Phase Equalization, Charge Transfer, Information Flows and Electron Communications in Donor–Acceptor Systems

Abstract

Subsystem phases and electronic flows involving the acidic and basic sites of the donor (B) and acceptor (A) substrates of chemical reactions are revisited. The emphasis is placed upon the phase–current relations, a coherence of elementary probability flows in the preferred reaction complex, and on phase-equalization in the equilibrium state of the whole reactive system. The overall and partial charge-transfer (CT) phenomena in alternative coordinations are qualitatively examined and electronic communications in A—B systems are discussed. The internal polarization (P) of reactants is examined, patterns of average electronic flows are explored, and energy changes associated with P/CT displacements are identified using the chemical potential and hardness descriptors of reactants and their active sites. The nonclassical (phase/current) contributions to resultant gradient information are investigated and the preferred current-coherence in such donor–acceptor systems is predicted. It is manifested by the equalization of equilibrium local phases in the entangled subsystems.

1. Introduction

The Information Theory (IT) [1,2,3,4,5,6,7,8] of Fisher [1] and Shannon [3] has been successfully applied in an entropic interpretation of the molecular electronic structure (e.g., [9,10,11]). Several information principles have been investigated [9,10,11,12,13,14,15,16] and pieces of molecular electron density attributed to Atoms-in-Molecules (AIM) have been approached [12,16,17,18,19,20], providing the IT basis for the intuitive stockholder division of Hirshfeld [21]. Patterns of entropic bond multiplicities have been extracted from electronic communications in molecules [9,10,11,22,23,24,25,26,27,28,29,30,31,32], information distributions in molecules have been explored [9,10,11,33,34], and the nonadditive Fisher (gradient) information [1,2,9,10,11,35,36] has been linked to Electron Localization Function (ELF) [37,38,39] of Density Functional Theory (DFT) [40,41,42,43,44,45]. This analysis has enabled a formulation of the novel Contragradience (CG) probe for localizing chemical bonds [9,10,11,46], while the Orbital Communication Theory (OCT) of the chemical bond using the “cascade” propagations in molecular information systems has identified the bridge interactions between AIM [11,47,48,49,50,51,52], realized through orbital intermediates.

In molecular Quantum Mechanics (QM), the wavefunction phase or its gradient determining the effective velocity of probability density and its current give rise to nonclassical information/entropy supplements to classical measures of Fisher [1] and Shannon [3]. In resultant IT descriptors of electronic states, the information/entropy content in the probability (wavefunction modulus) distribution is combined with the relevant complement due to the current density (wavefunction phase) [53,54,55,56,57,58,59,60,61,62]. The overall gradient information is then proportional to the expectation value of the system kinetic energy of electrons. Such combined descriptors are also required in the phase distinction between the bonded (entangled) and nonbonded (disentangled) states of molecular subsystems, e.g., the substrate fragments of reactive systems [63,64,65]. This generalized treatment then allows one to interpret the variational principle for electronic energy as equivalent information rule, and to use the molecular virial theorem [66] in general reactivity considerations [67,68,69,70,71].

The extremum principles for the global and local (gradient) measures of the state resultant entropy have determined the phase-transformed (“equilibrium”) states of molecular fragments, identified by their local “thermodynamic” phases [53,54,55,56,57]. The minimum-energy principle of QM has also been interpreted as physically-equivalent variational rule for the resultant gradient information, proportional to the state average kinetic energy [67,68,69,70,71]. In the grand-ensemble framework, they both determine the same thermodynamic equilibrium in an externally-open molecular system. This equivalence parallels identical predictions resulting from the minimum-energy and maximum-entropy principles of ordinary thermodynamics [72].

Elsewhere [63,73,74], the potential use of the DFT construction by Harriman, Zumbach, and Maschke (HZM) [75,76], of wavefunctions yielding the prescribed electron distribution, in a description of reactive systems has been examined. In such density-constrained Slater determinants, the defining Equidensity Orbitals (EO), of the Macke/Gilbert [77,78] type, exhibit the same molecular probability density, with the orbital orthogonality being assured by the EO local phases alone. These orbital states define the constrained multicomponent system composed of the mutually-closed orbital units, with each subsystem being characterized by its own phase and separate chemical-potential descriptors. Their simultaneous opening onto a common electron reservoir, and hence also onto themselves, generates the externally- and mutually-open orbital system, in which the EO fragments are effectively “bonded” (entangled) [63]. They then exhibit a common (molecular) phase descriptor and equalize their chemical potentials at the global reservoir level. This (mixed) equilibrium state is determined by the density operator corresponding to thermodynamic (grand-ensemble) probabilities of EO related to their orbital energies and average electron occupations.

In the present analysis, we focus on the phase component of electronic wavefunctions and the related current descriptor of molecular states, with special emphasis on the donor–acceptor reactive systems [63,67,68,69,70,71]. We examine the reactant phases and electronic flows involving the substrate acidic and basic active sites. The mutual relations between the state phase and current descriptors is explored, the phase-equalization in the equilibrium state of the whole reactive system is conjectured, and a coherence of probability flows in the preferred reaction complex is established. The overall and partial charge-transfer (CT) phenomena in alternative coordinations of the acidic (A) and basic (B) reactants are investigated, the electronic communications in alternative A—B complexes are qualitatively examined, and nonclassical contributions to the resultant gradient information are introduced. We also tackle the (internal) polarization (P) of reactants, induced by the (external) CT displacements between both substrates, and patterns of the resultant electronic flows on their active sites are qualitatively explored. The energy changes associated with specific P/CT fluxes in A—B systems are estimated using the familiar descriptors of the Charge Sensitivity Analysis (CSA) [79,80,81,82,83,84,85]: the global or regional chemical-potential/electronegativity [86,87,88,89,90] and hardness/softness [91] or Fukui function [92] descriptors of reactants and their acidic and basic active sites. The preferred coordination of reactants in such donor–acceptor systems is shown to exhibit a substantial current-coherence, which is also manifested by the equalization of the equilibrium local phases in the entangled (bonded) subsystems [63].

2. Molecular States and Their Phases

In QM, the state Ψ(N) of N-electrons is represented by its representative vector in the abstract (Hilbert) space,

|Ψ(N)〉 ≡ M|D(N)〉,

(1)

where M stands for its modulus (“length”),

M = 〈Ψ(N)|Ψ(N)〉^1/2 ≡ |Ψ(N)|,

(2)

and |D(N)〉 denotes the corresponding directional (“unit”) vector:

|D(N)| = 〈D(N)|D(N)〉^1/2 = 1.

(3)

A variety of quantum states is exhausted by all admissible orientations of the unit vector |D(N)〉 [93]. Thus, for the unity-normalized state vectors, when M = 1, |Ψ(N)〉 ≡ |D(N)〉.

The molecular state can be also identified by the spin-position representation of |Ψ(N)〉, called the wavefunction:

Ψ[Q^(N)] = 〈Q^(N)|Ψ(N)〉 ≡ Ψ(N) = D[Ψ(N)] exp{iΦ[Ψ(N)]} ≡ D(N) F(N).

