# Understanding Electronic Structure and Chemical Reactivity: Quantum-Information Perspective

## Abstract

**:**

## 1. Introduction

## 2. Physical Attributes of Quantum States and Generalized Information Descriptors

**r**, t) = 〈

**r**|ψ(t)〉 = R(

**r**, t) exp[iϕ(

**r**, t)] ≡ R(t) exp[iϕ(t)] ≡ ψ(t),

**r**, t) ≡ R(t) and ϕ(

**r**, t) ≡ ϕ(t) stand for its modulus and phase parts, respectively. It determines the state probability distribution at the specified time t,

**r**, t) = 〈ψ(t)|

**r**〉〈

**r**|ψ(t)〉 = ψ(

**r**, t)

^{*}ψ(

**r**, t) = R(t)

^{2}≡ p(t),

**j**(

**r**, t) = [ħ/(2mi)] [ψ(

**r**, t)

^{*}∇ψ(

**r**, t) − ψ(

**r**, t) ∇ψ(

**r**, t)

^{*}] = (ħ/m) Im[ψ(

**r**, t)

^{*}∇ψ(

**r**, t)]

= (ħ/m) p(

**r**, t) ∇ϕ(

**r**, t) ≡ p(t)

**V**(t) ≡

**j**(t).

**V**(

**r**, t) =

**j**(t)/p(t) ≡

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

**V**(t) =

**j**(t)/p(t) = (ħ/m) ∇ϕ(t).

**j**).

**r**), due to the “frozen” nuclei of the Born–Oppenheimer (BO) approximation determining the system geometry, described by the electronic Hamiltonian

**V**(t), the current-per-particle of the probability “fluid” [7,69,72]. The relevant continuity relations resulting from SE are summarized in Appendix A.

_{0}, let us suppress this parameter in the list of state arguments, e.g., ψ(

**r**, t

_{0}) ≡ ψ(

**r**) = 〈

**r**|ψ〉, etc. We, again, examine the mono-electron system in (pure) quantum state |ψ〉. The average Fisher’s measure [11,12] of the classical gradient information for locality events, called the intrinsic accuracy, which is contained in the molecular probability density p(

**r**) = R(

**r**)

^{2}is reminiscent of von Weizsäcker’s [86] inhomogeneity correction to the density functional for electronic kinetic energy:

**r**) [∇lnp(

**r**)]

^{2}d

**r**= 〈ψ|(∇lnp)

^{2}|ψ〉 = ∫[∇p(

**r**)]

^{2}/p(

**r**) d

**r**= 4∫[∇R(

**r**)]

^{2}d

**r**≡ I[R].

**r**) lnp(

**r**) d

**r**= − 〈ψ|lnp|ψ〉 = −2∫R(

**r**)

^{2}lnR(

**r**) d

**r**≡ S[R],

_{S}[p] ≡ S[p].

_{p}(

**r**) = [∇p(

**r**)/p(

**r**)]

^{2}and I

_{ϕ}(

**r**) = 4[∇ϕ(

**r**)]

^{2}.

**j**] thus combines the classical (probability) contribution I[p] of Fisher and the nonclassical (phase/current) supplement I[ϕ] = I[

**j**]. The positive sign of the latter expresses the fact that a nonvanishing current pattern introduces more structural determinicity (order information) about the system, which also implies less state indeterminicity (disorder information). This dimensionless measure is seen to reflect the average kinetic energy T[ψ] = 〈ψ|$\widehat{\mathrm{T}}$|ψ〉:

^{2}) T[ψ] ≡ σ T[ψ].

**r**) ϕ(

**r**) d

**r**≡ S[p] + S[ϕ] ≡ S[p, ϕ].

_{S}[p] ≡ S[p] and (negative) nonclassical supplement S[ϕ] reflecting the state average phase. These entropy contributions also reflect the real and imaginary parts of the associated complex-entropy concept [8], the quantum expectation value of the non-Hermitian entropy operator

**S**(

**r**) = − 2lnψ(

**r**),

**S**[p, ϕ] = − 2〈ψ|lnψ|ψ〉 ≡ S[p] + i S[ϕ].

^{2}− (2∇ϕ)

^{2}|ψ〉 = I[p] − I[ϕ] ≡ M[p] + M[ϕ] ≡ M[p, ϕ].

_{eq.}≥ 0 [7,63,64,65,66,67]:

^{*}(

**r**) = 0 or δM[ψ]/δψ

^{*}(

**r**) = 0} ⇒ ϕ

_{eq.}(

**r**) = − (1/2) lnp(

**r**).

**j**

_{eq.}(

**r**) = (ħ/m) p(

**r**) ∇ϕ

_{eq.}(

**r**) = − [ħ/(2m)] ∇p(

**r**).

**r**) = Np(

**r**), where p(

**r**) stands for the density probability (shape) factor. The corresponding N-electron information operator then combines terms due to each particle,

**r**) and exhibiting the density-dependent spatial phases,

**f**(

**r**) =

**f**[ρ;

**r**], which safeguard the MO orthogonality.

**ψ**= {ψ

_{s}} = (ψ

_{1}, ψ

_{2}, …, ψ

_{N}), {n

_{s}= 1},

Ψ(N) = |ψ

_{1}ψ

_{2}…ψ

_{N}|.

**χ**= (χ

_{1}, χ

_{2}, …, χ

_{k}, …),

**ψ**〉 = |

**χ**〉

**C**,

**C**= 〈

**χ**|

**ψ**〉 = {C

_{k}

_{,s}= 〈χ

_{k}|ψ

_{s}〉},

**n**= {n

_{s}δ

_{s}

_{,s′}} = {δ

_{s}

_{,s′}} of MO occupations, then reads

## 3. Probing Formation of the Chemical Bond

_{I}

^{eq.}= −σ

_{M}

^{eq.}∝ − ∫

**j**

_{eq.}(

**r**)⋅∇v(

**r**) d

**r**∝ ∫ ∇p(

**r**) ⋅∇v(

**r**) d

**r**,

**r**)⋅∇v(

**r**) < 0, thus confirming a derease of the longitudinal contribution to the average structure information, i.e., an increase in the axial component of the overall gradient entropy (Equation (A24)) as a result of the chemical bond formation: σ

_{I}

^{eq.}(axial) < 0 and σ

_{M}

^{eq}(axial)

^{.}> 0. This accords with the chemical intuition: electron delocalization in the covalent chemical bond at its equilibrium length R = R

_{e}should produce a higher indeterminicity (disorder, entropy) measure and a lower level of the determinicity (order, information) descriptor, particularly in the axial bond region between the two nuclei.

