuclear separation for which the potential is available, r1 and r2 are the classical turning points for the centrifugal and effective potentials, respectively. The potential should have a value of zero at rmax. Having calculated the scattering phase shifts several different types of cross section may be evaluated. The differential cross section in the center-of-mass reference frame for non-identical particles is given by: d CM E ,   22 d 4k

 2l  1e 

2 i l  E 



 1 Pl cos 

2

(3)

l 0

where Pl(cosθ) is the l-th Legendre polynomial, θ is the center-of-mass scattering angle and



k  2OAr E /  2



1/ 2

is the wave vector of the relative motion. The integral cross section (ICS) is obtained by

integrating Eq. (3) over all polar angles:

 int E   2 



0

d CM E , sind  42 d k



 2l  1sin  E  2

(4)

l

l 0

The transport (diffusion) cross section needed for calculating a diffusion coefficient in a gas is given by the formula26:

 d E   2 



0

d CM E , 1  cos sind  42 d k



 l  1sin  E    E  2

l 1

l

(5)

l 0

The viscosity cross section that is important for computing a viscosity coefficient can be written as26:

 v E   2 



0

d CM E ,  1  cos 2  sind  22 d k







l  1l  2 sin 2  E    E  l 2 l

 l  3 / 2

(6)

l 0

3. Results and discussion 3.1 Ar  ion – O atom interaction potentials The OAr  electron states and their associated potential energy curves (PECs) are essential to the present work. The shape of the PEC governs two important aspects of the outcome of the collisions. Firstly, the bound states supported by the potential give rise to the glory maxima arising in the integral collision cross section27. Secondly, analysis of the effective potentials defines the energy range within which shape resonances can appear. The computed PECs must therefore be carefully analyzed. We performed ab initio calculations at 90 different internuclear separations ranging from 0.75 Bohr (0.397 Ǻ) to 33 Bohr (17.5 Ǻ) for each of the a²Π and b²Π electronic states of the OAr+ molecule with two 4


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different atomic orbital basis sets. The minimum distance of 0.75 Bohr was chosen to satisfy the upper energy level of 500 eV needed for the elastic cross section calculations while at a separation of 33 Bohr the Ar+ ion and the O atom can be considered to be completely separated. The calculated energies are given in the Supporting Information provided with this paper. In order to provide a finer and regular mesh of energy points for our subsequent dynamical calculations we performed a cubic spline interpolation procedure to create a grid of 10751 points. The potential values between two neighboring points of the 10751 point grid were evaluated using linear interpolation. Figure 1 shows the calculated PECs. The two curves display an avoided crossing at an internuclear separation of 5.75 Bohr. As the two electronic states have the same symmetry, they are not allowed to cross28. To calculate the dissociation energy, D0, we must calculate the energy of the lowest vibrational level. This is done using the Fourier Grid Hamiltonian method29,30. The value of the dissociation energy computed in this way for the a²Π PEC is D0 = 2.594 eV, while the equilibrium separation is re = 1.666 Å. These quantities have previously been calculated by Frenking et. al.7 who obtained noticeably different values of D0 =2.133eV and re =1.665 Å using the MP4 method with a 6-311 basis set. These results are similar to those we obtain at the Hartree-Fock level with our larger basis set. The potential supports 81 bound vibrational levels. Our computed dissociation energy is greater than the value of D0 = 2.46 eV found from a thermodynamic cycle analysis13. The difference may be due to our neglect of the spin-orbit splitting, especially in Ar+ which has a splitting of 0.1774 eV. Inclusion of the spin-orbit splitting will therefore significantly lower the calculated value of the dissociation energy. Also no correction has been applied for the basis set superposition error. A correction for this will also lead to a smaller value for the dissociation energy. For the upper b²Π state we calculate values of D0 =0.145 eV and re = 2.961 Å. There are no previous computed values for these quantities. Both the potential energy curves should dissociate to the same Ar+(2Po) + O(3P) atomic asymptotes. In our calculations the two curves differ by 0.0001 a.u. (0.0027 eV) at the largest internuclear separation for which we have performed calculations. This is at the limit of the numerical accuracy of the ab initio calculations (see Supporting Information).

