Role of electronic kinetic energy and resultant gradient information in chemical reactivity

Abstract

The role of resultant gradient-information concept, reflecting the kinetic energy of electrons, in shaping the molecular electronic structure and reactivity preferences of open reactants is examined. This quantum-information descriptor combines contributions due to both the modulus (probability) and phase (current) components of electronic wavefunctions. The importance of resultant entropy/information concepts for distinguishing the bonded (entangled) and nonbonded (disentangled) states of molecular fragments is emphasized and variational principle for the minimum of ensemble-average electronic energy is interpreted as a physically equivalent rule for the minimum of resultant gradient-information, and the information descriptors of charge-transfer (CT) phenomena are introduced. The in situ reactivity criteria, represented by the populational CT derivatives of the ensemble-average values of electronic energy or resultant information, are mutually related, giving rise to identical predictions of electron flows in the acid(A) — base(B), reactive systems. The virial theorem decomposition of electronic energy is used to reveal changes in the resultant information content due to the chemical bond formation, and to rationalize the Hammond postulate of reactivity theory. The complementarity principle of structural chemistry is confronted with the regional hard (soft) acid and bases (HSAB) rule by examining the polarizational and relaxational flows in such acceptor–donor reactive systems, responses to the external potential and CT displacements, respectively. The frontier-electron basis of the HSAB principle is reexamined and the intra- and inter-reactant communications in A—B systems are explored.

Introduction

Thermodynamic principles for the minimum electronic energy in molecules can be interpreted as the variational rule for the minimum of the ensemble-average resultant gradient-information [1, 2], related to average kinetic energy of electrons in such (mixed) quantum states. In the grand-ensemble representation of the externally open molecular systems, they both determine the same set of the optimum probabilities of the system (pure) stationary states. This equivalence resembles identical predictions resulting from the minimum-energy and maximum-entropy principles in ordinary thermodynamics [3]. The energy and resultant gradient-information rules thus represent physically equivalent sources of reactivity criteria. Such an information transcription of the familiar energy principle allows one to reinterpret criteria for the charge transfer (CT) in reactive systems, the populational derivatives of electronic energy, as the associated derivatives of the overall measure of the quantum-information in molecular states, which combines the “classical” (modulus, probability) and “nonclassical” (phase, current) aspects of molecular wavefunctions. The proportionality between the resultant gradient-information and the system kinetic energy then allows one to use the molecular virial theorem [4] in general reactivity considerations [1, 2].

The classical information theory (IT) of Fisher and Shannon [5,6,7,8,9,10,11,12] has been successfully applied to interpret in chemical terms the molecular probability distributions, e.g., [13,14,15,16]. Information principles have been explored [17,18,19,20,21,22] and density pieces attributed to atoms in molecules (AIM) have been approached [13, 17, 21,22,23,24,25], providing the information basis for the intuitive (stockholder) division of Hirshfeld [26]. Patterns of chemical bonds have been extracted from molecular electronic communications [13,14,15,16, 27,28,29,30,31,32,33,34,35,36,37] and entropy/information distributions in molecules have been examined [13,14,15,16, 38, 39]. The nonadditive Fisher information [13,14,15,16, 40, 41] has been linked to electron localization function (ELF) [42,43,44] of modern density functional theory (DFT) [45,46,47,48,49,50]. This analysis has also formulated the contragradience (CG) probe for localizing chemical bonds [13,14,15,16, 51], and the orbital communication theory (OCT) of the chemical bond [27,28,29,30,31,32,33,34,35,36,37] has identified the bridge-bonds originating from the cascade propagations of information between AIM, which involve intermediate atomic orbitals [15, 16, 51,52,53,54,55,56,57].

In entropic theories of molecular electronic structure, one ultimately requires such quantum extensions of the complementary classical measures of Fisher [5, 6] and Shannon [7, 8], of the information/entropy content in probability distributions, which are appropriate for complex probability amplitudes (wavefunctions) of quantum mechanics (QM). The IT distinction between the bonded (entangled) and nonbonded (disentangled) states of molecular subsystems also calls for their generalized (resultant) information descriptors [16, 58,59,60,61,62,63,64,65,66,67,68,69,70], which combine the classical (probability) and nonclassical (current) contributions. Probability distributions generate the classical entropy/information descriptors of electronic states. These contributions reflect only the wavefunction modulus, while the wavefunction phase, or its gradient determining the current density, give rise to the corresponding nonclassical supplements in the resultant measure the overall information content in molecular electronic states [16, 58,59,60]. The variational principles of such generalized entropy concepts have been used to determine the phase-equilibria in molecules and their constituent fragments [16, 61,62,63,64,65].

Paraphrasing Prigogine [71], one could regard the molecular probability distribution as determining an instantaneous structure of “being”, while the system’s current pattern generates the associated structure of “becoming”. Both of these levels of the system electronic organization contribute to the state overall entropy/information content. In quantum information theory (QIT), the classical information term, conceptually rooted in DFT, probes the entropic content of the incoherent (disentangled) local “events”, while it is the nonclassical supplement that provides the information contribution due to coherence (entanglement) of such local events. For example, resultant measures combining the probability and phase/current contributions allow one to distinguish the information content of states generating the same electron density but differing in their phase/current distributions [47, 72, 73].

The resultant Fisher-type gradient-information in specified electronic state is proportional to the state average kinetic energy [1, 2, 13, 16, 18, 40]. This allows one to interpret the variational principle for electronic energy as equivalent quantum-information rule. The latter forms a basis for the novel, information-treatment of reactivity phenomena [1, 2]. Various DFT-based approaches to classical issues in reactivity theory [74,75,76,77,78,79,80] use the energy-centered arguments in justifying the observed reaction paths and relative yields of their products. Qualitative considerations on preferences in chemical reactions usually emphasize changes in energies of both reactants and of the whole reactive system, which are induced by displacements (perturbations) in parameters describing the relevant (real or hypothetical) electronic states. In such treatments, usually covering also the linear responses to these primary shifts, one explores reactivity implications of the electronic equilibrium and stability criteria [13, 15, 74, 75, 79]. For example, in charge sensitivity analysis (CSA) [74, 75], the energy derivatives with respect to the system external potential (v) and its overall number of electrons (N) and the associated charge responses of both the whole reactive systems and their constituent subsystems have been explored as potential reactivity descriptors. In R = acid(A) ← base(B) ≡ A--B complexes, consisting of the coordinated electron-acceptor and electron-donor reactants, respectively, such responses can be subsequently combined into the corresponding in situ indices characterizing the B → A CT [74, 75]. These difference characteristics of polarized subsystems can be expressed in terms of elementary (principal) charge sensitivities of reactants [74, 75, 78, 79]. The nonclassical IT descriptors of polarized subsystems can be similarly combined into the corresponding in situ properties describing the whole reactive system.

In this work, the role of resultant gradient-information/kinetic-energy in shaping the chemical reactivity preferences will be explored and variations of the kinetic energy of electrons in the bond-forming/bond-breaking processes will be examined. The continuities of the principal physical degrees-of-freedom of electronic states, the modulus/probability and phase/current distributions, respectively, will be summarized and the virial theorem will be used to interpret, in information terms, the bond-formation process. The theorem implications for the Hammond [81] postulate of reactivity theory will also be explored. The frontier-electron approximation to molecular interactions will be adopted to extract the information perspective on Pearson’s [82] hard (soft) acids and bases (HSAB) principle of structural chemistry (see also [83]), and physical equivalence of the energy and information reactivity descriptors in the grand-ensemble representation of molecular thermodynamic equilibria will be stressed. The populational derivatives of resultant gradient-information will be examined and advocated as alternative indices of chemical reactivity, adequate in predicting both the direction and magnitude of electron flows in reactive systems. The phase-description of hypothetical stages of reactants in chemical reactions will be explored, the activation (“promotion”) of molecular substrates will be examined, and the in situ populational derivatives of resultant-information will be applied to determine the optimum amount of CT in donor–acceptor reactive systems.

Classical and nonclassical sources of the structure-information in molecular states

Consider, for reasons of simplicity, a single electron moving in the external potential v(r) created by the fixed nuclei of the molecule. Its quantum state at time t, |ψ(t)〉 ≡ |ψ(t)〉, is then described by the generally complex wavefunction

$$ \psi \left(\boldsymbol{r},t\right)=\left\langle \boldsymbol{r}|\psi (t)\right\rangle =R\left(\boldsymbol{r},t\right)\ \exp \left[\mathrm{i}\phi \left(\boldsymbol{r},t\right)\right], $$

(1)

where (real) functions R(r, t) = p(r, t)^1/2 ≥ 0 and ϕ(r, t) ≥ 0 stand for its modulus and phase components, respectively. They generate the state two principal physical degrees-of-freedom: its instantaneous probability distribution,

$$ p\left(\boldsymbol{r},t\right)=\psi \left(\boldsymbol{r},t\right)\ \psi {\left(\boldsymbol{r},t\right)}^{\ast }=R{\left(\boldsymbol{r},t\right)}^2, $$

(2)

and the current density

$$ {\displaystyle \begin{array}{l}\boldsymbol{j}\left(\boldsymbol{r},t\right)=\left[\hslash /\left(2m\mathrm{i}\right)\right]\kern0.24em \left[\psi {\left(\boldsymbol{r},t\right)}^{\ast}\nabla \psi \left(\boldsymbol{r},t\right)-\psi \left(\boldsymbol{r},t\right)\nabla \psi {\left(\boldsymbol{r},t\right)}^{\ast}\right]\\ {}=\left(\hslash /m\right)p\left(\boldsymbol{r},t\right)\nabla \phi \left(\boldsymbol{r},t\right)\equiv p\left(\boldsymbol{r},t\right)\kern0.24em V\left(\boldsymbol{r},t\right),\end{array}} $$

(3)

where the current-per-particle V(r, t) = j(r, t)/p(r, t) determines an effective velocity field dr(t)/dt for the probability “fluid”. The probability descriptor of the (pure) molecular state thus measures a product of the conjugate states ψ and ψ^*, while the phase component reflects their ratio:

$$ \phi \left(\boldsymbol{r},t\right)={\left(2\mathrm{i}\right)}^{-1}\ \ln \left[\psi \left(\boldsymbol{r},t\right)/\psi {\left(\boldsymbol{r},t\right)}^{\ast}\right], $$

(4)

In the current definition of Eq. (3), the probability “velocity”

$$ \boldsymbol{V}\left(\boldsymbol{r},t\right)=\boldsymbol{V}\left[\boldsymbol{r}(t),t\right]=\left(\hbar /m\right)\ \nabla \phi \left(\boldsymbol{r},t\right) $$

(5)

reflects the state phase-gradient ∇ϕ(r, t).

