The effects of Heisenberg constraint on the classical cross sections in proton hydrogen collision

The interaction between a proton and a ground state hydrogen atom is studied using a standard three-body classical trajectory Monte Carlo (CTMC) and a quasi-classical trajectory Monte Carlo (QCTMC) model where the quantum feature of the collision system is mimicked using the model potential in the Hamiltonian as was proposed by Kirschbaum and Wilets (1980 Phys. Rev. A 21 834). The influence of the choice of the model potential parameters (α, ξ) on the initial radial and momentum distribution of the electron are analyzed and optimized. We found that although these distributions may not be as close to the quantum results as the distribution of standard CTMC results, we can find the combination of the (α, ξ) where the calculated cross sections are closer to the experimental data and closer to the results obtained quantum mechanically. We show that the choice of 3 < α < 5 is reasonable. To validate our observation, we present cross sections for ionization, excitation, charge exchange (CX), and state selective CX to the projectile bound state. Calculations are carried out in the projectile energy range between 10 and 1000 keV amu−1.


Introduction
The proton-hydrogen collision system is the simplest collision system in ion-atom collisions. This fundamental one-electron system has great significance for testing various theoretical descriptions where, luckily, a large number of experimental results are available for various channels like ionization, charge exchange (CX), excitation and state-selective CX cross sections. * Author to whom any correspondence should be addressed.
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We have also used the H + + H collision system to evaluate the effects of the Heisenberg correction term on cross sections in the a quasi-classical trajectory Monte Carlo (QCTMC) model where the quantum features of the collision system are mimicked using the model potential in the Hamiltonian as was proposed by Kirschbaum and Wilets [1].
We note here that the collision processes between ions and atomic hydrogen also have considerable interest for scientists working on fusion plasma research. Neutral beams of hydrogen atoms can be injected into tokamak plasmas to heat and fuel them [2], and they provide powerful plasma composition diagnostic tools through CX spectroscopy [3]. So the accurate cross sections data are essential.
The proton-hydrogen collision system has been widely studied both theoretically and experimentally for various collision channels. Theoretically, Cohen calculated cross sections for all possible electronic rearrangements in H + + H using a quasiclassical-trajectory Monte Carlo energy-bounded approach (QTMC-EB) [4]. This model is proposed to extend the classical trajectory Monte Carlo (CTMC) model. Avazbaev et al [5] used the semiclassical convergent close-coupling (SC-CCC) approach to study the cross sections for excitation, total and state-selective CX channels for a wide range of the proton impact energies from 1 keV to 1 MeV. The latest research on the subject is the development of the quantummechanical convergent-close-coupling [6,7].
The proton-hydrogen collision was also studied extensively experimentally. The ionization cross sections in H + + H collisions were obtained by Shah and Gilbody and Shah et al in 1981 and1987 [8, 9], where the authors claim that the experimental error on the level of 5% or better. The cross sections for CX between a proton and a hydrogen atom have been measured in the low projectile energies by McClure [10]. The collisional excitation of the proton-hydrogen system has been investigated at intermediate energies applying the optical method by Detleffsen et al [11]. The state-selective CX is another interesting collision channel that has been considered for many years. The experimental data for cross sections of the state-selective CX into 2s and 2p states of the projectile in H + + H(1s) collision published in references [12][13][14][15][16].
The CTMC method was introduced in the early 1960s [17][18][19]. In this method, classical equations of motions are solved numerically, and the initial conditions are chosen randomly [20][21][22]. Abrines and Percival [23] applied the CTMC method to calculate the electronic and nuclear motions in p + H collisions. It was quite surprising that the classical description was able to reproduce numerous experimental data. One of the advantages of the CTMC method is that many-body interactions are precisely taken into account during the collisions on a classical level.
The QCTMC method was proposed by Kirschbaum and Wilets in 1980 [1]. According to this model, for multi-electron atoms, a basic requirement for improving the standard CTMC method is that the electrons cannot collapse to the nucleus, and two identical electrons cannot occupy the same energy level. The effective potential was introduced to mimic the Heisenberg uncertainty principle and the Pauli constraint. The effectiveness of the QCTMC is justified in many studies. Zajfman [24] presented the ionization and CX cross sections for bare ions on helium atoms by applying the QCTMC model and obtained an excellent agreement with the experimental data. The ionization and CX in multiply charged ion-helium collisions have also been studied at intermediate energies by McKenzie [25]. Cohen evaluated various collision systems by considering the Heisenberg and Pauli constraints [26][27][28][29][30]. In recent years, the use of this model has been considered. Jorge et al demonstrated the reliability of this approach for H + H and H + + H − collisions [31]. In addition, the differential cross sections for CX in the collision of protons with He have been evaluated using CTMC and QCTMC methods by Bachi and Otranto [32]. Schematic illustration of the position vectors r e , r P and r T in the laboratory frame in three-body approximation. Reproduced from [22]. CC BY 4.0. The relative positions are defined as: A = r e − r T , B = r T − r p and C = r p − r e , in such way that A + B + C = 0. Also, R is the relative coordinate of the projectile with respect to the electron-target center of mass. r Te is the position vector of the center-of-mass of the target system, and b is the impact parameter.
In this work, the interaction between a proton and a ground state hydrogen atom is studied using a standard three-body CTMC and a QCTMC model where the quantum feature of the collision system is mimicking by the potential proposed by Kirschbaum and Wilets [1]. Although our calculation system is one of the simplest systems, we apply the constrained potential in the description of the hydrogen atom. We expect improvements in the cross section calculations compared with the standard CTMC model because with the model potential we include quantum behavior in the classical system. The paper is organized as follows. In section 2, we describe the calculation models. Section 3 presents our results. In section 3.1 we show the distance and momentum dependent behavior of the model potential. We define the initial conditions of r (radial distance between electron and hydrogen) and p (linear momentum of the electron) in the QCTMC in section 3.2. To find the appropriate constants (α · ξ) used in Heisenberg potential function, firstly, we study the electron radial and momentum distribution in the QCTMC method in section 3.3. Secondly, we obtain the cross sections for all possible channels, such as target ionization in section 3.5, projectile CX in section 3.6, projectile state-selective CX target in section 3.7, and target excitation in section 3.8 in p + H collisions. Finally, we predict the optimal combination of constants (α · ξ) by simultaneously analyzing and comparing the previous theoretical methods and experimental data with our theoretical calculations. The conclusions are summarized in section 4.