_{I}

^{eq.}(axial) < 0, particularly in the bond region between the two nuclei, while the (dominating) latter component implies an effective transverse contraction of the electron distribution, i.e., σ

_{I}

^{eq.}(transverse) > 0. The bonded system thus exhibits a net increase in the probability inhomogeneity, i.e., a higher gradient information compared to the separated-atoms limit (SAL). This is independently confirmed by a lowering of the system overall potential-energy displacement ΔW(R) = W(R) − W(SAL) at the equilibrium bond-length R

_{e},

_{e}) = 2ΔE(R

_{e}) = −2ΔT(R

_{e}) < 0

_{e}(R) + ΔU

_{n}(R)] ≡ ΔV(R) + ΔU(R)

_{e}+ U

_{n}) energies between electrons (U

_{e}) and nuclei (U

_{n}).

ΔW(R) = 2ΔE(R) + R [dΔE(R)/∂R] = R

^{−1}d[R

^{2}ΔE(R)]/dR.

_{e}the resultant information already rises above the SAL value, due to the dominating increase in transverse components of the kinetic-energy/information (corresponding to coordinates “x” and “y” perpendicular to the bond axis). Therefore, at the equilibrium separation R

_{e}between atoms the bond-formation results in a net increase of the resultant gradient-information relative to SAL, due to—on average—more compact electron distribution in the field of both nuclei.

**R**, or the associated progress variable P = |

_{c}**R**|, of the arc length along this trajectory, for which the virial relations also assume the diatomic-like form. The virial theorem decomposition of the energy profile E(P) along

_{c}**R**in bimolecular reaction

_{c}^{‡}→ C + D,

^{‡}denotes the transition-state (TS) complex, then generates the associated profile of its kinetic-energy component T(P), which also reflects the associated resultant gradient information I(P). Such an application of the molecular virial theorem to endo- and exo-ergic reactions is presented in the upper panel of Figure 3, while the energy-neutral case of such a chemical process, on a “symmetric” potential energy surface (PES), refers to a lower panel in the figure.

^{‡}to either its substrates α ∈ (A, B) or products β ∈ (C, D) to the reaction energy ΔE

_{r}= E(P

_{prod.}) − E(P

_{sub.}): in exo-ergic (ΔE

_{r}< 0) processes, R

^{‡}≈ α and in endo-ergic (ΔE

_{r}> 0) reactions, R

^{‡}≈ β. Accordingly, for the vanishing reaction energy ΔE

_{r}= 0, the position of TS complex is expected to be located symmetrically between the reaction substrates and products. A reference to Figure 3 indeed shows that the activation barrier appears “early” in exo-ergic reaction, e.g., H

_{2}+ F → H + HF, with the reaction substrates being only slightly modified in TS, R

^{‡}≈ [A–B]. Accordingly, in the endo-ergic bond-breaking–bond-forming process, e.g., H + HF → H

_{2}+ F, the barrier is “late” along the reaction coordinate P and the activated complex resembles more reaction products: R

^{‡}≈ [C–D]. This qualitative statement has been subsequently given several more quantitative formulations and theoretical explanations using both the energetic and entropic arguments [93,94,95,96,97,98,99,100]

_{sub.}) is again directly “translated” by the virial theorem into the associated displacement in its kinetic-energy contribution ΔT(P) = T(P) − T(P

_{sub.}), proportional to the corresponding change ΔI(P) = I(P) − I(P

_{sub.}) in the system resultant gradient information, ΔI(P) = σ ΔT(P),

_{‡}> 0 and (dT/dP)

_{‡}> 0, ΔE

_{r}> 0;

energy-neutral: (dI/dP)

_{‡}= 0 and (dT/dP)

_{‡}= 0, ΔE

_{r}= 0;

exo-direction: (dI/dP)

_{‡}< 0 and (dT/dP)

_{‡}< 0, ΔE

_{r}< 0.

_{‡}, can indeed serve as an alternative detector of the reaction energetic character: its positive/negative values respectively identify the endo/exo-ergic processes, exhibiting the late/early activation barriers, respectively, with the neutral case, ΔE

_{r}= 0 or dT/dP|

_{‡}= 0, exhibiting an “equidistant” position of TS between the reaction substrates and products on a symmetric PES, e.g., in the hydrogen exchange reaction H + H

_{2}→H

_{2}+ H.

_{r}determines the corresponding change in the resultant gradient information, ΔI

_{r}= I(P

_{prod.}) − I(P

_{sub.}) = σ ΔT

_{r}, proportional to ΔT

_{r}= T(P

_{prod.}) − T(P

_{sub.}) = −ΔE

_{r}. The virial theorem thus implies a net decrease of the resultant gradient information in endo-ergic processes, ΔI

_{r}(endo) < 0, its increase in exo-ergic reactions, ΔI

_{r}(exo) > 0, and a conservation of the resultant gradient information in the energy-neutral chemical processes: ΔI

_{r}(neutral) = 0. One also recalls that the classical part of this information displacement probes an average inhomogeneity of electronic density. Therefore, the endo-ergic processes, requiring a net supply of energy to R, give rise to more diffused electron distributions in the reaction products, compared to substrates. Accordingly, the exo-ergic transitions, which release the energy from R, generate a more compact electron distributions in products and no such change is predicted for the energy-neutral case.

