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

## Abstract

**:**

## 1. Introduction

## 2. Molecular States and Their Phases

^{1/2}≡ |Ψ(N)|,

^{1/2}= 1.

**Q**

^{(N)}] = 〈

**Q**

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

^{*}(N)]

^{1/2}≡ 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

**(N)|Ψ(N)〉,**

_{Q}**Q**

^{(N)}≡ 1,

_{Ψ}(N) and P

**(N) stand for the state and basis-set projection operators, respectively, while ∫d**

_{Q}**Q**

^{(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.

^{(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)}〉 = |

**q**

_{1},

**q**

_{2}, …,

**q**

_{N}〉 = {|

**q**

_{k}〉 = |σ

_{k},

**r**

_{k}〉}, k = 1, 2, …, N,

_{k}}) and position ({

**r**

_{k}}) variables of all N electrons. It includes the eigenvectors {|

**q**

_{k}〉} of the electronic spin (

**s**

_{k}=

**i**s

_{k}

_{,x}+

**j**s

_{k}

_{,y}+

**k**s

_{k}

_{,z}, s

_{k}= −|

**s**

_{k}|) and position (

**r**

_{k}) operators:

**s**

_{k}

^{2}|

**q**

_{k}〉 = ¾ ħ

^{2}|

**q**

_{k}〉 ≡ s

_{k}

^{2}|

**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}〉.

_{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:

_{w}〉 ≡ |ξ

_{w}〉|φ

_{w}〉.

**q**-representation,

_{w}(

**q**) = 〈

**q**|ψ

_{w}〉 = 〈σ|ξ

_{w}〉 〈

**r**|φ

_{w}〉 ≡ ξ

_{w}(σ) φ

_{w}(

**r**),

_{w}(σ) = 〈σ|ξ

_{w}〉, |ξ

_{w}| = (Σ

_{σ}|ξ

_{w}(σ)|

^{2})

^{1/2}= 1,

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

_{w}(

**r**) = 〈

**r**|φ

_{w}〉 = d

_{w}(

**r**) exp[iϕ

_{w}(

**r**)] ≡ d

_{w}(

**r**) f

_{w}(

**r**),

_{w}(

**r**)

^{*}φ

_{w}(

**r**) d

**r**= 1 or |φ

_{w}| = 1,

_{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:

_{v}|φ

_{w}〉 = ∫〈φ

_{v}|

**r**〉〈

**r**|φ

_{w}〉 d

**r**= ∫φ

_{v}(

**r**)

^{*}φ

_{w}(

**r**) d

**r**= δ

_{k}

_{,l}.

_{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}〉,

_{w}(

**r**) = |φ

_{w}(

**r**)|

^{2}= d

_{w}(

**r**)

^{2}≥ 0, ∫p

_{w}(

**r**) d

**r**= 1,

_{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

_{w}〉 = |φ

_{w}〉/|φ

_{w}| = |φ

_{w}〉, |φ

_{w}| = [∫|φ

_{w}(

**r**)|

^{2}d

**r**]

^{1/2}≡ 1,

_{w}(

**r**) = 〈

**r**|d

_{w}〉 = exp[iϕ

_{w}(

**r**)] = cosϕ

_{w}(

**r**) + i sinϕ

_{w}(

**r**).

**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**).

**ψ**= (ψ

_{1}, ψ

_{2}, …, ψ

_{N}),

**Q**

^{(N)}|Ψ〉 = (N!)

^{−1/2}|ψ

_{1}(Ψ)ψ

_{2}(Ψ) … ψ

_{N}(Ψ)| ≡ det[

**ψ**(Ψ)].

## 3. Electronic Communications

**χ**〉 = {|χ

_{j}〉, j = 1, 2, …, t},

_{i}|χ

_{j}〉 = ∫〈χ

_{i}|

**r**〉 〈

**r**|χ

_{j}〉 d

**r**= ∫χ

_{i}(

**r**)

^{*}χ

_{j}(

**r**) d

**r**= δ

_{i}

_{,j}.

**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**〉,

**C**= 〈

**χ**|

**φ**〉 = {C

_{i}

_{,w}}, satisfies the matrix orthonormality relations:

**CC**= {δ

^{†}_{i}

_{,j}} =

**C**= {δ

^{†}C_{w}

_{,w’}} ≡

**I**.

**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**)],

_{j}|χ

_{i}) = P(χ

_{i}, χ

_{j})/P(χ

_{i}) ≡ P(j|i), Σ

_{j}P(j|i) = 1,

_{j}|χ

_{i}) ≡ A(j|i):

^{2}}.

_{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:

_{i}[Σ

_{j}P(i, j)] = Σ

_{i}P(i) = 1.

**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:

**r**)

^{*}ζ(

**r**) d

**r**= ∫m

_{θ}(

**r**) m

_{ζ}(

**r**) exp{i[ϕ

_{ζ}(

**r**)−ϕ

_{θ}(

**r**)]} d

**r**

≡ ∫M(ζ|θ) exp{iΦ(ζ|θ)] d

**r**.

_{θ}(

**r**) m

_{ζ}(

**r**) and the phase component

_{ζ}(

**r**) − ϕ

_{θ}(

**r**).

^{2}= 〈ζ|θ〉 〈θ|ζ〉 ≡ 〈ζ|P

_{θ}|ζ〉

= 〈θ|ζ〉 〈ζ|θ〉 ≡ 〈θ|P

_{ζ}|θ〉,

P

_{ϑ}= |ϑ〉 〈ϑ|, P

_{ϑ}

^{2}= P

_{ϑ}, ϑ = (θ, ζ).

_{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’}},

_{w}(Ψ) = 1 (Ψ-occupied MO) or n

_{w}(Ψ) = 0 (Ψ-virtual MO),

_{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),

_{Ψ}(χ

_{j}|χ

_{i}) ≡ A(χ

_{j}|χ

_{i}‖Ψ) ≡ A

_{Ψ}(j|i):

_{Ψ}(χ

_{j}|χ

_{i}) = P(χ

_{j}|χ

_{i}‖Ψ) ≡ P

_{Ψ}(j|i) = |A

_{Ψ}(j|i)|

^{2}.

