物理化学学报  2017, Vol. 33 Issue (12): 2491-2509   (1033 KB)    
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  • Received: April 19, 2017
  • Revised: May 31, 2017
  • Published on Web: June 13, 2017
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    NALEWAJSKIRoman F
    Chemical Reactivity Description in Density-Functional and Information Theories
    NALEWAJSKIRomanF    
    Department of Theoretical Chemistry, Jagiellonian University, R. Ingardena 3, 30-060 Cracow, Poland
    摘要:In Quantum Information Theory (QIT) the classical measures of information content in probability distributions are replaced by the corresponding resultant entropic descriptors containing the nonclassical terms generated by the state phase or its gradient (electronic current). The classical Shannon (S[p]) and Fisher (I[p]) information terms probe the entropic content of incoherent local events of the particle localization, embodied in the probability distribution p, while their nonclassical phase-companions, S[φ] and I[φ], provide relevant coherence information supplements. Thermodynamic-like couplings between the entropic and energetic descriptors of molecular states are shown to be precluded by the principles of quantum mechanics. The maximum of resultant entropy determines the phase-equilibrium state, defined by “thermodynamic” phase related to electronic density, which can be used to describe reactants in hypothetical stages of a bimolecular chemical reaction. Information channels of molecular systems and their entropic bond indices are summarized, the complete-bridge propagations are examined, and sequential cascades involving the complete sets of the atomic-orbital intermediates are interpreted as Markov chains. The QIT description is applied to reactive systems R=A-B, composed of the Acidic (A) and Basic (B) reactants. The electronegativity equalization processes are investigated and implications of the concerted patterns of electronic flows in equilibrium states of the complementarily arranged substrates are investigated. Quantum communications between reactants are explored and the QIT descriptors of the A-B bond multiplicity/composition are extracted.
    关键词Density-functional theory    Donor-acceptor system    Electronegativity equalization and electron flows    Information theory    Markov chains    Phase-equilibria    

    1 Introduction

    The key issue in entropic theories of molecular electronic structure is the phase/current generalization of the original (probability) information concepts of Classical Information Theory (CIT) pioneered by Fisher1 and Shannon2, which are appropriate for the complex amplitudes (wavefunctions) of Quantum Mechanics (QM). The electron distribution alone generates the state average classical information, i.e., the information received from outcomes of the incoherent local events of the particle-position measurements. It has been argued elsewhere3-8 that in Quantum IT (QIT)3 both the particle probability distribution and the state phase/current densities ultimately contribute to the resultant information descriptors of molecular electronic states. The electron density alone determines only the classical part of the overall information content, while the wavefunction phase or its gradient (probability-current) generate its nonclassical supplement.

    Resultant measures of the entropy/information content of electronic states thus combine the classical contributions, due to wavefunction modulus, and their nonclassical supplements due to the state phase. Such descriptors allow one to distinguish the information content of states generating the same electron density but differing in their phase/current composition. The classical and nonclassical components of the quantum-entropy descriptors of molecular states have been also related to the real and imaginary parts of the complex entropy concept3, 9, a natural quantum extension of the classical Shannon entropy2. The classical information terms, conceptually related to functionals of modern Density Functional Theory (DFT)10-13, probe the entropic content of the incoherent ("disentangled") local events, the outcomes of the repeated measurements of the particle position, while their nonclassical companions provide the information supplement due to the coherence ("entanglement") of such local outcomes. Generalized variational principles for the resultant information measures determine the phase-equilibria in molecules and their constituent fragments3-8. The extrema of both the global and gradient measures of the resultant entropy have been shown to give rise to the same equilibrium-phase solution related to the system electron density, which defines the "thermodynamic" phase-transformation of the Schrödinger states.

    In phenomenological thermodynamics the energetic and entropic aspects are mutually coupled, with the equilibrium states resulting from alternative principles of the minimum internal energy for the fixed equilibrium entropy, or of the maximum entropy for the fixed equilibrium energy. We shall emphasize that in the molecular-state scenario such coupled energy/information principles are precluded by the variational principle of QM.

    In a bimolecular chemical reaction between the donor (basic) and acceptor (acidic) substrates both the system electron distribution and its geometry relax, when reactants interact at a finite separation between them, ultimately determining the equilibrium electron distribution and optimum intrinsic geometry for the current value of the reaction progress-variable. Within the electron-following perspective of the familiar Born-Oppenheimer approximation, each (open) subsystem, now in the external potential due to the nuclei of both subsystems, responds to a presence of the reaction partner by changing its electron density and the effective average number of electrons. This perturbation induces the effective polarization (promotion) of the mutually-closed reactants and generates a (fractional) charge transfer (CT) between them, when the hypothetical barrier preventing a flow of electrons between the two subsystems is lifted. Such spontaneous responses of molecules to displacements in their external potential and the average electron number are all grounded in DFT and explain gross features in reactivity preferences13-18. Such perturbation-response relations have been formulated in the DFT-based Charge Sensitivity Analysis (CSA)14-16 or Conceptual DFT13, 17, 18. The chemical indices of electronic electronegativity13, 19-23, hardness24, 25, softness (or Fukui function)26, and electrophilicity/nucleophilicity concepts18, 27 help to systematize the complex phenomenona of chemical reactivity. Their understanding calls for descriptors measuring both responses in the electronic structure of reactants13-27, and of its coupling to the system geometry28-34, which ultimately determines the reaction Minimum-Energy Path (MEP).