## 4. Reactivity Criteria

_{j}

^{i}(μ, T; v)} of the ensemble stationary states {|ψ

_{j}

^{i}〉}, eigenstates of Hamiltonians {$\hat{\mathrm{H}}({N}_{i},v)$}: $\hat{\mathrm{H}}({N}_{i},v)$|ψ

_{j}

^{i}〉 = E

_{j}

^{i}|ψ

_{j}

^{i}〉. These state probabilities correspond to the grand-potential minimum with respect to the ensemble density operator (see Equations (A29) and (A38)):

_{ens.}= $\mathcal{W}$, multiplied by the Lagrange multiplier

**N**, the chemical (μ), and information (ξ) potentials, in response to a perturbation created by an electron inflow (outflow) ΔN. This is in accordance with the familiar Le Châtelier and Le Châtelier–Braun principles of thermodynamics [4], that the secondary (spontaneous) responses in system intensities to an initial population displacement diminish effects of this primary perturbation.

## 5. Donor-Acceptor Systems

_{α}= N

_{α}

^{0}} in the isolated (separated) reactants {α

^{0}}, is symbolized by the solid vertical line, e.g., in the intermediate, polarized reactive system R

^{+}≡ (A

^{+}|B

^{+}) combining the internally polarized but mutually closed subsystems. It should be emphasized that only due to this mutual closure the substrate identity remains a meaningful concept. Their descriptors in the final, equilibrium-reactive system R

^{*}≡ (A

^{*}¦B

^{*}) ≡ R, combining the mutually open (bonded) fragments, as symbolized by the vertical broken line separating the two subsystems, can be inferred only indirectly [1,2,3], by externally opening the two mutually closed subsystems of R

^{+}with respect to their separate (macroscopic) electron reservoirs {$\mathcal{R}$

_{α}} in the composite polarized system

_{R}

^{+}= ($\mathcal{R}$

_{A}¦A

^{+}|B

^{+}¦$\mathcal{R}$

_{B}) ≡ [$\mathcal{M}$

_{A}

^{+}|$\mathcal{M}$

_{B}^{+}].

_{α}= N

_{α}p

_{α}}, with p

_{α}denoting the internal probability distribution in fragment α, are “frozen” in the promolecular reference R

^{0}= (A

^{0}|B

^{0}) consisting of the isolated-reactant distributions {ρ

_{α}

^{0}= N

_{α}

^{0}p

_{α}

^{0}} shifted to their actual positions in the “molecular” system R. The polarized reactive system R

^{+}combines the relaxed subsystem densities, modified in presence of the reaction partner at finite separation between both subsystems: {ρ

_{α}

^{+}= N

_{α}

^{+}p

_{α}

^{+}, N

_{α}

^{+}= ∫ρ

_{α}

^{+}d

**r**= N

_{α}

^{0}}. In the global equilibrium state of R as a whole, these polarized subsystem densities are additionally modified by the effective inter-reactant CT: {ρ

_{α}

^{*}= N

_{α}

^{*}p

_{α}

^{*}, N

_{α}

^{*}= ∫ρ

_{α}

^{*}d

**r**≠ N

_{α}

^{0}}.

^{+}is given by the sum of reactant densities, polarized due to “molecular” external potential v = v

_{A}+ v

_{B}combining contributions due to the fixed nuclei in both substrates at their final mutual separation,

_{R}

^{+}≡ N

_{R}p

_{R}

^{+}= ρ

_{A}

^{+}+ ρ

_{B}

^{+}≡ N

_{A}

^{+}p

_{A}

^{+}+ N

_{B}

^{+}p

_{B}

^{+}, N

_{α}

^{+}= ∫ρ

_{α}

^{+}d

**r**, ∑

_{α}N

_{α}

^{+}= N

_{R}.

_{α}

^{+}= ρ

_{α}

^{+}/N

_{α}

^{+}} stand for the internal probability densities in such promoted fragments, and the global probability distribution reflects the “shape” factor of the overall electron density,

_{R}

^{+}= ρ

_{R}

^{+}/N

_{R}= (N

_{A}

^{+}/N

_{R}) p

_{A}

^{+}+ (N

_{B}

^{+}/N

_{R}) p

_{B}

^{+}

≡ P

_{A}

^{+}p

_{A}

^{+}+ P

_{B}

^{+}p

_{B}

^{+}, ∫p

_{R}

^{+}d

**r**= P

_{A}

^{+}+ P

_{B}

^{+}= 1,

_{α}

^{+}= N

_{α}

^{+}/N

_{R}= N

_{α}

^{0}/N

_{R}= P

_{α}

^{0}} denote fragment shares in N

_{R}. At this polarization stage, both fragments exhibit internally equalized chemical potentials {μ

_{α}

^{+}= μ[N

_{α}

^{0}, v]}, different from the separate-reactant levels {μ

_{α}

^{0}= μ[N

_{α}

^{0}, v

_{α}]}.

^{*}, which allows for the inter-fragment (intra-R) flows of electrons. In such a global equilibrium each “part” effectively extends over the whole molecular system since the hypothetical boundary defining the fragment identity does not exists any more. Both subsystems then effectively exhaust the molecular electron distribution, their electron populations are equal to the global number of electrons,

_{A}

^{*}= ρ

_{B}

^{*}= ρ

_{R}, {μ

_{α}

^{*}= μ

_{R}≡ μ[N

_{R}, v], N

_{α}

^{*}= N

_{R}, p

_{α}

^{*}= ρ

_{R}/N

_{R}≡ p

_{R}},

_{R}: {μ

_{α}

^{*}= μ

_{R}}.

