Statistical-law formulas for zero- to two-electron-transfer probabilities in proton–molecule and proton cancer therapy reactions from electron nuclear dynamics theory

Abstract

We present the first quantum-mechanical derivation of statistical-law formulas to calculate zero- to two-electron transfers (ETs) in proton–molecule reactions. The original statistical derivation assumed that the n-ET probabilities of N electrons in a shell obey an N-trial binomial distribution with success probability equal to an individual one-ET probability; the latter was heuristically identified with the number of transferred electrons from the integrated charge density. The obtained formulas proved accurate to calculate ET cross sections in proton–molecule and proton cancer therapy (PCT) reactions. We adopt the electron nuclear dynamics (END) theory in our quantum-mechanical derivation due to its versatile description of ETs via a Thouless single-determinantal state. Since non-orthogonal Thouless dynamical spin-orbitals pose mathematical difficulties, we first present a derivation for a model system with N ≥ 2 electrons where only two with opposite spins are ET active; in that scheme, the Thouless dynamical spin-orbitals become orthogonal, a fact that facilitates a still intricate derivation. In the end, we obtain the number of transferred electrons from the Thouless state charge density and the ETs probabilities from the Thouless state resolution into projectile–molecule eigenstates describing ETs. We prove that those probabilities and numbers of electrons interrelate as in the statistical-law formulas via their common dependency on the Thouless variational parameters. We review past ET results of proton–molecule and PCT reactions obtained with these formulas in the END framework and present new results of H⁺ + N₂O. We will present the derivation for systems with N > 2 electrons all active for ETs in a sequel.

I. INTRODUCTION

There is a renewed interest in simulating proton–molecule reactions with quantum-mechanical methods, given the key roles played by these reactions in astrophysics, planetary atmospheric chemistry, plasma physics, and proton cancer therapy (PCT).^1–3 For example, in the case of PCT, proton beams can obliterate cancerous cells with higher precision and fewer side effects than photon beams in x-ray therapy.^2,3 Consequently, various quantum-mechanical studies of PCT reactions have been conducted to elucidate PCT microscopic mechanisms and establish its rational design.^3–7 An important type of process during proton–molecule reactions includes molecule-to-proton n electron transfers (ETs): H⁺ + M → H¹⁻ⁿ + M⁺ⁿ, where the integer n is, in fact, restricted to 0 ≤ n ≤ 2 with proton projectiles. ET scattering processes are usually quantified in terms of differential cross sections (DCSs) and integral cross sections (ICSs).⁸ These dynamical properties are important to characterize and control the chemical systems where proton–molecule reactions occur. For instance, in PCT, n-ET ICSs can be part of the input data of Monte Carlo method codes^9,10 employed to calculate radiation doses for patients. Thus, various efforts have been made to determine accurate n-ET DCSs and ICSs both experimentally and theoretically.^5,7,11–13

The quantum-mechanical prediction of n-ET DCSs and ICSs requires the calculation of ET probability amplitudes and corresponding probabilities.⁸ To obtain the former, the final-time (or asymptotic) wavefunction describing a reaction H⁺ + M → H¹⁻ⁿ + M⁺ⁿ is projected onto the eigenstates of the supramolecular system H¹⁻ⁿ + M⁺ⁿ.⁸ This projection procedure can become computationally onerous if a very accurate wavefunction is chosen for the simulations [e.g., a full configuration interaction (CI) or a multi-configuration electronic wavefunction to describe the large biomolecules in PCT]. Furthermore, the probabilities thus obtained are difficult to relate to more intuitive chemical descriptors (e.g., the number of transferred electrons, electronegativities, etc.) that can provide a conceptual interpretation of ET reactions. In part to circumvent those limitations, a set of statistical-law formulas to calculate n-ET probabilities in ion–molecule reactions was proposed several years ago.¹⁴ In that approach, one considers N electrons in the same shell of a molecule M and assumes that each of those electrons has the same independent probability P_e, 0 ≤ P_e ≤ 1, to transfer to a neighboring ion; then, under those assumptions, the molecule-to-ion n-ET probabilities P_n-ET, 0 ≤ n ≤ N, should be the successive terms of a N-trial binomial distribution with success probability P_e,¹⁴

$$\begin{array}{ll}\hfill 1& ={\left[P_e+\left(1-P_e\right)\right]}^N\hfill \\ \hfill & =\sum _{n=0}^N\left[\begin{array}{c}\hfill N\hfill \\ n\hfill \end{array}\right]P_e^n{\left(1-P_e\right)}^{N-n}\hfill \\ \hfill & =\sum _{n=0}^N{P}_{n-ET}\Rightarrow P_{n-ET}\hfill \\ \hfill & =\left[\begin{array}{c}\hfill N\hfill \\ \hfill n\hfill \end{array}\right]P_e^n{\left(1-P_e\right)}^{N-n},\hfill \end{array}$$ (1)

where $\left[\begin{array}{c}\hfill N\hfill \\ \hfill n\hfill \end{array}\right]=\frac{N!}{n!(N-n)!}$ are binomial coefficients. These formulas, and the those discussed below, only become practical if the individual electron probability P_e can be easily obtained from a chemical property or descriptor, as will be shown shortly. In most proton–molecule reactions H⁺ + M → H¹⁻ⁿ + M⁺ⁿ, one observes transfers of up to one α-spin electron and/or one β-spin electron from a closed-shell target molecule M to H⁺. In those situations, we can extend the previous statistical assumptions leading to Eq. (1) ¹⁴ and apply a binomial distribution to the transferring α-spin electron with N = N^α = 1 and P_e = P^α $\Rightarrow 1={\left[P^{\alpha }+\left(1-P^{\alpha }\right)\right]}^{N_{\alpha }=1}$ and another distribution to the transferring β-spin electron with N = N^β and $P_e=P^{\beta }\Rightarrow 1={\left[P^{\beta }+\left(1-P^{\beta }\right)\right]}^{N_{\beta }=1}$. If we assume that these α and β distributions are independent of each other, then the total molecule-to-proton n-ET probabilities P_n-ET, 0 ≤ n ≤ 2, are the successive terms of the distributions product,¹⁵

$$\begin{array}{c}1={\left[P^{\alpha }+\left(1-P^{\alpha }\right)\right]}^{N_{\alpha }=1}{\left[P^{\beta }+\left(1-P^{\beta }\right)\right]}^{N_{\beta }=1}\\ =P^{\alpha }{P}^{\beta }+P^{\alpha }\left(1-P^{\beta }\right)+P^{\beta }\left(1-P^{\alpha }\right)+\left(1-P^{\alpha }\right)\left(1-P^{\beta }\right)\\ \Rightarrow P_{0-ET}=\left(1-P^{\alpha }\right)\left(1-P^{\beta }\right),P_{1-ET}=P^{\alpha }\left(1-P^{\beta }\right)+P^{\beta }\left(1-P^{\alpha }\right),\\ P_{2-ET}=P^{\alpha }{P}^{\beta },\end{array}$$ (2)

where P^α and P^β are not necessarily identical. The number of α/β electrons $N_{H^q}^{\alpha /\beta }$, $0\le N_{H^q}^{\alpha /\beta }\le 1$, finally transferred to the projectile H^q when it is well separated from the target can be calculated as $N_{H^q}^{\alpha /\beta }=\int {\rho }_{H^q}^{\alpha /\beta }\left(\mathbf{r}\right)d\mathbf{r}$, where ${\rho }_{H^q}^{\alpha /\beta }\left(\mathbf{r}\right)$ is the α/β charge density of the projectile. If we intuitively identify the α/β probabilities P^α/β with the α/β numbers of electrons $N_{H^q}^{\alpha /\beta }$: $P^{\alpha /\beta }=N_{H^q}^{\alpha /\beta }$, then the n-ET probabilities P_n-ET of Eq. (2) adopt the practical form,¹⁵

$$\begin{array}{c}{P}_{0-ET}=\left(1-N_{H^q}^{\alpha }\right)\left(1-N_{H^q}^{\beta }\right),\\ P_{1-ET}=N_{H^q}^{\alpha }\left(1-N_{H^q}^{\beta }\right)+N_{H^q}^{\beta }\left(1-N_{H^q}^{\alpha }\right),P_{2-ET}=N_{H^q}^{\alpha }{N}_{H^q}^{\beta }.\end{array}$$ (3)

The above P_n-ET formulas are astonishingly simple, and their input data $N_{H^q}^{\alpha /\beta }$ can be easily calculated from the final wavefunction via the corresponding ${\rho }_{H^q}^{\alpha /\beta }\left(\mathbf{r}\right)$. These formulas can be applied in conjunction with many types of quantum dynamics methods and wavefunctions.

Despite their statistical origin and simple form, the P_n-ET formulas in Eqs. (2) and (3) have proven accurate to calculate ET properties.¹⁴ An example of such a success is their application in association with the electron nuclear dynamics (END) method.^4,16,17 END is a time-dependent, variational, direct, and non-adiabatic method to simulate chemical reactions.^4,16,17 The simplest-level (SL) END (SLEND) version^4,16,17 adopted herein describes the nuclei via classical mechanics and the electrons with a Thouless single-determinantal wavefunction.¹⁸ In that context, the n-ET DCSs and ICSs of proton–molecule reactions can be calculated via Eq. (3) with $N_{H^q}^{\alpha /\beta }$ obtained from the Thouless single-determinantal wavefunction at the end of a simulation (cf. Sec. III for details). This approach has rendered accurate molecule-to-ion n-ET DCSs and ICSs in various reactive systems, such as He^+/2+ + H₂O,¹⁹ H⁺ + H₂,¹¹ H⁺ + C₂H₄,¹³ ${\mathrm{H}}^{+}+{\left({\mathrm{H}}_2\mathrm{O}\right)}_{1-6}$,^7,12 and H⁺ + DNA/RNA bases,⁵ inter alia (cf. Sec. II A for more details about these previous calculations). Note that the last two systems are prototypes for water radiolysis and proton-induced DNA damage in PCT, respectively.

Despite this success, no rigorous quantum-mechanical derivation of the P_n-ET formulas in Eqs. (2) and (3) has been presented thus far. The previously outlined derivation via statistical-law arguments^14,15 is at best heuristic and calls for a more precise justification. This situation is unsettling because without a firm theoretical foundation, the application of those formulas remains questionable and their success might be suspected as fortuitous. To overcome this situation, we present herein the first quantum-mechanical derivation of the P_n-ET formulas within the SLEND framework. In this approach, the SLEND Thouless single-determinantal wavefunction describing a reaction H⁺ + M → H¹⁻ⁿ + M⁺ⁿ can be written in the end as a linear combination of the H¹⁻ⁿ + M⁺ⁿ eigenstates corresponding to all the n-ETs. In the Thouless representation, that linear combination has a particular form in terms of the electronic variational parameters that leads to the P_n-ET formulas, Eq. (3), after arduous algebraic manipulations. At the core of this derivation lies the fact that the probabilities P_n-ET and the numbers of electrons in the final projectile $N_{H^q}^{\alpha /\beta }$ are all functions of the final electronic variational parameters; then, through their common dependency on those parameters, P_n-ET and $N_{H^q}^{\alpha /\beta }$ turn out to be interrelated exactly as in Eq. (3).

The general derivation of the P_n-ET formulas for N-electron systems, 0 ≤ n ≤ N, becomes cumbersome when N > 2 because of the complex (but still tractable) expressions of the Thouless single-determinantal wavefunction. Specifically, the non-orthogonality of the spin-orbitals in the Thouless representation¹⁸ when N > 2 poses mathematical difficulties. Therefore, in this article, we present progressing derivations of the P_n-ET formulas with two model systems that differ in their total number of electrons, N, and in their number of electrons active for ETs and electron excitations (EEs), n. The first model system named the 1α1β model has N = n = 2. In contrast, the second model system named the 1α1β/core model has N > 2 and n = 2 so that the remaining N − 2 electrons stay inactive for ETs and EEs in a core. In both models, the n = 2 active electrons have α and β spins, respectively. As shown in Secs. III C and III E, the non-orthogonality of the Thouless spin-orbitals disappears in both models, a fact that simplifies one aspect of a still arduous derivation. The derivation with the 1α1β/core model is the main goal of this article because this model is more general than the 1α1β one. However, since it is hard to deal with 1α1β/core model directly, we will first present a derivation with the simpler 1α1β model and then extend its results to the derivation with the 1α1β/core model. The latter model is by no means unrealistic because its resulting P_n-ET formulas for 0 ≤ n ≤ 2 have proven to be accurate in the aforesaid ion–molecule and PCT reactions with N > 2^{19 5,7,11–13} (cf. Sec. II B for more details). Despite their assumptions, the present derivations for n = 2 ≤ N active electrons remain laborious and pave the way for their more demanding extension to systems with all the electrons active for ETs and EES. We will present such an extension in a subsequent publication. The extended P_n-ET formulas will be useful for n-ET reactions in ion–molecule collisions with n > 2 as those occurring in C⁶⁺ cancer therapy.²

