Testing Standard Basis Sets for Direct Ionizations: H⁺ + H at E_Lab = 0.1–100 keV

Abstract

With the simplest-level electron nuclear dynamics (SLEND) method, we test standard Slater-type-orbital/contracted-Gaussian-functions (STO/CGFs) basis sets for the simulation of direct ionizations (DIs), charge transfers (CTs), and target excitations (TEs) in H⁺ + H at E_Lab = 0.1–100 keV. SLEND is a time-dependent, variational, on-the-fly, and nonadiabatic method that treats nuclei and electrons with classical dynamics and a Thouless single-determinantal state, respectively. While previous tests for CTs and TEs exist, this is the first SLEND/STO/CGFs test for challenging DIs. Spin-orbitals with negative/positive energies are treated as bound/unbound states for bound-to-bound (CT and TE) and bound-to-unbound (DI) transitions. SLEND/STO/CGFs simulations correctly reproduce all the features of DIs, CTs and TEs over all the considered impact parameters and energies. SLEND/STO/CGFs simulations correctly predict CT integrals cross sections (ICSs) over all the considered energies and predict satisfactory DI and TE ICSs within some energy ranges. Strategies to improve SLEND/STO/CGFs for DI predictions are discussed.

Graphical Abstract

The simplest-level electron nuclear dynamics (SLEND) method in conjunction with standard quantum chemistry basis sets simulates direct ionizations, charge transfers, and target excitations in the H⁺ + H scattering system.

1. Introduction

There is a renovated interest in simulating high-energy ion-molecule reactions due to their relevance for atmospheric chemistry, astrochemistry, nuclear reactors technology, and ion cancer therapy^1,2. At high energies, ion-molecule collisions give rise to a wealth of chemical reactions, wherein both nuclei (atomic centers) and electrons undergo diverse modifications. In the case of nuclei, the colliding ion projectiles cause on the target molecules various types of fragmentation, rearrangement, and substitution reactions^3,4. In the case of electrons, the colliding ion projectiles cause three types of processes: Charge transfers (CTs), direct ionizations (DIs), and target excitations (TEs), wherein electrons initially located in the discrete bound states of the target migrate to the discrete bound states of the ion (CTs), to the continuous unbound states of the ion or target (DIs), and to other discrete bound states of the target (TEs), respectively. We symbolize these three electronic processes in the H⁺ + H scattering system with the following equations:

$${\text{H}}^{+}+{\text{ H}}^{•}\left(1s^1\right)\to \{\begin{array}{cc}{\text{H}}^{•}\left(1s^1,2s^1,\mathrm{\dots }\right)+{\text{ H}}^{+};& \left(\text{CT}\right)\\ {\text{H}}^{+}+{\text{ H}}^{+}{\text{ + e}}^{-};& \left(\text{DI}\right)\\ {\text{ H}}^{+}+{\text{ H}}^{•}\left(2s^1,2p_0^1,\mathrm{\dots }\right);& \left(\text{TE}\right)\end{array}$$ (1)

In the field of molecular physics, simulations of ion-molecule reactions mostly concentrated on electronic processes (CTs, DIs, and TEs), with little consideration, if not neglect, of nuclear processes (fragmentations, rearrangements, substitutions, etc.)^5–15. To treat electronic processes, numerous and diverse methods have been developed, such as the numerical solution of the time-dependent Schrödinger equation on a lattice⁵, the numerical solution of the time-dependent two-center Dirac equation with Dirac-Sturm basis functions⁶, various versions of the quantum close-coupling (CC) method [e.g., two-center atomic orbital CC (TCAO-CC)⁷, molecular orbital (MO) CC (MO-CC)⁸, semiclassical-atomic orbital CC (SC-AO-CC)⁹, quantum mechanical convergent (QMC) CC (QMC-CC)¹⁰, and one-center convergent (OCC) CC (OCC-CC)¹¹], the two-center basis generator method (TC-BGM)¹², the continuum distorted wave (CDW) scattering method¹³, the first-order distorted wave theory (FODWT)¹⁵, and classical and quasi-classical trajectory Monte Carlo methods (CTMC and QTMC, respectively)^9,14. These methods provided valuable insight into the investigated processes and predicted accurate properties. However, their way of representing the continuous and unbound states of electrons involved in DIs remains rather alien to quantum chemistry praxis (e.g., representations via numerical lattices⁵ and distorted scattering waves¹³).

In the field of chemical dynamics, ion-molecule reactions have been investigated with time-dependent direct-dynamics (TDDD) methods^3,4,16. In broad lines, TDDD methods treat nuclei and electrons with classical and quantum mechanics, respectively, and calculate the electronic wavefunction and potential energy terms on the fly, i.e., as the simulation proceeds. This approach circumvents the burdensome predetermination of full potential energy surfaces to run dynamics. TDDD wavefunctions are calculated with various electronic structure methods, e.g., Hartree-Fock (HF), Kohn-Sham density functional theory (KSDFT), many-body perturbation theory (MBPT), and multi-configuration self-consistent-field (MCSCF)^3,4,16; these wavefunctions are ultimately constructed with standard quantum chemistry basis sets^17–22. Born-Oppenheimer TDDD methods have accurately simulated ground-state nuclear processes, e.g., fragmentation and substitution reactions, most notably the S_N2 reactions: X⁻ + CH₃Y → Y⁻ + CH₃X³. Furthermore, non-adiabatic TDDD methods have accurately described those processes as well as CTs^4,16. In that regard, we have systematically applied the electron nuclear dynamics (END) method^4,16 —a variational, non-adiabatic TDDD method employed in this investigation— to simulate ion-molecule reactions of interest in astrochemistry and ion cancer therapy^4,23–26. For instance, we have simulated collision-induced fragmentations and CTs in the water radiolysis and DNA damage processes that underlie ion cancer therapy^4,23–26.