_{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)

**γ**(Ψ) = 〈

**χ**|

**Ψ**(Ψ)〉

**n**(Ψ) 〈

**Ψ**(Ψ)|

**χ**〉 =

**C**(Ψ)

**n**(Ψ)

**C**(Ψ)

^{†}= {γ

_{i},

_{j}(Ψ)}.

_{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)〉.

_{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),

_{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

_{j}P

_{b}(Ψ)|i〉 ≡ 〈i|P

_{j}

^{b}(Ψ)|i〉

= 〈j|P

_{b}(Ψ)|i〉 〈i|P

_{b}(Ψ)|j〉 = 〈j|P

_{b}(Ψ)P

_{i}P

_{b}(Ψ)|j〉 ≡ 〈j|P

_{i}

^{b}(Ψ)|j〉.

## 4. Phase–Current Relation

**q**) = φ(

**r**)ξ(σ),

**r**) = m(

**r**) exp[iϕ(

**r**)].

**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**).

**V**(

**r**) = {V

_{u}(

**r**) ≡ V

_{u}(u, {v≠u}), u = x, y, z},

**V**(

**r**) =

**j**(

**r**)/p(

**r**) = (ħ/m) ∇ϕ(

**r**).

_{u}(

**r**) = (ħ/m) [∂ϕ(

**r**)/∂u], u = x, y, z.

**(**

_{K}**r**) = A exp(i

**K**·

**r**) = A exp[i(K

_{x}x + K

_{y}y + K

_{z}z),

**K**= (m/ħ)

**V**,

**V**=

**i**V

_{x}+

**j**V

_{y}+

**k**V

_{z}= const.

**r**) =

**K**·

**r**= (m/ħ) (V

_{x}x + V

_{y}y + V

_{z}z),

**r**) [∇ϕ(

**r**)]

^{2}d

**r**≡ ∫p(

**r**) I

_{ϕ}(

**r**) d

**r**

= (2m/ħ)

^{2}∫p(

**r**) V(

**r**)

^{2}d

**r**= (2m/ħ)

^{2}∫p(

**r**)

^{−1}j(

**r**)

^{2}d

**r**.

**K**(

**r**) = (m/ħ)

**V**(

**r**) = ∇ϕ(

**r**).

**r**)

**K**(

**r**)

^{2}d

**r**≡ 4〈K

^{2}〉.

^{2}〉 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

_{B}|…| b

_{B}) ≡ (a

_{B}|b

_{B})

_{A}|…| b

_{A}) ≡ (a

_{A}|b

_{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.

_{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:

_{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}.

_{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.

_{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.

_{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**) d

**r**≡

**j**(λ), ∫p

_{λ}(

**r**) d

**r**= 1,

**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,

_{λ}= N

_{λ}〈

**j**〉

_{λ},

_{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:

_{λ}|ΔN

_{λ}| = Σ

_{λ}N

_{λ}|〈

**j**〉

_{λ}| ≡ Σ

_{λ}N

_{λ}〈j〉

_{λ}⇒ minimum.

_{λ}[ϕ

_{λ}] = 4∫p

_{λ}(

**r**) [∇ϕ

_{λ}(

**r**)]

^{2}d

**r**

= (2m/ħ)

^{2}∫p

_{λ}(

**r**) V

_{λ}(

**r**)

^{2}d

**r**

= (2m/ħ)

^{2}∫p

_{λ}(

**r**)

^{−1}j

_{λ}(

**r**)

^{2}d

**r**.

**ϕ**] = Σ

_{λ}I

_{λ}[ϕ

_{λ}] ⇒ minimum.

^{*}= (A

^{*}¦ B

^{*}) = (a

_{A}

^{*}¦ b

_{A}

^{*}¦ a

_{B}

^{*}¦ b

_{B}

^{*}),

_{λ}(

**r**)} = {p

_{κ}(

**r**; X) = ρ

_{κ}(

**r**; X)/N

_{κ}(X)}, ∫p

_{κ}(

**r**; X) d

**r**= 1,

_{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)}

**k**(X)〉 = Σ

_{κ}P

_{κ}(X) ∫p

_{κ}(

**r**; X)

**k**

_{κ}(

**r**; X) d

**r**≡ Σ

_{κ}P

_{κ}(X) 〈

**k**(X)〉

_{κ}

≡ Σ

_{κ}∫w

_{κ}(

**r**; X)

**k**

_{κ}(

**r**; X) d

**r**≡ ∫

**k**(

**r**; X) d

**r**,

**V**(X)〉 = Σ

_{κ}P

_{κ}(X) ∫p

_{κ}(

**r**; X)

**V**

_{κ}(

**r**; X) d

**r**≡ Σ

_{κ}P

_{κ}(X) 〈

**V**(X)〉

_{κ}

≡ Σ

_{κ}∫w

_{κ}(

**r**; X)

**V**

_{κ}(

**r**; X) d

**r**≡ ∫

**V**(

**r**; X) d

**r**,

**j**(X)〉 = Σ

_{κ}P

_{κ}(X) ∫p

_{κ}(

**r**; X)

**j**

_{κ}(

**r**; X) d

**r**≡ Σ

_{κ}P

_{κ}(X) 〈

**j**(X)〉

_{κ}

≡ Σ

_{κ}∫w

_{κ}(

**r**)

**j**

_{κ}(

**r**; X) d

**r**≡ ∫

**j**(

**r**; X) d

**r**.

_{κ}N

_{κ}(X) stands for the global number of electrons in reactant X,

_{κ}(X) = N

_{κ}(X)/N(X)

_{κ}P

_{κ}(X) = Σ

_{κ}∫w

_{κ}(

**r**; X) d

**r**= 1.

_{X}(R) = N(X)/N(R)}, N(R) = Σ

_{X}N(X);

**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)〉.

## 6. Overall Charge Transfer

^{+}= (a

_{A}

^{+}¦ b

_{A}

^{+}) and B

^{+}= (a

_{B}

^{+}¦ b

_{B}

^{+})

_{α}

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

_{α}

^{+}= [A

^{+}(α)|B

^{+}(α)] = [a

_{A}

^{+}(α) ¦ b

_{A}

^{+}(α) | a

_{B}

^{+}(α) ¦ b

_{B}

^{+}(α)].