    The phase/current degrees-of-freedom of electronic states have also implications for generalized communications in molecular information channels2, 3, 35, which generate the entropic measures of the bond multiplicity and descriptors of its ionic/covalent composition3, 36-38. We recall, that within CIT the network conditional-entropy (average "noise") has been linked to the system "covalent" bond component, while the mutual-information ("flow") descriptor has been identified as the complementary "ionic" index of all bonds in a molecular system under consideration. The phase-description and communication treatment can be also applied to reactive systems3, 39-41. The CIT descriptors generate the entropy representation of reaction mechanism and uncover the whole complexity of the process. Compared to the MEP profiles of potential-energy surface (PES)42, the corresponding sections of the entropy/information surfaces have uncovered a presence of additional features revealing the chemically important regions where the bond-breaking and bond-forming processes actually occur43, 44. The Orbital Communication Theory (OCT) of the chemical bond3, 36-38 examines the entropic bond descriptors generated by communication networks of the probability propagation between basis functions of quantum-mechanical calculations. It has identified the new bridge mechanism38, 45-51 of the bond formation, associated with the cascade scatterings involving intermediate orbitals. Although significance of this extra bond-component for reactive systems remains to be explored, it can be surmised that in transition-state complexes on PES, involving partly broken/formed bonds crucial for the chemical reaction in question, such indirect bonds should be of paramount importance. In the present analysis we reexamine the bridge propagations of the electronic conditional probabilities and their amplitudes. We shall also interpret the sequential-cascade scatterings in molecules as Markov processes.

    The DFT-based reactivity concepts an related principle of Electronegativity-Equalization (EE)20 allow one to diagnose the crucial polarizational and CT electron flows in reactive systems, which ultimately determine the observed reactivity preferences. In the present analysis we shall qualitatively examine such readjustments in electronic structure of the acidic and basic reactants at various hypothetical stages of a chemical reaction in the donor-acceptor system. The "classical" language of DFT, focusing on displacements in the electronic probability distributions rather than on the associated changes in the system many-electron wave functions, loses the phase ("entanglement") aspect of reactivity phenomena. It is retained in probability-amplitudes resulting from the Superposition Principle (SP) of QM52, being also reflected by the "nonclassical" entropy/information contributions3, 35. In this work we shall explore the QIT treatment of reactive systems at several hypothetical stages invoked in the theory of chemical reactivity. Although, for simplicity reasons, the one-electron case will often be assumed, the modulus (density) and phase (current) aspects of general many-electron states can be similarly separated using the Harriman-Zumbach-Maschke(HZM)53, 54 construction of Slater determinants yielding the specified electron density, formulated in terms of the Equidensity Orbitals (EO) of DFT3, 4.

    2 Resultant information in electronic states

    Let us assume the simplest case of a single (N = 1) electron described by the complex wavefunction specified by its (real) modulus (R) and phase (ϕ) parts: ψ(r) = R(r)exp[iϕ(r)]. In accordance with SP of QM52 the wavefunction ψ(r) = < r|ψ> measures the amplitude of position-probability p(r) = |ψ(r)|2 = R(r)2, i.e., the conditional probability P(r|ψ) of observing in |ψ> the localized-electron state |r>,

    $ \begin{array}{l} P(\mathit{\boldsymbol{r}}|y) = \left\langle {\mathit{\boldsymbol{r}}|y} \right\rangle \left\langle {y|\mathit{\boldsymbol{r}}} \right\rangle \equiv \left\langle {\mathit{\boldsymbol{r}}\left| {{{{\rm{\hat P}}}_\psi }} \right|\mathit{\boldsymbol{r}}} \right\rangle \\ \;\;\;\;\;\;\;\;\;\;\;\; = \left\langle {\psi |\mathit{\boldsymbol{r}}} \right\rangle \left\langle {\mathit{\boldsymbol{r}}|\psi } \right\rangle \equiv \left\langle {\psi \left| {{{{\rm{\hat P}}}_\mathit{\boldsymbol{r}}}} \right|\psi } \right\rangle , \end{array} $ (1)

    where ${{\rm{\hat P}}_\psi }$ and ${{\rm{\hat P}}_\boldsymbol{r}}$ denote the corresponding projection operators. The state phase-gradient similarly determines the probability-current density:

    $ \mathit{\boldsymbol{j}}\left\langle {\mathit{\boldsymbol{r}}|\psi } \right\rangle \equiv \mathit{\boldsymbol{j}}\left( \mathit{\boldsymbol{r}} \right) = \left( {\hbar /m} \right)p\left( \mathit{\boldsymbol{r}} \right)\nabla \phi \left( \mathit{\boldsymbol{r}} \right) \equiv p\left( \mathit{\boldsymbol{r}} \right)\mathit{\boldsymbol{V}}\left( \mathit{\boldsymbol{r}} \right), $ (2)

    with the current-per-particle V(r) = j(r)/p(r) = (ħ/m) ϕ(r) defining the effective velocity of the probability flux.

    The dynamics of a general wavefunction ψ(r, t) = < r|ψ(t)> is described by the Schrödinger equation (SE),

    $ {\rm{\hat H}}\left( \mathit{\boldsymbol{r}} \right)\psi (\mathit{\boldsymbol{r}};t) = i\hbar [\partial \psi (\mathit{\boldsymbol{r}};t)/\partial t], $ (3)

    where the Hamiltonian

    $ {\rm{\hat H}}\left( \mathit{\boldsymbol{r}} \right) = - \left( {\hbar /2m} \right){\nabla ^2} + v\left( \mathit{\boldsymbol{r}} \right) = {\rm{\hat T}}\left( \mathit{\boldsymbol{r}} \right) + v\left( \mathit{\boldsymbol{r}} \right) $ (4)

    combines the kinetic-energy operator ${\rm{\hat T}}(\boldsymbol{r})$ and the external potential v(r) due to the system fixed nuclei. It implies the (sourceless) continuity equation for the probability density,

