Electron nuclear dynamics with plane wave basis sets: complete theory and formalism

Abstract

Electron nuclear dynamics (END) is an ab initio quantum dynamics method that adopts a time-dependent, variational, direct, and non-adiabatic approach. The simplest-level (SL) END (SLEND) version employs a classical mechanics description for nuclei and a Thouless single-determinantal wave function for electrons. A higher-level END version, END/Kohn–Sham density functional theory, improves the electron correlation description of SLEND. While both versions can simulate various types of chemical reactions, they have difficulties to simulate scattering/capture of electrons to/from the continuum due to their reliance on localized Slater-type basis functions. To properly describe those processes, we formulate END with plane waves (PWs, END/PW), basis functions able to represent both bound and unbound electrons. As extra benefits, PWs also afford fast algorithms to simulate periodic systems, parametric independence from nuclear positions and momenta, and elimination of basis set linear dependencies and orthogonalization procedures. We obtain the END/PW formalism by extending the Thouless wave function and associated electron density to periodic systems, expressing the energy terms as functionals of the latter entities, and deriving the energy gradients with respect to nuclear and electronic variables. END/ PW has a great potential to simulate electron processes in both periodic (crystal) and aperiodic (molecular) systems (the latter in a supercell approach). Following previous END studies, END/PW will be applied to electron scattering processes in proton cancer therapy reactions.

1. Introduction

The study of chemical reactions with computational methods allows the interpretation of microscopic events and the prediction of mechanisms and selectivities without conducting experiments [1, 2]. Remarkably, some information obtained with computational methods remains inaccessible to experiments [1, 2]; such information provides a deep understanding of chemical reactions that assists in their interpretation and control. Due to a large number of options, the selection of an appropriate method to study a reactive system requires a careful evaluation of the mechanisms to simulate, the properties to calculate, and the computational power to use. For example, in many reactions involving electron transfers, it is necessary to calculate non-adiabatic couplings [3], terms neglected in widely used methods such as the single potential energy surface (PES) dynamics [4], Bohr–Oppenheimer molecular dynamics (BOMD) [5], and Car–Parrinello molecular dynamics (CPMD) [6]. In contrast, other methods properly include non-adiabatic couplings such as some types of Ehrenfest molecular dynamics [5] and the electron nuclear dynamics (END) [7–10]. In fact, END is the most embracing method regarding non-adiabatic couplings because it incorporates terms related not only to nuclear positions but also to nuclear momenta (cf. Sect. 2.1).

In the last 15 years [11, 12], our group has applied and further developed END to simulate a vast array of chemical reactions. END offers particular capabilities to describe chemical dynamics processes because it is (a) time-dependent, (b) variational, (c) direct, and (d) non-adiabatic. These four attributes are valuable for reactions’ simulations because they, respectively, provide (a) time-dependent detail, (b) a systematic way to select trial wave functions, (c) independence from pre-calculated PESs, and (d) a correct description of non-adiabatic processes. END adopts different versions according to the complexity of their nuclear and electronic descriptions [7–10], but only two END versions are currently implemented. One is the simplest-level (SL) END (SLEND) [7–10] that describes the nuclei in terms of classical mechanics and the electrons with a Thouless single-determinantal wave function [13]. The other implemented version [9, 14] is END in terms of the time-dependent Kohn–Sham density functional theory [15] (KSDFT: END/KSDFT); this version offers a better description of electron correlation than SLEND does [9, 14] (see Sect. 2.1).

SLEND and END/KSDFT have accurately described a great variety of reactions in the collision energy range, 10 eV ≤ E_Lab ≤ 25 keV, that includes ion–molecule [7, 9, 11, 12, 16–26], Diels–Alder [9], S_N2 [9], and intra-molecular [27] reactions as well as reactions under applied electromagnetic field [28–30]. However, one of the most fruitful areas of research with SLEND and END/KSDFT is the simulation of the high-energy reactions occurring during proton cancer therapy (PCT) [31]. That research line is illustrated by our pioneering studies of water radiolysis reactions [9, 31–33], electron capture by protons from DNA/ RNA bases [34], proton-induced damage on DNA bases [9, 31, 32, 34] and nucleotides [31], and electron-induced damage on DNA nucleotides [31].

In spite of their success, SLEND and END/KSDFT expose difficulties to describe the scattering/capture of electrons to/from the continuum, processes relevant in high-energy reactions [33, 35]. This limitation stems from the fact that both SLEND and END/KSDFT represent electrons with standard Slater-type orbitals (STO) as contractions of Gaussian-type orbitals (STO/CGTO) [36]. Those functions are localized around nuclear centers and are by design suitable to represent the bound and discrete part of the electronic spectrum but not the unbound and continuum counterpart. A very large basis sets of the described type can generate a quasi-continuum of close-lying virtual molecular orbitals, but that provides a limited description of the unbound part of the electronic spectrum [37, 38]. During PCT, electron scattering from the colliding reactants becomes a significant process at the highest end of the proton radiation energy [33, 35]; moreover, those scattered electrons are captured by DNA molecules that subsequently undergo electron-induced single- and double-strand breaks [35, 38]. Therefore, to describe those processes very accurately, it is peremptory to extend the SLEND and END/KSDFT formalisms beyond a STO/CGTO formulation. Such a development is not only valuable for PCT reactions but for all types of reactions where electron scattering/capture to/from the continuum is important.

An approach widely employed in the literature to describe unbound electrons relies on plane wave (PW) functions [5, 39, 40] (cf. Sect. 2.2); PWs by themselves describe free electrons and are therefore a natural choice to represent those particles in the unbound and continuum part of the spectrum. Consequently, we advocate herein the adoption of PWs in the END context to describe electron scattering processes. While achieving the latter is our main motivation, the advantages of using PWs in END are not limited to those scattering processes because PWs are appropriate to describe other phenomena and systems. For instance, due to their periodicity, PWs are ideal to describe crystalline solids [5, 39, 40]. Thus, in combination with PWs, END will be also able to describe electron-transfer reactions in crystals and regular polymers. In addition, PWs confer various numerical advantages to the SLEND and END/KSDFT algorithms that will improve their computer performance on several types of reactions (cf. Sect. 2.2).

Toward the stated goals, the following sections delineate the formulation of the END theory in terms of PWs, an endeavor leading to the novel END/PW method. In that formulation, END is first generalized to the case of periodic systems by taking advantage of the periodicity of PWs as is customarily done in solid state physics [5, 39, 40]. The resulting formalism can be applied to time-dependent processes in periodic systems or can be adapted to molecular reactions in a non-periodic setting via the supercell approach of CPMD [6]. The present formulation overcomes various theoretical challenges prompted by the generalization of the intricate END framework to periodic systems. Notable features in this formulation include (1) extending the END Thouless single-determinantal wave function to periodic systems, (2) formulating an electron density that satisfies lattice symmetries from the Thouless wave function, (3) expressing all the energy terms as functionals of the Thouless wave function and its associated density, and (4) obtaining the energy gradient terms with respect to both nuclear and electronic parameters.

2. Background

2.1. The END methodology, SLEND, END/KSDFT and the code PACE

In this section, we will explain the standard END theory [7–10] in considerable detail in order to make intelligible our subsequent development of END/PW. END is a time-dependent, variational, direct and non-adiabatic approach to simulate chemical reactions [7–10]. At its highest level, the END trial wave function originates from the Born-Huang [8] expansion and adopts a multi-configurational form:

$${\Psi }^{\text{END}}(\mathit{X},x;Y,z)=\sum _{\pi }{c}_{\pi }{\mathit{\Psi }}_{n,\pi }^{\text{END}}(X;Y){\mathit{\Psi }}_{e,\pi }^{\text{END}}(x;z,Y)$$ (1)

Here, X denotes the positions of the nuclei, x denotes simultaneously the spatial r and spin s coordinates of the electrons, γ and z are sets of nuclear and electronic variational parameters, ${\mathit{\Psi }}_{n,\pi }^{\text{END}}(X;Y)$ and ${\mathit{\Psi }}_{e,\pi }^{\text{END}}(x;z,\mathit{Y})$ are nuclear and electronic wave functions (configurations), respectively, and c_π are multi-configuration coefficients accounting for nucleus–nucleus, electron–electron, and nucleus–electron correlations. Specific examples of nuclear and electronic wave functions and their parameters will be presented in the following paragraphs. c _π, γ and z are the time-dependent variational parameters of Ψ^END(X, x;γ, z),, and their time evolutions are determined via the time-dependent variational principle (TDVP) [41] as explained below.

In theory, the END wave function becomes the exact time-dependent wave function in the infinite-basis-set limit that implies an infinite number of configurations [8]. While that limit is impractical, the END wave function can achieve high accuracy by adopting a large but finite basis set and by employing a few chemically significant configurations. However, for large systems, any multi-configuration approach entails a prohibitive computational cost. Fortunately, END exhibits a hierarchy of approximate models that converge to the exact solution as the complexity of their nuclear and electronic descriptions increases [7–10]. The first term in the ascending hierarchy of END models is the simplest-level (SL) END (SLEND) [7–9], whose wave function Ψ^SLEND(X, x;R, P, z) utilizes one nuclear ${\mathit{\Psi }}_n^{\text{SLEND}}(X;R,P)$ and one electronic ${\mathit{\Psi }}_e^{\text{SLEND}}(x;z,R,P)$ wave function in Eq. (1):

$${\Psi }^{\text{SLEND}}(X,x;R,P,z)={\Psi }_n^{\text{SLEND}}(X;R,P){\Psi }_e^{\text{SLEND}}(x;z,R,P)$$ (2)

where γ = R, P will be defined shortly. Since a single nuclear electronic configuration is used, only one configuration coefficient c_π of Eq. (1) is not null; then, reference to those coefficients becomes superfluous and will be omitted henceforth. The nuclear wave function ${\Psi }_n^{\text{SLEND}}(X;R,P)$ is the simple product of s-type Gaussian wave packets:

$${\Psi }_n^{\text{SLEND}}(X;R,P)=\prod _{l=1}^{N_{\text{at}}}\text{exp}\left[-\frac{{\left(X_l-R_l\right)}^2}{2{\alpha }_l^2}+i{P}_l\cdot \left(X_l-R_l\right)\right]$$ (3)

where N_at is the number of atoms (nuclei) in the molecular system, and R_l, P_l, and α_l are the Gaussian wave packets’ positions, momenta and widths, respectively. To reduce computational cost, the zero-width limit α_l → 0∀l, is applied in the last stages of the SLEND formulation, just after constructing the SLEND quantum Lagrangian [cf. the remarks preceding Eq. (10)]. Such a procedure renders a classical mechanics description of the nuclei in terms of positions R_l and momenta P_l that nevertheless retains nucleus–electron non-adiabatic coupling terms.

The electronic wave function ${\Psi }_e^{\text{SLEND}}(x;z,R,P)$ is a single-determinantal wave function in the Thouless representation [13]:

$$\begin{gathered} {\Psi }_e^{\text{SLEND}}(x;z,R,P)=\text{det}\left[{\chi }_h\left(x_g;z,R,P\right)\right];1\le h,g\le N_e \\ {\chi }_h(x;z,R,P)={\chi }_h(r,s;z,R,P)={\psi }_h^{*}(x;R,P)+\sum _{p=N_e+1}^{\mathit{K}}{z}_{ph}{\psi }_p^{°}(x;R,P) \end{gathered}$$ (4)

where K > N_e is the number of electrons in the system, is the total number of orthonormal molecular spin orbitals Ψ_i(x;R, P), χ_h(x;~, R, P) are the non-orthogonal dynamical spin orbitals as linear combinations of N_e occupied ${\psi }_h^{·}$ and K − N_e unoccupied (virtual) molecular spin orbitals ${\psi }_p^{°}$, and {z_ph} are the complex-valued Thouless parameters that act as the molecular coefficients of the χ_h(x;z, R, P). Following the particular notation of Ref. [7], the symbols ∙ and ° on spin orbitals and related properties denote their inclusion in the occupied (hole) and unoccupied (particle) spaces, respectively. The molecular spin orbitals are

$${\psi }_i(x;R,P)={\psi }_i(r,s;R,P)={\tilde{\psi }}_i(r;R,P){\sigma }_i(s)$$ (5)

where, again, x represents simultaneously the spatial r and spin s coordinates, and R and P coordinates, and momenta of the nuclei. σ_i(s) are the spin eigenfunctions σ_i(s) or β_i(s) and ${\tilde{\psi }}_i(r;R,P)$ the spatial orbitals. The latter are expressed in terms of non-orthogonal STO/CGTOs with contracted j-type (j = s, p_x, d_xy, etc.) primitive Gaussians centered on the moving nuclei l. Therefore, the spatial orbitals are

$${\tilde{\psi }}_i(r;R,P)=\sum _{j=1}^{\mathit{K}}{C}_{ji}{\left(r_x-R_{l(j),x}\right)}^{k_{l(j)}}{\left(r_y-R_{l(j)y}\right)}^{m_{l(j)}}{\left(r_z-R_{l(j),z}\right)}^{n_{l(j)}}\times \text{exp}\left[-\frac{{\left(\mathit{r}-R_{l(j)}\right)}^2}{2{\gamma }_{l(j)}^2}+\frac{i}{{\mathit{M}}_{l(j)}}{P}_{l(j)}\cdot \left(r-R_{l(j)}\right)\right]$$ (6)

where C_ij are the contraction coefficients with respect to the primitive Gaussians, and M_l(j), R_l(j), and P_l(j) are the mass, position and momentum of the nuclei l carrying the Gaussian j with width γ_l(j). These primitive Gaussians are those from standard electronic structure theory [36] but augmented with electron translation factors (ETFs) [the P_l(j) -terms in Eq. (6)] [42]; ETFs introduce the explicit kinetic effect on the electrons by the nuclei moving with velocity P_l(j)∕M_l(j). ETFs should be included in accurate calculations at very high collision energies. In SLEND, the molecular spin orbitals Ψ_i(x;R, P) are obtained via a time-independent Hartree–Fock (HF) optimization at initial time

$$\left[-\frac{1}{2}{\nabla }^2-\sum _{j=1}^{N_{at}}\frac{Z_j}{\left|r-R_j\right|}+\int \frac{\rho \left({\mathit{r}}^{\prime };R,P\right)}{\left|r-r^{\prime }\right|}\text{d}{r}^{\prime }-\widehat{K}\left(x^{\prime };R,P\right)\right]{\psi }_i(x;R,P)={\epsilon }_i{\psi }_i(x;R,P)$$ (7)

where Z_j are the nuclear charges, ρ(r′;R, P) is the electron density from ${\mathit{\Psi }}_n^{\text{SLEND}}(x;z,R,P)$ with z = 0 at initial time ε_i are the orbital energies, and $\widehat{K}\left(x^{\prime };R,P\right)$ is the total non-local HF exchange operator [36]. Notice that the HF molecular spin orbitals Ψ_i(x;R, P) are orthogonal, while the dynamical spin orbitals χ_h (x_g;R, P) are not as Eq. (4) reveals.