The physical descriptors p(r, t) and j(r, t) of a complex quantum state constitute independent sources of an overall information content of the molecular electronic structure: the probability distribution alone generates its classical contribution while the current (velocity) density determines its nonclassical complement in the resultant measure [16, 58,59,60].

The IT gradient descriptors extract the information contained in local inhomogeneities of these two principal physical distributions, reflected by their gradient and divergence, respectively. These gradient probes reflect the complementary facets of the state “structure” content: ∇p = 2R ⋅∇ R extracts the spatial inhomogeneity of the probability density, the structure of “being”, while ∇⋅ j =  ∇ p ⋅V  = (ℏ/m) ∇ p ⋅ ∇ ϕ uncovers the current structure of “becoming”. We have used above a direct implication of the probability-continuity,

$$ {\displaystyle \begin{array}{l} dp\left[\boldsymbol{r}(t),t\right]/ dt\equiv {\sigma}_p\left(\boldsymbol{r},t\right)=\partial p\left(\boldsymbol{r},t\right)/\partial t+\nabla \cdot \boldsymbol{j}\left(\boldsymbol{r},t\right)\\ {}=\partial p\left(\boldsymbol{r},t\right)/\partial t+\left[\partial p\left(\boldsymbol{r},t\right)/\partial \boldsymbol{r}\right]\ \left[d\boldsymbol{r}(t)/ dt\right]\\ {}=\partial p\left(\boldsymbol{r},t\right)/\partial t+\nabla p\left(\boldsymbol{r},t\right)\cdot \boldsymbol{V}\left(\boldsymbol{r},t\right)=0,\mathrm{or}\\ {}\partial p\left(\boldsymbol{r},t\right)/\partial t=-\nabla \cdot \boldsymbol{j}\left(\boldsymbol{r},t\right)\\ {}=-\left[\nabla p\left(\boldsymbol{r},t\right)\cdot \boldsymbol{V}\left(\boldsymbol{r},t\right)+p\left(\boldsymbol{r},t\right)\nabla \cdot \boldsymbol{V}\left(\boldsymbol{r},t\right)\right]\\ {}=-\nabla p\left(\boldsymbol{r},t\right)\cdot \boldsymbol{V}\left(\boldsymbol{r},t\right),\end{array}} $$

(6)

that divergence of the effective velocity field V(r, t), determined by the state phase-Laplacian, identically vanishes:

$$ \nabla \cdot \boldsymbol{V}\left(\boldsymbol{r},t\right)=\left(\hbar /m\right)\ \Delta \phi \left(\boldsymbol{r},t\right)=0. $$

(7)

Here, dp/dt ≡ σ_p and ∂p/∂t denote the total and partial time-derivatives of probability density p(r, t) = p[r(t), t], respectively. The local probability “source” (“production”) σ_p is reflected by the total derivative dp/dt, which measures the time rate of change in an infinitesimal volume element of the probability fluid flowing with the probability current, while the partial derivative ∂p/∂t represents the corresponding rate at the specified (fixed) point in space. One observes that the total time derivative of Eq. (6) expresses the sourceless continuity relation for electronic probability distribution: σ_p(r, t) = 0.

In a molecular scenario, one envisages the system electrons moving in the external potential v(r) due to the “frozen” nuclei of the familiar Born–Oppenheimer (BO) approximation. The mono-electronic system is then described by the Hamiltonian

$$ \hat{\mathrm{H}}\left(\boldsymbol{r}\right)=-\left({\hslash}^2/2m\right){\nabla}^2+v\left(\boldsymbol{r}\right)\kern0.40em \equiv \kern0.40em \hat{\mathrm{T}}\left(\boldsymbol{r}\right)+v\left(\boldsymbol{r}\right) $$

(8)

where $\hat{\mathrm{T}}\left(\boldsymbol{r}\right)$ stands for its kinetic part. The dynamics of electronic wavefunction ψ(r, t) is determined by the Schrödinger equation (SE) of molecular quantum mechanics (QM),

$$ \mathrm{i}\hslash \kern0.28em \partial \psi \left(\boldsymbol{r},t\right)/\partial t=\hat{\mathrm{H}}\left(\boldsymbol{r}\right)\psi \left(\boldsymbol{r},t\right), $$

(9)

which further implies specific temporal evolutions of p(r, t) and ϕ(r, t).

The probability-velocity descriptor should be also attributed to the current concept associated with the state phase-component:

$$ \boldsymbol{J}\left(\boldsymbol{r},t\right)=\phi \left(\boldsymbol{r},t\right)\ \boldsymbol{V}\left(\boldsymbol{r},t\right). $$

(10)

The phase field ϕ(r, t) and its current J(r, t) then determine a nonvanishing source term σ_ϕ(r, t) in the phase-continuity equation:

$$ {\displaystyle \begin{array}{l}{\sigma}_{\phi}\left(\boldsymbol{r},t\right)\equiv d\phi \left(\boldsymbol{r},t\right)/ dt=\partial \phi \left(\boldsymbol{r},t\right)/\partial t+\nabla \cdot \boldsymbol{J}\left(\boldsymbol{r},t\right)\kern0.5em \mathrm{or}\\ {}\partial \phi \left(\boldsymbol{r},t\right)/\partial t=-\nabla \cdot \boldsymbol{J}\left(\boldsymbol{r},t\right)+{\sigma}_{\phi}\left(\boldsymbol{r},t\right).\end{array}} $$

(11)

Using Eq. (7) gives the following expression for this phase-source:

$$ {\displaystyle \begin{array}{l} d\phi \left[\boldsymbol{r}(t),t\right]/ dt=\partial \phi \left[\boldsymbol{r}(t),t\right]/\partial t+d\boldsymbol{r}(t)/ dt\cdot \partial \phi \left[\boldsymbol{r}(t),t\right]/\partial \boldsymbol{r}\\ {}=\partial \phi \left(\boldsymbol{r},t\right)/\partial t+\boldsymbol{V}\left(\boldsymbol{r},t\right)\cdot \nabla \phi \left(\boldsymbol{r},t\right)\\ {}=\partial \phi \left(\boldsymbol{r},t\right)/\partial t+\left(\hslash /m\right)\kern0.28em {\left[\nabla \phi \left(\boldsymbol{r},t\right)\right]}^2.\end{array}} $$

(12)

The phase-dynamics from SE,

$$ \partial \phi /\partial t=\left[\hslash /(2m)\right]\left[{R}^{-1}\varDelta R-{\left(\nabla \phi \right)}^2\right]-v/\hslash, $$

(13)

ultimately identifies the state phase-production of Eq. (11):

$$ {\sigma}_{\phi }=\left[\hbar /(2m)\right]\ \left[{R}^{-1}\Delta R+{\left(\nabla \phi \right)}^2\right]-v/\hbar . $$

(14)

To summarize, the classical continuity relation of QM expresses a sourceless character of the electron probability distribution, while its nonclassical companion introduces a nonvanishing phase-source combining both the classical (modulus) and nonclassical (phase) inputs.

Resultant information and kinetic energy

Let us consider the fixed time t = t₀ and for simplicity suppress this parameter in the list of state arguments. The average Fisher’s measure [5, 6] of the classical gradient information for locality events contained in probability density p(r) = R(r)² is reminiscent of von Weizsäcker’s [84] inhomogeneity correction to the kinetic-energy functional:

(15)

Here, denotes functional’s overall density and I_p(r) stands for the associated density-per-electron. The amplitude form I[R] reveals that this classical descriptor reflects a magnitude of the state modulus-gradient. It characterizes an effective “narrowness” of the particle spatial probability distribution, i.e., a degree of determinicity in the particle position.

This classical functional of the gradient information in probability distribution generalizes into the corresponding resultant descriptor, functional of the quantum state |ψ(t)〉 itself, which combines the modulus (probability) and phase (current) contributions [40]. It is defined by the quantum expectation value of the Hermitian operator of the overall gradient information [16, 40], related to the kinetic energy operator $\hat{T}\left(\boldsymbol{r}\right)$ of Eq. (8),

$$ \hat{I}\left(\boldsymbol{r}\right)=-4\varDelta ={\left(2\mathrm{i}\mathrm{\nabla}\right)}^2=\left(8m/{\hslash}^2\right)\kern0.28em \hat{T}\left(\boldsymbol{r}\right). $$

(16)

The integration by parts then gives the following expression for the state average (resultant) gradient-information

(17)

with again denoting its overall density and I_ψ(r) standing for the corresponding density-per-electron. This quantum-information concept, I[ψ] = I[p, ϕ] = I[p, j], is seen to combine the classical (probability) contribution I[p] of Fisher and the corresponding nonclassical (phase/current) supplement I[ϕ] = I[j]. It also reflects the particle average (dimensionless) kinetic energy T[ψ]:

$$ I\left[\psi \right]=\left(8m/{\hslash}^2\right)\left\langle \psi |\hat{\mathrm{T}}|\psi \right\rangle \equiv \left(8m/{\hslash}^2\right)\ T\left[\psi \right]\equiv \sigma\ T\left[\psi \right]. $$

(18)

This one-electron development can be straightforwardly generalized into general N-electron systems in the corresponding quantum state |Ψ(N)〉 exhibiting electron density ρ(r) = Np(r), where p(r) stands for its probability (shape) factor. The corresponding N-electron information operator then combines terms due to each particle,

$$ \hat{\mathrm{I}}(N)=\sum \limits_{i=1}^N\kern0.28em \hat{\mathrm{I}}\left({\boldsymbol{r}}_i\right)=\left(8m/{\hslash}^2\right)\kern0.28em \sum \limits_{i=1}^N\hat{\mathrm{T}}\left({\boldsymbol{r}}_i\right)\equiv \left(8m/{\hslash}^2\right)\kern0ex \hat{\mathrm{T}}(N), $$

(19)

and determines the state overall gradient-information,

$$ I(N)=\left\langle \Psi (N)|\hat{\mathrm{I}}(N)|\Psi (N)\right\rangle =\left(8m/{\hslash}^2\right)\left\langle \Psi (N)|\hat{\mathrm{T}}(N)|\Psi (N)\right\rangle =\left(8m/{\hslash}^2\right)T(N), $$

(20)

proportional to the associated expectation value T(N) of the system kinetic-energy operator $\hat{\mathrm{T}}(N)$.