While successful with the above processes, TDDD practitioners have refrained to extend their simulations to DIs. This avoidance is grounded on the real properties and perceived limitations of the atomic basis sets utilized in TDDD²⁷ . The most employed quantum chemistry basis sets contains Slater-type orbitals (STOs) $\left\{{\varphi }_{\mu }^{\text{STO}}\left(\mathbf{\text{r}}-{\mathbf{\text{R}}}_A\right)\right\}$ centered on the nuclear positions ${\mathbf{\text{R}}}_A$ and expressed as contracted Gaussian functions (CGFs): ${\varphi }_{\mu }^{\text{STO/GF}}\left(\mathbf{\text{r}}-{\mathbf{\text{R}}}_A\right)={\sum }_{p=1}^L{d}_{p\mu }{\varphi }_p^{\text{GF}}\left(\mathbf{\text{r}}-{\mathbf{\text{R}}}_A;{\alpha }_{p\mu }\right)$, where ${\alpha }_{p\mu }$ and $d_{p\mu }$ are the orbital exponents and contraction coefficients of the primitive Gaussian functions $\left\{{\varphi }_p^{\text{GF}}\left(\mathbf{\text{r}}-{\mathbf{\text{R}}}_A;{\alpha }_p\right)\right\}$, respectively²⁷. STOs²⁷ are approximate bound hydrogenic orbitals^27,28 and have a discrete spectrum of negative energy eigenvalues. STOs are fitted to primitive Gaussians to facilitate the calculation of their corresponding atomic integrals²⁷. Thus, there are currently several fast and freely available packages to evaluate STO/CGFs integrals²⁹. By construction, STO/CGFs are bound, normalizable, and with a discrete energy spectrum, and any function expressed in their terms will inherit these properties. In HF and post-HF theories, the molecular spin-orbitals ${\psi }_i\left(\mathbf{\text{r}}\right)$ in the wavefunctions are expressed as linear combinations of STO/CGFs: ${\psi }_i\left(\mathbf{\text{x}}\right)={\sum }_{\mu =1}^K{c}_{\mu }{\varphi }_{\mu }^{\text{STO/GF}}\left(\mathbf{\text{r}}-{\mathbf{\text{R}}}_A\right)\sigma \left(s\right)$, where the atomic coefficients $c_{\mu }$ are determined by the self-consistent-field (SCF) HF procedure²⁷. For $N_e$ electrons described with a basis set of size $K>N_e$, the SCF HF procedure yields $N_e$ occupied $\left\{{\psi }_h\left(\mathbf{\text{x}}\right)\right\}$ and $K-N_e$ virtual $\left\{{\psi }_h\left(\mathbf{\text{x}}\right)\right\}$ spin-orbitals with discrete orbital energies $\left\{{\epsilon }_i\right\}=\left\{{\epsilon }_h,{\epsilon }_p\right\}$, respectively. In practice, occupied spin-orbitals always have negative orbital energies ${\epsilon }_h<0$. On the other hand, virtual spin-orbitals in most cases have positive orbital energies ${\epsilon }_p>0$, but, in some circumstances, some virtual spin-orbitals can have negative orbital energies ${\epsilon }_p<0$; the latter can happen with large basis sets as we will demonstrate it for the H⁺ + H system in Sect. 4. Spin-orbitals with negative energies ${\epsilon }_i<0$ (either occupied or virtual) are bound and normalizable, and exhibit a discrete spectrum of energy eigenvalues; due to these properties, spin-orbitals with ${\epsilon }_i<0$ are considered appropriate to represent one-electron bound states, which also have those properties. On the other hand, spin-orbitals with positive energy orbitals ${\epsilon }_i>0$ (always virtual) are also bound and normalizable, and exhibit a discrete spectrum of energy eigenvalues; due to those properties, spin-orbitals with ${\epsilon }_i>0$ are considered inappropriate to represent one-electron unbound states, which, in sheer contrast, are unbound and non-normalizable, and exhibit a continuum spectrum of energy eigenvalues. For those reasons, TDDD methods based on STO/CGFs have been used to simulate electronic processes from and to spin-orbitals with ${\epsilon }_i<0$, i.e., CTs and TEs^4,16, but not to processes from spin-orbitals with ${\epsilon }_i<0$ to others with ${\epsilon }_i>0$ , i.e., DIs.

A way to surmount this situation may involve the incorporation of unbound basis functions into STO/CGFs sets³⁰. Plane waves (PWs), which are the exact unbound eigenfunctions of a free electron, would be the most obvious candidates for such an addition. Thus, in a mixed STO/CGFs + PWs basis set^31,32, bound STO/CGFs will predominantly represent bound spin-orbitals and unbound PWs will predominantly represent unbound ones. However, the slow convergence of PWs and the scarcity of mixed STO/CGFs + PWs integral algorithms hinder this approach. Other related approaches mixing STO/CGFs with other unbound continuum representations (e.g., with a numerical lattice) will face similar complications (however, for other solutions, cf. Sect. 5 of this article).

In view of the above difficulties, it would be pragmatical to put aside the reservations against employing STO/CGFs basis sets for DIs and proceed to examine their accuracy (or lack thereof) to describe those processes. If successful, a STO/CGFs approach for DIs will present several advantages, e.g., an already proven accuracy for other nuclear and CT processes, an immediate adaptability to the many quantum chemistry methods based on STO/CGFs, and the availability of fast and free packages to evaluate STO/CGFs integrals²⁹. Then, herein, we will test the performance of various STO/CGFs basis sets for the simulation of DI and related electronic processes in H⁺ + H with the END method^4,16. We select the H⁺ + H scattering system due to its simplicity, its relevance for astrochemical processes, and the availability of corresponding CT, DI and TE data from experiments and alternative theories to assess our results. We employ the END method^4,16 because of its proven accuracy to simulate many ion-molecule reactions as discussed previously. Through these efforts, END will also be able to describe several types of DI processes, such as those involved in the electron-induced DNA damage of cancerous cells during ion cancer therapy. However, the results of this investigation will not only be useful for END but also for the numerous quantum chemistry methods based on STO/CGFs basis sets.

On extending STO/CGFs basis sets to DIs, we should avoid spoiling their acquired accuracy for the prediction of molecular properties (geometries, energies, etc.) and of nuclear and CT processes. Then, in this first attempt, we will follow a conservative approach³³ that will attain DI descriptions with standard STO/CGFs basis sets not by altering their critical parameters but by playing with their construction rules; we will consider re-optimizations of the basis sets’ parameters in a second stage, cf. our discussion in Sect. 5 of this article. Out of an astonishing number of atomic basis sets^17–22, we will concentrate on Pople^17,18 and Dunning correlation-consistent (cc) basis sets^19,20 due to their relative simplicity, their widespread use, and their proven accuracy in the prediction of molecular properties and chemical reactions. Virtual spin-orbitals constructed with STO/CGFs and having energies ${\epsilon }_p>0$ will always be bound and discrete. However, we can force those spin-orbitals to resemble more closely the unbound and continuum states for DIs by decreasing their degree of localization (so to approach non-boundness²⁸), and by increasing their number (so to approach a continuum of states²⁸). We will achieve delocalization by increasing the number of diffuse basis functions in the sets according to the even-tempered scheme^19,20; those functions have low exponent coefficients ${\alpha }_{p\mu }$ and were originally designed for highly charged anions. We will increase the number of virtual spin-orbitals with ${\epsilon }_p>0$ by increasing the total number of basis functions (i.e., single-zeta core, multiple-zeta valence, polarization, and diffuse functions) following the construction rules of the basis sets.

This article is organized as follows. In Sect. 2, we will describe the END theory as our chemical dynamics method to simulate CT, DI, and TE processes. In Sect. 3, we will provide the computational details of our END simulations of CT, DI, and TE processes in H⁺ + H at collision energies $E_{Lab}$ = 0.1–100 keV. In Sect. 4, we will present and discuss the results of those END simulations. Finally, in Sect. 5, we will summarize the conclusions of this study and discuss some strategies for the further development of this DI approach.