The less conventional Thouless representation [13] is adopted in SLEND because it provides several advantages to simulate time-dependent non-adiabatic dynamics [7, 9]. First of all, the Thouless representation provides a minimum and non-redundant parameterization of a general single-determinant state in terms of N_e · K Thouless parameters. In contrast, a straightforward parameterization in terms of the atomic orbitals coefficients leads to K · K, K > N_e, parameters, among which K · K − N_e are redundant. Since in K ≫ N_e, accurate calculations the Thouless representation considerably reduces the number of electronic parameters, thus lowering the computational cost. More importantly, the non-redundancy of this representation provides a one-to-one, invertible mapping between independent single-determinantal states and electronic parameters that prevents numerical singularities in the END equations [7, 9]. Finally, the Thouless representation provides a convenient way to represent non-adiabatic processes because it holds that [13]

$${\Psi }_e^{\text{SLEND}}(x;z,R,P)=\text{det}\left[{\psi }_h^{*}\right]+\sum _{\underset{p=N_e+1}{h=1,}}^{N_e,K}{z}_{ph}\text{det}\left[{\psi }_h^{*}\to {\psi }_p^{°}\right]+\sum _{\underset{p\text{, }q=N_e\text{+1(}p<q\text{)}}{h\text{, }g\text{=1(}h<g\text{)};}}^{N_e;K}{z}_{ph}{z}_{qg}\text{det}\left[{\psi }_h^{*}{\psi }_g^{*}\to {\psi }_p^{°}{\psi }_q^{°}\right]+\cdots$$ (8)

where the first term is the reference state of the occupied spin orbitals and the second, third, etc., terms contain the single-, double-, etc., excited states from that reference. At initial time, a system can be prepared in its ground state ${\Psi }_e^{\text{SLEND}}=\text{det}\left[{\psi }_h^{*}\right]$ with z = 0. After a time, if the system reaches the non-adiabatic regime, then z(t) ≠ 0, and the dynamical orbital χ_h(x_g;R, P) will include virtual orbitals contributions [cf. Equation (4)] and ${\Psi }_e^{\text{SLEND}}$ will become a coherent superposition of ground and excited HF states [cf. Eq. (8)]. Furthermore, the linear combination of single excitations in Eq. (8) is related to the random phase approximation excited states [13].

The SLEND dynamical equations are obtained via the TDVP [41] as follows. The quantum Lagrangian is constructed as [7, 9, 41]:

$$L_{\text{SLEND}}=\frac{\langle {\Psi }^{\text{SLEND}}|\frac{i}{2}\left(\frac{\partial }{\partial t}-\frac{\tilde{\partial }}{\partial t}\right)-\widehat{H}|{\Psi }^{\text{SLEND}}\rangle }{\langle {\Psi }^{\text{SLEND}}\mid {\Psi }^{\text{SLEND}}\rangle }$$ (9)

where $f(\tilde{\partial }/\partial t)=(\partial /\partial t)f$ and $\widehat{H}$ is the total (nuclear and electronic) Hamiltonian operator. The SLEND dynamical equations for all the variational parameters q = R, P, z, z* are obtained as Euler–Lagrange equations $\partial L/\partial q=\text{d}(\partial L/\partial \dot{q})\text{d}t$ just after applying the zero-width limit to the nuclear wave packets. The resulting dynamical equations in matrix form are [7, 9]:

$$\left(\begin{array}{cccc}iC& 0& i{C}_{\mathit{R}}& 0\\ 0& -i{C}^{*}& -i{C}_R^{*}& -i{C}_P^{*}\\ i{C}_R^{†}& i{C}_R^T& C_{RR}& -I+C_{RP}\\ i{C}_P^{†}& i{C}_P^T& I+C_{PR}& C_{PP}\end{array}\right)\left(\begin{array}{l}\dot{z}\hfill \\ z^{*}\hfill \\ \dot{R}\hfill \\ \dot{P}\hfill \end{array}\right)=\left(\begin{array}{l}\partial E_{\text{T}}/\partial z^{*}\hfill \\ \partial E_{\text{T}}/\partial z\hfill \\ \partial E_{\text{T}}/\partial R\hfill \\ \partial E_{\text{T}}/\partial P\hfill \end{array}\right)$$ (10)

where E_T is the total energy of the system:

$$E_{\text{T}}\left(z,z^{*};R,P\right)=T_i(P)+T_e\left(z,z^{*};R,P\right)+E_{\text{nn}}(R)+E_{\text{H}}\left[\rho \left(r;z,z^{*},R,P\right)\right]+V_{\text{ext }}\left[\rho \left(\mathit{r};z,z^{*},R,P\right),R\right]-K_x\left(z,z^{*};R,P\right)$$ (11)

Above, T_i(P) and T_e(z, z*, R, P) are the total nuclear (classical) and electronic (quantum) kinetic energies, respectively, E_nn(R) is the nuclear repulsion energy, E_H[ρ(r;z, z*, R, P)] is the electronic Hartree (self-repulsion) energy, V_ext[ρ(r;z, z*, R, P),R] is the external potential energy exerted by the moving nuclei, and K_x(z, z*, R, P) is the electronic HF exchange energy. Further details of these terms are given in Sects. 3.4–3.8. In Eq. (10), the electron–electron C and nucleus–electron C_X and C_XY(X, Y = R or P) coupling terms are [7, 9, 43]:

$$\begin{gathered} {(C)}_{qg;ph}={\left.\frac{{\partial }^2\text{ ln }S\left(z^{\prime *},R^{\prime },P^{\prime },z,R,P\right)}{\partial z_{qg}^{*}\partial z_{ph}}\right|}_{{\mathit{R}}^{\prime }=\mathit{R}{\mathit{P}}^{\prime }=\mathit{P}}; \\ {\left(C_X\right)}_{qg;ik}={\left.\frac{{\partial }^2\text{ ln }S\left(z^{\prime *},R^{\prime },P^{\prime },z,R,P\right)}{\partial z_{qg}^{*}\partial X_{ik}}\right|}_{R^{\prime }=R,P^{\prime }=P} \\ {\left(C_{XY}\right)}_{ik;jl}=-{\left.2Im\frac{{\partial }^2\text{ ln ln }S\left(z^{\prime *},R^{\prime },P^{\prime },z,R,P\right)}{\partial X_{ik}^{*}\partial Y_{jl}}\right|}_{R^{\prime }=R,P^{\prime }=P}; \\ S\left(z^{\prime },R^{\prime },P^{\prime },z,R,P\right)=〈{\Psi }_e^{\text{SLEND}}\left(X,x;z^{\prime },R^{\prime },P^{\prime }\right)\mid {\Psi }_e^{\text{SLEND}}(X,x;z,R,P)〉 \end{gathered}$$ (12)

C_R and C_RR are the END equivalents to the standard non-adiabatic coupling terms [3]. C_P and C_PP are terms unique to END; they are an extension of the previous coupling terms to account for the explicit effect of the nuclear momenta on the electrons via the ETFs. The SLEND Eqs. (10)–(12) present the coupled nuclear and electronic dynamics in a generalized quantum symplectic form.

The next version in the ascending END hierarchy is the END/Kohn–Sham density functional theory (END/ KSDFT) [9, 14], which improves upon SLEND with a more accurate treatment of electron correlation effects. To formulate END/KSDFT, ${\Psi }_e^{\text{SLEND}}(x;z,R,P)$ is transformed into an auxiliary KSDFT wave function ${\Psi }_e^{\text{KSDFT}}(x;z,R,P)$ that would provide the exact electron density if the exact exchange–correlation functional was known [15, 44]. Specifically, ${\Psi }_e^{\text{KSDFT}}(x;z,R,P)$ is a Thouless single-determinantal wave function whose dynamical orbitals χ_h(x_g;z, R, P)are now based on KSDFT molecular spin orbitals [15, 44] instead of HF ones. Similarly, $-\widehat{K}\left(x\text{'};R,P\right)$ and − K_x(z, z*, R, P) in Eqs. (7) and (11), respectively, are transformed into the local KSDFT exchange–correlation potential v_KS[ρ(r′; z, z*, R, P)] and the exchange–correlation energy E_xc[ρ(r′; z, z*, R, P)], respectively [15, 44]. Moreover, the linear combination of single-excited states in Eq. (8) is now related to the time-dependent DFT excited states [45].

An important aspect of END is its relationship with the coherent states theory [9, 46, 47]. For instance, the nuclear wave packets, Eq. (3), and the Thouless single-determinantal wave function, Eq. (4), are examples of the canonical and Thouless coherent states, respectively [9, 46, 47]. Other types of coherent states related to END include the Morse oscillator [26], rotational [48], electronic valence-bond [49], and vector coherent states [50]. Those coherent states are useful to parameterize trial wave functions [9, 47, 50] and to relate classical and quantum mechanics descriptions [9, 26, 47–49].

SLEND and END/KSDFT are the only two END versions implemented in computer codes and available for research use [7, 9]. Both methods are implemented in our advanced END code PACE (Python-Accelerated Coherent states Electron nuclear dynamics) [9, 51]. PACE exploits various state-of-the-art computer science techniques such as a mixed programming language (Python for logic flow and FORTRAN and C++ for calculations), intra- and internode parallelization, and the fast OED/ERD atomic integral package [52] of the ACES III/IV program of the Bartlett group [53]. Thanks to PACE, SLEND, and END/KSDFT could be applied for the first time to reactions involving large molecules such as Diels–Alder [9] and S_N2 [9] reactions and to various types of PCT reactions involving large biomolecules [31].

2.2. Plane waves

As discussed previously, the use of localized STO/CGTOs in SLEND and END/KSDFT precludes these methods from properly describing electrons in the unbound part of their spectrum. This limitation will be overcome by reformulating these methods with a PW basis set (END/PW). Moreover, aside from allowing unbound electron descriptions, PWs bring extra advantages to all types of SLEND and END/ KSDFT calculations whether or not involving electron scattering processes [5, 39, 40]. Those advantages are [5, 39, 40] (1) unlike traditional STO/CGTO basis sets, PWs form an orthogonal basis set that is therefore free of linear dependency problems and does not require overlap matrix evaluations and orthogonalization procedures; (2) since PWs are independent of nuclear positions, Pulay forces (i.e., the energy gradient terms arising from the parametric dependence of a wave function or density on nuclear positions) vanish, a fact that simplifies force calculations; (3) due to their periodicity, algorithms based on PWs become very efficient in the case of periodic (e.g., crystalline solids, regular polymers) and quasi-periodic (e.g., DNA fragments) systems; (4) PWs form an unbiased basis set, i.e., all regions in the space has the same accuracy; (5) solids and molecules can be represented with the same PW basis set; and (6) since real and reciprocal spaces with PWs are connected efficiently via fast Fourier transforms (FFT), the calculation of energy terms can be accelerated by evaluating each term in the space that is numerically more advantageous; for example, in one-electron effective Hamiltonians, the kinetic and potential energy terms are diagonal and, therefore, easier to evaluate in reciprocal and real space, respectively.

Certainly, PW basis sets also exhibit some drawbacks discussed in the specialized literature [5, 39, 40]; two of these drawbacks and their solutions are pertinent in the present context. First, the main applications intended herein are on aperiodic systems such as colliding molecules during PCT reactions. The efficient PWs algorithms based on the periodicity of this basis set seem unsuitable for aperiodic systems. This apparent difficulty will be overcome by adopting the supercell approach customarily used in CPMD [6]; the reacting molecules are placed in a large supercell that satisfies periodic boundary conditions and has a large peripheral vacuum region to avoid interactions among the replicas of the system in each supercell. Second, core electrons are particularly localized near the nuclei and are rapidly varying in space; therefore, core electrons require a prohibitive large number of PWs to represent them. This difficulty will be overcome by adopting the pseudopotential approach [54]; the core electrons are represented implicitly with pseudopotentials, while the valence electrons, both bound and unbound, are treated explicitly with PWs. In part, this approach works well in terms of accuracy because core electrons are far less relevant for chemical reactions than valence ones. Another solution for the core electron treatment is to adopt a mixed PW/STO/CGTO basis set [55, 56], where PW functions will mostly describe bound and unbound valence electrons and STO/CGTO functions will mostly described core electrons; those are the best roles by each basis set, respectively. This combined basis set formulation will be considered in the future.

There exist several program suites implementing DFT methodologies in terms of PWs such as the well-known codes: WIEN2k [57], VASP [58], Qbox [59], and CPMD [6], inter alia. However, none of them implements a non-adiabatic molecular dynamics method like END.

3. END/PW

3.1. END/PW dynamical equations

The END/PW dynamical equations are obtained from the SLEND Eq. (10) by expressing all its components in terms of PWs. A prominent feature of PWs is their independence from the nuclear positions and momenta R and P. A first consequence of this property is that, unlike STO/CGTOs centered on nuclei, ETFs [42] are not required with PWs [cf. Equation (6)]; the absence of those terms confers a considerable saving of computational cost. Moreover, with PWs, ${\Psi }_e^{\text{SLEND}}(x;z,R,P)$, Eq. (4), transforms into its END/PW counterpart ${\Psi }_e^{\text{END}/\text{PW}}(x;z)$ that is now independent of R and P; this situation brings about advantageous simplifications in various terms. For instance, from Eq. (12), all the nucleus–electron non-adiabatic coupling terms C_X and C_XY(X, Y = R or P) become null. Thus, by incorporating all the mentioned simplifications, the END/PW equations become:

$$\left(\begin{array}{cccc}iC& 0& 0& 0\\ 0& -i{C}^{*}& 0& 0\\ 0& 0& 0& -\mathit{I}\\ 0& 0& \mathit{I}& 0\end{array}\right)\left(\begin{array}{c}\dot{z}\\ {\dot{z}}^{*}\\ \dot{R}\\ \dot{P}\end{array}\right)=\left(\begin{array}{l}\partial E_{\text{T}}/\partial z^{*}\hfill \\ \partial E_{\text{T}}/\partial z\hfill \\ \partial E_{\text{T}}/\partial R\hfill \\ \partial E_{\text{T}}/\partial P\hfill \end{array}\right)$$ (13)

where C is defined as before in Eq. (12) but with the R and P dependencies omitted. The challenge in the development of the END/PW formalism is to express the various components in Eq. (13) in terms of PWs, and that endeavor is accomplished in the next sections.