II. BACKGROUND: THE SLEND THEORY AND ITS CROSS SECTION CALCULATIONS WITH THE P_n-ET FORMULAS

A. The SLEND theory

The SLEND method provides the appropriate framework to derive the P_n-ET formulas. Therefore, we will review SLEND in some detail here (full accounts of SLEND are presented in Refs. 4, 16, and 17). SLEND is a time-dependent, variational, direct, and non-adiabatic method to simulate chemical reactions. SLEND employs a total trial wavefunction $\left|{\mathrm{\Psi }}_{SLEND}\left(t\right)\right.⟩$ that is the product of nuclear $\left|\mathbf{R}\left(t\right),\mathbf{P}\left(t\right)\right.⟩$ and electronic $\left|\mathbf{z}\left(t\right);\mathbf{R}\left(t\right),\mathbf{P}\left(t\right)\right.⟩$ wavefunctions: $\left|{\mathrm{\Psi }}_{SLEND}\left(t\right)\right.⟩=\left|\mathbf{R}\left(t\right),\mathbf{P}\left(t\right)\right.⟩\left|\mathbf{z}\left(t\right);\mathbf{R}\left(t\right),\mathbf{P}\left(t\right)\right.⟩$. In a system with N_N nuclei, $\left|\mathbf{R}\left(t\right),\mathbf{P}\left(t\right)\right.⟩$ is the product of 3N_N, 1D, frozen, narrow, Gaussian wave packets,

$$\begin{array}{ll}\hfill ⟨\mathbf{X}\left|\mathbf{R}\left(t\right),\mathbf{P}\left(t\right)\right.⟩& =\prod _{i=1}^{3N_N}⟨X_i\left|R_i\left(t\right),P_i\left(t\right)\right.⟩\hfill \\ \hfill & =\prod _{i=1}^{3N_N}\exp \left\{-{\left[\frac{X_i-R_i\left(t\right)}{2\mathrm{\Delta }{R}_i}\right]}^2+i{P}_i\left[X_i-R_i\left(t\right)\right]\right\},\hfill \end{array}$$ (4)

where $R_i\left(t\right)$, $P_i\left(t\right)$, and ΔR_i are the average positions, average momenta, and widths of the wave packets, respectively. $\left|\mathbf{z}\left(t\right);\mathbf{R}\left(t\right),\mathbf{P}\left(t\right)\right.⟩$ is a single-determinantal state in the Thouless representation.¹⁸ Having N_e electrons and an atomic basis set of size K > N_e, N_e hole (occupied) and K − N_e particle (virtual) molecular spin-orbitals (MSOs), ψ_h and ψ_p, respectively, are calculated at the Hartree–Fock (HF) level. Then, the Thouless single-determinantal state $\left|\mathbf{z}\left(t\right);\mathbf{R}\left(t\right),\mathbf{P}\left(t\right)\right.⟩$ from the HF reference state $\left|0\right.⟩=\left|{\psi }_1\dots {\psi }_h\dots {\psi }_{N_e}\right.⟩$ is^4,16,18

$$\begin{array}{cc}\hfill \left|\mathbf{z}\left(t\right);\mathbf{R}\left(t\right),\mathbf{P}\left(t\right)\right.⟩& =\prod _{h=1}^{N_e}\left(1+\sum _{p=N_e+1}^K{z}_{ph}\left(t\right)b_p^{†}{b}_h\right)\left|0\right.⟩\hfill \\ \hfill & =\left|{\chi }_1\dots {\chi }_h\dots {\chi }_{N_e}\right.⟩,\hfill \\ \hfill \left.⟨\mathbf{x}\right|\mathbf{z}\left(t\right);\mathbf{R}\left(t\right),\mathbf{P}\left(t\right)⟩& =\det \left\{{\chi }_h\left[{\mathbf{x}}_h;\mathbf{z}\left(t\right),\mathbf{R}\left(t\right)\right]\right\},1\le h\le N_e,\hfill \end{array}$$ (5)

where χ_h are the non-orthogonal dynamical spin-orbit als (DSOs),

$$\begin{array}{ll}\hfill {\chi }_h\left[\mathbf{x};\mathbf{z}\left(t\right),\mathbf{R}\left(t\right),\mathbf{P}\left(t\right)\right]& ={\psi }_h\left[\mathbf{x};\mathbf{R}\left(t\right),\mathbf{P}\left(t\right)\right]+\sum _{p=N_e+1}^K{\psi }_p\hfill \\ \hfill & \times \left[\mathbf{x};\mathbf{R}\left(t\right),\mathbf{P}\left(t\right)\right]z_{ph},1\le h\le N_e,\hfill \end{array}$$ (6)

in terms of the complex-valued Thouless parameters $z_{ph}\left(t\right)$; these parameters are the DSO molecular coefficients in the MSO basis. The atomic basis set functions to construct MSOs, DSOs, and $\left|\mathbf{z}\left(t\right);\mathbf{R}\left(t\right),\mathbf{P}\left(t\right)\right.⟩$ are centered on the wave packets’ positions $R_i\left(t\right)$ that move with momenta $P_i\left(t\right)$; for that reason, MSOs, DSOs, and $\left|\mathbf{z}\left(t\right);\mathbf{R}\left(t\right),\mathbf{P}\left(t\right)\right.⟩$ parametrically depend on $R_i\left(t\right)$ and $P_i\left(t\right)$. Originally, SLEND adopted the Thouless representation¹⁸ because it provides a non-redundant parameterization of a general single-determinantal state from a reference $\left|0\right.⟩$; that parameterization preempts numerical singularities during time evolution.^4,16 In this investigation, we will illustrate an additional advantage of the Thouless representation: its suitable formal structure to describe ETs and derive the P_n-ET formulas (cf. Sec. III).

The SLEND dynamical equations are obtained by subjecting the total trial wavefunction $\left|{\mathrm{\Psi }}_{SLEND}\left(t\right)\right.⟩$ to the time-dependent variational principle.²⁰ Specifically, the SLEND action $A_{SLEND}\left[{\mathrm{\Psi }}_{SLEND}^{*}\left(t\right),{\mathrm{\Psi }}_{SLEND}\left(t\right)\right]$ is made stationary with respect to all the variational parameters: $R_i\left(t\right)$,$P_i\left(t\right)$, $z_{ph}\left(t\right)$, and $z_{ph}^{*}\left(t\right)$;^4,16,17 the resulting SLEND dynamical equations for those parameters are^4,16,17

$$\left[\begin{array}{cccc}\hfill i\mathbf{C}\hfill & \hfill \mathbf{0}\hfill & \hfill i{\mathbf{C}}_{\mathbf{R}}\hfill & \hfill i{\mathbf{C}}_{\mathbf{P}}\hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill \\ \hfill \mathbf{0}\hfill & \hfill -i{\mathbf{C}}^{*}\hfill & \hfill -i{\mathbf{C}}_{\mathbf{R}}^{*}\hfill & \hfill -i{\mathbf{C}}_{\mathbf{P}}^{*}\hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill \\ \hfill i{\mathbf{C}}_{\mathbf{R}}^{†}\hfill & \hfill -i{\mathbf{C}}_{\mathbf{R}}^T\hfill & \hfill {\mathbf{C}}_{\mathbf{R}\mathbf{R}}\hfill & \hfill -\mathbf{I}+{\mathbf{C}}_{\mathbf{R}\mathbf{P}}\hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill \\ \hfill i{\mathbf{C}}_{\mathbf{P}}^{†}\hfill & \hfill -i{\mathbf{C}}_{\mathbf{P}}^T\hfill & \hfill \mathbf{I}+{\mathbf{C}}_{\mathbf{P}\mathbf{R}}\hfill & \hfill {\mathbf{C}}_{\mathbf{R}\mathbf{P}}\hfill \end{array}\right]\left[\begin{array}{c}\hfill \frac{d\mathbf{z}}{dt}\hfill \\ \hfill \frac{d{\mathbf{z}}^{*}}{dt}\hfill \\ \hfill \frac{d\mathbf{R}}{dt}\hfill \\ \hfill \frac{d\mathbf{P}}{dt}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill \frac{\partial E_{Total}}{\partial {\mathbf{z}}^{*}}\hfill \\ \hfill \frac{\partial E_{Total}}{\partial \mathbf{z}}\hfill \\ \hfill \frac{\partial E_{Total}}{\partial \mathbf{R}}\hfill \\ \hfill \frac{\partial E_{Total}}{\partial \mathbf{P}}\hfill \end{array}\right].$$ (7)

Above, E_Total is the total (nuclear and electronic) energy, C_ph,qg are the electron–electron couplings, and ${\mathbf{C}}_{\mathbf{X}}\left(\mathbf{X}=\mathbf{R}\mathrm{o}\mathrm{r}\mathbf{P}\right)$ and ${\mathbf{C}}_{\mathbf{X}\mathbf{Y}}\left(\mathbf{X},\mathbf{Y}=\mathbf{R}\mathrm{o}\mathrm{r}\mathbf{P}\right)$ are the first- and second-order nucleus–electron non-adiabatic couplings, respectively.^4,16 To accelerate calculations, the SLEND dynamical equations [Eq. (7)] are expressed in the zero-width limit of the nuclear wave packets $\left|R_i\left(t\right),P_i\left(t\right)\right.⟩$, ΔR_i → 0∀i; that limit leads to a classical nuclear dynamics in terms of positions $R_i\left(t\right)$ and momenta $P_i\left(t\right)$ coupled to a quantum electronic dynamics in terms of the Thouless parameters $z_{ph}\left(t\right)$ and $z_{ph}^{*}\left(t\right)$. Reaction properties, including our considered P_n-ET formulas, are obtained from the final SLEND wavefunction $\left|{\mathrm{\Psi }}_{SLEND}\left(t_f\right)\right.⟩$ (cf. Sec. II B just below and also Secs. III C and III D).

B. Review of SLEND calculations of cross sections with the P_n-ET formulas

Having discussed the SLEND theory, we can now explain the calculations of n-ET DCSs and ICSs with the P_n-ET formulas within the SLEND framework. In addition, we will review previous^5,7,11–13 and new results obtained with that approach. Figure 1 shows the initial laboratory-frame conditions of the nuclei in a typical SLEND simulation of a H⁺ + M → H¹⁻ⁿ + M⁺ⁿ reaction with a linear molecule M as the target. At the initial time, M is in its equilibrium geometry, placed at rest with its center of mass (CM) on the coordinate center and with an orientation determined by two spherical angles: 0 ≤ α ≤ π and 0 ≤ β < 2π. (If the target M is not linear, its orientation is determined with three Euler angles; cf. Ref. 3.) The initial M is in its ground HF reference state $\left|0\right.⟩$. Figure 1 also shows the initial conditions of the H⁺ projectile: it is placed on the x–z plane, well separated from M (z ≥ −20 a.u.), with an impact parameter b ≥ 0 from the z axis and with the initial momentum ${\mathbf{P}}_{H^{+}}^i$ along the +z direction. Numerous SLEND simulations (trajectories) are calculated from varying initial conditions $\left(\alpha ,\beta \right)$ and b. Upon completing those simulations, the target-to-projectile n-ET DCSs in the CM frame $d{\sigma }_{\left(\alpha ,\beta \right)}^{n-ET}\left({\theta }_{CM}\right)/d\mathrm{\Omega }$ are^4,8,21

$$\begin{array}{ll}\hfill \frac{d{\sigma }_{\left(\alpha ,\beta \right)}^{n-ET}\left({\theta }_{CM}\right)}{d\mathrm{\Omega }}& =\frac{1}{4k_i^2}\left|\sum _{l=0}^{\infty }\left(2l+1\right)P_l\left(\cos {\theta }_{CM}\right)\right.\hfill \\ \hfill & \times {\left.{\left\{\sqrt{P_{n-ET}{\left(l\right)}_{\left(\alpha ,\beta \right)}}\exp \left[i2\eta \left(l\right)\right]\right\}}_{\left(\alpha ,\beta \right)}\right|}^2,\hfill \end{array}$$ (8)

where θ_CM, k_i, l, and $\eta \left(l\right)$ are the projectile’s scattering angle, initial wave vector modulus, final orbital angular momentum quantum number, and phase shift, respectively, and $P_l\left(\cos {\theta }_{CM}\right)$ are Legendre polynomials. SLEND employs classical nuclear dynamics; therefore, l and $\eta \left(l\right)$ are obtained from nuclear variables via semi-classical relationships; for instance, l = k_ib.^4,8,21 The CM DCSs are calculated via Eq. (8) with the SLEND data converted into the CM frame; the obtained CM DCSs are transformed back to the laboratory frame and averaged over the initial orientations $\left(\alpha ,\beta \right)$ to obtain the DCSs discussed below. The n-ET ICSs are obtained by integrating the n-ET DCSs over the solid angle Ω of the scattering direction θ. A key component in n-ET DCS and ICS calculations via Eq. (8) is the final n-ET probabilities $P_{n-ET}{\left(l\right)}_{\left(\alpha ,\beta \right)}$ from each trajectory with the initial conditions $\left(\alpha ,\beta \right)$ and b = l/k_i; these probabilities are calculated with the P_n-ET formulas, Eq. (3), using the number of electrons transferred to the projectile, $N_{H^q}^{\alpha /\beta }=N_B^{\alpha /\beta }$, given by the final SLEND electronic wavefunction. It is important to emphasize that SLEND does not assume any model description, such as the $1\alpha 1\beta \left(/core\right)$ model or any other, to describe a whole reaction. Specifically, SLEND does not enforce a core of inactive electrons but allows all of them to participate in ETs and EEs. Thus, in these n-ET DCS and ICS calculations with SLEND, the $1\alpha 1\beta \left(/core\right)$ model description is assumed only at the final time just to calculate the n-ET probabilities with the derived P_n-ET formulas. Use of those formulas is numerically advantageous in the case of PCT systems involving large biomolecules.^5,7

FIG. 1.

Nuclear initial conditions of the H⁺ + M reactants involving a linear molecule target M; a yellow sphere represents an incoming H⁺ projectile with momentum P and impact parameter b; the black and red spheres represent the nuclei of the target molecule with projectile–target orientation (α, β).

The described scheme to calculate n-ET DCSs and ICSs with the P_n-ET formulas in conjunction with SLEND provided accurate zero-ET and one-ET DCSs for H⁺ + H₂,¹¹ + H₂O,¹² and + C₂H₄¹³ and accurate one-ET ICSs for H⁺ + H₂,¹¹ + (H₂O)_1–6,^7,12 and + DNA/RNA bases,⁵ inter alia; note that the last two systems are prototypes for water radiolysis and proton-induced DNA damage in PCT, respectively. The reader can find the details of those calculations and the comparison of their results with experimental data in the cited references. As a further illustration, we present herein new unpublished results of H⁺ + N₂O at E_Lab = 30 eV. Figure 2 shows the zero-ET and one-ET DCSs of that reaction vs the projectile scattering angle θ in comparison with their experimental counterparts.²² (Experimental two-ET DCSs are unavailable for most proton–molecule reactions.) The theoretical DCSs are calculated with the P_n-ET formulas on SLEND simulations with the 6-311G atomic basis set (SLEND/6-311G). The raw experimental data were reported in arbitrary units.²² Then, to allow for comparisons in Fig. 2, we follow this standard procedure: we first normalize the experimental zero-ET DCS so that it has a maximum overlap with the SLEND zero-ET DCS; then, we normalize the experimental one-ET DCS with the same factor and adjust the SLEND one-ET DCS with the experimental detector H⁰/H⁺ sensitivity ratio of 3.75%.²² (The same detector measured the H⁰ and H⁺ scattered projectiles with the indicated sensitivity ratio.) Figure 2 shows an excellent agreement between experimental and SLEND zero-ET and one-ET DCSs over the whole range of scattering angles θ_Lab examined in the experiment. Noticeably, the damped experimental DCSs barely reveal the expected rainbow scattering angle peaks at θ_Lab ≈ 9°–10°;²² in contrast, the SLEND DCSs clearly show those expected peaks. (A rainbow peak corresponds to a maximum projectile–target attractive scattering.^4,8,21) The reader can find additional examples of the P_n-ET formulas’ utilization in n-ET DCS and ICS SLEND calculations in Refs. 5, 7, and 11–13.