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Figure 1. Ab initio interaction potentials as functions of the interatomic separation of argon ion and oxygen atom calculated using a relativistically corrected MRCI method followed by extrapolation to the Complete Basis set (CBS) limit. The solid curve indicates the exited a²Π state of the OAr+ molecule; the dotted one stands for the exited b²Π potential. The insert shows a close-up of the avoided crossing region between the two PECs.

Figures 2a and 2b present the effective potentials for the a²Π and b²Π electronic states versus the interatomic separation of the Ar+ ion and O atom for four orbital quantum numbers l = 0, 200, 300,500 and l = 0, 100, 150, 200, respectively. The figures demonstrate that there can be no quasi-bond states at relative collision energies higher than ≈ 0.05 a.u. (1.4 eV) for a²Π and ≈ 0.012 a.u. (0.32 eV) for b²Π as the effective potentials do not possess any wells above these energies.

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Figure 2. Effective potentials for the a²Π and b²Π electronic states of ArO+ as a function of interatomic separation of Ar+ and O (A) Effective potentials for the a²Π state for angular momentum, quantum numbers l = 0(solid line), 200(short dashes), 300(long dashes) and 500(dot-dash). (B) Effective potentials for the a²Π state for angular momentum, quantum numbers l = 0(solid line), 100(short dashes), 150(long dashes) and 200(dot-dash).

Figure 3 shows the PECs for the a2Π and b2Π electronic states in a region where they approach each other very closely. They are not allowed to cross as the states are of the same symmetry. We denote the internuclear separation at the point where the curves approach most closely to each other by R=Rc . As this separation is traversed there is a rapid change of the adiabatic eigenfunctions. At this point the transition probability between these two molecular states can be estimated using the Landau-Zener (LZ) approximation31:







P  2 exp(  * /  ) 1  exp   * /  ,

(7)

with

  *

2 H12

2

'  H11'  H 22

(8) R  Rc

where  is the Plank constant divided by 2π,   2( E  V1 ) / OAr 

1/ 2

is the radial component of the

relative velocity of the colliding particles, E is the kinetic energy of their relative motion, V1 is a potential of the a2Π state at the R=Rc relative to its asymptotic point and H'11 and H'22 are derivatives of diagonal elements of the potential matrix. The energy separation of the two adiabatic PECs at Rc is ΔE=2H12. Our analysis of the area of avoided crossing led to the following values of the parameters mentioned: Rc= 5.75a0, H12 = 0.0075Eh, |H'11 - H'22| = 0.0017 Eh/a0 and V1 = -0.00685Eh. These parameters yield a  * value 7



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Figure 3. The avoided crossing at R=Rc between the a2Π and b²Π PECs is shown. The energy separation ΔE of two adiabatic curves at Rc is equal to 2H12.

of 454.6 km/s, which is very high. Because of this the probability the transition calculated using Eq. (7) is negligible for all energies below 50 eV when it starts to grow from 3×10-5 to 0.014 at E=500 eV. For this reason it is permissible to calculate the elastic cross sections for Ar+ collisions with neutral oxygen on the two 2Π state PECs separately ignoring coupling between the two PECs. In this work we consider scattering only on the a²Π PEC. We have investigated the electronic nature of the a²Π and b²Π states on either side of the crossing point by examining their natural molecular orbitals and their occupation numbers. Our two key observations are that: a) The crossing does not arise from a charge transfer process as the dipole moments associated with both states are similar and are both of the same sign, with the positive charge being mainly located on the Ar ion, and b) The molecular orbital description of the upper state in the internal region and the lower state at internuclear separations greater than the crossing point involves a singly occupied anti-bonding molecular orbital (8σ*) formed from the Ar(3pz) and the O(2pz) atomic orbitals. This antibonding molecular orbital is unoccupied in the other electronic state. Thus the excitation process from the a²Π to the b²Π state in the inner region can be represented as a molecular orbital excitation of the form 7σ→8σ*. The combination of an Ar+(²P) ion and a O(³P) atom gives rise to 12 different molecular electronic states and 54 independent Ω levels. All these levels are coupled by the spin-orbit coupling. In order to 8


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estimate the validity of using just the a²Π PEC for our scattering calculations, we have also calculated PECs at the less accurate CASSCF level for all 12 electronic states that dissociate asymptotically to Ar +(²P) and O(³P). The dissociation energy of the CASSCF a²Π PEC is 1.67eV. Further work will be required to consider the effect of the spin-orbit coupling in the entrance channel32.