    $ \begin{array}{l} {\rm{d}}p(\mathit{\boldsymbol{r}};t)/dt \equiv {\sigma _p}(\mathit{\boldsymbol{r}};t) = \partial p(\mathit{\boldsymbol{r}};t)/\partial t + \nabla \cdot \mathit{\boldsymbol{j}}(\mathit{\boldsymbol{r}};t) = 0\;\;\;\;{\rm{or}}\\ \partial p(\mathit{\boldsymbol{r}};t)/\partial t = - \nabla \cdot \mathit{\boldsymbol{j}}(\mathit{\boldsymbol{r}};t), \end{array} $ (5)

    and the phase-dynamics equation:

    $ \begin{array}{l} \partial \phi (\mathit{\boldsymbol{r}};t)/\partial t = \left( {\hbar /2m} \right)\left\{ {R{{(\mathit{\boldsymbol{r}};t)}^{ - 1}}{\nabla ^2}R(\mathit{\boldsymbol{r}};t) - {{\left[ {\nabla \phi (\mathit{\boldsymbol{r}};t)} \right]}^2}} \right\}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - v\left( \mathit{\boldsymbol{r}} \right)/\hbar . \end{array} $ (6)

    It should be observed that the elementary particle-localization states {|r>} are regarded in CIT as being incoherent in character, i.e., having no prior, common molecular "ancestor". Indeed, they represent the "disentangled" (phase-unrelated) measurements of the particle position. However, in Eq.(1) the outcomes of the localization experiments are conditional on the quantum state |ψ>, which thus determines the common molecular "ancestor" of the underlying ("entangled", coherent) conditional events {(r|ψ)=(r|R, ϕ)}.Onefurther notices that probabilities (square moduli of wavefunctions) loose the information about phases/currents of such elementary event-states, which is preserved in the corresponding probability amplitudes themselves. Therefore, the amplitude (wavefunction) data should be used to determine the coherency ("entanglement") descriptors of such elementary events in the electronic state in question. In QIT the coherent entropy/information concepts combine the classical (probability) and nonclassical (phase/current) contributions in the corresponding resultant measures.

    The average Fisher1 measure of the classical gradient-information content in the probability density p(r) is reminiscent of von Weizs cker's55 inhomogeneity correction to the kinetic energy functional,

    $ \begin{array}{l} I\left[ p \right] = \int {\nabla p(\mathit{\boldsymbol{r}}){]^2}} /p(\mathit{\boldsymbol{r}}){\rm{d}}\mathit{\boldsymbol{r}} = \int {p(\mathit{\boldsymbol{r}})} {[\nabla \ln p(\mathit{\boldsymbol{r}})]^2}{\rm{d}}r \equiv \int {p(\mathit{\boldsymbol{r}}){I_p}(\mathit{\boldsymbol{r}})dr} \\ \;\;\;\;\;\;\; = 4\int {{{[\nabla R\left( \mathit{\boldsymbol{r}} \right)]}^2}} {\rm{d}}\mathit{\boldsymbol{r}} \equiv I\left[ R \right] \end{array} $ (7)

    The amplitude form I[R] reveals that it measures the average length of the state modulus-gradient. This classical descriptor characterizes an effective sharpness (determinicity, "narrowness") of the particle probability distribution.

    The complementary classical descriptor of the Shannon2 entropy,

    $ \begin{array}{l} S\left[ p \right] = - \int {p(\mathit{\boldsymbol{r}})} {\rm{ln}}p(\mathit{\boldsymbol{r}}){\rm{d}}\mathit{\boldsymbol{r}} \equiv \int {p(\mathit{\boldsymbol{r}})} {S_p}(\mathit{\boldsymbol{r}}){\rm{d}}\mathit{\boldsymbol{r}}\\ \;\;\;\;\;\;\;\; = - 2\int {{R^2}\left( \mathit{\boldsymbol{r}} \right)} {\rm{ln}}R\left( \mathit{\boldsymbol{r}} \right){\rm{d}}\mathit{\boldsymbol{r}} \equiv S\left[ R \right] \end{array} $ (8)

    reflects the average uncertainty (indeterminacy, "spread") in the random position variable. It also provides the amount of information received, when this uncertainty is removed by an appropriate localization experiment: IS[p] ≡ S[p]. The densities-per-electron of these probability/modulus functionals satisfy the classical relation

    $ {I_p}\left( \mathit{\boldsymbol{r}} \right) = {\left[ {\nabla {S_p}\left( \mathit{\boldsymbol{r}} \right)} \right]^2} $ (9)

    The resultant entropy/information descriptors of the molecular electronic state |ψ> combine these familiar classical contributions and the associated nonclassical supplements due to the state (spatial) phase or probability current3-9:

    $ \begin{array}{l} I\left( \psi \right) = - 4\left\langle {\psi \left| {{\nabla ^2}} \right|\psi } \right\rangle \equiv \left\langle {\psi \left| {{\rm{\hat I}}} \right|\psi } \right\rangle = 4\int {{{\left| {\nabla \psi \left( \mathit{\boldsymbol{r}} \right)} \right|}^2}{\rm{d}}\mathit{\boldsymbol{r}}} \equiv \int {p(\mathit{\boldsymbol{r}})} I(\mathit{\boldsymbol{r}}){\rm{d}}\mathit{\boldsymbol{r}}\\ \;\;\;\;\;\;\;\; = I\left[ p \right] + 4\int {p(\mathit{\boldsymbol{r}})} {[\nabla \phi (\mathit{\boldsymbol{r}})]^2}{\rm{d}}\mathit{\boldsymbol{r}} \equiv \int {p(\mathit{\boldsymbol{r}})} \left[ {{I_p}\left( \mathit{\boldsymbol{r}} \right) + {I_\phi }\left( \mathit{\boldsymbol{r}} \right)} \right]{\rm{d}}\mathit{\boldsymbol{r}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \equiv I\left[ p \right] + I\left[ \phi \right]\\ \;\;\;\;\;\;\;\; = I\left[ p \right] + {\left( {2m/\hbar } \right)^2}\int {{\mathit{\boldsymbol{j}}^2}\left( \mathit{\boldsymbol{r}} \right)/p\left( \mathit{\boldsymbol{r}} \right){\rm{d}}\mathit{\boldsymbol{r}}} = \int {p(\mathit{\boldsymbol{r}})} \left[ {{I_p}\left( \mathit{\boldsymbol{r}} \right) + {I_j}\left( \mathit{\boldsymbol{r}} \right)} \right]{\rm{d}}\mathit{\boldsymbol{r}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = I\left[ p \right] + I\left[ \mathit{\boldsymbol{j}} \right]\;\;\;\;\;\;{\rm{or}} \end{array} $ (10)