FIG. 2.

SLEND/6-311G and experimental²² zero-ET and one-ET DCSs of H⁺ + N₂O vs the H⁺ scattering angle at collision energy = 30 eV. Theoretical DCS calculations employ the derived zero-ET and one-ET probability formulas [Eq. (3)] (cf. the main text for more details).

III. THEORY DEVELOPMENT: DERIVATION OF THE n-ET PROBABILITY FORMULAS

A. Models for the derivation: The 1α1β and 1α1β/core models

As stated in the Introduction, we will derive the P_n-ET equations (3) for N-electron systems having up to n = 2 electrons active for ETs and EEs, N ≥ n = 2. Toward that goal, we consider a scattering system consisting of target A and projectile B; A and B can be atoms and/or molecules. In a first stage (Secs. III B–III D), we will consider the simple 1α1β model system A + B having only two electrons in total: N = n = 2, with α and β spins, respectively. At the initial time, we will prepare the 1α1β model system with the A and B reactants well separated (i.e., non-interacting), with B moving toward A and with the two electrons located on A. In the end, we will have post-collision fragments A and B well separated again, moving away from each other and with the two electrons distributed over A and B according to the P_n-ET probabilities. It is worth noting that some real systems (e.g., H₂ + H⁺ or He²⁺, H⁻ + H⁺ or He²⁺, and similar ones) correspond exactly to the 1α1β model. The ET reactions in this model system can be written as

$$A\left[\mathrm{2}{e}^{-}\right]+B\left[\mathrm{0}{e}^{-}\right]\to A^{+n}\left[\left(2-n\right)e^{-}\right]+B^{-n}\left[n{e}^{-}\right],0\le n\le 2,N=n.$$ (9)

In a second stage (Sec. III E), we will generalize the 1α1β model derivation for cases with N > 2 electrons, where n = 2 electrons with α and β spins, respectively, will be active for ETs and EEs and the remaining N − 2 ones will be inactive for those processes,

$$\begin{array}{ll}\hfill & A\left[\left(2+n_{Acore}\right)e^{-}\right]+B\left[n_{Bcore}{e}^{-}\right]\to A^{+n}\left[\left(2+n_{Acore}-n\right)e^{-}\right]\hfill \\ \hfill & +B^{-n}\left[\left(n_{Bcore}+n\right)e^{-}\right],0\le n\le 2,N=n_{Acore}+n_{Bcore}+2>2.\hfill \end{array}$$ (10)

This 1α1β/core model can accurately describe ETs in various ion–molecule systems, as mentioned in Sec. I, and we will further discuss in Sec. IV.

B. Thouless state for the 1α1β model

In SLEND, the electron dynamics description primarily comes from the Thouless single-determinantal state, Eq. (5); thus, we will concentrate on that state to describe ETs. The DSOs of the Thouless state in Eq. (5) are general because each of them combines α and β MSOs, a spin arrangement equivalent to that of the generalized unrestricted HF (GUHF) method.²³ However, for our chemical systems, we only need that each DSO combines MSOs with the same spin, either α or β, so that z = z^α ⊕ z^β; this spin arrangement is equivalent to that of the unrestricted HF (UHF) method.²³ The required DSOs are

$$\begin{array}{ll}\hfill {\mathrm{\chi }}_h^{\sigma }(\mathbf{x};{\mathbf{z}}^{\sigma })& ={\psi }_h^{\sigma }(\mathbf{x})+\sum _{p=N^{\sigma }+1}^K{z}_{ph}^{\sigma }{\psi }_p^{\sigma }(\mathbf{x})\hfill \\ \hfill & =\left({\tilde{\psi }}_h^{\sigma }(\mathbf{r})+\sum _{p=N^{\sigma }+1}^K{z}_{ph}^{\sigma }{\tilde{\psi }}_p^{\sigma }(\mathbf{r})\right)\sigma (\omega ),1\le h\le N^{\sigma },\hfill \end{array}$$ (11)

where x comprises the spatial r and spin ω variables, $\mathbf{x}=\left(\mathbf{r},\omega \right)$, ${\psi }_h^{\sigma }\left(\mathbf{x}\right)$ and ${\psi }_p^{\sigma }\left(\mathbf{x}\right)$ are hole (occupied) and particle (virtual) UHF MSOs, ${\tilde{\psi }}_h^{\sigma }\left(\mathbf{r}\right)$ and ${\tilde{\psi }}_p^{\sigma }\left(\mathbf{r}\right)$ are their respective spatial parts, σ(ω) = α(ω) or β(ω) is the MSO common spin eigenfunction, N^σ is the number of electrons with spin σ, and K is the atomic basis set size. In order to simplify the notation in Eq. (11) and subsequent ones, we omit writing the dependence of the Thouless coefficients $z_{ph}^{\sigma }$ on time and the dependence of the MSOs on the nuclear variables $\mathbf{R}\left(t\right)$ and $\mathbf{P}\left(t\right)$, as originally shown in Eqs. (5) and (6); the reader should bear in mind those implicit dependencies that we will make explicit again in future expressions when they play a crucial role. The overlap between two DSOs from Eq. (11) is given as

$$\begin{array}{cc}\hfill ⟨{\chi }_g^{{\sigma }^{\prime }}\left(\mathbf{x};{\mathbf{z}}^{{\sigma }^{\prime }}\right)\left|\right.{\chi }_h^{\sigma }\left(\mathbf{x};{\mathbf{z}}^{\sigma }\right)⟩& =⟨{\tilde{\psi }}_g^{{\sigma }^{\prime }}(\mathbf{r})+\sum _{q=N^{{\sigma }^{\prime }}+1}^K{z}_{qg}^{{\sigma }^{\prime }}{\tilde{\psi }}_q^{{\sigma }^{\prime }}(\mathbf{r})\left|\right.{\tilde{\psi }}_h^{\sigma }(\mathbf{r})+\sum _{p=N^{\sigma }+1}^K{z}_{ph}^{\sigma }{\tilde{\psi }}_p^{\sigma }(\mathbf{r})⟩⟨{\sigma }^{\prime }(\omega )\left|\right.\sigma (\omega )⟩\hfill \\ \hfill & =\int \left({\tilde{\psi }}_g^{{\sigma }^{\prime }*}(\mathbf{r}),{\tilde{\psi }}_h^{\sigma },(\mathbf{r}),+,\sum _{p=N^{\sigma }+1}^K,z_{ph}^{\sigma },{\tilde{\psi }}_g^{{\sigma }^{\prime }*},(\mathbf{r}),{\tilde{\psi }}_p^{\sigma },(\mathbf{r}),+,\sum _{q=N^{{\sigma }^{\prime }}+1}^K,z_{qg}^{{\sigma }^{\prime }*},{\tilde{\psi }}_q^{{\sigma }^{\prime }*},(\mathbf{r}),{\tilde{\psi }}_h^{\sigma },(\mathbf{r})\right.\hfill \\ \hfill & \left.+\sum _{q=N^{{\sigma }^{\prime }}+1}^K\sum _{p=N^{\sigma }+1}^K{z}_{qg}^{{\sigma }^{\prime }*}{z}_{ph}^{\sigma }{\tilde{\psi }}_q^{{\sigma }^{\prime }*}(\mathbf{r}){\tilde{\psi }}_p^{\sigma }(\mathbf{r})\right)d\mathbf{r}\int {\sigma }^{\prime }(\omega )\sigma (\omega )d\omega \hfill \\ \hfill & =\left({\delta }_{gh}+\sum _{p=N^{\sigma }+1}^K{z}_{pg}^{{\sigma }^{*}}{z}_{ph}^{\sigma }\right){\delta }_{{\sigma }^{\prime }\sigma },\hfill \end{array}$$ (12)

where the orthonormality of spin eigenfunctions, $\int {\sigma }^{\prime }(\omega )\sigma (\omega )d\omega ={\delta }_{\sigma {\sigma }^{\prime }}$, and of the MSO spatial parts within spin blocks, $\int {\tilde{\psi }}_i^{{\sigma }^{*}}(\mathbf{r}){\tilde{\psi }}_j^{\sigma }(\mathbf{r})d\mathbf{r}={\delta }_{ij}$, was applied in the last step. Equation (12) shows that the DSOs ${\chi }_h^{\sigma }$ are un-normalized and non-orthogonal within spin blocks. Normalized DSOs ${\overline{\chi }}_h^{\sigma }$ are easily obtained as

$$\begin{array}{cc}\hfill {\overline{\chi }}_h^{\sigma }\left(\mathbf{x};{\mathbf{z}}^{\sigma }\right)& ={\left[f_h^{\sigma }\left({\mathbf{z}}^{\sigma },{\mathbf{z}}^{{\sigma }^{*}}\right)\right]}^{-1/2}{\chi }_h^{\sigma }\left(\mathbf{x};{\mathbf{z}}^{\sigma }\right),f_h^{\sigma }\left({\mathbf{z}}^{\sigma },{\mathbf{z}}^{{\sigma }^{*}}\right)=⟨{\chi }_h^{{\sigma }^{\prime }}\left(\mathbf{x};{\mathbf{z}}^{\sigma }\right)\left|\right.{\chi }_h^{\sigma }\left(\mathbf{x};{\mathbf{z}}^{\sigma }\right)⟩=1+\sum _{p=N^{\sigma }+1}^K{\left|z_{ph}^{\sigma }\right|}^2,\hfill \end{array}$$ (13)

where $f_h^{\sigma }$ are the DSO normalization factors obtained via Eq. (12); while normalized, ${\overline{\chi }}_h^{\sigma }$ still remain non-orthogonal within spin blocks.

In the 1α1β model, the un-normalized Thouless state ${\left|\mathbf{z}\right.⟩}^{1\alpha 1\beta }$ is [cf. Eq. (5)] given as

$$\begin{array}{ll}\hfill {\left|\mathbf{z}\right.⟩}^{1\alpha 1\beta }& =\left|{\chi }_h^{\alpha }{\chi }_{h^{\prime }}^{\beta }\right.⟩=\left(1+\sum _{p=2}^K{z}_{ph}^{\alpha }{b}_p^{\alpha †}{b}_h^{\alpha }\right)\left(1+\sum _{p^{\prime }=2}^K{z}_{p^{\prime }{h}^{\prime }}^{\beta }{b}_{p^{\prime }}^{\beta †}{b}_{h^{\prime }}^{\beta }\right)\left|h{h}^{\prime }\right.⟩\hfill \\ \hfill & =\left(1+\sum _{p=2}^K{z}_{ph}^{\alpha }{b}_p^{\alpha †}{b}_h^{\alpha }+\sum _{p^{\prime }=2}^K{z}_{p^{\prime }{h}^{\prime }}^{\beta }{b}_{p^{\prime }}^{\beta †}{b}_{h^{\prime }}^{\beta }\right.\left.+\sum _{p=2}^K\sum _{p^{\prime }=2}^K{z}_{ph}^{\alpha }{z}_{p^{\prime }{h}^{\prime }}^{\beta }{b}_p^{\alpha †}{b}_h^{\alpha }{b}_{p^{\prime }}^{\beta †}{b}_{h^{\prime }}^{\beta }\right)\left|h{h}^{\prime }\right.⟩\hfill \\ \hfill & =\left|h{h}^{\prime }\right.⟩+\sum _{p=2}^K{z}_{ph}^{\alpha }\left|p{h}^{\prime }\right.⟩+\sum _{p^{\prime }=2}^K{z}_{p^{\prime }{h}^{\prime }}^{\beta }\left|h{p}^{\prime }\right.⟩+\sum _{p=2}^K\sum _{p^{\prime }=2}^K{z}_{ph}^{\alpha }{z}_{p^{\prime }{h}^{\prime }}^{\beta }\left|p{p}^{\prime }\right.⟩.\hfill \end{array}$$ (14)