3.2 Differential cross sections

The differential cross section (DCS) for the MRCI potential of the a2Π state have been calculated for 3600 angular and 2588 energy points. The grid of energy points is attached in one of our files of Supporting Information. Figure 4a shows the differential cross section as a function of the center-of-mass scattering angle calculated using the a²Π PEC for a relative collision energy of 0.03 eV. The cross section is plotted on a logarithmic scale to allow for the presence of both a large forward (at θ = 0o) and a large backward (at θ = 180o) glory maximum. The huge number and magnitude of the oscillations arises from the shape of the potential and the bound states that it supports. If only the repulsive part of the potential is used, only a small number of oscillations occur and the backward glory maximum is entirely absent24.

Figure 4. The elastic center-of-mass differential cross sections for an argon positive ion collision with an oxygen atom on the a²Π state PEC for relative collision energies of (a) 0.03 eV, (B) 5 eV and (C) 100 eV

Figures 4b and 4c show the differential cross sections, in atomic units squared per steradian, computed using the same potential but for relative collision energies of 5 eV and 100 eV, respectively. In Fig. 4b we see the main rainbow maximum at around 50o. It is preceded by many so called supernumerary rainbows at smaller angles. There are also many quantum oscillations underlying the rainbow maxima. Figure 4c shows a decreased cross section and predominantly forward scattering (i.e. little deflection). There is however still a lot of quantum interference structure in the DCS. 9



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Figure 5. The elastic differential scattering cross sections for an argon positive ion collision with an oxygen atom on the a²Π state PEC in a Laboratory coordinate system with the O atom taken to be initially at rest for relative collision energies of (a) 0.03 eV, (B) 5 eV and (C) 100 eV

When modelling the sputtering process involving argon positive ions impinging on oxygencontaining low-k films, the collision takes place in a laboratory or space-fixed coordinate system in which the oxygen atom is initially at rest33. In fig. 5 we plot the DCS’s in this laboratory reference frame for the same three energies used in fig. 4. Note that for any given collision energy there are now two different collision cross sections. The oxygen atom, being the lighter of the two collision partners, is scattered in all possible directions, while the Ar+ ion can only be scattered into a limited range of angles. If θ1 and θ2 are the laboratory scattering angles of the Ar+ and O atom respectively, the laboratory and center-of-mass differential cross sections are related by the equations:



d CM d 1 1  2 x cos  x 2 E ,1   E ,  d1 d 1  x cos



3/ 2

d 2 E , 2   d CM E ,  4 sin 1  d 2 d 2

(9)

where tan 1  sin  / x  cos   and  2      / 2 , θ1 and θ2 are the scattering angles of the Ar+ ion and the O atom relatively to direction of the impact and x=mAr/mO. The maximum scattering angle of the Ar+ ion is given by α=arcsin(1/x) while the maximum scattering angle of the O atom is π/2, i.e. the Ar+ ion is scattered mainly in the direction of its movement and the O atom flies away at all angles ranging from the initial direction of motion of the Ar+ ion to an angle perpendicular to this direction. Note, the DCS of the Ar+ ion is a sum of two branches originating from the small and large scattering angles in the center-of-mass reference frame24.

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To summarize the calculated differential cross section results we present a three dimensional plot of the DCS as a function of relative collision energy and scattering angle in the center-of-mass reference frame over the energy range 0.03-20 eV in Figure 6. It shows a nice global view of the DCS, especially of the ridge which maps out the locus of the main rainbow maximum as a function of scattering angle and energy.

Figure 6. Elastic differential cross section for collisions between an argon atom positive ion and an oxygen atom on the a²Π state PEC plotted as a function of the relative collision energy and center-of-mass scattering angle over ranges of 0.03-20 eV and 20o – 180o, respectively. The results above have been obtained using only the a2Π PEC computed at the MRCI level. When an Ar+(2P) ion collides with a O(3P) atom the system can access any of 12 different electronic states (2Σ+, a2Π, b2Π, 2Σ-(2), 2Δ, 4Σ+, 4Π(2), 4Σ-(2) and 4Δ) with their associated PECs. In order to estimate the effect of this additional complexity we have computed the PECs for all 12 electronic states at the lower CASSCF level and have computed DCS for each of them. The blue curve in figure 7 shows the DCS in the center-of-mass reference frame at a collision energy of 1eV obtained for the a2Π PEC on its own, while the red curve shows the DCS averaged over all 12 states using their statistical weights (0.037037,2x0.074074, 2x0.037037, 11



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2x0.074074, 2x0.1481481,2x0.074074 and 0.1481481). The form and magnitude of the DCS for the a2Π PEC on its own and the averaged DCS are very similar. The difference lies in the large oscillations present in the former curve, that are almost totally absent in the averaged curve.