(4)

The squared modulus component,

D[Ψ(N)] = [Ψ(N)Ψ^*(N)]^1/2 ≡ D(N),

(5)

determines the normalized probability distribution of N electrons:

P(N) = D(N)² = 〈Q^(N)|Ψ(N)〉 〈Ψ(N)|Q^(N)〉 ≡ 〈Q^(N)|P_Ψ(N)|Q^(N)〉
= 〈Ψ(N)|Q^(N)〉 〈Q^(N)|Ψ(N)〉 ≡ 〈Ψ(N)|P_Q(N)|Ψ(N)〉,

∫P(N) dQ^(N) ≡ 1,

(6)

Here, P_Ψ(N) and P_Q(N) stand for the state and basis-set projection operators, respectively, while ∫dQ^(N) denotes the integrations over spatial positions {r_k} and summations over spin variables {σ_k} of all N electrons. The state exponential factor F(N) involves the N-electron phase function Φ[Ψ(N)]} ≡ Φ(N), which generates the state current density.

The wave function Ψ(N) thus reflects projections (directional “cosines”) of the state vector |Ψ^(N)〉 onto the basis vectors {|Q^(N)〉} of the adopted representation. It contains all essential “orientation” information about |Ψ(N)〉. This specific representation corresponds to the vector basis of N-electron states,

|Q^(N)〉 = |q₁, q₂, …, q_N〉 = {|q_k〉 = |σ_k, r_k〉},   k = 1, 2, …, N,

(7)

identified by the spin ({σ_k}) and position ({r_k}) variables of all N electrons. It includes the eigenvectors {|q_k〉} of the electronic spin (s_k = is_k,x + js_k,y + ks_k,z, s_k = −|s_k|) and position (r_k) operators:

s_k²|q_k〉 = ¾ ħ²|q_k〉 ≡ s_k²|q_k〉, s_k.z|q_k〉 = σ_k ħ|q_k〉 ≡ s_z|q_k〉,  σ_k = ± ½,  and

r_k|q_k〉 = r_k|q_k〉.

(8)

In what follows, we also examine (normalized) molecular orbital (MO) states {|ψ_w〉} of a single electron, |ψ_w| = 1. The state vector |ψ_w〉 is then given by the product of the spin (|ξ_w〉) and spatial (|φ_w〉) states:

|ψ_w〉 ≡ |ξ_w〉|φ_w〉.

(9)

It generates the associated spin-orbital (SO) function in q-representation,

ψ_w(q) = 〈q|ψ_w〉 = 〈σ|ξ_w〉 〈r|φ_w〉 ≡ ξ_w(σ) φ_w(r),

(10)

the product of its normalized spin-function of an electron,

ξ_w(σ) = 〈σ|ξ_w〉,  |ξ_w| = (Σ_σ |ξ_w(σ)|²)^1/2 = 1,

(11)

ξ_w(σ) ∈ {α(σ), spin-up; β(σ), spin-down},

(12)

and the normalized (complex) spatial MO component,

φ_w(r) = 〈r|φ_w〉 = d_w(r) exp[iϕ_w(r)] ≡ d_w(r) f_w(r),

(13)

∫φ_w(r)^*φ_w(r) dr = 1   or   |φ_w| = 1,

(14)

with p_w(r) = d_w(r)² defining the MO probability distribution. One customarily requires the spin-orbitals defining the N-electron configuration in a molecule to form the independent (orthonormal) set:

〈φ_v|φ_w〉 = ∫〈φ_v|r〉〈r|φ_w〉 dr = ∫φ_v(r)^*φ_w(r) dr = δ_k,l.

(15)

To summarize, the complex MO wavefunction φ_w(r) fully reflects the orientation properties of |φ_w〉 in the position representation. The square of its modulus d_w(r) = |φ_w(r)| determines the (normalized) spatial probability distribution in |ψ_w〉,

p_w(r) = |φ_w(r)|² = d_w(r)² ≥ 0,  ∫p_w(r) dr = 1,

(16)

while |ξ_w(σ)|² similarly determines the probability density of observing the specified spin component s_z = σ ħ. The phase factor f_w(r) = exp[iϕ_w(r)] identifies the orientation of the (normalized) state vector |φ_w〉 in the complex plane. It constitutes the r-representation of the directional (unit) vector

|d_w〉 = |φ_w〉/|φ_w| = |φ_w〉,  |φ_w| = [∫|φ_w(r)|² dr]^1/2 ≡ 1,

(17)

d_w(r) = 〈r|d_w〉 = exp[iϕ_w(r)] = cosϕ_w(r) + i sinϕ_w(r).

(18)

The MO phase gradient ultimately determines the orbital current

j_w(r) = [ħ/(2mi)] [φ_w(r)^* ∇φ_w(r) − φ_w(r) ∇φ_w(r)^*] = (ħ/m)] p_w(r) ∇ϕ_w(r).

(19)

As an illustration, we summarize in Appendix A the components of the quantum state describing a single electron, explore its probability/current descriptors, and summarize the relevant continuity relations. The latter result directly from the Schrödinger equation (SE) of molecular QM (see Appendix B).

One finally recalls that in the one-electron (MO) approximation the N-electron wavefunctions are defined as Slater determinants constructed from the configuration occupied SO, ψ = (ψ₁, ψ₂, …, ψ_N),

Ψ(N) = 〈Q^(N)|Ψ〉 = (N!)^−1/2 |ψ₁(Ψ)ψ₂(Ψ) … ψ_N(Ψ)| ≡ det[ψ(Ψ)].

(20)

3. Electronic Communications

In OCT of the chemical bond [9,10,11,22,23,24,25,26,27,28,29,30,31,32,62], one explores the entropy/information descriptors of molecular information channels, e.g., the electron probability networks in the orthonormal Atomic Orbital (AO) resolution |χ〉 = {|χ_j〉, j = 1, 2, …, t},

〈χ_i|χ_j〉 = ∫〈χ_i|r〉 〈r|χ_j〉 dr = ∫χ_i(r)^* χ_j(r) dr = δ_i,j.

(21)

Such electron communications are reflected by the conditional probabilities of observing in the molecular bond system the monitored “output” orbitals, given the specified “input” orbitals.

In this standard LCAO MO approach, the optimum MO |φ〉 = {|φ_w〉, w = 1, 2, …, t} are represented as Linear Combinations of the AO basis functions:

φ(r) = {φ_w(r) = Σ_j χ_j(r) C_j,w} = χ(r) C,   χ(r) = 〈r|χ〉,

(22)

where the square matrix of the expansion coefficients, C = 〈χ|φ〉 = {C_i,w}, satisfies the matrix orthonormality relations:

CC^† = {δ_i,j} = C^†C = {δ_w,w’} ≡ I.

(23)

The AO-resolved communication theory explores the entropy/information descriptors of the electronic-signal propagation network between “input” and “output” AO states.

These scattering connections in the given electron configuration of Equation (20) are determined by its occupied MO, ψ(Ψ). The representative probability of observing in Ψ the specified output AO χ_j(r) = 〈r|χ_j〉 = m_j(r) exp[iϕ_j(r)], given the input AO χ_i(r) = 〈r|χ_i〉= m_i(r) exp[iϕ_i(r)],

P(χ_j|χ_i) = P(χ_i, χ_j)/P(χ_i) ≡ P(j|i),  Σ_j P(j|i) = 1,

(24)

is generated by the squared modulus of the associated conditional amplitude A(χ_j|χ_i) ≡ A(j|i):

{P(j|i) ≡ |A(j|i)|²}.