    $ \begin{array}{l} \tilde I\left[ \psi \right] = I\left[ p \right] - I\left[ \phi \right] \equiv \tilde I\left[ p \right] + \tilde I\left[ \phi \right] = \int {p(\mathit{\boldsymbol{r}})} \left[ {{{\tilde I}_p}\left( \mathit{\boldsymbol{r}} \right) + {{\tilde I}_\phi }\left( \mathit{\boldsymbol{r}} \right)} \right]{\rm{d}}\mathit{\boldsymbol{r}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \int {p(\mathit{\boldsymbol{r}})} \tilde I\left( \mathit{\boldsymbol{r}} \right){\rm{d}}\mathit{\boldsymbol{r}},\;\;\;{\rm{and}} \end{array} $ (11)

    $ \begin{array}{l} S\left[ \psi \right] = - \left\langle {\psi \left| {\ln p + 2\phi } \right|\psi } \right\rangle \equiv \left\langle {\psi \left| {{\rm{\hat S}}} \right|\psi } \right\rangle = \int {\psi {{\left( \mathit{\boldsymbol{r}} \right)}^ * }{\rm{\hat S}}\left( \mathit{\boldsymbol{r}} \right)\psi \left( \mathit{\boldsymbol{r}} \right){\rm{d}}\mathit{\boldsymbol{r}}} \\ \;\;\;\;\;\;\;\; = \int {p\left( \mathit{\boldsymbol{r}} \right)\left[ {{S_p}\left( \mathit{\boldsymbol{r}} \right) - 2\phi \left( \mathit{\boldsymbol{r}} \right)} \right]{\rm{d}}\mathit{\boldsymbol{r}}} \equiv \int {p\left( \mathit{\boldsymbol{r}} \right)\left[ {{S_p}\left( \mathit{\boldsymbol{r}} \right) + {S_\phi }\left( \mathit{\boldsymbol{r}} \right)} \right]{\rm{d}}\mathit{\boldsymbol{r}}} \\ \;\;\;\;\;\;\;\; = \int {p(\mathit{\boldsymbol{r}})} S\left( \mathit{\boldsymbol{r}} \right){\rm{d}}\mathit{\boldsymbol{r}} \equiv S\left[ p \right] + S\left[ \phi \right]. \end{array} $ (12)

    We have introduced above the resultant densities-per-electron of the gradient determinicity-information I[ψ],

    $ I\left( \mathit{\boldsymbol{r}} \right) = \left[ {\nabla \ln p{{\left( \mathit{\boldsymbol{r}} \right)}^2}} \right] + 4{\left[ {\nabla \phi \left( \mathit{\boldsymbol{r}} \right)} \right]^2}, $ (13)

    of the indeterminicity-information $\tilde I[\psi]$, also called the gradient-entropy,

    $ \tilde I\left( \mathit{\boldsymbol{r}} \right) = {\left[ {\nabla \ln p\left( \mathit{\boldsymbol{r}} \right)} \right]^2} - 4{\left[ {\nabla \phi \left( \mathit{\boldsymbol{r}} \right)} \right]^2} $ (14)

    and of the global-entropy S[ψ]:

    $ S\left( \mathit{\boldsymbol{r}} \right) = - \left[ {\ln p\left( \mathit{\boldsymbol{r}} \right) + 2\phi \left( \mathit{\boldsymbol{r}} \right)} \right]. $ (15)

    The resultant gradient-information of Eq.(10), the expectation value of the Hermitian operator

    $ \rm{\hat I} = - 4{\nabla ^2} = \left( {8m/{\hbar ^2}} \right){\rm{\hat T,}} $

    is proportional to the average kinetic energy T[ψ] = < ψ|${\rm{\hat T}}$|ψ> corresponding to the operator ${\rm{\hat T}}(\boldsymbol{r}) =- \, [{\hbar ^2}/(2m)]\, \Delta $,

    $ \tilde I\left[ \psi \right] \equiv \left\langle {\psi \left| {{\rm{\hat I}}} \right|\psi } \right\rangle = \left( {8m/{\hbar ^2}} \right)T\left[ \psi \right], $

    and reflects the state gradient-deterministic aspect. Notice that the operator in Eq.(12) generating the resultant global-entropy is also Hermitian. The nonclassical contributions to densities-per-electron of the resultant gradient-information [Eq.(10)] and the global-entropy [Eq.(12)] also obey Eq.(9):

    $ {I_\phi }\left( \mathit{\boldsymbol{r}} \right) = {\left[ {\nabla {S_\phi }\left( \mathit{\boldsymbol{r}} \right)} \right]^2}, $ (16)

    while the nonclassical entropy densities of the Shannon and Fisher type satisfy the modified relation:

    $ {{\tilde I}_\varphi }\left( \mathit{\boldsymbol{r}} \right) = - {\left[ {\nabla {S_\phi }\left( \mathit{\boldsymbol{r}} \right)} \right]^2}. $ (17)