CTMC model
In the present work, we performed our calculations based on the standard three-body CTMC model. The electron e of mass m e , the projectile ion P of mass m P and the target nucleus T of mass m T form a three-body approximation. The r e , r P and r T represent the position vectors for the electron, projectile ion and target nucleus, respectively (see figure 1).
The Hamiltonian equation for the three particles can be written as: where are total kinetic energy and Coulomb potential of interaction system. Also, r, p, Z and m are the position, momentum vector, the charge and the mass of the given particles p; projectile, e; electron, T; target, respectively. The equations of motion taking into account the Hamiltonian mechanics is given as follows: The velocities of objects according to Hamiltonian mechanics are obtained as follows: where we define N i = 1 m i . These differential equations were integrated with respect to the time by the 4th order Runge-Kutta method with an adaptive step size, starting from a given configuration of the collision for a given set of initial conditions [33]. The center-of-mass of the target atom is the origin of our coordinate system in the laboratory frame, and the velocity vector of the projectile is parallel to the z-axis (see figure 1).
The total cross sections are calculated by: and the statistical uncertainty of the cross sections is given by: where T N is the total number of trajectories calculated for impact parameters less than b max , T (i) N is the number of trajectories that satisfy the criteria for the corresponding final channels, and b j (i) is the actual impact parameter for the trajectory corresponding electron capture processes.
In the classical calculations, the classical principal quantum number (n c ) and the classical orbital angular momentum (l c ) are given by where μ Te is the reduced mass of the target nucleus and the target electron. x, y, and z are the Cartesian coordinates of the electron relative to the nucleus andẋ,ẏ, andż are the corresponding velocities. The classical values of n c are 'quantized' to a specific n level if they satisfy the relation: Since l c is uniformly distributed for a given n level [34], the quantal statistical weights are reproduced by choosing bin sizes such that l n n c l c l + 1, where l is the quantum-mechanical orbital angular momentum.