(25)

Here, P(χ_i) ≡ P(i) denotes the probability of detecting AO state |χ_i〉 ≡ |i〉 in the chemical bond system of the electron configuration in question, while P(χ_i, χ_j) ≡ P(i, j) stands for the joint-probability of these two AO events, of simultaneously observing the two AO states |i〉 and |j〉 in Ψ. These probabilities satisfy the relevant normalizations:

Σ_i [Σ_j P(i, j)] = Σ_i P(i) = 1.

(26)

In accordance with the Superposition Principle (SP) of QM [93], the scattering amplitude A(ζ|θ) between states θ(r) ≡ 〈r|θ〉 = m_θ(r) exp[iϕ_θ(r)] and ζ(r) ≡ 〈r|ζ〉 = m_ζ(r) exp[iϕ_ζ(r)] is defined by the mutual projection of the two state-vectors involved:

A(ζ|θ) = 〈θ|ζ〉 ≡ ∫θ(r)^* ζ(r) dr = ∫m_θ(r) m_ζ(r) exp{i[ϕ_ζ(r)−ϕ_θ(r)]} dr
      ≡ ∫M(ζ|θ) exp{iΦ(ζ|θ)] dr.

(27)

This conditional amplitude is thus generated by the (local) effective modulus function M(ζ|θ) = m_θ(r) m_ζ(r) and the phase component

Φ(ζ|θ) = ϕ_ζ(r) − ϕ_θ(r).

(28)

Therefore, the conditional probability P(θ|ζ) can be interpreted as the expectation value in one state of the (idempotent) projection operator onto another state:

P(θ|ζ) ≡ |A(θ|ζ)|² = 〈ζ|θ〉 〈θ|ζ〉 ≡ 〈ζ|P_θ|ζ〉
        = 〈θ|ζ〉 〈ζ|θ〉 ≡ 〈θ|P_ζ|θ〉,
P_ϑ = |ϑ〉 〈ϑ|,  P_ϑ² = P_ϑ,  ϑ = (θ, ζ).

(29)

In molecular electron configuration of Equation (20), each AO event is additionally conditional on the molecular state Ψ(N), identified either by the N-electron projection operator P_Ψ(N) = |Ψ(N)〉〈Ψ(N)| or the configuration (idempotent) bond-projection P_b(N) involving only the occupied MO selected by their finite occupation numbers

n(Ψ) = {n_w(Ψ)δ_w,w’},

n_w(Ψ) = 1 (Ψ-occupied MO) or n_w(Ψ) = 0 (Ψ-virtual MO),

P_b(Ψ) = |Ψ(Ψ)〉 n(Ψ) 〈Ψ(Ψ)| = Σ_w |ψ_w(Ψ)〉 n_w(Ψ) 〈ψ_w(Ψ)|
= Σ_s |Ψ_s(Ψ)〉 〈Ψ_s(Ψ)| ≡ Σ_s P_s(Ψ),

(30)

where w = 1, 2, …, t (all SO, occupied, and virtual) and s = 1, 2, …, N (occupied SO only).

The conditional probability of observing |j〉 given |i〉 in electronic configuration |Ψ〉 thus involves the doubly-conditional (coherent) amplitude A_Ψ(χ_j|χ_i) ≡ A(χ_j|χ_i‖Ψ) ≡ A_Ψ(j|i):

P_Ψ(χ_j|χ_i) = P(χ_j|χ_i‖Ψ) ≡ P_Ψ(j|i) = |A_Ψ(j|i)|².

(31)

Its amplitude

A_Ψ(j|i) = 〈i|P_b(Ψ)|j〉 = Σ_w 〈i|ψ_w(Ψ)〉 n_w(Ψ) 〈ψ_w(Ψ)|j〉
       = Σ_s 〈i|Ψ_s(Ψ)〉 〈Ψ_s(Ψ)|j〉 = Σ_s C_i,s(Ψ) C_j,s(Ψ)^*≡ γ_i,j(Ψ)

(32)

thus defines the relevant element of the (idempotent) Charge-and-Bond−Order (CBO) matrix of LCAO MO theory:

γ(Ψ) = 〈χ|Ψ(Ψ)〉 n(Ψ) 〈Ψ(Ψ)|χ〉 = C(Ψ) n(Ψ) C(Ψ)^† = {γ_i,_j(Ψ)}.

(33)

This molecular communication amplitude is thus characterized by the effective modulus and phase components of γ_i,j(Ψ) resulting from corresponding descriptors of (complex) LCAO MO coefficients

C(Ψ) = {C_k,w(Ψ) = 〈χ_k|ψ_w(Ψ)〉 ≡ 〈k|ψ_w(Ψ)〉.

For the given electron configuration |Ψ(N)〉, it determines the molecular conditional probabilities between the specified pair of AO states. These scattering probabilities are now defined by the AO expectation values of the (nonidempotent) basis set projections in the occupied MO subspace,

P_l^b(Ψ) = P_b(Ψ) |l〉 〈l| P_b(Ψ) = P_b(Ψ) P_l P_b(Ψ),  [P_l^b(Ψ)]² ≠ P_l^b(Ψ),

(34)

of the system chemical bonds in |Ψ(N)〉:

P_Ψ(j|i) ≡ |A_Ψ(j|i)|² = γ_i,_j(Ψ) γ_j,_i(Ψ)
= 〈i|P_b(Ψ)|j〉 〈j|P_b(Ψ)|i〉 = 〈i|P_b(Ψ)P_jP_b(Ψ)|i〉 ≡ 〈i|P_j^b(Ψ)|i〉
= 〈j|P_b(Ψ)|i〉 〈i|P_b(Ψ)|j〉 = 〈j|P_b(Ψ)P_iP_b(Ψ)|j〉 ≡ 〈j|P_i^b(Ψ)|j〉.

(35)

4. Phase–Current Relation

In this section, we briefly explore the mutual relation between the phase and current descriptors [see Equation (19)] of the quantum state of an electron,

ψ(q) = φ(r)ξ(σ),

where

φ(r) = m(r) exp[iϕ(r)].

(36)

Its probability distribution reflects the square of the MO modulus factor, p(r) = m(r)², while its phase component ϕ(r) determines the state probability current of Equation (19):

j(r) = [ħ/(2mi)] [φ(r)^* ∇φ(r) − φ(r) ∇φ(r)^*]
= (ħ/m) p(r) ∇ϕ(r) ≡ p(r) V(r).

(37)

Here, the effective velocity of the probability “fluid”,

V(r) = {V_u(r) ≡ V_u(u, {v≠u}), u = x, y, z},

measures the local current-per-particle and reflects the state phase gradient:

V(r) = j(r)/p(r) = (ħ/m) ∇ϕ(r).

(38)

Therefore, velocity components of the probability flux are determined by the corresponding partials of the state phase function:

V_u(r) = (ħ/m) [∂ϕ(r)/∂u], u = x, y, z.

(39)

It follows from this equation that the phase of electronic state can be formally reconstructed by an indefinite integration of the corresponding components of the velocity field of the probability flux:

$$\varphi (u,\{v\ne u\})=(m/\hslash ){\displaystyle \underset{-\infty }{\overset{u}{\int }}{V}_u}(\vartheta ,\{v\ne u\})d\vartheta ,u=x,y,z.$$

(40)

As an illustrative example, consider the fixed-momentum (free-particle) state

φ_K(r) = A exp(iK·r) = A exp[i(K_xx + K_yy + K_zz),

K = (m/ħ)V, V = iV_x + jV_y + kV_z = const.

(41)

One indeed recovers its phase components,

ϕ(r) = K·r = (m/ħ) (V_xx + V_yy + V_zz),

(42)

by straightforward unidimensional integrations of Equation (40).