    The explanation of this sign change calls for the complex-entropy concept3, 9, with the phase-entropy component Sϕ(r) being attributed to its imaginary part:

    $ H\left( \mathit{\boldsymbol{r}} \right) = {S_p}\left( \mathit{\boldsymbol{r}} \right) + {\rm{i}}{S_\phi }\left( \mathit{\boldsymbol{r}} \right) = - \left[ {\ln p\left( \mathit{\boldsymbol{r}} \right) + 2{\rm{i}}\phi \left( \mathit{\boldsymbol{r}} \right)} \right]. $ (18)

    This generalized entropy follows naturally from the classical concept when one refers to the logarithmic function of the complex argument z = |z|exp(iα),

    $ {\rm{ln}}z = {\rm{ln}}\left| z \right| + {\rm{i}}\alpha . $ (19)

    The resultant complex-entropy now reflects the expectation value of the non-Hermitian (multiplicative) logarithmic operator

    $ \begin{array}{l} \hat {\mathcal{H}}\left( \mathit{\boldsymbol{r}} \right) = - 2\ln \psi \left( \mathit{\boldsymbol{r}} \right) = - \left[ {\ln p\left( \mathit{\boldsymbol{r}} \right) + 2{\rm{i}}\phi \left( \mathit{\boldsymbol{r}} \right)} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \equiv {H_p}\left( \mathit{\boldsymbol{r}} \right) + {H_\phi }\left( \mathit{\boldsymbol{r}} \right), \end{array} $ (20)

    $ \begin{array}{l} H\left[ \psi \right] \equiv \left\langle {\psi \left| {\hat {\mathcal{H}}} \right|\psi } \right\rangle = \left\langle {\psi \left| { - 2\ln \psi } \right|\psi } \right\rangle = S\left[ p \right] + {\rm{i}}S\left[ \phi \right]\\ \;\;\;\;\;\;\;\;\;\; = S\left[ R \right] + {\rm{i}}S\left[ \phi \right] \equiv H\left[ p \right] + H\left[ \phi \right]. \end{array} $ (21)

    The complex entropy3, 9 thus provides a natural complex-amplitude generalization of the familiar Shannon measure2 of the entropy content in the probability distribution.

    To summarize, the Hermitian information operator Î = -4▽2=(8m/ħ2)${\rm{\hat T}}$ gives rise to the real expectation value of the state resultant Fisher-type determinicity-information content I[ψ], while the non-Hermitian entropy operator ${\hat{\mathcal{H}}}$(r)= -2lnψ(r) generates the complex average quantity of the gradient indeterminicity-information H[ψ]. The densities of these functionals now obey the classical relation between the information and entropy components:

    $ \begin{array}{l} {{\tilde I}_p}\left( \mathit{\boldsymbol{r}} \right) = {\left[ {\nabla {H_p}\left( \mathit{\boldsymbol{r}} \right)} \right]^2}\;\;\;{\rm{and}}\\ {{\tilde I}_\phi }\left( \mathit{\boldsymbol{r}} \right) = {\left[ {\nabla {H_\phi }\left( \mathit{\boldsymbol{r}} \right)} \right]^2} = {\left[ {{\rm{i}}\nabla {S_\phi }\left( \mathit{\boldsymbol{r}} \right)} \right]^2} = - {\left[ {\nabla {S_\phi }\left( \mathit{\boldsymbol{r}} \right)} \right]^2}. \end{array} $ (22)

    3 Equilibrium states

    The phase-equilibrium state of an electron corresponds to the maximum of resultant measures of the global-or gradient-entropy content in |ψ>:

    $ \begin{align} & {{\delta }_{\psi }}\left\{ \left\langle \psi \left| {\rm{\hat{S}}} \right|\psi \right\rangle -\lambda \left\langle \psi \left| \psi \right. \right\rangle \right\}=0\ \ \ \ \ \ \ \ \ \ \rm{or} \\ & {{\delta }_{\psi }}\left\{ \left\langle \psi \left| {\rm{\hat{\tilde{I}}}} \right|\psi \right\rangle -\kappa \left\langle \psi \left| \psi \right. \right\rangle \right\}=0, \\ \end{align} $ (23)

    where λ and κ denote the relevant Lagrange multipliers associated with the ("geometric") constraint of the wavefunction normalization3-8. These information principles have been applied to the Schrödinger states ψ(r) giving the specified probability distribution p(r), ψp, to determine the optimum ("thermodynamic") phase

    $ {\phi _{eq}}\left[ {p;\mathit{\boldsymbol{r}}} \right] = - \left( {1/2} \right)\ln p\left( \mathit{\boldsymbol{r}} \right) \equiv {\phi _{eq}}\left( \mathit{\boldsymbol{r}} \right), $ (24)

    which defines the "thermodynamic" (phase-transformed) equilibrium state:

    $ {\psi _{eq}}\left[ {p;\mathit{\boldsymbol{r}}} \right] = \psi \left( \mathit{\boldsymbol{r}} \right)\exp \left[ {{\rm{i}}{\phi _{eq}}\left( \mathit{\boldsymbol{r}} \right)} \right] \equiv {\psi _{eq}}\left( \mathit{\boldsymbol{r}} \right). $ (25)

    In general N-electron states of the HZM construction the resultant equilibrium phase of the occupied EO also contains the orthogonality-phase term, which assures the independence of these molecular orbitals (MO)3-8 besides thermodynamic contribution of Eq.(24).