For example, in one-determinantal representation, of a single electron (orbital) configuration Ψ(N) = |ψ₁ψ₂ …ψ_N|, e.g., in the familiar Hartree–Fock of Kohn–Sham theories, these N-electron descriptors combine the additive contributions due to the (singly) occupied, {n_s = 1}, spin molecular orbitals (MO) ψ = (ψ₁, ψ₂, …, ψ_N) = {ψ_s}:

$$ T(N)={\sum}_s\ {n}_s\ \left\langle {\psi}_s|\hat{\mathrm{T}}|{\psi}_s\right\rangle \equiv {\sum}_s\ {n}_s\ {T}_s=\left({\hbar}^2/8m\right){\sum}_s\ {n}_s\ \left\langle {\psi}_s|\hat{\mathrm{I}}|{\psi}_s\right\rangle \equiv \left({\hbar}^2/8m\right){\sum}_s\ {n}_s\ {I}_s. $$

(21)

In the analytical LCAO MO representation, with the occupied MO expressed as linear combinations of (orthogonalized) atomic orbitals (AO) χ = (χ₁, χ₂, …, χ_k, …),

$$ \mid \boldsymbol{\psi} \left\rangle =\kern0.5em \mid \boldsymbol{\chi} \right\rangle\ \mathbf{C},\kern1.5em \mathbf{C}=\left\langle \boldsymbol{\chi} |\boldsymbol{\psi} \right\rangle =\left\{{C}_{k,s}=\left\langle {\chi}_k|{\psi}_s\right\rangle \right\}, $$

(22)

the average gradient information in the orbital configuration Ψ(N), for the unit matrix of MO occupations, n = {n_s δ_s,s’ = δ_s,s’}, reads:

$$ I(N)={\sum}_s\ {n}_s\left\langle {\psi}_s|\hat{\mathrm{I}}|{\psi}_s\right\rangle ={\sum}_k{\sum}_l\left\{{\sum}_s\ {C}_{k,s}\ {n}_s\ {C_{s,l}}^{\ast}\right\}\left\langle {\chi}_l|\hat{\mathrm{I}}|{\chi}_k\right\rangle \equiv {\sum}_k{\sum}_l\ {\gamma}_{k,l}\ {I}_{l,k}=\mathrm{tr}\left(\boldsymbol{\upgamma} \mathbf{I}\right). $$

(23)

Here, the AO representation of the resultant gradient-information operator,

$$ \mathbf{I}=\left\{{I}_{k,l}=\left\langle {\chi}_k|\hat{I}|{\chi}_l\right\rangle \propto \left\langle {\chi}_k|\hat{\mathrm{T}}|{\chi}_l\right\rangle ={T}_{k,l}\right\}, $$

(24)

and the charge/bond-order (CBO) (density) matrix of LCAO MO theory,

$$ \boldsymbol{\upgamma} ={\mathbf{CnC}}^{\dagger }=\left\langle \boldsymbol{\chi} |\boldsymbol{\psi} \right\rangle\ \mathbf{n}\ \left\langle \boldsymbol{\psi} |\boldsymbol{\chi} \right\rangle \equiv \left\langle \boldsymbol{\chi} |{\hat{\mathrm{P}}}_{\psi }|\boldsymbol{\chi} \right\rangle, $$

(25)

is seen to provide the AO-representation of the projection onto the occupied MO-subspace,

$$ {\hat{\mathrm{P}}}_{\psi }=N\left[{\sum}_s\mid {\psi}_s\right\rangle \left({n}_s/N\right)\left\langle {\psi}_s\mid \right]\equiv N\left[{\sum}_s\mid {\psi}_s\right\rangle {p}_s\left\langle {\psi}_s\mid \right]\equiv N\hat{\mathrm{d}}, $$

(26)

proportional to the density operator $\hat{\mathrm{d}}$ of the configuration MO “ensemble”.

This average overall information thus assumes thermodynamic-like form, as the trace of the product of CBO matrix, the AO representation of the (occupation-weighted) MO projector, which establishes the configuration density operator, and the corresponding AO matrix of the Hermitian operator for the resultant gradient information, related to the system electronic kinetic energy. In this MO “ensemble”-averaging, the AO information matrix I constitutes the quantity-matrix, while the CBO (density) matrix γ provides the “geometrical” weighting factors in this MO “ensemble”, reflecting the system electronic state. It has been argued elsewhere [16, 27,28,29,30,31,32,33,34,35,36,37] that elements of the CBO matrix generate amplitudes of electronic communications between molecular AO “events”. This observation thus adds a new angle to interpreting this average-information expression: it is seen to represent the communication-weighted (dimensionless) kinetic energy of the system electrons.

The relevant separation of the modulus- and phase-components of general N-electron states calls for wavefunctions yielding the specified electron density [47]. It can effected using the Harriman–Zumbach–Maschke (HZM) [83, 84] construction of DFT. It uses N (complex) equidensity orbitals, each generating the molecular probability distribution p(r) and exhibiting the density-dependent spatial phases, which safeguard the MO orthogonality.

Bonded (entangled) and nonbonded (disentangled) states of reactants

The resultant entropy/information concepts of QIT have been applied to describe the substrate current activation and to distinguish the bonded and nonbonded states of molecular fragments [1, 2, 66,67,68,69,70]. In the course of a chemical reaction, one conventionally recognizes several of its hypothetical stages [1, 2, 13, 16, 72, 73] involving either the mutually closed [nonbonded (n), disentangled] or open [bonded (b), entangled] reactants, e.g., the electron acceptor and donor substrates in a bimolecular reactive system R = A----B involving the acidic (A) and basic (B) partners, respectively. The nonbonded status of such closed (polarized) subsystems in R⁺ ≡ (A⁺|B⁺), conserving the initial numbers of electrons of isolated reactants {α⁰}, {N_α⁺ = N_α⁰}, and - at a finite separation - relaxed in a presence of the reaction partner, {ρ_α⁺ ≠ ρ_α⁰}, is symbolized by the solid vertical line separating the two reactants at this polarization (⁺) stage. Only due to this mutual closure the identity of the two substrates remains a meaningful concept. The overall electron density of R⁺ as a whole then reads:

$$ {\displaystyle \begin{array}{l}{\rho_{\mathrm{R}}}^{+}={\rho_{\mathrm{A}}}^{+}+{\rho_{\mathrm{B}}}^{+}\kern0.40em \equiv \kern0.40em {N_{\mathrm{A}}}^{+}{p_{\mathrm{A}}}^{+}+{N_{\mathrm{B}}}^{+}{p_{\mathrm{B}}}^{+}={N}_{\mathrm{R}}{p_{\mathrm{R}}}^{+},\\ {}{p_{\alpha}}^{+}={\rho_{\alpha}}^{+}/{N_{\alpha}}^{+},{N_{\alpha}}^{+}=\int {\rho_{\alpha}}^{+}d\boldsymbol{r}={N_{\alpha}}^0,\alpha =\mathrm{A},\mathrm{B},\end{array}} $$

(27)

where

$$ {p_{\mathrm{R}}}^{+}=\left({N_{\mathrm{A}}}^{+}/{N}_{\mathrm{R}}\right)\ {p_{\mathrm{A}}}^{+}+\left({N_{\mathrm{B}}}^{+}/{N}_{\mathrm{R}}\right)\ {p_{\mathrm{B}}}^{+}\equiv {P_{\mathrm{A}}}^{+}\ {p_{\mathrm{A}}}^{+}+{P_{\mathrm{B}}}^{+}\ {p_{\mathrm{B}}}^{+} $$

(28)

denotes the system global probability distribution, the shape-factor of ρ_R⁺, and the condensed probabilities {P_α⁺ = N_α⁺/N_R = N_α⁰/N_R = P_α⁰} reflect reactant “shares” in the overall number of electrons: N_R⁺ = ∑_α N_α⁺ = ∑_α N_α⁰ = N_R⁰ ≡ N_R. These subsystems lose their “identity” in the bonded status, as mutually open parts of the externally closed (isoelectronic) reactive system R ≡ (A^*¦B^*) conserving N_R, where the absence of a barrier for internal electron flows between the two substrates has been symbolically represented by the broken vertical line separating the two reactants. Indeed, in absence of the dividing “wall”, each “part” physically exhausts the whole reactive system.

However, one can also contemplate the external flows of electrons, between the mutually nonbonded reactants and their separate (external, macroscopic) reservoirs of electrons . The formal mutual-closure then implies the relevancy of subsystem identities, while the external-openness in now macroscopic (composite) subsystems of the whole composite reactive system

(29)

allows one to independently manipulate the chemical potentials of both parts, , and hence also their ensemble-average electron densities {ρ_α^* = N_α^*p_α^*} and populations N_α^* = ∫ρ_α^* dr. In particular, the substrate chemical potentials equalized at the molecular level in both composite subsystems, {μ_α^* = μ_R ≡ μ_R(N_R)}, thus conserving the overall (ensemble-average) electron number 〈N_R〉_ens. = N_R(μ_R), define the equilibrium macroscopic system

(30)

in which one observes the equilibrium reactant distributions {ρ_α^* = ρ_α(μ_R)} and the associated populations {N_α^* = N_α(μ_R)} of the “bonded” molecular fragments {α^*} in . They must also characterize the equilibrium “bonded” (entangled) substrates in a hypothetical reactive system R^* = (A^*¦B^*) ≡ R, corresponding to the equalized fragment chemical potentials, at molecular level, and the conserved (ensemble-average) number of electrons: 〈N_R〉_ens. = N_R(μ_R) = N_R. These effective electron populations thus exhibit the equilibrium amount of the inter-reactant CT:

$$ {N}_{\mathrm{CT}}={N_{\mathrm{A}}}^{\ast }-{N_{\mathrm{A}}}^0={N_{\mathrm{B}}}^0-{N_{\mathrm{B}}}^{\ast }>0, $$

(31)

in the globally isoelectronic reaction:

$$ {N}_{\mathrm{R}}\equiv {N_{\mathrm{A}}}^{\ast }+{N_{\mathrm{B}}}^{\ast }={N_{\mathrm{A}}}^{+}+{N_{\mathrm{B}}}^{+}\equiv {N_{\mathrm{R}}}^{+}={N_{\mathrm{A}}}^0+{N_{\mathrm{B}}}^0\equiv {N_{\mathrm{R}}}^0 $$

(32)

To summarize, the fragment identity can be retained only for the mutually closed (nonbonded) status of the acidic and basic reactants, e.g., in the polarized reactive system R_n⁺ or in the equilibrium composite system . The subsystem electron densities {ρ_α = N_α p_α} can be either “frozen”, e.g., in the promolecular reference R⁰ = (A⁰|B⁰) ≡ R_n⁰ consisting of the isolated-reactant distributions shifted to their actual positions in the molecular reactive system R, or “polarized” in R⁺ or R, i.e., relaxed in presence of the reaction partner. The final equilibrium in R as a whole, combining the bonded subsystems {α^*} after CT, accounts for the extra CT-induced polarization of reactants compared to R⁺. As we have argued above, descriptors of this state, of the mutually bonded (formally open) reactants, can be inferred only indirectly, by examining the chemical potential equalization in the composite system . Similar external reservoirs are involved, when one examines the independent population displacements on reactants, e.g., in defining the fragment chemical potentials and their hardness tensor.