_{A}(α) + v

_{B}(α),

_{α}

^{+}: the reactant chemical potentials and hardness descriptors.

_{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,

_{α}(X

^{+}) ≡ ∂E

_{α}

^{+}[

**N**(R

_{α}), v(α)]/∂N

_{α}(X

^{+})]

_{v}

_{(α)}, α = (c, HSAB), X

^{+}∈ (A

^{+}, B

^{+}).

_{α}(X

^{+}) ≡ ∂E

_{α}

^{+}[

**Q**(R

_{α}

^{+}), v(α)]/∂Q

_{α}(X

^{+})]

_{v}

_{(α)}= −μ

_{α}(X

^{+}),

_{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

^{+}).

**μ**(R

_{α}

^{+}) = [∂E

_{α}

^{+}/∂

**N**(R

_{α}

^{+})]

_{v}

_{(α)}= {μ

_{α}(X

^{+}) = ∂E

_{α}

^{+}/∂N

_{α}(X

^{+})}

_{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}

^{+},

_{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(CT

_{1}) > N(CT

_{2}), so that B

^{+}net donates and A

^{+}accepts electrons.

^{+}and B

^{+}substrates in R

_{α}

^{+},

_{α}(CT) = N

_{α}(A

^{*}) − N

_{α}(A

^{0}) = N

_{α}(B

^{0}) − N

_{α}(B

^{*}) > 0,

_{α}(X

^{0})} 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,

_{α}

^{*}= [A

^{*}(α) ¦ B

^{*}(α)] = [a

_{A}

^{*}(α) ¦ b

_{A}

^{*}(α) ¦ a

_{B}

^{*}(α) ¦ b

_{B}

^{*}(α)],

_{α}(CT) = ∂E

_{α}

^{+}[N

_{α}(CT), v(α)]/∂N

_{α}(CT) = μ

_{α}(A

^{+}) − μ

_{α}(B

^{+}) < 0,

_{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,

**η**(R

_{α}

^{+}) = [∂

^{2}E

_{α}

^{+}/∂

**N**(R

_{α}

^{+}) ∂

**N**(R

_{α}

^{+})]

_{v}

_{(α)}= [∂

**μ**(R

_{α}

^{+})/∂

**N**(R

_{α}

^{+})]

_{v}

_{(α)}= {η

_{α}(X

^{+},Y

^{+})},

_{α}(X

^{+},Y

^{+}) ≡ {∂

^{2}E

_{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

^{+}).

_{α}(A

^{+}) < μ

_{α}(B

^{+}) < 0.

_{α}(CT) = −μ

_{α}(CT)/η

_{α}(CT),

^{nd}-order stabilization energy due to CT,

_{α}(CT) = μ

_{α}(CT) N

_{α}(CT)/2 = − [μ

_{α}(CT)]

^{2}/[2η

_{α}(CT)] < 0.

## 7. Partial Electronic Flows

_{A}| b

_{A}) and B = (a

_{B}| b

_{B}) (see Figure 1), for the external potential v(α) of Equation (64),

_{α}= [A(α) | B(α)] = [a

_{A}(α) | b

_{A}(α) | a

_{B}(α) | b

_{B}(α)].

**λ**(R

_{α}) = {

**λ**

_{α}(X) ≡ [a

_{X}(α), b

_{X}(α)]}

≡ {a

_{A}(α), b

_{A}(α), a

_{B}(α), b

_{B}(α)} ≡ {λ

_{1}, λ

_{2}, λ

_{3}, λ

_{4}},

**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}},

**n**(R

_{α}) = {n

_{α}(λ) ≡ n

_{λ}} = {

**n**

_{α}(X) = [n

_{α}(a

_{X}), n

_{α}(b

_{X})]}.

**h**(R

_{α}) = {h

_{λ}

_{,λ’}= ∂

^{2}E

_{α}/∂n

_{λ}∂n

_{λ}

_{’}= ∂u

_{α}(λ)/∂n

_{λ}

_{’}}.

_{λ}= −I

_{λ}) or its negative electron affinity (u

_{λ}= −A

_{λ}), when this fragment acts as an external electron donor or acceptor, respectively.

_{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}.

_{α}(1) = −I

_{λ}, u

_{α}(2) = −A

_{λ}, and u

_{α}(3) = −(I

_{λ}+ A

_{λ})/2.

_{α}(a

_{A}) < u

_{α}(a

_{B}) < u

_{α}(b

_{A}) < u

_{α}(b

_{B}) < 0.

_{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:

_{α}

^{(1)}= Σ

_{λ}u

_{α}(λ) Δn

_{λ}≡ Σ

_{λ}ΔE

_{α}

^{(1)}(λ).

_{α}

^{P}(λ) + N

_{α}

^{CT}(λ) ≡ Δ

_{α}(λ),

_{α}

^{P}(λ) + N

_{α}

^{CT}(λ) ≡ δ

_{α}(λ).

_{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].

_{c}the dominating CT

_{1}process b

_{B}(λ

_{4})→a

_{A}(λ

_{1}) is described by the following in situ gradient and Hessian descriptors:

_{c}(CT

_{1}) = u

_{c}(a

_{A}) − u

_{c}(b

_{B}) = u

_{1}− u

_{4}and

h

_{c}(CT

_{1}) = h

_{1,1}− h

_{1,4}+ h

_{4,4}− h

_{4,1}.

_{c}(CT

_{1}) = − u

_{c}(CT

_{1})/h

_{c}(CT

_{1}),

_{c}(CT

_{1}) = − [u

_{c}(CT

_{1})]

^{2}/[2h

_{c}(CT

_{1})].

_{2}process b

_{A}(λ

_{2})→a

_{B}(λ

_{3}) in R

_{c}accordingly read:

_{c}(CT

_{2}) = u

_{c}(a

_{B}) − u

_{c}(b

_{A}) = u

_{3}(c) − u

_{2}(c) and

h

_{c}(CT

_{2}) = h

_{3,3}(c) − h

_{2,3}(c) + h

_{2,2}(c) − h

_{3,2}(c).