3.2. END/PW wave function

PW basis sets and their associated algorithms are appropriate to describe infinite periodic systems, e.g., a crystal or a regular polymer, with a specific unit cell. However, if the system under study is finite and aperiodic, e.g., two colliding molecules participating in a chemical reaction, a periodic system can be constructed by a series of supercells, each having a replica of the original system surrounded by a large vacuum space to avoid supercells interactions. This is a calculation construct because the interest does not lie in the period system itself but in the reaction happening inside any supercell. The following formalism is valid for both cell and supercell approaches (i.e. for both genuine periodic systems and for those constructed as such).

The first step in the development of the END/PW formalism is to define the END/PW total wave function ${\Psi }^{\text{END}/\text{PW}}(X,x,R,P)={\Psi }_n^{\text{END}/\text{PW}}(X;R,P){\Psi }_e^{\text{END}/\text{PW}}(x;z)$ in analogy to its SLEND and END/KSDFT counterparts, Eqs. (2)–(4). In the present approach, PWs are used to describe only electrons; therefore, the END/PW nuclear wave function ${\Psi }_n^{\text{END}/\text{PW}}(X;R,P)$ is identical to the SLEND one in Eq. (3). In contrast, the END/PW Thouless electronic wave function ${\Psi }_e^{\text{END}/\text{PW}}(x;z)$ and its building blocks, the END/PW molecular spin orbitals ${\psi }_i^{(\mathit{k})}(x)$ and dynamical ${\chi }_h^{(\mathit{k})}\left(x_g;z\right)$ spin orbitals, should be formulated in terms of PW basis functions ϕ(r, G). In that scheme, the END/PW ${\psi }_i^{(\mathit{k})}(x)$ are

$$\begin{gathered} {\psi }_i^{(k)}(x)={\tilde{\psi }}_i^{(\mathit{k})}(r){\sigma }_i^{(\mathit{k})}(s)=\sum _{G=0}^{\infty }{C}_i^{(\mathit{k})}(G){\varphi }^{(\mathit{k})}(r,G){\sigma }_i^{(\mathit{k})}(s) \\ {\varphi }^{(\mathit{k})}(r,G)=\frac{1}{\sqrt{\Omega }}{e}^{i(\mathit{k}+G)\cdot r}=e^{i\mathit{k}\cdot r}\varphi (r,G) \\ \varphi (r,G)=\frac{1}{\sqrt{\Omega }}{e}^{iG\cdot r} \end{gathered}$$ (14)

where k is a wave vector within the first Brillouin zone (BZ) [5, 39, 40] of the reciprocal cell, i ∈ { 1 … K_k} is a band index spanning a total of K_k electronic states for a given k, G is a reciprocal lattice vector, $C_i^{(\mathit{k})}(G)$ are the Fourier coefficients of the PW expansion, and Ω is the volume of the unit cell. ϕ^(k)(r, G)are regular PWs ϕ(r, G) augmented with the phase factors e^ik·r that arise from the translational symmetry of the periodic system [5, 39, 40]. In that scheme, each ${\psi }_i^{(\mathit{k})}(x)$ properly satisfies Bloch theorem for periodic systems [5, 39, 40]

$${\psi }_i^{(k)}\left(x+{\mathit{a}}_j\right)=e^{ik\cdot a_j}{\psi }_i^{(k)}(x)$$ (15)

where a_j is one of the Bravais lattice vectors (unit vectors of the real-space cell). The proof of the above equation ultimately relies on the Born-Von Karman periodic boundary conditions [5, 39, 40], $e^{iG\cdot a_j}=1$, and can be found in the standard literature on solid state physics [5, 39, 40]. END/ PW is developed in the KSDFT electronic framework. Therefore, the reference spin orbitals ${\psi }_i^{(\mathit{k})}(x)$ and their Fourier coefficients $C_i^{(\mathit{k})}(G)$ at initial time are obtained via Eq. (7) in KSDFT form and applied to periodic systems; that procedure leads to separated equations of ${\tilde{\psi }}_i^{(\mathit{k})}(r)$ for each k [40]. The resulting ${\psi }_i^{(k)}(x)$ satisfy the orthogonality condition $〈{\psi }_i^{(\mathit{k})}\mid {\psi }_j^{(l)}〉={\delta }_{ij}{\delta }_{\mathit{k}l}$.

The END/PW dynamical spin orbitals ${\chi }_h^{(\mathit{k})}\left(x_g^{\left(\mathit{k}\right)};z^{(\mathit{k})}\right)$ are constructed in analogy to Eq. (4) as:

$${\chi }_h^{(\mathit{k})}\left(x;z^{(k)}\right)={\chi }_h^{(k)}\left(r,s;z^{(k)}\right)={\psi }_h^{(k)\cdot }(r,s)+\sum _{p=N_k+1}^{K_k}{z}_{ph}^{(k)}{\psi }_p^{(k)°}(r,s)=e^{i\mathit{k}\cdot \mathit{r}}\left[\sum _{\mathit{G}=0}^{\infty }{C}_h^{(\mathit{k})}(\mathit{G})\varphi (\mathit{r},\mathit{G}){\sigma }_i^{(\mathit{k})}(s)+\sum _{p=N_{\mathit{k}}+1}^{{\mathit{K}}_{\mathit{k}}}{z}_{ph}^{(\mathit{k})}\sum _{\mathit{G}=0}^{\infty }{C}_p^{(\mathit{k})}(\mathit{G})\varphi (\mathit{r},\mathit{G}){\sigma }_p^{(\mathit{k})}(s)\right]$$ (16)

where the hole index h ∈ {1 … N_k} spans N_k occupied states for the given k, and the particle index p ∈ {N_k + 1 … K_k} spans K_k − N_k unoccupied (virtual) states for the same spans k. Notice that three limits of the h and p ranges, N_k, N_k + 1, and K_k, depend on the selected k vector. The ${\chi }_h^{(k)}\left(x_g;z\right)$ have all its constituent spin orbitals ${\psi }_h^{(k)·}$ and ${\psi }_p^{(k)°}$ with the same k vector. With that condition, the END/PW spin orbitals ${\chi }_h^{(k)}\left(x_g;z^{(k)}\right)$ also satisfy Bloch theorem since

$$\begin{gathered} {\chi }_h^{(\mathit{k})}\left(r+a_i,s;z^{(\mathit{k})}\right)={\psi }_h^{(\mathit{k})*}\left(r+a_i,s\right)+\sum _{p=N_{\mathit{k}}+1}^{{\mathit{K}}_{\mathit{k}}}{z}_{ph}^{(\mathit{k})}{\psi }_p^{(\mathit{k})°}\left(r+a_i,s\right) \\ =e^{ik\cdot a_i}{\psi }_h^{(k)•}(r,s)+\sum _{p=N_k+1}^{K_k}{z}_{ph}^{(\mathit{k})}{e}^{ik\cdot a_i}{\psi }_p^{(\mathit{k})°}(r,s) \\ =e^{ik\cdot a_i}\left({\psi }_h^{(\mathit{k})\cdot }(r,s)+\sum _{p=N_k+1}^{k_k}{z}_{ph}^{(\mathit{k})}{\psi }_p^{(\mathit{k})°}(r,s)\right) \\ =e^{ik\cdot a_i}{\chi }_h^{(\mathit{k})}\left(r,s;z^{(\mathit{k})}\right) \end{gathered}$$ (17)

where Eq. (15) was used from the first to the second line. The symmetry properties in Eq. (17) determine the correct symmetry of the electron density as shown in Sect. 3.3.

Having defined the END/PW ${\chi }_h^{(k)}\left(x_g;z^{(k)}\right)$, the END/PW Thouless electronic wave function ${\Psi }_e^{\text{END}/\text{PW}}(x;z)$ for a system with N_e electron is [cf. Eq. (4)]

$${\Psi }_e^{\text{END}/\text{PW}}(x;z)=x\mid {\chi }_1^{(\mathit{k})}\dots {\chi }_{N_{\mathit{k}}}^{(\mathit{k})}{\chi }_1^{(l)}\dots {\chi }_{N_l}^{(l)}\dots {\chi }_1^{(n)}\dots {\chi }_{N_n}^{(n)}=M\text{det}\left[{\chi }_1^{(\mathit{k})}\left(x_{1(\mathit{k})};z^{(\mathit{k})}\right)\dots {\chi }_{N_{\mathit{k}}}^{(\mathit{k})}\left(x_{N_{\mathit{k}}};z^{(\mathit{k})}\right)\dots {\chi }_1^{(n)}\left(x_{1(n)};z^{(n)}\right)\dots {\chi }_{N_n}^{(n)}\left(x_{N_n};z^{(n)}\right)\right]$$ (18)

where k, l, … n are N_BZ wave vectors in the BZ, $N_e={\sum }_{k\in BZ}^{N_{BZ}}{N}_k$ and M is a normalization constant. Notice that the ${\chi }_h^{(k)}\left(x_g;z^{(k)}\right)$ are not orthonormal, and therefore M ≠ (N_e!)^−1/2 as would be the case with a standard Slater determinant having orthogonal spin orbitals [36]. M is determined in Sect. 3.3. Like in SLEND and END/KSDFT, at initial time, z = 0 and in that situation, ${\Psi }_e^{\text{END}/\text{PW}}(x;z)$ becomes into a standard Slater determinant with the orthogonal hole spin orbitals ${\psi }_h^{\left(k\right)}$; that determinant represents the reference ground state of the system

$${\Psi }_e^{\text{END}/\text{PW}}(x;z=0)=x\mid {\psi }_1^{(\mathit{k})}\cdot \dots {\psi }_{N_{\mathit{k}}}^{(\mathit{k})}{\psi }_1^{(l)}\cdot \dots {\psi }_{N_l}^{(l)}\dots {\psi }_1^{(n)}\cdot \dots {\psi }_{N_n}^{(n)}={\left(N_e!\right)}^{-1/2}\text{det}\left[{\psi }_1^{(\mathit{k})*}\left(x_{1(\mathit{k})}\right)\dots {\psi }_{N_{\mathit{k}}}^{(\mathit{k})\cdot }\left(x_{N_{\mathit{k}}}\right)\dots {\psi }_1^{(n)*}\left(x_{1(n)}\right)\dots {\psi }_{N_n}^{(n)*}\left(x_{N_n}\right)\right]$$ (19)

3.3. END/PW electronic density

In the present KSDFT context, the electron density ρ(r) plays a central role because many END/PW terms are functionals of it; therefore, ρ(r) should be formulated first. Taking the one-electron density operator $\widehat{\rho }(r)={\sum }_i^{N_e}\delta \left(r_i-r\right)$ [36], the END/PW electron density ρ(r;z, z*) is the average value of $\widehat{\rho }(r)$ over ${\Psi }_e^{\text{END}/\text{PW}}(x;z)$ [36]

$$\rho \left(r;z,z^{*}\right)={\Psi }_e^{\text{END}/\text{PW}}(x;z)\left|\widehat{\rho }(r)\right|{\Psi }_e^{\text{END}/\text{PW}}(x;z)=\langle {\chi }_1^{(k)}\left(x_{1(k)};z^{(k)}\right)\dots {\chi }_{N_n}^{(n)}\left(x_{N_n};z^{(n)}\right)\left|\sum _{i=1}^{N_e}\delta \left(r_i-r\right)\right|{\chi }_1^{(k)}\left(x_{1(k)};z^{(k)}\right)\dots {\chi }_{N_n}^{(n)}\left(x_{N_n};z^{(n)}\right)\rangle$$ (20)

where ${\Psi }_e^{\text{END}/\text{PW}}(x;z)$ is normalized [cf. Eq. (18)]. Due to the non-orthogonality of the ${\chi }_h^{(\mathit{k})}\left(x_g;z^{(\mathit{k})}\right)$, the above operator average cannot be evaluated with the standard Slater–Condon rules [36]. Instead, the more general Löwdin rules involving non-orthogonal spin orbitals should be used [60]. The matrix element in Eq. (20) corresponds to a diagonal matrix element—i.e., an element with the same state on the bra and the ket sides, ${\Psi }_e^{\text{END}/\text{PW}}(x;z)$ —and with an operator $\widehat{\rho }(r)={\sum }_i^{N_e}\delta \left(r_i-r\right)$ that is a sum of one-electron operators. For such a case, Löwdin rules first requires constructing the N_e × N_e overlap matrix S_χχ of the ${\chi }_h^{(\mathit{k})}\left(x_g;z^{(\mathit{k})}\right)$ [60]

$$S_{\chi \chi }=\left[\begin{array}{ccccccc}〈{\chi }_1^{(\mathit{k})}|{\chi }_1^{(\mathit{k})}〉& \cdots & \langle {\chi }_1^{(\mathit{k})}|{\chi }_{N_{\mathit{k}}}^{(\mathit{k})}\rangle & \cdots & 〈{\chi }_1^{(\mathit{k})}|{\chi }_1^{(n)}〉& \cdots & 〈{\chi }_1^{(\mathit{k})}|{\chi }_{N_n}^{(n)}〉\\ \langle {\chi }_2^{(\mathit{k})}|{\chi }_1^{(\mathit{k})}\rangle & \cdots & \langle {\chi }_2^{(\mathit{k})}\left|{\chi }_{N_{\mathit{k}}}^{(\mathit{k})}\right.〉& \cdots & 〈{\chi }_2^{(\mathit{k})}|{\chi }_1^{(n)}〉& \cdots & 〈{\chi }_2^{(\mathit{k})}|{\chi }_{N_n}^{(n)}〉\\ ⋮& \ddots & ⋮& \cdots & ⋮& \ddots & ⋮\\ 〈{\chi }_{Nn}^{(n)}|{\chi }_1^{(\mathit{k})}〉& \cdots & 〈{\chi }_{N_n}^{(n)}|{\chi }_{N_{\mathit{k}}}^{(\mathit{k})}〉& \cdots & 〈{\chi }_{N_n}^{(n)}|{\chi }_1^{(n)}〉& \cdots & 〈{\chi }_{N_n}^{(n)}|{\chi }_{N_n}^{(n)}〉\end{array}\right]$$ (21)

where, again, $N_e={\sum }_{\mathit{k}\in BZ}^{N_{BZ}}{N}_{\mathit{k}}$. Then, Löwdin rules provides the normalization constant M of ${\Psi }_e^{\text{END}/\text{PW}}(x;z)$ in Eq. (18) as

$$M={\left[\left(N_e\right)!D_{\chi \chi }\right]}^{-1/2};D_{\chi \chi }=\text{det}\left(S_{\chi \chi }\right)$$ (22)

and ρ(r;z, z*) from Eq. (20) as

$$\rho \left(r;z,z^{*}\right)=\sum _{k,I\in BZ}^{N_{BZ}{N}_{BZ}}\sum _{g,h=1}^{N_l,N_{\mathit{k}}}\langle {\chi }_g^{(l)}\left(r_i,s\right)\left|\delta \left(r_i-r\right)\right|{\chi }_h^{(\mathit{k})}\left(r_i,s\right)\rangle {\left(S_{\chi \chi }^{{(\mathit{k}l)}^{-1}}\right)}_{hg}$$ (23)