_{α}}. The mutual closure then implies the relevancy of subsystem identities established at the polarization stage of R

^{+}, while the external openness in the composite subsystems {$\mathcal{M}$

_{α}

^{+}= (α

^{+}¦$\mathcal{R}$

_{α}

^{+})} allows one to independently “regulate” the external chemical potentials of both parts, {μ

_{α}

^{+}= μ($\mathcal{R}$

_{α})}, and hence also their average densities {ρ

_{α}

^{+}= N

_{α}

^{+}p

_{α}

^{+}} and electron populations N

_{α}

^{+}= ∫ρ

_{α}

^{+}d

**r**. In particular, the substrate chemical potentials equalized at the molecular level in both subsystems, {μ

_{α}

^{+}≡ μ[N

_{α}

^{*}, v] = μ

_{R}≡ μ[N

_{R}, v]}, which then also describes a common molecular reservoir {$\mathcal{R}$

_{α}(μ

_{R}) = $\mathcal{R}$(μ

_{R})} coupled to both reactants, ($\mathcal{R}$(μ

_{R})¦A

^{*}¦B

^{*}), formally define the global equilibrium state in the molecular part R

^{*}= (A

^{*}¦B

^{*}) of the composite system which (indirectly) represents the global equilibrium in R as a whole:

_{R}

^{*}= $\mathcal{M}$

_{R}

^{+}(μ

_{R}) ≡ [$\mathcal{M}$

_{A}

^{*}(μ

_{R})|$\mathcal{M}$

_{B}^{*}(μ

_{R})]

= [$\mathcal{R}$(μ

_{R})¦A

^{+}(μ

_{R})|B

^{+}(μ

_{R})¦$\mathcal{R}$(μ

_{R})] ≡ [$\mathcal{R}$(μ

_{R})¦A

^{*}(μ

_{R})¦B

^{*}(μ

_{R})] ≡ ($\mathcal{R}$

^{*}¦R

^{*}).

_{R}). It allows for an effective donor(B)→acceptor(A) flow of electrons, between subsystems of the molecular fragment R

^{*}of $\mathcal{M}$

_{R}

^{*}, while retaining the reactant identities assured by the direct mutual closeness of the polarized molecular part R

^{+}in $\mathcal{M}$

_{R}

^{+}.

_{α}

^{*}= ρ

_{α}

^{+}(μ

_{R})} in R

^{*}and the associated electron populations {N

_{α}

^{*}= N

_{α}

^{+}(μ

_{R})} are then modified by effects of the inter-reactant CT

_{CT}= N

_{A}

^{*}− N

_{A}

^{0}= N

_{B}

^{0}− N

_{B}

^{*}> 0,

_{A}

^{*}+ N

_{B}

^{*}≡ N

_{R}

^{*}= N(μ

_{R}) = N

_{R}= N

_{A}

^{+}+ N

_{B}

^{+}≡ N

_{R}

^{+}= N

_{A}

^{0}+ N

_{B}

^{0}≡ N

_{R}

^{0}.

**f**

^{+}(

**r**) = {f

_{α}

_{,}

_{β}(

**r**) = [∂ρ

_{β}

^{+}(

**r**)/∂N

_{α}]

_{v}},

_{α}

^{CT}(

**r**) = ∂ρ

_{α}

^{+}(

**r**)/∂N

_{CT}= f

_{α}

_{,}

_{α}(

**r**) − f

_{β}

_{,}

_{α}(

**r**); (α, β≠α) ∈ {A, B}.

_{R}

^{CT}= ∂ρ

_{R}

^{+}/∂N

_{CT}= ∑

_{α}

_{=A,B}∑

_{β}

_{=A,B}(∂ρ

_{β}

^{+}/∂N

_{α}) (∂N

_{α}/∂N

_{CT})

= (f

_{A,A}− f

_{B,A}) − (f

_{B,B}− f

_{A,B}) ≡ f

_{A}

^{CT}− f

_{B}

^{CT},

^{*}with respect to the effective internal CT, between the externally open but mutually closed reactants.

^{+}or in the R

^{+}part of $\mathcal{M}$

_{R}

^{+}. The global equilibrium in R as a whole, R = R

^{*}, combining the effectively “bonded”, externally open but mutually closed subsystems {α

^{*}} in $\mathcal{M}$

_{R}

^{*}, accounts for the extra CT-induced polarization of reactants compared to R

^{+}. Descriptors of this state, of the mutually “bonded” reactants, can be inferred only indirectly, by examining the chemical potential equalization in the equilibrium composite system $\mathcal{M}$

_{R}

^{*}. Similar external reservoirs are involved when one examines independent population displacements on reactants, e.g., in defining the fragment chemical potentials and their hardness tensor in the substrate fragment of $\mathcal{M}$

_{R}

^{+}. In this hypothetical chain of reaction “events”, the polarized system R

^{+}thus appears as the launching stage for the subsequent CT and the accompanying induced polarization, after the hypothetical barrier for the flow of electrons between subsystems has been effectively lifted.

_{R}

^{*}of$\mathcal{M}$

_{R}

^{+}also represents the effectively open reactants in R

^{*}. The equilibrium substrates {α

^{*}} indeed display the final equilibrium densities {ρ

_{α}

^{*}= ρ

_{α}(μ

_{R})} after the B→A CT, giving rise to molecular electron distribution

_{A}

^{*}+ ρ

_{B}

^{*}= ρ

_{A}

^{+}(μ

_{R}) + ρ

_{B}

^{+}(μ

_{R}) = ρ

_{R}(μ

_{R}) ≡ ρ

_{R}

_{α}

^{+}(μ

_{R}) d

**r**= N

_{α}

^{+}(μ

_{R}) = N

_{α}

^{*}} corresponding to the chemical potential equalization in R = R

^{*}as a whole: μ

_{A}

^{*}= μ

_{B}

^{*}= μ

_{R}. One observes that the reactant chemical potentials have not been equalized at the preceding polarization stage in the molecular part R

_{n}

^{+}= (A

^{+}|B

^{+}) of a general composite system

_{R}

^{+}= ($\mathcal{M}$

_{A}

^{+}|$\mathcal{M}$

_{B}

^{+}) = ($\mathcal{R}$

_{A}

^{+}¦A

^{+}|$\mathcal{R}$

^{+}¦R

_{B}

^{+}), {$\mathcal{R}$

_{α}

^{+}= $\mathcal{R}$

_{α}[μ

_{α}

^{+}]},

_{A}

^{+}[ρ

_{A}

^{+}] < μ

_{B}

^{+}[ρ

_{B}

^{+}].