    The entropic phase-transformation has been added on top of the preceding energy-optimization stage, which generates SE, with the entropy/information and energy principles remaining totally decoupled in this two-stage approach. Although the two states ψ and ψeq generate the same probability distribution,

    $ p\left( \mathit{\boldsymbol{r}} \right) \equiv {\left| {\psi \left( \mathit{\boldsymbol{r}} \right)} \right|^2} = {p_{eq}}\left( r \right) \equiv {\left| {{\psi _{eq}}\left( \mathit{\boldsymbol{r}} \right)} \right|^2}, $ (26)

    their electronic energies differ by the nonclassical, (phase/current)-dependent term in the average kinetic energy of an electron:

    $ \begin{array}{l} E\left[ {{\psi _{eq}}} \right] = \left\langle {{\psi _{eq}}\left| {{\rm{\hat H}}} \right|{\psi _{eq}}} \right\rangle \\ \;\;\;\;\;\;\;\;\;\;\;\; = \left\langle {\psi \left| {{\rm{\hat H}}} \right|\psi } \right\rangle + \left[ {{\hbar ^2}/\left( {2m} \right)} \right]\int {p\left( \mathit{\boldsymbol{r}} \right){{\left[ {\nabla {\phi _{eq}}\left( \mathit{\boldsymbol{r}} \right)} \right]}^2}{\rm{d}}\mathit{\boldsymbol{r}}} \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \equiv E\left[ \psi \right] + T\left[ {{\phi _{eq}}} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\; = E\left[ \psi \right] + \left( {m/2} \right)\int {p\left( \mathit{\boldsymbol{r}} \right){{\left[ {{\mathit{\boldsymbol{j}}_{eq}}\left( \mathit{\boldsymbol{r}} \right)/p\left( \mathit{\boldsymbol{r}} \right)} \right]}^2}{\rm{d}}\mathit{\boldsymbol{r}}} \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \equiv E\left[ \psi \right] + T\left[ {{\mathit{\boldsymbol{j}}_{eq}}} \right], \end{array} $ (27)

    with the equilibrium probability-current being determined by the distribution gradient:

    $ {\mathit{\boldsymbol{j}}_{eq}}\left[ p \right] = \left( {\hbar /m} \right)p\nabla {\phi _{eq}}\left[ p \right] = - \left[ {\hbar /\left( {2m} \right)} \right]\nabla p. $ (28)

    In the ground-state ψ0(N) of N-electron systemthe electronic energy is determined by the functional of electronic density ρ0(N) = Np0(N),

    $ {E_v}\left[ {{\Psi _0}\left( N \right)} \right] = {E_v}\left[ {{\rho _0}\left( N \right)} \right] \equiv \int {p\left( \mathit{\boldsymbol{r}} \right)v\left( \mathit{\boldsymbol{r}} \right){\rm{d}}\mathit{\boldsymbol{r}}} + F\left[ {{\rho _0}} \right] = E\left[ {N,v} \right], $ (29)

    where the universal functional F[ρ0] generates the sum of electron kinetic and repulsion energies. This electron distribution also marks the equalized local chemical potential μ0(r) at the global level μ[N, v],

    $ \begin{array}{l} {\mu _0}\left( \mathit{\boldsymbol{r}} \right) \equiv {\left( {\frac{{\delta {E_v}\left[ \rho \right]}}{{\delta \rho \left( \mathit{\boldsymbol{r}} \right)}}} \right)_{{\rho _0}}} = v\left( \mathit{\boldsymbol{r}} \right) + {\left( {\frac{{\delta F\left[ \rho \right]}}{{\delta \rho \left( \mathit{\boldsymbol{r}} \right)}}} \right)_{{\rho _0}}} = \mu \left[ {{\rho _0}} \right]\\ \;\;\;\;\;\;\;\;\; = \mu \left[ {N,v} \right] \equiv {\left( {\partial E\left[ {N,v} \right]/\partial N} \right)_v}. \end{array} $ (30)

    The phase transformation of Eq.(25) modifies this local energy-intensity:

    $ \begin{array}{l} {\mu _{eq}}\left( \mathit{\boldsymbol{r}} \right) = \mu \left[ {{\rho _0}} \right] + {\left( {\frac{{\delta T\left[ {{\phi _{eq}}\left[ \rho \right]} \right]}}{{\delta \rho \left( \mathit{\boldsymbol{r}} \right)}}} \right)_{{\rho _0}}}\\ = \mu \left[ {N,v} \right] + \left[ {{\hbar ^2}/\left( {8mN} \right)} \right]\left\{ {{{\left[ {{\rho _0}{{\left( \mathit{\boldsymbol{r}} \right)}^{ - 1}}\nabla {\rho _0}\left( \mathit{\boldsymbol{r}} \right)} \right]}^2} - 2{\rho _0}{{\left( \mathit{\boldsymbol{r}} \right)}^{ - 1}}\Delta {\rho _0}\left( \mathit{\boldsymbol{r}} \right)} \right\}\\ \equiv \mu \left[ {N,v} \right] + {\mu ^{nclass}}\left[ {{\rho _0},\mathit{\boldsymbol{r}}} \right]. \end{array} $ (31)

    Therefore, the phase-equilibrium of the electron configuration ψ0eq(N) corresponding to the Slater determinant constructed from N lowest (occupied) equilibrium Kohn-Sham (KS)11 MO of Eq.(25), exhibiting a common thermodynamic phase of Eq.(24) for the ground-state electron distribution ρ0 = Np0, differs from the original energy-equilibrium ground-state configuration ψ0(N) = ψ0[ρ0] constructed from the original KS orbitals. Indeed, the latter corresponds to the vanishing spatial phase, ϕ[r0] = 0, and hence to the zero electronic current, j[ρ0] = 0, while the phase-displaced ground-state exhibits a finite equilibrium current jeq[ρ0] = Njeq[p0] = -[ħ/(2m)]▽ρ0, and hence also the nonvanishing density-source term σeq(r) = σeq[p0; r],