QCTMC model
In one electron collision systems, the effective potential is introduced to mimic the Heisenberg uncertainty principle. This approach was proposed by Kirschbaum and Wilets [1]. In this case, the Hamiltonian consists of constraining potential, V H , motivated by the Heisenberg principle is added to the pure Coulomb inter-particle potentials describing the atom. Thus where H 0 is the usual Hamiltonian containing the total kinetic energy of all bodies and Coulomb potential terms between all pairs of electrons and between the nucleus and electrons, respectively. The extra term is where a and b denote the nuclei and the i index the electrons. Also, r αβ = r β − r α and relative momenta are: The Heisenberg correction function is defined as [1]: In our simulations, the correction term is taken into account between the target electron and both of target nucleus and projectile. This calculation schema is called a target-projectile centered calculation. We note that we also did calculations using other calculation schemas to determine the influence of correction terms on the cross sections (see section 3.4). These are the following: (1) projectile-centered, when the correction term is taken into account between target electron and projectile, (2) target-centered, when the correction term is taken into account between target electron and the target nucleus. The correction term according to projectile-centered and target centered are expressed as: (21) It is worth noting that the vector direction r Te is from the target to electron and r pe is from the projectile to electron, respectively. The equations of motion taking into account the Hamiltonian mechanics and the Heisenberg correction term is given by: For the case of the target-centered schema C T = 1, C P = 0, for the case of the projectile-centered schema C T = 0, C P = 1 and for the case of the target-projectile centered schema C T = C P = 1. We note that by using C T = C P = 0 the equation of motions naturally gives back the case of standard three-body equation of motions.
The Heisenberg correlation potential is sensitive to the choice of the two constants α and ξ. ξ H and α H are the dimensionless constant and adjustable hardness parameters, respectively, which must be determined. ξ H is determined by our requiring r i p i = ξh for the ground state of an atom. By considering atomic unitsh = m e = e = 1, Hamiltonian of hydrogen atom is defined in a form of In the ground state or stationary configuration at the lowest energy, ∂H ∂ p = 0 and ∂H ∂r = 0 [1]. This gives In the case of the hydrogen atom, the binding energy of the electron in the ground state is −0.5 a.u. Figure 2 shows the variations of ξ H according to α H for ground state hydrogen atom.

Results and discussion
To study the collision between H + and hydrogen atom, we used the standard three-body CTMC and QCTMC methods. We performed a classical simulation with an ensemble of 1 × 10 6 primary trajectories for each energy. The calculations are carried out in the projectile energy range between 10 and 1000 keV amu −1 when the target hydrogen atom is in the ground state. According to a large number of primary histories, the estimated uncertainties (see equation (11)) of the cross sections are in general around 0.6%.

Heisenberg constraining potential function dependence on r and p
The correction term is a function of the radial (r) and linear momentum (p) of the target electron (see equation (19)). To define the initial conditions for r and p in the QCTMC model, we changed the initial conditions used in the standard threebody CTMC model. In the first step, we showed the dependence of the Heisenberg correction term on the parameters r and p. Figure 3 shows the correction term dependence as a function of r (with some typical constant p) and p (with some typical constant r), respectively. Figure 3, shows the exponential dependence of the Heisenberg correction term function as a function of p, when r is assumed to be constant, (see figure 3(a)) and as a function of r, when p is assumed to be constant (see figure 3(b)). The correction function defined by equation (19) shows a stronger dependence on r than on p. Therefore, the Hamiltonian of the system is more sensitive to changes of r than to the changes of p. Thereby in the definition of the initial conditions of the QCTMC model we select first the p values and later the corresponding r values satisfying that the total energy of the system is the ground state energy of the hydrogen atom, i.e. E = 0.5 a.u.