However, with the present choice of ${\chi }_h^{(\mathit{k})}\left(x_g^{\left(\mathit{k}\right)};z^{(\mathit{k})}\right)$ in Eq. (16), S_χχ adopts a relatively simpler form due to the orthogonality with respect to k of the ${\psi }_i^{(\mathit{k})}$ composing the ${\chi }_h^{(\mathit{k})}$, Eq. (14),

$${\left(S_{\chi \chi }^{(\mathit{k}l)}\right)}_{hg}=〈{\chi }_h^{(k)}\mid {\chi }_g^{(l)}〉={\delta }_{\mathit{k}l}〈{\chi }_h^{(k)}\mid {\chi }_g^{(k)}〉={\delta }_{\mathit{k}l}{\left(S_{\chi \chi }^{(\mathit{k}\mathit{k})}\right)}_{hg}$$ (24)

where $S_{\chi \chi }^{(\mathit{k}\mathit{k})}$ are the N_BZ diagonal-block submatrices of S_χχ for each k and of dimensions N_k × N_k, $S_{\chi \chi }=S_{\chi \chi }^{(\mathit{k}\mathit{k})}\oplus S_{\chi \chi }^{(ll)}\oplus S_{\chi \chi }^{(mm)}$ … Then, from Eq. (16),

$$\begin{gathered} {\left(S_{\chi \chi }^{(\mathit{k}\mathit{k})}\right)}_{hg}=〈{\chi }_h^{(k)}\mid {\chi }_g^{(k)}〉=〈{\psi }_h^{(k)•}(r,s)+\sum _{p=N_k+1}^{K_k}{z}_{ph}^{(k)}{\psi }_p^{(k)°}(r,s)\mid {\psi }_g^{(k)•}(r,s)+\sum _{q=N_{\mathit{k}}+1}^{K_k}{z}_{qg}^{(k)}{\psi }_q^{(k)°}(r,s)〉={\delta }_{hg}^{(k)}+\sum _{p=N_k+1}^{K_k}\sum _{q=N_k+1}^{K_k}{z}_{ph}^{(k)*}{z}_{qg}^{(k)}{\delta }_{pq}^{(k)} \\ ={\delta }_{hg}^{(k)}+\sum _{p=N_k+1}^{k_k}{z}_{hp}^{(k)†}{z}_{pg}^{(k)} \\ ={\left({\mathit{I}}^{(k)•}+z^{(k)†}{z}^{(k)}\right)}_{hg} \end{gathered}$$ (25)

where I^(k)∙ = (δ_ij) and $z^{(\mathit{k})}=\left(z_{ph}^{(\mathit{k})}\right)$ are N_k × N_k and (K_k − N_k) × N_k matrices, respectively. Under these conditions, the normalization constant M of ${\Psi }_e^{\text{END}/\text{PW}}$ in Eq. (22) becomes

$$\begin{gathered} M={\left[\left(N_e\right)!\prod _{k\in BZ}^{{\mathit{K}}_{BZ}}{D}_{\chi \chi }^{(\mathit{k}\mathit{k})}\right]}^{-1/2}; \\ {D}_{\chi \chi }^{(\mathit{k}\mathit{k})}=\text{det}\left(S_{\chi \chi }^{(\mathit{k}\mathit{k})}\right)=\text{det}\left(I^{(k)\cdot }+z^{(k)†}{z}^{(k)}\right.) \end{gathered}$$ (26)

and ρ(r;z, z*) from Eq. (23) with the Löwdin rules becomes

$$\begin{gathered} \rho \left(r;z,z^{*}\right)=\sum _{k,l\in BZ}^{N_{BZ},N_{BZ}}\sum _{g,h=1}^{N_l{N}_k}{\chi }_g^{(l)}\left(r_i,s\right)\left|\delta \left(r_i-r\right)\right|{\chi }_h^{(k)}\left(r_i,s\right){\delta }_{kl}{\left(S_{\chi \chi }^{{(kk)}^{-1}}\right)}_{hg} \\ =\sum _{\mathit{k}\in BZ}^{N_{BZ}}\sum _{g,h=1}^{N_l,N_{\mathit{k}}}\left(\iint {\chi }_g^{(\mathit{k})*}\left(r_i,s\right)\delta \left(r_i-r\right){\chi }_h^{(\mathit{k})}\left(r_i,s\right)\text{d}{r}_i\text{d}s\right){\left(S_{\chi \chi }^{{(\mathit{k}\mathit{k})}^{-1}}\right)}_{hg} \\ =\sum _{\mathit{k}\in BZ}^{N_{BZ}}\sum _{g,h=1}^{N_{\mathit{k}},N_{\mathit{k}}}\left(\int {\chi }_g^{(\mathit{k})*}(r,s){\chi }_h^{(\mathit{k})}(r,s)\text{d}s\right){\left[{\left(I^{(\mathit{k})}\cdot +z^{(\mathit{k})†}{z}^{(\mathit{k})}\right)}^{-1}\right]}_{hg} \end{gathered}$$ (27)

where the integration properties of the Dirac delta δ(r_i – r) were used from the second to the third line, and ${\left(S_{\chi \chi }^{{(\mathit{k}\mathit{k})}^{-1}}\right)}_{gh}$ from Eq. (25) was introduced in the last line. Then, introducing the expressions of the ${\chi }_h^{(\mathit{k})}\left(x_g;z^{(\mathit{k})}\right)$ from Eq. (16),

$$\begin{gathered} \rho \left(r;z,z^{*}\right) \\ =\int \left(\sum _{\mathit{k}\in BZ}^{N_{BZ}}\sum _{g,h=1}^{N_{\mathit{k}},N_{\mathit{k}}}{\left[{\left(I^{(\mathit{k})•}+z^{(\mathit{k})†}{z}^{(\mathit{k})}\right)}^{-1}\right]}_{hg}\left[{\psi }_h^{(\mathit{k})\cdot }(r,s){\psi }_g^{(\mathit{k})\cdot *}(r,s)\right.\right. \\ +\sum _{q=N_{\mathit{k}}+1}^{{\mathit{k}}_{\mathit{k}}}{\psi }_h^{(\mathit{k})•}(r,s)z_{qg}^{(\mathit{k})*}{\psi }_q^{(\mathit{k})°*}(r,s) \\ +\sum _{p=N_{\mathit{k}}+1}^{{\mathit{k}}_{\mathit{k}}}{\psi }_p^{(\mathit{k})°}(r,s)z_{ph}^{(\mathit{k})}{\psi }_g^{(\mathit{k})•*}(r,s) \\ +\left.\sum _{q=N_{\mathit{k}}+1}^{{\mathit{k}}_{\mathit{k}}}\sum _{p=N_{\mathit{k}}+1}^{{\mathit{k}}_{\mathit{k}}}{\psi }_p^{(\mathit{k})°}(r,s)z_{ph}^{(\mathit{k})}{z}_{qg}^{(\mathit{k})*}{\psi }_q^{(\mathit{k})°*}(r,s)\right]\text{d}s \\ =\int \left(\sum _{\mathit{k}\in BZ}^{N_{BZ}}\sum _{i,j=1}^{{\mathit{K}}_{\mathit{k}},{\mathit{K}}_{\mathit{k}}}{\psi }_i^{(\mathit{k})}(r,s){\Gamma }_{ij}^{(\mathit{k}\mathit{k})}\left(z^{*},z\right){\psi }_j^{(\mathit{k})*}(r,s)\right)\text{d}s \end{gathered}$$ (28)

where the one-electron density matrix ${\Gamma }_{ij}^{(\mathit{k}\mathit{k})}\left(z^{*},z\right)$ is

$${\Gamma }_{ij}^{(\mathit{k}\mathit{k})}\left(z^{*},z\right)=\sum _{h,g=1}^{N_k,N_k}{\left(\begin{array}{c}{I}^{(k)•}\\ z^{(k)÷}\end{array}\right)}_{ih}{\left[{\left(I^{(\mathit{k})•}+z^{(\mathit{k})†}{z}^{(\mathit{k})}\right)}^{-1}\right]}_{hg}{\left(I^{(\mathit{k})•}{z}^{(\mathit{k})†}\right)}_{gj}$$ (29)

By performing spin integration in Eq. (28), one obtains

$$\begin{gathered} \rho \left(r;z,z^{*}\right)=\sum _{\mathit{k}\in BZ}^{N_{BZ}}\sum _{i,j=1}^{{\mathit{K}}_{\mathit{k}},{\mathit{K}}_{\mathit{k}}}{\tilde{\psi }}_i^{(\mathit{k})}(r){\Gamma }_{ij}^{(\mathit{k}\mathit{k})}\left(z^{*},z\right){\tilde{\psi }}_j^{(\mathit{k})*}(r)\int {\sigma }_i^{(\mathit{k})}(s){\sigma }_j^{(\mathit{k})*}(s)\text{d}s \\ =\sum _{\mathit{k}\in BZ}^{N_{BZ}}\sum _{i,j=1}^{{\mathit{K}}_{\mathit{k}},{\mathit{K}}_{\mathit{k}}}{\tilde{\psi }}_i^{(\mathit{k})}(r){\Gamma }_{ij}^{(\mathit{k}\mathit{k})}\left(z^{*},z\right){\tilde{\psi }}_j^{(\mathit{k})*}(r){\delta }_{{\sigma }_i^{(\mathit{k})}{\sigma }_j^{(\mathit{k})}} \end{gathered}$$ (30)

Since k is a continuous variable, the above discrete sum in k should be substituted for an integration over the wave vectors in the BZ [5, 39, 40]

$$\rho \left(r;z,z^{*}\right)\propto \int \sum _{i,j=1}^{{\mathit{k}}_k,{\mathit{K}}_{\mathit{k}}}{\tilde{\psi }}_i^{(\mathit{k})}(r){\Gamma }_{ij}^{(\mathit{k}\mathit{k})}\left(z^{*},z\right){\tilde{\psi }}_j^{(\mathit{k})*}(r){\delta }_{{\sigma }_i^{(\mathit{k})}{\sigma }_j^{(\mathit{k})}}\text{d}k$$ (31)

In practice, such an integration is accomplished numerically by a BZ average [5, 39, 40]

$$\rho \left(r,z,z^{*}\right)=\sum _{\mathit{k}\in BZ}^{N_{BZ}}{\omega }_{\mathit{k}}\sum _{i,j=1}^{{\mathit{K}}_{\mathit{k}},{\mathit{K}}_{\mathit{k}}}{\tilde{\psi }}_i^{(\mathit{k})}(r){\Gamma }_{ij}^{(\mathit{k}\mathit{k})}\left(z^{*},z\right){\tilde{\psi }}_j^{(\mathit{k})*}(r){\delta }_{{\sigma }_i^{(\mathit{k})}{\sigma }_j^{(\mathit{k})}}$$ (32)

that involves some selected k values with weights ω_k, ${\sum }_{\mathit{k}\in BZ}^{N_{BZ}}{\omega }_{\mathit{k}}{N}_{\mathit{k}}=N$. In some cases, it is computationally more efficient to work with the END/PW electron density in reciprocal space ρ(G;z, z*) (cf. Sect. 2.2). Accordingly, from Eq. (32)

$$\begin{gathered} \rho \left(G;z,z^{*}\right)=\frac{1}{\Omega }{\int }_{\Omega }\rho \left(r;z,z^{*}\right)e^{-iG\cdot r}\text{d}r \\ =\sum _{\mathit{k}\in BZ}^{N_{BZ}}{\omega }_{\mathit{k}}\sum _{i,j=1}^{{\mathit{K}}_{\mathit{k}},{\mathit{K}}_{\mathit{k}}}{\Gamma }_{ij}^{(\mathit{k}\mathit{k})}\left(z^{*},z\right){\delta }_{{\sigma }_i^{(\mathit{k})}{\sigma }_j^{(\mathit{k})}} \\ \times \sum _{G^{\prime },G^{\prime \prime }=0}^{\infty ,\infty }{C}_j^{(\mathit{k})*}\left(G^{\prime \prime }\right)C_i^{(\mathit{k})}\left(G^{\prime }\right)\frac{1}{\Omega }{\int }_{\Omega }{\varphi }^{(\mathit{k})*} \\ \left(r,G^{\prime \prime }\right){\varphi }^{(\mathit{k})}\left(r,G^{\prime }\right)e^{-iG\cdot r}\text{d}r \\ =\frac{1}{\Omega }\sum _{\mathit{k}\in BZ}^{N_{BZ}}{\omega }_{\mathit{k}}\sum _{i,j=1}^{{\mathit{K}}_{\mathit{k}},{\mathit{K}}_{\mathit{k}}}{\Gamma }_{ij}^{(\mathit{k}\mathit{k})}\left(z^{*},z\right){\delta }_{{\sigma }_i^{(\mathit{k})}{\sigma }_j^{(\mathit{k})}}\sum _{G^{\prime }=0}^{\infty ,\infty }{C}_j^{(\mathit{k})} \\ \left(G^{\prime }-G\right)C_i^{(\mathit{k})}\left(G^{\prime }\right) \end{gathered}$$ (33)

where the ${\tilde{\psi }}_i^{(\mathit{k})}$ were expanded in terms of the generalized PWs ϕ^(k)(r, G), Eq. (14), from the first to the second line, and the PWs orthogonality properties were used from the second to the third line:

$$\begin{gathered} {\int }_{\Omega }{\varphi }^{(\mathit{k})*}\left(r,G^{\prime \prime }\right){\varphi }^{(\mathit{k})}\left(r,G^{\prime }\right)e^{-iG\cdot r}\text{d}r= \\ \frac{1}{\Omega }{\int }_{\Omega }{e}^{-i\left(G+G^{\prime \prime }-G^{\prime }\right)\cdot r}\text{d}r={\delta }_{G+G^{\prime \prime },G^{\prime }} \end{gathered}$$ (34)