In the third line above, the first state $\left|h{h}^{\prime }\right.⟩=\left|{\psi }_h^{\alpha }{\psi }_h^{\beta }\right.⟩$ is the HF reference state $\left|0\right.⟩$ [cf. Eq. (5)], while the remaining states $\left|p{h}^{\prime }\right.⟩$, $\left|h{p}^{\prime }\right.⟩$, and $\left|p{p}^{\prime }\right.⟩$ are all the possible excited states out of $\left|0\right.⟩$. The presence of all these states in Eq. (14) illustrates the non-adiabatic character of the Thouless representation. The normalization condition of ${\left|\mathbf{z}\right.⟩}^{1\alpha 1\beta }$ in Eq. (14) is given as

$$\begin{array}{ll}\hfill {⟨\mathbf{z}\left|\right.\mathbf{z}⟩}^{1\alpha 1\beta }& =1+\sum _p{\left|z_{ph}^{\alpha }\right|}^2+\sum _{p^{\prime }}{\left|z_{p^{\prime }{h}^{\prime }}^{\beta }\right|}^2+\sum _{p,p^{\prime }}{\left|z_{ph}^{\alpha }\right|}^2{\left|z_{p^{\prime }{h}^{\prime }}^{\beta }\right|}^2\hfill \\ \hfill & =\left(1+\sum _{p=2}^K{\left|z_{ph}^{\alpha }\right|}^2\right)\left(1+\sum _{p=2}^K{\left|z_{p^{\prime }{h}^{\prime }}^{\beta }\right|}^2\right)=f_h^{\alpha }{f}_{h^{\prime }}^{\beta },\hfill \end{array}$$ (15)

which is equal to the product of the DSO normalization factors $f_h^{\alpha }$ and $f_{h^{\prime }}^{\beta }$; cf. Eq. (13). Due to their opposite spins, the two DSOs ${\chi }_h^{\alpha }$ and ${\chi }_{h^{\prime }}^{\beta }$ become orthogonal; cf. Eq. (12). However, ${\left|\mathbf{z}\right.⟩}^{1\alpha 1\beta }$ is not a standard Slater determinant state because the DSOs and $\left|{\mathbf{z}}^{1\alpha 1\beta }\right.⟩$ itself are not normalized. Normalization of ${\left|\mathbf{z}\right.⟩}^{1\alpha 1\beta }$ with the factor in Eq. (15), or, equivalently, through normalization of each DSOs with Eq. (13), leads to a standard Slater determinant state $\left|{\mathrm{\Psi }}_e^{1\alpha 1\beta }\right.⟩$,

$$\begin{array}{ll}\hfill {\mathrm{\Psi }}_e^{1\alpha 1\beta }\left(\mathbf{x};\mathbf{z}\right)& =\frac{⟨\mathbf{x}{\left|\mathbf{z}\right.⟩}^{1\alpha 1\beta }}{{\left({⟨\mathbf{z}\left|\right.\mathbf{z}⟩}^{1\alpha 1\beta }\right)}^{1/2}}\hfill \\ \hfill & =\frac{⟨\mathbf{x}\left|{\chi }_h^{\alpha }{\chi }_{h^{\prime }}^{\beta }\right.⟩}{{\left({⟨\mathbf{z}\left|\right.\mathbf{z}⟩}^{1\alpha 1\beta }\right)}^{1/2}}\hfill \\ \hfill & =\frac{1}{2^{1/2}}det\left[{\overline{\chi }}_h^{\alpha }\left({\mathbf{x}}_1;{\mathbf{z}}^{\alpha }\right){\overline{\chi }}_{h^{\prime }}^{\beta }\left({\mathbf{x}}_2;{\mathbf{z}}^{\beta }\right)\right].\hfill \end{array}$$ (16)

Molecular properties (e.g., energy, charge density, etc.) are readily calculated from $\left|{\mathrm{\Psi }}_e^{1\alpha 1\beta }\right.⟩$ via the standard Slater–Condon rules;²³ thus, we will utilize $\left|{\mathrm{\Psi }}_e^{1\alpha 1\beta }\right.⟩$ instead of its equivalent ${\left|\mathbf{z}\right.⟩}^{1\alpha 1\beta }$ for property calculations in Secs. III C and III D. We should note that in the general case with N > 2 electrons all active for ETs and EEs, the SLEND electronic wavefunction is not equivalent to a Slater determinant but remains as a Thouless determinantal state with various non-orthogonal DSOs; in that case, molecular properties should be calculated with the more involving Löwdin rules.²³

Before considering property calculations, it is instructive to analyze how ${\left|\mathbf{z}\right.⟩}^{1\alpha 1\beta }$ describes ETs and EEs in the A + B system. At the initial and final times, A and B are well separated, so each MSO gets localized on either A or B. At the initial time t_i, the 1α1β model starts with the two electrons on target A; then, the hole MSOs ψ_h and ${\psi }_{h^{\prime }}$ of the HF reference are localized on A, h, h′ ∈ A, and the initial Thouless state is ${\left|\mathbf{z}\left(t_i\right)\right.⟩}^{1\alpha 1\beta }=\left|{\psi }_h^{\alpha }{\psi }_h^{\beta }\right.⟩=\left|h{h}^{\prime }\right.⟩=\left|0\right.⟩$ with all the Thouless coefficients $z_{ph}^{\sigma }\left(t_i\right)=0$. As the reaction proceeds, $z_{ph}^{\sigma }\left(t>t_i\right)$ will take non-zero values as dictated by the SLEND dynamical Eq. (7); therefore, ${\left|\mathbf{z}\left(t>t_i\right)\right.⟩}^{1\alpha 1\beta }$ adopts the full form in Eq. (14). Thus, at the final time t_f, the Thouless state ${\left|\mathbf{z}\left(t_f\right)\right.⟩}^{1\alpha 1\beta }$ can be expressed as

$$\begin{array}{ll}\hfill {\left|\mathbf{z}\left(t_f\right)\right.⟩}^{1\alpha 1\beta }& =\left|h{h}^{\prime }\right.⟩+\sum _{p\in \mathrm{A}}{z}_{ph}^{\alpha }\left(t_f\right)\left|p{h}^{\prime }\right.⟩+\sum _{p^{\prime }\in \mathrm{A}}{z}_{p^{\prime }{h}^{\prime }}^{\beta }\left(t_f\right)\left|h{p}^{\prime }\right.⟩\hfill \\ \hfill & +\sum _{p\in \mathrm{A},p^{\prime }\in \mathrm{A}}{z}_{ph}^{\alpha }\left(t_f\right)z_{p^{\prime }{h}^{\prime }}^{\beta }\left(t_f\right)\left|p{p}^{\prime }\right.⟩\hfill \\ \hfill & +\sum _{p\in \mathrm{B}}{z}_{ph}^{\alpha }\left(t_f\right)\left|p{h}^{\prime }\right.⟩+\sum _{p^{\prime }\in \mathrm{B}}{z}_{p^{\prime }{h}^{\prime }}^{\beta }\left(t_f\right)\left|h{p}^{\prime }\right.⟩\hfill \\ \hfill & +\sum _{p\in \mathrm{A},p^{\prime }\in \mathrm{B}}{z}_{ph}^{\alpha }\left(t_f\right)z_{p^{\prime }{h}^{\prime }}^{\beta }\left(t_f\right)\left|p{p}^{\prime }\right.⟩\hfill \\ \hfill & +\sum _{p\in \mathrm{B},p^{\prime }\in \mathrm{A}}{z}_{ph}^{\alpha }\left(t_f\right)z_{p^{\prime }{h}^{\prime }}^{\beta }\left(t_f\right)\left|p{p}^{\prime }\right.⟩\hfill \\ \hfill & +\sum _{p\in \mathrm{B},p^{\prime }\in \mathrm{B}}{z}_{ph}^{\alpha }\left(t_f\right)z_{p^{\prime }{h}^{\prime }}^{\beta }\left(t_f\right)\left|p{p}^{\prime }\right.⟩,h,h^{\prime }\in \mathrm{A}\cdot \hfill \end{array}$$ (17)

Formally, the above expression contains the full-CI HF states up to double-excitations within the 1α1β model. Specifically, the first term accounts for the zero-ET and zero-EEs between/in A and B; the second through fourth terms account for the one- and two-EEs within target A, respectively; the fifth through eighth terms account for all the one-ETs from A to B with (seventh and eighth) and without (fifth and sixth) concurrent one-EES within target A; and finally, the last term accounts for the two-ETs from A to B. We will utilize this analysis of the ET and EE components of $\left|{\mathbf{z}}^{1\alpha 1\beta }\left(t_f\right)\right.⟩$ to determine their probabilities in Sec. III D.

C. Charge density and number of electrons per fragment in the 1α1β model

At this point, we need to obtain expressions for the charge density and numbers of electrons per fragment from ${\mathrm{\Psi }}_e^{1\alpha 1\beta }\left(\mathbf{x};\mathbf{z}\right)$ as functions of the Thouless parameters z; these properties are useful to describe ETs in the 1α1β model. Since ${\mathrm{\Psi }}_e^{1\alpha 1\beta }\left(\mathbf{x};\mathbf{z}\right)$ is a Slater determinant state, Eq. (16), its total charge density ${\rho }^T\left(\mathbf{r};{\mathbf{z}}^{*},\mathbf{z}\right)$ is the average of the one-electron density operator $\widehat{\rho }\left(\mathbf{r}\right)={\sum }_{i=1}^{N^{\alpha }+N^{\beta }}\delta \left({\mathbf{r}}_i-\mathbf{r}\right)$ in ${\mathrm{\Psi }}_e^{1\alpha 1\beta }\left(\mathbf{x};\mathbf{z}\right)$ that can be evaluated with the Slater–Condon rules,²³

$$\begin{array}{ll}\hfill {\rho }^T\left(\mathbf{r};{\mathbf{z}}^{*},\mathbf{z}\right)& =⟨{\mathrm{\Psi }}_e^{1\alpha 1\beta }\left(\mathbf{x};\mathbf{z}\right)\left|\widehat{\rho }(\mathbf{r})\right|{\mathrm{\Psi }}_e^{1\alpha 1\beta }\left(\mathbf{x};\mathbf{z}\right)⟩\hfill \\ \hfill & =⟨{\mathrm{\Psi }}_e^{1\alpha 1\beta }\left(\mathbf{x};\mathbf{z}\right)\left|\sum _{i=1}^{N^{\alpha }+N^{\beta }=2}\delta \left({\mathbf{r}}_i-\mathbf{r}\right)\right|{\mathrm{\Psi }}_e^{1\alpha 1\beta }\left(\mathbf{x};\mathbf{z}\right)⟩\hfill \\ \hfill & =⟨{\overline{\chi }}_h^{\alpha }(\mathbf{x};{\mathbf{z}}^{\alpha })\left|\delta \left({\mathbf{r}}_1-\mathbf{r}\right)\right|{\overline{\chi }}_h^{\alpha }(\mathbf{x};{\mathbf{z}}^{\alpha })⟩\hfill \\ \hfill & +⟨{\overline{\chi }}_{h^{\prime }}^{\beta }(\mathbf{x};{\mathbf{z}}^{\beta })\left|\delta \left({\mathbf{r}}_2-\mathbf{r}\right)\right|{\overline{\chi }}_{h^{\prime }}^{\beta }(\mathbf{x};{\mathbf{z}}^{\beta })⟩\hfill \\ \hfill & =\sum _{\sigma =\alpha ,\beta }{\rho }^{\sigma }(\mathbf{r};{\mathbf{z}}^{{\sigma }^{*}},{\mathbf{z}}^{\sigma }),\hfill \end{array}$$ (18)

where the α and β charge densities ${\rho }^{\sigma }(\mathbf{r};{\mathbf{z}}^{{\sigma }^{*}},{\mathbf{z}}^{\sigma })$ are

$$\begin{array}{ll}\hfill {\rho }^{\sigma }(\mathbf{r};{\mathbf{z}}^{\sigma *},{\mathbf{z}}^{\sigma })& =⟨{\overline{\chi }}_h^{\sigma }({\mathbf{r}}_i,\omega ;{\mathbf{z}}^{\sigma })\left|\delta \left({\mathbf{r}}_i-\mathbf{r}\right)\right|{\overline{\chi }}_h^{\sigma }({\mathbf{r}}_i,\omega ;{\mathbf{z}}^{\sigma })⟩\hfill \\ \hfill & ={\left[f_h^{\sigma }\left({\mathbf{z}}^{\sigma },{\mathbf{z}}^{{\sigma }^{*}}\right)\right]}^{-1}\left({\tilde{\psi }}_h^{{\sigma }^{*}}(\mathbf{r}){\tilde{\psi }}_h^{\sigma }(\mathbf{r})\right.\hfill \\ \hfill & +\sum _p{z_{ph}^{\sigma }}^{*}{\tilde{\psi }}_p^{*\sigma }(\mathbf{r}){\tilde{\psi }}_h^{\sigma }(\mathbf{r})\sum _q{z}_{qh}^{\sigma }{\tilde{\psi }}_h^{{\sigma }^{*}}(\mathbf{r}){\tilde{\psi }}_q^{\sigma }(\mathbf{r})\hfill \\ \hfill & \left.+\sum _{p,q}{z_{ph}^{\sigma }}^{*}{z}_{qh}^{\sigma }{\tilde{\psi }}_p^{{\sigma }^{*}}(\mathbf{r}){\tilde{\psi }}_q^{\sigma }(\mathbf{r})\right)\hfill \end{array}$$ (19)

(cf. the Appendix for the mathematical details to prove the above equation). Equation (19) expresses ρ^σ(r; z^σ∗, z^σ) in the spatial molecular orbital basis, ${\tilde{\psi }}_h^{\sigma }(\mathbf{r})$ and ${\tilde{\psi }}_p^{\sigma }(\mathbf{r})$. We can express ${\rho }^{\sigma }\left(\mathbf{r};{\mathbf{z}}^{\sigma *},{\mathbf{z}}^{\sigma }\right)$ in the atomic orbital basis, $\left\{\left.{\varphi }_{\mu }\left(\mathbf{r}\right)\right|\mu =1,\dots ,K\right\}$, by expanding ${\tilde{\psi }}_h^{\sigma }(\mathbf{r})$ and ${\tilde{\psi }}_p^{\sigma }(\mathbf{r})$ in terms of their atomic counterparts, ${\tilde{\psi }}_i^{\sigma }\left(\mathbf{r}\right)={\sum }_{\mu =1}^K{C}_{\mu i}^{\sigma }{\varphi }_{\mu }\left(\mathbf{r}\right)$,

$$\begin{array}{ll}\hfill {\rho }^{\sigma }(\mathbf{r};{\mathbf{z}}^{\sigma *},{\mathbf{z}}^{\sigma })& ={\left[f_h^{\sigma }\left({\mathbf{z}}^{\sigma },{\mathbf{z}}^{{\sigma }^{*}}\right)\right]}^{-1}\sum _{\nu ,\mu }\left(C_{\nu h}^{{\sigma }^{*}}{C}_{\mu h}^{\sigma }+\sum _p{z_{ph}^{\sigma }}^{*}{C}_{\nu p}^{{\sigma }^{*}}{C}_{\mu h}^{\sigma }\right.\hfill \\ \hfill & \left.+\sum _q{z}_{qh}^{\sigma }{C}_{\nu h}^{{\sigma }^{*}}{C}_{\mu q}^{\sigma }+\sum _{p,q}{z_{ph}^{\sigma }}^{*}{z}_{qh}^{\sigma }{C}_{\nu p}^{{\sigma }^{*}}{C}_{\mu q}^{\sigma }\right){\varphi }_{\nu }^{*}\left(\mathbf{r}\right){\varphi }_{\mu }\left(\mathbf{r}\right).\hfill \end{array}$$ (20)

