Targeted Energy Transfer Dynamics and Chemical Reactions

Abstract

Ultrafast reaction processes take place when resonant features of nonlinear model systems are taken into account. In the targeted energy or electron transfer dimer model this is accomplished through the implementation of nonlinear oscillators with opposing types of nonlinearities, one attractive while the second repulsive. In the present work, we show that this resonant behavior survives if we take into account the vibrational degrees of freedom as well. After giving a summary of the basic formalism of chemical reactions we show that resonant electron transfer can be assisted by vibrations. We find the condition for this efficient transfer and show that in the case of additional interaction with noise, a distinct non-Arrhenius behavior develops that is markedly different from the usual Kramers-like activated transfer.

1. Introduction

Ever since the famous Kramers paper [1] that cast the process of chemical reactions into a stochastic process picture, there have been numerous important contributions to the topic [2]. Among the many questions that may arise in the statistical picture of chemical reactions one very important one is what is the effect of the Gaussianity in the stochastic noise in the overall reaction process and its speed [3]. The statistical aspects of the noise as well as the specific entropic contribution to the free energy may be very important in determining the reaction specifics. The concept of the entropic barrier is very important in this direction and relates to statistical and thermodynamical efficiency in general reaction processes [4]. We should also note that quantum effects may play a significant role in reactions and more specifically in electron transfer processes between molecular species [5]. These effects may be affected also by long-range interactions specifically in donor–acceptor systems [6]. A molecular system of this type, i.e., where an electron transfers energy between a donor and an acceptor, appears in the processes of photosynthesis [7]. An important aspect of this transfer in light harvesting in chlorophyll molecules is the speed of the electron and energy transfer across the donor–acceptor system that emerges due to exciton coherence [8]. To address the unique aspects of this process a number of theoretical models have been introduced that aim at explaining the experimental results [9]. In the environmentally assisted quantum transfer model the acquired coherence from the environment assists in the donor–acceptor energy transfer [10]. An alternative, but similar in spirit approach uses a detailed parameter model to show the coherent as well as statistical aspect of the enhanced transfer [11]. While these models contain a realistic interaction of the transfer processes to the environment, they do not take into account directly possible resonant transfer features that may arise from the nonlinearity in the model itself. In the present work, we focus on a type of resonant process induced by nonlinearity termed targeted energy transfer [12]. This model may play a very important role in the energetics of the transfer and also include the aspects of coherence that are seen experimentally in chlorophyll.

The Targeted Energy Transfer (TET) model considers two weakly coupled nonlinear oscillators that have soft and hard nonlinearity, respectively, and finds a parameter regime where the perfect resonant transfer occurs between the oscillators [13]. There is by now extended literature on this model, both in the classical, and quantum but also in the engineering regime with a more practical application of the concept [14]. Recently, a machine learning approach was used in both classical [15] and quantum [16] regimes that explore the possibility of finding precise transfer resonance through learning processes. The machine learning method such as a hybrid quantum-classical neural network is capable of providing ground state energies for simple molecules [17]. This method is quite promising, since it, in principle, enables the extension of the TET concept to arbitrary-size systems, a feat that cannot easily be accomplished through more traditional numerical techniques.

The concept of targeted transfer is motivated by biology and, in particular, by the specificity and efficiency that certain electron transfer processes have in biological systems [18]. In chlorophyll, in particular, there is an ultra-fast transfer electronic path [8], and, as a result, it is bound to involve certain resonant transfer features. The TET mechanism provides a simplified framework for this ultra-fast transfer through the generation of a specific transfer between non-identical nonlinear oscillators that is practically perfect.

One feature that is missing in the original TET formulation, if one wants to address the electron transfer processes, is the presence and interaction with additional degrees of freedom and specifically phonons. While this is conducted in an indirect way through the use of the Discrete Nonlinear Schrödinger (DNLS) Equation model, for a more complete and realistic analysis one needs to include explicitly these degrees of freedom. This is accomplished through the use of a model that includes both electronic and vibrational degrees of freedom as well as their coupling. The main target of the present work is to focus precisely on this microscopic case, viz. a complete electron–phonon model, and address the TET condition in this more general and mode realistic case. This study involves a deeper understanding of the chemical reaction processes and aspects that may affect their efficiency. In order to explore the complete power of the chemistry under TET we first provide a description of chemical reactions and explore the differences of TET-driven reactions compared to more standard ones.

The plan of this paper is then the following: In the next section, we discuss chemical reactions and introduce the diabatic approximation for their description. We describe the Born–Oppenheimer framework and detail Marcus’ theory of chemical reactions. We work explicitly with a two-state model and present the reaction dynamics in the semiclassical approximation. We use adiabatic surfaces in order to discuss conical intersections. Subsequently, we introduced targeted electron/energy transfer through a simplified two-state model that also includes semiclassically vibrational degrees of freedom. We show that this model has radically different behavior compared to the standard Marcus law and investigate the temperature dependence of the transfer process. In the presence of noise, we observe a clear non-Arrhenius behavior that is quite distinct from the typical standard chemical reactions. In the concluding section, we summarize our findings and give an outlook for the generalization of this work.

2. Chemical Reactions

Formulas are widely used in chemistry for describing the organization of molecules, radicals, and complexes [19,20]. In their most detailed forms, i.e., the “condensed formulas”, they describe schematically how the nuclei constituents of a chemical species are spatially organized and bounded by covalent single, double, triple bonds, hydrogen bonds, Van de Waals bonds [21]. They also specify where the charged atoms or radicals are located, the pending bonds, etc., while they also give spatial information about the organization of a single molecule or an aggregate of molecules. Thus, the condensed formula of a molecule or radical is nothing but a characterization of its electronic state as precisely as possible.

Chemical reactions are processes in which one or more substances, known as reactants, are chemically transformed into one or more new substances, called products [22,23]. These changes involve the creation or the breaking of chemical bonds or a charge transfer that generates relative nuclei displacements and thus molecular reorganization. These changes are schematically represented by a symbolic reaction formula that describes the changes that occur in the substances. In many cases, they consist of a collection of subsequent elementary processes, referred to as elementary steps or elementary reactions, that describe how the overall reaction proceeds.

We focus essentially on Elementary Chemical Reactions (ECR) introduced through the Diabatic representation. Diabatic states are defined empirically as an implicit consequence of the theory of chemical bonds pioneered by L. Pauling [24]. They are based on orbital occupation, non-occupation, and related overlaps. This theory is remarkably successful because it provides the basis for the concept of a chemical formula and valence rules, which suffer only rare exceptions, such as for the intermediate valence materials.

Next, we discuss the validity of the Diabatic approximation within the Born–Oppenheimer representation, which has well-defined foundations. It is also widely used in quantum chemistry although its flaw is that it does not suggest any intuitive representation of the molecular arrangements and their chemical formula.

2.1. Elementary Chemical Reactions in the Diabatic Representation

The following exposition on chemical reactions draws on a more extended earlier report [25]. The work has been conducted to describe long-range inter-molecular photo-initiated electron transfer using Potential Energy Surfaces (PES) [26,27]. The calculation of the PES can be produced using various approaches [28]. Here we use the semiclassical approximation from the beginning; in other words, we describe the dynamics of the nuclei by classical variables instead of quantum operators. The electrons still remain quantum; however, we project their wave function into a two-dimensional subspace in order to deal with only two complex variables with the unit norm.

An ECR is defined as a chemical transformation process whereby an initial state, described by a specific chemical formula, is converted into a final state, represented by a different chemical formula. We consider two species, that can be reacting molecules, radicals, or clusters that are designated each with the chemical formula D and A, respectively, and that are supposed to exchange, for instance, an electron; this process is represented by the chemical reaction $D+A\to D^{+}{A}^{-}$.

Subsequently, we assume the existence of two diabatic states ${\chi }_D(\left\{r_{\nu }\right\};\left\{R_n\right\})$ and ${\chi }_A(\left\{r_{\nu }\right\};\left\{R_n\right\})$ corresponding to the initial and the final states, respectively. More precisely, the quantity ${\chi }_D(\left\{r_{\nu }\right\};\left\{R_n\right\})$ represents the real global electronic wave function of the set of electronic variables $\left\{r_{\nu }\right\}$ for the system in the initial state $D+A$. This wave function also depends on the nuclei coordinates $\left\{R_n\right\}$ which are supposed to be classical variables, and therefore, appear as parameters. The second diabatic state ${\chi }_A(\left\{r_{\nu }\right\};\left\{R_n\right\})$ is defined identically, but for the global system in the final state $D^{+}{A}^{-}$. These two wave functions a priori are not orthogonal. It is assumed that they can be orthogonalized with slight perturbations. We thus assume for all nuclei coordinates $\left\{R_n\right\}$${〈{\chi }_D(\left\{r_{\nu }\right\};\left\{R_n\right\})|{\chi }_A(\left\{r_{\nu }\right\};\left\{R_n\right\})〉}_e=0$, where the index e specifies that the scalar product involves integration only over the electronic degrees of freedom. The nonadiabatic theory of ET is described in [18]. Since we study the transition between the initial state and the final state, we suggest that during that transition the electronic wave function has the form

$$\chi (\left\{r_{\nu }\right\};\left\{R_n\right\})={\phi }_D\left(t\right){\chi }_D(\left\{r_{\nu }\right\};\left\{R_n\right\})+{\phi }_A\left(t\right){\chi }_A(\left\{r_{\nu }\right\};\left\{R_n\right\})$$

(1)

where ${\phi }_D\left(t\right)$ and ${\phi }_A\left(t\right)$ are time dependant complex coefficients which fulfill the normalization condition $|{\phi }_D{\left(t\right)|}^2+{\left|{\phi }_A\left(t\right)\right|}^2=1$.