_{2}and the associated stabilization energy:

_{c}(CT

_{2}) = −u

_{c}(CT

_{2})/h

_{c}(CT

_{2}), E

_{c}(CT

_{2}) = − [u

_{c}(CT

_{2})]

^{2}/[2h

_{c}(CT

_{2})].

_{HSAB}:

_{1}: b

_{B}(λ

_{4})→b

_{A}(λ

_{2}) and CT

_{2}: a

_{B}(λ

_{3})→a

_{A}(λ

_{1}).

_{HSAB}(CT

_{1}) = u

_{HSAB}(b

_{A}) − u

_{HSAB}(b

_{B}) ≡ u

_{2}(HSAB) − u

_{4}(HSAB) and

h

_{HSAB}(CT

_{1}) = h

_{2,2}(HSAB) − h

_{2,4}(HSAB) + h

_{4,4}(HSAB) − h

_{4,2}(HSAB).

_{HSAB}(CT

_{1}) = −u

_{HSAB}(CT

_{1})/h

_{HSAB}(CT

_{1}) and

E

_{HSAB}(CT

_{1}) = −[u

_{HSAB}(CT

_{1})]

^{2}/[2h

_{HSAB}(CT

_{1})].

_{2}displacement in R

_{HSAB}, one similarly finds the relevant chemical in situ gradient and Hessian descriptors,

_{HSAB}(CT

_{2}) = u

_{HSAB}(a

_{A}) − u

_{HSAB}(a

_{B}) ≡ u

_{1}(HSAB) − u

_{3}(HSAB),

_{HSAB}(CT

_{2}) = h

_{1,1}(HSAB) − h

_{1,3}(HSAB) + h

_{3,3}(HSAB) − h

_{3,1}(HSAB),

_{HSAB}(CT

_{2}) = −u

_{HSAB}(CT

_{2})/h

_{HSAB}(CT

_{2}), E

_{HSAB}(CT

_{2}) = −[u

_{HSAB}(CT

_{2})]

^{2}/[2h

_{HSAB}(CT

_{2})].

_{c}, the initial charge transfers modify the site chemical potentials, initially equalized in the equilibrium reactants [see Equation (67)],

_{1}: δu

_{c}(a

_{A}) = [h

_{1,1}(c) − h

_{1,4}(c)] N

_{c}(CT

_{1}) + [h

_{1,3}(c) − h

_{1,2}(c)] N

_{c}(CT

_{2}),

λ

_{2}: δu

_{c}(b

_{A}) = [h

_{2,1}(c) − h

_{2,4}(c)] N

_{c}(CT

_{1}) + [h

_{2,3}(c) − h

_{2,2}(c)] N

_{c}(CT

_{2}),

λ

_{3}: δu

_{c}(a

_{B}) = [h

_{3,1}(c) − h

_{3,4}(c)] N

_{c}(CT

_{1}) + [h

_{3,3}(c) − h

_{3,2}(c)] N

_{c}(CT

_{2}),

λ

_{4}: δu

_{c}(b

_{B}) = [h

_{4,1}(c) − h

_{4,4}(c)] N

_{c}(CT

_{1}) + [h

_{4,3}(c) − h

_{4,2}(c)] N

_{c}(CT

_{2}).

_{1}: δu

_{HSAB}(a

_{A}) = [h

_{1,2}(HSAB) − h

_{1,4}(HSAB)] N

_{HSAB}(CT

_{1})

+ [h

_{1,1}(HSAB) − h

_{1,3}(HSAB)] N

_{HSAB}(CT

_{2}),

λ

_{2}: δu

_{HSAB}(b

_{A}) = [h

_{2,2}(HSAB) − h

_{2,4}(HSAB)] N

_{HSAB}(CT

_{1})

+ [h

_{2,1}(HSAB) − h

_{2,3}(HSAB)] N

_{HSAB}(CT

_{2}),

λ

_{3}: δu

_{HSAB}(a

_{B}) = [h

_{3,2}(HSAB) − h

_{3,4}(HSAB)] N

_{HSAB}(CT

_{1})

+ [h

_{3,1}(HSAB) − h

_{3,3}(HSAB)] N

_{HSAB}(CT

_{2}),

λ

_{4}: δu

_{HSAB}(b

_{B}) = [h

_{4,2}(HSAB) − h

_{4,4}(HSAB)] N

_{HSAB}(CT

_{1})

+ [h

_{4,1}(HSAB) − h

_{4,3}(HSAB)] N

_{HSAB}(CT

_{2}).

_{c}, these internal flows are defined in Figure 1 and the corresponding current pattern in Equation (48). In the complementary complex, one finds:

_{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}),

_{A}and P

_{B}flows define {b

_{X}→a

_{X}} flows in each reactant,

_{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}).

_{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).

_{α},

_{α}(P

_{X}) = −u

_{α}(P

_{X})/h

_{α}(P

_{X}),

_{α}(P

_{X}) = −[u

_{α}(P

_{X})]

^{2}/[2h

_{α}(P

_{X})], α = (c, HSAB), X = A, B.

_{α}(CT) = [E

_{α}(CT

_{1}) + E

_{α}(CT

_{2})] + Σ

_{X=A,B}E

_{α}(P

_{X}), α = (c, HSAB).

## 8. Communication Considerations

_{α}the “output” (“receiver”) fragment λ’, given the “input” (“source”) site λ:

_{α}(λ’|λ) = P

_{α}(λ’, λ)/P

_{α}(λ) ≡ P

_{α}(λ→λ’), Σ

_{λ}

_{’}P

_{α}(λ’|λ) = 1.

_{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:

_{λ}[Σ

_{λ}

_{’}P

_{α}(λ’, λ)] = Σ

_{λ}P

_{α}(λ) = 1.

_{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,

_{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}).

_{HSAB}coordination Figure 3b generates two indirect (single-cascade) scatterings between the crucial sites a

_{A}and b

_{B}:

_{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}).

_{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.

_{α}(b

_{X}→λ) > P

_{α}(a

_{X}→λ) and P

_{α}(λ→b

_{X}) > P

_{α}(λ→a

_{X}).

_{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:

_{B}→b

_{A}) >> P(a

_{B}→a

_{A}).

_{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}.

**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].

_{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

^{0}= (A

^{0}| B

^{0}), describing the “frozen” (molecularly placed) electron distributions in the ground-states of separate reactants {X

^{0}}, 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.