In this chain of hypothetical reaction “events”, the polarized system R⁺ appears as the launching stage for the subsequent CT and the accompanying induced polarization, after the hypothetical barrier for the flow of electrons between the two subsystems has been effectively lifted. This density polarization is also accompanied by the subsystem current-promotion, reflected by the modified electron flow patterns in both substrates, compared to promolecule R⁰, in accordance with their current equilibrium-phase distributions [16, 66,67,68,69]. This nonclassical (current) activation of both subsystems complements the classical (probability) polarization of reactants in presence of their reaction partners. The phase aspect is thus vital for accounting for the mutual coherence (entanglement) of reactants in the reactive system as a whole.

The fragment chemical potentials μ_R⁺ = {μ_α⁺} and elements of the hardness matrix η_R⁺ = {η_α,β} of the polarized reactants represent the populational derivatives of the ensemble-average electronic energy in the reactant resolution, reflecting the mutually closed but externally open substrates {α(μ_α⁺)} in composite subsystems of the macroscopic polarized system

(33)

calculated for the fixed external potential of the whole system, v = v_A + v_B, reflecting the “frozen” molecular geometry in R⁺:

(34)

The associated global properties of R = (A^*¦B^*) are defined by the corresponding derivatives with respect to the overall (ensemble-average) number of electrons N_R in the R fragment of the combined system ,

(35)

The optimum amount of the (fractional) CT is then determined by the difference in chemical potentials of the (equilibrium) polarized reactants in R⁺, the CT gradient

(36)

and the effective in situ hardness (η_CT) or softness (S_CT) for this process,

$$ {\displaystyle \begin{array}{c}{\eta}_{\mathrm{CT}}=\partial {\mu}_{\mathrm{CT}}/\partial {N}_{\mathrm{CT}}\\ {}=\left({\eta}_{\mathrm{A},\mathrm{A}}-{\eta}_{\mathrm{A},\mathrm{B}}\right)+\left({\eta}_{\mathrm{B},\mathrm{B}}-{\eta}_{\mathrm{B},\mathrm{A}}\right)\equiv \kern0.35em {\eta_{\mathrm{A}}}^{\mathrm{R}}+{\eta_{\mathrm{B}}}^{\mathrm{R}}={S_{\mathrm{CT}}}^{-1},\end{array}} $$

(37)

representing the CT Hessian and its inverse, respectively; here η_X^R denotes the effective chemical hardness of the “embedded” reactant X in R [72, 73]. The optimum amount of CT,

$$ {N}_{\mathrm{CT}}=-{\mu}_{\mathrm{CT}}\ {S}_{\mathrm{CT}}, $$

(38)

then generates the associated (second-order) stabilization energy:

$$ {E}_{\mathrm{CT}}={\mu}_{\mathrm{CT}}\ {N}_{\mathrm{CT}}/2=-{\mu_{\mathrm{CT}}}^2\ {S}_{\mathrm{CT}}/2<0. $$

(39)

Grand-ensemble principle for thermodynamic equilibrium

The populational derivatives of the average electronic energy or of the resultant gradient-information call for the grand-ensemble representation [1, 2, 85, 86]. Indeed, only the average overall number of electrons in the externally open molecular part 〈M(v)〉_ens., identified by the system external potential v, of the equilibrium combined (macroscopic) system including the external electron reservoir ,

(40)

exhibits the continuous (fractional) spectrum of values, thus justifying the very concept of the populational () derivative itself. Here,

$$ \hat{\mathrm{N}}={\sum}_i\mid {\psi}_i\left\rangle\;{N}_i\right\langle {\psi}_i\mid ={\sum}_i{N}_i\left[{\sum}_j\mid {\psi_j}^i\right\rangle\;\left\langle {\psi_j}^i\mid \right] $$

stands for the particle-number operator in Fock’s space and the density operator

$$ \hat{\mathrm{D}}={\sum}_i{\sum}_j\mid {\psi_j}^i\left\rangle\;{P_j}^i\;\right\langle {\psi_j}^i\mid $$

identifies the statistical mixture of the system (pure) states {|ψ_i〉 ≡ |ψ(N_i)〉 = (ψ_jⁱ, j = 0, 1, …)}, defined for different (integer) numbers of electrons {N_i}, which appear in the ensemble with the external (thermodynamic) probabilities {P_jⁱ}. Such -derivatives are involved in definitions of the system reactivity criteria, e.g., its chemical potential (negative electronegativity) [74, 75, 85,86,87,88,89] or the chemical hardness (softness) [74, 75, 90] and Fukui function (FF) [74, 75, 91] descriptors.

These -derivatives are thus definable only in the mixed electronic states, e.g., those corresponding to thermodynamic equilibria in the externally open molecule 〈M(v)〉_ens.. In the grand-ensemble, this state is determined by the equilibrium density operator specified by the corresponding thermodynamic (externally imposed) intensive parameters: the chemical potential of electron reservoir , and the absolute temperature T of heat bath . These intensities ultimately determine the relevant Legzendre-transform

(41)

of the ensemble-average energy

(42)

which minimizes at the optimum ensemble probabilities for these thermodynamic conditions, {(P_jⁱ)^opt. = P_jⁱ(μ, T; v)},

(43)

The grand-potential corresponds to replacing the “extensive” state parameters of the particle number and thermodynamic entropy [92].

$$ S\left[\hat{D}\right]=\mathrm{tr}\left(\hat{\mathrm{D}}\hat{\mathrm{S}}\right)=-{k}_{\mathrm{B}}{\sum}_i{\sum}_j{P_j}^i\kern0.28em \ln {P}_j^i, $$

(44)

where k_B denotes the Boltzmann constant, by their “intensive” conjugates, the chemical potential μ and absolute temperature T, respectively. The Legendre-transform (41) includes these “intensities” as Lagrange multipliers enforcing at the grand-potential minimum constraints of the specified values of the system ensemble-average values of the conjugate “extensive” parameters: the overall number of electrons,

(45)

and average thermodynamic (von Neumann’s [92]) entropy,

$$ {\displaystyle \begin{array}{c}{\left\langle S\right\rangle}_{ens.}=S\left[{\hat{\mathrm{D}}}_{eq.}\right]={\left\langle S\left(\mu, T\right)\right\rangle}_{ens.}\\ {}=-{k}_B{\sum}_i{\sum}_j{P_j}^i\left(\mu, T;v\right)\kern0.28em \ln\ {P_j}^i\left(\mu, T;v\right)\\ {}=S\left[\mu, T;v\right]\equiv S.\end{array}} $$

(46)

This allows one to formally identify the (external) “intensive” parameters as partial derivatives of the average energy,

(47)

with respect to the corresponding constraint-values:

(48)

The externally imposed parameters μ and T thus determine the associated optimum probabilities of the (pure) stationary states {|ψ_jⁱ〉 ≡ |ψ_j[N_i,v]〉}, eigenstates of partial Hamiltonians,

$$ {\displaystyle \begin{array}{l}\hat{\mathrm{H}}\left({N}_i,v\right)\mid {\psi}_j\left[{N}_i,v\right]\left\rangle ={E}_j^i\mid {\psi}_j\left[{N}_i,v\right]\right\rangle, \\ {}{P}_j^i\left(\mu, T;v\right)={\varXi}^{-1}\exp \left[\beta \left(\mu {N}_i-{E}_j^i\right)\right],\end{array}} $$

(49)

which define the associated density operator of the (mixed) equilibrium state in the grand-ensemble:

$$ \hat{\mathrm{D}}\left(\mu, T;v\right)={\sum}_i{\sum}_j\mid {\psi}_j^i\left\rangle {P}_j^i\left(\mu, T;v\right)\right\langle {\psi}_j^i\mid \equiv {\hat{\mathrm{D}}}_{eq.} $$

(50)

Here, Ξ stands for the grand-ensemble partition-function and β = (k_BT)⁻¹. In the limit T → 0 such a mixture of molecular ground-states {|ψ_i〉 = ψ[N_i, v]} corresponding to integer numbers of electrons {N_i} and energies

$$ {E_j}^i=\left\langle {\psi}_j^i|\hat{\mathrm{H}}\left({N}_i,v\right)|{\psi}_j^i\right\rangle ={E}_j\left[{N}_i,v\right], $$

appearing in the grand-ensemble with probabilities {P_jⁱ(μ, T → 0; v)}, represents an externally open molecule 〈M(μ, T → 0; v)〉_ens. in these thermodynamic conditions.

Information descriptors of chemical reactivity

The ensemble-average value of the resultant gradient-information, given by the weighted expression in terms of the equilibrium probabilities in this thermodynamic (mixed) state,

(51)

is related to the ensemble-average kinetic energy T:

(52)

Therefore, the thermodynamic rule of Eq. (43), for the minimum of the constrained average electronic energy, can be alternatively interpreted as the corresponding principle for the constrained average content of resultant gradient-information:

(53)

Here, the ensemble-average value of the system overall potential energy,

(54)

combines contributions due to electron-nuclear attraction () as well as the electron and nuclear repulsions ().

The information principle of Eq. (53) is seen to contain an additional constraint of the fixed overall potential energy, , multiplied by the Lagrange multiplier

(55)

It also includes the “scaled” intensities associated with the remaining constraints:

(56)

(57)

It should be stressed that the two conjugate thermodynamic principles, for the constrained minima of the ensemble-average energy

(58)

and overall gradient-information

(59)

have the same optimum-probability solutions of Eq. (49). This manifests the physical equivalence of the energetic and entropic principles for determining the equilibrium states in thermodynamics [3].

Several -derivatives of the ensemble-average electronic energy or resultant gradient-information define useful reactivity criteria [74, 75]. The physical equivalence of the energy and information principles in molecular thermodynamics indicates that such concepts are mutually related, being both capable of describing the CT phenomena in donor–acceptor systems [1, 2]. The ensemble interpretation also applies to diagonal and mixed second derivatives of the electronic energy, which involve the differentiation with respect to electron population variable . For example, in the electronic energy representation the chemical hardness reflects derivative of the chemical potential,

(60)

while the information “hardness” measures the derivative of information “potential”:

(61)

The positive signs of these “diagonal” populational derivatives assure the external stability of 〈M(v)〉_ens. with respect to hypothetical electron flows between the molecular system and its reservoir [74, 75]. Indeed, they imply an increase (a decrease) of the global energetic and information “intensities” (μ and ξ), which are coupled to , in response to the perturbation created by the primary electron inflow (outflow). This is in accordance with the familiar Le Châtelier and Le Châtelier-Braun principles of thermodynamics [3] that spontaneous responses in system intensities to the initial population displacements diminish effects of such primary perturbations.