The global quantum Hamiltonian of the system can be expressed as follows

$$\mathcal{H}={\mathcal{H}}_e(\left\{r_{\nu }\right\},\left\{p_{\nu }\right\};\left\{R_n\right\})+{\mathcal{H}}_K$$

(2)

The term ${\mathcal{H}}_e(\left\{r_{\nu }\right\},\left\{p_{\nu }\right\};\left\{R_n\right\})$ denotes the portion of the Hamiltonian that concerns exclusively electrons $\nu$ considered as fermions. The kinetic energy, given by $1/\left(2m_e\right)p_{\nu }^2$ for all the electrons (with mass $m_e$) functions of the momentum operators ($p_{\nu }$) and all potential Coulomb interactions, including those between electrons, electrons and nuclei, and nuclei themselves (which is generated only by the Coulomb potential of the nuclei with coordinates $\left\{R_n\right\}$ and does not affect $\left\{r_{\nu }\right\}$ and their conjugate operators $\left\{p_{\nu }\right\}$). ${\mathcal{H}}_K={\sum }_{n}1/\left(2M_n\right)P_n^2$ is the total kinetic energy operator of the nuclei with coordinates $R_n$, conjugate variables $P_n$ and mass $M_n$. $P_n=(\hslash /i)\partial ./\partial R_n$ is the conjugate quantum operator associated with the nuclei variable $R_n$. Then, we can define electronic energy.

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill V({\phi }_D,{\phi }_A,\left\{R_n\right\})& ={〈\chi (\left\{r_{\nu }\right\};\left\{R_n\right\})\left|{\mathcal{H}}_e(\left\{r_{\nu }\right\},\left\{p_{\nu }\right\};\left\{R_n\right\})\right|\chi (\left\{r_{\nu }\right\};\left\{R_n\right\})〉}_e\hfill \\ \hfill & =\left(\begin{array}{cc}{\phi }_D& {\phi }_A\end{array}\right)·\left(\begin{array}{cc}{E}_D\left(\left\{R_n\right\}\right)& \Lambda \left(\left\{R_n\right\}\right)\\ \Lambda \left(\left\{R_n\right\}\right)& E_A\left(\left\{R_n\right\}\right)\end{array}\right)\left(\begin{array}{c}{\phi }_D\\ {\phi }_A\end{array}\right)\hfill \end{array}\end{array}$$

(3)

where

$$\begin{array}{cc}\hfill & E_D\left(\left\{R_n\right\}\right)={〈{\chi }_D\left|{\mathcal{H}}_e\right|{\chi }_D〉}_e\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & E_A\left(\left\{R_n\right\}\right)={〈{\chi }_A\left|{\mathcal{H}}_e\right|{\chi }_A〉}_e\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & \Lambda \left(\left\{R_n\right\}\right)={〈{\chi }_D\left|{\mathcal{H}}_e\right|{\chi }_A〉}_e={〈{\chi }_A\left|{\mathcal{H}}_e\right|{\chi }_D〉}_e\hfill \end{array}$$

(6)

are real functions of the nuclei coordinates.

We can obtain an effective classical Hamiltonian written as

$${\mathcal{H}}_{diab}={\mathcal{H}}_K+V({\phi }_D,{\phi }_A,\left\{R_n\right\})$$

(7)

The Hamiltonian in Equation (7) describes the dynamics of the electronic state projected in the two-dimensional subspace defined by Equation (1) coupled now with the set of nuclei coordinates. The corresponding Hamilton equations for the nuclei and the electronic state are

$$\begin{array}{cc}\hfill i\hslash {\dot{\phi }}_D& =E_D\left(\left\{R_n\right\}\right){\phi }_D+\Lambda \left(\left\{R_n\right\}\right){\phi }_A\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill i\hslash {\dot{\phi }}_A& =\Lambda \left(\left\{R_n\right\}\right){\phi }_D+E_A\left(\left\{R_n\right\}\right){\phi }_A\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill M_n{\ddot{R}}_n& +{\displaystyle \frac{\partial E_D}{\partial R_n}}|{\phi }_D{|}^2+{\displaystyle \frac{\partial E_A}{\partial R_n}}{\left|{\phi }_A\right|}^2+{\displaystyle \frac{\partial \Lambda }{\partial R_n}}({\phi }_D^{★}{\phi }_A+{\phi }_A^{★}{\phi }_D)=0\hfill \end{array}$$

(10)

It is convenient to rewrite this Hamiltonian Equation (7) as a spin boson system. For that, we expand the $2\times 2$ in Equation (3) on the base Pauli matrices

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\sigma }_0& =\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\hfill \\ \hfill {\sigma }_x=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\sigma }_y& =\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),{\sigma }_z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right),\hfill \end{array}\end{array}$$

(11)

which yields a Hamiltonian for nuclei coupled anharmonically to a spin operator, where ${\phi }_D|\uparrow \rangle +{\phi }_A|\uparrow \rangle$ represents the general state including the spin. This is expressed as follows:

$$\begin{array}{c}\hfill {\mathcal{H}}_{diab}={\mathcal{H}}_K+V_0\left(\left\{R_n\right\}\right)+h_x\left(\left\{R_n\right\}\right){\sigma }_x+h_z\left(\left\{R_n\right\}\right){\sigma }_z\end{array}$$

(12)

where

$$\begin{array}{cc}\hfill V_0\left(\left\{R_n\right\}\right)& ={\displaystyle \frac{1}{2}}(E_D\left(\left\{R_n\right\}\right)+E_A\left(\left\{R_n\right\}\right)\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill h_x\left(\left\{R_n\right\}\right)& =\Lambda \left(\left\{R_n\right\}\right)\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill h_z\left(\left\{R_n\right\}\right)& ={\displaystyle \frac{1}{2}}(E_D\left(\left\{R_n\right\}\right)-E_A\left(\left\{R_n\right\}\right)\hfill \end{array}$$

(15)

We note that the z-coupling terms $h_z\left(\left\{R_n\right\}\right)$ (Equation (15)) in the Hamiltonian Equation (12) favors ionicity, i.e., an eigenvector of the spin component ${\sigma }_z$ which are $|\uparrow \rangle$ or $|\downarrow \rangle$ (corresponding to the initial diabatic state and the final diabatic state). On the other hand, the x-coupling term $h_x\left(\left\{R_n\right\}\right)$ (Equation (14)) favors covalence, i.e., an eigenvector of the spin component ${\sigma }_x$, which is $1/\sqrt{2}(|\uparrow \rangle +|\downarrow \rangle )$ or $1/\sqrt{2}(|\uparrow \rangle -|\downarrow \rangle )$ transverse to the z component. In the context of electron transfer theory, the transverse term is typically regarded as a relatively minor contributor.

2.2. The Marcus Theory

We consider the example of an elementary reaction which consists of an electron transfer between a donor and an acceptor. The outer sphere electron transfer (OET) has been intensively studied by R. Marcus [29,30]. The reaction process may be interpreted as the thermally activated jump of a single electron from an orbital near a donor site $D^{-}$ to another orbital near an acceptor site A symbolically represented by the chemical reaction $D^{-}A\to D{A}^{-}$ [31]. The Marcus model for OET is based on the (empirical) Diabatic representation of elementary chemical reactions. It assumes that the covalent interactions favoring hybridization between the donor and acceptor are small compared to the ionic interactions favoring the localization of the electron either on the donor or acceptor sites. This covalent interaction plays a possible role only near the transition state where it can favor the jump of the electron between two potential energy surfaces. There are other forms of electron transfer (ET) called inner sphere electron transfer (IET) with more complex descriptions involving transient covalent binding.

Although the diabatic states may look rather well defined far enough from the intersection between the two diabatic energy surfaces, they are only empirically defined especially in the vicinity of their intersection. Despite the fact that it is known that generally the diabatic representation cannot be defined rigorously [32], it is nevertheless commonly used in quantum chemistry. The main possibilities considered for the donor–acceptor system are illustrated in Figure 1 and Figure 2. In Figure 1, two intersecting surfaces lead to the so-called normal (upper) and inverted (lower) Marcus regimes. An energy gap opens up in these two cases, which leads to specific Arrhenius-type exchanges. In Figure 2, we consider another possibility of the intersection of the energy surface that is not connected to a specific chemical reaction. In the following subsection, we discuss how the diabatic curves can be related to the PES defined within the Born–Oppenheimer representation.

An extension of the original Marcus model was already introduced where the assumption was that the covalent interactions are stronger and comparable to the ionic interactions and an interesting intermediate regime was discovered where coherent Ultrafast Electron Transfer (UET) may occur [25].

3. The Born–Oppenheimer Representation

The Born–Oppenheimer (BO) representation has a rigorous definition that does not explicitly include terms that lead to chemical formulas. It was initially developed for a non-relativistic model consisting of a finite collection of electrons (fermions) and nuclei considered spinless particles where magnetic interactions are neglected. The whole system obeys a Schrödinger equation with a complex global Hamiltonian $\mathcal{H}$ Equation (2).

As commented previously, the possible interactions due to the magnetic fields which could be generated by the spin of the electrons and/or their orbitals are neglected as well as many BO theories. If one neglects the nuclei kinetic energy ${\mathcal{H}}_K$, the nuclei coordinates $\left\{R_n\right\}$ appear as real parameters in ${\mathcal{H}}_e\left(\left\{R_n\right\}\right)$. We may then proceed to diagonalize this electronic component formally, namely ${\mathcal{H}}_e(\left\{r_{\nu }\right\},\left\{p_{\nu }\right\};\left\{R_n\right\})$. The sequence of electronic levels, $E_i\left(\left\{R_n\right\}\right)$, is defined in increasing order with $i=0,1,\dots +\infty$ associated with electronic eigenfunctions ${\chi }_i(\left\{r_{\nu }\right\};\left\{R_n\right\})$ which are normalized and have to be antisymmetric under electron exchange (with the same spin) since the electrons are fermions. These eigenfunctions are real since no magnetic interaction is involved in the electronic Hamiltonian Equation (2). The smallest electronic energy $E_0\left(\left\{R_n\right\}\right)$ corresponds to the electronic ground state.