^{*}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.

_{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.

## Funding

## Conflicts of Interest

## Appendix A. Continuity Relations Revisited

**r**, t) = 〈

**r**|Ψ(t)〉 = R(

**r**, t) exp[iϕ(

**r**, t)], ϕ(

**r**, t) ≥ 0.

**r**, t) = ψ(

**r**, t)

^{*}ψ(

**r**, t) = R(

**r**, t)

^{2},

**j**(

**r**, t) = [ħ/(2mi)] [ψ(

**r**, t)

^{*}∇ψ(

**r**, t) − ψ(

**r**, t)∇ψ(

**r**, t)

^{*}]

= (ħ/m) p(

**r**, t) ∇ϕ(

**r**, t)

≡ p(

**r**, t)

**V**(

**r**, t).

**V**(

**r**, t) of the probability “fluid” measures the current-per-particle and reflects the state phase-gradient:

**V**(

**r**, t) =

**j**(

**r**, t)/p(

**r**, t) = (ħ/m) ∇ϕ(

**r**, t).

**r**, t) = lnR(

**r**, t) + iϕ(

**r**, t),

**r**

≡ S[p] + i S[ϕ],

^{2}− (i∇ϕ)

^{2}] = 4[(∇lnR)

^{2}+ (∇ϕ)

^{2}] = 4|∇lnψ| and

M = 4[(∇lnR)

^{2}+ (i∇ϕ)

^{2}],

^{2}[(∇lnR)

^{2}+ (∇ϕ)

^{2}] d

**r**= 4∫[(∇R)

^{2}+ (R ∇ϕ)

^{2}] d

**r**≡ I[R] + I[ϕ]

= ∫p[(∇lnp)

^{2}+ 4(∇ϕ)

^{2}] d

**r**= ∫p

^{−1}(∇p)

^{2}d

**r**+ 4∫p (∇ϕ)

^{2}d

**r**≡ I[p] + I[ϕ],

^{2}|ψ〉 + 〈ψ|(i∇ϕ)

^{2}|ψ〉} = I[p] − I[ϕ]

= 4∫[(∇R)

^{2}− (∇ϕ)

^{2}] d

**r**≡ M[R] + M[ϕ]

= ∫p[(∇lnp)

^{2}− 4(∇ϕ)

^{2}] d

**r**= M[p] + M[ϕ].

^{2}/(2m) Δ = [ħ

^{2}/(8m)] I:

^{2}) T[ψ] ≡ σT[ψ].

**r**) due to the “frozen” nuclear frame of the familiar Born–Oppenheimer approximation. The electronic Hamiltonian

**r**) = − [ħ

^{2}/(2m)] Δ + v(

**r**) ≡ T(

**r**) + v(

**r**),

_{0}) ≡ U(τ) = exp(−iħ

^{−1}τ H),

_{0}).

**V**,

^{−1}ΔR − (∇ϕ)

^{2}] − v/ħ.

**V**= − ∇p·

**V**= − ∇·

**j**or

σ

_{p}≡ dp/dt = ∂p/∂t + ∇·

**j**= ∂p/∂t + ∇p ·

**V**= 0.

**j**= ∇p ·

**V**+ p ∇·

**V**= ∇p ·

**V**,

**V**related to ∇

^{2}ϕ = Δϕ:

**V**= (ħ/m) Δϕ = 0 or Δϕ = 0.

**r**)/dt determines the vanishing local probability “source”: σ

_{p}(

**r**) = 0. It measures the time rate of change in an infinitesimal volume element of probability fluid moving with velocity

**V**= d

**r**/dt, while the partial derivative ∂p[

**r**(t), t]/∂t refers to volume element around the fixed point in space. Indeed, separating the explicit time dependence of p(

**r**, t) from its implicit dependence through the particle position

**r**(t), p(

**r**, t) = p[

**r**(t), t], gives:

_{p}(

**r**, t) = ∂p[

**r**(t), t]/∂t + (d

**r**/dt) · ∂p(

**r**, t)/∂

**r**

= ∂p(

**r**, t)/∂t +

**V**(

**r**, t) ·∇p(

**r**, t) = ∂p(

**r**, t)/∂t + ∇·

**j**(

**r**, t) = 0.

**V**of the probability-current

**j**= p

**V**also determines the phase-flux and its divergence:

**J**= ϕ

**V**and ∇·

**J**= ∇ϕ ·

**V**= (ħ/m) (∇ϕ)

^{2}.

_{ϕ}≡ dϕ/dt = ∂ϕ/∂t + ∇·

**J**= ∂ϕ/∂t +

**V**· ∇ϕ ≠ 0.

_{ϕ}= [ħ/(2m)][R

^{−1}ΔR + (∇ϕ)

^{2}] − v/ħ.

**j**= p

**V**is also of interest,

**≡ d**

_{j}**j**/dt = σ

_{p}

**V**+ p (d

**V**/dt) = p (d

**V**/dt),

**= (ħ/m) p d/dt(∇ϕ) = (ħ/m) p ∇(dϕ/dt) = (ħ/m) p ∇σ**

_{j}_{ϕ}.

**= [ħ**

_{j}^{2}/(2m

^{2})] [R ∇

^{3}R − (ΔR) ∇R] − (p/m) ∇v.

## Appendix B. Schrödinger Equation and Wavefunction Components

= Hψ = {−[ħ

^{2}/(2m)] [ΔR + 2i∇R·∇ϕ − R(∇ϕ)

^{2}] + vR} exp(iϕ),

^{−1}ΔR + 2i(∇lnR)·∇ϕ − (∇ϕ)

^{2}] + v/ħ.

**V**·∇lnR,

**j**or σ

_{p}= dp/dt = 0.

^{−1}ΔR − (∇ϕ)

^{2}] − v/ħ.

_{ϕ}= dϕ/dt [Equation (A23)] for its flux definition of Equation (A21), in the phase continuity relation:

**J**+ σ

_{ϕ}.