By the Maxwell cross-differentiation relation the mixed second derivative of the system ensemble-average energy,

(62)

measuring its global FF [91], can be alternatively interpreted as either the density response per unit populational displacement or the response in global chemical potential per unit displacement in the external potential. The associated mixed derivative of the average resultant gradient information then reads:

(63)

Use of virial-theorem partitioning

It is of interest to examine the ground-state variations of the electronic resultant gradient-information in specific geometrical displacements ΔQ of the molecular or reactive systems. Its proportionality to the system kinetic-energy component calls for using the virial theorem [4] in the BO approximation of molecular QM,

$$ {\displaystyle \begin{array}{l}2\Delta T\left(\boldsymbol{Q}\right)+\Delta W\left(\boldsymbol{Q}\right)+\boldsymbol{Q}\cdot \left[\mathrm{\partial \Delta }E\left(\boldsymbol{Q}\right)/\partial \boldsymbol{Q}\right]=0,\\ {}\Delta E\left(\boldsymbol{Q}\right)=\Delta T\left(\boldsymbol{Q}\right)+\Delta W\left(\boldsymbol{Q}\right),\end{array}} $$

(64)

which allows one to extract changes in the kinetic, ΔT(Q), and potential, ΔW(Q), components of the overall electronic energy ΔE(Q) for the system current geometrical structure Q:

$$ {\displaystyle \begin{array}{l}\Delta T\left(\boldsymbol{Q}\right)=-\Delta E\left(\boldsymbol{Q}\right)-\boldsymbol{Q}\cdot \left[\mathrm{\partial \Delta }E\left(\boldsymbol{Q}\right)/\partial \boldsymbol{Q}\right]\ \mathrm{and}\\ {}\Delta W\left(\boldsymbol{Q}\right)=2\Delta E\left(\boldsymbol{Q}\right)+\boldsymbol{Q}\cdot \left[\mathrm{\partial \Delta }E\left(\boldsymbol{Q}\right)/\partial \boldsymbol{Q}\right].\end{array}} $$

(65)

These virial relations assume a particularly simple form in diatomics, for which the internuclear distance R uniquely specifies the molecular geometry,

$$ {\displaystyle \begin{array}{l}\Delta E(R)=\Delta T(R)+\Delta W(R),\\ {}2\Delta T(R)+\Delta W(R)+R\left[d\Delta E(R)/ dR\right]=0,\\ {}\mathrm{or}\\ {}\Delta T(R)=-\Delta E(R)-R\left[d\Delta E(R)/ dR\right]=-d\left[R\Delta E(R)\right]/ dR\kern0.28em \mathrm{and}\\ {}\Delta W(R)=2\Delta E(R)+R\left[d\Delta E(R)/\partial R\right]={R}^{-1}d\left[{R}^2\Delta E(R)\right]/ dR.\end{array}} $$

(66)

Figure 1 presents qualitative plots for a diatomic molecule: of the BO potential ΔE(R) and its kinetic-energy component ΔT(R), which also reflects the ground-state resultant gradient-information ΔI(R). It follows from the figure that during a mutual approach by both atoms the kinetic-energy/gradient-information is first diminished relative to the separated-atom limit (SAL), due to the longitudinal Cartesian component of the kinetic energy associated with the “z” direction along the bond axis [93,94,95,96]. At the equilibrium distance R_e the resultant information rises above the SAL value, due to an increase in transverse components of the kinetic energy, corresponding to “x” and “y” directions perpendicular to the bond axis. Therefore, at the equilibrium separation R_e, for which ΔT(R_e) = − ΔE(R_e), the bond-formation results in a net increase of the molecular resultant gradient-information relative to SAL, due to generally more compact electron distribution in the field of both nuclei.

Fig. 1

Qualitative diagram of variations in electronic energy ΔE(R) (solid line) with the internuclear distance R in a diatomic molecule, and of its kinetic energy component ΔT(R) = −d/dR[RΔE(R)] (broken line) reflecting also the state resultant gradient-information ΔI(R) ∝ ΔT(R)

Another interesting case of variations in molecular geometry is the (intrinsic) reaction coordinate R_c, or the associated progress variable P of the arc-length along this trajectory, for which the virial relations also assume the diatomic-like form [4]. Let us examine the virial theorem decomposition of the energy profile along R_c in typical bimolecular reaction

$$ \mathrm{A}+\mathrm{B}\to {\mathrm{R}}^{\ddagger}\to \mathrm{C}+\mathrm{D}, $$

(67)

where R^‡ denotes the transition-state (TS) complex, to which the qualitative Hammond postulate [81] of the chemical reactivity theory applies (see Fig. 2). The virial-theorem application to extract qualitative plots of the resultant gradient-information from energy profiles in the endo- and exo-ergic reactions (upper panel), and in the energy-neutral chemical processes on symmetric potential energy surfaces (PES) (lower panel) has been reported elsewhere [1, 2, 4]. These analyses have shown that this qualitative rule of chemical reactivity is fully explained by the sign of the P-derivative of the overall gradient-information measure at TS complex.

Fig. 2

Variations of the electronic total (E) and kinetic (T) energies in the exo-ergic (ΔE_r < 0) and endo-ergic (ΔE_r > 0) reactions (upper panel). The lower panel provides qualitative plots for the symmetrical PES (ΔE_r = 0)

The qualitative Hammond postulate emphasizes a relative resemblance of the reaction TS complex R^‡ to its substrates (products) in the exo-ergic (endo-ergic) reactions, while for the vanishing reaction energy the position of TS complex is predicted to be located symmetrically between substrates and products. The activation barrier thus appears “early” in exo-ergic reactions, e.g., H₂ + F → H + HF, with the reaction substrates being only slightly modified in TS, R^‡ ≈ [A---B], both electronically and geometrically. Accordingly, in endo-ergic bond-breaking-bond-forming process, e.g., H + HF → H₂ + F, the barrier is “late” along the reaction-progress coordinate P and the activated complex resembles more the reaction products: R^‡ ≈ [C---D]. This qualitative statement has been subsequently given several more quantitative formulations and theoretical explanations using both the energetic and entropic arguments [20, 97,98,99,100,101,102,103].

The energy profile along the reaction “progress” coordinate P, ΔE(P) = E(P) - E(P_sub.) is directly “translated” by the virial theorem into the associated displacement ΔT(P) = T(P) - T(P_sub.) in its kinetic-energy contribution, proportional to the corresponding change ΔI(P) = I(P) - I(P_sub.) in the system resultant gradient-information, ΔI(P) = σ ΔT(P),

$$ \Delta T(P)=-\Delta E(P)-P\ \left[d\Delta E(P)/ dP\right]=-d\left[P\Delta E(P)\right]/ dP $$

(68)

The energy profile ΔE(P) in the endo- or exo-direction, for the positive and negative reaction energy ΔE_r = E(P_prod.) - E(P_sub.), respectively, thus uniquely determines the associated profiles of the kinetic-energy or resultant-information: ΔT(P) ∝ ΔI(P). A reference to qualitative plots in Fig. 2 shows that the latter distinguishes these two directions by the sign of its derivative at TS:

$$ {\displaystyle \begin{array}{l} endo- direction:{\left( dI/ dP\right)}_{\ddagger }>0\ and\ {\left( dT/ dP\right)}_{\ddagger }>0,\varDelta {E}_r>0;\\ {} energy- neutral:{\left( dI/ dP\right)}_{\ddagger }=0\ and\ {\left( dT/ dP\right)}_{\ddagger }=0,\varDelta {E}_r=0;\\ {} exo- direction:{\left( dI/ dP\right)}_{\ddagger }<0\ and\ {\left( dT/ dP\right)}_{\ddagger }<0,\varDelta {E}_r<0.\end{array}} $$

(69)

This observation demonstrates that RC derivative of the resultant gradient-information at TS complex, dI/dP|_‡, proportional to dT/dP|_‡, can serve as an alternative detector of the reaction energetic character: its positive/negative values respectively identify the endo/exo-ergic reactions exhibiting the late/early activation energy barriers, with the neutral case (ΔE_r = 0 or dT/dP|_‡ = 0) exhibiting an equidistant position of TS between the reaction substrates and products on a symmetrical potential energy surface, e.g., in the hydrogen exchange reaction H + H₂ → H₂ + H.

The reaction energy ΔE_r determines the corresponding change in the resultant gradient-information, ΔI_r = I(P_prod.) - I(P_sub.), proportional to ΔT_r = T(P_prod.) - T(P_sub.) = -ΔE_r. The virial theorem thus implies a net decrease of the resultant gradient information in endo-ergic processes, ΔI_r(endo) < 0, its increase in exo-ergic reactions, ΔI_r(exo) > 0, and a conservation of the overall gradient-information in the energy-neutral chemical rearrangements: ΔI_r(neutral) = 0. One also recalls that the classical part of this information displacement probes an average spatial inhomogeneity of the electronic density. Therefore, the endo-ergic processes, requiring a net supply of energy to R, give rise to relatively less compact electron distributions in the reaction products, compared with the substrates. Accordingly, the exo-ergic transitions, which net release the energy from R, generate a relatively more concentrated electron distributions in products, compared to substrates, and no such an average change is predicted for the energy-neutral case.

Regional HSAB versus complementary coordinations

Some subtle preferences in chemical reactivity result from the induced (polarizational or relaxational) electron-flows in reactive systems, reflecting responses to the primary or induced displacements in the electronic structure of the reaction complex, e.g., [16, 104]. Such flow patterns can be diagnosed, estimated, and compared by using either the energetical or information reactivity criteria defined above. One such still-problematic issue is the best mutual arrangement of the acidic and basic parts of molecular reactants in the donor–acceptor systems [16, 104, 105].

Consider the reactive complex A—B consisting of the basic reactant B = (a_B|…|b_B) ≡ (a_B|b_B) and the acidic substrate A = (a_A|…|b_A) ≡ (a_A|b_A), where a_X and b_X denote the acidic and basic parts of subsystem X, respectively. The acidic (electron acceptor) part is relatively harder, i.e., less responsive to external perturbation, exhibiting lower values of the fragment FF descriptor, while the basic (electron donor) fragment is relatively softer, more polarizable, as reflected by its higher density/population responses. The acidic part a_X exerts an electron-accepting (stabilizing) influence on the neighboring part of the other reactant Y, while the basic fragment b_X produces an electron-donor (destabilizing) effect on a fragment of Y in its vicinity.