Then we can expand the electronic Hamiltonian on its base of eigenstates

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathcal{H}}_e(\left\{r_{\nu }\right\},\left\{p_{\nu }\right\};\left\{R_n\right\})=\sum _i{E}_i\left(\left\{R_n\right\}\right)·|{\chi }_i(\left\{r_{\nu }\right\};\left\{R_n\right\})\rangle \langle {\chi }_i(\left\{r_{\nu }\right\};\left\{R_n\right\})|\end{array}\end{array}$$

(16)

Similarly, the global wave function $\Phi (\left\{R_n\right\},\left\{r_{\nu }\right\})$ of the system described by Hamiltonian Equation (2) can be also expanded in this electronic base $|{\chi }_i(\left\{r_{\nu }\right\};\left\{R_n\right\})\rangle$ as

$$\begin{array}{c}\hfill \Phi (\left\{R_n\right\},\left\{r_{\nu }\right\})=\sum _i{\phi }_i\left(\left\{R_n\right\}\right){\chi }_i(\left\{r_{\nu }\right\};\left\{R_n\right\})\end{array}$$

(17)

which defines a vector of wave functions $\overline{\phi }\left(\left\{R_n\right\}\right)=\left\{{\phi }_i\left(\left\{R_n\right\}\right)\right\}$ for the nuclei variables.

If we eliminate the electronic variables $\left\{x_{\nu }\right\}$ the initial global Hamiltonian Equation (2) becomes formally a new Hamiltonian acting on vectorial wave functions $\overline{\phi }\left(\left\{R_n\right\}\right)$. This Hamiltonian takes the form of a matrix $\mathcal{H}(\left\{R_n\right\},\left\{P_n\right\})$ of operators ${\widehat{\mathcal{H}}}_{i,j}(\left\{R_n\right\},\left\{P_n\right\})$

$$\begin{array}{c}\hfill 〈\overline{\phi }\left(\left\{R_n\right\}\right)\left|\mathcal{H}\right|\overline{\phi }\left(\left\{R_n\right\}\right)〉=\sum _{i,j}〈{\phi }_i\left(\left\{R_n\right\}\right)\left|{\widehat{\mathcal{H}}}_{i,j}\right|{\phi }_j\left(\left\{R_n\right\}\right)〉\end{array}$$

(18)

where the matrix elements are defined as

$$\begin{array}{c}\hfill {\widehat{\mathcal{H}}}_{i,j}=E_i\left(\left\{R_n\right\}\right){\delta }_{i,j}+{〈{\chi }_i(\left\{r_{\nu }\right\};\left\{R_n\right\})\left|{\mathcal{H}}_K\right|{\chi }_j(\left\{r_{\nu }\right\};\left\{R_n\right\})〉}_e\end{array}$$

The Hermitian product ${\langle .|.\rangle }_e$ in this equation involves only the integration of the electronic variables. We first calculate $P_n·{\chi }_i$, which is involved in the calculation of $〈{\chi }_i\left|{\mathcal{H}}_K\right|{\chi }_j〉$. Then, $P_n·{\chi }_i=i/\hslash \partial {\chi }_i(\left\{x_{\nu }\right\};\left\{R_n\right\})/\partial R_n$ considered as a function of electronic variables $\left\{x_{\nu }\right\}$ can be expanded on the base of electronic wave functions ${\chi }_j$ where $\left\{R_n\right\}$ are considered as parameters.

$$\begin{array}{c}\hfill P_n·{\chi }_i(\left\{x_{\nu }\right\};\left\{R_n\right\})={\displaystyle \frac{\hslash }{i}}\sum _j{b}_{i,j}^{\left(n\right)}\left(\left\{R_n\right\}\right){\chi }_j(\{x_{\nu };\left\{R_n\right\})\end{array}$$

(19)

The coefficients $b_{i,j}^{\left(n\right)}\left(\left\{R_n\right\}\right)$ are defined as

$$\begin{array}{c}\hfill b_{i,j}^{\left(n\right)}\left(\left\{R_n\right\}\right)={〈{\chi }_j\left(\left\{R_n\right\}\right)|{\displaystyle \frac{\partial }{\partial R_n}}{\chi }_i\left(\left\{R_n\right\}\right)〉}_e\end{array}$$

(20)

These coefficients are real and obtained from standard perturbation theory for $i\ne j$

$$\begin{array}{c}\hfill b_{i,j}^{\left(n\right)}\left(\left\{R_n\right\}\right)={\displaystyle \frac{{〈{\chi }_j\left(\left\{R_n\right\}\right)\left|\frac{\partial {\mathcal{H}}_e}{\partial R_n}\right|{\chi }_i\left(\left\{R_n\right\}\right)〉}_e}{E_i-E_j}}\end{array}$$

(21)

while for $i=j$, the normalization of ${\chi }_i$ implies $b_{i,i}^{\left(n\right)}\left(\left\{R_n\right\}\right)=0$. We note that $b_{i,j}^{\left(n\right)}\left\{R_n\right\})=-b_{j,i}^{\left(n\right)}\left(\left\{R_n\right\}\right)$. Taking into account that variable $R_n$ and its conjugate variable $P_n$ considering quantum operators do not commute with each other, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\widehat{\mathcal{H}}}_{i,i}={\mathcal{H}}_K+V_i\left(\left\{R_n\right\}\right)\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\hfill \\ \hfill & V_i\left(\left\{R_n\right\}\right)=E_i\left(\left\{R_n\right\}\right)-\sum _n{\displaystyle \frac{{\hslash }^2}{2M_n}}\sum _k{b}_{i,k}^{\left(n\right)}\left(\left\{R_n\right\}\right)b_{k,i}^{\left(n\right)}\left(\left\{R_n\right\}\right)\hfill \end{array}\end{array}$$

(22)

and for $i\ne j$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\widehat{\mathcal{H}}}_{i,j}=C_{i,j}\left(\left\{R_n\right\}\right)+i{D}_{i,j}\left(\left\{R_n\right\}\right)\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\hfill \\ \hfill & C_{i,j}\left(\left\{R_n\right\}\right)=-\sum _n{\displaystyle \frac{{\hslash }^2}{2M_n}}\sum _k{b}_{i,k}^{\left(n\right)}\left(\left\{R_n\right\}\right)b_{k,j}^{\left(n\right)}\left(\left\{R_n\right\}\right)\hfill \\ \hfill & D_{i,j}(\left\{R_n\right\},\left\{P_n\right\})=\sum _n{\displaystyle \frac{\hslash }{2M_n}}(b_{i,j}^{\left(n\right)}\left(\left\{R_n\right\}\right)P_n+P_n{b}_{i,j}^{\left(n\right)}\left(\left\{R_n\right\}\right)\hfill \end{array}\end{array}$$

(23)

Finally, the dynamics of the nuclei coupled to the electrons take the usual simple form

$$i\hslash \dot{\overline{\phi }}=\widehat{\mathcal{H}}·\overline{\phi }$$

(24)

where $\overline{\phi }$ is a multicomponent wave functions defined by Equation (17). There are no approximations to the initial global Hamiltonian (2) at this stage.

3.1. The Born–Oppenheimer Approximation

The Born–Oppenheimer (BO) approximation assumes that all off-diagonal terms ${\widehat{\mathcal{H}}}_{i,j}$ ($i\ne j$) of the matrix $\mathcal{H}(\left\{R_n\right\},\left\{P_n\right\})$ of operators are negligible, such that the operator $\widehat{\mathcal{H}}$ becomes diagonal. Consequently, we obtain a collection of independent BO Hamiltonians associated with each electronic eigenstate i:

$$\begin{array}{c}\hfill {\mathcal{H}}_{BO}={\mathcal{H}}_K+V_i\left(\left\{R_n\right\}\right)\end{array}$$

(25)

where the effective nuclei potentials are defined as $V_i\left(\left\{R_n\right\}\right)=E_i\left(\left\{R_n\right\}\right)+{\Delta }_{i,i}\left(\left\{R_n\right\}\right)$. Subsequently, the global wave function of the system when the system is located on the $i^{th}$ PES can be expressed in a simple BO form

$$\begin{array}{c}\hfill {\Phi }_i(\left\{R_n\right\},\left\{r_{\nu }\right\})={\phi }_i\left(\left\{R_n\right\}\right){\chi }_i(\left\{r_{\nu }\right\};\left\{R_n\right\})\end{array}$$

(26)

The electronic eigenenergy i is defined in terms of the nuclei coordinates as a function of a PES, which is denoted by the symbol $V_i\left(\left\{R_n\right\}\right)$. The PES are commonly observed in spectroscopy experiments and are studied in physics and chemistry. Additionally, their values have been calculated numerically for simple molecules using ab initio methods or for the electronic ground state, $i=0$, employing Density Functional Theory (DFT).

The minimum of the lowest PES $V_0\left(\left\{R_n\right\}\right)$ yields the spatial arrangement of the nuclei system in its ground state. When approximating $V_0\left(\left\{R_n\right\}\right)$ near its minimum by a harmonic expansion, the BO Hamiltonian can be diagonalized, thereby yielding the phonon spectrum and their associated modes possibly observable. The same procedure can be applied to the electronically excited states corresponding to different PES which yield different spatial nuclei arrangements and phonon spectrum near their minimum.

The pioneering Kramer’s Transition State theory of chemical reactions [1,33] describes the dynamics under thermal fluctuations between two local minima of the electronic ground state PES $V_0\left(\left\{R_n\right\}\right)$ assumed to correspond to the two different chemical species before and after the chemical reaction. The kinetics of the chemical reaction are simply represented by the random path of a diffusive particle moving on the PES with friction and subjected to a random thermal force. The lowest saddle point of the PES (called in mathematics minimax) between the two minima corresponds to the transition state near which the most probable reaction paths should pass. It yields the energy barrier involved in the Arrhenius law.