It is important to examine the initial conditions of ${\Gamma }_{ij}^{(\mathit{k}\mathit{k})}\left(z^{*},z\right)$ and ρ(r;z, z*). At initial time,, and from Eq. (29) one obtains

$${\Gamma }_{ij}^{(\mathit{k}\mathit{k})}\left(z^{*}=0,z=0\right)={\delta }_{ij}{f}_i^{(k)}$$ (35)

where $f_h^{(k)}=1$ (holes) and $f_p^{(k)}=0$ (particles) are the occupation numbers of the molecular spin orbitals ${\psi }_i^{(k)}(x)$ in the reference ground state. Under these conditions, the END/PW ρ(r;z, z*) correctly reverts to the standard time-independent electron density in solid state physics that corresponds to the reference ground state of the periodic system [5, 39, 40]

$$\rho \left(r;z=0,z^{*}=0\right)=\sum _{\mathit{k}\in BZ}^{N_{BZ}}{\omega }_{\mathit{k}}\sum _{i=1}^{{\mathit{k}}_{\mathit{k}}}{f}_i^{(\mathit{k})}{\left|{\tilde{\psi }}_i^{(\mathit{k})}(r)\right|}^2$$ (36)

The END/PW density ρ(r;z, z*) in Eq. (32) correctly reproduces the symmetry of the periodic system since

$$\begin{gathered} \rho \left(r+a_m;z,z^{*}\right) \\ =\sum _{\mathit{k}\in BZ}^{N_{BZ}}{\omega }_{\mathit{k}}\sum _{i,j=1}^{{\mathit{K}}_{\mathit{k}},{\mathit{K}}_{\mathit{k}}}{e}^{+i\mathit{k}\cdot a_m}{\tilde{\psi }}_i^{(\mathit{k})}(r){\Gamma }_{ij}^{(\mathit{k}\mathit{k})}\left(z^{*},z\right)e^{-ik\cdot a_m}{\tilde{\psi }}_j^{(\mathit{k})*}(r){\delta }_{{\sigma }_i^{(\mathit{k})}{\sigma }_j^{(\mathit{k})}} \\ =\sum _{\mathit{k}\in BZ}^{N_{BZ}}{\omega }_{\mathit{k}}\sum _{i,j=1}^{{\mathit{K}}_{\mathit{k}},{\mathit{K}}_{\mathit{k}}}{\tilde{\psi }}_i^{(\mathit{k})}(r){\Gamma }_{ij}^{(\mathit{k}\mathit{k})}\left(z^{*},z\right){\tilde{\psi }}_j^{(\mathit{k})*}(r){\delta }_{{\sigma }_i^{(\mathit{k})}{\sigma }_j^{(\mathit{k})}} \\ =\rho \left(r;z,z^{*}\right) \end{gathered}$$ (37)

where Eq. (15) was directly applied in the first line. The symmetry property of the END/PW ρ(r;z, z*) revealed by Eq. (37) is appropriate to simulate a reactive molecular system A + B + ⋯→ P +Q ⋯in the supercell approach because the electron density should be the same in equivalent points of the supercells. The same holds for genuine periodic system such as crystals and regular polymers. Therefore, the END/PW ρ(r;z, z*) from Eq. (32) will be used henceforth.

3.4. END/PW total energy

The next term to derive in the END/PW formalism is its total energy E_T per unit (super-) cell in analogy to the END/KSDFT total energy in Eq. (11). As discussed in Sect. 2.2 END/PW will be numerically efficient if the PWs explicitly represent valence electrons while pseudopotentials implicitly represent core electrons. In that approach, E_T with the END/PW electron density from ρ=ρ(r;z, z*) Eq. (32) is [40]:

$$\begin{gathered} E_{\text{T}}\left(z,z^{*},R,P\right)=T_{\text{i}}(P)+T_{\text{e}}\left(z,z^{*}\right)+E_{\text{ii}}(R)+E_{\text{H}}\left[\rho \left(r;z,z^{*}\right)\right] \\ +E_{\text{PS}}^{\text{loc}}\left[\rho \left(r;z,z^{*}\right),R\right]+E_{\text{xc}}\left[\rho \left(r;z,z^{*}\right)\right]+E_{\text{PS}}^{\text{nl}}\left(z,z^{*},R\right) \\ =T_{\text{i}}(P)+T_{\text{e}}\left(z,z^{*}\right)+E_{\text{es}}\left[\rho \left(r;z,z^{*}\right),R\right]+E_{\text{xc}}\left[\rho \left(r;z,z^{*}\right)\right]+E_{\text{PS}}^{\text{nl}}\left(z,z^{*},R\right) \end{gathered}$$ (38)

where T_i(P) is the classical mechanics kinetic energy of the ions, T_e(z, z*) the KSDFT kinetic energy of the electrons, E_ii(R) the ion–ion repulsion energy, E_H[ρ(r;z, z*)] is the Hartree (self-repulsion) energy of the electrons, $E_{\text{PS}}^{\text{loc}}\left[\rho \left(r;z,z^{*}\right),R\right]$ the electron–ion interaction energy involving local pseudopotentials, E_xc [ρ(r;z, z*)] the KSDFT exchange–correlation energy of the electrons, and $E_{\text{PS}}^{\text{nl}}\left(z,z^{*},R\right)$ the electron–ion interaction energy involving non-local pseudopotentials. The ions in the above terms are just the aggregates of the bare nuclei with their respective core electrons. These ions result from the use of pseudopotentials and replace the bare nuclei of END/KSDFT. Consequently, E_ii(R) replaces the END/KSDFT nucleus–nucleus repulsion energy E_nn(R) and $E_{\text{PS}}^{\text{loc}}\left[\rho \left(r;z,z^{*}\right),R\right]$ and $E_{\text{PS}}^{\text{nl}}\left(z,z^{*},R\right)$ replace the END/KSDFT external potential energy V_ext[ρ, R] exerted by the bare nuclei [cf. Equation (11)]. For numerical reasons discussed in Sect. 3.6, E_ii(R), E_H[ρ(r;z, z*)], and $E_{\text{PS}}^{\text{loc}}\left[\rho \left(r;z,z^{*}\right),R\right]$ are evaluated together as the electrostatic energy $E_{\text{es}}\left[\rho \left(r;z,z^{*}\right),R\right]=E_{\text{ii}}(R)+E_{\text{H}}\left[\rho \left(r;z,z^{*}\right)\right]+E_{\text{PS}}^{\text{loc}}\left[\rho \left(r;z,z^{*}\right),R\right]$. The expressions of all of the terms in E_T are derived below.

3.5. END/PW nuclear and electronic kinetic energies

Similarly to its SLEND and END/KSDFT counterparts, the END/PW kinetic energy of the classical ions Ti(P) is

$$T_{\text{i}}(P)=\sum _{I=1}^{N_I}\frac{P_I^2}{2M_I}$$ (39)

where M₁ and P₁ are the mass and momentum of the ion I and N₁ is the total number of ions in the (super-)cell. On the other hand, the END/PW kinetic energy of the electrons T_e(z, z*) is the average value of the total kinetic operator ${\widehat{T}}_e=-1/2{\sum }_{i=1}^{N_e}{\nabla }_i^2\text{ over }{\Psi }_e^{\text{END}/\text{PW}}(x;z)$

$$\begin{gathered} T_{\text{e}}\left(z,z^{*}\right)=\langle {\Psi }_e^{\text{END}/\text{PW}}(x;z)\left|{\widehat{T}}_e\right|{\Psi }_e^{\text{END}/\text{PW}}(x;z)\rangle \\ =\langle {\chi }_1^{(\mathit{k})}\left(x_{1(\mathit{k})};z^{(\mathit{k})}\right)\dots {\chi }_{N_n}^{(n)}\left(x_{N_n};z^{(n)}\right)\left|-\frac{1}{2}\sum _{i=1}^{N_e}{\nabla }_i^2\right|{\chi }_1^{(\mathit{k})}\left(x_{1(k)};z^{(\mathit{k})}\right)\dots {\chi }_{N_n}^{(n)}\left(x_{N_n};z^{(n)}\right)\rangle \end{gathered}$$ (40)

The above expression is analogous to that of the END/ PW density ρ(r;z, z*), Eq. (20), with ${\widehat{T}}_e$ replacing the density operator $\widehat{\rho }(r)$. Therefore, the application of Löwdin rules [60] to T_e(z, z*) as was done for ρ(r;z, z*) in Eqs. (20)–(32) renders

$$T_{\text{e}}\left(z,z^{*}\right)=-\frac{1}{2}\sum _{\mathit{k}\in BZ}{\omega }_{\mathit{k}}\sum _{i,j=1}^{{\mathit{K}}_{\mathit{k}}{\mathit{K}}_{\mathit{k}}}{\Gamma }_{ij}^{(\mathit{k}\mathit{k})}\left(z^{*},z\right){\delta }_{{\sigma }_i^{(\mathit{k})}{\sigma }_j^{(\mathit{k})}}\int {\tilde{\psi }}_j^{(\mathit{k})*}(r){\nabla }^2{\tilde{\psi }}_i^{(\mathit{k})}(r)\text{d}r$$ (41)

The above expression is in real space but it can be computed more efficiently in reciprocal space where the kinetic operator is diagonal. To accomplish that one should expand the spatial orbitals ${\tilde{\psi }}_i^{(\mathit{k})}(r)$ in terms of the generalized PWs ϕ^(k)(r, G), Eq. (14), set ∇²ϕ^(k)(r,G) = −|k + G|²ϕ^(k)(r,G), and perform the resulting integrals involving the orthogonal ²ϕ^(k)(r,G). Following those steps, T_e(z, z*) in reciprocal space results

$$T_{\text{e}}\left(z,z^{*}\right)=\frac{1}{2}\sum _{\mathit{k}\in BZ}{\omega }_{\mathit{k}}\sum _{i,j=1}^{{\mathit{K}}_{\mathit{k}}{\mathit{K}}_{\mathit{k}}}{\Gamma }_{ij}^{(\mathit{k}\mathit{k})}\left(z^{*},z\right){\delta }_{{\sigma }_i^{(\mathit{k})}{\sigma }_j^{(\mathit{k})}}\sum _{G=0}^{\infty }{C}_j^{(\mathit{k})*}(G)C_i^{(\mathit{k})}(G)|\mathit{k}+G{|}^2$$ (42)

3.6. END/PW electrostatic energy

Unlike previous terms, the END/PW electrostatic energy E_es[ρ, R], Eq. (38), can be obtained directly by setting the END/PW density ρ = ρ(r;z, z*) from Eq. (32) into the E_es[ρ, R] expression routinely used in solid state physics [40]. Since the derivation of E_es[ρ, R] for periodic systems is well-known [40], only its final expression and a few remarks about its components are presented herein. The evaluation of E_es[ρ, R] over periodic systems involves a series of standard techniques designed to prevent divergent results. One component of those techniques is a neutralizing, continuous, auxiliary density ρ_i(r) that is added to the actual density ρ(r;z, z*) to produce a neutral total density ρ_T (r;z, z*) = ρ_i(r) + ρ(r;z, z*). ρ_i(r) is a sum of Gaussian charge distributions centered on the ion positions R_I and with exponent coefficient η². In that scheme, E_es[ρ, R] is [40]

$$\begin{gathered} E_{\text{es}}\left[\rho \left(G;z,z^{*}\right),R\right]=2\Omega \pi \sum _{G\ne 0}\frac{{\left|{\rho }_T\left(G;z,z^{*}\right)\right|}^2}{G^2} \\ +\Omega \sum _{G\ne 0}\sum _{s=1}^{N_s}{S}_s(G;R)u_{\text{PS}}^{\text{loc},s}(G)\rho \left(-G;z,z^{*}\right) \\ +\frac{1}{2}\sum _{I=1}^{N_I}\sum _{J\ne I}^{N_I}{Z}_I{Z}_J\left[\sum _{L=-L_{\text{max}}}^L\frac{\text{erfc}\left(\left|R_I+L-R_J\right|\eta \right)}{\left|R_I+L-R_J\right|}\right] \\ -\frac{\eta }{\sqrt{\pi }}\sum _{I=1}^{N_I}{Z}_I^2-\frac{Q}{\Omega }\left(\sum _{s=1}^{N_s}{P}_s\Delta v_{\text{PS}}^{1\text{lc},s}+\frac{\pi Q}{2{\eta }^2}\right) \end{gathered}$$ (43)

where ρ_T (G;z, z*) and ρ (G;z, z*), Eq. (33), are the total and real electron densities in reciprocal space, respectively, s labels the N_s atomic species (i.e., the chemical identities of the ions) for the pseudopotentials, S_s(G;R) is the atomic structure factor of the species s, $u_{\text{PS}}^{\text{loc},s}(G)$ the reciprocal space local pseudopotential $v_{\text{PS}}^{\text{loc},s}$ for the species s including the ρ_i(r) contribution, Z_I are the charges of the N_I ions, L_max is the maximum number of periodically repeated boxes in the real-space Ewald sum part of E_ii(R) [third term on the right-hand side of Eq. (43)], the complementary error function, Q = N_e the total electronic charge in the cell, P_s the number of ion positions R_I for the species s, and $\Delta v_{\text{PS}}^{\text{loc},s}$ a pseudopotential integral for G = 0. Further details of these terms are given in Ref. [40] and references cited therein.