_{ψ}= 〈ψ|H|ψ〉 = −[ħ

^{2}/(2m)] ∫[RΔR − R

^{2}(∇ϕ)

^{2}] d

**r**+ ∫R

^{2}v d

**r**

= [ħ

^{2}/(2m)] ∫[(∇R)

^{2}+ R

^{2}(∇ϕ)

^{2}]d

**r**+ ∫R

^{2}v d

**r**

**≡**〈T〉

_{ψ}+ 〈V

_{ne}〉

_{ψ},

_{ne}〉

_{ψ}denotes the state average electron-nuclei attraction energy. The phase-dependent part of the average kinetic energy 〈T〉

_{ψ}identically vanishes in the stationary electronic state

_{s}(

**r**, t) = R

_{s}(

**r**) exp[iϕ

_{s}(t)],

_{s}= R

_{s}(

**r**)

^{−1}H(

**r**) R

_{s}(

**r**) = − [ħ

^{2}/(2m)] R

_{s}(

**r**)

^{−1}ΔR

_{s}(

**r**) + v(

**r**) = const.,

_{s}(t) = − (E

_{s}/ħ) t ≡ − ω

_{s}t and hence ∇ϕ

_{s}(t) = 0.

_{R}

^{*}(N,t

_{0}) of the whole externally-closed but internally-open reactive complex R

^{*}= (A

^{*}¦ B

^{*}) at time t

_{0}≡ 0, defined by its overall external potential v = v

_{A}+ v

_{B}and number of electrons N = N

_{A}+ N

_{B}(integer), can be expanded

_{R}

^{*}(N, t

_{0}) = Σ

_{s}C

_{s}(t

_{0}) R

_{s}(N), C

_{s}(t

_{0}) = 〈R

_{s}(N)|Ψ

_{R}

^{*}(N, t

_{0})〉,

_{s}(N) = E

_{s}R

_{s}(N)

_{ne}(N) + U

_{ee}(N),

T(N) = Σ

_{k}T(k), V

_{ne}(N) = Σ

_{k}v(k), U

_{ee}(N) = Σ

_{k}

_{<l}g(k, l).

_{s}(N) denotes the time-independent amplitude (modulus) function of N electrons, while T(N), V

_{ne}(N) and U

_{ee}(N) stand for the quantum operators of electronic kinetic, attraction and repulsion energies, respectively. Given the state Ψ(N, t

_{0}) at the initial time t

_{0}= 0, its form at time τ [see Equations (A13) and (A14)] is determined by the interval evolution operator

^{−1}τ H(N)],

_{0}) = Σ

_{s}C

_{s}(t

_{0}) {exp[iϕ

_{s}(τ)] R

_{s}(N)}

≡ Σ

_{s}C

_{s}(t

_{0}) Ψ

_{s}(N, τ),

_{s}(N, τ) = R

_{s}(N) exp[iϕ

_{s}(τ)] is the full stationary state of N-electrons corresponding to energy E

_{s}.

^{+}= (A

^{+}| B

^{+}), however, involving the mutually- and externally-closed substrates, each subsystem X

^{+}conserves the initial (integer) number of electrons N

_{X}

^{0}so that the interacting-fragment Hamiltonians {H

_{X}(N

_{X}

^{0})} for the overall external potential v of the whole system are well defined, e.g.,

_{A}(N

_{A}

^{0}) = T(N

_{A}

^{0}) + V

_{ne}(N

_{A}

^{0}) + U

_{A}(N

_{A}

^{0}) + U

_{AB}(N

_{A}

^{0}, N

_{B}

^{0}),

U

_{A}(N

_{A}

^{0}) = Σ

_{(k<l)}

_{∈ A}g(k, l), U

_{AB}(N

_{A}

^{0}, N

_{B}

^{0}) = Σ

_{k}

_{∈ A}Σ

_{l}

_{∈ B}g(k, l), etc.

_{X}(N

_{X}

^{0}) φ

_{u}(N

_{X}

^{0}) = E

_{u}(X

^{+}) φ

_{u}(N

_{X}

^{0}), X = A, B,

_{0}of general states in the polarized subsystems:

_{X}

^{+}(N

_{X}

^{0}, t

_{0}) = Σ

_{u}C

_{u}(X

^{+}, t

_{0}) φ

_{u}(N

_{X}

^{0}), C

_{u}(X

^{+}, t

_{0}) = 〈φ

_{u}(N

_{X}

^{0})|Ψ

_{X}

^{+}(N

_{X}

^{0}, t

_{0})〉.

_{X}(τ, N

_{X}

^{0}) = exp[−iħ

^{−1}τ H

_{X}(N

_{X}

^{0})], X = A, B,

_{X}

^{+}(N

_{X}

^{0}, τ) = U

_{X}(τ, N

_{X}

^{0}) Ψ

_{X}

^{+}(N

_{X}

^{0}, t

_{0})

= Σ

_{u}C

_{u}(X

^{+}, t

_{0}) {exp[iϕ

_{u}(X

^{+}, τ)] φ

_{u}(N

_{X}

^{0})}

≡ Σ

_{u}C

_{u}(X

^{+}, t

_{0}) Φ

_{u}(N

_{X}

^{0}, τ),

_{u}(X

^{+},τ) = − [E

_{u}(X

^{+})/ħ] τ ≡ − ω

_{u}(X

^{+})τ

^{+}, while

_{u}(N

_{X}

^{0}, τ) = φ

_{u}(N

_{X}

^{0}) exp[iϕ

_{u}(X

^{+}, τ)]

_{u}(X

^{+}).

_{u}

_{,w}(A

^{+}, B

^{+}; τ) = Φ

_{u}(N

_{A}

^{0}, τ) Φ

_{w}(N

_{B}

^{0}, τ)

= φ

_{u}(N

_{A}

^{0}) φ

_{w}(N

_{B}

^{0}) exp{i[ϕ

_{u}(A

^{+},τ) + ϕ

_{w}(B

^{+},τ)]}

≡ ϑ

_{u}

_{,w}(N

_{A}

^{0}, N

_{B}

^{0}) exp[iθ

_{u}

_{,w}(τ)]},

_{u}

_{,w}(τ) = −ħ

^{−1}[E

_{u}(A

^{+}) + E

_{w}(B

^{+})]τ = −[ω

_{u}(A

^{+}) + ω

_{w}(B

^{+})] τ = −ω

_{u}

_{,w}τ,

^{+}[compare Equation (A39)]:

_{R}

^{+}(N, τ) = Σ

_{u,w}D

_{u,v}(t

_{0}) {exp[iθ

_{u}

_{,w}(τ)]ϑ

_{u}

_{,w}(N

_{A}

^{0}, N

_{B}

^{0})}

= Σ

_{u,w}D

_{u,v}(t

_{0}) Φ

_{u}(N

_{A}

^{0}, τ) Φ

_{w}(N

_{B}

^{0}, τ)

= Σ

_{u,w}D

_{u,v}(t

_{0}) Θ

_{u}

_{,w}(A

^{+}, B

^{+}; τ),

D

_{u,v}(t

_{0}) = 〈ϑ

_{u}

_{,w}(N

_{A}

^{0}, N

_{B}

^{0})|Ψ

_{R}

^{+}(N, t

_{0})〉.