Integration of ρ^σ(r; z^σ∗, z^σ) with respect to the electron position r leads to the total numbers of σ electrons N^σ over the whole system that is equal to 1 for σ = α and β in the 1α1β model,

$$\begin{array}{ll}\hfill N^{\sigma }& =\int {\rho }^{\sigma }(\mathbf{r};{\mathbf{z}}^{\sigma *}{,\mathbf{z}}^{\sigma })d\mathbf{r}\hfill \\ \hfill & ={\left[f_h^{\sigma }\left({\mathbf{z}}^{\sigma },{\mathbf{z}}^{{\sigma }^{*}}\right)\right]}^{-1}\sum _{\nu ,\mu }\left(C_{\nu h}^{{\sigma }^{*}}{C}_{\mu h}^{\sigma }+\sum _p{z_{ph}^{\sigma }}^{*}{C}_{\nu p}^{{\sigma }^{*}}{C}_{\mu h}^{\sigma }\right.\hfill \\ \hfill & +\left.\sum _q{z}_{qh}^{\sigma }{C}_{\nu h}^{{\sigma }^{*}}{C}_{\mu q}^{\sigma }+\sum _{p,q}{z_{ph}^{\sigma }}^{*}{z}_{qh}^{\sigma }{C}_{\nu p}^{{\sigma }^{*}}{C}_{\mu q}^{\sigma }\right)S_{\nu \mu }\hfill \\ \hfill & ={\left(1+\sum _p{\left|z_{ph}^{\sigma }\right|}^2\right)}^{-1}\left(1+\sum _p{\left|z_{ph}^{\sigma }\right|}^2\right)=1\hfill \end{array}$$ (21)

(cf. the Appendix for the mathematical details to prove the above equation). N^σ in Eq. (21) can be written in the matrix form as²³

$$N^{\sigma }=\sum _{\nu ,\mu }{P}_{\mu \nu }^{\sigma }{S}_{\nu \mu }=\sum _{\mu }{\left({\mathbf{P}}^{\sigma }\mathbf{S}\right)}_{\mu \mu }=\mathrm{Tr}\left({\mathbf{P}}^{\sigma }\mathbf{S}\right),$$ (22)

where S_νμ are the atomic overlap matrix elements and $P_{\mu \nu }^{\sigma }$ are the density matrix elements

$$\begin{array}{ll}\hfill P_{\mu \nu }^{\sigma }& ={\left[f_h^{\sigma }\left({\mathbf{z}}^{\sigma },{\mathbf{z}}^{{\sigma }^{*}}\right)\right]}^{-1}\left(C_{\nu h}^{{\sigma }^{*}}{C}_{\mu h}^{\sigma }+\sum _p{z_{ph}^{\sigma }}^{*}{C}_{\nu p}^{{\sigma }^{*}}{C}_{\mu h}^{\sigma }\right.\hfill \\ \hfill & +\left.\sum _q{z}_{qh}^{\sigma }{C}_{\nu h}^{{\sigma }^{*}}{C}_{\mu q}^{\sigma }+\sum _{p,q}{z_{ph}^{\sigma }}^{*}{z}_{qh}^{\sigma }{C}_{\nu p}^{{\sigma }^{*}}{C}_{\mu q}^{\sigma }\right).\hfill \end{array}$$ (23)

Note that if z = 0, then $f_h^{\sigma }\left({\mathbf{z}}^{\sigma },{\mathbf{z}}^{{\sigma }^{*}}\right)=1+{\sum }_{p=N^{\sigma }+1}^K{\left|z_{ph}^{\sigma }\right|}^2$ = 1 and the above $P_{\mu \nu }^{\sigma }$ become the standard UHF density matrix elements $P_{\mu \nu }^{\sigma }=C_{\nu h}^{\sigma }{C}_{\mu h}^{\sigma }$.²³ This is consistent with ${\left|\mathbf{z}\right.⟩}^{1\alpha 1\beta }$ in Eq. (17) because that equation shows that ${\left|\mathbf{z}=\mathbf{0}\right.⟩}^{1\alpha 1\beta }=\left|h{h}^{\prime }\right.⟩$, where $\left|h{h}^{\prime }\right.⟩$ is a standard UHF Slater determinant state.

In order to describe ETs between fragments A and B, we should rearrange the atomic orbital summations of N^σ in Eq. (22) in terms of fragment contributions,

$$\begin{array}{ll}\hfill N^{\sigma }& =\sum _{\nu ,\mu }{P}_{\mu \nu }^{\sigma }{S}_{\nu \mu }\hfill \\ \hfill & =\sum _{\nu \in \mathrm{A},\mu \in \mathrm{A}}{P}_{\mu \nu }^{\sigma }{S}_{\nu \mu }+\sum _{\nu \in \mathrm{A},\mu \in \mathrm{B}}{P}_{\mu \nu }^{\sigma }{S}_{\nu \mu }+\sum _{\nu \in \mathrm{B},\mu \in A}{P}_{\mu \nu }^{\sigma }{S}_{\nu \mu }+\sum _{\nu \in \mathrm{B},\mu \in \mathrm{B}}{P}_{\mu \nu }^{\sigma }{S}_{\nu \mu }.\hfill \end{array}$$ (24)

The above equation is valid throughout the whole course of the A + B reaction. We are interested in applying Eq. (24) in the pre- and post-collision situations to characterize the initial and final states corresponding to ETs. In those situations, Eq. (24) simplifies. First, at the initial and final times, fragments A and B are at a long separation, R_AB → ∞, where the atomic overlap matrix elements between fragments become null: $S_{\nu \mu }\left(R_{AB}\to \infty \right)=0$ for ν ∈ A/B and μ ∈ B/A. At those times, Eq. (24) becomes

$$\begin{array}{ll}\hfill N^{\sigma }& =\sum _{\nu \in \mathrm{A},\mu \in \mathrm{A}}{P}_{\mu \nu }^{\sigma }{S}_{\nu \mu }+\sum _{\nu \in \mathrm{B},\mu \in \mathrm{B}}{P}_{\mu \nu }^{\sigma }{S}_{\nu \mu }\hfill \\ \hfill & =N_A^{\sigma }+N_B^{\sigma }\hfill \\ \hfill & =\sum _{F=\mathrm{A},\mathrm{B}}{N}_F^{\sigma }\left(R_{AB}\to \infty \right),\hfill \end{array}$$ (25)

where $N_F^{\sigma }$ is the number of electrons in fragment F = A or B,

$$\begin{array}{ll}\hfill N_F^{\sigma }& =\int {\rho }_F^{\sigma }(\mathbf{r};{\mathbf{z}}^{\sigma *},{\mathbf{z}}^{\sigma })d\mathbf{r}=\sum _{\nu \in F,\mu \in F}{P}_{\mu \nu }^{\sigma }{S}_{\nu \mu }\hfill \\ \hfill & ={\left[f_h^{\sigma }\left({\mathbf{z}}^{\sigma },{\mathbf{z}}^{{\sigma }^{*}}\right)\right]}^{-1}\sum _{\nu \in F,\mu \in F}\left(C_{\nu h}^{{\sigma }^{*}}{C}_{\mu h}+\sum _{p,q\in A,B}{z_{ph}^{\sigma }}^{*}{z}_{qh}^{\sigma }{C}_{\nu p}^{{\sigma }^{*}}{C}_{\mu q}^{\sigma }\right)\hfill \\ \hfill & \times S_{\nu \mu }\left(h\in A,R_{AB}\to \infty \right)\hfill \end{array}$$ (26)

(cf. the Appendix for the mathematical details to prove the above equation). Proper fragment separations at the initial and final times imply that each spatial molecular orbital ${\tilde{\psi }}_i\left(\mathit{r}\right)$ localizes on fragment A or B as ${\tilde{\psi }}_{i\in A}\left(\mathit{r}\right)$ or ${\tilde{\psi }}_{i\in B}\left(\mathit{r}\right)$; these orbitals have null molecular coefficients C_μ∈A,i∈B and C_μ∈B,i∈A from atomic orbitals on the opposite fragment,

$$\begin{array}{c}{C}_{\mu \in A,i\in B}=⟨\left.{\varphi }_{\mu \in A}\left(\mathit{r}\right)\right|{\tilde{\psi }}_{i\in B}\left(\mathit{r}\right)⟩=0,\\ C_{\mu \in B,i\in A}=⟨\left.{\varphi }_{\mu \in B}\left(\mathit{r}\right)\right|{\tilde{\psi }}_{i\in A}\left(\mathit{r}\right)⟩=0\left(R_{AB}\to \infty \right).\end{array}$$ (27)

Application of the above condition into Eq. (26) leads to

$$\begin{array}{ll}\hfill N_F^{\sigma }& ={\left[f_h^{\sigma }\left({\mathbf{z}}^{\sigma },{\mathbf{z}}^{{\sigma }^{*}}\right)\right]}^{-1}\sum _{\nu \in F,\mu \in F}\left(C_{\nu h}^{{\sigma }^{*}}{C}_{\mu h}+\sum _{p,q\in A,B}{z_{ph}^{\sigma }}^{*}{z}_{qh}^{\sigma }{C}_{\nu p}^{{\sigma }^{*}}{C}_{\mu q}^{\sigma }\right)S_{\nu \mu }\hfill \\ \hfill & ={\left[f_h^{\sigma }\left({\mathbf{z}}^{\sigma },{\mathbf{z}}^{{\sigma }^{*}}\right)\right]}^{-1}\left({\delta }_{AF}{\delta }_{hh}+\sum _{p,q\in F}{z_{ph}^{\sigma }}^{*}{z}_{qh}^{\sigma }{\delta }_{pq}\right)\hfill \\ \hfill & =\frac{{\delta }_{AF}+\sum _{p\in F}{\left|z_{pi}^{\sigma }\right|}^2}{1+{\sum }_p{\left|z_{ph}^{\sigma }\right|}^2}\left(h\in A,R_{AB}\to \infty \right)\hfill \end{array}$$ (28)

(cf. the Appendix for the mathematical details to prove the above equation). Equation (28) specialized for fragments A and B is given as

$$N_A^{\sigma }=\frac{1+\sum _{p\epsilon A}{\left|z_{ph}^{\sigma }\right|}^2}{1+{\sum }_p{\left|z_{ph}^{\sigma }\right|}^2},N_B^{\sigma }=\frac{\sum _{p\in B}{\left|z_{ph}^{\sigma }\right|}^2}{1+{\sum }_p{\left|z_{ph}^{\sigma }\right|}^2}\Rightarrow N_A^{\sigma }+N_B^{\sigma }=1,$$ (29)

where $N_A^{\sigma }$ and $N_B^{\sigma }$ correctly add up to the total number of electrons with σ spin in the 1α1β model. By tracing back the numerators terms 1 = δ_AA and $\sum _{p\epsilon A}{\left|z_{ph}^{\sigma }\right|}^2{N}_A^{\sigma }$ in the detailed derivation of Eq. (28), Eq. (A5), we find that they correspond to contributions from hole (occupied) and particle (virtual) MSOs, respectively. We can therefore write

$$N_A^{\sigma }=N_A^{\sigma ,\cdot }+N_A^{\sigma ,◦},N_A^{\sigma ,\cdot }=\frac{1}{1+{\sum }_p{\left|z_{ph}^{\sigma }\right|}^2},N_A^{\sigma ,◦}=\frac{{\sum }_{p\in A}{\left|z_{ph}^{\sigma }\right|}^2}{1+{\sum }_p{\left|z_{ph}^{\sigma }\right|}^2},$$ (30)

where $N_A^{\sigma ,\cdot }$ and $N_A^{\sigma ,◦}$ are the number of σ electrons in fragment A located in its hole and particle MSOs, respectively. We indicate those particle and hole contributions with the superscripts ⋅ and ◦ on $N_A^{\sigma ,\cdot }$ and $N_A^{\sigma ,◦}$, respectively. A similar analysis on $N_B^{\sigma ,◦}$ provides

$$N_B^{\sigma }=N_B^{\sigma ,◦}=\frac{\sum _{p\in B}{\left|z_{ph}^{\sigma }\right|}^2}{1+{\sum }_p{\left|z_{ph}^{\sigma }\right|}^2},$$ (31)

where the numerator term ${\sum }_{p\in B}{\left|z_{ph}^{\sigma }\right|}^2$ corresponds to contributions from particle MSOs; for that reason, we simply write $N_B^{\sigma }=N_B^{\sigma ,◦}$.

Before finishing this section, we would like to emphasize that the numbers of electrons of well-separated fragments A and B, $N_A^{\sigma }$ and $N_B^{\sigma }$ in Eqs. (30) and (31), are justified on physical grounds because from Eq. (26), we have that

$$N_A^{\sigma }=\int {\rho }_A^{\sigma }(\mathbf{r};{\mathbf{z}}^{\sigma *},{\mathbf{z}}^{\sigma })d\mathbf{r};N_B^{\sigma }=\int {\rho }_B^{\sigma }(\mathbf{r};{\mathbf{z}}^{\sigma *},{\mathbf{z}}^{\sigma })d\mathbf{r}\left(R_{AB}\to \infty \right),$$ (32)

i.e., $N_A^{\sigma }$ and $N_B^{\sigma }$ come from the integration of the well-separated electron densities on A and B. We can get further insight into $N_A^{\sigma }$, $N_A^{\sigma ,\cdot }$, $N_A^{\sigma ,◦}$, $N_B^{\sigma }$, and $N_B^{\sigma ,◦}$ in Eqs. (30) and (31) by analyzing their dependencies on the Thouless parameters z. If z^σ = 0, $N_A^{\sigma }$ = $N_A^{\sigma ,\cdot }$ = 1 and $N_A^{\sigma ,◦}$ = $N_B^{\sigma }$ = $N_B^{\sigma ,◦}$ = 0; this is consistent with the 1α1β model pre-collision condition with ${\mathbf{z}}^{\sigma }\left(t_i\right)=\mathbf{0}$ where each σ electron occupies a hole MSO of fragment A; this also corresponds to a post-collision situation with ${\mathbf{z}}^{\sigma }\left(t_f\right)=\mathbf{0}$ where neither ET no EE occurs between fragments: P_0-ET = 1, P_n-ET = 0 for n = 1, 2, and the electrons remain on the hole MSOs of A. On the other hand, if $\left|{\mathbf{z}}^{\sigma }\left(t_f\right)\right|\to \infty$, $N_A^{\sigma ,\cdot }$ = 0 and $N_A^{\sigma ,◦}$ + $N_B^{\sigma ,◦}$ = 1; this corresponds to a post-collision situation where each electron leaves its hole MSO on fragment A and ends up either excited on A or transferred to B. Situations with $0<\left|{\mathbf{z}}^{\sigma }\left(t_f\right)\right|<\infty$ correspond to cases with zero-, one-, and two-ETs from A to B with concurrent EEs at varying probabilities P_n-ET.