2. Theory, The END Method

The END method has been discussed in detail in various review articles^4,16,34; therefore, herein, we will present a succinct outline of it. END is a time-dependent, variational, on-the-flight, and non-adiabatic method to simulate chemical reactions^4,16,34. In the simplest-level END (SLEND) version utilized herein, a total trial wavefunction ${|\Psi \left(t\right)\rangle }_{\mathit{\text{SLEND}}}=|\mathbf{\text{R}}\left(t\right),\mathbf{\text{P}}\left(t\right)\rangle |\mathbf{\text{z}}\left(t\right),\mathbf{\text{R}}\left(t\right),\mathbf{\text{P}}\left(t\right)\rangle$ contains nuclear $|\mathbf{\text{R}}\left(t\right),\mathbf{\text{P}}\left(t\right)\rangle$ and electronic $|\mathbf{\text{z}}\left(t\right),\mathbf{\text{R}}\left(t\right),\mathbf{\text{P}}\left(t\right)\rangle$ parts^4,16. The nuclear part $|\mathbf{\text{R}}\left(t\right),\mathbf{\text{P}}\left(t\right)\rangle$ is a product of $N_N$ frozen Gaussian wave packets representing $N_N$ nuclei:

$$\langle \mathbf{\text{X}}|\mathbf{\text{R}}\left(t\right),\mathbf{\text{P}}\left(t\right)\rangle =\prod _{A=1}^{3N_N}\text{exp}\left\{-{\left[\frac{X_A-R_A\left(t\right)}{2\Delta R_A}\right]}^2+i{P}_A\left(t\right)\left[X_A-R_A\left(t\right)\right]\right\};$$ (2)

where ${\mathbf{\text{R}}}_A\left(t\right)$, ${\mathbf{\text{P}}}_A\left(t\right)$, and $\Delta R_A$ are the wave packets’ positions, momenta, and widths, respectively.

These wave packets are ultimately employed in the zero-width limit, a procedure that renders a classical dynamics for the nuclei in terms of $\mathbf{\text{R}}\left(t\right)$ and $\mathbf{\text{P}}\left(t\right)$. The electronic part $|\mathbf{\text{z}}\left(t\right),\mathbf{\text{R}}\left(t\right),\mathbf{\text{P}}\left(t\right)\rangle$ is a Thouless single-determinantal wavefunction³⁵ in terms of $N_e$ dynamical spin-orbitals $\left\{{\chi }_h\left(\mathbf{\text{x}}\right)\right\}$:

$$\langle \mathbf{\text{x}}|\mathbf{\text{z}}\left(t\right);\mathbf{\text{R}}\left(t\right),\mathbf{\text{P}}\left(t\right)\rangle =\det \left\{{\chi }_h\left[{\mathbf{\text{x}}}_h;\mathbf{\text{z}}\left(t\right),\mathbf{\text{R}}\left(t\right),\mathbf{\text{P}}\left(t\right)\right]\right\},1\le h\le N_e.$$ (3)

Each $\left\{{\chi }_h\left(\mathbf{\text{x}}\right)\right\}$ is a linear combination of one occupied ${\psi }_h\left(\mathbf{\text{x}}\right)$ and the $K-N_e$ unoccupied molecular spin-orbitals $\left\{{\psi }_p\left(\mathbf{\text{x}}\right)\right\}$ with complex-valued coefficients $\mathbf{\text{z}}\left(t\right)$³⁵:

$${\chi }_h\left[\mathbf{\text{x}};\mathbf{\text{R}}\left(t\right),\mathbf{\text{P}}\left(t\right)\right]={\psi }_h\left[\mathbf{\text{x}};\mathbf{\text{R}}\left(t\right),\mathbf{\text{P}}\left(t\right)\right]+\sum _{p=N_e+1}^K{z}_{ph}\left(t\right){\psi }_p\left[\mathbf{\text{x}};\mathbf{\text{R}}\left(t\right),\mathbf{\text{P}}\left(t\right)\right].$$ (4)

The $\left\{{\psi }_h\left(\mathbf{\text{x}}\right),{\psi }_p\left(\mathbf{\text{x}}\right)\right\}$ are expressed in terms of travelling STO/CGFs $\left\{{\varphi }_{\mu }^{\text{STO/GF}}\left(\mathbf{\text{r}}-{\mathbf{\text{R}}}_A\right)\right\}$ centered on the nuclei, and are calculated at the HF or DFT level at initial time³⁶. While the $\left\{{\psi }_h\left(\mathbf{\text{x}}\right),{\psi }_p\left(\mathbf{\text{x}}\right)\right\}$ are orthonormal, the $\left\{{\chi }_h\left(\mathbf{\text{x}}\right)\right\}$ are not as one can discern from Eq. (4). The non-standard Thouless single-determinantal wavefunction confers various advantages to SLEND, e.g., avoidance of singularities during evolution^4,16, inducement of time-dependent symmetry breaking^37,38, and analytical formulae for electron-transfer probabilities³⁹.

The SLEND equations of motion are obtained by subjecting ${|\Psi \left(t\right)\rangle }_{\mathit{\text{SLEND}}}$ to the time-dependent variational principle⁴⁰, which enforces the stationarity of the quantum action $A\left[\Psi \left(t\right)\right]$; that produces a set of Euler-Lagrange equations for the variational variables $\mathbf{\text{R}}\left(t\right)$, $\mathbf{\text{P}}\left(t\right)$, $\mathbf{\text{z}}\left(t\right)$, and ${\mathbf{\text{z}}}^{*}\left(t\right)$^4,16:

$$\left[\begin{array}{cccc}i\mathbf{\text{C}}& \mathbf{0}& i{\mathbf{\text{C}}}_{\mathbf{\text{R}}}& i{\mathbf{\text{C}}}_{\mathbf{\text{P}}}\\ \mathbf{0}& -i{\mathbf{\text{C}}}^{*}& -i{\mathbf{\text{C}}}_{\mathbf{\text{R}}}^{*}& -i{\mathbf{\text{C}}}_{\mathbf{\text{P}}}^{*}\\ i{\mathbf{\text{C}}}_{\mathbf{\text{R}}}^{†}& -i{\mathbf{\text{C}}}_{\mathbf{\text{R}}}^T& {\mathbf{\text{C}}}_{\mathbf{\text{RR}}}& -\mathbf{\text{I}}+{\mathbf{\text{C}}}_{\mathbf{\text{RP}}}\\ i{\mathbf{\text{C}}}_{\mathbf{\text{P}}}^{†}& -i{\mathbf{\text{C}}}_{\mathbf{\text{P}}}^T& \mathbf{\text{I}}+{\mathbf{\text{C}}}_{\mathbf{\text{PR}}}& {\mathbf{\text{C}}}_{\mathbf{\text{RP}}}\end{array}\right]\left[\begin{array}{c}\frac{d\mathbf{\text{z}}}{dt}\\ \frac{d{\mathbf{\text{z}}}^{\mathbf{*}}}{dt}\\ \frac{d\mathbf{\text{R}}}{dt}\\ \frac{d\mathbf{\text{P}}}{dt}\end{array}\right]=\left[\begin{array}{c}\frac{\partial E_{\mathit{\text{Total}}}}{\partial {\mathbf{\text{z}}}^{*}}\\ \frac{\partial E_{\mathit{\text{Total}}}}{\partial \mathbf{\text{z}}}\\ \frac{\partial E_{\mathit{\text{Total}}}}{\partial \mathbf{\text{R}}}\\ \frac{\partial E_{\mathit{\text{Total}}}}{\partial \mathbf{\text{P}}}\end{array}\right];$$ (5)

where $E_{\mathit{\text{Total}}}$ is the total energy, and the metric matrices $\mathbf{\text{C}}$, ${\mathbf{\text{C}}}_{\mathbf{\text{R}}}$, ${\mathbf{\text{C}}}_{\mathbf{\text{RR}}}$, etc., represent various electron-electron and nucleus-electron coupling terms (${\mathbf{\text{C}}}_{\mathbf{\text{R}}}$ and ${\mathbf{\text{C}}}_{\mathbf{\text{RR}}}$ are equivalent to the standard non-adiabatic coupling terms)^4,16. We solve the SLEND Eqs. (5) with our END code PACE⁴.