^{*}in R

^{*}= (A

^{*}¦B

^{*}), identified by their partial densities {ρ

_{X}

^{*}= N

_{X}

^{*}p

_{X}

^{*}, N

_{X}

^{*}= ∫ρ

_{X}

^{*}d

**r**(fractional)}, pieces of molecular electron density

_{R}

^{*}= ρ

_{A}

^{*}+ ρ

_{B}

^{*},

_{X}

^{*}} on subsystems gives rise to finite electronic currents on reactants. This current pattern in both substrates or on their acidic and basic sites manifests the valence-state activation of such open fragments, which generates nonvanishing contributions to the associated (nonclassical) entropy/information descriptors.

## Appendix C. Information Principle

**j**

_{α}(

**r**, λ)} and probability distributions {p

_{α}(

**r**, λ)} in active fragments

_{A}=1, b

_{A}= 2, a

_{B}= 3, b

_{B}= 4)

_{α}, α = (c, HSAB), determine the additive site contributions to the resultant (nonclassical) gradient-information [see Equations (43) and (A8)] in R

_{α}:

_{α}[{

**j**

_{α}(λ)}] = Σ

_{λ}I

_{α}

^{λ}[

**j**

_{α}(λ)], I

_{α}

^{λ}[

**j**

_{α}(λ)] = (2m/ħ)

^{2}∫p

_{α}(

**r**, λ)

^{−1}

**j**

_{α}(

**r**, λ)

^{2}d

**r**.

_{α}[{

**j**

_{α}(λ)}] implies the maximum of the complementary nonclassical current-indeterminicity descriptor, of the site gradient “entropy” [Equation (A9)] containing negative nonclassical contribution [62,63]. Consider such a nonclassical information principle subject to the local constraint of preserving the resultant local current in R

_{α},

**j**

_{α}(

**r**) = Σ

_{λ}

**j**

_{α}(

**r**, λ),

_{α}[{

**j**

_{α}(λ)}] − ∫

**ξ**

_{α}(

**r**)·

**j**

_{α}(

**r**) d

**r**} = 0,

**ξ**

_{α}(

**r**) enforces the local constraint of Equation (A53). The Euler equations determining the optimum site-currents then read:

_{α}[{

**j**

_{α}(λ)}]/δ

**j**

_{α}(

**r**, λ) = (8m

^{2}/ħ

^{2}) [

**j**

_{α}(

**r**, λ)/p

_{α}(

**r**, λ)] ≡ (8m

^{2}/ħ

^{2})

**V**

_{α}(

**r**, λ) =

**ξ**

_{α}(

**r**).

**V**

_{α}(

**r**, λ) = [ħ

^{2}/(8m

^{2})]

**ξ**

_{α}(

**r**) ≡

**V**

_{α}

^{eq.}(

**r**).

_{α}(

**r**, λ)} of the site “states” can differ only by a constant, irrelevant in QM, thus containing the same (site “equalized”) local contribution. The minimization of the nonclassical gradient information thus gives rise to the site phase-equalization in the information-equilibrium state of the reactive system as a whole [63].

## Appendix D. Reactant Entanglement

^{*}= (A

^{*}¦ B

^{*}) containing the mutually- and internally-open substrates {X

^{*}}. Therefore, when brought into temporary interaction and then infinitely separated in R

^{*}(∞) = A

^{*}+ B

^{*}, such dissociated fragments can no longer be described by the individual wavefunctions for each reactant, even after the interaction has utterly ceased and the “molecular” Hamiltonian is given by the sum of subsystem Hamiltonians:

**r**) [62,63], ϕ

_{eq.}(

**r**) = − (½) lnp(

**r**), thus predicting the equilibrium current proportional to the negative gradient of probability density [see Equation (37)]:

**j**

_{eq.}(

**r**) = (ħ/m) p(

**r**) ∇ϕ

_{eq.}(

**r**) ≡ p(

**r**)

**V**

_{eq.}(

**r**) = − [ħ/(2m)] ∇p(

**r**).

**r**) of the past (interacting) subsystems at a finite separation between reactants, contained in the equilibrium phase for this “molecular” separated reactive system.

^{*}= (A

^{*}¦ B

^{*}), and polarized, R

^{+}= (A | B), complexes, respectively. For the same probability distribution p

_{R}(

**r**) in both these hypothetical states, the two subsystems are free to exchange electrons in the former, while in the latter this flow of electrons is forbidden. The equilibrium phase of Ψ(R

^{*}) then reflects the negative of lnp

_{R}, ϕ

_{eq.}(R

^{*}) = ϕ

_{eq.}[p

_{R}], while at a finite separation R

_{AB}between reactants ϕ

_{eq.}(R

^{+}) is proportional to the negative of ln[(p

_{A}p

_{B)}

^{1/2}]. Therefore, it is also the phase component which distinguishes the bonded (entangled) state of reactants in R

^{*}from their nonbonded (disentangled) status in R

^{+}, also at finite separations between the two reactants.

_{eq.}[R

^{*}(x

_{AB})], where x

_{AB}denotes the separation coordinate, which is naturally transformed in the dissociation limit R

_{AB}→∞ into the equilibrium phases of reactants,

_{eq.}[R

^{*}(R

_{AB}→∞)]→{ϕ

_{eq.}[A

^{*}(x

_{AB}→ −∞)] or ϕ

_{eq.}[B

^{*}(x

_{AB}→ +∞)]},

_{R}[R

^{*}(R

_{AB}→∞)] → {ρ

_{A}

^{*}(x

_{AB}→ −∞) or ρ

_{B}

^{*}(x

_{AB}→ +∞)}.