Figure 7. Differential cross section for an argon positive ion collision with an oxygen atom in the center-of mass reference frame calculated at a relative collision energy of 1 eV based on the CASSCF potentials. DCS calculated using only the a2Π PEC (blue curve). DCS averaged over PECs for all 12 electronic states (red curve).

3.3 Integral and momentum transfer cross sections Figure 8 displays the ICS, diffusion and viscosity cross sections for argon ion – oxygen atom collisions as a function of the relative collision energy calculated using the a²Π ab initio potential. The energy range of 0.03 – 500 eV was chosen to cover possible applications in arc welding and plasma processing.

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Figure 8. Integral, transport (diffusion) and viscosity cross sections as functions of relative collision energy for Ar+-O system obtained from calculations based on the a²Π ab initio PEC.

We see from the figure that the ICS increases with decreasing collision energy from just below 160 Å2 at 500 eV to around 1800 Å2 at 0.03 eV.

Figure 9 shows a small portion of the ICS curve versus collision energy on an expanded scale. The main features of the ICS, namely resonance features arising from shape resonances and oscillations arising from glory maxima are clearly visible24 in the two figures. Bernstein27 in his pioneering papers regarding this phenomenon found the number of glory oscillations in the plot of the ICS is equal to the number of bound states supported by the potential. It has been mentioned above that the total number of bound states supported by our a²Π ab initio potential is 81. We should therefore observe 81 glory maxima in the plot of the ICS.

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Figure 9. Details of the ICS for Ar+-O atom collisions computed using the a²Π state PEC. The ICS is shown over the collision energy range 1.04-1.14 eV. The plot shows several shape resonances and one glory oscillations. The ICS is plotted over an extended energy range in Fig. 8.

It is interesting to note that the calculated viscosity cross section is greater than the transport (diffusion) one at relative collision energies higher than the well depth of the potential. At the low energies arising in arc welding applications the scattering is more isotropic and the diffusion cross section dominates over the viscosity one.

In order to examine the possible effect of altering the dissociation energy (D0) from our computed value of 2.594 eV to the experimentally determined value of 2.46 eV, we have performed calculations of the integral cross section using a potential which has been scaled so that its value of D0 corresponds to the experimental one. Calculations were performed for three energy ranges, namely: 0.1-1 eV, 10-50 eV and 200-300 eV.

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Table 1. The comparison of the ICS calculated using our ab initio potential and one that was scaled to yield a Do value equal to the experimental value of 2.46 eV.

Energy, a.u. 0.003675 0.018375 0.036751 0.367512 0.735024 5.512679 7.350238

Energy, eV 0.1 0.5 1.0 10.0 20.0 150.0 200.0

ICS, ao2 ab initio potential 5701.1 4379.6 3684.9 2586.8 2222.7 1127.8 965.2

ICS, ao2 scaled potential 5651.4 4188.1 3612.4 2517.5 2171.6 1061.2 934.1

Variation, % -0.9 -4.3 -2.0 -2.7 -2.3 -5.9 -3.2

The ICS’s computed using the scaled potential are compared with those obtained using our ab initio potential in table 1 for some selected scattering energies. There is good agreement between the values of the two ICS’s, the largest deviation being less than 6%.

In order to check the validity of the mixed, quantum mechanical -- semi-classical, approach (MA) we have used in our calculations for energies exceeding 5 eV and orbital angular quantum numbers higher than 500 we have carried out some comparison calculations where the exact quantum mechanical calculations (QM) were used throughout. Table 2 shows the results of these calculations.