3.7. END/PW exchange–correlation energy

The END/PW exchange–correlation energy E_xc[ρ] is its standard KSDFT expression [44] specialized with the END/PW density ρ = ρ(r;z, z*), Eq. (32). The general form of E_xc[ρ] in real space for exchange–correlation functionals up to the generalized gradient approximation (GGA) level is [5], [44]

$$E_{\text{xc}}\left[\rho \left(r;z,z^{*}\right)\right]=\int \rho \left(r;z,z^{*}\right){\epsilon }_{\text{xc}}\left[\rho \left(r;z,z^{*}\right),\nabla \rho \left(r;z,z^{*}\right)\right]\text{d}r$$ (44)

where ε_xc[ρ(r;z, z*). ∇ρ(r;z, z*)] is the exchange–correlation energy density functional with gradient corrections [5]; the explicit form of the latter depends on the type of exchange–correlation functional chosen.

3.8. END/PW non‑local electron–ion interaction energy

The END/PW non-local electron–ion interaction $E_{\text{PS}}^{\text{nl}}\left(z,z^{*},R\right)$ involves a total pseudopotential operator ${\widehat{V}}_{\text{PS}}^{\text{nl Total }}$

$${\widehat{V}}_{\text{PS}}^{\text{nl Total }}=\sum _{i=1}^{N_e}{\widehat{V}}_{\text{PS}}^{\text{nl}}\left(r_i;R\right)=\sum _{i=1}^{N_e}\left(\sum _{s=1}^{N_s}\sum _{l=0}^{l_{max}}\Delta {\widehat{V}}_{\text{PS}}^{l,s}\left(r_i;R\right){\widehat{P}}_l\right)$$ (45)

where ${\widehat{V}}_{\text{PS}}^{\text{nl}}\left(r_i;R\right)$ is the one-electron pseudopotential operator for the electron i. $\Delta {\widehat{V}}_{\text{PS}}^{l,s}\left(r_i;R\right){\widehat{P}}_l$. is the pseudopotential operator component of the atomic species for the angular momenta l of the non-local projector operator ${\widehat{P}}_l$ of the spherical harmonic |Y_lm〉 [40]. $E_{\text{PS}}^{\text{nl}}\left(z,z^{*},R\right)$ is the average value of ${\widehat{V}}_{\text{PS}}^{\text{nl Total}}$ over ${\Psi }_e^{\text{END}/\text{PW}}(x;z)$

$$\begin{gathered} E_{\text{PS}}^{\text{nl}}\left(z,z^{*},R\right)=\langle {\Psi }_e^{\text{END}/\text{PW}}(x;z)\left|{\widehat{V}}_{\text{PS}}^{\text{nl Total }}\right|{\Psi }_e^{\text{END}/\text{PW}}(x;z)\rangle \\ =\langle {\chi }_1^{(\mathit{k})}\left(x_{1(\mathit{k})};z^{(\mathit{k})}\right)\dots {\chi }_{N_n}^{(n)}\left(x_{N_n};z^{(n)}\right)\left|\sum _{i=1}^{N_e}{\widehat{V}}_{\text{PS}}^{\text{nl}}\left(r_i;R\right)\right|{\chi }_1^{(\mathit{k})}\left(x_{1(\mathit{k})};z^{(\mathit{k})}\right)\dots {\chi }_{N_n}^{(n)}\left(x_{N_n};z^{(n)}\right)\rangle \end{gathered}$$ (46)

This expression is analogous to that of T_e(z, z*), Eq. (40). Therefore, through a procedure similar to that performed in Eqs. 40–41, one obtains

$$E_{\text{PS}}^{\text{nl}}\left(z,z^{*},R\right)=\sum _{\mathit{k}\in BZ}{\omega }_{\mathit{k}}\sum _{i,j=1}^{{\mathit{K}}_{\mathit{k}},{\mathit{K}}_{\mathit{k}}}{\Gamma }_{ij}^{(\mathit{k}\mathit{k})}\left(z^{*},z\right){\delta }_{{\sigma }_i^{(\mathit{k})}{\sigma }_j^{(\mathit{k})}}\int {\tilde{\Psi }}_j^{(\mathit{k})*}(r){\widehat{V}}_{\text{PS}}^{\text{nl}}(r;R){\tilde{\psi }}_i^{(\mathit{k})}(r)\text{d}r$$ (47)

By expanding the spatial orbitals ${\tilde{\psi }}_i^{(\mathit{k})}$ in terms of the generalized PWs ϕ^(k)(r,G) = 〈r|k + G〉, Eq. (14), and using Eq. (45), the integrals in Eq. (47) are

$$\begin{gathered} \int {\tilde{\psi }}_j^{(\mathit{k})*}(r){\widehat{V}}_{\text{PS}}^{\text{nl}}(r;R){\tilde{\psi }}_i^{(\mathit{k})}(r)\text{d}r \\ =\sum _{s=1}^{N_s}\sum _{l=0}^{l\text{max}}\sum _{G=0}^{\infty }{C}_j^{(\mathit{k})*}(G)C_i^{(\mathit{k})}(G)k+G\left|\Delta {\widehat{V}}_{\text{PS}}^{l,s}(r;R){\widehat{P}}_l\right|k+G \end{gathered}$$ (48)

Specific expressions for various kinds of operators $\Delta {\widehat{V}}_{\text{PS}}^{l,s}(r;R){\widehat{P}}_l$ and their integrals $\mathit{k}+G\left|\Delta {\widehat{V}}_{\text{PS}}^{l,s}(r;R){\widehat{P}}_l\right|\mathit{k}+G$ are given in the standard literature on solid state physics [40]. For instance, by adopting the numerically efficient pseudopotential by Kleiman and Bylander [40, 61], $E_{\text{PS}}^{\text{nl}}\left(z,z^{*},R\right)$ becomes

$$\begin{gathered} E_{\text{PS}}^{\text{nl}}\left(z,z^{*},R\right)=\sum _{\mathit{k}\in BZ}{\omega }_{\mathit{k}}\cdot \sum _{s=1}^{N_s}\sum _{l=0}^{l\text{max}}\sum _{m=-l}^l\sum _{I=1}^{N_I}\sum _{i,j=1}^{{\mathit{K}}_{\mathit{k}},{\mathit{K}}_{\mathit{k}}}{F}_{I,i}^{lm,s(\mathit{k})}{\Gamma }_{ij}^{(\mathit{k}\mathit{k})}\left(z^{*},z\right){\delta }_{{\sigma }_i^{(\mathit{k})}{\sigma }_j^{(\alpha )}}{\alpha }_{lm}^s{F}_{I,j}^{lm,s{(\mathit{k})}^{*}} \\ {F}_{I,i}^{lm,s(\mathit{k})}=\sum _G{e}^{iG\cdot R_l}{f}_{lm}^s(\mathit{k}+G)C_i^{(\mathit{k})}(G)\#\#\#\#f_{lm}^s(\mathit{k}+G)=\int r^2{\Phi }_{PS}^{lm,s*}(r)\Delta v_{PS}^{ls}(r)j_l(|\mathit{k}+G|r)\text{d}r \\ {\alpha }_{lm}^s={\left(\int r^2{\Phi }_{\text{PS}}^{lm,s*}(r)\Delta v_{PS}^{l,s}(r){\Phi }_{PS}^{lm,s}(r)\text{d}r\right)}^{-1} \\ \Delta v_{\text{PS}}^{l,s}(r)=v_{\text{PS}}^{l,s}(r)-v_{\text{PS}}^{\text{loc},s}(r) \end{gathered}$$ (49)

where l is the occupied angular momentum of the core states, m the magnetic quantum number, I labels each ionic core, ${\Phi }_{\text{PS}}^{lm,s}$ is the atomic pseudo-wave function, j_l a spherical Bessel function, $v_{PS}^{l,s}(r)$ the pseudopotential for angular momentum l, and $\Delta v_{\text{PS}}^{\text{loc},s}$ the local component of the pseudopotential for the atomic species. Further details of these terms are given in Refs. [40, 61].

3.9. Energy gradients with respect to the electronic Thouless parameters and

The next step in the derivation of the END/PW dynamical equations is to obtain their energy gradients with respect to all the variational parameters, i.e., to obtain the terms ∂E_T/∂z, ∂E_T/∂z*, ∂E_T/∂R and ∂E_T/∂P appearing on the right-hand side of Eq. (13). The energy gradients with respect to the electronic Thouless parameters z and z*, ∂E_T/∂z and ∂E_T/∂z*, will be considered first. These gradients are unique to the END framework becasue they do not have equivalents in any alternative electronic structure theory or chemical dynamics method; furthermore, these gradients are more intricate than the remaining ones with respect to the nuclear parameters. Therefore, the derivation of ∂E_T/∂z and ∂E_T/∂z* will be presented in detail. For sake of brevity, only the expressions of ∂E_T/∂z will be reported because those of ∂E_T/∂z can be obtained immediately as the Hermitian adjoints of the former ones. Direct differentiation of the total END/PW energy E_T in Eq. (38) provides

$$\frac{\partial E_{\text{T}}\left(z,z^{*},R,P\right)}{\partial z_{\text{ph}}^{(\mathit{k})}}=\frac{\partial T_{\text{e}}\left(z,z^{*}\right)}{\partial z_{\text{ph}}^{(\mathit{k})}}+\frac{\partial E_{\text{es}}\left[\rho \left(r;z,z^{*}\right),R\right]}{\partial z_{\text{ph}}^{(\mathit{k})}}+\frac{\partial E_{\text{xc}}\left[\rho \left(r;z,z^{*}\right)\right]}{\partial z_{\text{ph}}^{(\mathit{k})}}+\frac{\partial E_{\text{PS}}^{\text{nl}}\left(z,z^{*},R\right)}{\partial z_{\text{ph}}^{(\mathit{k})}}$$ (50)

The dependence with respect to $z_{\text{ph}}^{(k)}$ of all the above terms: ρ(r;z, z*), T_e(z, z*), E_es[ρ(r;z, z*), R], E_xc[ρ(r;z, z*)] and $E_{\text{PS}}^{\text{nl}}\left(z,z^{*},R\right.)$, Eqs. (32), (41)–(44) and (47), respectively, come through the one-electron density matrix ${\Gamma }_{ij}^{(\mathit{k}\mathit{k})}\left(z^{*},z\right)$, Eq. (29). Therefore, the derivation of the $\partial {\Gamma }_{ij}^{(\mathit{k}\mathit{k})}\left(z^{*},z\right)/\partial z_{\text{ph}}^{(\mathit{k})}$ gradients should be accomplished first. That derivation is laborious and is therefore presented in “Appendix A”. The final result is

$$\begin{gathered} \frac{\partial {\Gamma }_{ij}^{(\mathit{k}\mathit{k})}\left(z^{*},z\right)}{\partial z_{\text{ph}}^{(\mathit{k})}}={\left[\left(\begin{array}{c}-z^{(\mathit{k})†}\\ I^{(\mathit{k})°}\end{array}\right){\left(I^{(\mathit{k})°}+z^{(\mathit{k})†}{z}^{(\mathit{k})}\right)}^{-1}\right]}_{ip} \\ {\left[\left[{\left(I^{(\mathit{k})•}+z^{(\mathit{k})†}{z}^{(\mathit{k})}\right)}^{-1}\right]\left(I^{(\mathit{k})•}{z}^{(\mathit{k})†}\right)\right]}_{hj} \end{gathered}$$ (51)

where I^(k)° = (δ_ij) is a (K_k − N_k) × (K_k − N_k) matrix and the other matrices were already defined after Eq. (25). The gradients of the terms in Eq. (50) that depend explicitly on ${\Gamma }_{ij}^{(\mathit{k}\mathit{k})}\left(z^{*},z\right)$, i.e., ρ(r;z, z*), T_e(z, z*), and $E_{\text{PS}}^{\text{nl}}\left(z,z^{*},R\right)$ cf. Eqs. (32), (41)–(42), and (47), adopt a common form

$$\frac{\partial A\left(z^{*},z\right)}{\partial z_{\text{ph}}^{(\mathit{k})}}=\sum _{\mathit{k}\in BZ}{\omega }_{\mathit{k}}\sum _{i,j=1}^{{\mathit{K}}_{\mathit{k}},{\mathit{K}}_{\mathit{k}}}\frac{\partial {\Gamma }_{ij}^{(\mathit{k}\mathit{k})}\left(z^{*},z\right)}{\partial z_{\text{ph}}^{(\mathit{k})}}{B}_{ij}^{(\mathit{k}\mathit{k})}{\delta }_{{\sigma }_i^{(\mathit{k})}{\sigma }_j^{(\mathit{k})}}$$ (52)

where for A(z*, z) = ρ(r;z, z*), T_e(z, z*), and $E_{\text{PS}}^{\text{nl}}\left(z,z^{*},R\right)$ then $B_{ij}^{(\mathit{k}\mathit{k})}={\tilde{\psi }}_i^{(\mathit{k})}(r){\tilde{\psi }}_j^{(\mathit{k})*}(r)$, $\left(G^{\prime }\right)C_j^{(\mathit{k})*}\left(G^{\prime }-G\right)$, $(-1/2)\int {\tilde{\psi }}_j^{(\mathit{k})*}(r){\nabla }^2{\tilde{\psi }}_i^{(\mathit{k})}(r)\text{d}r$, and $\int {\tilde{\psi }}_j^{(\mathit{k})*}(r){\widehat{V}}_{\text{PS}}^{\text{nl}}(r;R){\tilde{\psi }}_i^{(\mathit{k})}(r)\text{d}r$, respectively. For instance,

$$\frac{\partial \rho \left(r;z,z^{*}\right)}{\partial z_{\text{ph}}^{(\mathit{k})}}=\sum _{\mathit{k}\in BZ}{\omega }_{\mathit{k}}\sum _{i,j=1}^{{\mathit{k}}_{\mathit{k}}{\mathit{k}}_{\mathit{k}}}{\tilde{\psi }}_i^{(\mathit{k})}(r)\frac{\partial {\Gamma }_{ij}^{(\mathit{k}\mathit{k})}\left(z^{*},z\right)}{\partial z_{\text{ph}}^{(\mathit{k})}}{\tilde{\psi }}_j^{(\mathit{k})*}(r){\delta }_{{\sigma }_i^{(\mathit{k})}{\sigma }_j^{(\mathit{k})}}$$ (53)