Definition of initial conditions of QCTMC model
Finding the pairs of r 0 and p 0 for the initial condition of the electron is an essential issue in classical calculations. For example, if the potential between hydrogen and electron is net Coulomb potential (without correction term), the initial r 0 and p 0 values are given as follows: where E b and μ te denote the electron binding energy and reduced mass between target and electron, respectively. In the QCTMC model, the Heisenberg correction term is added to the net Coulomb potential which influences the total energy of the target electron and target nucleus. The target electron must be in the ground state energy level and thereby must fulfill relations defined in the equations (34) and (35): where f H (r· ) is the Heisenberg correction term and 0.5 is the binding energy of the electron in ground state hydrogen. Therefore in this case we need an allowed interval in the combination of the initial conditions (r 0 , p 0 ) that provide the ground state energy. Figure 4 shows the allowed interval for r 0 and p 0 in which satisfy equations (34) and (35) by considering the constants α H = 3.5 and ξ H = 0.9354, respectively. By fixing the p parameter and finding the root of function F(r, p) (see equation (36)), we specified the new initial conditions in QCTMC model for r parameter in the allowed interval Figure 5 shows the dependence of function F(r· ) according to r. We also consider that the obtained values for r and p in this part must be in the allowed interval (see figure 4). According to figure 5, we found that the initial conditions for the r parameter are around p = 1 a.u.

Radial and momentum distribution
In the QCTMC model, Hamiltonian equations are numerically solved using a fourth-order Runge-Kutta integration method with an adaptive step size. For initialization procedures for the standard microcanonical QCTMC model, we obtained the  radial and momentum distribution of the electron. Figure 6 shows these distributions applying various combinations of α H and ξ H in comparison with the corresponding quantummechanics results.
According to figure 6, we can see that the radial and momentum distributions are highly influenced by α H and ξ H . As discussed in section 3.2, we fixed the p and changed the r to find the initial condition in the QCTMC model. Therefore, it is expected that the momentum distribution is not matched to the classical and quantum distributions (see figure 6(b)). Figure 6(a) shows that the standard classical radial distribution is terminated around r = 2. Since the Heisenberg constraint is one of the quantum-mechanics concepts we use in the classical simulation, we expect to see the quasi-classical radial distributions in the special quantum zone (beyond r = 2). We see this condition for α H 3.5. Therefore the suitable and reasonable range of α H for the H + + H(1s) collision system is expected to be α H 3.5. Furthermore, we note that the distributions are started from non-zero values because due to the Heisenberg constraint, the electron is not allowed to collapse to the nucleus. To find the adequate combination of α H and ξ H for proton and ground state hydrogen collision system in the QCTMC model, we calculated the cross sections of ionization, CX, state-selective CX, and excitation channels as a function of incident energies and compared with the available quantummechanics approaches and experimental data.

Effects of Heisenberg correction term on three different calculation schemes
In the QCTMC model, the effect of Heisenberg correlation potential on the particles in the collision is interesting. Therefore we tested three calculations schemes as described in section 2.2 as follows: (a) Projectile-centered, where the correction term is considered between target electron and projectile (b) Target-centered, where the correction term is considered between the target electron the target nucleus (c) Combined one, i.e., target and projectile centered where the correction term is taken into account between target electron and both the target nucleus and projectile   Figure 7 shows our CTMC and QCTMC results corresponding to the three calculation schemes of the ionization and total CX cross sections in H + + H(1s) collision as a function of the impact energy. We compared our results with the QTMC-EB method and experimental data.
According to figure 7(a), the present CTMC and projectilecentered QCTMC results are almost the same. Furthermore, it can be seen that the target-centered QCTMC results and the target-projectile-centered ones are almost the same. It means that due to the large distance between the electron and the projectile relative to the distance from the electron to the target nucleus, the Heisenberg correction term effect between the electron and the projectile is practically negligible. Therefore, we conclude that the correction term between the target electron and the projectile is ineffective in the ionization channel. According to the previous literature, both CTMC and projectile-centered QCTMC results are in good agreement with the QTMC-EB results of Cohen [4], and the experimental data are shown by Shah and Gilbody [8] at intermediate and high impact energies. Also, the target-centered QCTMC cross sections and target-projectile-centered QCTMC ones are in excellent agreement with the experimental data reported by Shah et al [9] at low impact energies.
On the other hand, according to figure 7(b), the effects of correction term on three different calculation schemes are evidence in the CX channel. The correction term provides the repulsive force between the electron and both target and projectile causes to better transmitting the electron to the projectile states in the CX channel. A better agreement is seen between experimental data and the QCTMC model, where the correction term is taken into account between the target electron and both the target nucleus and projectile at intermediate energies. It is worth noting that we used only the combination scheme as a justifiable cause in our calculations.   [4]. Experimental results are due to; black circles: Shah and Gilbody [8], black triangles: Shah et al [9].