There are two ways in which both reactants can mutually coordinate in the corresponding reactive complexes [16, 104, 105]. In the complementary (c) arrangement of Fig. 3,

$$ {\mathrm{R}}_c\equiv \left[\begin{array}{c}{a}_{\mathrm{A}}-{b}_{\mathrm{B}}\\ {}{b}_{\mathrm{A}}-{a}_{\mathrm{B}}\end{array}\right] $$

(70)

the reactants orient themselves in such a way that geometrically accessible a-fragment of one reactant faces the geometrically accessible b-fragment of the other substrate. This pattern follows from the maximum complementarity (MC) rule [104] of chemical reactivity, which reflects a simple electrostatic preference that electron-rich (repulsive, basic) fragment of one reactant prefers to face the electron-deficient (attractive, acidic) part of the reaction partner. In the alternative regional HSAB-type structure of Fig. 4, the acidic (basic) fragment of one reactant faces the like-fragment of the other substrate:

$$ {\mathrm{R}}_{\mathrm{HSAB}}\equiv \left[\begin{array}{c}{a}_{\mathrm{A}}-{a}_{\mathrm{B}}\\ {}{b}_{\mathrm{A}}-{b}_{\mathrm{B}}\end{array}\right] $$

(71)

Fig. 3

Polarizational {P_α = (a_α → b_α)}, (α, β) ∈ {A, B}, and charge-transfer, CT₁ = (b_B → a_A) and CT₂ = (b_A → a_B), electron flows involving the acidic A = (a_A|b_A) and basic B = (a_B|b_B) reactants in the complementary arrangement R_c of their acidic (a) and basic (b) parts, with the chemically “hard” (acidic) fragment of one substrate facing the chemically “soft” (basic) fragment of its reaction partner. The polarizational flows {P_α} (black arrows) in the mutually closed substrates, relative to the substrate “promolecular” references, preserve the overall numbers of electrons in isolated reactants {α⁰}, while the two partial CT fluxes (white arrows), from the basic fragment of one reactant to the acidic part of the other reactant, generate a substantial resultant B → A transfer of N_CT = CT₁ - CT₂ electrons between the mutually open reactants. These electron flows in the “complementary complex” are seen to produce an effective concerted (“circular”) flux of electrons between the four fragments invoked in this regional “functional” partition, which precludes an exaggerated depletion or concentration of electrons on any fragment of this reactive system

Fig. 4

Polarizational {P_α = (b_α → a_α)}, (α, β) ∈ {A, B}, and charge-transfer, CT₁ = (b_B → b_A) and CT₂ = (a_B → a_A), electron flows involving the acidic A = (a_A|b_A) and basic B = (a_B|b_B) reactants in the HSAB complex R_HSAB, in which the chemically hard (acidic) and soft (basic) fragments of one reactant coordinate to the like-fragments of the other substrate. The two partial CT fluxes (white arrows) now generate a moderate overall B → A transfer of N_CT = CT₁ + CT₂ electrons between the mutually open reactants. These electron flows in the regional-HSAB complex are seen to produce a disconcerted pattern of four elementary fluxes, producing an exaggerated outflow of electrons from b_B and their accentuated inflow to a_A. This electron removal/accumulation pattern of the charge reconstruction is predicted to be energetically less favorable compared to the concerted-flow model of Fig. 3

The complementary complex, in which the “excessive” electrons of b_X are in the attractive field generated by the electron “deficiency” of a_Y, is expected to be electrostatically preferred since the other arrangement produces the regional repulsion either between two acidic or two basic sites of both reactants.

An additional rationale for this complementary preference over the regional HSAB alignment of reactants comes from examining the charge flows created by the dominating shifts in the site chemical potential due to the presence of the (“frozen”) coordinated site of the nearby part of the reaction partner. At finite separations between the two subsystems, these displacements trigger the polarizational flows {P_X} shown in Figs. 3 and 4, which restore the internal equilibria in both subsystems, initially displaced by the presence of the other reactant.

In R_c, the harder (acidic) site a_Y initially lowers the chemical potential of the softer (basic) site b_X, while b_Y rises the chemical potential level of a_X. These shifts trigger the internal (polariaztional) flows {a_X → b_X}, which enhance the acceptor capacity of a_X and donor ability of b_X, thus creating more favorable conditions for the subsequent inter-reactant CT of Fig. 3. A similar analysis of R_HSAB (Fig. 4) predicts the b_X → a_X polarizational flows, which lowers the acceptor capacity of a_X and donor ability of b_X, i.e., the electron accumulation on a_X and electron depletion on b_X, thus creating less favorable conditions for the subsequent inter-reactant CT.

The complementary preference also follows from the electronic stability considerations, in spirit of the familiar Le Châtelier-Braun principle of the ordinary thermodynamics [3]. In contrast to analysis of Figs. 3 and 4, where the CT responses follow the internal polarizations of reactants, the equilibrium responses to displacements {Δv_X = v_Y} in the external potential on subsystems, one now assumes the primary (inter-reactant) CT displacements {ΔCT₁, ΔCT₂} of Figs. 3 and 4, in the internally closed but externally open reactants, and then examines the induced (secondary) relaxational responses {I_X} to these perturbations.

Let us first examine the CT-displaced complementary complex R_c of Fig. 3,

(72)

defined by the initial populational shifts:

$$ \left[\Delta \left({\mathrm{CT}}_1\right)=\Delta N\left({a}_{\mathrm{A}}\right)=-\Delta N\left({b}_{\mathrm{B}}\right)\right]>\left[\Delta \left({\mathrm{CT}}_2\right)=\Delta N\left({a}_{\mathrm{B}}\right)=-\Delta N\left({b}_{\mathrm{A}}\right)\right]. $$

(73)

In accordance with the Le Châtelier stability principle [3], an inflow (outflow) of electrons from the given site c increases (decreases) the site chemical potential, as indeed reflected by the positive value of the site (diagonal) hardness descriptor

$$ {\eta}_{c,c}=\partial {\mu}_c/\partial {N}_c\equiv {\eta}_c>0. $$

(74)

The initial CT flows {ΔCT_k} thus create the following shifts in the site chemical potentials, compared to the equalized levels in isolated reactants A⁰ = (a_A⁰¦b_A⁰) and B⁰ = (a_B⁰¦b_B⁰),

$$ {\displaystyle \begin{array}{l}\left[\varDelta {\mu}_{a_{\mathrm{A}}}\left({\mathrm{CT}}_1\right)>0\right]>\left[\varDelta {\mu}_{b_{\mathrm{A}}}\left({\mathrm{CT}}_2\right)<0\right]\;\mathrm{and}\\ {}\left[\varDelta {\mu}_{a_{\mathrm{B}}}\left({\mathrm{CT}}_2\right)>0\right]>\left[\varDelta {\mu}_{b_{\mathrm{B}}}\left({\mathrm{CT}}_1\right)<0\right]\end{array}}. $$

(75)

These CT-induced shifts in the fragment electronegativities thus trigger the following secondary, induced flows {I_X} in R_c^CT:

$$ {a}_{\mathrm{A}}\overset{{\mathrm{I}}_{\mathrm{A}}}{\to }{b}_{\mathrm{A}}\kern0.53em \mathrm{and}\ \kern0.28em {a}_{\mathrm{B}}\overset{{\mathrm{I}}_{\mathrm{B}}}{\to }{b}_{\mathrm{B}} $$

(76)

which diminish effects of the initial CT perturbations by reducing the extra charge accumulations/depletions created by these primary CT displacements.

In the CT-displaced HSAB complex R_HSAB of Fig. 4,

(77)

the primary CT perturbations,

$$ \left[\Delta \left({\mathrm{CT}}_1\right)=\Delta N\left({b}_{\mathrm{A}}\right)=-\Delta N\left({b}_{\mathrm{B}}\right)\right]<\left[\Delta \left({\mathrm{CT}}_2\right)=\Delta N\left({a}_{\mathrm{A}}\right)=-\Delta N\left({a}_{\mathrm{B}}\right)\right], $$

(78)

where the inequality sign reflects magnitudes of the associated in situ chemical potentials,

$$ \left[|\mu \left({\mathrm{CT}}_1\right)|=\mu \left({b}_{\mathrm{B}}\right)-\mu \left({b}_{\mathrm{A}}\right)\right]<\left[|\mu \left({\mathrm{CT}}_2\right)|=\mu \left({a}_{\mathrm{B}}\right)-\mu \left({a}_{\mathrm{A}}\right)\right], $$

(79)

induce the internal relaxations in reactants,

$$ {a}_{\mathrm{A}}\overset{{\mathrm{I}}_{\mathrm{A}}}{\to }{b}_{\mathrm{A}}\kern0.53em \mathrm{and}\ \kern0.28em {b}_{\mathrm{B}}\overset{{\mathrm{I}}_{\mathrm{B}}}{\to }{a}_{\mathrm{B}}, $$

(80)

which further exaggerate the charge depletions/accumulations created by the primary perturbation, thus giving rise to a less stable reactive complex compared to R_c^CT.

The global CT equilibrium in the reactive complex as a whole is reached when reactants are both internally and mutually open, as a result of the hypothetical barrier for inter-subsystem flows of electrons being lifted,

$$ \mathrm{R}=\left({\mathrm{A}}^{\ast}\brokenvert {\mathrm{B}}^{\ast}\right)=\left({a_{\mathrm{A}}}^{\ast}\brokenvert {b_{\mathrm{A}}}^{\ast}\brokenvert {a_{\mathrm{B}}}^{\ast}\brokenvert {b_{\mathrm{B}}}^{\ast}\right)=\left({i_{\mathrm{A}}}^{\ast}\brokenvert {i_{\mathrm{A}}^{\prime}}^{\ast}\brokenvert \dots \brokenvert {j_{\mathrm{B}}}^{\ast}\brokenvert {j_{\mathrm{B}}^{\prime}}^{\ast}\brokenvert \dots \right), $$

(81)

as symbolized by the broken vertical lines separating the two reactants and their constituent parts. This global equilibrium marks the chemical potential equalization throughout R:

$$ {\displaystyle \begin{array}{l}{\mu_{\mathrm{A}}}^{\ast }=\partial E\left(\mathrm{R}\right)/\partial {N_{\mathrm{A}}}^{\ast }=\left\{{\mu_i}^{\ast}\left(\mathrm{R}\right)=\partial E\left(\mathrm{R}\right)/\partial {N_i}^{\ast}\right\}\\ {}={\mu_{\mathrm{B}}}^{\ast }=\partial E\left(\mathrm{R}\right)/\partial {N_{\mathrm{B}}}^{\ast }=\left\{{\mu_j}^{\ast}\left(\mathrm{R}\right)=\partial E\left(\mathrm{R}\right)/\partial {N_j}^{\ast}\right\}\\ {}={\mu}_{\mathrm{R}}=\partial E\left(\mathrm{R}\right)/\partial {N}_{\mathrm{R}}.\end{array}} $$

(82)

The final electron densities {ρ_X^*} of reactants, marking the equilibrium distributions in the reactive system as a whole, then account for the extra, CT-induced substrate polarizations {Δρ_X^CT = ρ_X^* - ρ_X⁺} in the resultant displacements {Δρ_X^* = ρ_X^* - ρ_X⁰ = Δρ_X⁺ + Δρ_X^CT} relative to isolated reactants {X⁰}. They integrate to fractional, CT-displaced changes in electron populations of reactants {ΔN_X^* = N_X^* - N_X⁰}. Here, Δρ_X⁺ = ρ_X⁺ − ρ_X⁰ stands for the density shift of X⁺ due to the substrate internal polarization, in the polarized complex R⁺ = (A⁺|B⁺) consisting of the internally open but externally closed reactants.