**μ**

_{R}

^{+}= {μ

_{α}

^{+}= μ

_{α}[N

_{α}

^{0}, v]} and the resulting elements of the hardness matrix

**η**

_{R}

^{+}= {η

_{α}

_{,}

_{β}} represent populational derivatives of the system average electronic energy in reactant resolution, E

_{v}({N

_{β}}). They are properly defined in $\mathcal{M}$

_{R}

^{+}, calculated for the fixed external potential v reflecting the “frozen” molecular geometry. These quantities represent the corresponding partials of the system ensemble-average energy with respect to ensemble-average populations {N

_{α}} on subsystems in the mutually closed (externally open) composite subsystems {$\mathcal{M}$

_{α}

^{+}= (α

^{+}¦$\mathcal{R}$

_{α}

^{+})} in $\mathcal{M}$

_{R}

^{+}= ($\mathcal{M}$

_{A}

^{+}|$\mathcal{M}$

_{B}

^{+}):

_{α}≡ ∂E

_{v}({N

_{γ}})/∂N

_{α}, η

_{α}

_{,}

_{β}= ∂

^{2}E

_{v}({N

_{γ}})/∂N

_{α}∂N

_{β}= ∂μ

_{α}/∂N

_{β}.

^{*}¦B

^{*}) ≡ R

^{*}part of $\mathcal{M}$

_{R}

^{*}, similarly involve differentiations with respect to the average number of electrons of R in the combined system $\mathcal{M}$

_{R}

^{*}= (R

^{*}¦$\mathcal{R}$

^{*}) = (A

^{*}¦B

^{*}¦$\mathcal{R}$

^{*}):

_{R}= ∂E

_{v}(N

_{R})/∂N

_{R}, η

_{R}= ∂

^{2}E

_{v}(N

_{R})/∂N

_{R}

^{2}= ∂μ

_{R}/∂N

_{R}.

_{CT}= ∂E

_{v}(N

_{CT})/∂N

_{CT}= μ

_{A}

^{+}− μ

_{B}

^{+}= σ

^{−1}[ξ

_{A}

^{+}− ξ

_{B}

^{+}] ≡ σ

^{−1}ξ

_{CT}< 0,

_{CT}) or softness (S

_{CT}) for this process,

_{CT}= ∂μ

_{CT}/∂N

_{CT}= (η

_{A,A}−η

_{A,B}) + (η

_{B,B}− η

_{B,A+}) ≡ η

_{A}

^{R}+ η

_{B}

^{R}= S

_{CT}

^{−1}

= σ

^{−1}∂ξ

_{CT}/∂N

_{CT}= σ

^{−1}ω

_{CT},

_{CT}. The optimum amount of the inter-reactant CT,

_{CT}= −μ

_{CT}/η

_{CT}= −ξ

_{CT}/ω

_{CT}> 0,

_{CT}= μ

_{CT}N

_{CT}/2 = − μ

_{CT}

^{2}/(2η

_{CT}) = σ

^{−}

^{1}[− ξ

_{CT}

^{2}/(2ω

_{CT})] ≡ σ

^{−}

^{1}I

_{CT}< 0,

_{CT}in the resultant gradient information.

## 6. HSAB Principle Revisited

**γ**= {γ

_{k}

_{,l}} of Equation (26), weighting factors in Equation (24), determine amplitudes of conditional probabilities defining molecular (direct) communications between AO. Entropic descriptors of such information channel generate the entropic bond orders and measures of their covalent/ionic components, which ultimately facilitate an IT understanding of molecular electronic structure in chemical terms [7,19,20,21,32,33,34,35,36,37,38,39,40,41,42]. The communication “noise” (orbital indeterminicity) in this network, measured by the channel conditional entropy, is due to the electron delocalization in the bond system of a molecule. It represents the system overall bond “covalency”, while the channel information “capacity” (orbital determinicity), reflected by the mutual information of the molecular communication network, measures its resultant bond “iconicity”. The more scattering (indeterminate) is the molecular information system, the higher its covalent character; a more deterministic (less noisy) channel thus represents a more ionic molecular system. These two bond attributes thus compete with one another.

_{2}or carbons in ethylene, when the interacting AO in the familiar MO diagrams of chemistry exhibit the same AO energies. The bond ionicity accompanies large differences in electronegativities, generating a substantial CT between the interacting atoms. Such bonds correspond to a wide separation of AO energies in MO diagrams. The ionic bond component diminishes noise and introduces more determinicity into AO communications, thus representing the bond mechanism competitive with the bond covalency.

_{ion.}= |E

_{CT}| = μ

_{CT}

^{2}/(2η

_{CT}) > 0,

_{CT}and η

_{CT}stand for the effective chemical potential and hardness descriptors of R involving the FE of reactants. Since the donor/acceptor properties of reactants are already implied by their known relative acidic or basic character, one applies the biased estimate of the CT chemical potential. In FE approximation, the chemical potential difference μ

_{CT}for the effective internal B→A CT then reads (see Figure 4):

_{CT}(B→A) = μ

_{A}

^{(}

^{−)}− μ

_{B}

^{(+)}= ε

_{A}(LUMO) − ε

_{B}(HOMO) ≈ I

_{B}− A

_{A}> 0.

_{CT}electrons:

_{B}

_{→A}(N

_{CT}) = μ

_{CT}(B→A) N

_{CT}> 0.

_{B}= E(B

^{+1}) − E(B

^{0}) > 0,

_{B}

^{(+)}= ε

_{B}(HOMO) ≈ − I

_{B},

_{A}= E(A

^{0}) − E(A

^{−1}) > 0,

μ

_{A}

^{(}

^{−)}= ε

_{A}(LUMO) ≈ −A

_{A}.

^{−1}–––B

^{+1}] + [A

^{+1}–––B

^{−1}],

_{CT}= (I

_{A}− A

_{A}) + (I

_{B}− A

_{B})

≈ [ε

_{A}(LUMO) − ε

_{A}(HOMO)] + [ε

_{B}(LUMO) − ε

**(HOMO)]**

_{B}= η

_{A}+ η

_{B}> 0.