3. Computational Details

In Fig. 1, we depict the initial conditions of the current H⁺ + H simulations. The H atom target is prepared in its 1s ground state at the HF level and placed at rest on the center of the laboratory coordinates system. The H⁺ ion projectile is placed at position $\left(-30\text{ a.u.},\text{ 0, }b\right)$, where $b\ge 0$ is the impact parameter, and with momenta $\left(\text{+}{p}_x\text{, 0},0\right)$ corresponding to collision energies $E_{Lab}$ = 0.1, 1.0, 10.0, 20.0, 40.0, 60.0, 80.0, and 100 keV. For a given $E_{Lab}$ , b varies from 0.0 to 10.0 a.u. in increments $\Delta b=$ 0.1 a.u. A selected travelling STO/CGFs basis set is assigned to both target and projectile. From these initial conditions, each simulation runs for a total time ranging from 500 a.u. (for $E_{Lab}$ = 100 keV) to 5,000 a.u. (for $E_{Lab}$ = 0.1 keV); these runtimes ensure a final projectile-target separation equal to or bigger than the initial one. In this scheme, the simulations start with a SLEND electronic part $|{\chi }_1\left(t=0\right)\rangle =|{\psi }_{1s}\left(t=0\right)\rangle$, where the H-atom state ${\psi }_{1s}$ is centered on the target, and end up with an electronic part $|{\chi }_1\left(t\to +\infty \right)\rangle$ that is a superposition of H-atom states on both projectile and target, cf. Eqs. (3) and (4). The state-resolved probabilities $P_{i=1s\to j}^{\text{CT or TE}}$ of CT or TE processes from the initial target state $i$ = 1s to the final ones $j$ = 1s, 2s, 2p₀, etc., are:

$$P_{i=1s\to j}^{\text{CT or TE}}={\left|\langle {\Psi }_j|{\Psi }_{\mathit{\text{SLEND}}}\left(t\to +\infty \right)\rangle \right|}^2;$$ (6)

where ${\Psi }_j$ is a H-atom state with orbital energy ${\epsilon }_j<0$ centered on the outgoing projectile for CT or on the final target for TE. The total CT, TE, and DI probabilities, $P_{\mathit{\text{Total}}}^{\text{CT}}$, $P_{\mathit{\text{Total}}}^{\text{TE}}$ and $P_{\mathit{\text{Total}}}^{\text{DI}}$, are:

$$P_{\mathit{\text{Total}}}^{\text{CT}}={\sum }_j{P}_{i=1s\to j}^{\text{CT }};P_{\mathit{\text{Total}}}^{\text{TE}}={\sum }_k{P}_{i=1s\to k}^{\text{TE }};P_{\mathit{\text{Total}}}^{\text{DI}}=1-P_{\mathit{\text{Total}}}^{\text{CT}}-P_{\mathit{\text{Total}}}^{\text{TE}};$$ (7)

where the sums run over all the H-atom states with ${\epsilon }_j<0$ on the outgoing projectile (CT) or the final target (TE) generated with a given basis set. $P_{\mathit{\text{Total}}}^{\text{DI}}$ is equal to the total transition probability = 1 minus the total CT and TE ones and contains contributions from H-atom states with ${\epsilon }_j>0$ on both the outgoing projectile and the final target. Finally, the total integral cross sections (ICSs) ${\sigma }_{\mathit{\text{Total}}}^X$ for $X=$ CT, TE or DI are:

$${\sigma }_{\mathit{\text{Total}}}^X=2\pi \underset{0}{\overset{+\infty }{\int }}{P}_{\mathit{\text{Total}}}^X\left(b\right)bdb,X=\text{CT, DI, TE,}$$ (8)

with analogous expressions for state-resolved ICSs.

Figure 1.

Initial conditions of the H⁺ + H simulations. The ground-state H atom target is prepared at rest on the center of the laboratory coordinates system. The H⁺ ion projectile initially travels on the x-z plane, parallel to the x-axis, and toward the target with initial momentum $\overrightarrow{P}$ and impact parameter $b$.

4. Results and Discussion

The prediction of reaction properties depends on the description of the H-atom states with the STO/CGFs basis sets. In this study, we test the People^17,18 basis sets: 6–31G**, 6–311G**, 6–31++G**, and 6–311++G**; and the Dunning^19,20 basis sets: cc-pVDZ, aug-cc-pVDZ, d-aug-cc-pVDZ, q-aug-cc-pVDZ, cc-pVTZ, aug-cc-pVTZ, d-aug-cc-pVTZ and cc-pVQZ. Fig. 2 shows the orbital energy levels ${\epsilon }_i$ of the 1-electron Hamiltonian of an H atom calculated at the SCF HF level with the selected basis sets. Orbital energy levels are depicted with dashed lines: green lines for orbital energies ${\epsilon }_i<0$ and red lines for orbital energies ${\epsilon }_i>0$; the 1, 3, 5, or 7 dashes on a given line correspond to 1-, 3-, 5- or 7-fold degeneracies of the s, p, d, or f orbital symmetry, respectively. In the present approach, spin-orbitals with ${\epsilon }_i<0$ and ${\epsilon }_i>0$ aim to represent bound and unbound H-atom states, respectively. The orbital energies ${\epsilon }_i<0$ agree very well with the exact energy eigenvalues of the bound H-atom states, $E_n=-1/2n^2$, $n=1,2,\mathrm{3\dots }$²⁸. As expected, the number of calculated spin-orbitals and energy levels grows with the size of the basis set. As the latter increases, the negative energy levels tend to successively appear in their increasing energy order: 1s, 2s, 2p’s, 3s, etc., packing themselves closely below the Fermi energy level = 0.0 a.u.; in contrast, the positive energy levels successively appear without such an orderly pattern. In general, basis set augmentations by increasing multiple-zeta valence descriptions (6–31 to 6–311 in Pople sets; the series: DZ, TZ and QZ in Dunning ones) and/or diffuse functions (no ++ to ++ in Pople sets; the series: no aug, aug, d-aug, and q-aug in Dunning ones) increases the number of both negative and positive energy levels, but the last ones tend to increase in a higher proportion. The series: cc-pVTZ, aug-cc-pVTZ, d-aug-cc-pVTZ clearly shows that effect. The biggest considered basis set, cc-pVQZ, shows the highest positive/negative energy levels ratio. Nearly infinite basis sets will generate numerous negative and positive energy levels, with the latter forming a near continuous spectrum of values. Those basis sets may become computationally costly so we will investigate them in a sequel to this study.

Figure 2.