The Frank–Condon principle [34,35], a well-known concept in photochemistry, explicitly incorporates the existence of PES. Subsequently, the absorption of a photon with a frequency of $\hslash \omega =V_j\left(\left\{R_n\right\}\right)-V_i\left(\left\{R_n\right\}\right)>0$ ($j>i$) initiates a direct transition from the initial PES i to an upper PES j at the same nuclear coordinates $\left\{R_n\right\}$. The next step is for the nuclei configuration to relax near the minimum of the new PES. Subsequently, the electronic state will also relax back to a lower PES, resulting in the emission of a photon with a frequency smaller than the initial value.

The BO approximation may break for a given PES in some domain of nuclei coordinates $\left\{R_n\right\}$ when the electronic gap between this PES and the nearest ones above and/or below [36]. This is indicated by the fact that the energy differences $E_{i+1}\left(\left\{R_n\right\}\right)-E_i\left(\left\{R_n\right\}\right)$ and/or $E_i\left(\left\{R_n\right\}\right)-E_{i-1}\left(\left\{R_n\right\}\right)$ become sufficiently small to be within the range of phonon frequencies. The largest phonon frequency $\hslash {\omega }_c$ (frequency cut-off) in the phonon spectrum for realistic materials is usually about a fraction of eV. Then, phonon quanta may have enough energy to trigger direct electronic transitions (in the lowest order) between two PES. The consequence is that the electronic state of the system cannot remain invariant as assumed in the BO approximation.

Note, that the electromagnetic spectrum does not exhibit any frequency cut-off (unlike the phonon spectrum). Consequently, direct photonic transitions may be induced between different PESs, provided that the photon frequency corresponds to the electronic gap. Since the external electromagnetic fields are weak compared to the microscopic fields, these transitions are usually described with the Fermi Golden rule. Then, the lifetime of the excited electronic state is sufficiently long at the scale of phonon frequencies so that the system has time to relax while staying on its PES (Frank–Condon principle [37]). The standard situations where the BO approximation holds concern insulators or semiconductors when the lowest electronic gaps are at least of the order of 1 eV or much larger while the phonon excitations are at most fractions of eV. Thus, note that BO approximation is, in principle, not valid when the considered system is a gapless metal or nearly metallic. Then, corrections to the BO approximation are generally taken into account by extra electron–phonon interactions. However, keep in mind that in this paper we are considering transitions between localized electronic excitations in an infinite system. This assumption obviously requires that the global system is not metallic, but a dielectric insulator. Then, the involved PES are still well described considering only a finite subsystem corresponding to the nearby environment of the electronic excitation large enough but still finite. Nevertheless, assuming that the global system is infinite remains essential for explaining the energy dissipation of the chemical reaction by phonon transportation. However, in the case of periodic systems with electronic bands but with strong electron–phonon coupling, we may recover localized excitations such as polarons or excitons, the mobility of which could be described by an N-state model extending the two-state model we study here but on a lattice.

Since ${\mathcal{H}}_e\left(\left\{R_n\right\}\right)$ has infinitely many eigenenergies which become denser and denser at high energies, the spacing between two consecutive eigenenergies cannot always remain small for high-energy electronic states so that BO approximation does not hold anymore for high-energy PES. Of course, it is also not valid in the case of degenerate electronic states, but this situation is rare because, in a system with N degrees of freedom, we generally have avoided crossing between the different PES due to level repulsion Figure 1 according to a Von Neumann–Wigner theorem also called avoided crossing theorem [38] which is valid only for finite systems.

This theorem implies that the PES cannot be degenerate except at intersections between two PES but with dimension $N-2$ (instead of the expected dimension $N-1$) generally because of special symmetries. Such intersections are called conical; indeed in the case of electronic degeneracy, most distortions of the nuclei configuration raise linearly this degeneracy (except on a $N-2$ manifold).

3.2. Extended Two-State BO Approximation

Kramers’ theory may not be sufficient to describe the kinetics of all elementary reactions because it may involve a jump between two PES. The latter is a forbidden process within the BO approximation since it assumes that the system should remain on the same PES, see, for example, the lower scheme Figure 1 corresponds to the inverted regime in the Marcus theory.

Generally, it is more realistic to consider that the electronic subspace visited by the elementary chemical reaction path has dimension two since we expect the existence of two diabatic states, the initial and final ones, as previously explained. Figure 1 illustrates the approximate generation of two PES in the standard literature in chemistry. This is achieved by simply opening a gap at the intersection which are, for example, the two lowest possible energy surfaces $V_0\left(\left\{R_n\right\}\right)$ and $V_1\left(\left\{R_n\right\}\right)$. It should be noted that photochemically induced reactions may involve different PES.

This extended BO approximation becomes useful when the two PES come close enough to each other with a gap ranging in the phonon frequencies spectrum. It is equivalent to assuming that the global wave function has the form

$$\begin{array}{c}\hfill \Phi (\left\{R_n\right\},\left\{r_{\nu }\right\})={\varphi }_0\left(\left\{R_n\right\}\right){\chi }_0\left(\left\{R_n\right\}\right)+{\varphi }_1\left(\left\{R_n\right\}\right){\chi }_1\left(\left\{R_n\right\}\right)\end{array}$$

(27)

where ${\chi }_0\left(\left\{R_n\right\}\right)$ and ${\chi }_1\left(\left\{R_n\right\}\right)$ are the two consecutive eigenstates of the electronic Hamiltonian involved in the avoided PES crossing.

The matrix of operators $\widehat{\mathcal{H}}$ is defined by Equations (22) and (23) restricted in this 2d electronic subspace defined by $i=0$ and $i=1$ becomes a $2\times 2$ matrix. The corrective term in Equation (22) is the same for both PES so that

$$\begin{array}{c}\hfill \begin{array}{c}\hfill V_0\left(\left\{R_n\right\}\right)=E_0\left(\left\{R_n\right\}\right)+\sum _n{\displaystyle \frac{{\hslash }^2}{2M_n}}{b}_{0,1}^{\left(n\right)2}\left(\left\{R_n\right\}\right)\\ \hfill V_1\left(\left\{R_n\right\}\right)=E_1\left(\left\{R_n\right\}\right)+\sum _n{\displaystyle \frac{{\hslash }^2}{2M_n}}{b}_{0,1}^{\left(n\right)2}\left(\left\{R_n\right\}\right)\end{array}\end{array}$$

(28)

This $2\times 2$ matrix can be expanded based on the Pauli matrices Equation (11) and the identity so that it obtains the form

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \widehat{\mathcal{H}}={\mathcal{H}}_K+A_0\left(\left\{R_n\right\}\right)+A_x\left(\left\{R_n\right\}\right){\sigma }_x+A_y(\left\{R_n\right\},\left\{P_n\right\}){\sigma }_y+A_z\left(\left\{R_n\right\}\right){\sigma }_z\end{array}\end{array}$$

(29)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill A_0\left(\left\{R_n\right\}\right)& ={\displaystyle \frac{1}{2}}(V_0\left(\left\{R_n\right\}\right)+V_1\left(\left\{R_n\right\}\right))\hfill \\ \hfill A_x\left(\left\{R_n\right\}\right)& =C_{0,1}\left(\left\{R_n\right\}\right)=C_{1,0}\left(\left\{R_n\right\}\right)=+\sum _n{\displaystyle \frac{{\hslash }^2}{2M_n}}{b}_{0,1}^{\left(n\right)2}\left(\left\{R_n\right\}\right)\hfill \\ \hfill A_y(\left\{R_n\right\},\left\{P_n\right\})& =i{D}_{0,1}(\left\{R_n\right\},\left\{P_n\right\})=i\sum _n{\displaystyle \frac{\hslash }{2M_n}}(b_{0,1}^{\left(n\right)}\left(\left\{R_n\right\}\right)P_n+P_n{b}_{0,1}^{\left(n\right)}\left(\left\{R_n\right\}\right))\hfill \\ \hfill A_z\left(\left\{R_n\right\}\right)& ={\displaystyle \frac{1}{2}}(V_0\left(\left\{R_n\right\}\right)-V_1\left(\left\{R_n\right\}\right))\hfill \end{array}\end{array}$$

Thus, the projection of the global Hamiltonian Equation (2) in the subspace of two electronic eigenstates may be viewed as a Spin-Boson model where a fictitious spin $1/2$ describing the electronic state is coupled in all spin directions to the collection of nuclei. Note, that the transverse fields $A_x\left(\left\{R_n\right\}\right)$ and $A_y\left(\left\{R_n\right\}\right)$ favor some hybridization of the electronic states associated with the two PES. Although this formulation is exact within the assumption of a two-dimensional subspace of electronic wave functions, it is useful to redefine diabatic states because they closely represent the chemical structure. It does not change the model to rotate the fictitious quantum spin $\sigma$, that is, to rotate the BO base of electronic state ${\chi }_0\left(\left\{R_n\right\}\right),{\chi }_1\left(\left\{R_n\right\}\right)$. This rotation is accomplished by a unitary $2\times 2$ matrix, $\mathcal{U}\left(\left\{R_n\right\}\right)$. The general form of this transformation may be expressed as the product of three rotations, each expressed in terms of the Pauli matrices Equation (11).

$$\mathcal{U}\left(\left\{R_n\right\}\right)=e^{i\gamma }(cos\alpha +isin\alpha ·{\sigma }_z)·(cos\theta +i{\sigma }_{y}sin\theta )·(cos\beta +isin\beta ·{\sigma }_z)$$

(30)

where angles $\alpha ,\beta ,\gamma$ and $\theta$ are functions of $\left\{R_n\right\}$.