The two partial CT-responses in R_c generate the overall CT between reactants,

$$ {\displaystyle \begin{array}{l}{N}_{\mathrm{CT}}=\mathrm{C}{\mathrm{T}}_1-\mathrm{C}{\mathrm{T}}_2={N_{\mathrm{A}}}^{\ast }-{N_{\mathrm{A}}}^0\equiv \Delta {N}_{\mathrm{A}}\\ {}={N_{\mathrm{B}}}^0-{N_{\mathrm{B}}}^{\ast}\equiv -\Delta {N}_{\mathrm{B}}>0,\end{array}} $$

(83)

where {N_X^* = N_X⁰ + ΔN_X} stand for the equilibrium electron populations in the molecular complex R = (A^*¦B^*) = (a_A^*¦b_A^*¦a_B^*¦b_B^*) consisting of the mutually open reactants {X^*} and their constituent parts {a_X^* and b_X^*}. Its magnitude is determined by in situ chemical potential of this reactive system, measured by the difference of chemical potentials of the mutually closed (internally polarized) acidic and basic reactants in R⁺ = (A⁺|B⁺) = (a_A⁺¦b_A⁺|a_B⁺¦b_B⁺) [see Eq. (36)]:

$$ \Delta {\mu_{\mathrm{R}}}^{+}={\mu_{\mathrm{A}}}^{+}-{\mu_{\mathrm{B}}}^{+}\equiv {\mu}_{\mathrm{CT}}<0, $$

(84)

and the reactant-resolved hardness tensor of the polarized reactive system R⁺,

$$ {{\boldsymbol{\upeta}}_{\mathrm{R}}}^{+}=\left\{{\eta}_{\mathrm{X},\mathrm{Y}}=\partial {\mu}_{\mathrm{X}}/\partial {N}_{\mathrm{Y}};\kern1.0em \mathrm{X},\mathrm{Y}\in \left(\mathrm{A},\mathrm{B}\right)\right\}, $$

(85)

which generates the in situ hardness (η_CT) and softness (S_CT) descriptors [74, 75] for this CT process [see Eq. (37)]:

$$ {\eta}_{\mathrm{CT}}=\partial {\mu}_{\mathrm{CT}}/\partial {N}_{\mathrm{CT}}=\left({\eta}_{\mathrm{A},\mathrm{A}}-{\eta}_{\mathrm{A},\mathrm{B}}\right)+\left({\eta}_{\mathrm{B},\mathrm{B}}-{\eta}_{\mathrm{B},\mathrm{A}}\right)\equiv {\eta_{\mathrm{A}}}^{+}+{\eta_{\mathrm{B}}}^{+}={S_{\mathrm{CT}}}^{-1}>0. $$

(86)

The resultant CT,

$$ {N}_{\mathrm{CT}}=-{\mu}_{\mathrm{CT}}/{\eta}_{\mathrm{CT}}=-{\mu}_{\mathrm{CT}}\ {S}_{\mathrm{CT}}>0, $$

(87)

then generates the associated (second–order) CT stabilization energy of Eq. (39):

$$ {E}_{\mathrm{CT}}={\mu}_{\mathrm{CT}}\ {N}_{\mathrm{CT}}/2=-{\left({\mu}_{\mathrm{CT}}\right)}^2\ {S}_{\mathrm{CT}}/2<0. $$

(88)

Frontier-electron and communication outlooks on HSAB principle

The physical equivalence of reactivity concepts formulated in the energy and resultant gradient-information representations has also direct implications [1, 2] for the communication theory of the chemical bond (CTCB) [13,14,15,16, 27,28,29,30,31,32,33,34,35,36,37]. In OCT, the theory orbital realization, one treats a molecule as an information channel propagating signals of the AO origins of electrons in the bond system determined by the system occupied MO. It has been argued elsewhere [14,15,16] that elements of the CBO matrix γ = {γ_k,l} [Eq. (25)], the weighting factors in expression of Eq. (23) for the average resultant gradient-information, determine amplitudes of conditional probabilities defining the direct communications between AO. Entropic descriptors of this channel then generate the information bond orders and their covalent/ionic components, which ultimately facilitate an IT understanding of molecular electronic structure in chemical terms.

The communication noise (orbital indeterminicity) in this network, measured by the channel conditional entropy, is due to the electron delocalization in the bond system of a molecule. It represents the overall bond “covalency”, while the channel information capacity (orbital determinicity), reflected by the mutual information of this communication network, measures the resultant bond “ionicity”. Therefore, the more scattering (indeterminate) the molecular information system, the higher its covalent character. Accordingly, a less noisy (more deterministic) channel represents a more ionic molecular system [13,14,15,16, 27,28,29,30,31,32,33,34,35,36,37].

In chemistry the bond covalency, a common possession of electrons by interacting atoms, is indeed synonymous with the electron delocalization generating the communication noise. A classical example is provided by bonds connecting identical atoms, e.g., hydrogens in H₂ or carbons in ethylene, when the interacting AO in the familiar MO diagrams of chemistry exhibit the same levels of AO energies. The bond ionicity accompanies large differences in atomic electronegativities generating a substantial CT. Such bonds correspond to a wide separation of the interacting AO energies in the familiar MO diagrams of chemistry. The ionic bond component introduces more determinicity (less noise) into molecular AO communications, thus representing a bond mechanism competitive with bond covalency [29, 106,107,108,109,110].

One of the celebrated (qualitative) rules of chemistry deals with stability preferences in molecular coordinations. The HSAB principle of Pearson [82] predicts that chemically hard (H) acids (A) prefer to coordinate hard bases (B) in the [HH]-complexes, and soft (S) acids prefer to coordinate soft bases in [SS]-complexes, whereas the “mixed” [HS]- or [SH]-complexes, of hard acids with soft bases or of soft acids with hard bases, respectively, are relatively unstable [83, 90]. As we have emphasized in the preceding section, this global preference is no longer valid regionally, between fragments of reactants, where the complementarity principle [104, 105] dictates the preferred arrangement between the acidic and basic sites of both reactants.

Little is known about the communication implications of the HSAB principle [1, 2]. The following questions arise in the reactivity context:

• How does the [HH] or [SS] preference shape the intra- and inter-reactant communications in the whole reactive complex?

• How is the H or S character of a substrate reflected by its internal communications?

• How does the HSAB preference influence the inter-reactant propagations of information?

In the communication perspective on reactive systems [16, 111], the H and S reactants correspond to the internally ionic (deterministic) and covalent (noisy) reactant channels, respectively. The former involves localized orbital communications between chemically bonded atoms, while the latter corresponds to strongly delocalized information scatterings between AO basis states. A natural question then arises: what is the overall character of communications responsible for the mutual interaction between reactants? Do the S-substrates in [SS]-complex predominantly interact “covalently”, and H substrates of the [HH]-complex “ionically”?

In the frontier electron (FE) approach [112,113,114] to molecular interactions and CT phenomena, the orbital energy of the substrate highest occupied MO (HOMO) determines its donor (basic) level of the chemical potential, while the lowest unoccupied MO (LUMO) energy establishes its acceptor (acidic) capacity (see Fig. 5). The HOMO-LUMO energy gap then reflects the molecular hardness. One also recalls that the interaction between the reactant MO of comparable orbital energies is predominantly covalent (chemically “soft”) in character, while that between the subsystem MO of distinctly different energies becomes mostly ionic (chemically “hard”). A qualitative diagram of Fig. 5 summarizes the alternative, relative positions of the donor (HOMO) levels of the basic reactant, relative to the acceptor (LUMO) levels of its acidic partner, for all admissible hardness combinations in the R = A---B reactive system. In view of the proportionality relations between the energetic and information reactivity criteria, these relative MO energy levels also reflect the corresponding information potential and hardness quantities of subsystems, including the in situ derivatives driving the information transfer between reactants.

Fig. 5

Schematic diagram of the in situ chemical potentials μ_CT(B → A) ≡ μ_R(B → A), determining the effective internal CT from the basic (B) reactant to its acidic (A) partner in R = [A----B] complex, for their alternative hard (H) and soft (S) combinations. The subsystem hardnesses, measured by the HOMO-LUMO gaps in their MO energies, are also indicated

A magnitude of the ionic (CT) stabilization energy in A---B systems is then determined by the corresponding in situ populational derivatives in R,

$$ \Delta {\varepsilon}_{ion.}=\mid {E}_{\mathrm{CT}}\mid ={\mu_{\mathrm{CT}}}^2/\left(2{\eta}_{\mathrm{CT}}\right), $$

(88)

where μ_CT and η_CT stand for the effective chemical potential and hardness descriptors of R involving the relevant FE of reactants. Since the donor/acceptor properties of reactants are already implied by their (known) relative acidic or basic character, one applies the biased estimate of the CT chemical potential.