Energy level diagrams of the H atom with the selected Pople and Dunning basis sets. Energy levels with orbital energies < 0 and > 0 are depicted with green and red lines, respectively. The 1, 3, 5 or 7 dashes on a line represent the 1-, 3-, 5-, or 7-fold degeneracy of the s, p, d, or f orbital symmetry.

Fig. 3 displays 3-D plots of the SLEND weighted probabilities $b{P}_{Total}^{\text{CT}}$, $b{P}_{Total}^{DI}$, and $b{P}_{1s\to 2s}^{\text{TE}}$ of the total CT, total DI, and 1s→2s TE processes, respectively, vs. the impact parameter $b$ and the collision energy$E_{Lab}$, calculated with the aug-cc-pVTZ basis set. The plotted surfaces are also projected on the $b-E_{Lab}$ and $b-b{P}_{Total}^{DI}$ planes for a better visualization. The analysis of these weighted probabilities is relevant because they quantify the likelihood of the investigated processes as a function of the collision variables, and determine the values of the ICSs via Eq. (8). At low $E_{Lab}$‘s, the $b{P}_{Total}^{\text{CT}}$ curve shows a series of conspicuous peaks as a function of $b$ that corresponds to an oscillatory behavior of the CT probabilities with respect to$b$; this phenomenon has been experimentally observed in H⁺ + H ^41–43 and is correctly predicted herein (cf. Ref. ⁴³ for an earlier END prediction of this effect). In general, $b{P}_{Total}^{\text{CT}}$decreases as $E_{Lab}$ increases at constant$b$, with a considerable decline in value far from the peaked region.$b{P}_{Total}^{\text{CT}}$ reaches its highest values at the lowest $E_{Lab}$‘s and intermediate $b$‘s (about$2\le b\le 8\text{ a.u}.$), with its global maximum appearing at $E_{Lab}$ = 0.1 keV and $b$ = 5.7 a.u. Unlike$b{P}_{Total}^{\text{CT}}$, the $b{P}_{Total}^{DI}$ curve shows no prominent features and reaches its highest values at intermediate $E_{Lab}$‘s and low $b$‘s; this finding is expected because the transfer of a 1s electron to unbound states requires more energy than its transfer to bound ones. The $b{P}_{1s\to 2s}^{\text{TE}}$ curve also shows some peaks at low $E_{Lab}$ ‘s and low $b$‘s, although they are less prominent than the CT peaks. $b{P}_{1s\to 2s}^{\text{TE}}$reaches its highest values at low (peaked region) and intermediate (non-peaked region) $E_{Lab}$‘s in conjunction with low $b$‘s. The features and trends observed in the weighted probabilities calculated with the aug-cc-pVTZ basis set appear analogously in the weighted probabilities with the remaining basis sets.

Figure 3.

SLEND/aug-cc-pVTZ weighted probabilities $b{P}_Y^X\left(b,E_{Lab}\right)$ of H⁺ + H vs. impact parameter $b$ and collision energy $E_{Lab}$ for $X$ = total CT (frame a), total DI (frame b), and 1s→2s TE (frame c) processes.

Table 1 presents the values of the SLEND ICSs for the total CT, total DI, and 1s→2s TE processes of H⁺ + H calculated with the selected STO/CGFs basis sets in the energy range $E_{Lab}$ = 0.1–100 keV. To assess the accuracy of these results, Table 2 presents the mean absolute percentage errors (MAPEs) of all the SLEND ICSs with respect to their experimental counterparts by McClure et al.⁴⁴ (CT), Shah et al.^45–47 (DI), and Higgins et al.⁴⁸ (TE). In all our comparisons, the ICSs MAPEs are:

$$\text{MAPE}\left(IC{S}^X\right)=\frac{1}{N}\sum _{n=1}^N\left|\frac{{\left(IC{S}_{\mathit{\text{Expt}}.}^X\right)}_n-{\left(IC{S}_{\mathit{\text{Theor}}.}^X\right)}_n}{{\left(IC{S}_{\mathit{\text{Expt}}.}^X\right)}_n}\right|;$$ (9)

where $IC{S}_{\mathit{\text{Expt}}.}^X$ and $IC{S}_{\mathit{\text{Theor}}.}^X$ are the experimental and theoretical ICSs for the processes $X$ = total CT, total DI, and 1→2s TE, and $N$ is the number of considered ICSs values (sample size). In addition, Table 3 presents the ICSs MAPEs of the same three processes from SLEND and from alternative theoretical methods: TC-BGM by Leung et al.¹², OCC-CC by Abdurakhmanov et al.¹¹, FODWT by Das et al.¹⁵, and CTMC and QCTMC by Ziaeian et al.¹⁴, all with respect to the aforesaid experimental data^44–48. In Table 3, we report the lowest SLEND ICSs MAPEs per process and energy range; these MAPEs correspond to the most accurate SLEND calculations and are identified by the utilized basis set. All the MAPEs are calculated over three ranges of collision energy: full energy ~1–100 keV, low energy ~1–40 keV, and high energy ~41–100 keV ranges, cf. Table 3 for the precise values of the ranges. We consider low and high energy ranges (in addition to the full one) because some sets of experimental data lie entirely in either of them. Finally, Figs. 4, 5, and 6 plot the best SLEND ICSs values for the total CT (SLEND/aug-cc-pVTZ), total DI (SLEND/aug-cc-pVDZ and /q-aug-cc-pVDZ) and 1s→2s TE (SLEND/6–311++G**) processes, respectively, vs. the collision energy $E_{Lab}$ and along with their counterparts from the theoretical methods considered in Table 3^11,12,14,15 and from numerous experiments^44–48.

Table 1.

Total CT, total DI, and 1s→2s TE integral cross sections of H⁺ + H from SLEND calculations with various basis sets and at various collision energies in the range $E_{Lab}$ = 0.1–100 keV.