    $ \begin{array}{l} {\mathit{\boldsymbol{\sigma }}_{eq}}\left( \mathit{\boldsymbol{r}} \right) = {\rm{d}}\rho \left( \mathit{\boldsymbol{r}} \right)/{\rm{d}}t\left| {_{eq}} \right. = \partial \rho \left( \mathit{\boldsymbol{r}} \right)/\partial t\left| {_{eq}} \right. + \nabla \cdot {\mathit{\boldsymbol{j}}_{eq}}\left( \mathit{\boldsymbol{r}} \right) = \nabla \cdot {\mathit{\boldsymbol{j}}_{eq}}\left( \mathit{\boldsymbol{r}} \right)\\ \;\;\;\;\;\;\;\;\;\; = - \left[ {{\hbar ^2}/\left( {2m} \right)} \right]\Delta {\rho _0}, \end{array} $

    in the density-continuity relation for the phase-equilibrium state:

    $ \partial \rho \left( \mathit{\boldsymbol{r}} \right)/\partial t\left| {_{eq}} \right. = - \nabla \cdot {\mathit{\boldsymbol{j}}_{eq}}\left( \mathit{\boldsymbol{r}} \right) + {\mathit{\boldsymbol{\sigma }}_{eq}}\left( \mathit{\boldsymbol{r}} \right) = \partial \rho \left( \mathit{\boldsymbol{r}} \right)/\partial t\left| {_0} \right. = - \nabla \cdot \mathit{\boldsymbol{j}}\left( \mathit{\boldsymbol{r}} \right) = 0. $ (32)

    The current jeq(r), "promoted" by the phase transformation in response to the chemical-potential (electronegativity) differentiation of Eq.(31), thus preserves in time the original (stationary) electron distribution.

    A truly "thermodynamic" way56 of combining the energetic and entropic aspects of molecular electronic structure calls for the maximum-entropy principle globally-coupled to the electronic energy. For example, in the single-electron system such variational rules read:

    $ \begin{align} & \delta \left\{ \left\langle \psi \left| {\rm{\hat{S}}} \right|\psi \right\rangle -\tau _{S}^{-1}\left\langle \psi \left| {\rm{\hat{H}}} \right|\psi \right\rangle -\lambda \left\langle \psi \left| \psi \right. \right\rangle \right\}=0\ \ \ \ \ \ \ \text{or} \\ & \delta \left\{ \left\langle \psi \left| {\rm{\hat{\tilde{I}}}} \right|\psi \right\rangle -\tau _{I}^{-1}\left\langle \psi \left| {\rm{\hat{H}}} \right|\psi \right\rangle -\kappa \left\langle \psi \left| \psi \right. \right\rangle \right\}=0, \\ \end{align} $ (33)

    with the Lagrange multipliers of the energy-constraint reflecting inverses of the associated information "temperatures". The latter are defined by thermodynamic-like derivatives of the average electronic energy with respect to the resultant entropy measure S[ψ] or $\tilde{I}[\psi]$,

    $ {\tau _S} = \partial E\left[ \psi \right]/\partial S\left[ \psi \right]\left| {_{eq}} \right.\;\;\;{\rm{or}}\;\;\;\;{\tau _I} = \partial E\left[ \psi \right]/\partial \tilde I\left[ \psi \right]\left| {_{eq}} \right.. $ (34)

    These (energy-constrained) variational principles, with respect to the (complex) trial state |ψ> for the adopted measure of the resultant entropy and specified probability distribution p0, give the associated Euler equations determining the equilibrium state |ψeq[ p0]> ≡ |ψeq>:

    $ \begin{align} & \left( \rm{\hat{S}}-\tau _{\text{S}}^{\text{-1}}\rm{\hat{H}} \right)\left| {{\psi }^{eq}} \right\rangle =\lambda \left| {{\psi }^{eq}} \right\rangle \ \ \ \ \ \ \ \ \text{and} \\ & \left( \rm{\hat{\tilde{I}}}-\tau _{\text{I}}^{\text{-1}}\rm{\hat{H}} \right)\left| {{\psi }^{eq}} \right\rangle =\kappa \left| {{\psi }^{eq}} \right\rangle . \\ \end{align} $ (35)

    They can be fulfilled only when ${\rm{\hat S}}$ (or$\widehat {\widetilde {\text{I}}}$) and ${\rm{\hat H}}$ have the common set of eigenvectors. However, since these operators do not commute, one ultimately concludes that such global "thermodynamic" relations cannot be satisfied within the pure-state QM.

    Indeed, by the familiar variational principle of QM any attempt to conserve energy, while modifying the system (exact) wavefunction ψ0(r) of the nondegenerate ground-state,

    $ {\rm{\hat H}}\left( \mathit{\boldsymbol{r}} \right){\psi _0}\left( \mathit{\boldsymbol{r}} \right) = {E_0}{\psi _0}\left( \mathit{\boldsymbol{r}} \right) \equiv {\varepsilon _0}\left( \mathit{\boldsymbol{r}} \right){\psi _0}\left( \mathit{\boldsymbol{r}} \right), $ (36)

    where the state local energy

    $ {\varepsilon _0}\left( \mathit{\boldsymbol{r}} \right) = \left[ {{\rm{\hat H}}\left( \mathit{\boldsymbol{r}} \right){\psi _0}\left( \mathit{\boldsymbol{r}} \right)} \right]/{\psi _0}\left( \mathit{\boldsymbol{r}} \right), $ (37)

    e.g., as a result of the "thermodynamic" phase-transformation of Eq.(25), must fail since such manipulations always raise the energy above E0: E[ψeq] > E0. This implies that the global variational principles of Eq.(33) can be satisfied only for the exactly vanishing "thermodynamic" phase ϕeq(r) = 0 which follows directly from the stationary SE (36).