Consider now the phase/current dependent term of the overall content of MO resultant gradient-information [28,53,54,55,56,57,58,59,60,61,62] in Quantum Information Theory (QIT) [62]:

I[ϕ] = 4∫p(r) [∇ϕ(r)]²dr ≡ ∫p(r) I_ϕ(r) dr
= (2m/ħ)² ∫p(r) V(r)² dr = (2m/ħ)² ∫p(r)⁻¹ j(r)² dr.

(43)

By analogy to the free-electron wavefunction of Equation (41), one introduces the wave-vector distribution in molecular systems, defined by the local phase gradient

K(r) = (m/ħ) V(r) = ∇ϕ(r).

(44)

In terms of this local momentum measure, the nonclassical information functional simplifies:

I[ϕ] = 4 ∫p(r) K(r)²dr ≡ 4〈K²〉.

(45)

Therefore, in such an inhomogeneous (molecular) electron density, the nonclassical MO information reflects the state average square 〈K²〉 of the wave-vector K(r). It thus follows from the preceding equation that this nonclassical information measure, related to the current contribution to the state resultant kinetic energy, depends only on the magnitude of the local probability flow, being independent of its direction.

5. Current Coherence in Donor–Acceptor Systems

Electron flows in molecules are determined by the energy conditions. The molecular energetics is reflected by SE of QM. Indeed, the entropy/information flows in molecular systems are carried by the state electronic flows. The QIT description thus offers only a supplementary, alternative framework for exploring and ultimately better understanding the rules and structures of chemistry. The energetics of a nondegenerate ground-state of the whole molecular system is devoid of any phase aspect, while subtle preferences involving reactants may already involve local phases of electronic states in the substrate subsystems, reflecting their entanglement in the whole reactive complex [63,64,65,66,67,68,69,70,71].

As an illustrative example let us now recall the energy-preferred complementary reactive complex A—B [67,70,94,95] shown in Figure 1, consisting of the basic subsystem

B = (a_B |…| b_B) ≡ (a_B|b_B)

and its reaction companion—the acidic substrate

A = (a_A |…| b_A) ≡ (a_A|b_A),

where a_X and b_X denote the active acidic and basic sites of X, respectively. The four molecular fragments, λ ∈{(a_A, b_A), (a_B, b_B)}, define the active parts in the system charge reconstruction. The acidic (electron acceptor) part is relatively harder, i.e., less responsive to external perturbations, thus exhibiting lower values of the fragment Fukui function or chemical softness descriptor, while the basic (electron donor) fragment is relatively more polarizable, as indeed reflected by higher response descriptors of its electron density or site populations. The acidic part a_X exerts an electron-accepting (stabilizing) influence on the neighboring part of another reactant Y, while the basic fragment b_X produces an electron-donor (destabilizing) effect on the fragment Y in its vicinity.

In the most stable complementary (c) complex of Figure 1 [94,95], a geometrically accessible a-fragment of one reactant faces the geometrically accessible b-fragment of the other substrate

$$R_c\equiv \left[\begin{array}{l}{a}_{\mathrm{A}}-b_{\mathrm{B}}\hfill \\ b_{\mathrm{A}}-a_{\mathrm{B}}\hfill \end{array}\right],$$

(46)

while in a less favorable regional HSAB-type coordination the acidic (basic) fragment of one reactant faces the like-fragment of the other substrate:

$${\mathrm{R}}_{\mathrm{HSAB}}\equiv \left[\begin{array}{l}{a}_{\mathrm{A}}-a_{\mathrm{B}}\hfill \\ b_{\mathrm{A}}-b_{\mathrm{B}}\hfill \end{array}\right].$$

(47)

A relative stability of R_c reflects an electrostatic preference: an electron-rich (repulsive, basic) fragment of one reactant indeed prefers to face an electron-deficient (attractive, acidic) part of the reaction partner. As shown in Figure 1, at finite separations between the two subsystems, spontaneous (primary) CT displacements between reactants trigger the induced (secondary) polarizational flows {P_X} within each reactant, which restore the intra-substrate equilibria initially displaced by the presence of the other fragment and the inter-reactant CT:

$${\mathrm{R}}_c:\hspace{1em}\hspace{1em}\begin{array}{ccc}{a}_{\mathrm{B}}& \leftarrow & b_{\mathrm{A}}\\ \downarrow & & \uparrow \\ b_{\mathrm{B}}& \to & a_{\mathrm{A}}\end{array},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\mathrm{R}}_{\mathrm{HSAB}}:\hspace{1em}\hspace{1em}\begin{array}{ccc}{a}_{\mathrm{B}}& \to & a_{\mathrm{A}}\\ \uparrow & & \uparrow \\ b_{\mathrm{B}}& \to & b_{\mathrm{A}}\end{array}$$

(48)

It has been inferred from the (energetic) Electronegativity Equalization principle [95] that the electronic CT and P flows in R_c generate the concerted pattern shown in Figure 1 and Equation (48), which exhibits the maximum current (phase-gradient) coherence. It implies the least population activation on both reactants, which also energetically favors the complementary complex relative to the regional HSAB arrangement. Indeed, in R_HSAB coordination the energy-preferred disconcerted flow pattern implies a more exaggerated depletion of electrons on b_B and their more accentuated accumulation on a_A. In fact, these partial flows of electrons signify the “bridge” CT between the key (“diagonal”) sites b_B and a_A, via the intermediate (“off-diagonal”) sites a_B and b_A.

In the complementary arrangement of both reactants, the partial fluxes involving the four active sites indicate the “flow-through” behavior of all these fragments, which combines the external (CT) and internal (P) currents. One observes that the acidic sites accept electrons as a result of charge transfer between reactants and donate electrons through the substrate polarization. The opposite behavior of the basic sites is observed: they receive electrons due to P-mechanism and lose electrons as a result of CT. In the HSAB complex, one observes a similar flow-through pattern only on the a_B and b_A sites, b_B exclusively donates electrons both internally (P) and externally (CT), while a_A only accepts electrons from its complementary fragment b_A (P) and a_B (CT) part of the other reactant.

To summarize, in R_c, one observes the concerted flow of electrons involving all four active sites of both reactants, with small net changes in electron populations on these fragments, while the charge reconstruction in R_HSAB can be regarded as a transfer of electrons from b_B to a_A through the remaining (intermediate) sites a_B and b_A. The energetical preference of the complementary arrangement of reactants [94,95] also signifies the maximum phase-gradient coherence on each site, with its P or CT inflow part being accompanied by the associated outflow flux. Such a flow pattern thus corresponds to the least populational displacements on all constituent active sites. The above “activation” perspective then provides a natural physical explanation of the observed preference of the complementary coordination.

The energetically favored, concerted pattern of electronic fluxes in R_c can be realized only by an appropriate coherence of the site effective phase-gradients, of the pure or mixed states {Ψ_λ} describing the mutually-closed fragments λ ∈ {a_A =1, b_A = 2, a_B = 3, b_B = 4} of both reactants. These states define the average resultant currents on each site, weighted by the site electron probability distribution p_λ(r),

〈j〉_λ = ∫p_λ(r) j_λ(r) dr ≡ j(λ),   ∫p_λ(r) dr = 1,

(49)

already accounting for the average outflows-from and inflows-to the part in question, caused by the CT or P displacements in the system electronic structure.