SLEND Integral Cross Sections: Total CT / Total DI / 1s → 2s TE (cm−18)
Basis Set	$E_{Lab}$= 0.1 keV	1 keV	10 keV	20 keV
6–31G**	2368.14/0.5/-	1596.95/5.73/-	763.31/82.28/-	505.68/163.84/-
6–311G**	2547.07/1.36/-	1641.83/16.62/-	723.19/152.86/-	426.27/223.15/-
6–31++G**	2546.51/0.53/0.01	1653.49/4.64/1.05	782.66/60.12/22.44	520.18/125.69/44.67
6–311++G**	2554.84/1.32/0.03	1646.69/12.18/1.44	756.12/102.48/20.6	437.96/192.95/21.58
cc-pVDZ	2515.25/1.41/-	1637.16/17.61/-	713.91/162.95/-	421.79/216.41/-
aug-cc-pVDZ	2649.64/0.45/0.21	1724.42/3.85/0.09	825.78/33.73/32.06	507.42/79.89/24.44
d-aug-cc-pVDZ	2649.8/4.9/0.15	1693.44/31.1/0.33	733.31/139.34/26.92	427.6/184.72/16.26
q-aug-cc-pVDZ	2650.45/4.23/0.15	1697.79/26.39/0.34	739.66/119.09/26.5	432.59/170.06/15.66
cc-pVTZ	2556.48/5.49/-	1617.67/53.37/-	638.8/249.27/-	333.07/262.92/-
aug-cc-pVTZ	2671.24/1.33/0.04	1713.94/10.19/0.2	746.14/110.31/14.49	354.71/180.15/11.28
d-aug-cc-pVTZ	2663.92/1.05/0.03	1698.31/31.03/0.27	696.61/176.61/6.17	346.01/209.35/14.25
cc-pVQZ	2579.49/12.72/0.25	1629.16/72.07/1.83	634.99/226.41/31.37	337.66/230.15/16.86
Basis Set	$E_{Lab}$= 40 keV	60 keV	80 keV	100 keV
6–31G**	201.85/185.29/-	100.49/166.7/-	61.98/142.23/-	74.65/90.41/-
6–311G**	155.1/211.94/-	75.0/177.03/-	52.77/140.74/-	52.83/105.89/-
6–31++G**	231.36/156.04/20.84	122.69/140.94/15.82	80.78/117.25/13.44	63.13/95.23/11.7
6–311++G**	178.31/184.63/13.34	98.0/150.48/12.1	68.01/121.19/10.87	61.05/92.93/9.64
cc-pVDZ	143.21/216.76/-	80.26/169.51/-	63.35/129.5/-	14.54/143.63/-
aug-cc-pVDZ	201.92/100.07/11.19	102.49/77.82/9.17	67.94/54.73/8.33	46.42/45.02/7.6
d-aug-cc-pVDZ	154.88/188.47/11.72	99.53/136.32/10.64	52.29/130.71/9.21	36.22/114.96/8.02
q-aug-cc-pVDZ	174.28/160.61/11.48	89.91/135.7/10.41	52.4/120.26/8.99	36.29/105.12/7.84
cc-pVTZ	107.66/229.4/-	46.23/202.86/-	20.49/187.93/-	23.57/160.8/-
aug-cc-pVTZ	101.22/180.73/21.4	57.67/131.29/18.25	26.23/112.78/14.63	13.04/96.91/11.89
d-aug-cc-pVTZ	109.62/208.43/20.98	56.94/178.16/17.18	28.12/155.26/13.47	18.04/133.81/11.21
cc-pVQZ	97.29/253.95/17.02	42.59/237.26/18.43	34.65/209.92/17.35	18.28/197.17/15.71

Table 2.

Mean absolute percentage errors (MAPEs) of total CT, total DI, and 1s→2s TE integral cross sections of H⁺ + H from SLEND calculations with various basis sets and with respect to their experimental counterparts by McClure et al.⁴⁴ (CT), Shah et al.^45–47 (DI), and Higgins et al.⁴⁸ (TE). MAPEs are calculated over three ranges of collision energy: full energy ~2–100 keV, low energy ~2–40 keV, and high energy ~41–100 keV ranges. The lowest and highest MAPEs for each process and in each energy range are marked with subscripts “a” and “b”, respectively.

Total CT ICS MAPEs
	Full Energy Range (0–100 keV)	Low Energy Range (0–40 keV)	High Energy Range (41–100 keV)
6–31G**	89.50	11.52	140.19
6–311G**	55.63	9.56	85.58
6–31++G**	108.12b	16.46	167.70b
6–311++G**	81.11	7.97a	128.65
cc-pVDZ	41.03	11.43	60.26
aug-cc-pVDZ	77.00	12.97	118.63
d-aug-cc-pVDZ	52.70	9.76	80.61
q-aug-cc-pVDZ	50.44	8.55	77.67
cc-pVTZ	25.54	24.99	25.89
aug-cc-pVTZ	14.66a	22.83	9.35a
d-aug-cc-pVTZ	18.26	23.29	14.99
cc-pVQZ	27.61	25.88b	28.74
Total DI ICS MAPEs
	Full Energy Range (0–100 keV)	Low Energy Range (0–40 keV)	High Energy Range (41–100 keV)
6–31G**	125.77	293.64	16.52
6–311G**	257.71	621.72	20.58
6–31++G**	82.83	199.87	6.82
6–311++G**	175.22	427.22	11.27
cc-pVDZ	270.99	654.56	21.38
aug-cc-pVDZ	71.62a	107.60a	48.60
d-aug-cc-pVDZ	284.85	712.83	6.31
q-aug-cc-pVDZ	239.19	599.32	5.04a
cc-pVTZ	530.55	1271.23	48.91
aug-cc-pVTZ	171.36	418.24	10.72
d-aug-cc-pVTZ	341.28	822.97	27.87
cc-pVQZ	593.13b	1395.94b	71.30b
1s→ 2s TE ICS MAPEs
	Full Energy Range (10–100 keV)	Low Energy Range (10–40 keV)	High Energy Range (41–100 keV)
6–31G**	-	-	-
6–311G**	-	-	-
6–31++G**	94.99b	213.13b	33.95
6–311++G**	31.63a	83.92	4.61a
cc-pVDZ	-	-	-
aug-cc-pVDZ	54.76	121.85	20.10
d-aug-cc-pVDZ	33.03	74.14	11.79
q-aug-cc-pVDZ	33.17	70.73	13.76
cc-pVTZ	-	-	-
aug-cc-pVTZ	44.61	42.91	45.49
d-aug-cc-pVTZ	36.34	35.05a	37.01
cc-pVQZ	70.92	98.05	56.91b

Table 3.

Mean absolute percentage errors (MAPEs) of total CT, total DI, and 1s→2s TE integral cross sections (ICSs) of H⁺ + H from various theoretical methods: TC-BGM by Leung et al.¹², OCC-CC by Abdurakhmanov et al.¹¹, FODWT by Das et al.¹⁵, CTMC and QCTMC by Ziaeian et al.¹⁴, and SLEND, with respect to their experimental counterparts by McClure et al.⁴⁴ (CT), Shah et al.^45–47 (DI), and Higgins et al.⁴⁸ (TE). MAPEs are calculated over three ranges of collision energy: full energy ~2–100 keV, low energy ~2–40 keV, and high energy ~41–100 keV ranges; the specific energy ranges in keV are listed in parentheses by the MAPEs. For SLEND, the lowest MAPEs for each process and energy range are listed with the corresponding basis sets.

Total CT ICS MAPEs
TC-BGM, Leung et al.12	6.36 (2–100)	3.15 (2–40)	8.45 (41–100)
OCC-CC, Abdurakhmanov et al.11	5.63 (11–100)	4.27 (11–40)	6.30 (41–100)
FODWT, Das et al.15	33.98 (45–100)	-	33.98 (45–100)
CTMC, Ziaeian et al.14	15.82 (10–100)	27.22 (10–40)	9.93 (41–100)
QCTMC, Ziaeian et al.14	10.1 (10–100)	16.12 (10–40)	6.99 (41–100)
SLEND	aug-cc-pVTZ 14.66 (2–100)	6–311++G** 7.97 (2–40)	aug-cc-pVTZ 9.35 (41–100)
Total DI ICS MAPEs
TC-BGM, Leung et al.12	18.4 (10–100)	7.01 (10–40)	24.28 (41–100)
OCC-CC, Abdurakhmanov et al.11	19.8 (10–100)	14.57 (10–40)	22.56 (41–100)
SLEND	aug-cc-pVDZ 27.40 (10–100)	aug-cc-pVDZ 28.56 (10–100)	q-aug-cc-pVDZ 5.04 (41–100)
1s→ 2s TE ICS MAPEs
TC-BGM, Leung et al.12	28.1 (10–100)	34.24 (10–40)	24.94 (41–100)
OCC-CC, Abdurakhmanov et al.11	23.5 (10–100)	30.31 (10–40)	20.2 (41–100)
SLEND	6–311++G** 31.63	d-aug-cc-pVTZ 35.05 (10–40)	6–311++G** 4.61 (41–100)

Figure 4.