_{ion.}= μ

_{CT}

^{2}/(2η

_{CT}) = [ε

_{A}(LUMO) − ε

_{B}(HOMO)]

^{2}/[2(η

_{A}+ η

_{B})].

_{A}(LUMO)−ε

_{B}(HOMO) between the interacting orbitals reaches the minimum value, implies the strongest covalent stabilization in the reactive complex. Indeed, the lowest (bonding) energy level ε

_{b}of this FE interaction, corresponding to the bonding combination of the (positively overlapping) frontier MO of subsystems,

_{b}= N

_{b}[φ

_{B}(HOMO) + λφ

_{A}(LUMO)], S = 〈φ

_{A}(LUMO)|φ

_{B}(HOMO)〉 > 0,

_{cov.}= ε

_{B}(HOMO) − ε

_{b}> 0.

_{cov.}≅ (β − ε

_{b}S)

^{2}/[ε

_{A}(LUMO) − ε

_{B}(HOMO)],

_{α}

_{,}

_{β}} + {[α→β] (1 − δ

_{α}

_{,}

_{β})} = {intra} + {inter}.

## 7. Regional HSAB versus Complementary Coordinations

_{B}|…|b

_{B}) ≡ (a

_{B}|b

_{B}) and the acidic substrate A = (a

_{A}|…|b

_{A}) ≡ (a

_{A}|b

_{A}), where a

_{X}and b

_{X}denote the acidic and basic parts of X, respectively. The acidic (electron acceptor) part is relatively hard, i.e., less responsive to external perturbations, exhibiting lower values of the fragment FF descriptor, while the basic (electron donor) fragment is relatively soft and more polarizable, as reflected by its higher density or population response descriptors. 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 a fragment of Y in its vicinity.

_{X}are placed in the attractive field generated by the electron “deficient” a

_{Y}, is expected to be favored electrostatically since the other arrangement produces regional repulsions between two acidic and two basic sites of reactants. Additional rationale for this complementary preference over the regional HSAB alignment comes from examining charge flows created by the dominating shifts in the site chemical potential due to the presence of the (“frozen”) coordinated site of the nearby part of the reaction partner. At finite separations between the two subsystems, these displacements trigger the polarizational flows {P

_{X}} shown in Figure 5 and Figure 6, which restore the internal equilibria in subsystems, initially displaced by the presence of the other reactant.

_{c}, the harder (acidic) site a

_{Y}initially lowers the chemical potential of the softer (basic) site b

_{X}, while b

_{Y}rises the chemical potential level of a

_{X}. These shifts trigger the internal polariaztional flows {a

_{X}→b

_{X}} in both reactants, which enhance the acceptor capacity of a

_{X}and donor ability of b

_{X}, thus creating more favourable conditions for the subsequent inter-reactant CT of Figure 5. A similar analysis of R

_{HSAB}(Figure 6) predicts the b

_{X}→a

_{X}polarizational flows, which lower the acceptor capacity of a

_{X}and donor ability of b

_{X}, i.e., produce an excess electron accumulation on a

_{X}and stronger electron depletion on b

_{X}, thus creating less favourable conditions for the subsequent inter-reactant CT.

_{X}} in the external potential on subsystems. In stability considerations one first assumes the primary (inter-reactant) CT displacements {ΔCT

_{1}, ΔCT

_{2}} of Figure 5 and Figure 6, in the internally “frozen” but externally open reactants, and then examines the induced secondary (intra-reactant) relaxational responses {I

_{X}} to these populational shifts in the CT-perturbed substrates.

_{c},

_{1}) = ΔN(a

_{A}) = −ΔN(b

_{B})] > [Δ(CT

_{2}) = ΔN(a

_{B}) = −ΔN(b

_{A})].

_{x}

_{,x}= ∂μ

_{x}/∂N

_{x}≡ η

_{x}> 0.

_{k}} thus create the following shifts in the site chemical potentials, compared to the initially equalized levels in isolated reactants A

^{0}= (a

_{A}

^{0}¦b

_{A}

^{0}) and B

^{0}= (a

_{B}

^{0}¦b

_{B}

^{0}),

_{X}} in R

_{c}

^{CT},

_{HSAB},

_{1}) = ΔN(b

_{A}) = −ΔN(b

_{B})] < [Δ(CT

_{2}) = ΔN(a

_{A}) = −ΔN(a

_{B})],

_{1})| = μ(b

_{B}) − μ(b

_{A})] < [|μ(CT

_{2})| = μ(a

_{B}) − μ(a

_{A})],

## 8. Conclusions

## Conflicts of Interest

## Nomenclature

**A**is the row or column vector,

**A**represents a square or rectangular matrix, and the dashed symbol $\widehat{\mathrm{A}}$ stands for the quantum-mechanical operator of the physical property A. The logarithm of Shannon’s information measure is taken to an arbitrary but fixed base: log = log

_{2}corresponds to the information content measured in bits (binary digits), while log = ln expresses the amount of information in nats (natural units): 1 nat = 1.44 bits.

## Appendix A. Continuities of Probability and Phase Distributions

**r**(t), t] reads

**r**/dt) ⋅ ∂p(t)/∂

**r**= ∂p(t)/∂t +

**V**(t) ⋅∇p(t) ≡ σ

_{p}(t).

_{p}(

**r**, t) ≡ σ

_{p}(t), which measures the time rate of change in an infinitesimal volume element around

**r**in the probability fluid, moving with the local velocity d

**r**/dt =

**V**(t), while the partial derivative ∂p(t)/∂t refers to the volume element around the fixed point in space:

_{p}(t) −

**V**(t) ⋅∇p(t)

= σ

_{p}(t) − (ħ/2mi)[ψ(t)

^{*}Δψ(t) − ψ(t) Δψ(t)

^{*}] = σ

_{p}(t) − ∇⋅

**j**(t)

= σ

_{p}(t) − [

**V**(t) ⋅∇p(t) + p(t) ∇⋅

**V**(t)]

= σ

_{p}(t) − (ħ/m) [∇ϕ(t) ⋅∇p(t) + p(t) ∇

^{2}ϕ(t)].