The gradients of the terms in Eq. (50) that depend implicitly on $\partial {\rho }_T\left(G;z,z^{*}\right)/\partial z_{\text{ph}}^{(\mathit{k})}=\partial \rho \left(G;z,z^{*}\right)/\partial z_{\text{ph}}^{\left(\mathit{k}\right)}$ through the functional argument ρ(r;z, z*), i.e., E_es[ρ(r;z, z*), R] and E_xc[ρ(r;z, z*)], cf. Equations (43) and (44), are obtained by chain-rule differentiation through $\partial \rho \left(G;z,z^{*}\right)/\partial z_{\text{ph}}^{(\mathit{k})}$, Eq. (53). Thus, for E_es[ρ(r;z, z*), R] in Eq. (43), one obtains

$$\begin{gathered} \frac{\partial E_{\text{es}}\left[\rho \left(r;z,z^{*}\right),R\right]}{\partial z_{\text{ph}}^{(\mathit{k})}} \\ =\frac{\Omega }{2}\sum _{G\ne 0}\frac{4\pi }{G^2}\left[\frac{\partial {\rho }_T\left(G;z,z^{*}\right)}{\partial z_{ph}^{(\mathit{k})}}{\rho }_T^{*}\left(G;z,z^{*}\right)+{\rho }_T\left(G;z,z^{*}\right)\frac{\partial {\rho }_T^{*}\left(G;z,z^{*}\right)}{\partial z_{ph}^{(\mathit{k})}}\right] \\ +\Omega \sum _{G\ne 0}\left[\sum _{s=1}^{N_s}{S}_s(G;R)v_{\text{PS}}^{\text{loc}s}(G)\right]\frac{\partial {\rho }^{*}\left(G;z,z^{*}\right)}{\partial z_{\text{ph}}^{(\mathit{k})}} \end{gathered}$$ (54)

where $\partial {\rho }_T\left(G;z,z^{*}\right)/\partial z_{\text{ph}}^{(\mathit{k})}=\partial \rho \left(G;z,z^{*}\right)/\partial z_{\text{ph}}^{(\mathit{k})}$ from ρ_T (G;z, z*) = ρ_i(G;R)+ ρ (G;z, z*), was employed in the first and the second lines; $\partial \rho \left(G;z,z^{*}\right)/\partial z_{\text{ph}}^{(\mathit{k})}$ is given via Eq. (53). Finally, for E_xc[ρ(r;z, z*)] in Eq. (44), one obtains

$$\begin{gathered} \frac{\partial E_{\text{xc}}\left[\rho \left(r;z,z^{*}\right)\right]}{\partial z_{\text{ph}}^{(\mathit{k})}}=\int \frac{\partial E_{\text{xc}}[\rho ]}{\partial \rho (r)}\frac{\partial \rho (r)}{\partial z_{\text{ph}}^{(\mathit{k})}}\text{d}r \\ =\int \left[\frac{\partial \left(\rho (r){\epsilon }_{\text{xc}}[\rho (r),\nabla \rho (r)]\right)}{\partial \rho (r)}-\sum _{s=x,y,z}\frac{\partial }{\partial r_s}\left(\frac{\partial \left(\rho (r){\epsilon }_{\text{xc}}[\rho (r),\nabla \rho (r)]\right)}{\partial \left({\partial }_s\rho (r)\right)}\right)\right]\frac{\partial \rho (r)}{\partial z_{\text{ph}}^{(\mathit{k})}}\text{d}r \end{gathered}$$ (55)

where the rules of differentiation of the functional E_xc[ρ(r;z, z*)] with respect to the parameter $z_{\text{ph}}^{(\mathit{k})}$ in its functional argument ρ(r;z, z*) were employed in the first line [44]. The specific form of Eq. (55) depends on the chosen exchange–correlation energy density functional ε_xc[ρ(r), ∇ρ(r)][5].

3.10. Energy gradients with respect to the nuclear parameters and

Finally, the energy gradients with respect to the nuclear parameters R and P, ∂E_T/∂R and ∂E_T/∂P, should be obtained. The fact that the PWs do not depend on R and P provides a substantial simplification in these END/PW gradients in contrast to their more complex SLEND and END/KSDFT counterparts [7, 9, 14]. From the total energy E_T in Eq. (38), one obtains by direct differentiation with respect to the ion position R_I

$$\begin{gathered} \frac{\partial E_{\text{T}}\left(z,z^{*},R,P\right)}{\partial R_I}= \\ \frac{\partial \left\{T_{\text{i}}(P)+T_{\text{e}}\left(z,z^{*}\right)+E_{\text{es}}[\rho ,R]+E_{\text{xc}}\left[\rho \left(r;z,z^{*}\right)\right]+E_{\text{PS}}^{\text{nl}}\left(z,z^{*},R\right)\right\}}{\partial R_I} \\ =\frac{\partial E_{\text{es}}\left[\rho \left(r;z,z^{*}\right),R\right]}{\partial R_I}+\frac{\partial E_{\text{PS}}^{\text{nl}}\left(z,z^{*},R\right)}{\partial R_I} \end{gathered}$$ (56)

Then, from E_es[ρ(r;z, z*), R] in Eqs. (43), one obtains by direct differentiation [40]

$$\frac{\partial E_{\text{es}}[\rho ,R]}{\partial R_I}=\frac{Z_I}{2}\sum _{J\ne 1}^{N_J}{Z}_J\sum _{L=-L_{\text{max}}}^{L_{\text{max}}}\left(R_I+L-R_J\right)\times \left[\frac{\text{erfc}\left(\left|R_I+L-R_J\right|\eta \right)}{{\left|R_I+L-R_J\right|}^3}+\frac{\eta \text{exp}\left(-{\eta }^2{\left|R_I+L-R_J\right|}^2\right)}{\left|R_I+L-R_J\right|}\right]+\Omega \sum _{G\ne 0}iG\text{exp}\left(-iG{R}_I\right)u_{\text{PS}}^{\text{loc}s}(G){\rho }^{*}(G)$$ (57)

where the species index s of $u_{\text{PS}}^{\text{loc},s}(G)$ in the last term corresponds to the chemical species of the ion with position R_I. Similarly, from $E_{\text{PS}}^{\text{nl}}\left(z,z^{*},R\right)$ with the Kleiman and Bylander pseudopotential in Eq. (49), one obtains [40, 61]

$$\begin{gathered} \frac{\partial E_{\text{PS}}^{\text{nl}}\left(z,z^{*},R\right)}{\partial R_I}=\sum _{k\in BZ}{\omega }_{\mathit{k}}\sum _{l=0}^{l\text{max}}\sum _{m=-l}^l\sum _{i,j=1}^{{\mathit{k}}_{\mathit{k}}{\mathit{k}}_{\mathit{k}}}{\Gamma }_{ij}^{(\mathit{k}\mathit{k})}\left(z^{*},z\right){\delta }_{{\sigma }_i^{(\mathit{k})}{\sigma }_j^{(\mathit{k})}}\times {\alpha }_{lm}^s\left(F_{I,i}^{lm,s{(\mathit{k})}^{*}}{D}_{I,j}^{lm,s(\mathit{k})}+F_{I,i}^{lm,s(\mathit{k})}{D}_{I,j}^{lm,s{(\mathit{k})}^{*}}\right) \\ {D}_{I,i}^{lm,s(\mathit{k})}=\sum _{G}iG{e}^{iG\cdot R_I}{f}_{lm}^s(k+G)C_i^{(\mathit{k})}(G) \end{gathered}$$ (58)

where all the terms in the above equation were already defined after Eq. (49). Finally, it follows straightforwardly from E_T in Eq. (38) that ∂E_T/∂P_I is simply the gradient of the kinetic energy of the classical ions with respect to the ion momentum P_I

$$\frac{\partial E_{\text{T}}\left(z,z^{*},R,P\right)}{\partial P_I}=\frac{\partial T_{\text{i}}(P)}{\partial P_I}=\frac{\partial \left(\sum _{J=1}^{N_J}\frac{p_J^2}{2M_J}\right)}{\partial P_I}=\frac{P_J}{M_J}$$ (59)

4. Conclusions, implementation, and applications

The complete formalism of the novel END/PW method has been presented in detail. Special attention has been paid to the theoretical aspects of the END framework [7–10] and of the PWs methodology for periodic systems [5, 39, 40] in order to achieve a rigorous derivation of END/PW. Except for some standard PWs topics, the presented formulation is self-contained so that the reader can find all the necessary concepts and mathematical details to entirely follow the derivation and reasoning. As discussed in Sects. 1 and 2.2, the development of END/PW is strongly motivated by its potential applications to electron scattering processes and to phenomena in periodic systems. However, applications aside, the presented derivation possesses a great theoretical value in itself. First of all, the derived END/PW method shows how the Thouless single-determinantal wave function [13] and its associated electron density expressed in PWs can be extended to periodic systems and satisfy lattice symmetry conditions. Secondly, the derived END/PW method also shows how to obtain all the END energy and energy gradient terms from the Thouless single-determinantal wave function [13] and its associated electron density expressed in PWs. The provided END/PW dynamical equations are also the working equations of this method and are, therefore, ready for its computer implementation. Currently, we are implementing the END/PW dynamical equations into our END code PACE [9, 51] employing the KSDFT and fast Fourier transform capabilities of the massively parallel code Qbox [59].

As discussed in Sects. 1 and 2.2, the novel END/PW method can be applied to two main types of systems. The first type of systems includes periodic systems undergoing dynamical processes. In this case, either one unit cell or a supercell containing a number of unit cells will be used according to the nature and extension of the simulated phenomenon. A certain number of k vectors will be used in the END/PW basis set according to the size of the cell or supercell and to the selected sampling method for k [5, 39, 40]. The second type of systems includes finite, non-periodic systems such as those involving a few molecules participating in reactive collisions. In this case, a constructed periodic system of large supercells each containing a replica of the system under study will be used; a large vacuum space will be allowed around each supercell to avoid spurious interactions among replicas. Since these supercells are large, the BZ will shrink to the single point k = 0 (the central Γ-point in the BZ) and only that single wave vector will be utilized [5, 39, 40]. The numerical advantages of PWs discussed in Sect. 2.2 will considerably accelerate the simulations of various types of molecular reactions with END/PW in the supercell approach. While future END/PW applications are multiple, our main interest lies in applying END/PW to reactions involving electron scattering to/from the continuum. While those reactions can happen in numerous scenarios, they are predominant in some PCT reactions such as the scattering of electrons induced by colliding protons in cell water (water radiolysis) and the capture of secondary electrons by DNA that leads to DNA single- and double-strand breaks [31, 33, 35]. Various PCT reactions have been intensively and successfully studied by our group using the preexisting SLEND and END/KSDFT methods [9, 31–34].

However, the novel END/PW method will allow to extend those studies to additional PCT reactions where electron scattering plays a greater role. In that way, END/PW will continue making a positive impact on PCT research, a topic of increasing importance in contemporaneous biophysics and medical physics [9, 31–34].

Appendix A: Gradients of the END/PW density matrix ${\Gamma }_{ij}^{(\mathit{k}\mathit{k})}\left(z^{*},z\right)$ with respect to the electronic Thouless parameters $z_{\text{ph}}^{(\mathit{k})}$, $\partial {\Gamma }_{ij}^{(\mathit{k})}\left(z^{*},z\right)/\partial z_{ph}^{(\mathit{k})}$

To obtain the $\partial {\Gamma }_{ij}^{(\mathit{k}\mathit{k})}\left(z^{*},z\right)/\partial z_{ph}^{(\mathit{k})}$ gradients, the one-electron density matrix ${\Gamma }_{ij}^{(\mathit{k}\mathit{k})}\left(z^{*},z\right)$ in Eq. (29) is first differentiated with respect to $z_{\text{ph}}^{(\mathit{k})}$

$$\begin{gathered} \frac{\partial {\Gamma }_{ij}^{(\mathit{k}\mathit{k})}\left(z^{*},z\right)}{\partial z_{ph}^{(\mathit{k})}}=\sum _{l,l^{\prime }=1}^{N_{\mathit{k}},N_{\mathit{k}}}\frac{\partial {\left(\begin{array}{c}{I}^{(\mathit{k})}\cdot \\ z^{(\mathit{k})}\end{array}\right)}_{il}}{\partial z_{ph}^{(\mathit{k})}}{\left[{\left(I^{(\mathit{k})}\cdot +z^{(\mathit{k})†}{z}^{(\mathit{k})}\right)}^{-1}\right]}_{ll}{\left(I^{(\mathit{k})}\cdot z^{(\mathit{k})†}\right)}_{l\text{'}j} \\ +\sum _{l,l^{\prime }=1}^{N_{\mathit{k}},N_{\mathit{k}}}{\left(\begin{array}{c}{I}^{(\mathit{k})\cdot }\\ z^{(\mathit{k})}\end{array}\right)}_{il}\frac{\partial {\left[{\left(I^{(\mathit{k})•}+z^{(\mathit{k})†}{z}^{(\mathit{k})}\right)}^{-1}\right]}_{l{l}^{\prime }}}{\partial z_{\text{ph}}^{(\mathit{k})}}{\left(I^{(\mathit{k})\cdot }{z}^{(\mathit{k})†}\right)}_{l^{\prime }j} \end{gathered}$$ (60)

where all the matrices in the above equation are defined just after Eq. (25). In order to obtain an expression appropriate for coding, it is necessary to apply the following matrix relationships [7]:

$$\frac{\partial A_{bc}}{\partial A_{de}}={\delta }_{bd}{\delta }_{ec};\frac{\partial X{(x)}^{-1}}{\partial x}=-X{(x)}^{-1}\frac{\partial X(x)}{\partial x}X{(x)}^{-1}$$ (61)

where A and X(x) are matrices and x is a variable. By applying these relationships to the derivative matrix terms in the first sum of Eq. (60), one obtains

$$\frac{\partial {\left(\begin{array}{l}{I}^{(\mathit{k})·}\hfill \\ z^{(\mathit{k})}\hfill \end{array}\right)}_{il}}{\partial z_{\text{ph}}^{(\mathit{k})}}={\delta }_{ip}^{(\mathit{k})}{\delta }_{hl}^{(\mathit{k})}\text{ with }{\delta }_{ip}^{(\mathit{k})}={\left(\begin{array}{c}{0}^{(\mathit{k})}\\ I^{(\mathit{k})°}\end{array}\right)}_{ip}\text{ and }{\delta }_{hl}^{(\mathit{k})}=I_{hl}^{(\mathit{k})·}$$ (62)