SLEND/aug-cc-pVTZ total CT integral cross sections of H⁺ + H vs. collision energy in comparison with their available theoretical (TC-BGM by Leung et al.¹², OCC-CC by Adbdurakhmanov et al.¹¹, FODWT by Das et al.¹⁵, and CTMC and QCTMC by Ziaeian et al.¹⁴) and experimental (McClure et al.⁴⁴, Fite et al.⁵¹ and Gilbody et al.⁵²) counterparts. Experimental error bars are plotted when available.

Figure 5.

SLEND/aug-cc-pVDZ and /q-aug-cc-pVDZ total DI integral cross sections of H⁺ + H vs. collision energy in comparison with their available theoretical (TC-BGM by Leung et al.¹² and OCC-CC by Abdurakhmanov et al.¹¹) and experimental (Shah et al.^45–47) counterparts. Experimental error bars are plotted as well.

Figure 6.

SLEND/6–311+G** 1s→2s TE integral cross sections of H⁺ + H vs. collision energy in comparison with their available theoretical (TC-BGM by Leung et al.¹² and OCC-CC by Abdurakhmanov et al.¹¹) and experimental (Higgins et al.⁴⁸ and Morgan et al.⁵³) counterparts. Experimental error bars are plotted as well.

Before proceeding to analyze the obtained SLEND ICSs, we should point out once again that the tested STO/CGFs basis sets were originally designed for the time-independent prediction of bound-state molecular properties, and not for the time-dependent simulation of electronic processes; this is particularly the case for the demanding DI processes involving unbound states. Then, one should bear in mind these limiting facts when appraising the obtained results in their qualitative and quantitative aspects.

Inspection of the data in Table 1 and in Figs. 4–6 demonstrates that SLEND in conjunction with STO/CGFs basis sets correctly predicts the qualitative behaviors of the considered ICSs with respect to the collision energy $E_{Lab}$. These behaviors are a monotonically decreasing pattern in the total CT ICSs vs. $E_{Lab}$, and an increasing pattern with a constant asymptotic trend in the total DI and 1s→2s TE ICSs vs. $E_{Lab}$. These results are unsurprising in the CT and TE cases given the documented success of SLEND to simulate these processes with STO/CGFs basis sets^4,16. However, these results are remarkable and significant in the new DI case in view of the perceived limitations of the STO/CGFs basis sets to describe DIs.

Inspection of the statistics in Tables 2–3 and of the plots in Figs. 4–6 permits judging the quantitative accuracy of the calculated SLEND/STO/CGFs ICSs. In Table 2, the SLEND ICSs MAPEs with respect to the experimental data reveal that the SLEND/STO/CGFs ICSs values appreciably depend on the employed basis sets, especially in the DI case. This result is not completely unexpected because, in standard quantum chemistry, predictions of molecular properties can also appreciably depend on the employed basis sets²⁷. It would be instructive to compare the basis set dependence of the SLEND/STO/CGFs ICSs with the dependences of the other theoretical ICSs with respect to their built-in parameters and special functions^11,12,14,15; however, to the best of our knowledge, such information has not been reported. Table 2 shows that there is not a single STO/CGFs basis set that produces ICSs with a uniform high level of accuracy for the three considered processes at once. Therefore, one should choose a different type of basis set for a given process and energy range to obtain the best accuracy. For the total CT ICSs, the best SLEND result in the high and full energy ranges is from SLEND/aug-cc-pVTZ, which in terms of the MAPEs is more accurate than the ICSs from FODWT and CTMC and less accurate than the ICSs from QCTMC, TC-BGM and OCC-CC. Moreover, for the total CT ICSs, the best result in the low energy range is from SLEND/6–311++G**, which in terms of the MAPEs is more accurate than the ICSs from CTMC and QCTMC and less accurate than the ICSs from TC-BGM and OCC-CC. Overall, the data in Table 3 and Fig. 4 show that SLEND/aug-cc-pVTZ and SLEND/6–311++G** predict well the total CT ICS over the full range of the considered collision energies $E_{Lab}$= 0.1–100 keV; this is expected due to the documented success of SLEND to simulate CT processes with STO/CGFs basis sets^4,16. For the total DI ICSs, the best SLEND result in the low and full energy ranges is from SLEND/aug-cc-pVDZ, which in terms of the MAPEs is less accurate than the ICSs from TC-BGM and OCC-CC. Moreover, for the total DI ICSs, the best result in the high energy range is from SLEND/q-aug-cc-pVDZ, which in terms of the MAPEs is more accurate than the ICSs from TC-BGM and OCC-CC; this SLEND/q-aug-cc-pVDZ result suggests that a relatively high number of diffuse basis functions can lead to better DI predictions at high energies, as expected. Overall, the data in Table 3 and Figs. 4–5 show that the SLEND/STO/CGFs predictions of total DI ICSs are quantitatively less satisfactory than those of total CT ICSs. This is unsurprising because the STO/CGFs basis sets were not originally designed for DI predictions. Nevertheless, the quantitative performance of SLEND/STO/CGFs for DIs is much better than expected, especially if one observes the acceptable accuracies of SLEND/aug-cc-pVDZ for $E_{Lab}\le$ 10 keV and of SLEND/q-aug-cc-pVDZ for $E_{Lab}\ge$ 40 keV, cf. Fig, 5. The SLEND/STO/CGFs ICSs for DIs are qualitatively correct over all the considered energies, and are quantitatively correct within some energy ranges; then, they should be improvable by a re-optimization of the STO/CGFs basis sets for DIs as outlined in Sect. 5. For the 1s→2s TE ICSs, the best SLEND result in the high and full energy ranges is from SLEND/6–311++G**, which in terms of the MAPEs is more (less) accurate than the ICSs from TC-BGM and OCC-CC in the high (full) energy range. Moreover, for the 1s→2s TE ICSs, the best SLEND result in the low energy range is from SLEND/d-aug-cc-pVTZ, which in terms of the MAPEs is less accurate than but not far from the ICSs from TC-BGM and OCC-CC. The SLEND/6–311++G** 1s→2s TE ICS agrees well with the experimental data at $E_{Lab}\ge$ 40 keV but agrees far less satisfactorily at 5 keV $\le E_{Lab}\le$ 30 keV. Overall, the SLEND/STO/CGFs predictions of the 1s→2s TE ICS are more accurate than that those of the total DI ICS and less accurate than those of the total CT ICS. Nevertheless, one should bear in mind that predictions of the detailed state-to-state 1s→2s TE ICS are more difficult than those of the undetailed total CT and DI ICSs.