Table 2. The comparison of the ICS made with the MA and QM calculation approaches. Energy, a.u. 1.837560 3.675119 7.350238

Energy, eV 50.0 100.0 200.0

ICS (MA), ao2 1640.1 1321.7 965.2

ICS (QM), ao2 1594.4 1261.1 955.4

Variation, % 2.85 4.81 1.03

The comparisons show that the mixed quantum mechanical -- semi-classical ICS’s differ by less than 5% from those evaluated using exact quantum mechanical calculations

In order to examine the possible effect of including of all 12 molecular states on the ICS we used the DCS calculated using the CASSCF PECs at four different collision energies to compute ICSs. In table 3 we present the ICSs computed using only the a2Π PEC and compare it with the ICSs averaged, using the appropriate statistical weights, over that obtained from all 12 PECs. The cross sections at 0.1 eV and 1 eV were obtained using Numerov's algorithm while those at 10 eV and 100 eV were calculated using the JWKB method.

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Table 3. Comparison of the ICS calculated for the CASSCF potentials of Ar+-O molecule at four energies. The average cross sections are obtained by averaging over all 12 states using the appropriate statistical weights.

Energy, eV

 a  , a.u.2

 average , a.u.2

0.1 1.0 10.0 100.0

9221 2501 324 193

12006 4724 295 138

2

F=  a 2  /  average 0.77 0.53 1.10 1.40

It is interesting to note that the difference between the ICS calculated using only the a2Π state and the averaged cross section do not exceed a factor 2. The ICS obtained on the CASSCF PEC clearly differs from that obtained using the MRCI PEC and coupling between states is not considered, but the comparison permits an approximate estimation to be made as to the likely error we make in using only the PEC of the a2Π electronic state. In the Supporting Information we supply several supplementary data files. These tabulate the ab initio potentials, integral, diffusion and viscosity cross sections. These should prove useful for further research in the field of gas shielded arc welding and plasma processing. All phase shifts and cross sections were calculated with use of our own Numerov computer code on the supercomputer “Lomonosov”34 at Moscow State University, Russia. The use of the supercomputer for the scattering calculation was necessitated by the need to obtain results on a very dense energy grid so as to resolve all shape resonances and glory maxima in the very large energy range covered in the work.

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4. Conclusion In this paper we have reported high level ab initio calculations of the interaction potentials for the two lowest lying electronic states, a²Π and b²Π, for the collision of Ar+ with and O atom. These potentials correlate at large internuclear separations with the respective ground state of the collision partners, Ar+(²P) and O(³P). The MRCI method used in the electronic structure calculations accounted for relativistic effects through the inclusion of the fourth-order Douglas–Kroll Hamiltonian approximation. The calculations were performed using the two largest orbital basis sets available and were then extrapolated to the complete basis set limit. These computed PECs display an avoided crossing at an internuclear separation of 5.75 Bohr. Quantum mechanical calculations of the differential, integral, transport (diffusion) and viscosity cross sections for Ar+ ion - O atom collisions were then performed using the computed a²Π potential for a very wide range of relative collision energies (0.03 – 500 eV) for the first time. In order to estimate the error which might arise through using only the single PEC of the a2Π electronic state, we have also computed the PECs of all 12 electronic states that correlate asymptotically with the ground states of the collision partners at the lower CASSCF level of accuracy. These PECs have been used to compute cross sections averaged over all the possible electronic states. Comparisons between the cross sections computed using just a single PEC and the averaged cross sections indicate that the ICSs should be uncertain by no more than a factor of 2. The cross sections obtained in this paper are expected to be applied in the fields of nanotechnology and gas shielded arc welding: i.e. in the modelling of interaction between argon plasma and porous SiOCH low-k films and in the calculation of the motion ions through a neutral gas during the welding process. Supporting Information: This consists of two files: 1) a file “OAr+V(Q,5)Z.xls” containing the results of our ab initio calculations of the potential energy curves. 2) a file “Integral Cross Section.xls” containing the integral, transport (diffusion) and viscosity cross sections.

ACKNOWLEDGMENTS A.A. Sycheva and A.P. Palov thank the Russian Science Foundation (RSF) for financial support (Grant №14-12-01012).

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References

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(34)Sadovnichy, V.; Tikhonravov, A.; Voevodin, Vl.; Opanasenko, V. “Lomonosov”: Supercomputing at Moscow State University. In Contemporary High Performance Computing: From Petascale to Exacsale; CRC Press, Boca Raton, USA, 2013, pp. 283-307.

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Collision Cross Sections for O + Ar+

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