The standard schemes shown in Figure 1 suggest that when $\left\{R_n\right\}$ is near the first minimum of $V_0\left(\left\{R_n\right\}\right)$ corresponding to the initial state, the electronic state ${\chi }_D\left(\left\{R_n\right\}\right)$ is nearly identical to ${\chi }_0\left(\left\{R_n\right\}\right)$ while ${\chi }_A\left(\left\{R_n\right\}\right)$ is nearly identical to ${\chi }_1\left(\left\{R_n\right\}\right)$. Therefore, the unitary matrix should be the identity matrix, corresponding to $\alpha =\beta =\gamma =\theta =0$ in Equation (30). Similarly when $\left\{R_n\right\}$ is near the second minimum of $V_0\left(\left\{R_n\right\}\right)$ corresponding to the final state where the electron is on the acceptor site while ${\chi }_A\left(\left\{R_n\right\}\right)$, whereby ${\chi }_D\left(\left\{R_n\right\}\right)$ should be nearly identical to ${\chi }_1\left(\left\{R_n\right\}\right)$ and ${\chi }_A\left(\left\{R_n\right\}\right)$ should be approximately identical to ${\chi }_0\left(\left\{R_n\right\}\right)$ so that the unitary matrix should be equal to ${\sigma }_x$ which corresponds to $\alpha =-\pi /4$, $\beta =\pi /4$, $\gamma =\pi /2$ and $\theta =-\pi /2$ in Equation (30). To generate diabatic states, the unitary transformation should vary from unity when $\left\{R_n\right\}$ is close to the first minimum of $V_0\left(\left\{R_n\right\}\right)$, corresponding to a state with a well-defined chemical formula to ${\sigma }_x$ when $\left\{R_n\right\}$ is close to the second minimum of $V_0\left(\left\{R_n\right\}\right)$, corresponding to a state $V_0\left(\left\{R_n\right\}\right)$ with another well defined chemical formula.

Such a transformation determines new states called diabatic as

$$\begin{array}{c}\hfill \left(\begin{array}{c}{\phi }_D\\ {\phi }_A\end{array}\right)=\mathcal{U}\left(\left\{R_n\right\}\right)·\left(\begin{array}{c}{\phi }_0\\ {\phi }_1\end{array}\right)\end{array}$$

(31)

However since the conjugate operators $R_n$ and $P_n$ do not commute, the terms in $\widehat{\mathcal{H}}$ are modified since the unitary matrix $\mathcal{U}$ depends on $\mathcal{U}\left(\left\{R_n\right\}\right)$. Finally, the Hamiltonian retains the form of a spin-boson Hamiltonian with the form

$$\begin{array}{c}\hfill \widehat{\mathcal{H}}=\sum _{\alpha =0,x,y,z}{\tilde{A}}_{\alpha }(\left\{R_n\right\},\left\{P_n\right\}){\sigma }_{\alpha }\end{array}$$

(32)

where ${\sigma }_{\alpha }$ are the Pauli matrices Equation (11). It is not necessary to explicitly show the complex result of the calculation of ${\tilde{A}}_{\alpha }(\left\{R_n\right\},\left\{P_n\right\})$. The main issue is to find a unitary transformation $\mathcal{U}\left(\left\{R_n\right\}\right)$ that is physically realistic. For this purpose, the new coefficients ${\tilde{A}}_{\alpha }(\left\{R_n\right\},\left\{P_n\right\})$ should have a smooth variation so that they can be approximated reasonably well by their lowest-order expansions. It is not clear whether this is possible in all cases, for instance, when three diabatic states are involved in the same elementary chemical reaction. Then, we should extend this analysis to a three-dimensional electronic subspace. The existence of a diabatic representation is often assumed a priori in chemistry, for example, in the Marcus theory of electron transfer and yields qualitatively correct results [30]. This representation is very convenient for describing the electronic organization in terms of occupied or unoccupied electron orbitals and/or their quantum hybridization as conducted in chemistry.

3.3. Semiclassical Approximation

Paul Dirac [39] proposed that classical mechanics could be derived from quantum mechanics as a consequence of destructive interference among paths in the classical phase space that do not externalize the Lagrangian action. Those with constructive interference are merely close to the classical trajectories. Dirac’s ideas were later elaborated by R. Feynman in his Path Integral Representation of Quantum Mechanics [40]. In contrast, the Heisenberg uncertainty principle asserts that the measurement accuracies for two conjugate variables, $R_n$ and $P_n$, must obey the rule $\Delta R_n,\Delta P_n\ge \hslash /2$. This rule gives the order of accuracy of the classical trajectory. Consequently, classical dynamics are only applicable to large displacements in the phase space, which are much larger than phonon quantum fluctuations.

In our system, electrons are very light and are fermions so they should be treated quantumly. However, the dynamics of the nuclei may be well studied within a semiclassical approximation subsequently to the Born–Oppenheimer approximation, Equation (25) where the electrons are treated quantum mechanically and the nuclei classically.

However, the standard BO approximation may not be valid when two PES become too close to each other, that is, when some electronic frequencies $E_1\left(\left\{R_n\right\}\right)-E_0\left(\left\{R_n\right\}\right)\hslash {\omega }_e$ get into the phonon spectrum which extends up to some upper cut-off frequency ${\omega }_p$. We have shown above that such situations may be described by a Spin-Boson Hamiltonian Equation (29) which takes into account extra quantum interactions between these two PESs and keeps a part of the electron dynamics.

3.4. Conical Intersection of Two Potential Energy Surfaces

Conical intersections are critical to the photochemical processes of essential biological molecules, impacting their stability, energy dissipation, and evolutionary trajectories [41]. The PES obtained with the BO approximation are generally difficult to calculate and, moreover, uneasy to use to understand the corresponding nuclei configurations as well as the possible chemical reactions [27,42]. The semiclassical approximation is similar to a gauge transformation

$$\begin{array}{c}\hfill \left(\begin{array}{c}{\chi }_D\left(\left\{R_n\right\}\right)\\ {\chi }_A\left(\left\{R_n\right\}\right)\end{array}\right)=\left(\begin{array}{cc}cos\alpha \left(\left\{R_n\right\}\right)& sin\alpha \left(\left\{R_n\right\}\right)\\ -sin\alpha \left(\left\{R_n\right\}\right)& cos\alpha \left(\left\{R_n\right\}\right)\end{array}\right)·\left(\begin{array}{c}{\chi }_0\left(\left\{R_n\right\}\right)\\ {\chi }_1\left(\left\{R_n\right\}\right)\end{array}\right)\end{array}$$

(33)

To obtain a good diabatic base, the new coefficients of the Spin-Boson system Equation (29) should be a function of the nuclei coordinate which is as smooth as possible and well approximated by low order expansion since this is not true. However, there are no standard criteria for optimizing the choice of this rotation angle $\alpha \left(\left\{R_n\right\}\right)$ for obtaining the best diabatic base.

We assume that there exist two orthogonal electronic states so that ${\chi }_D\left(\left\{R_n\right\}\right)$ should well represent the initial state of the system where the electron only occupies the orbital on the donor (which is deformable when $\left\{R_n\right\}$ vary) while ${\chi }_A\left(\left\{R_n\right\}\right)$ represents the final state of the system where the electron only occupies the orbital on the acceptor.

$$\begin{array}{c}\hfill \widehat{\mathcal{H}}={\mathcal{H}}_K+A_0\left(\left\{R_n\right\}\right)+A_x^{\prime }\left(\left\{R_n\right\}\right){\sigma }_x+A_z^{\prime }\left(\left\{R_n\right\}\right){\sigma }_z\end{array}$$

(34)

where

$$\begin{array}{c}\hfill \left(\begin{array}{c}{A}_x^{\prime }\left(\left\{R_n\right\}\right)\\ A_z^{\prime }\left(\left\{R_n\right\}\right)\end{array}\right)=\left(\begin{array}{cc}cos\alpha \left(\left\{R_n\right\}\right)& sin\alpha \left(\left\{R_n\right\}\right)\\ -sin\alpha \left(\left\{R_n\right\}\right)& cos\alpha (\left\{R_n\right\}\end{array}\right)·\left(\begin{array}{c}{A}_x\left(\left\{R_n\right\}\right)\\ A_z\left(\left\{R_n\right\}\right)\end{array}\right)\end{array}$$

Assuming that we obtain a good diabatic representation, we see that when neglecting $A_x^{\prime }\left(\left\{R_n\right\}\right)$, the coefficients $A_z^{\prime }\left(\left\{R_n\right\}\right)$ favor a spin $|\uparrow \rangle$ or $|\downarrow \rangle$ corresponding to the electronic state ${\chi }_D\left(\left\{R_n\right\}\right)$ or ${\chi }_A\left(\left\{R_n\right\}\right)$. We obtain the purely ionic case. In contrast, when neglecting $A_z^{\prime }\left(\left\{R_n\right\}\right)$, the coefficients $A_x^{\prime }\left(\left\{R_n\right\}\right)$ favor a purely covalent state $1/2(|\uparrow \rangle +|\downarrow \rangle )$ or $1/2(|\uparrow \rangle +|\downarrow \rangle )$

More precisely, potential $A_0(\left\{R_n\right\})$ is supposed to have a single minimum which may be assumed to be the origin for the nuclear coordinates as well as for the energy scale. Then, it is assumed to be well approximated by its expansion as a quadratic function

$$\begin{array}{c}\hfill A_0\left(\right\{R_n\left\}\right)\approx {\displaystyle \frac{1}{2}}\sum _{n,m}{a}_{n,m}{R}_n{X}_m\end{array}$$

(35)

Next, we can expand to the lowest order

$$\begin{array}{c}\hfill A_x^{\prime }\left(\left\{R_n\right\}\right)\approx A_x^{\prime }\left(\left\{0\right\}\right)+\sum _n{t}_n{R}_n\end{array}$$

(36)

$$\begin{array}{c}\hfill A_z^{\prime }\left(\left\{R_n\right\}\right)\approx A_z^{\prime }\left(\left\{0\right\}\right)+\sum _n{h}_n{R}_n\end{array}$$

(37)

When the electron remains in the donor orbital that corresponds to the spin $|\uparrow \rangle$, we obtain a PES we call Diabatic PES $|\uparrow \rangle$ the Diabatic PES is $V_D\left(\left\{R_n\right\}\right)=A_0\left(\left\{R_n\right\}\right)+A_z^{\prime }\left(\left\{R_n\right\}\right)$ while when the electron remains in the acceptor orbital represented by $|\downarrow \rangle$, the corresponding Diabatic PES is $V_A\left(\left\{R_n\right\}\right)=A_0\left(\left\{R_n\right\}\right)-A_z^{\prime }\left(\left\{R_n\right\}\right)$. The description used in [25] assumes the existence of this base of orbitals as already conducted in the Marcus theory of Electron Transfer.