By doing the same on the more complex derivate matrix terms in the second sum of Eq. (60), one obtains

$$\begin{gathered} \frac{\partial {\left[{\left(I^{(\mathit{k})•}+z^{(\mathit{k})†}{z}^{(\mathit{k})}\right)}^{-1}\right]}_{l{l}^{\prime }}}{\partial z_{\text{ph}}^{(\mathit{k})}} \\ =\sum _{a,b=1}^{N_{\mathit{k}},N_{\mathit{k}}}-{\left[{\left(I^{(\mathit{k})•}+z^{(\mathit{k})†}{z}^{(\mathit{k})}\right)}^{-1}\right]}_{la}\frac{\partial {\left[I^{(\mathit{k})•}+z^{(\mathit{k})†}{z}^{(\mathit{k})}\right]}_{ab}}{\partial z_{ph}^{(\mathit{k})}}{\left[{\left(I^{(\mathit{k})•}+z^{(\mathit{k})†}{z}^{(\mathit{k})}\right)}^{-1}\right]}_{b{l}^{\prime }} \\ =\sum _{a,b=1}^{N_{\mathit{k}},N_{\mathit{k}}}-{\left[{\left(I^{(\mathit{k})•}+z^{(\mathit{k})†}{z}^{(\mathit{k})}\right)}^{-1}\right]}_{la}\left[\frac{\partial {\left[I^{(\mathit{k})•}\right]}_{ab}}{\partial z_{\text{ph}}^{(\mathit{k})}}+\frac{\partial {\left[z^{(\mathit{k})†}{z}^{(\mathit{k})}\right]}_{ab}}{\partial z_{\text{ph}}^{(\mathit{k})}}\right]\times {\left[{\left(I^{(\mathit{k})•}+z^{(\mathit{k})†}{z}^{(\mathit{k})}\right)}^{-1}\right]}_{b{l}^{\prime }} \\ =\sum _{a,b=1}^{N_{\mathit{k}},N_{\mathit{k}}}-{\left[{\left(I^{(\mathit{k})•}+z^{(\mathit{k})†}{z}^{(\mathit{k})}\right)}^{-1}\right]}_{la}\left[\frac{\partial \left(\sum _{q=N_{\mathit{k}}+1}^{{\mathit{k}}_{\mathit{k}}}{\left(z^{(\mathit{k})†}\right)}_{aq}{z}_{qb}^{(\mathit{k})}\right)}{\partial z_{ph}^{(\mathit{k})}}\right]{\left[{\left(I^{(\mathit{k})•}+z^{(\mathit{k})†}{z}^{(\mathit{k})}\right)}^{-1}\right]}_{b{l}^{\prime }} \\ =\sum _{a,b=1}^{N_{\mathit{k}},N_{\mathit{k}}}-{\left[{\left(I^{(\mathit{k})}{}^{·}+z^{(\mathit{k})†}{z}^{(\mathit{k})}\right)}^{-1}\right]}_{la}\left\{\sum _{q=1}^{{\mathit{k}}_{\mathit{k}}}\left[\frac{\partial {\left(z^{(\mathit{k})†}\right)}_{aq}}{\partial z_{\text{ph}}^{(\mathit{k})}}\cdot z_{qb}^{(\mathit{k})}+{\left(z^{(\mathit{k})†}\right)}_{aq}\frac{\partial \left(z_{qb}^{(\mathit{k})}\right)}{\partial z_{ph}^{(\mathit{k})}}\right]\right\}\times {\left[{\left(I^{(\mathit{k})}{}^{·}+z^{(\mathit{k})†}{z}^{(\mathit{k})}\right)}^{-1}\right]}_{b{l}^{\prime }} \\ =\sum _{a,b=1}^{N_{\mathit{k}},N_{\mathit{k}}}-{\left[{\left(I^{(\mathit{k})•}+z^{(\mathit{k})†}{z}^{(\mathit{k})}\right)}^{-1}\right]}_{la}\left\{\sum _{q=1}^{{\mathit{k}}_{\mathit{k}}}\left[{\left(z^{(\mathit{k})†}\right)}_{aq}{\delta }_{qp}^{(\mathit{k})}{\delta }_{hb}^{(\mathit{k})}\right]\right\}\times {\left[{\left(I^{(\mathit{k})}{}^{·}+z^{(\mathit{k})†}{z}^{(\mathit{k})}\right)}^{-1}\right]}_{b{l}^{\prime }} \\ =\sum _{a,b=1}^{N_{\mathit{k}},N_{\mathit{k}}}-{\left[{\left(I^{(\mathit{k})•}+z^{(\mathit{k})†}{z}^{(\mathit{k})}\right)}^{-1}\right]}_{la}\left[{\left(z^{(\mathit{k})†}\right)}_{ap}\cdot {\delta }_{hb}^{(\mathit{k})}\right]{\left[{\left(I^{(\mathit{k})•}+z^{(\mathit{k})†}{z}^{(\mathit{k})}\right)}^{-1}\right]}_{b{l}^{\prime }} \\ =-{\left[{\left(I^{(\mathit{k})•}+z^{(\mathit{k})†}{z}^{(\mathit{k})}{I}^{*}+z^{†}z\right)}^{-1}{z}^{(\mathit{k})†}\right]}_{lp}{\left[{\left(I^{(\mathit{k})•}+z^{(\mathit{k})†}{z}^{(\mathit{k})}\right)}^{-1}\right]}_{h{l}^{\prime }} \end{gathered}$$ (63)

By setting Eqs. (62) and (63) into Eq. (60), one obtains

$$\begin{gathered} \frac{\partial {\Gamma }_{ij}^{(kk)}\left(z^{*},z\right)}{\partial z_{\text{ph}}^{(k)}}=\sum _{l,l^{\prime }=1}^{N_k,N_k}{\left(\begin{array}{c}{0}^{(k)°}\\ I^{(k)°}\end{array}\right)}_{ip}{I}_{hl}^{(k)·}{\left[{\left(I^{(k)·}+z^{(k)†}{z}^{(k)}\right)}^{-1}\right]}_{l{l}^{\prime }}{\left(I^{(k)·}{z}^{(k)†}\right)}_{l\text{'}j}-\sum _{l,l^{\prime }=1}^{N_k{N}_k}{\left(\begin{array}{c}{I}^{(k)·}\\ z^{(k)}\end{array}\right)}_{il}{\left[{\left(I^{(k)·}+z^{(k)†}{z}^{(k)}\right)}^{-1}{z}^{(k)+}\right]}_{lp}\times {\left[{\left(I^{(k)\cdot }+z^{(k)†}{z}^{(k)}\right)}^{-1}\right]}_{h^l}{\left(I^{(k)\cdot }{z}^{(k)†}\right)}_{l\text{'}j} \\ =\sum _{l=1}^{N_k}{\left(\begin{array}{l}{0}^{(k)°}\hfill \\ I^{(k)°}\hfill \end{array}\right)}_{ip}{\left[{\left(I^{{}^{(k)•}}+z^{(k)†}{z}^{(k)}\right)}^{-1}\right]}_{hl}{\left(I^{(k)*}{z}^{(k)†}\right)}_{lj} \\ -{\left[\left(\begin{array}{c}{I}^{(k)\cdot }\\ z^{(k)}\end{array}\right)\left[{\left(I^{(k)•}+z^{(k)†}{z}^{(k)}\right)}^{-1}{z}^{(k)+}\right]\right]}_{ip}{\left[\left[{\left(I^{{}^{(k)•}}+z^{(k)†}{z}^{(k)}\right)}^{-1}\right]\left(I^{(k)\cdot }{z}^{(k)†}\right)\right]}_{hj} \\ ={\left(\begin{array}{l}{0}^{(k)°}\hfill \\ I^{(k)°}\hfill \end{array}\right)}_{ip}{\left[\left[{\left(I^{(k)*}+z^{(k)†}{z}^{(k)}\right)}^{-1}\right]\left(I^{(k)·}{z}^{(k)†}\right)\right]}_{hj} \\ -{\left[\left(\begin{array}{c}{I}^{(k)\cdot }\\ z^{(k)}\end{array}\right)\left[{\left(I^{(k)\cdot }+z^{(k)†}{z}^{(k)}\right)}^{-1}{z}^{(k)†}\right]\right]}_{ip}{\left[\left[{\left(I^{(k)·}+z^{(k)†}{z}^{(k)}\right)}^{-1}\right]\left(I^{(k)·}{z}^{(k)†}\right)\right]}_{hj} \\ ={\left[\left(\begin{array}{l}{0}^{(k)°}\hfill \\ I^{(k)°}\hfill \end{array}\right)-\left(\begin{array}{c}{I}^{(k)·}\\ z^{(k)}\end{array}\right)\left[{\left(I^{(k)\cdot }+z^{(k)†}{z}^{(k)}\right)}^{-1}{z}^{(k)†}\right]\right]}_{ip} \\ \times {\left[\left[{\left(I^{(k)•}+z^{(k)†}{z}^{(k)}\right)}^{-1}\right]\left(\left(I^{(k)•}{z}^{(k)†}\right)\right)\right]}_{hj} \\ ={\left[\left(\begin{array}{l}{0}^{(k)°}\hfill \\ I^{(k)°}\hfill \end{array}\right)-\left(\begin{array}{c}{\left(I^{(k)·}+z^{(k)†}{z}^{(k)}\right)}^{-1}{z}^{(k)†}\\ z^{(k)}{\left(I^{(k)·}+z^{(k)÷}{z}^{(k)}\right)}^{-1}{z}^{(k)†}\end{array}\right)\right]}_{ip} \\ \times {\left[\left[{\left(I^{(k)•}+z^{(k)†}{z}^{(k)}\right)}^{-1}\right]\left(I^{(k)*}{z}^{(k)†}\right)\right]}_{hj} \\ ={\left.\left(\begin{array}{l}{0}^{(k)°}\hfill \\ I^{(k)°}\hfill \end{array}\right)-\left(\begin{array}{c}{\left(I^{(k)·}+z^{(k)†}{z}^{(k)}\right)}^{-1}{z}^{(k)†}\\ z^{(k)}{\left(I^{(k)*}+z^{(k)†}{z}^{(k)}\right)}^{-1}{z}^{(k)†}\end{array}\right)\right]}_{ip} \\ \times {\left[\left[{\left(I^{(k)•}+z^{(k)†}{z}^{(k)}\right)}^{-1}\right]\left(I^{(k)·}{z}^{(k)†}\right)\right]}_{hj}={\left(\begin{array}{c}-{\left(I^{(k)*}+z^{(k)†}{z}^{(k)}\right)}^{-1}{z}^{(k)†}\\ I^{(k)°}-z^{(k)}{\left(I^{(k)*}+z^{(k)†}{z}^{(k)}\right)}^{-1}{z}^{(k)†}\end{array}\right)}_{ip} \\ \times {\left[\left[{\left(I^{(k)•}+z^{(k)†}{z}^{(k)}\right)}^{-1}\right]\left(I^{(k)•}{z}^{(k)†}\right)\right]}_{hj} \end{gathered}$$ (64)

The matrix of the first element in the last line satisfies the identity (cf. Eq. 2.28 of Ref. [7])

$$\left(\begin{array}{c}-z^{(\mathit{k})†}{\left(I^{(\mathit{k})°}+z^{(\mathit{k})}{z}^{(\mathit{k})+}\right)}^{-}\\ {\left(I^{(\mathit{k})°}+z^{(\mathit{k})}{z}^{(\mathit{k})†}\right)}^{-1}\end{array}\right)=\left[\left(\begin{array}{c}-z^{(\mathit{k})†}\\ I^{(\mathit{k})°}\end{array}\right){\left(I^{(\mathit{k})°}+z^{(\mathit{k})†}{z}^{(\mathit{k})}\right)}^{-1}\right]$$ (65)

By setting the above identity into Eq. (64), one finally obtains

$$\begin{gathered} \frac{\partial {\Gamma }_{ij}^{(\mathit{k}\mathit{k})}\left(z^{*},z\right)}{\partial z_{\text{ph}}^{(k)}} \\ ={\left(\begin{array}{c}-z^{(\mathit{k})†}{\left(I^{(\mathit{k})°}+z^{(\mathit{k})}{z}^{(\mathit{k})†}\right)}^{-1}\\ {\left(I^{(\mathit{k})°}+z^{(\mathit{k})}{z}^{(\mathit{k})†}\right)}^{-1}\end{array}\right)}_{ip}{\left[\left[{\left(I^{(\mathit{k})·}+z^{(\mathit{k})†}{z}^{(\mathit{k})}\right)}^{-1}\right]\left(I^{(\mathit{k})·}{z}^{(\mathit{k})†}\right)\right]}_{hj} \\ ={\left[\left(\begin{array}{c}-z^{(\mathit{k})†}\\ I^{(\mathit{k})°}\end{array}\right){\left(I^{(\mathit{k})°}+z^{(\mathit{k})†}{z}^{(\mathit{k})}\right)}^{-1}\right]}_{ip}{\left[\left[{\left(I^{(\mathit{k})\cdot }+z^{(\mathit{k})†}{z}^{(\mathit{k})}\right)}^{-1}\right]\left(I^{(\mathit{k})·}{z}^{(\mathit{k})†}\right)\right]}_{hj} \end{gathered}$$ (66)