D. ET probabilities in the 1α1β model

We can now derive the expressions of the n-ET probabilities P_n-ET, 0 ≤ n ≤ 2, in the 1α1β model in terms of the numbers of electrons on the fragments. As discussed in Sec. III B, the initial un-normalized electronic wavefunction is ${\left|\mathbf{z}\left(t_i\right)=\mathbf{0}\right.⟩}^{1\alpha 1\beta }=\left|h{h}^{\prime }\right.⟩$, i.e., the reference HF state; cf. Eq. (17). The final electronic wavefunction ${\left|\mathbf{z}\left(t_f\right)\right.⟩}^{1\alpha 1\beta }$ is in terms of the reference $\left|h{h}^{\prime }\right.⟩$ and all the excited states $\left|h\to p\mathrm{a}\mathrm{n}\mathrm{d}/\mathrm{o}\mathrm{r}{h}^{\prime }\to p^{\prime }\right.⟩$, as shown in Eq. (17). Then, from that equation, we can obtain the state-to-state probabilities $P\left(\left|h{h}^{\prime }\right.⟩\to \left|ij\right.⟩\right)$ of the transitions from the initial $\left|h{h}^{\prime }\right.⟩$ to the final states $\left|ij\right.⟩$ as

$$P\left(\left|h{h}^{\prime }\right.⟩\to \left|ij\right.⟩\right)=\frac{{\left|⟨ij{\left|\mathbf{z}\left(t_f\right)\right.⟩}^{1\alpha 1\beta }\right|}^2}{{⟨\mathbf{z}\left(t_f\right)\left|\right.\mathbf{z}\left(t_f\right)⟩}^{1\alpha 1\beta }},i,j=h,h^{\prime },p,p^{\prime },$$ (33)

where $⟨ij{\left|\mathbf{z}\left(t_f\right)\right.⟩}^{1\alpha 1\beta }$ is the projection of ${\left|\mathbf{z}\left(t_t\right)\right.⟩}^{1\alpha 1\beta }$ onto the state $\left|ij\right.⟩$ and ${\left({⟨\mathbf{z}\left(t_f\right)\left|\right.\mathbf{z}\left(t_f\right)⟩}^{1\alpha 1\beta }\right)}^{+1/2}$ is the normalization factor of ${\left|\mathbf{z}\left(t_t\right)\right.⟩}^{1\alpha 1\beta }$, Eq. (15). The hole MSOs ${\psi }_h^{\sigma }$ and ${\psi }_{h^{\prime }}^{\sigma }$ are on fragment A, while the particle MSOs ${\psi }_p^{\sigma }$ and ${\psi }_{p^{\prime }}^{\sigma }$ can be on fragment A or B. Each electron in the state $\left|ij\right.⟩$ can be assigned to fragment A or B according to the location of its MSO ψ_i or ψ_j. Then, the n-ET probability P_n-ET, 0 ≤ n ≤ 2, is the sum of all the state-to-state probabilities $P\left(\left|h{h}^{\prime }\right.⟩\to \left|ij\right.⟩\right)$ compatible with a final electron localization ${\mathrm{A}}^{+n}\left[\left(2-n\right)e^{-}\right]+{\mathrm{B}}^{-n}\left(n{e}^{-}\right)$. For instance, P_0-ET is given as

$$P_{0-ET}=P_{0-ET}^0+P_{0-ET}^X,$$ (34)

where $P_{0-ET}^0$ is the probability that both electrons end up unexcited in the hole MSOs ψ_h and ${\psi }_{h^{\prime }}$ ∈ A and $P_{0-ET}^X$ is the probability that one or both electrons end up excited in the particle MSOs ψ_p ∈ A. Specifically, $P_{0-ET}^0$ is given as

$$\begin{array}{ll}\hfill P_{0-ET}^0& =P\left(\left|h{h}^{\prime }\right.⟩\to \left|h{h}^{\prime }\right.⟩\right)\hfill \\ \hfill & =\frac{1}{\left(1+\sum _{p=2}^K{\left|z_{ph}^{\alpha }\right|}^2\right)\left(1+\sum _{p=2}^K{\left|z_{p^{\prime }{h}^{\prime }}^{\beta }\right|}^2\right)}\hfill \\ \hfill & =N_A^{\alpha ,\cdot }{N}_A^{\beta ,\cdot },h\in A,h^{\prime }\in A,\hfill \end{array}$$ (35)

where $N_A^{\alpha ,\cdot }$ and $N_A^{\beta ,\cdot }$ from Eq. (30) were introduced in the last step. Similarly, $P_{0-ET}^X$ is given as

$$\begin{array}{ll}\hfill P_{0-ET}^X& =\sum _{p\in A}P\left(\left|h{h}^{\prime }\right.⟩\to \left|h{p}^{\prime }\right.⟩\right)+\sum _{p^{\prime }\in A}P\left(\left|h{h}^{\prime }\right.⟩\to \left|p{h}^{\prime }\right.⟩\right)\hfill \\ \hfill & +\sum _{p\in A,p^{\prime }\in A}P\left(\left|h{h}^{\prime }\right.⟩\to \left|p{p}^{\prime }\right.⟩\right)\left(h,h^{\prime }\in A\right)\hfill \\ \hfill & =\frac{\sum _{p\in A}{\left|z_{ph}^{\alpha }\right|}^2+\sum _{p^{\prime }\in A}{\left|z_{p^{\prime }{h}^{\prime }}^{\beta }\right|}^2+\sum _{p\in A,p^{\prime }\in A}{\left|z_{ph}^{\alpha }\right|}^2{\left|z_{p^{\prime }{h}^{\prime }}^{\beta }\right|}^2}{\left(1+\sum _p{\left|z_{ph}^{\alpha }\right|}^2\right)\left(1+\sum _{p^{\prime }}{\left|z_{p^{\prime }{h}^{\prime }}^{\beta }\right|}^2\right)}\hfill \\ \hfill & =\frac{\sum _{p\in A}{\left|z_{ph}^{\alpha }\right|}^2}{\left(1+\sum _p{\left|z_{ph}^{\alpha }\right|}^2\right)}\frac{1}{\left(1+\sum _{p^{\prime }}{\left|z_{p^{\prime }{h}^{\prime }}^{\beta }\right|}^2\right)}+\frac{1}{\left(1+\sum _p{\left|z_{ph}^{\alpha }\right|}^2\right)}\hfill \\ \hfill & \times \frac{\sum _{p^{\prime }\in A}{\left|z_{p^{\prime }{h}^{\prime }}^{\beta }\right|}^2}{\left(1+\sum _{p^{\prime }}{\left|z_{p^{\prime }{h}^{\prime }}^{\beta }\right|}^2\right)}+\frac{\sum _{p\in A}{\left|z_{ph}^{\alpha }\right|}^2}{\left(1+\sum _p{\left|z_{ph}^{\alpha }\right|}^2\right)}\frac{\sum _{p^{\prime }\in A}{\left|z_{p^{\prime }{h}^{\prime }}^{\beta }\right|}^2}{\left(1+\sum _{p^{\prime }}{\left|z_{p^{\prime }{h}^{\prime }}^{\beta }\right|}^2\right)}\hfill \\ \hfill & =N_A^{\alpha ,◦}{N}_A^{\beta ,\cdot }+N_A^{\alpha ,\cdot }{N}_A^{\beta ,◦}+N_A^{\alpha ,◦}{N}_A^{\beta ,◦},\hfill \end{array}$$ (36)

where $N_A^{\alpha ,◦}$, $N_A^{\beta ,\cdot }$, etc., from Eq. (30) were introduced in the last line. By setting $P_{0-ET}^0$ and $P_{0-ET}^X$ from Eqs. (35) and (36) into Eq. (34), we finally obtain P_0-ET as

$$P_{0-ET}=N_A^{\alpha ,\cdot }{N}_A^{\beta ,\cdot }+N_A^{\alpha ,◦}{N}_A^{\beta ,\cdot }+N_A^{\alpha ,\cdot }{N}_A^{\beta ,◦}+N_A^{\alpha ,◦}{N}_A^{\beta ,◦}=N_A^{\alpha }{N}_A^{\beta },$$ (37)

where Eq. (30) was used to obtain the last line. Similarly, P_1-ET is given as

$$\begin{array}{ll}\hfill P_{1-ET}& =\sum _{p\in B}P\left({\left|h{h}^{\prime }\right.⟩}_{\to \left|h{p}^{\prime }\right.⟩}\right)+\sum _{p^{\prime }\in B}P\left(\left|h{h}^{\prime }\right.⟩\to \left|p{h}^{\prime }\right.⟩\right)+\sum _{p\in A,p^{\prime }\in B}P\left(\left|h{h}^{\prime }\right.⟩\to \left|p{p}^{\prime }\right.⟩\right)+\sum _{p\in B,p^{\prime }\in A}P\left(\left|h{h}^{\prime }\right.⟩\to \left|p{p}^{\prime }\right.⟩\right)\left(h,h^{\prime }\in A\right)\hfill \\ \hfill & =\frac{\sum _{p\in B}{\left|z_{ph}^{\alpha }\right|}^2+\sum _{p^{\prime }\in B}{\left|z_{p^{\prime }{h}^{\prime }}^{\beta }\right|}^2+\sum _{p\in A,p^{\prime }\in B}{\left|z_{ph}^{\alpha }\right|}^2{\left|z_{p^{\prime }{h}^{\prime }}^{\beta }\right|}^2+\sum _{p\in B,p^{\prime }\in A}{\left|z_{ph}^{\alpha }\right|}^2{\left|z_{p^{\prime }{h}^{\prime }}^{\beta }\right|}^2}{\left(1+\sum _p{\left|z_{ph}^{\alpha }\right|}^2\right)\left(1+\sum _{p^{\prime }}{\left|z_{p^{\prime }{h}^{\prime }}^{\beta }\right|}^2\right)}\hfill \\ \hfill & =N_B^{\alpha ,◦}{N}_A^{\beta ,\cdot }+N_A^{\alpha ,\cdot }{N}_B^{\beta ,◦}+N_A^{\alpha ,◦}{N}_B^{\beta ,◦}+N_B^{\alpha ,◦}{N}_A^{\beta ,◦}=N_A^{\alpha }{N}_B^{\beta }+N_A^{\beta }{N}_B^{\alpha },\hfill \end{array}$$ (38)

and P_2-ET is given as

$$\begin{array}{ll}\hfill P_{2-ET}& =P\left(\left|h{h}^{\prime }\right.⟩\to \left|p{p}^{\prime }\right.⟩\right)\hfill \\ \hfill & =\frac{\sum _{p\in B,p^{\prime }\in B}{\left|z_{ph}^{\alpha }\right|}^2{\left|z_{p^{\prime }{h}^{\prime }}^{\beta }\right|}^2}{\left(1+\sum _p{\left|z_{ph}^{\alpha }\right|}^2\right)\left(1+\sum _{p^{\prime }}{\left|z_{p^{\prime }{h}^{\prime }}^{\beta }\right|}^2\right)}\hfill \\ \hfill & =N_B^{\alpha ,◦}{N}_B^{\beta ,◦}=N_B^{\alpha }{N}_B^{\beta }.\hfill \end{array}$$ (39)

As required, the sum of all the n-ET probabilities P_n-ET is equal to one because

$$\begin{array}{c}{P}_{0-ET}+P_{1-ET}+P_{2-ET}=N_A^{\alpha }{N}_A^{\beta }+N_A^{\alpha }{N}_B^{\beta }+N_A^{\beta }{N}_B^{\alpha }+N_B^{\alpha }{N}_B^{\beta }\\ =N_A^{\alpha }\left(N_A^{\beta }+N_B^{\beta }\right)+N_B^{\alpha }\left(N_A^{\beta }+N_B^{\beta }\right)\\ =N_A^{\alpha }+N_B^{\alpha }=1,\end{array}$$ (40)

where the total number σ of electrons equal to 1 in the 1α1β model was applied three times. The expressions of P_n-ET in Eqs. (37)–(39) can be rewritten in terms of the number of electrons on the projectile B via $N_A^{\sigma }=1-N_B^{\sigma }$ as

$$\begin{array}{c}{P}_{0-ET}=\left(1-N_B^{\alpha }\right)\left(1-N_B^{\beta }\right),\\ P_{1-ET}=N_B^{\alpha }\left(1-N_B^{\beta }\right)+\left(1-N_B^{\alpha }\right)N_B^{\beta },P_{2-ET}=N_B^{\alpha }{N}_B^{\beta }.\end{array}$$ (41)

If we consider projectile B as an outgoing H⁺ projectile, then we can set $N_B^{\alpha /\beta }=N_{H^q}^{\alpha /\beta }$ and the above P_n-ET expressions become identical to those in Eq. (3) inferred from statistical arguments. This completes our quantum-mechanical derivation of the P_n-ET formulas for the 1α1β model system under SLEND dynamics.