From our calculated data, it is not easy to discern any set of rules that straightforwardly correlate the accuracy of the SLEND ICSs with the number and types of functions in the basis sets. Nevertheless, in the case of the Pople basis sets, the largest considered 6–311++G** set, which has the highest multiple-zeta description and includes polarization and diffuse functions, tends to produce the best results. In addition, in the case of the Dunning basis sets, those with an intermediate level of multiple-zeta descriptions (DZ and TZ) and with an intermediate or high number of diffuse functions (aug to q-aug) tend to produce the best results.

5. Conclusions and Future Work

In the framework of the SLEND method^4,16, we have tested various standard STO/CGFs basis sets^17–20 for the prediction of DI and related electronic processes in H⁺ + H at $E_{Lab}$ =0.1–100 keV. STO/CGFs basis sets are constructed with bound and normalizable primitive Gaussians, and are optimized for the prediction of time-independent molecular properties of bound wavefunctions²⁷; therefore, in principle, these basis sets are not designed for the simulation of time-dependent dynamical processes, and even less so of time-dependent DI processes involving unbound and non-normalizable states. Nevertheless, TDDD methods including SLEND^3,4,16 have demonstrated that STO/CGFs basis sets are accurate for the simulation of fragmentations, rearrangements, and substitutions in ion-molecule reactions^3,4,16; furthermore, SLEND have demonstrated that those basis sets are also accurate for the simulation of CT and TE processes^4,16. Then, in this study, we endeavored to determine as proof of concept the feasibility of SLEND/STO/CGFs simulations of DI processes for the first time.

For our SLEND simulations, we utilized a collection of standard Pople^17,18 and Dunning^19,20 basis sets. The main obstacle toward DI treatments with bound and normalizable STO/CGFs is their intrinsic limitation to represent the unbound, non-normalizable and continuous scattering states involved in DIs. Therefore, we arranged these basis sets so that: 1) Their increasing number of diffuse functions augments the delocalization degree of virtual spin-orbitals with energies ${\epsilon }_i>0$ —in this way, those spin-orbitals approximate the unbound and non-normalizable character of the scattering states; and 2) their increasing total number of basis functions generate a considerable number of virtual spin-orbitals with energies ${\epsilon }_p>0$—in this way, those spin-orbitals approximate the continuum of scattering states. At the HF level, the employed basis sets generate a finite and discrete set of one occupied and some virtual spin-orbitals all with energies ${\epsilon }_i<0$ that exactly reproduce the energies and symmetries of the bound orbitals of the H atom; similarly, these basis sets generate a finite and discrete set of virtual spin-orbitals with energies ${\epsilon }_p>0$ that exactly reproduce the energies and symmetries of the unbound orbitals of the H atom. The SLEND/STO/CGFs weighted probabilities $b{P}_Y^X\left(b,E_{Lab}\right)$ and integral cross sections ${\text{ICS}}^X\left(E_{Lab}\right)$ for $X$ = total CT, total DI, and 1s→2s TE, exhibit correct qualitative behaviors over the full range of considered impact parameters $b$ = 0.0–10.0 a.u. and/or collision energies $E_{Lab}$ = 0.1–100 keV. In particular, the SLEND/STO/CGFs $b{P}_{\mathit{\text{Total}}}^{CT}\left(b,E_{Lab}\right)$ correctly display an oscillatory behavior (a series of peaks) vs. $b$ at low constant energies in agreement with experimental data^41–43. All these findings are expected for the CT and TE cases given the documented success of SLEND to predict these processes with STO/CGFs basis sets^4,16. However, these findings are surprising and significant for the new DI case. In quantitative terms, we found that the SLEND ICSs values noticeably vary with the employed basis sets, especially in the DI case. We also found that no single STO/CGFs basis set produces ICSs with a uniform high level of accuracy for the three considered processes at once. Therefore, one should choose a different type of basis set for a given process and energy range to achieve the best accuracy. For the total CT process, SLEND/aug-cc-pVDZ predicts the best ICSs that are in good agreement with their counterparts from experiments⁴⁴ and alternative theories^11,12,14,15 over all the considered collision energies $E_{Lab}$ = 0.1–100 keV. For the total DI process, SLEND/aug-cc-pVDZ and SLEND/q-aug-cc-pVDZ predict the best ICSs for $E_{Lab}\le$ 10 keV and $E_{Lab}\ge$ 40 keV, respectively, but predict less satisfactory results in the remaining energy ranges. The SLEND/q-aug-cc-pVDZ results suggest that basis sets with a relatively high number of diffuse functions tend to produce better DI predictions at high energies as originally assumed. For the 1s→2s TE process, SLEND/6–311++G** predicts the best ICSs results at $E_{Lab}\ge$ 40 but predicts less satisfactory results in the remaining energy range.

The current results show for the first time that SLEND in conjunction with standard STO/CGFs basis sets correctly reproduce the qualitative features of DI processes in H⁺ + H over a large range of impact parameters and collision energies. In addition, these results also show that SLEND/STO/CGFs satisfactorily predicts total DI ICSs in H⁺ + H within some energy ranges. Despite some numerical shortcomings, the SLEND results for DIs are significant and encouraging in view of the structural limitations of the STO/CGFs basis sets for DI descriptions and of their absolute lack of parameterization for those purposes. These results also suggest that the SLEND/STO/CGFs accuracy for DI predictions can be improved by further modification and re-parameterization of the basis sets. We are currently working on those tasks following various strategies. One approach is to increase even further the number of diffuse functions in the basis sets to exceedingly increase the number and delocalization of virtual spin-orbitals with energy ${\epsilon }_p>0$; in that way, those spin-orbitals will capture more effectively the continuum of scattering states. A more effective and challenging approach involves the complete re-parametrization of the STO/CGFs basis sets for optimal DI predictions without impairing their acquired accuracy for nuclear and CT processes. In that regard, recent machine learning techniques to efficiently optimize the parameters of semi-empirical methods⁴⁹ can be adapted to the present task. However, even with the best re-parameterization, the bound and non-normalizable character of the STO/CGFs may constitute an insurmountable obstacle to achieving very high accuracy for DI predictions. If so, the incorporation of unbound and non-normalizable basis functions into the STO/CGFs basis sets will be necessary. In Sect. 1, we discussed the incorporation of PWs into STO/CGFs basis sets and expressed our reservations against it due to numerical difficulties. However, a better approach involves fitting some unbound scattering states into a set of complex Gaussian functions inside a box⁵⁰. The new atomic integrals from this fitting are in terms of Gaussians and can therefore be evaluated with standard atomic integral packages with minor modifications.

Finally, as explained in the Introduction, we study the H⁺ + H scattering system due to its simplicity, its relevance for astrochemical processes, and the availability of its corresponding CT, DI and TE data from experiments and alternative theories for comparison. However, we intend to apply our SLEND/STO/CGFs methodology to DI processes in larger systems. For instance, in continuation of previous efforts^4,23–26, we plan to investigate collision-induced DIs in proton (H⁺) and carbon (C⁶⁺) cancer therapy reactions, specifically, in the water radiolysis reactions: H⁺/C⁶⁺ + H₂O , and in the DNA damage reactions: H⁺/C⁶⁺ + DNA nucleobase. These investigations will further enhance the understanding of ion cancer therapy processes at the microscopic level.