The elementary P and CT currents of the flow-patterns in Equation (48) give rise to the corresponding vector sums {j(λ)}, the site resultant currents schematically drawn in Figure 2. They are seen to generate a “conrotatory” pattern in the complementary complex R_c and a collective “translational” pattern in R_HSAB. The magnitude |〈j〉_λ| of this average (directional) site descriptor of the probability-flow ultimately determines the size |ΔN_λ| of the net change in the fragment electron population in unit time,

ΔN_λ = N_λ 〈j〉_λ,

(50)

where N_λ stands for the fragment average number of electrons in the separate reactant. The energy-favored (complementary) complex, which represents the least population-activation process, then corresponds to the lowest overall population displacement:

Σ_λ |ΔN_λ| = Σ_λ N_λ |〈j〉_λ| ≡ Σ_λ N_λ 〈j〉_λ ⇒ minimum.

(51)

A reference to Equation (43) shows that the nonclassical gradient information of site λ also probes the average lengths of the directional phase-related properties:

I_λ[ϕ_λ] = 4∫p_λ(r) [∇ϕ_λ(r)]²dr
     = (2m/ħ)² ∫p_λ(r) V_λ(r)² dr
      = (2m/ħ)² ∫p_λ(r)⁻¹ j_λ(r)² dr.

(52)

Therefore, the least site-activation of Equation (51), which occurs for the current-coherence in reactive system, also implies the minimum of the (additive) overall nonclassical information content:

I[ϕ] = Σ_λ I_λ[ϕ_λ] ⇒ minimum.

(53)

Its absolute minimum is reached for the mutually-open (entangled) sites in the (bound) nondegenerate quantum state of the whole reactive system,

R^* = (A^* ¦ B^*) = (a_A^* ¦ b_A^* ¦ a_B^* ¦ b_B^*),

when the local phase contribution identically vanishes. Moreover, in the information equilibrium state of the entangled molecular fragments, corresponding to the minimum of the system nonclassical gradient-information, this also implies the equalization of subsystem phases at the global phase of the wavefunction describing the reactive system as a whole (see also Appendix C).

The normalized probability distributions on sites {κ = (a, b)} in reactants {X = (A, B)},

{p_λ(r)} = {p_κ(r; X) = ρ_κ(r; X)/N_κ(X)},  ∫p_κ(r; X) dr = 1,

(54)

the shape factors of the fragment electronic densities {ρ_κ(r; X)}, can be used as weights in determining the corresponding fragment (internal) averages of physical properties. For example, the site vector (directional) densities

{k_κ(r; X)},  {V_κ(r; X)}  and  {j_κ(r; X)}

(55)

generate the associated average descriptors on each substrate:

〈k(X)〉 = Σ_κ P_κ(X) ∫p_κ(r; X) k_κ(r; X) dr ≡ Σ_κ P_κ(X) 〈k(X)〉_κ
≡ Σ_κ ∫w_κ(r; X) k_κ(r; X) dr ≡ ∫k(r; X) dr,

(56)

〈V(X)〉 = Σ_κ P_κ(X) ∫p_κ(r; X) V_κ(r; X) dr ≡ Σ_κ P_κ(X) 〈V(X)〉_κ
≡ Σ_κ ∫w_κ(r; X) V_κ(r; X) dr ≡ ∫V(r; X) dr,

(57)

〈j(X)〉 = Σ_κ P_κ(X) ∫p_κ(r; X) j_κ(r; X) dr ≡ Σ_κ P_κ(X) 〈j(X)〉_κ
≡ Σ_κ ∫w_κ(r) j_κ(r; X) dr ≡ ∫j(r; X) dr.

(58)

Here, N(X) = Σ_κ N_κ(X) stands for the global number of electrons in reactant X,

P_κ(X) = N_κ(X)/N(X)

(59)

denotes the site probability in X, and the fragment and local weighting factors obey the usual (internal) normalizations for each reactant:

Σ_κ P_κ(X) = Σ_κ ∫w_κ(r; X) dr = 1.

(60)

The corresponding quantities in the whole reactive system then result from weighting these substrate averages with their probabilities in R as a whole:

{P_X(R) = N(X)/N(R)},   N(R) = Σ_X N(X);

(61)

〈k(R)〉 = Σ_X P_X(R) 〈k(X)〉,  〈V(R)〉 = Σ_X P_X(R) 〈V(X)〉,  〈j(R)〉 = Σ_X P_X(R) 〈j(X)〉.

(62)

6. Overall Charge Transfer

Let us now briefly examine some energetic consequences of such displacements in electron populations on these functional sites of the acidic and basic subsystems [79,80,81,82,83,84,85]. The known (external) CT action of these substrates in the reactive complex R, the net inflow of electrons to A and an outflow from B, suggests the use of the “biased” estimates of the chemical potentials of the equilibrium reactant subsystems

A⁺ = (a_A⁺ ¦ b_A⁺)   and   B⁺ = (a_B⁺ ¦ b_B⁺)

in the polarized reactive complex R_α⁺, α = (c, HSAB), defining its substrate-resolution. It combines the internally-open but mutually-closed reactants in presence of each other:

R_α⁺ = [A⁺(α)|B⁺(α)] = [a_A⁺(α) ¦ b_A⁺(α) | a_B⁺(α) ¦ b_B⁺(α)].

(63)

Each coordination α defines its specific external potential due to the fixed nuclei in both subsystems,

v(α) = v_A(α) + v_B(α),

(64)

which is constrained in definitions of the substrate populational derivatives of R_α⁺: the reactant chemical potentials and hardness descriptors.

The chemical potentials (negative electronegativities) of reactant subsystems in R_α⁺, μ(R_α⁺) = {μ_α(X⁺)}, represent partial derivatives of the system energy E_α⁺[N(R_α⁺)}, v(α)] with respect to the substrate electronic populations N(R_α⁺) = {N_α(X⁺)} for the fixed external potential v(α), i.e., the “frozen” geometry of the whole reactive system,

μ_α(X⁺) ≡ ∂E_α⁺[N(R_α), v(α)]/∂N_α(X⁺)]_v(α),   α = (c, HSAB),   X⁺ ∈ (A⁺, B⁺).

(65)

These population “potentials” also reflect the subsystem electronegativities,

χ_α(X⁺) ≡ ∂E_α⁺[Q(R_α⁺), v(α)]/∂Q_α(X⁺)]_v(α) = −μ_α(X⁺),

(66)

measuring the related partials of electronic energy E_α⁺[Q(R_α⁺), v(α)] with respect to the reactant net-charges Q(R_α⁺) = {Q_α(X⁺)}, dQ_α(X⁺) = −dN_α(X⁺).

The internally-equalized chemical potentials

μ(R_α⁺) = [∂E_α⁺/∂N(R_α⁺)] _v(α) = {μ_α(X⁺) = ∂E_α⁺/∂N_α(X⁺)}

of the basic and acidic reactants, when acting as donor (B) and acceptor (A) of electrons, respectively, then read:

μ_α(A⁺) = μ_α(a_A⁺) = μ_α(b_A⁺) = −I_A⁺   and
  μ_α(B⁺) = μ_α(a_B⁺) = μ_α(b_B⁺) = −A_B⁺,

(67)

where I_X⁺ and A_X⁺ denote the ionization potential and electron affinity of X⁺, respectively. These biased substrate descriptors apply to both R_c⁺ and R_HSAB⁺ complexes. Indeed, in the complementary arrangement, the amount of the first partial charge transfer dominates the second one (see Figure 1), N(CT₁) > N(CT₂), so that B⁺ net donates and A⁺ accepts electrons.