## Appendix E. Density Matrices for Interacting Subsystems

^{+}= [A

^{+}(x)|B

^{+}(ξ)], where x and ξ denote their internal coordinates, respectively. For example, in the topological, physical-space partitioning [104] of the molecular electron density, into pieces belonging to separate basins of the physical space, these coordinates describe the disjoint sets of the position variables in these regions of space, while in the functional-space division schemes, e.g., the stockholder partition [21], each of these coordinates explores the whole physical space of the allowed positions of the separate groups of electrons.

^{+}as a whole is assumed to represent an isolated system, it is described by the specific wavefunction Ψ(x, ξ), the pure quantum state of the reactive system. For example, the interacting-fragment Hamiltonians in R

^{+}, {H

_{X}(N

_{X}

^{0})} act on the position variables of N

_{X}

^{0}electrons belonging to X

^{+}. The stationary states of H

_{A}(N

_{A}

^{0}) ≡ H

_{A}(x),

_{A}(x) φ

_{s}(x) = E

_{s}(A

^{+}) φ

_{s}(x),

_{s}Φ

_{s}(ξ) φ

_{s}(x), Φ

_{s}(ξ) = ∫φ

_{s}

^{*}(x) Ψ(x, ξ) dx.

^{+}will act only on variables x: L(A) = L

_{x}. Its average value in the general state of Equation (A62), of the reactive system as a whole, then reads:

_{Ψ}= ∫∫Ψ

^{*}(x, ξ) L

_{x}Ψ(x, ξ) dx dξ

= Σ

_{s}Σ

_{s}

_{’}[∫Φ

_{s}

_{’}

^{*}(ξ) Φ

_{s}(ξ) dξ] [∫φ

_{s}

_{’}

^{*}(x) L

_{x}φ

_{s}(x) dx]

≡ Σ

_{s}Σ

_{s}

_{’}ρ

_{s}

_{,s’}(A) L

_{s}

_{’,s}(A) = tr

_{A}[

**ρ**(A)

**L**(A)].

**ρ**(A) = {ρ

_{s}

_{’,s}} stands for the effective density matrix of subsystem A

^{+}(x), already integrated over coordinates of electrons in the complementary subsystem B

^{+}(ξ). The partial trace of the preceding equation thus enables one to calculate the ensemble average of the subsystem quantity L(A) as if this part of R

^{+}were isolated, being in the effective mixed state defined by subsystem density matrix

**ρ**(A) in representation {φ

_{s}(x)}.

_{Ψ}= ∫∫ρ

_{A}(x, x’) 〈x’|L

_{x}|x〉 dx dx’,

_{x}|x〉 = L

_{x}δ(x’ − x), one obtains the following expression for the subsystem density matrix:

_{A}(x, x’) = ∫Ψ

^{*}(x’, ξ) Ψ(x, ξ) dξ = Σ

_{s}Σ

_{s}

_{’}ρ

_{s}

_{,s’}(A) φ

_{s}

_{’}

^{*}(x’) φ

_{s}(x)

≡ Σ

_{s}Σ

_{s}

_{’}ρ

_{s}

_{,s’}(A) Ω

_{s}

_{’,s}(x, x’) = tr

_{A}[

**ρ**(A)

**Ω**

_{A}(x, x’)].

^{+}} in R

^{+}cannot be described by a single wavefunction of the pure quantum state. They have to instead be characterized by the density matrix reflecting an incoherent mixture of subsystem states, weighted by the ensemble probability factors and corresponding to the substrate mixed quantum state.

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**Figure 1.**The Charge-Transfer (CT) {b

_{X}→a

_{Y}} and Polarizational (P

_{X}) {a

_{X}→b

_{X}} electron flows involving 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) sites in reactive complex R = (A|B). These electronic flows are seen to produce an effective concerted (circular) flux of electrons in the equilibrium reactive system R

^{*}= (A

^{*}¦ B

^{*}) = (a

_{A}

^{*}¦b

_{A}

^{*}¦a

_{B}

^{*}¦b

_{B}

^{*}) as a whole, with all fragments exhibiting the “flow-through” current pattern, which precludes an exaggerated depletion or accumulation of electrons on any site in reactive system.

**Figure 2.**Qualitative diagrams of the conrotatory pattern of the resultant site-currents {

**j**(λ)} in the complementary complex R

_{c}(

**a**) and their translational pattern in the R

_{HSAB}arrangement (

**b**) of the acidic A = (a

_{A}|b

_{A}) and basic B = (a

_{B}|b

_{B}) reactants.

**Figure 3.**The dominating (nearest neighbor) communications between the chemically bonded sites of the Acid–Base complexes R

_{c}(

**a**) and R

_{HSAB}(

**b**), respectively. The former involves a strong (direct)

**b**→

_{B}**a**propagation between the key sites

_{A}**a**and

_{A}**b**, which determine the overall chemical behavior of reactants, and a weak (intermediate) double-bridge communication

_{B}**b**→[a

_{B}_{B}→b

_{A}]→

**a**, while the latter exhibits only two indirect (single-bridge) communications:

_{A}**b**→a

_{B}_{B}→

**a**and

_{A}**b**→b

_{B}_{A}→

**a**.

_{A}© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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Nalewajski, R.F.
Phase Equalization, Charge Transfer, Information Flows and Electron Communications in Donor–Acceptor Systems. *Appl. Sci.* **2020**, *10*, 3615.
https://doi.org/10.3390/app10103615

**AMA Style**

Nalewajski RF.
Phase Equalization, Charge Transfer, Information Flows and Electron Communications in Donor–Acceptor Systems. *Applied Sciences*. 2020; 10(10):3615.
https://doi.org/10.3390/app10103615

**Chicago/Turabian Style**

Nalewajski, Roman F.
2020. "Phase Equalization, Charge Transfer, Information Flows and Electron Communications in Donor–Acceptor Systems" *Applied Sciences* 10, no. 10: 3615.
https://doi.org/10.3390/app10103615