4. A Simple Prototype Model with Ultrafast Targeted Electron Transfer

The phenomenon of exceptional chemical reactions that do not obey the Arrhenius law is ubiquitous in biological systems and plays an essential role in the functioning of life. A lot of work has been conducted to describe fast electron transfer in the different systems [43,44]. The investigation of classical and quantum-targeted energy transfer between nonlinear oscillators was studied in the paper [45]. For example, UET at the Photosynthetic Reaction Center (PRC) allows living photosynthetic cells to capture sunlight energy with great efficiency, which is then converted into chemical energy for subsequent use in biological processes [12]. We believe that the novel approach will prove beneficial in elucidating a multitude of puzzling phenomena observed in living cells, thereby stimulating further research to advance these new concepts.

The extension of electron transfer occurs in the vicinity of the situation where the PES exhibits a degenerate ground state, which continuously connects the state with the electron on the donor state. This phenomenon is influenced by the competition between ionic and covalent interactions.

Our approach for electron transfer consists of studying the quantum dynamics of the wave function of an electron (or any other kind of quantum excitation) from a donor site D to an acceptor site A. The wave function of the electron at time t has the form ${\phi }_D\left(t\right)|D\rangle +{\phi }_A\left(t\right)|A\rangle$ where $|D\rangle$ is the orbital of the electron localized at the donor site D and $|A\rangle$ the orbital of the electron localized at the acceptor site A. The Hamiltonian of such a model has the simple form

$$\begin{array}{c}\hfill H=E_D|{\phi }_D{|}^2+E_A{\left|{\phi }_A\right|}^2+\Gamma ({\phi }_D^{★}{\phi }_A+{\phi }_A^{★}{\phi }_D)\end{array}$$

(38)

where $E_D$ and $E_A$ are the onsite energies on the donor and acceptor and $\Gamma$ is the transfer integral which depends on the overlap between the two orbitals on the donor and acceptor. However, this model is not isolated. It should be taken into account for its interaction with the complex environment in which the electron transfer couples.

Our purpose is to produce an example of a model where TET could occur, it is convenient to assume for simplicity that the model we consider is nearly symmetric which implies the reaction energy $E_D-E_A$ is relatively small as it is in elementary biochemical reactions. ATP hydrolysis reaction energy 0.3 eV may be considered a good unit for scaling much smaller than those of most chemical reactions in inorganic chemistry.

More precisely, the onsite energies $E_D$ and $E_A$ depend on this environment because a charge transfer between donor and acceptor changes the local electric field which polarizes the environment and consequently generates nuclei displacement. Otherwise, since organic molecules are easily deformable, the spatial distance between the donor and acceptor may also vary during electron transfer, which changes the orbital overlap and consequently the transfer integral $\Gamma$.

It can be assumed that the environment is described as a collection of infinitely many coupled harmonic oscillators. Two classical linear oscillators are selected, with variables $u_z$ and $u_x$, which are coupled to the other oscillators. These oscillators will be considered as a Langevin bath. The mode $u_z$ describes the deformation of the environment essentially due to charge transfer. We assume that it is linearly coupled to both charge densities $|{\phi }_D{|}^2$ and $|{\phi }_A{|}^2$, but since $|{\phi }_D{|}^2+{\left|{\phi }_A\right|}^2=1$ is constant, it turns out to be only coupled to $C=|{\phi }_A{|}^2-{\left|{\phi }_D\right|}^2$. Thus, the variation of $u_z$ changes linearly the effective electronic levels $E_D$ and $E_A$. The exchange of donor and acceptor, C is changed to $-C$, hence it may be argued that this mode is antisymmetric in relation to the electron exchange between donor and acceptor. Mode $u_x$ represents the deformation of the environment when the spatial distance between the donor and the acceptor sites varies which thus changes the overlap integral $\Gamma$. We again assume that this oscillator is linearly coupled to $({\phi }_D^{★}{\phi }_A+{\phi }_A^{★}{\phi }_D)$. Unlike mode $u_z$, this mode $u_x$ is symmetric when exchanging the donor and acceptor, and thus must be different from the symmetric mode $u_z$. The Hamiltonian of our model (when mode $u_z$ and $u_x$ are decoupled from the Langevin bath)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill H=E_D{\left|{\phi }_D\right|}^2& +E_A|{\phi }_D{|}^2+k_z{u}_z\left(\right|{\phi }_A{|}^2-|{\phi }_D{|}^2)\hfill \\ \hfill & +({ϵ}_x+k_x{u}_x)({\phi }_D^{*}{\phi }_A+{\phi }_A^{*}{\phi }_D)\hfill \\ \hfill & +\frac{1}{2}{p}_z^2+\frac{1}{2}{\Omega }_z^2{u}_z^2+\frac{1}{2}{p}_x^2+\frac{1}{2}{\Omega }_x^2{u}_x^2\hfill \end{array}\end{array}$$

(39)

and its Hamilton equations are

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & i\hslash {\dot{\phi }}_D=(E_D-k_z{u}_z){\phi }_D+({ϵ}_x+k_x{u}_x){\phi }_A\hfill \\ \hfill & i\hslash {\dot{\phi }}_A=(E_A+k_z{u}_z){\phi }_A+({ϵ}_x+k_x{u}_x){\phi }_D\hfill \\ \hfill & {\ddot{u}}_z+{\Omega }_z^2{u}_z+k_z({\left|{\phi }_A\right|}^2-{\left|{\phi }_D\right|}^2)=0\hfill \\ \hfill & {\ddot{u}}_x++{\Omega }_x^2{u}_x+k_x({\phi }_D^{*}{\phi }_A+{\phi }_A^{*}{\phi }_D)=0\hfill \end{array}\end{array}$$

(40)

The quantity ℏ will be dropped from these equations. Consequently, $1/\hslash$ will be chosen as the unit of time, with ℏ as the unit of energy, and the variables $u_z$, $u_x$, as well as the energies $E_D$, $E_A$, and frequencies ${\Omega }_z$, ${\Omega }_x$ will be rescaled. If we now presume that phonon modes $u_x$ and $u_z$ are coupled to two independent Langevin baths (symmetric and antisymmetric) which add damping and Langevin noise to the oscillator equation Hamilton equations.

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & i{\dot{\phi }}_D=(E_D-k_z{u}_z){\phi }_D+({ϵ}_x+k_x{u}_x){\phi }_A\hfill \\ \hfill & i{\dot{\phi }}_A=(E_A+k_z{u}_z){\phi }_A+({ϵ}_x+k_x{u}_x){\phi }_D\hfill \\ \hfill & {\ddot{u}}_z+{\gamma }_z{\dot{u}}_z+{\Omega }_z^2{u}_z+k_z({\left|{\phi }_A\right|}^2-{\left|{\phi }_D\right|}^2)={\eta }_z\left(t\right)\hfill \\ \hfill & {\ddot{u}}_x+{\gamma }_x{\dot{u}}_x+{\Omega }_x^2{u}_x+k_x({\phi }_D^{*}{\phi }_A+{\phi }_A^{*}{\phi }_D)={\eta }_x\left(t\right)\hfill \end{array}\end{array}$$

(41)

Since we assumed above a quasi symmetry for the donor–acceptor system, we should also split the surrounding phonon collection into symmetric or antisymmetric modes thus generating two almost non-interacting Langevin baths, the antisymmetric one interacting only with the antisymmetric ionic mode $u_z$ and the symmetric one with the covalent mode $u_x$. ${\gamma }_z$ is a constant that depends on the coupling of this oscillator z which is the Langevin bath and the same for the x component. ${\eta }_x(t+\tau )$ and ${\eta }_z\left(t\right)$ are random Gaussian white noise at temperature T with the standard correlation:

$$\begin{array}{cc}\hfill & \langle {\eta }_x(t+\tau ){\eta }_x\left(t\right)\rangle =2{\gamma }_x{k}_{B}T\delta \left(\tau \right)\hfill \\ \hfill & \langle {\eta }_z(t+\tau ){\eta }_z\left(t\right)\rangle =2{\gamma }_z{k}_{B}T\delta \left(\tau \right)\hfill \end{array}$$

This model is equivalent to a Spin-Boson model which corresponds to a quantum spin $1/2$ with quantum state ${\phi }_D|\uparrow \rangle +{\phi }_A|\downarrow \rangle$ coupled to a phonon bath through variables $u_z$ and $u_x$. If we vanish the coupling $k_x$ and keep ${ϵ}_x$ small, our model becomes equivalent to the standard Marcus model, which is well-known in chemistry for describing electron transfer. We then recover the Arrhenius law. The covalent coupling described by the parameter $k_x$ is usually dropped. This model from the first principles was described in detail [25].

At zero degree where ${\eta }_z\left(t\right)={\eta }_x\left(t\right)=0$, Equation (41) readily yield

$$\begin{array}{c}\hfill \dot{H}=-{\gamma }_z{\dot{u}}_z^2-{\gamma }_x{\dot{u}}_x^2\end{array}$$

(42)

proving that the system energy decays as a function of time because the oscillators are damped. By substitution in Equation (40), we obtain a DNLS dimer Hamiltonian $H_{aa}$ which may be viewed as the anti-adiabatic approximation of the general Hamiltonian valid when the atoms are very light, that is, when the frequencies ${\Omega }_z$ and ${\Omega }_x$ are very large (so that the nuclei follow adiabatically the electronic variables).