The optimum amount of the resultant CT between A⁺ and B⁺ substrates in R_α⁺,

N_α(CT) = N_α(A^*) − N_α(A⁰) = N_α(B⁰) − N_α(B^*) > 0,

(68)

where {N_α(X⁰)} denote electron numbers in separate reactants and {N(X^*)} stand for the average electron populations in the coordination final, equilibrium reactive system with the mutually-open subsystems,

R_α^* = [A^*(α) ¦ B^*(α)] = [a_A^*(α) ¦ b_A^*(α) ¦ a_B^*(α) ¦ b_B^*(α)],

(69)

is determined by the corresponding in situ CT “force”, the effective chemical-potential for this process (populational “gradient”), measuring the difference between chemical potentials of the polarized acidic and basic reactants,

μ_α(CT) = ∂E_α⁺[N_α(CT), v(α)]/∂N_α(CT) = μ_α(A⁺) − μ_α(B⁺) < 0,

(70)

and the coordination in situ hardness descriptor η(CT) for this electron transfer,

η_α(CT) = ∂μ_α(CT)/∂N_α(CT)
= η_α(A⁺,A⁺) − η_α(A⁺,B⁺) + η_α(B⁺,B⁺) − η_α(B⁺,A⁺) > 0,

(71)

representing the effective CT-hardness (populational “Hessian”). Here, the elements of the hardness tensor in reactant resolution measure the second populational partials

η(R_α⁺) = [∂²E_α⁺/∂N(R_α⁺) ∂N(R_α⁺)]_v(α) = [∂μ(R_α⁺)/∂N(R_α⁺)]_v(α) = {η_α(X⁺,Y⁺)},

η_α(X⁺,Y⁺) ≡ {∂²E_α⁺[{N(R_α⁺)}, v(α)]/∂N_α(X⁺) ∂N_α(Y⁺)}_v(α)
       = [∂μ_α(X⁺)/∂N_α(Y⁺)]_v(α),  X⁺, Y⁺ ∈ (A⁺, B⁺).

(72)

Since the acidic (basic) reactants are identified by their chemically hard (soft) character, as substrates of relatively small (high) polarizability, their chemical potentials obey the following inequality: μ_α(A⁺) < μ_α(B⁺) < 0.

Finally, the coordination equilibrium amount of the inter-reactant CT [79,80,82],

N_α(CT) = −μ_α(CT)/η_α(CT),

(73)

generates the associated 2^nd-order stabilization energy due to CT,

E_α(CT) = μ_α(CT) N_α(CT)/2 = − [μ_α(CT)]²/[2η_α(CT)] < 0.

(74)

It should be emphasized that this energy estimate already contains the implicit polarization contributions, due to internal charge adjustments within each internally-open reactant. Indeed, all spontaneous electron flows, of both of CT and P origins, stabilize the system.

7. Partial Electronic Flows

It is also of interest to establish energy contributions due to all partial electronic flows shown in the flux patterns in Equation (48), of both P and CT origins. As observed above, all such spontaneous responses of the reactive system contribute the stabilizing (negative) contributions to the reaction energy. The P-currents represent the intra-reactant responses created by a presence of the other substrate and the external CT.

The P/CT resolved flow patterns in Equation (48) and the associated energy terms call for the site-resolution of the polarized reactive system, now corresponding to the mutually-closed reaction sites on the internally- and mutually-closed reactants A = (a_A | b_A) and B = (a_B | b_B) (see Figure 1), for the external potential v(α) of Equation (64),

R_α = [A(α) | B(α)] = [a_A(α) | b_A(α) | a_B(α) | b_B(α)].

(75)

Their functional fragments, the acidic and basic sites in both reactants,

λ(R_α) = {λ_α(X) ≡ [a_X(α), b_X(α)]}
≡ {a_A(α), b_A(α), a_B(α), b_B(α)} ≡ {λ₁, λ₂, λ₃, λ₄},

(76)

which contain n(R_α) = {n_λ(α)} electrons, are then characterized by different levels of the site chemical potentials

u(R_α) = ∂E_α[n(R_α); v(α)]/∂n = {u_α(λ) = ∂E_α/∂n_λ ≡ u_λ},

(77)

measured by the partial energy derivatives with respect to average electron populations:

n(R_α) = {n_α(λ) ≡ n_λ} = {n_α(X) = [n_α(a_X), n_α(b_X)]}.

(78)

The site-hardness descriptors are similarly defined by the corresponding matrix of the second populational derivatives of the energy in this resolution level:

h(R_α) = {h_λ,λ’ = ∂²E_α/∂n_λ ∂n_λ’ = ∂u_α(λ)/∂n_λ’}.

(79)

In this more resolved perspective, one also applies the biased estimates of the fragment chemical potentials, measured either by the site negative ionization potential (u_λ = −I_λ) or its negative electron affinity (u_λ = −A_λ), when this fragment acts as an external electron donor or acceptor, respectively.

A reference to Figure 1 and Equation (48) shows that the given site λ can act in the following three ways:

       (1) As internal (P) and external (CT) donor of electrons, e.g., b_B site in R_HSAB;
(2) As internal and external acceptor, e.g., a_A site in R_HSAB;
        (3) As flow-through fragment, e.g., all sites in R_c and (a_B, b_A) parts of R_HSAB.

(80)

In the latter category the inflow/outflow currents, either of CT or P origins, compete with one another, thus minimizing the site net electron accumulation or depletion. The corresponding chemical potentials for these types of behavior, the biased measures for Groups (1) and (2) and the unbiased estimates for Group (3), then read:

u_α(1) = −I_λ,  u_α(2) = −A_λ,  and  u_α(3) = −(I_λ + A_λ)/2.

(81)

These chemical potentials are expected to display the following hierarchy reflecting the site softnesses (polarizabilities):

u_α(a_A) < u_α(a_B) < u_α(b_A) < u_α(b_B) < 0.

(82)

The sum of the associated first-order changes in electronic energy, {ΔE_α⁽¹⁾(λ)}, of the site energies {E_α(λ)} following displacements {Δn_λ} in their electron populations, then generates the following overall energy displacement:

ΔE_α⁽¹⁾ = Σ_λ u_α(λ) Δn_λ ≡ Σ_λ ΔE_α⁽¹⁾(λ).

(83)

One next observes that the population displacements are large in Groups (1) and (2) of the list (80), where the P and CT displacement enhance one another,

N_α^P(λ) + N_α^CT(λ) ≡ Δ_α(λ),

(84)

and small in Group (3), when they cancel one another:

N_α^P(λ) + N_α^CT(λ) ≡ δ_α(λ).

(85)

Therefore, the energy change in R_HSAB is determined by two large population displacements Δ_HSAB(λ), due to charge activations of the key sites a_A (strongly acidic) and b_B (strongly basic), and two remaining (small) flow-through displacements δ_HSAB(λ) of the mixed-character fragments a_B and b_A. In R_c, the collective charge displacement includes four flow-through δ_c(λ) contributions reflecting the charge activation on all sites. This observation further justifies the complementary preference in such donor-acceptor coordinations [94,95].