E. Generalization of the 1α1β model derivation to the 1α1β/core model

We will now generalize the 1α1β model for two-electron systems to the 1α1β/core model for N-electron systems, N > 2, where n = 2 electrons with α and β spins, respectively, are active for ETs and EEs, while the remaining N − 2 electrons form a core inactive for those processes [cf. Sec. III A and Eq. (10)]. To describe target-to-projectile ETs, the n = 2 active electrons are located at the initial time on target A, while the remaining N − 2 core electrons are distributed over target A and projectile B in any proportion; cf. Eq. (10). The Thouless state ${\left|\mathbf{z}\right.⟩}^{1\alpha 1\beta /core}$ of the 1α1β/core model is given as

$$\begin{array}{ll}\hfill {\left|\mathbf{z}\right.⟩}^{1\alpha 1\beta /core}& =\left|\left[core\right]{\chi }_h^{\alpha }{\chi }_{h^{\prime }}^{\beta }\right.⟩\hfill \\ \hfill & =\left(1+\sum _{p=N+1}^K{z}_{ph}^{\alpha }{b}_p^{†}{b}_h\right)\left(1+\sum _{p^{\prime }=N+1}^K{z}_{p^{\prime }{h}^{\prime }}^{\beta }{b}_{p^{\prime }}^{†}{b}_{h^{\prime }}\right)\left|\left[core\right]{\psi }_h{\psi }_{h^{\prime }}\right.⟩\hfill \\ \hfill & =\left|\left[core\right]h{h}^{\prime }\right.⟩+\sum _{p\in A}{z}_{ph}^{\alpha }\left|\left[core\right]p{h}^{\prime }\right.⟩+\sum _{p^{\prime }\in A}{z}_{p^{\prime }{h}^{\prime }}^{\beta }\left|\left[core\right]h{p}^{\prime }\right.⟩+\sum _{p\in A,p^{\prime }\in A}{z}_{ph}^{\alpha }{z}_{p^{\prime }{h}^{\prime }}^{\beta }\left|\left[core\right]p{p}^{\prime }\right.⟩+\sum _{p\in B}{z}_{ph}^{\alpha }\left|\left[core\right]p{h}^{\prime }\right.⟩+\sum _{p^{\prime }\in B}{z}_{p^{\prime }{h}^{\prime }}^{\beta }\left|\left[core\right]h{p}^{\prime }\right.⟩\hfill \\ \hfill & +\sum _{p\in A,p^{\prime }\in B}{z}_{ph}^{\alpha }{z}_{p^{\prime }{h}^{\prime }}^{\beta }\left|\left[core\right]p{p}^{\prime }\right.⟩+\sum _{p\in B,p^{\prime }\in A}{z}_{ph}^{\alpha }{z}_{p^{\prime }{h}^{\prime }}^{\beta }\left|\left[core\right]p{p}^{\prime }\right.⟩+\sum _{p\in B,p^{\prime }\in B}{z}_{ph}^{\alpha }{z}_{p^{\prime }{h}^{\prime }}^{\beta }\left|\left[core\right]p{p}^{\prime }\right.⟩,h,h^{\prime }\in A,\hfill \end{array}$$ (42)

where $\left[core\right]$ = ${\psi }_1\dots {\psi }_g\dots {\psi }_{\left(N-2\right)}$ contains the N − 2 MSOs of the core electrons and ${\chi }_h^{\alpha }$ and ${\chi }_{h^{\prime }}^{\beta }$ are the DSOs of the active electrons, Eq. (11). In Eq. (42), there are particle–hole pair operators, $b_p^{†}{b}_h$ and $b_{p^{\prime }}^{†}{b}_{h^{\prime }}$; Thouless parameters, $z_{ph}^{\alpha }$ and $z_{p^{\prime }{h}^{\prime }}^{\beta }$; and DSOs, ${\chi }_h^{\alpha }$ and ${\chi }_{h^{\prime }}^{\beta }$, only for the n = 2 active electrons because only they undergo ETs and EEs. Concomitantly, there are no particle–hole operators, Thouless coefficients, and DSOs for the core electrons in Eq. (42) because they remain in their original MSOs ψ_g∈core inactively for ETs and EEs. Note that the ${\left|\mathbf{z}\right.⟩}^{1\alpha 1\beta /core}$ expansion in Eq. (42) is equivalent term by term to the ${\left|\mathbf{z}\right.⟩}^{1\alpha 1\beta }$ expansion in Eq. (17), including the ET and EE interpretation of the analogous terms; the only relevant difference between ${\left|\mathbf{z}\right.⟩}^{1\alpha 1\beta /core}$ and ${\left|\mathbf{z}\right.⟩}^{1\alpha 1\beta }$ is the presence of $\left[core\right]$ part in all the terms of ${\left|\mathbf{z}\right.⟩}^{1\alpha 1\beta /core}$. In analogy with Eq. (16), the standard Slater determinant state $\left|{\mathrm{\Psi }}_e^{1\alpha 1\beta /core}\right.⟩$ corresponding to ${\left|\mathbf{z}\right.⟩}^{1\alpha 1\beta /core}$ is given as

$$\begin{array}{ll}\hfill {\mathrm{\Psi }}_e^{1\alpha 1\beta /core}\left(\mathbf{x};\mathbf{z}\right)& ={\left({⟨\mathbf{z}\left|\right.\mathbf{z}⟩}^{1\alpha 1\beta /core}\right)}^{-1/2}⟨\mathbf{x}{\left|\mathbf{z}\right.⟩}^{1\alpha 1\beta /core}\hfill \\ \hfill & ={\left(N!\right)}^{-1/2}\det \left[\left[core\right]{\overline{\chi }}_h^{\alpha }\left({\mathbf{x}}_1;{\mathbf{z}}^{\alpha }\right){\overline{\chi }}_{h^{\prime }}^{\beta }\left({\mathbf{x}}_2;{\mathbf{z}}^{\beta }\right)\right],\hfill \end{array}$$ (43)

where ${\overline{\chi }}_h^{\alpha }\left({\mathbf{x}}_1;{\mathbf{z}}^{\alpha }\right)$ and ${\overline{\chi }}_{h^{\prime }}^{\beta }\left({\mathbf{x}}_2;{\mathbf{z}}^{\beta }\right)$ are the normalized DSOs of the active electrons, Eq. (13). All the spin-orbitals in Eq. (43) are orthonormal, viz. $⟨{\psi }_{g\in core}\left|{\psi }_{f\in core}\right.⟩$ = δ_gf because they are regular HF MSOs, $\left.⟨{\overline{\chi }}_h^{\alpha }\right|{\overline{\chi }}_{h^{\prime }}^{\beta }⟩$ = 0 because they have opposite spin, and $⟨{\psi }_{g\in core}\left|{\overline{\chi }}_h^{\alpha /\beta }\right.⟩$ = 0 because ${\overline{\chi }}_h^{\alpha /\beta }$ contains no ψ_g∈core. By using again the Slater–Condon rules,²³ we obtain the σ charge densities ${\rho }^{\sigma }(\mathbf{r};{\mathbf{z}}^{{\sigma }^{*}},{\mathbf{z}}^{\sigma })$ of ${\mathrm{\Psi }}_e^{1\alpha 1\beta /core}\left(\mathbf{x};\mathbf{z}\right)$ as

$$\begin{array}{ll}\hfill {\rho }^{\sigma }(\mathbf{r};{\mathbf{z}}^{{\sigma }^{*}},{\mathbf{z}}^{\sigma })& =\sum _g^{N_{core}^{\sigma }}\int {\psi }_g^{{\sigma }^{*}}(\mathbf{r}){\psi }_g^{\sigma }(\mathbf{r})d\omega +\int {\overline{\chi }}_h^{{\sigma }^{*}}(\mathbf{r};{\mathbf{z}}^{{\sigma }^{*}}){\overline{\chi }}_h^{\sigma }(\mathbf{r};{\mathbf{z}}^{\sigma })d\omega \hfill \\ \hfill & ={\rho }_{core}^{\sigma }(\mathbf{r})+{\rho }_{active}^{\sigma }(\mathbf{r};{\mathbf{z}}^{\sigma *},{\mathbf{z}}^{\sigma }),\hfill \end{array}$$ (44)

where ${\rho }_{core}^{\sigma }(\mathbf{r})$ and ${\rho }_{active}^{\sigma }(\mathbf{r};{\mathbf{z}}^{\sigma *},{\mathbf{z}}^{\sigma })$ are the σ charge densities of the core and active electrons, respectively. Both densities can change due to their implicit dependency on the moving nuclear positions $\mathbf{R}\left(t\right)$; however, ${\rho }_{active}^{\sigma }(\mathbf{r};{\mathbf{z}}^{\sigma *},{\mathbf{z}}^{\sigma })$ can also change due to the evolution of the Thouless parameters ${\mathbf{z}}^{\sigma }\left(t\right)$, thus reflecting ETs and EEs. We can obtain the number of transferred electrons in the 1α1β/core model from ${\rho }^{\sigma }(\mathbf{r};{\mathbf{z}}^{{\sigma }^{*}},{\mathbf{z}}^{\sigma })$, as we did in the 1α1β model, Eqs. (21)–(31). The total number of σ electrons $N_F^{\sigma }$ in the fragment F = A or B is now given as follows:

$$N_F^{\sigma }=N_{F,core}^{\sigma }+N_{F,active}^{\sigma }\left({\mathbf{z}}^{{\sigma }^{*}},{\mathbf{z}}^{\sigma }\right),$$ (45)

where $N_{F,core}^{\sigma }$ and $N_{F,active}^{\sigma }\left({\mathbf{z}}^{{\sigma }^{*}},{\mathbf{z}}^{\sigma }\right)$ are the numbers of core and active σ electrons on the considered fragment, respectively. $N_{F,active}^{\sigma }\left({\mathbf{z}}^{{\sigma }^{*}},{\mathbf{z}}^{\sigma }\right)$ is identical to $N_F^{\sigma }\left({\mathbf{z}}^{{\sigma }^{*}},{\mathbf{z}}^{\sigma }\right)$ in the 1α1β model, Eqs. (26)–(31). Aside from changes due to the moving nuclear positions, $N_{F,active}^{\sigma }\left({\mathbf{z}}^{{\sigma }^{*}},{\mathbf{z}}^{\sigma }\right)$ can change in the range $0\le N_{F,active}^{\sigma }\left({\mathbf{z}}^{{\sigma }^{*}},{\mathbf{z}}^{\sigma }\right)\le 1$ due to the evolution of the Thouless coefficients ${\mathbf{z}}^{\sigma }\left(t\right)$; in contrast, $N_{F,core}^{\sigma }$ remains constant during dynamics. In analogy with the 1α1β model treatment, the evaluation of the state-to-state probabilities $P\left(\left|\left[core\right]h{h}^{\prime }\right.⟩\to \left|\left[core\right]ij\right.⟩\right)$ involves projecting the final state ${\left|\mathbf{z}\left(t_f\right)\right.⟩}^{1\alpha 1\beta /core}$ onto the HF states $\left|\left[core\right]ij\right.⟩$; cf. Eq. (33). Since ${\left|\mathbf{z}\left(t_f\right)\right.⟩}^{1\alpha 1\beta /core}$ and ${\left|\mathbf{z}\left(t_f\right)\right.⟩}^{1\alpha 1\beta }$ are equivalent term by term and the $\left[core\right]$ part in all the terms of ${\left|\mathbf{z}\left(t_f\right)\right.⟩}^{1\alpha 1\beta /core}$ does not alter the projection results, $P\left(\left|\left[core\right]h{h}^{\prime }\right.⟩\to \left|\left[core\right]ij\right.⟩\right)$ and the subsequent probabilities P_0-ET, P_1-ET, and P_2-ET in the 1α1β/core model result all identical to their counterparts in the 1α1β model, Eqs. (34)–(39). This completes our quantum-mechanical derivation of the P_n-ET formulas for the 1α1β/core model system under SLEND dynamics.

IV. CONCLUSIONS AND FUTURE RESEARCH

We present the first part of the first quantum-mechanical derivation of statistical-law formulas to predict zero-, one-, and two-ETs in proton–molecule reactions.¹⁴ The original formula derivation assumed that the ET probabilities of N electrons in a shell obey an N-trial binomial distribution.¹⁴ The transfer probability of an individual electron was equated to the success probability of the distribution¹⁴ and was heuristically identified with the number of electrons transferred to the projectile as given by the integrated charge density. Despite their astonishing simplicity and tenuous grounds, the formulas proved accurate to calculate ET DCSs and ICSs in various proton–molecule reactions.^{5,7,11–13,19} This success extended to projectile-to-target ETs in PCT reactions involving large water clusters and biomolecules;^5,7 in the latter calculations, the computational feasibility of these formulas offers substantial numerical advantages over more costly quantum-mechanical alternatives. For our quantum-mechanical derivation, we adopt the SLEND theory^4,16,17 to describe projectile–molecule reactions. SLEND is a time-dependent, variational, direct, and non-adiabatic method that describes the nuclei via classical dynamics and the electrons with a single-determinantal state in the Thouless representation.¹⁸ This state plays a crucial role in our derivation because it can be suitably resolved into the final projectile–target eigenstates corresponding to all the ETs and EEs. The non-orthogonality of the dynamical spin-orbitals of the Thouless state poses a mathematical challenge for this derivation. Thus, in this first part of our investigation, we present a derivation for a model system with N > 2 electrons where two electrons with opposite spins are active for ETs and EEs, while the remaining N − 2 electrons remain inactive for those processes in a core. In this scheme, the Thouless dynamical spin-orbitals become orthogonal so that the Thouless state can be treated with the standard Slater–Condon rules;²³ such a fact simplifies one aspect of our derivation that, nevertheless, remains arduous. We express the n-ET probabilities projected out of the Thouless state and the number of transferred electrons from the integrated charge density as functions of the electronic variational parameters. We further demonstrate that, through this common parametric dependency, the ET probabilities are functions of the number of transferred electrons as the statistical-law formulas assert from less rigorous grounds. In addition to our derivation, we review the performance of the investigated formulas in the calculations of ET DCSs and ICSs with the SLEND method in various proton–molecule and PCT reactions.^5,7,11–13 Furthermore, we present new results of zero- and one-ET DCSs of H⁺ + N₂O. We will present the second and final part of our derivation in a subsequent publication. Therein, we will address the general case of reactive systems with N > 2 electrons all active for ETs and EEs. In those systems, the Thouless dynamical spin-orbitals are inescapably non-orthogonal; we will overcome such a mathematical difficulty by treating the corresponding Thouless states with the Löwdin rules.²³