Ionization
Trajectories of particles in collisions are an instructive and exciting way to show the behavior of the projectile, target, and electrons. Figure 8 shows trajectories of the ionization channel (see equation (37)) in the y-z lab frame coordinate system without and with considering the Heisenberg correction term, respectively. The trajectories were obtained at 70 keV amu −1 impact energy. The target electron experiences the attractive Coulomb force from both of target and projectile positive charge. On the other hand, the presence of correction term in the QCTMC model causes continuous changes in the sum of the forces acting on the electron. Therefore it can be seen the distortion in electron trajectory (see figure 8(b)). Also, target repulsion becomes more apparent with the addition of the Heisenberg correction term (37) Figure 9 shows the present CTMC and QCTMC results of ionization cross sections in H + + H(1s) as a function of impact energy. We considered α H = 3, 3.5, 4, 4.5, 5 with corresponding ξ H in the QCTMC model. The Comparison was made with the QTMC-EB method used by Cohen [4], and the experimental data have shown come from Shah and Gilbody [8] and Shah et al [9].
It can be seen that the CTMC results are in good agreement with QTMC-EB results and experimental data of Shah and Gilbody [8] at intermediate and high energies. The Heisenberg correction term exerts a repulsive force on the electron. Due to the small distance between the electron and the target nucleus, the repulsive force between the electron and the target nucleus is much larger than the repulsive force between the electron and the projectile. Therefore, the ionization cross sections in the QCTMC model are higher than the CTMC ones in the whole range of impact energy. According to figure 9, the QCTMC (α H = 3, ξ H = 0.9258) and QCTMC (α H = 3.5, ξ H = 0.9354) results match the experimental data of Shah et al [9] at low energies. Also, the QCTMC (α H = 3, ξ H = 0.9258) results are in good agreement with the experimental data of Shah and Gilbody [8] at high energies. However, one can see the close agreement between the QCTMC (α H = 3.5, ξ H = 0.9354) results and the experimental results due to Shah and Gilbody [8] at high energies.

Charge exchange
For the illustration of the CX channel (see equation (38)), figure 10 represents the trajectories in the x-z projectile frame for electron, target, and projectile without and with considering the Heisenberg correction term. The presence of the Heisenberg repulsive force (as described in section 3.5) causes  slight distortion in the electron trajectory before the electron is placed into the states of the projectile Furthermore, we calculated the CTMC and QCTMC results of total CX cross sections in H + + H(1s) collision as a function of impact energy. We considered α H = 3.5, 4, 4.5, 5 with ξ H correspondence in the QCTMC model. We compared our results with the QTMC-EB method used by Cohen [4], and the experimental data shown come from McClure [10] (see figure 11).
By considering the correction term to mimic the Heisenberg constrain, the repulsive force reduces the effects of the attractive Coulomb force between the electron and target nuclei. Therefore, the tendency of the electron to be placed at the states of the projectile increases. According to figure 11, the QCTMC model increases the CX cross sections compared with CTMC and QTMC-EB methods at low and intermediate impact energies. The QCTMC (α H = 5, ξ H = 0.9535) results are very close to the experimental data.