Consider now the second-order energetic consequences of the partial charge transfer (CT) flows, shown in the diagrams of Equation (48). A reference to Equations (70)–(74), (77), and (79) indicates that in R_c the dominating CT₁ process b_B(λ₄)→a_A(λ₁) is described by the following in situ gradient and Hessian descriptors:

  u_c(CT₁) = u_c(a_A) − u_c(b_B) = u₁ − u₄  and
h_c(CT₁) = h_1,1 − h_1,4 + h_4,4 − h_4,1.

(86)

They predict the optimum amount of this partial inter-reactant population displacement,

N_c(CT₁) = − u_c(CT₁)/h_c(CT₁),

(87)

and the associated stabilization energy:

E_c(CT₁) = − [u_c(CT₁)]²/[2h_c(CT₁)].

(88)

The associated chemical potential and hardness descriptors for the CT₂ process b_A(λ₂)→a_B(λ₃) in R_c accordingly read:

u_c(CT₂) = u_c(a_B) − u_c(b_A) = u₃(c) − u₂(c)   and
h_c(CT₂) = h_3,3(c) − h_2,3(c) + h_2,2(c) − h_3,2(c).

(89)

They determine the optimum amount of CT₂ and the associated stabilization energy:

N_c(CT₂) = −u_c(CT₂)/h_c(CT₂),   E_c(CT₂) = − [u_c(CT₂)]²/[2h_c(CT₂)].

(90)

Consider now the associated inter-reactant fluxes in R_HSAB:

CT₁: b_B(λ₄)→b_A(λ₂)   and   CT₂: a_B(λ₃)→a_A(λ₁).

The first of these transfers of electrons generates the following in situ descriptors:

u_HSAB(CT₁) = u_HSAB(b_A) − u_HSAB(b_B) ≡ u₂(HSAB) − u₄(HSAB)   and
h_HSAB(CT₁) = h_2,2(HSAB) − h_2,4(HSAB) + h_4,4(HSAB) − h_4,2(HSAB).

(91)

They predict the following population displacement and energy change:

N_HSAB(CT₁) = −u_HSAB(CT₁)/h_HSAB(CT₁)     and
E_HSAB(CT₁) = −[u_HSAB(CT₁)]²/[2h_HSAB(CT₁)].

(92)

For the second CT₂ displacement in R_HSAB, one similarly finds the relevant chemical in situ gradient and Hessian descriptors,

u_HSAB(CT₂) = u_HSAB(a_A) − u_HSAB(a_B) ≡ u₁(HSAB) − u₃(HSAB),

h_HSAB(CT₂) = h_1,1(HSAB) − h_1,3(HSAB) + h_3,3(HSAB) − h_3,1(HSAB),

(93)

and the resulting population and energy displacements:

N_HSAB(CT₂) = −u_HSAB(CT₂)/h_HSAB(CT₂),   E_HSAB(CT₂) = −[u_HSAB(CT₂)]²/[2h_HSAB(CT₂)].

(94)

These (primary) CT displacements in reactive complexes can be regarded as perturbations creating conditions for subsequent (secondary) polarization (P) responses in reactants.

In R_c, the initial charge transfers modify the site chemical potentials, initially equalized in the equilibrium reactants [see Equation (67)],

λ₁:  δu_c(a_A) = [h_1,1(c) − h_1,4(c)] N_c(CT₁) + [h_1,3(c) − h_1,2(c)] N_c(CT₂), 
λ₂:  δu_c(b_A) = [h_2,1(c) − h_2,4(c)] N_c(CT₁) + [h_2,3(c) − h_2,2(c)] N_c(CT₂),
λ₃:  δu_c(a_B) = [h_3,1(c) − h_3,4(c)] N_c(CT₁) + [h_3,3(c) − h_3,2(c)] N_c(CT₂),
λ₄:  δu_c(b_B) = [h_4,1(c) − h_4,4(c)] N_c(CT₁) + [h_4,3(c) − h_4,2(c)] N_c(CT₂).

(95)

The corresponding displacements in the regional HSAB coordination read:

λ₁: δu_HSAB(a_A) = [h_1,2(HSAB) − h_1,4(HSAB)] N_HSAB(CT₁)
        + [h_1,1(HSAB) − h_1,3(HSAB)] N_HSAB(CT₂),
λ₂: δu_HSAB(b_A) = [h_2,2(HSAB) − h_2,4(HSAB)] N_HSAB(CT₁)
        + [h_2,1(HSAB) − h_2,3(HSAB)] N_HSAB(CT₂),
λ₃: δu_HSAB(a_B) = [h_3,2(HSAB) − h_3,4(HSAB)] N_HSAB(CT₁)
        + [h_3,1(HSAB) − h_3,3(HSAB)] N_HSAB(CT₂),
λ₄: δu_HSAB(b_B) = [h_4,2(HSAB) − h_4,4(HSAB)] N_HSAB(CT₁)
        + [h_4,1(HSAB) − h_4,3(HSAB)] N_HSAB(CT₂).

(96)

These shifts in site electronegativities generate the associated in situ populational gradients for the subsequent P relaxation of reactants. For R_c, these internal flows are defined in Figure 1 and the corresponding current pattern in Equation (48). In the complementary complex, one finds:

R_c: u_c(P_A) = δu_c(b_A) − δu_c(a_B),  u_c(P_B) = δu_c(b_B) − δu_c(a_B),

(97)

while, in the HSAB coordination, where the internal P_A and P_B flows define {b_X→a_X} flows in each reactant,

R_HSAB: u_HSAB(P_A) = δu_HSAB(a_A) − δu_HSAB(b_A),
    u_HSAB(P_B) = δu_HSAB(a_B) − δu_HSAB(b_B).

(98)

Finally, to estimate magnitudes of these polarizational relaxations, one applies the following in situ hardness descriptors for the polarizational flows in reactants:

h_α(P_A) = h_1,1(α) − h_1,2(α) + h_2,2(α) − h_2,1(α),
      h_α(P_B) = h_3,3(α) − h_3,4(α) + h_4,4(α) − h_4,3(α),   α = (c, HSAB).

(99)

They ultimately determine the optimum sizes of these polarization transfers in R_α,

N_α(P_X) = −u_α(P_X)/h_α(P_X),

(100)

and the associated polarization-energies:

E_α(P_X) = −[u_α(P_X)]²/[2h_α(P_X)],   α = (c, HSAB),   X = A, B.

(101)

To summarize, the overall stabilization energy in the reactant-resolution [Equation (74)] contains all four partial P/CT contributions in the site-resolution:

E_α(CT) = [E_α(CT₁) + E_α(CT₂)] + Σ_X=A,BE_α(P_X),   α = (c, HSAB).

(102)

8. Communication Considerations

These alternative coordinations in reactive systems can be also qualitatively approached within the communication theory of the chemical bond [9,10,11,22,23,24,25,26,27,28,29,30,31,32,62]. To directly connect to the discussion of the preceding section, one again adopts the site-resolution on both reactants [see Equation (76)], in which the system communication (information) channel is defined by the network of conditional probabilities of observing in R_α the “output” (“receiver”) fragment λ’, given the “input” (“source”) site λ:

P_α(λ’|λ) = P_α(λ’, λ)/P_α(λ) ≡ P_α(λ→λ’),   Σ_λ’ P_α(λ’|λ) = 1.

(103)

Here, P_α(λ) denotes the site-probability and P_α(λ’, λ) stands for the joint-probability of the occurrence of the two-site event in the bond system of R_α. They must satisfy the usual normalizations:

Σ_λ [Σ_λ’ P_α(λ’, λ)] = Σ_λ P_α(λ) = 1.