_{p}(t) = 0, or

**j**(t) = ∂p(t)/∂t + ∇p ⋅

**V**(t) + p ∇⋅

**V**(t) = 0.

**V**(t) = (ħ/m) ∇

^{2}ϕ(t) = 0.

**V**(t) ⋅∇p(t) = − (ħ/m) ∇ϕ(t) ⋅∇p(t)

**j**(t) = ∂p(t)/∂t +

**V**(t) ⋅∇p(t) = 0.

**r**) =

**k**⋅

**f**(

**r**) + C,

**r**= {x

_{α}}:

**V**(

**r**) = (ħ/m)

**k**∇⋅

**f**(

**r**). Its vanishing divergence then implies a local condition

**k**⋅∇[∇⋅

**f**(

**r**)] ≡

**k**⋅∇

^{2}

**f**(

**r**) = ∑

_{α}∑

_{β}k

_{α}[∂

^{2}f

_{β}(

**r**)/∂x

_{α}∂x

_{β}] = 0.

**V**(t) also determines the phase current, the flux concept associated with the state phase:

**J**(t) = ϕ(t)

**V**(t). The scalar field ϕ(t) and its conjugate current density

**J**(t) then generate a nonvanishing phase source in the associated continuity equation:

_{ϕ}(t) ≡ dϕ(t)/dt = ∂ϕ(t)/∂t + ∇⋅

**J**(t) = ∂ϕ(t)/∂t +

**V**(t) ⋅ ∇ϕ(t) ≠ 0 or

∂ϕ(t)/∂t − σ

_{ϕ}(t) = −∇⋅

**J**(t) = − (ħ/m) [∇ϕ(t)]

^{2}.

^{−1}ΔR − (∇ϕ)

^{2}] − v/ħ,

_{ϕ}= [ħ/(2m)] [R

^{−1}∇

^{2}R + (∇ϕ)

^{2}] − v/ħ.

_{s},

_{s}(t) = R

_{s}(

**r**) exp[iϕ

_{s}(t)], ϕ

_{s}(t) = − (E

_{s}/ħ) t = − ω

_{s}t,

_{s}(t) ≡ R

_{s}(

**r**)

^{2}= p

_{s}(

**r**), the purely time-dependent phase ϕ

_{s}(t),

**V**

_{s}(t) =

**0**and hence also

**j**

_{s}(t) =

**J**

_{s}(t) =

**0**. The phase-dynamics Equations (A11) and (A12) then recover the preceding stationary SE and identify the constant phase source (see Equation (A13)):

_{ϕ}[ψ

_{s}] = [ħ/(2m)] (R

_{s}

^{−1}∇

^{2}R

_{s}) − v/ħ = − ω

_{s}= − (E

_{s}/ħ) = const.

## Appendix B. Information Dynamics

^{†}ψ(t)|ψ(t)〉 or ∇

^{†}= −∇.

_{I}(t) ≡ dI(t)/dt = (4i/ħ) {〈ψ(t)|∇v ⋅ |∇ψ(t)〉 − 〈∇ψ(t)| ⋅∇v|ψ(t)〉}

= − (8/ħ) Im 〈ψ(t)|∇v ⋅ |∇ψ(t)〉

= − (8/ħ) Im[∫ψ(t)

^{*}∇v ⋅∇ψ(t) d

**r**]

= − (8/ħ) ∫p(t) ∇ϕ(t) ⋅∇v d

**r**= − σ ∫

**j**(t) ⋅∇v d

**r**.

_{I}(t) ≡ dI(t)/dt = dI[p]/dt + dI[ϕ]/dt = dI[ϕ]/dt,

_{p}(

**r**) = dp(

**r**)/dt = 0,

**r**)/dt] [δI[p]/δp(

**r**)] d

**r**= 0.

_{M}(t) ≡ dM(t)/dt = dI[p]/dt − dI[ϕ]/dt = −dI[ϕ]/dt = −σ

_{I}(t).

_{I}(t) > 0, implies the associated decrease in its structural indeterminicity (disorder) information (entropy): σ

_{M}(t) < 0.

## Appendix C. Ensemble Representation of Thermodynamic Conditions

_{ens.}≡ $\mathcal{N}$ of the (open) molecular part M(v), identified by the external potential v of the specified (microscopic) system, in the composite (macroscopic) system $\mathcal{M}$ = [M(v)¦$\mathcal{R}$] including the external electron-reservoir $\mathcal{R}$,

_{i}N

_{i}(∑

_{j}|ψ

_{j}

^{i}〉 〈ψ

_{j}

^{i}|) stands for the particle-number operator in Fock’s space and the density operator $\hat{\mathrm{D}}$ = ∑

_{i}∑

_{j}|ψ

_{j}

^{i}〉 P

_{j}

^{i}〈ψ

_{j}

^{i}| identifies the statistical mixture of the system (pure) stationary states {|ψ

_{j}

^{i}〉 ≡ |ψ

_{j}(N

_{i})〉} defined for different (integer) numbers of electrons {N

_{i}}, which appear with the (external) probabilities {P

_{j}

^{i}} in the ensemble. Such $\mathcal{N}$-derivatives are involved in definitions of familiar CT criteria of chemical reactivity, e.g., the chemical potential (negative electronegativity) [53,101,102,103,104,105] or the chemical hardness/softness [106], and Fukui function (FF) [107] descriptors (see also References [53,73,74]).