In this FE approximation the chemical potential difference μ_CT for the effective internal B → A CT thus reads (see Fig. 5):

$$ {\displaystyle \begin{array}{c}{\mu}_{\mathrm{CT}}\left(\mathrm{B}\to \mathrm{A}\right)={\mu_{\mathrm{A}}}^{\left(-\right)}-{\mu_{\mathrm{B}}}^{\left(+\right)}\\ {}={\varepsilon}_{\mathrm{A}}\left(\mathrm{LUMO}\right)-{\varepsilon}_{\mathrm{B}}\left(\mathrm{HOMO}\right)\approx {I}_{\mathrm{B}}-{A}_{\mathrm{A}}>0.\end{array}} $$

(89)

It determines the associated first-order energy change for this electron-transfer process:

$$ \Delta {E}_{\mathrm{B}\to \mathrm{A}}\left({N}_{\mathrm{CT}}\right)={\mu}_{\mathrm{CT}}\left(\mathrm{B}\to \mathrm{A}\right)\ {N}_{\mathrm{CT}}<0. $$

(90)

The CT chemical potential of Eq. (89) combines the electron-removal potential of the basic reactant, i.e., its negative ionization potential I_B = E(B⁺¹) - E(B⁰) > 0,

$$ {\mu_{\mathrm{B}}}^{\left(+\right)}={\varepsilon}_{\mathrm{B}}\left(\mathrm{HOMO}\right)\approx -{I}_{\mathrm{B}}, $$

(91)

and the electron-insertion potential of the acidic substrate, i.e., its negative electron affinity A_A = E(A⁰) - E(A⁻¹) > 0,

$$ {\mu_{\mathrm{A}}}^{\left(-\right)}={\varepsilon}_{\mathrm{A}}\left(\mathrm{LUMO}\right)\approx -{A}_{\mathrm{A}}. $$

(92)

The energy of the CT disproportionation process,

$$ {\displaystyle \begin{array}{c}\left[\mathrm{A}---\mathrm{B}\right]+\left[\mathrm{A}---\mathrm{B}\right]\kern1.50em \\ {}\to \left[{\mathrm{A}}^{-1}---{\mathrm{B}}^{+1}\right]+\left[{\mathrm{A}}^{+1}---{\mathrm{B}}^{-1}\right],\end{array}} $$

(93)

then generates the (unbiased) finite-difference measure of the effective hardness descriptor for this implicit CT [48, 74, 75]:

$$ {\displaystyle \begin{array}{l}{\eta}_{\mathrm{CT}}=\left({I}_{\mathrm{A}}-{A}_{\mathrm{A}}\right)+\left({I}_{\mathrm{B}}-{A}_{\mathrm{B}}\right)\\ {}\approx \left[{\varepsilon}_{\mathrm{A}}\left(\mathrm{LUMO}\right)-{\varepsilon}_{\mathrm{A}}\left(\mathrm{LUMO}\right)\right]+\left[{\varepsilon}_{\mathrm{B}}\left(\mathrm{LUMO}\right)-{\varepsilon}_{\mathrm{B}}\left(\mathrm{HOMO}\right)\right]\\ {}={\eta}_A+{\eta}_B>0.\end{array}} $$

(94)

These in situ populational derivatives ultimately determine a magnitude of the CT stabilization energy of Eq. (88), the ionic part of the overall interaction energy,

$$ \Delta {\varepsilon}_{ion.}={\mu_{\mathrm{CT}}}^2/\left(2{\eta}_{\mathrm{CT}}\right)={\left[{\varepsilon}_{\mathrm{A}}\left(\mathrm{LUMO}\right)-{\varepsilon}_{\mathrm{B}}\left(\mathrm{HOMO}\right)\right]}^2/\left[2\left({\eta}_{\mathrm{A}}+{\eta}_{\mathrm{B}}\right)\right]. $$

(95)

In the FE framework of Fig. 5, the CT (ionic) interaction energy is thus proportional to the squared gap between the LUMO orbital energy of the acidic reactant and the HOMO level of the basic substrate. This ionic interaction is thus predicted to be strongest in the [HH] pair of subsystems and weakest in the [SS]-arrangement, with the mixed [HS]- and [SH]-combinations representing the intermediate magnitudes of the ionic-stabilization effect [83].

It should be realized, however, that the ionic and covalent energy contributions complement each other in the resultant bond energy. Therefore, the [SS]-complex, for which the energy gap between the interacting orbitals, ε_A(LUMO) - ε_B(HOMO), reaches the minimum value, implies the strongest covalent-stabilization of the reactive complex. Indeed, the lowest (bonding) energy level ε_b of this FE interaction, corresponding to the bonding combination of the (positively overlapping), S = 〈φ_A(LUMO)|φ_B(HOMO)〉 > 0, frontier MO of subsystems,

$$ {\varphi}_b={N}_b\ \left[{\varphi}_{\mathrm{B}}\left(\mathrm{HOMO}\right)+{\lambda \varphi}_{\mathrm{A}}\left(\mathrm{LUMO}\right)\right], $$

(96)

then exhibits the maximum bonding energy due to covalent effect:

$$ \Delta {\varepsilon}_{\operatorname{cov}.}={\varepsilon}_{\mathrm{B}}\left(\mathrm{HOMO}\right)-{\varepsilon}_b>0. $$

(97)

It follows from the familiar secular equations of the Ritz method that this covalent energy can be approximated by the limiting MO expression

$$ \Delta {\varepsilon}_{\operatorname{cov}.}\cong {\left(\beta -{\varepsilon}_b\ S\right)}^2/\left[{\varepsilon}_{\mathrm{A}}\left(\mathrm{LUMO}\right)-{\varepsilon}_{\mathrm{B}}\left(\mathrm{HOMO}\right)\right], $$

(98)

where the coupling matrix element of the system electronic Hamiltonian,

$$ \beta =\left\langle {\varphi}_{\mathrm{A}}\left(\mathrm{LUMO}\right)|\hat{H}|{\varphi}_{\mathrm{B}}\left(\mathrm{HOMO}\right)\right\rangle, $$

(99)

is expected to be proportional to the overlap integral S between the frontier MO.

It follows from Eq. (97) that the maximum covalent component of the inter-reactant chemical bond is expected in interactions between soft, strongly overlapping reactants [83], since then the numerator assumes the highest value while the denominator reaches its minimum. For the same reason one predicts the smallest covalent stabilization in interactions between the hard, weakly overlapping substrates, with the mixed hardness combinations giving rise to intermediate bond covalencies.

To summarize, the [HH]-complex exhibits the maximum ionic-stabilization, the [SS]-complex the maximum covalent-stabilization, while the mixed combinations of reactant hardnesses in [HS]- and [SH]-coordinations exhibit a mixture of moderate covalent and ionic interactions between the acidic and basic subsystems [83]. Therefore, communications representing the inter-reactant bonds between the chemically soft (covalent) reactants are also expected to be predominantly “soft” (delocalized, indeterministic) in character, while those between the chemically hard (ionic) subsystems are predicted to be dominated by the “hard” (localized, deterministic) propagations in the communication system for R as a whole [1, 2].

The electron communications between reactants {α = A, B} in the acceptor–donor reactive system R = A----B are determined by the corresponding matrix of conditional probabilities in AO-resolution (or of their amplitudes), which can be partitioned into the corresponding intra-reactant (diagonal) parts, combining internal communications within individual substrates, and the inter-reactant (off-diagonal) blocks of external communications, between different subsystems,

$$ \left[\mathrm{R}\to \mathrm{R}\right]=\left\{\left[\alpha \to \beta \right]\right\}=\left\{\left[\alpha \to \alpha \right]{\delta}_{\alpha, \beta}\right\{+\left\{\left[\alpha \to \beta \right]\ \left(1-{\delta}_{\alpha, \beta}\right)\right\}=\left\{ intra\right\}+\left\{ inter\right\} $$

(100)

The [SS] complexes combining the “soft” (noisy), delocalized (internal) blocks of such probability propagations imply similar covalent character of the external blocks of electron AO communications between reactants, i.e., strongly indeterministic scatterings between subsystems:

$$ \left\{ intra-\mathrm{S}\right\}\Rightarrow \left\{ inter-\mathrm{S}\right\} $$

The “hard” (ionic) internal channels are similarly associated with the ionic (localized) external communications:

$$ \left\{ intra-\mathrm{H}\right\}\Rightarrow \left\{ inter-\mathrm{H}\right\} $$

This observation adds a communication angle to the classical HSAB principle of chemistry.

Conclusions

In this work, we have attempted the QIT description of the bimolecular donor–acceptor reactive system, including all hypothetical processes that accompany the bond-breaking/bond-forming processes of chemical reactions. The present (resultant) information analysis of reactivity phenomena complements earlier (classical) DFT-IT approaches, e.g., [115,116,117,118,119,120,121]. It should be emphasized, however, that the present resultant-information analysis has followed the standard thermodynamic approach to open microscopic systems, which does not imply any new “thermodynamic” transcription of DFT, see, e.g., [120, 121]. The continuities of the classical (modulus/probability) and nonclassical (phase/current) state parameters have been examined and contributions, that these molecular degrees-of-freedom generate in the resultant gradient-information descriptor of a quantum state, have been identified. The need for nonclassical (phase/current) complements of the classical (probability) measures of the information content in molecular electronic states has been reemphasized. It has been argued that the electron density alone reflects only the structure of “being”, missing the structure of “becoming” contained in the current distribution. Both of these manifestations of the molecular “organization” ultimately contribute to the overall information content in generally complex electronic wavefunctions, reflected by the resultant QIT concepts. Their importance in describing the mutual bonding and nonbonding status of reactants has been stressed and the in situ populational derivatives in the energy and information representations have been examined.

The DFT-based theory of chemical reactivity distinguishes several intuitive, hypothetical stages involving either the mutually bonded (entangled) or nonbonded (disentangled) states of reactants for the same electron distribution in constituent subsystems. These two categories are discerned only by the phase aspect of the quantum entanglement between molecular fragments. The equilibrium phases and currents of reactants can be related to the relevant electron densities using the entropic principles of QIT. This generalized approach deepens our understanding of the molecular promotions of constituent fragments and provides a more precise framework for monitoring the reaction progress.

The grand-ensemble description of thermodynamic equilibria in externally open molecular systems has been used to demonstrate the physical equivalence of the energy and resultant gradient-information principles. The populational derivatives of the resultant gradient-information, related to the system average kinetic energy, have been suggested as reliable reactivity criteria. They were shown to predict both the direction and magnitude of the electron flows in reactive systems. The grand-ensemble description of thermodynamic equilibria in the externally open molecular systems has been outlined and the physical equivalence of variational principles for the electronic energy and resultant gradient-information has been emphasized. The virial theorem has been used to explain the qualitative Hammond postulate of the theory of chemical reactivity, and the information production in chemical reactions has been addressed. The ionic and covalent interactions between frontier MO of the acidic and basic reactants have been examined to justify the HSAB principle of chemistry and to provide the communication perspective on interaction between reactants. It has been argued that the internally soft and hard reactants prefer to externally communicate in the like manner, consistent with their internal communications. This preference should be also reflected by the predicted character of the inter-reactant bonds/communications in stable coordinations: covalent in [SS] and ionic in [HH] complexes.