APPENDIX: MATHEMATICAL DETAILS OF SOME EQUATIONS

1. Equation section (next)

In this appendix, we provide the detailed derivations of Eqs. (19), (21), (26), and (28) presented in the main text. First, the full derivation of Eq. (19) is given as

$$\begin{array}{ll}\hfill {\rho }^{\sigma }(\mathbf{r};{\mathbf{z}}^{\sigma *},{\mathbf{z}}^{\sigma })& =⟨{\overline{\chi }}_h^{\sigma }({\mathbf{r}}_i,\omega ;{\mathbf{z}}^{\sigma })\left|\delta \left({\mathbf{r}}_i-\mathbf{r}\right)\right|{\overline{\chi }}_h^{\sigma }({\mathbf{r}}_i,\omega ;{\mathbf{z}}^{\sigma })⟩\hfill \\ \hfill & ={\left[f_h^{\sigma }\left({\mathbf{z}}^{\sigma },{\mathbf{z}}^{{\sigma }^{*}}\right)\right]}^{-1}\iint \underset{h}{\overset{{\sigma }^{*}}{\chi }}({\mathbf{r}}_i,\omega ;{\mathbf{z}}^{\sigma })\delta \left({\mathbf{r}}_i-\mathbf{r}\right)\underset{h}{\overset{\sigma }{\chi }}({\mathbf{r}}_i,\omega ;{\mathbf{z}}^{\sigma })d{\mathbf{r}}_{i}d\omega \hfill \\ \hfill & ={\left[f_h^{\sigma }\left({\mathbf{z}}^{\sigma },{\mathbf{z}}^{{\sigma }^{*}}\right)\right]}^{-1}\int {\chi }_h^{{\sigma }^{*}}(\mathbf{r},\omega ;{\mathbf{z}}^{\sigma }){\chi }_h^{\sigma }(\mathbf{r},\omega ;{\mathbf{z}}^{\sigma })d\omega \hfill \\ \hfill & ={\left[f_h^{\sigma }\left({\mathbf{z}}^{\sigma },{\mathbf{z}}^{{\sigma }^{*}}\right)\right]}^{-1}\left({\tilde{\psi }}_h^{{\sigma }^{*}}(\mathbf{r})+\sum _p{z_{ph}^{\sigma }}^{*}{\tilde{\psi }}_p^{{\sigma }^{*}}(\mathbf{r})\right)\left({\tilde{\psi }}_h^{\sigma }(\mathbf{r})+\sum _q{z}_{qh}^{\sigma }{\tilde{\psi }}_q^{\sigma }(\mathbf{r})\right)\int {\sigma }^{*}(\omega )\sigma (\omega )d\omega \hfill \\ \hfill & ={\left[f_h^{\sigma }\left({\mathbf{z}}^{\sigma },{\mathbf{z}}^{{\sigma }^{*}}\right)\right]}^{-1}\left({\tilde{\psi }}_h^{{\sigma }^{*}}(\mathbf{r}){\tilde{\psi }}_h^{\sigma }(\mathbf{r})+\sum _p{z_{ph}^{\sigma }}^{*}{\tilde{\psi }}_p^{*\sigma }(\mathbf{r}){\tilde{\psi }}_h^{\sigma }(\mathbf{r})\right.\left.\sum _q{z}_{qh}^{\sigma }{\tilde{\psi }}_h^{{\sigma }^{*}}(\mathbf{r}){\tilde{\psi }}_q^{\sigma }(\mathbf{r})+\sum _{p,q}{z_{ph}^{\sigma }}^{*}{z}_{qh}^{\sigma }{\tilde{\psi }}_p^{{\sigma }^{*}}(\mathbf{r}){\tilde{\psi }}_q^{\sigma }(\mathbf{r})\right),\hfill \end{array}$$ (A1)

where the expressions of ${\overline{\chi }}_h^{\sigma }$ and ${\chi }_h^{\sigma }$, Eqs. (11) and (13), respectively, were employed from first to second and from third to fourth lines, respectively. Next, the full derivation of Eq. (21) is

$$\begin{array}{ll}\hfill N^{\sigma }& =\int {\rho }^{\sigma }(\mathbf{r};{\mathbf{z}}^{\sigma *}{,\mathbf{z}}^{\sigma })d\mathbf{r}\hfill \\ \hfill & ={\left[f_h^{\sigma }\left({\mathbf{z}}^{\sigma },{\mathbf{z}}^{{\sigma }^{*}}\right)\right]}^{-1}\sum _{\nu ,\mu }\int \left[\left(C_{\nu h}^{{\sigma }^{*}}{C}_{\mu h}^{\sigma }+\sum _p{z_{ph}^{\sigma }}^{*}{C}_{\nu p}^{{\sigma }^{*}}{C}_{\mu h}^{\sigma }\right.\right.\left.+\left.\sum _q{z}_{qh}^{\sigma }{C}_{\nu h}^{{\sigma }^{*}}{C}_{\mu q}^{\sigma }+\sum _{p,q}{z_{ph}^{\sigma }}^{*}{z}_{qh}^{\sigma }{C}_{\nu p}^{{\sigma }^{*}}{C}_{\mu q}^{\sigma }\right){\varphi }_{\nu }^{*}\left(\mathbf{r}\right){\varphi }_{\mu }\left(\mathbf{r}\right)\right]d\mathbf{r}\hfill \\ \hfill & ={\left[f_h^{\sigma }\left({\mathbf{z}}^{\sigma },{\mathbf{z}}^{{\sigma }^{*}}\right)\right]}^{-1}\sum _{\nu ,\mu }\left(C_{\nu h}^{{\sigma }^{*}}{C}_{\mu h}^{\sigma }+\sum _p{z_{ph}^{\sigma }}^{*}{C}_{\nu p}^{{\sigma }^{*}}{C}_{\mu h}^{\sigma }\right.+\left.\sum _q{z}_{qh}^{\sigma }{C}_{\nu h}^{{\sigma }^{*}}{C}_{\mu q}^{\sigma }+\sum _{p,q}{z_{ph}^{\sigma }}^{*}{z}_{qh}^{\sigma }{C}_{\nu p}^{{\sigma }^{*}}{C}_{\mu q}^{\sigma }\right)S_{\nu \mu }\hfill \\ \hfill & ={\left[f_h^{\sigma }\left({\mathbf{z}}^{\sigma },{\mathbf{z}}^{{\sigma }^{*}}\right)\right]}^{-1}\left(\sum _{\nu ,\mu }{C}_{h\nu }^{\sigma †}{S}_{\nu \mu }{C}_{\mu h}^{\sigma }+\sum _p{z_{ph}^{\sigma }}^{*}\sum _{\nu ,\mu }{C}_{p\nu }^{\sigma †}{S}_{\nu \mu }{C}_{\mu h}^{\sigma }+\sum _q{z}_{qh}^{\sigma }\sum _{\nu ,\mu }{C}_{h\nu }^{\sigma †}{S}_{\nu \mu }{C}_{\mu q}^{\sigma }+\sum _{p,q}{z_{ph}^{\sigma }}^{*}{z}_{qh}^{\sigma }\sum _{\nu ,\mu }{C}_{p\nu }^{\sigma †}{S}_{\nu \mu }{C}_{\mu q}^{\sigma }\right)\hfill \\ \hfill & ={\left[f_h^{\sigma }\left({\mathbf{z}}^{\sigma },{\mathbf{z}}^{{\sigma }^{*}}\right)\right]}^{-1}\left({\delta }_{hh}+\sum _p{z_{ph}^{\sigma }}^{*}{\delta }_{ph}+\sum _q{z}_{qh}^{\sigma }{\delta }_{hq}+\sum _{p,q}{z_{ph}^{\sigma }}^{*}{z}_{qh}^{\sigma }{\delta }_{pq}\right)\hfill \\ \hfill & ={\left[f_h^{\sigma }\left({\mathbf{z}}^{\sigma },{\mathbf{z}}^{{\sigma }^{*}}\right)\right]}^{-1}\left(1+\sum _p{\left|z_{ph}^{\sigma }\right|}^2\right)={\left(1+\sum _p{\left|z_{ph}^{\sigma }\right|}^2\right)}^{-1}\left(1+\sum _p{\left|z_{ph}^{\sigma }\right|}^2\right)=1,\hfill \end{array}$$ (A2)

where we applied the orthonormality of the spatial molecular orbitals in the fourth line,

$${\delta }_{ij}=⟨\left.{\tilde{\psi }}_i^{\sigma }\left(\mathit{r}\right)\right|{\tilde{\psi }}_j^{\sigma }\left(\mathit{r}\right)⟩=\sum _{\nu ,\mu }{C}_{iv}^{\sigma †}{S}_{\nu \mu }{C}_{\mu j}^{\sigma },i,j=h,g,\dots p,q\dots ,$$ (A3)

and inserted the expression of $f_h^{\sigma }\left({\mathbf{z}}^{\sigma },{\mathbf{z}}^{{\sigma }^{*}}\right)$, Eq. (13), in the last line. Next, the full derivation of Eq. (27) is

$$\begin{array}{ll}\hfill N_F^{\sigma }& =\int {\rho }_F^{\sigma }(\mathbf{r};{\mathbf{z}}^{\sigma *},{\mathbf{z}}^{\sigma })d\mathbf{r}=\sum _{\nu \in F,\mu \in F}{P}_{\mu \nu }^{\sigma }{S}_{\nu \mu }\hfill \\ \hfill & ={\left[f_h^{\sigma }\left({\mathbf{z}}^{\sigma },{\mathbf{z}}^{{\sigma }^{*}}\right)\right]}^{-1}\sum _{\nu \in F,\mu \in F}\left(C_{\nu h}^{{\sigma }^{*}}{C}_{\mu h}+\sum _{p\in A,B}{z_{ph}^{\sigma }}^{*}{C}_{\nu p}^{{\sigma }^{*}}{C}_{\mu h}^{\sigma }+\sum _{q\in A,B}{z}_{qh}^{\sigma }{C}_{\nu h}^{{\sigma }^{*}}{C}_{\mu q}^{\sigma }+\sum _{p,q\in A,B}{z_{ph}^{\sigma }}^{*}{z}_{qh}^{\sigma }{C}_{\nu p}^{{\sigma }^{*}}{C}_{\mu q}^{\sigma }\right)S_{\nu \mu }\hfill \\ \hfill & ={\left[f_h^{\sigma }\left({\mathbf{z}}^{\sigma },{\mathbf{z}}^{{\sigma }^{*}}\right)\right]}^{-1}\left[\sum _{\nu \in F,\mu \in F}\left(C_{\nu h}^{{\sigma }^{*}}{C}_{\mu h}+\sum _{p,q\in A,B}{z_{ph}^{\sigma }}^{*}{z}_{qh}^{\sigma }{C}_{\nu p}^{{\sigma }^{*}}{C}_{\mu q}^{\sigma }\right)S_{\nu \mu }\right.\hfill \\ \hfill & \left.+\sum _{p\in A,B}{z_{ph}^{\sigma }}^{*}\sum _{\nu \in F,\mu \in F}{C}_{p\nu }^{\sigma †}{S}_{\nu \mu }{C}_{\mu h}^{\sigma }+\sum _{q\in A,B}{z}_{qh}^{\sigma }\sum _{\nu \in F,\mu \in F}{C}_{h\nu }^{\sigma †}{S}_{\nu \mu }{C}_{\mu q}^{\sigma }\right]\hfill \\ \hfill & ={\left[f_h^{\sigma }\left({\mathbf{z}}^{\sigma },{\mathbf{z}}^{{\sigma }^{*}}\right)\right]}^{-1}\sum _{\nu \in F,\mu \in F}\left(C_{\nu h}^{{\sigma }^{*}}{C}_{\mu h}+\sum _{p,q\in A,B}{z_{ph}^{\sigma }}^{*}{z}_{qh}^{\sigma }{C}_{\nu p}^{{\sigma }^{*}}{C}_{\mu q}^{\sigma }\right)S_{\nu \mu }\left(h\in A,R_{AB}\to \infty \right),\hfill \end{array}$$ (A4)

where Eq. (24) was employed from the first to the second line and the orthonormality condition from Eq. (A3) in the last two terms of the third line. Finally, the full derivation of Eq. (29) is

$$\begin{array}{ll}\hfill N_F^{\sigma }& ={\left[f_h^{\sigma }\left({\mathbf{z}}^{\sigma },{\mathbf{z}}^{{\sigma }^{*}}\right)\right]}^{-1}\sum _{\nu \in F,\mu \in F}\left(C_{\nu h}^{{\sigma }^{*}}{C}_{\mu h}+\sum _{p,q\in A,B}{z_{ph}^{\sigma }}^{*}{z}_{qh}^{\sigma }{C}_{\nu p}^{{\sigma }^{*}}{C}_{\mu q}^{\sigma }\right)S_{\nu \mu }\hfill \\ \hfill & ={\left[f_h^{\sigma }\left({\mathbf{z}}^{\sigma },{\mathbf{z}}^{{\sigma }^{*}}\right)\right]}^{-1}\sum _{\nu \in F,\mu \in F}\left({\delta }_{AF}{C}_{\nu h}^{{\sigma }^{*}}{C}_{\mu h}+\sum _{p,q\in F}{z_{ph}^{\sigma }}^{*}{z}_{qh}^{\sigma }{C}_{\nu p}^{{\sigma }^{*}}{C}_{\mu q}^{\sigma }\right)S_{\nu \mu }\hfill \\ \hfill & ={\left[f_h^{\sigma }\left({\mathbf{z}}^{\sigma },{\mathbf{z}}^{{\sigma }^{*}}\right)\right]}^{-1}\left({\delta }_{AF}\sum _{\nu \in F,\mu \in F}{C}_{h\nu }^{\sigma †}{S}_{\nu \mu }{C}_{\mu h}+\sum _{p,q\in F}{z_{ph}^{\sigma }}^{*}{z}_{qh}^{\sigma }\sum _{\nu \in F,\mu \in F}{C}_{p\nu }^{\sigma †}{S}_{\nu \mu }{C}_{\mu q}^{\sigma }\right)\hfill \\ \hfill & ={\left[f_h^{\sigma }\left({\mathbf{z}}^{\sigma },{\mathbf{z}}^{{\sigma }^{*}}\right)\right]}^{-1}\left({\delta }_{AF}{\delta }_{hh}+\sum _{p,q\in F}{z_{ph}^{\sigma }}^{*}{z}_{qh}^{\sigma }{\delta }_{pq}\right)=\frac{{\delta }_{AF}+\sum _{p\in F}{\left|z_{pi}^{\sigma }\right|}^2}{1+{\sum }_p{\left|z_{ph}^{\sigma }\right|}^2}\left(h\in A,R_{AB}\to \infty \right),\hfill \end{array}$$ (A5)

where we employed the orthonormality condition in Eq. (A3) from the third to the fourth line.

DATA AVAILABILITY

The data that support the findings of this study are available within the article.