State selective CX
We calculated the CTMC and QCTMC results of CX cross sections into 2s and 2p states of the projectile bound state (see equation (39)). The calculations were obtained in the QCTMC model according to α H = 3, 3.5, 4, 4.5 with ξ H correspondence, respectively (see figures 12 and 13) The comparison was made with the SC-CCC method used by Avazbaev et al [5], and the experimental data are due to Bayfield [12], Morgan et al [13], Hill et al [14], Ryding et al [15], kondow et al [16] and Stebbings et al [35].
According to figure 12, the QCTMC model improves the results at low energies significantly. At the low energies, the QCTMC (α H = 3, ξ H = 0.9258) and QCTMC (α H = 3.5, ξ H = 0.9354) results agree with the experimental data of Morgan et al [13]. Also, between impact energies 100-1000 keV, the QCTMC (α H = 3, ξ H = 0.9258) and QCTMC (α H = 3.5, ξ H = 0.9354) results match the SC-CCC [5] method. Figure 13 shows that the QCTMC (α H = 3, ξ H = 0.9258) and QCTMC (α H = 3.5, ξ H = 0.9354) results are close to the experimental data at low energies. Also, good agreement is seen with the SC-CCC theoretical method [5] at high energies. Figures 12 and 13 show that, the QCTMC cross sections are higher compared to CTMC ones at lower incident energies. This difference gradually decreases for higher energies. To explain this behavior physically, we focus on two factors: (1) force between the electron and the hydrogen nucleus, (2) interaction time. Heisenberg correction term creates a repulsive force in the opposite direction to the Coulomb force in the QCTMC model. Therefore, the attraction force between the electron target and the target nucleus decreases and the  [5]. Experimental results are due to; red circles: Bayfield [12], black squares: Ryding et al [15], black triangles: Morgan et al [13], crosses: Hill et al [14]. electron is more easily captured by the projectile. On the other hand, the passing projectile ion at low energies causes the extension of the interaction time. Therefore, the effect of these factors increases the cross section at low energies in the QCTMC model.

Excitation
To better understand the excitation channel (see equation (40)) without and with considering the Heisenberg correction term, we have shown the electron, projectile, and target trajectories in the x-z lab frame coordinate system in figure 14. The trajectories were obtained at 70 keV amu −1 impact energy. The effect of the correction term (as described in section 3.5) adds a repulsive force into the classical calculations. Competition between attractive Coulomb force and repulsive Heisenberg force distorts the trajectory of the electron in the excitation channel H + + H(1s) → H + + H(n, l) * .
In the following, we show cross sections for H + induced 1s → 2p transition in atomic hydrogen by using CTMC and QCTMC methods for α H = 3, 3.5, 4, 4.5, 5 with ξ H correspondence, respectively (see figure 15). In addition, we compared our results with the experimental data are shown by Morgan et al [13], Detleffsen et al [11], and kondow et al [16]. Also, the comparison is made with the SC-CCC method as a benchmark used by Avazbaev et al [5]. The SC-CCC method based on the impact parameter description is one semiclassical method in which the projectile motion with respect to the target is considered classically by linear trajectories. This classical treatment is valid when the associated de Broglie wavelength, λ proj = 2π mv , is smaller than the radius of the atomic target (a 0 = 1 a.u. for H(1s)). According to the relation, λ proj a 0 1, the SC-CCC method is valid at intermediate and high impact energies for comparison with CTMC and QCTMC models. The main difference between our classical models and the SC-CCC model is based on the treatment of the electronic motion which is described quantum mechanically in SC-CCC and classically in CTMC and QCTMC models.
According to figure 15, we realized that the QCTMC cross sections are higher than CTMC ones at the whole range of impact energies. This behavior can be understood by focusing on the force between the electron and hydrogen nucleus.  [5]. Experimental results are due to; red circles: Kondow [16], black triangles: Morgan et al [13], crosses: Stebbings et al [35].   [5]. Experimental results are due to; red circles: Detleffsen et al [11], black triangles: Morgan et al [13], crosses: Kondov et al [16].
good agreement with experimental data and SC-CCC results of Avazbaev et al [5]. The similarity of the QCTMC results at (α H = 3, ξ H = 0.9258) and at (α H = 3.5, ξ H = 0.9354) with the experimental data and with the semiclassical close-coupling approach are very appropriate.
As seen in figures 9, 11-13 and 15, we conclude that the results by considering the QCTMC method significantly improve the cross sections as a function of the impact energies. We also observed the effects of α H and ξ H on the displacement of cross sections for proton and ground state hydrogen collision systems. Since one combination of α H and ξ H in Heisenberg potential function should be chosen for collision systems, by investigating the electron radial and momentum distribution and analyzing the experimental data for various final channels, we found that the constants α H = 3.5 and ξ H = 0.9354 in the QCTMC model are reasonable in H + + H(1s) collisions.