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill H_{aa}=& -{\displaystyle \frac{1}{2}}{\displaystyle \frac{k_z^2}{{\Omega }_z^2}}\left(\right|{\phi }_A{|}^2-|{\phi }_D{{|}^2)}^2-{\displaystyle \frac{1}{2}}{\displaystyle \frac{k_x^2}{{\Omega }_x^2}}{({\phi }_D^{★}{\phi }_A+{\phi }_A^{★}{\phi }_D)}^2\hfill \\ \hfill & +E_D|{\phi }_D{|}^2+E_A{\left|{\phi }_A\right|}^2+{ϵ}_x({\phi }_D^{★}{\phi }_A+{\phi }_A^{★}{\phi }_D)\hfill \end{array}\end{array}$$

(43)

Since $|{\phi }_D{|}^2+{\left|{\phi }_A\right|}^2=1$ is a time-invariant of the first two Equation (40), we can redefine the variable $\rho =|{\phi }_A{|}^2$ that only varies between 0 and 1 (charge transfer) so that ${\phi }_A=\sqrt{\rho }{e}^{i{\alpha }_A}$ and ${\phi }_D=\sqrt{1-\rho }{e}^{i{\alpha }_D}$ where ${\alpha }_A$ and ${\alpha }_D$ are phase variables. Then, the equilibrium state are minimum of

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill H_{aa}& =-{\displaystyle \frac{1}{2}}{\displaystyle \frac{k_z^2}{{\Omega }_z^2}}{(2\rho -1)}^2-{\displaystyle \frac{2k_x^2}{{\Omega }_x^2}}\rho (1-\rho ){cos}^2({\alpha }_A-{\alpha }_D)\hfill \\ \hfill & +E_D(1-\rho )+E_A\rho +2{ϵ}_x\sqrt{\rho (1-\rho )}cos({\alpha }_A-{\alpha }_D)\hfill \end{array}\end{array}$$

(44)

Minimizing $H_{aa}$ with respect to the phase yields ${\alpha }_A-{\alpha }_D=0mod2\pi$ when ${ϵ}_x<0$ and ${\alpha }_A-{\alpha }_D=\pi mod2\pi$ so that the minimum of $H_{aa}$ is obtained by minimizing

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill F\left(\rho \right)& =-{\displaystyle \frac{1}{2}}{\displaystyle \frac{k_z^2}{{\Omega }_z^2}}{(2\rho -1)}^2-{\displaystyle \frac{2k_x^2}{{\Omega }_x^2}}\rho (1-\rho )+E_D(1-\rho )+E_A\rho -2\left|{ϵ}_x\right|\sqrt{\rho (1-\rho )}\hfill \\ \hfill & =\left({\displaystyle \frac{2k_x^2}{{\Omega }_x^2}}-{\displaystyle \frac{2k_z^2}{{\Omega }_z^2}}\right)({\rho }^2-\rho )+(E_A-E_D)\rho -2\left|{ϵ}_x\right|\sqrt{\rho (1-\rho )}+E_D-{\displaystyle \frac{1}{2}}{\displaystyle \frac{k_z^2}{{\Omega }_z^2}}\hfill \end{array}\end{array}$$

(45)

The ground state of $H_{aa}$ is degenerate when

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill E_A& =E_D\hfill \\ \hfill {\displaystyle \frac{k_z^2}{{\omega }_z^2}}& ={\displaystyle \frac{k_x^2}{{\omega }_x^2}}\hfill \\ \hfill {ϵ}_x& =0\hfill \end{array}\end{array}$$

(46)

TET should be searched in the vicinity of this set of parameters Equation (46). This model exhibits a physical flaw due to the fact that the transfer integral $\Gamma ={ϵ}_{x}exp(k_x{u}_x/{ϵ}_x)$ varies exponentially as a function of the spatial distance $u_x$ between the two orbitals. The linear expansion $\Gamma \left(u_x\right)={ϵ}_x+k_x{u}_x$ is physically acceptable under the condition that it does not change sign when $u_x$ varies. Otherwise, the resulting artifacts may be unacceptable. In order to obtain realistic scenarios, it is necessary to verify that the transfer integral does not undergo a change of sign during the time evolution of the system. This situation arises in the overdamped regime when the damping coefficients ${\gamma }_z$, and ${\gamma }_x$ are sufficiently large. Alternatively, the exponential form may be employed.

The first problem is to detect irreversible TET at zero temperature around this set of parameters and to explore its domain of existence. We expect that the fastest TET is obtained in the crossover region between the underdamped region where TET oscillates a long time between the donor and acceptor and the overdamped region where the large damping slows down the TET. The second problem is investigating the effect of temperature. When TET does not occur at zero temperature, it might take place at an optimized temperature which obeys the Marcus theory, and the Arrhenius law can be observed.

5. Results

5.1. Potential Energy Surface

First, it is crucial to comprehend the landscape of the reaction by calculating the potential energy surfaces. By employing this methodology, it is possible to ascertain the molecular dynamics, chemical reaction pathways, and other chemistry, physics, and biology-related processes and phenomena at the atomic scale through the use of an accurate PES. The PES describes the variation in the energy of a molecule as a function of the nuclear coordinates.

The accurate PES can be derived by minimizing the Hamiltonian Equation (43). The solution of the dynamical Equation (41) always converges to a stationary point of the Hamiltonian for any initial condition, which represents time-independent solutions of Equation (40). As a result, the initial values for both modes are as follows.

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & u_x=-{\displaystyle \frac{k_x}{{\Omega }_x^2}}({\phi }_D^{★}{\phi }_A+{\phi }_A^{★}{\phi }_D)=0.0\hfill \\ \hfill & u_z=-{\displaystyle \frac{k_z}{{\Omega }_z^2}}({\left|{\phi }_A\right|}^2-{\left|{\phi }_D\right|}^2)\hfill \end{array}\end{array}$$

(47)

If we consider the dimer problem, there are two scenarios to examine: one involving electron transfer and the other aligned with the traditional framework of Marcus’ theory for both normal and inverted cases. Achieving ultrafast electron transfer requires careful satisfaction of TET conditions as defined in Equation (46) along with initial parameters meeting Equation (47). Figure 3 demonstrates the PES with conical intersection under various TET regime parameters and projection of ground state into modes $u_x$ and $u_z$. The red and black lines represent the reaction pathway in the ground and excited states, respectively. The initial system parameters ($k_x$, $k_z$, ${\omega }_x$, ${\omega }_z$, ${\gamma }_x$, ${\gamma }_z$) remain consistent across each PES, while only $E_A$, $E_D$, ${ϵ}_x$ vary. In this instance, a temperature equal to 0 K (without random force) was utilized for these results obtained through Python calculations.

In the first diagram Figure 3a ($E_A=E_D$, ${ϵ}_x=0$) the PES exhibits a characteristic symmetric conical intersection, allowing for ultrafast and efficient electron transfer between the donor and acceptor sites, resulting in a degenerate reaction path. The accompanying small plot demonstrates the independence of difference between the ground and upper states along this degenerate path along the rotation angle ($\alpha$). The projection of the ground state (Figure 3d is symmetric around the conical intersection. This circular reaction path allows transfer in both directions around the conical intersection using a minimal PES pathway. The same phenomenon takes place when $E_A\ne E_D$ and other TET conditions are satisfied (Figure 3b), there is a breakdown of symmetry with a shift over the $u_z$ axis, leading to an almost linear dependence between levels and rotation angle. Despite this asymmetry, the conical intersection survives. However, the electron encounters greater resistance when moving to other sides of the system. The external field ${ϵ}_x\ne 0$ and $E_A=E_D$, depicted in Figure 3c, contributes to breaking the symmetry in mode $u_x$, which causes environmental deformation due to overlap integral. In this scenario, where $E_A\ne E_D$ and ${ϵ}_x\ne 0$ (Figure 3b,c), electron transfer can still occur. Nevertheless, the transition from donor to acceptor will not be as rapid as observed under pure TET conditions. The most frequent scenario, the normal Marcus case, is illustrated in Figure 4a. The greater the exergonic nature of the reaction, the smaller the obstacle for electron transfer. When both curves intersect before reaching the equilibrium position of the reactants, the activation energy rises with increasing exothermicity, which has been referred to as the Marcus inversion region (Figure 4b). The regime and conditions range the rate of the electron transfer. Eliminating the $u_x$ axis yields a 2D representation of Marcus’ theory [31]. For both regimes, the electron needs to overcome the barrier to be able to transfer from one side of the system (donor) to another site (Acceptor).

5.2. Temperature Influence

The temperature has a gradual impact on the transfer of electrons. In pure TET, the transition from one state to another can occur at zero temperature due to the conical intersection. However, introducing Gaussian white noise as a result of temperature for both degrees of freedom (${\eta }_x$ and ${\eta }_z$) accelerates the transfer process. Noise correlation is implemented by using a Gaussian distribution with a mean of 0 and a variance of $\sqrt{2\gamma k_{B}T/\delta t}$ for both variables. To demonstrate the influence of noise, we solve the system of differential equations (Equation (41)) using the Runge–Kutta 5(4) method [46,47,48]. The integration time step is set at $\delta t=0.001$. The noise strength is scaled to the units of $k_{B}T$ where $k_B$ is the Boltzmann constant.

To demonstrate the trajectory of the electron, we investigated two scenarios: when operating in the TET regime and when close to it ($0<E_D-E_A<=1$, ${ϵ}_x=-0.001$, $k_x^2/{\omega }_x^2=k_z^2/{\omega }_z^2\to 1=1$). We aim to depict the probability evolution at both the donor and acceptor sites under low-temperature conditions ($k_{B}T=0.01$) as well as high-temperature conditions ($k_{B}T=1.5$) Figure 5. A solid line denotes the mean for each scenario, while the shaded area indicates the standard deviation. A total of 50 independent simulations were conducted with varied noise seeds to calculate these averages.