(104)

The key CT sites in both reactants determine the overall chemical character of these subsystems in reactive complexes: the acceptor (acidic) site in A, a_A, and the donor (basic) site in B, b_B. These crucial fragments are accentuated by the bold symbols in Figure 3, where the dominating communications between the nearest neighbors in both coordinations are shown. The R_c diagram in the Figure 3a shows that only the complementary arrangement exhibits the direct communication b_B→a_A, reflected by a high conditional probability P_c(b_B→a_A), besides the double-cascade propagation b_B→[a_B→b_A]→a_A of a relatively low probability,

P_c(b_B→[a_B→b_A]→a_A) = P_c(b_B→a_B) P_c(a_B→b_A) P_c(b_A→a_A) << P_c(b_B→a_A).

(105)

The R_HSAB coordination Figure 3b generates two indirect (single-cascade) scatterings between the crucial sites a_A and b_B:

P_HSAB(b_B→a_B→a_A) = P_HSAB(b_B→a_B) P_HSAB(a_B→a_A)  and
P_HSAB(b_B→b_A→a_A) = P_HSAB(b_B→b_A) P_HSAB(b_A→a_A).

(106)

As indicated above, in communication theory the indirect scatterings involve products of the relevant direct propagations. Therefore, the conditional probability of the direct step b_B→a_A in R_c must dominate over all indirect communications between these sites. This provides additional rationale for the observed complementary preference in the acid-base coordination.

Moreover, the soft (basic) fragment is predicted to overlap with the neighboring sites more strongly compared to the hard (acidic) fragment, thus giving rise to stronger electron communications:

P_α(b_X→λ) > P_α(a_X→λ)   and   P_α(λ→b_X) > P_α(λ→a_X).

(107)

The strongest communications are thus expected for most covalently-bonded neighboring fragments b_A and b_B in the HSAB complex, while the weakest propagations are predicted between its two hard (acidic) sites a_A and a_B, which bind more ionically:

P(b_B→b_A) >> P(a_B→a_A).

(108)

In R_HSAB, this qualitative analysis thus suggests a covalent character of the chemical bond between the basic sites of both reactants, and the ionic bond between their acidic groups. In R_c, one similarly predicts a strong coordination bond between a_A and b_B, and a predominantly covalent bond between a_B and b_A.

The current density j(r) [see Equation (37)] reflects the flow of electronic probability density p(r) with an effective velocity measuring the current-per-particle: V(r) = j(r)/p(r). It implies the associated flux of the system resultant gradient-information, J_I(r) = I(r)V(r). Here, I(r) = I_p(r) + I_ϕ(r) denotes the density-per-electron of the overall information combining the classical contribution I_p(r) = [∇lnp(r)]² and its nonclassical supplement I_ϕ(r) = [2∇ϕ(r)]² due to the state phase component [see Equation (43)]. The information continuity equation then predicts the vanishing classical contribution to the resultant information source and a finite production of its nonclassical part [62,63,64,67,73].

The shifts in electron populations on active sites of donor–acceptor systems [Equation (48)] also imply the associated redistributions of the system resultant gradient-information. The concerted flows in R_c redistribute the information density more evenly, compared to a more localized redistribution in R_HSAB, where electronic flows net transport the information between the key sites of both reactants: from b_B to a_A. The complementary coordination thus corresponds to a lower level of the overall determinicity-information [see Equation (53)] compared to that in HSAB-type coordination. Therefore, the former reactive system represents more information uncertainty in comparison to the latter complex, thus exhibiting a higher resultant gradient-entropy (indeterminicity-information).

9. Conclusions

To paraphrase Progogine [96], the classical, modulus component of molecular wavefunction determines the electronic probability distribution—the state structure of “being”—while the gradient of the nonclassical phase variable generates the current density—the system structure of “becoming”. In this work, we have explored in some detail the mutual relation between the phase component of electronic wavefunction and its current descriptor, expressing the former as an indefinite integral over the probability velocity field. The probability and current descriptors of electronic states are both related by the quantum continuity relations implied by SE of molecular QM, which have also been summarized. The optimum distribution of the state electronic density is determined by density variational principle for the system electronic energy, while the equilibrium current density results from the subsidiary extremum rule for the nonclassical entropy or information measure, which determines the state optimum phase for the given electron density. The elementary chemical processes have also been monitored using the classical entropy/information descriptors [97,98,99,100].

In phenomenological approaches to chemical reactivity, it is customary to distinguish the mutually bonded (open, entangled) and nonbonded (closed, disentangled) status of the of electronic distributions in subsystems. The former reflects the quantum state of the whole reactive system, while the latter refers to states of the mutually closed reactants. The bonded or nonbonded/frozen character of molecular fragments at such hypothetical reaction stages also delineates the allowed types of electronic communications between substrates, which are responsible for the interreactant chemical bond: the promolecular reference state R⁰ = (A⁰ | B⁰), describing the “frozen” (molecularly placed) electron distributions in the ground-states of separate reactants {X⁰}, does not allow for any communications between the system constituent AIM or their basis functions; the polarized reactive system R⁺ = (A⁺ | B⁺) opens internal communications within each reactant, while the final equilibrium, fully “relaxed” molecular system R^* = (A^* ¦ B^*) already accounts for all intra- and inter-substrate propagations.

We have also demonstrated that the global equilibrium in R^* establishes a common phase component of the fragment states (see Appendix C). In other words, the bonded status of molecular fragments implies the phase equalization of the effective subsystem states, at the equilibrium phase related to the “molecular” electron distribution, in the interacting system as a whole. Therefore, the bonding (entangling) of molecular fragments represents the phase-phenomenon reflecting their common (molecular) electronic state, past or present (see Appendix D). This phase equalization is independent of the actual distance between subsystems, thus representing a long-range correlation effect.

The coherence of electronic currents has also been shown to play a crucial role in establishing the energetic preference of the least-activated complementary reactive complex combining the donor and acceptor substrates. Alternative coordinations of such reactants in R_c and R_HSAB complexes, respectively, imply different patterns of the dominating electronic currents and communications. The specific redistributions of the system electronic and information densities, via P and/or CT, channels, have been qualitatively discussed and the resultant pattern of the site currents have been established. We have also examined energetic implications of the overall and partial P/CT electron flows in A---B complexes using the chemical potential (electronegativity) and hardness/softness descriptors of reactants and their active sites, defined in the DFT based reactivity theory.

The phase equalization in the mutually open subsystems provides a consistent theoretical framework for distinguishing the mutually bonded and nonbonded states of their electron distributions. This IT phase criterion of electronic equilibria in molecular systems complements the familiar energetic principle of the chemical-potential (electronegativity) equalization. The complete set of requirements for the quantum equilibrium in the bonded reactive system thus calls for equalizations of both the modulus (probability) and phase (current) related (local) intensities, the chemical potentials and phases of molecular substrates, at the corresponding global descriptors characterizing the reactive system as a whole.

Thus, for the given probability distributions of subsystems, it is the equalized, “molecular” phase of the whole reactive system which marks the bonded (equilibrium) state in the reaction substrates. The phase component of molecular electronic states thus emerges as an important “association” fabric in chemical systems, which “glues” reactants in their “bonded” (entangled) condition. It keeps the “memory” of the present or past interactions between subsystems, and it shapes the probability fluxes between the substrate active sites, which effect the information flows in the reactive system.