Conclusions
We have shown an intensive study of the interaction between a proton and a ground state hydrogen atom. Calculations were performed employing a standard three-body CTMC and a QCTMC models where the Heisenberg correction term is added to the Hamiltonian of the collision system to mimic the Heisenberg uncertainty principle. The projectile energy range was between 10 and 1000 keV amu −1 . To increase the accuracy of the calculations, we considered one million trajectories for each impact parameter. Firstly, the initial conditions for distance (r) and linear momentum (p) of the target electron were obtained for the QCTMC model. Secondly, we presented ionization and total CX cross sections in three calculation schemes in H + + H(1s) collision. These were the followings: (1) projectile-centered, when the correction term is taken into account between target electron and projectile, (2) target-centered, when the correction term is taken into account between target electron and target nucleus, (3) combined one, i.e., when the correction term is taken into account between the target electron and both the target nucleus and projectile. Our QCTMC results in different schemes were compared with the results of the three-body CTMC model. We found that the effect of the correction term between the target electron and projectile is not noticeable in the ionization channel. However, the correction term between the particles plays an important role in calculations in the CX channel.
Finally, we obtained the relevant range for two important constants in the Heisenberg constraining function, i.e., α H and ξ H , by analyzing the radial and momentum distributions of the target electron. By comparing the present radial distribution with the quantum-mechanics ones, we found that the reasonable range of α H for ground state hydrogen as a target is expected to be α H 3.5 [36]. To select the most reasonable parameters we performed test calculations to obtain excitation, ionization and CX cross sections for various (α H , ξ H ) combination. We take the advance in the parameter selection that according to figure 4. We have a little freedom in the selection of (α H , ξ H ) for the correction potential. We used this option to obtain the reasonable parameters based on the comparison of calculated cross sections with experimental and theoretical data. We note, however, that the obtained parameters are used for all reaction channels (like excitation, ionization and CX). The combination of (α H , ξ H ) may element dependent. We found that according to the comparison of our calculated cross sections with the previous experimental and theoretical data, for hydrogen target the α H = 3.5, and ξ H = 0.9354 combination provide good numbers for all possible reaction channels. We note, that naturally, these parameters are not universal, for other elements the optimization procedure must be repeated. The parameters we obtained is valid strictly for hydrogen target.
Generally, the CTMC model is a well-known classical treatment for modeling atomic collisions to calculate various cross sections. But due to the lack of quantum features in the standard and original model, the CTMC model is not able to describe accurately the cross sections mostly at lower impact energies when the quantum mechanical character of the collision may dominant. The reason for this is that at lower projectile energies the interaction time is longer in the collision region, therefore the Coulomb interaction between the target electron and projectile should be more significant than at higher energies. Thus, a quantum mechanical description is required to calculate accurate cross sections. In the QCTMC description of the scattering problem, a model potential is introduced to mimic some quantum features of the collision.
Accurate cross section calculations between low charged fully striped ions (such as H + ) with ground state hydrogen is very essential in fusion research. On the other hand, treating atomic collisions with quantum mechanics approaches in many aspects are very complicated or unfeasible. Therefore, the QCTMC model with its simplicity, may have an alternative of the quantum-mechanical models providing the same results with relatively low computation efforts.

Acknowledgments
This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom research and training program (Grant Agreement No. 101052200-EUROfusion). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).