_{$\mathcal{R}$}, and the absolute temperature T of a heat bath $\mathcal{B}$, T = T

_{$\mathcal{B}$}: ${\hat{\mathrm{D}}}_{eq.}$ ≡ $\hat{\mathrm{D}}(\mu ,T;v)$. The optimum state probabilities {P

_{j}

^{i}(μ, T; v)} correspond to the minimum of the associated thermodynamic potential, the Legendre transform

**$\mathcal{N}$**and thermodynamic entropy

_{B}denotes the Boltzmann constant, by their respective “intensive” conjugates defining the applied thermodynamic conditions: μ = μ

_{$\mathcal{R}$}and T = T

_{$\mathcal{B}$}. The grand potential of Equation (A26) includes these externally imposed intensities as Lagrange multipliers enforcing constraints of the specified values of system’s average number of electrons, 〈N〉

_{ens.}= $\mathcal{N}$, and its thermodynamic-entropy, 〈$\mathcal{S}$〉

_{ens.}= $\mathcal{S}$, at the grand-potential minimum:

_{j}

^{i}(μ, T; v) = Ξ

^{−1}exp[β(μN

_{i}− E

_{j}

^{i})],

_{B}T)

^{−1}. The equilibrium probabilities of Equation (A30) represent eigenvalues of the grand-canonical statistical operator acting in Fock’s space:

_{0}

^{i}〉 and |ψ

_{0}

^{i}

^{+1}〉, corresponding to the neighboring integers “bracketing” the given (fractional) 〈N〉

_{ens.}= $\mathcal{N}$,

_{i}≤ $\mathcal{N}$ ≤ N

_{i}+ 1,

_{ens.}= i P

_{i}(T→0) + (i + 1)[1 − P

_{i}(T→0)] = $\mathcal{N}$

_{i}(T→0) = 1 + i − $\mathcal{N}$ ≡ 1 − ω and P

_{i}

_{+1}(T→0) = $\mathcal{N}$ − i ≡ ω.

_{i}}.

_{0}

^{i}〉 = ψ[N

_{i}, v]}, defined for the integer number of electrons N

_{i}= 〈ψ

_{i}|$\hat{\mathrm{N}}$|ψ

_{i}〉 and corresponding to energies

_{i}(μ, T→0; v), represents an externally open molecule 〈M(μ, T→0; v)〉

_{ens.}in these thermodynamic conditions.

_{j}

^{i}, when P

_{j}

^{i}= 1 for the vanishing remaining state probabilities.

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**Figure 1.**Schematic diagram of the axial (bond) profiles, in section containing the “z” direction of the coordinate system (along the bond axis), of the external potential (v) and electron probability (p) in a diatomic molecule A–B demonstrating a negative character of the scalar product ∇p(

**r**) ⋅∇v(

**r**). It confirms the negative equilibrium contribution σ

_{I}

^{eq.}(axial) of the resultant gradient information (Equations (A20) and (A21)) and positive source σ

_{M}

^{eq.}(axial) of the resultant gradient entropy (Equation (A24)) in the bond formation process, due to the equilibrium current of Equation (18),

**j**

_{eq.}(

**r**) ∝ −∇p(

**r**).

**Figure 2.**Variations of the electronic energy ΔE(R) (solid line) with the internuclear distance R in a diatomic molecule and of its kinetic energy component ΔT(R) (broken line) determined by the virial theorem partition.4. Reactivity Implications of Molecular Virial Theorem.

**Figure 3.**Variations of the electronic total (E) and kinetic (T) energies in exo-ergic (ΔE

_{r}< 0) or endo-ergic (ΔE

_{r}> 0) reactions (upper Panel (

**a**)), and on the symmetrical BO potential energy surface (PES) (ΔE

_{r}= 0) (lower Panel (

**b**)).

**Figure 4.**Schematic diagram of the in situ chemical potentials μ

_{CT}(B→A), determining the effective internal charge transfer (CT) from basic (B) reactant to its acidic (A) partner in A–B complexes, for their alternative hard (H) and soft (S) combinations. The subsystem hardnesses reflect the HOMO-LUMO gaps in their orbital energies.

**Figure 5.**Polarizational {P

_{α}= (a

_{α}→b

_{α})} and charge-transfer {CT

_{α}= (b

_{α}→a

_{β})} electron flows, (α, β≠α) ∈ {A, B}, involving the acidic A = (a

_{A}|b

_{A}) and basic B = (a

_{B}|b

_{B}) reactants in the complementary arrangement R

_{c}of their acidic (a) and basic (b) fragments, with the chemically “hard” (acidic) fragment of one substrate facing the chemically “soft” (basic) part of its reaction partner. The polarizational flows {P

_{α}} (black arrows) in the mutually closed substrates, relative to the substrate “promolecular” references, preserve the overall numbers of electrons of isolated reactants {α

^{0}}, while the two partial {CT

_{i}} fluxes (white arrows), from the basic fragment of one reactant to the acidic part of the other reactant, generate a substantial resultant B→A transfer of N

_{CT}= CT

_{1}− CT

_{2}electrons between the mutually open reactants. These hypothetical electron flows in such a “complementary complex” are seen to produce an effective concerted (“circular”) flux of electrons between the four fragments invoked in this regional “functional” partition, which precludes an exaggerated depletion or concentration of electrons on any fragment of reactive system.

**Figure 6.**Polarizational {P

_{α}= (b

_{α}→a

_{α})} and charge-transfer, CT

_{1}= (b

_{B}→b

_{A}) and CT

_{2}= (a

_{B}→a

_{A}), electron flows, involving the acidic A = (a

_{A}|b

_{A}) and basic B = (a

_{B}|b

_{B}) reactants in the regional HSAB complex R

_{HSAB}, in which the chemically hard (acidic) and soft (basic) fragments of one reactant coordinate to the like fragment of the other substrate. The two partial {CT

_{i}} fluxes (white arrows) now generate a moderate overall B→A transfer of N

_{CT}= CT

_{1}+ CT

_{2}electrons between the mutually open reactants. These hypothetical electron flows in the regional HSAB complex are seen to produce a disconcerted pattern of fluxes producing an exaggerated outflow of electrons from b

_{B}and and their accentuated inflow to a

_{A}.

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Nalewajski, R.F. Understanding Electronic Structure and Chemical Reactivity: Quantum-Information Perspective. *Appl. Sci.* **2019**, *9*, 1262.
https://doi.org/10.3390/app9061262

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Nalewajski, Roman F. 2019. "Understanding Electronic Structure and Chemical Reactivity: Quantum-Information Perspective" *Applied Sciences* 9, no. 6: 1262.
https://doi.org/10.3390/app9061262