In a pure TET regime, noise has a significant impact, Figure 5a,b. The mean value stopped at 0.8 for temperature $k_{B}T=0.01$ (Figure 5a), which shows that the transition for the different noise random seeds takes place in the same region but with a slight shift. The particle is oscillating back-and-forth between two sites. When noise levels increase $k_{B}T=1.5$ Figure 5b, transfer from donor to acceptor is observed with high frequency. More oscillations occur in the system, resulting in an average of 0.5 and a higher standard deviation. The system exhibits similar characteristics of TET but with some equilibrium conditions being disrupted as shown in the PES Figure 3c,d. This configuration maintains greater stability where the transfer occurs only once and minor fluctuations are visible for the $k_{B}T=0.01$ temperature. With a noise level of $k_{B}T=1.5$, the number of transfers is increased and the probability of acceptor averaging around 0.6–0.7 with a high standard deviation which is greater than in the pure TET case.

If we compare two scenarios involving pure TET with those that are away from TET, Figure 6 illustrates the PES and the projection of the PES onto the additional mode $u_x$, which is not included in Marcus’ theory. Figure 6a displays the case that satisfies the TET conditions Equation (46), the conical intersection and degeneracy are preserved and the reaction path is independent of rotation. Conversely, breaking the equality condition for parameters ($k_x$, $k_z$, ${\omega }_x$, and ${\omega }_z$) and introducing a significant inequality leads to a visualization of the inverted regime of the Marcus theory (Figure 6c). The intersection of PES occurs only when $u_x$ equals zero for both cases Figure 6b,d. We can remove the $u_x$ axis and see the usual case. A gap exists for every other value for both cases.

The Arrhenius equation is another important result of chemical kinetics. According to this principle, the rate constant of electron transfer ($k_{ET}$) is reduced as the temperature decreases and establishes a linear relationship between the natural logarithm of the reaction rate $ln\left(k_{ET}\right)$ and the reciprocal temperature $1/k_{B}T$ [49].

$$\begin{array}{c}\hfill k_{ET}=A_{ET}{exp}^{{\displaystyle \frac{E_a}{k_{B}T}}}\end{array}$$

(48)

where $A_{ET}$ and $E_a$ represent the pre-exponential factor and activation energy, respectively.

Figure 7a demonstrates the average maximum probability of the acceptor site across different temperatures for TET (navy line) and away from TET (blue line). Figure 7b displays the transfer time for the maximum probability. These results were obtained from 70 runs for the specific noise level, the colored areas indicating the standard deviation. The analyzed time interval was ten times longer than the transition time in the TET system. It is evident that in the case of TET Figure 7a (navy line), there is a transition between sites at every temperature value where the standard deviation is equal to zero. In contrast, when away from the TET (blue line), complete transfer occurs only after the temperature 1.5. The standard deviation shows the ability of full transfer at temperature $k_{B}T=1.0$. A smaller diagram represents the fulfillment of the Arrhenius law Equation (48). The maximal probability at the acceptor site can be interpreted as representing the activation energy of the reaction. It is clear to see the linear dependency on the inverted temperature which is always observed in the Marcus theory.

The transition from the donor to the acceptor happens rapidly under TET conditions, as illustrated in Figure 7b. Transfer always depends on the parameters and the transfer time varies for each system. Although temperature does not significantly increase electron transfer, it does promote oscillation across system sites for the pure TET regime. Temperature serves as a parameter driving transfer and reduces transfer time in the case far from the TET conditions, Figure 7b (blue line). However, there is a much larger standard deviation compared to the TET regime. The rate of transfer is four times faster in the system under TET conditions than in those situated away from such conditions.

It is important to note that a similar challenge arises in the SSH model, where a linear approximation is utilized for the transfer integral. In the absence of damping, TET manifests itself as an oscillation between the donor and the acceptor. To induce irreversibility in TET at absolute zero temperature, non-zero damping, and a positive reaction energy $E_D-E_A$ are necessary. When there is a significant disparity among the parameters in Equation (46), the model aligns with the standard Marcus model commonly employed in chemistry to elucidate electron transfer, particularly if coupling $k_x$ is disregarded and ${ϵ}_x$ remains small. Typically, a covalent coupling specified by parameter $k_x$ is not taken into account.

6. Conclusions

Chemical reaction processes are ubiquitous and fundamental for life. The theory of chemical reactions is formulated in terms of adiabatic energy surfaces that are useful both for visualization but also quantitative understanding. In this work, we first summarized the basic theory of chemical reactions and explored some of their intricacies. The Marcus theory provides a basic tool for addressing chemical dynamics through Born–Oppenheimer adiabatic surfaces.

We focused on the TET model that utilizes non-linear resonances and leads to efficient energy and/or electron transfer in a simple donor–acceptor molecular system. This model was originally introduced as a nonlinear system that has desired engineered transfer properties. In the present work, we show that this resonant transfer idea can be readily extended to cases where the electronic degrees of freedom are coupled to vibrational ones. In this context, the TET model becomes a model for special types of chemical reactions.

Starting from the first principles used in quantum chemistry, we have proven the possibility of a new kind of chemical reaction called TET; the latter are coherent, ultra-fast at the usual scale of chemical reactions, and still persist at moderate temperatures under thermal fluctuations. This new type of chemical reaction does not obey the standard Arrhenius law. The TETs concern situations of chemical reactions with moderate reaction energy and thus release little energy which is the general situation in biochemistry. Such a TET chemical reaction may be found near the border of the parameter region which operates between ionic and covalent reactions. The TET occurs when ionic and covalent interactions balance one with the other. When the TET conditions are fulfilled, there is a conical intersection between two PES which has almost cylindrical symmetry so that there is a rather flat groove connecting the initial and final state of the chemical reaction and corresponding to the reaction path. Reversely assuming that such a conical intersection exists within a Born–Oppenheimer representation, an expansion at the lowest significant order around the conical point of intersection allows us to recover our simple model directly without using any diabatic representation, which remains physically more intuitive. Indeed though conical intersections are a priori rare because of the Von Newmann–Wigner avoided crossing theorem, chemists were recently interested in these conical intersections for explaining anomalous photo-induced decay. The are now claims that from ab initio calculations, most complex molecules involving many nuclei exhibit many conical intersections between PES.

Having a conical intersection is not sufficient for having TET; it is also required that this intersection have at least roughly a cylindrical symmetry. In that case around the conical intersection, we have a quasi-continuum of states (nuclei configurations) with almost the same energy at the biological scale that is with low energy barriers not exceeding the energy range of 0.3 eV. These conditions require a tuning of the model parameters, which implies that the TET should be highly sensitive to small perturbations in the environment of the reaction. The consequence is that relatively small perturbations in the environment may sharply slow down the reaction or even block it in the biochemical temperature range. Otherwise, it may also reverse the TET process from the acceptor to the donor. Such situations sometimes occur in biology depending on changes in the environmental conditions such as pH concentrations of other chemical compounds, or the presence of specific molecules poisoning the biological system. Thus, in general, TET requires exceptional conditions in order to operate that, a priori, are very rare in inorganic chemistry. But the biochemistry of living beings also seems to be very rare in the universe, except of course on the earth. We believe that Darwinian processes over long periods of time have slowly selected and optimized this kind of chemical reaction to improve the efficiency of the living cells that use it.

We should also mention the problem of mixed valence in inorganic chemistry. Mixed valence complexes contain an element that is present in more than one oxidation state. The Robin–Day classification distinguishes three classes. Class 1 consists of complexes in which the oxidation state does not change over a long time. The change in oxidation occurs when the complex passes through a thermally activated energy barrier. Such changes can be described by the Marcus theory. Class III consists of complexes where the oxidation state is a quantum combination of the two possible oxidation states and may be viewed as a covalent state. According to the principles of quantum mechanics, the lifetime of an oxidation state is related to the quantum tunneling time which is usually short. For intermediate Class II, the time scale of the change in oxidation is much faster than that of the ionic complex of Class I but much slower than those of Class III which is at the time scale of purely electronic transitions. Then, these mixed-valence complexes are very labile, that is, they seem to exhibit a continuum of intermediate states between the two oxidation states. We may claim that this situation is favorable for having TET between the two oxidation states. Because TET is much faster than standard chemical reactions involving an energy barrier, it prevails over any other possible reaction affecting the donor or the acceptor because these are too slow.

The present analysis shows a path for further work. We considered here the simplest situation of elementary chemical reactions. We believe that in biological systems, TET is a ubiquitous chemical reaction that works coherently in living cells. The simpler model beyond the present dimer system is the trimer model which is a three-state model. Let us consider a reaction donor–acceptor which involves a large energy barrier but with a small reaction energy that would be improbable to occur spontaneously in the range of biological temperatures. It is then possible to choose the parameters of a third molecule C (we call catalyst) which is designed especially to exhibit TET with D and next, after it receives the electron from the donor, exhibits TET with the acceptor. When this enzyme binds with D and A and then forms a bridge, electron transfer occurs quickly, whereas this electron transfer would not occur directly. Such kinds of chemical reactions are easily affected and modulated by the environment. We already found such toy models which will be discussed in further works.

More generally, we believe it is possible to build intelligent networks of chemical reactions using elementary TET modules that accomplish well-defined complex tasks. We already presented some toy models in presentations, but further studies are needed. In conclusion, we believe there is some empirical analogy with the theory of semiconductors, where a simple TET dimer system would correspond to a simple Diode, the trimer catalytic model to a transistor while the global biochemical organizations of living cells would correspond to complex integrated circuits. Further work in this direction could show the generality of the ideas revolving around the TET mechanism.