    One can also contemplate the local-coupling between densities of the information entropy and electronic energy:

    $ \begin{align} & \delta \left\{ \int{{{\psi }^{*}}\left( \mathit{\boldsymbol{r}} \right)\left[ \rm{\hat{S}}\left( \mathit{\boldsymbol{r}} \right)-{{\tau }_{S}}{{\left( \mathit{\boldsymbol{r}} \right)}^{-1}}\rm{\hat{H}}\left( \mathit{\boldsymbol{r}} \right) \right]\psi \left( \mathit{\boldsymbol{r}} \right)\rm{d}\mathit{\boldsymbol{r}}-\lambda \int{{{\psi }^{*}}\left( \mathit{\boldsymbol{r}} \right)\psi \left( \mathit{\boldsymbol{r}} \right)\rm{d}\mathit{\boldsymbol{r}}}} \right\} \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \equiv \delta \left\{ \int{p\left( \mathit{\boldsymbol{r}} \right)\left[ S\left( \mathit{\boldsymbol{r}} \right)-{{\tau }_{S}}{{\left( \mathit{\boldsymbol{r}} \right)}^{-1}}\varepsilon \left( \mathit{\boldsymbol{r}} \right) \right]\rm{d}\mathit{\boldsymbol{r}}}-\lambda \int{p\left( \mathit{\boldsymbol{r}} \right)\rm{d}\mathit{\boldsymbol{r}}} \right\}=0\ \ \ \ \rm{or} \\ & \delta \left\{ \int{{{\psi }^{*}}\left( \mathit{\boldsymbol{r}} \right)\left[ \rm{\hat{\tilde{I}}}\left( \mathit{\boldsymbol{r}} \right)-{{\tau }_{I}}{{\left( \mathit{\boldsymbol{r}} \right)}^{-1}}\rm{\hat{H}}\left( \mathit{\boldsymbol{r}} \right) \right]\psi \left( \mathit{\boldsymbol{r}} \right)\rm{d}\mathit{\boldsymbol{r}}-\kappa \int{{{\psi }^{*}}\left( \mathit{\boldsymbol{r}} \right)\psi \left( \mathit{\boldsymbol{r}} \right)\rm{d}\mathit{\boldsymbol{r}}}} \right\} \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \equiv \delta \left\{ \int{p\left( \mathit{\boldsymbol{r}} \right)\left[ \tilde{I}\left( \mathit{\boldsymbol{r}} \right)-{{\tau }_{I}}{{\left( \mathit{\boldsymbol{r}} \right)}^{-1}}\varepsilon \left( \mathit{\boldsymbol{r}} \right) \right]\rm{d}\mathit{\boldsymbol{r}}}-\kappa \int{p\left( \mathit{\boldsymbol{r}} \right)\rm{d}\mathit{\boldsymbol{r}}} \right\}=0. \\ \end{align} $ (38)

    The local "temperature" descriptors τS(r) and τI(r) now represent ratios of the corresponding functional derivatives of E[ψeq], S[ψeq] or $\tilde I[{\psi _{eq}}]$,

    $ \begin{array}{l} {\tau _S}\left( \mathit{\boldsymbol{r}} \right) = {\left( {\frac{{\delta E\left[ \psi \right]}}{{\delta {\psi ^ * }\left( \mathit{\boldsymbol{r}} \right)}}} \right)_{{\psi ^{eq}}}}/{\left( {\frac{{\delta S\left[ \psi \right]}}{{\delta {\psi ^ * }\left( \mathit{\boldsymbol{r}} \right)}}} \right)_{{\psi ^{eq}}}}\\ \;\;\;\;\;\;\;\; = \left[ {{\rm{\hat H}}\left( \mathit{\boldsymbol{r}} \right){\psi ^{eq}}\left( \mathit{\boldsymbol{r}} \right)} \right]/\left[ {{\rm{\hat S}}\left( \mathit{\boldsymbol{r}} \right){\psi ^{eq}}\left( \mathit{\boldsymbol{r}} \right)} \right] \equiv {\varepsilon _0}\left( \mathit{\boldsymbol{r}} \right)/{S_0}\left( \mathit{\boldsymbol{r}} \right), \end{array} $ (39)

    $ \begin{align} & {{\tau }_{I}}\left( \mathit{\boldsymbol{r}} \right)={{\left( \frac{\delta E\left[ \psi \right]}{\delta {{\psi }^{*}}\left( \mathit{\boldsymbol{r}} \right)} \right)}_{{{\psi }^{eq}}}}/{{\left( \frac{\delta \tilde{I}\left[ \psi \right]}{\delta {{\psi }^{*}}\left( \mathit{\boldsymbol{r}} \right)} \right)}_{{{\psi }^{eq}}}} \\ & \ \ \ \ \ \ \ \ =\left[ \rm{\hat{H}}\left( \mathit{\boldsymbol{r}} \right){{\psi }^{eq}}\left( \mathit{\boldsymbol{r}} \right) \right]/\left[ \rm{\hat{\tilde{I}}}\left( \mathit{\boldsymbol{r}} \right){{\psi }^{eq}}\left( \mathit{\boldsymbol{r}} \right) \right]\equiv {{\varepsilon }_{0}}\left( \mathit{\boldsymbol{r}} \right)/{{{\tilde{I}}}_{0}}\left( \mathit{\boldsymbol{r}} \right), \\ \end{align} $ (40)

    which reflect the effective densities ε0(r), S0(r) or ${\tilde I_0}(\boldsymbol{r})$ of these properties:

    $ \begin{align} & \left[ \rm{\hat{S}}\left( \mathit{\boldsymbol{r}} \right){{\psi }_{0}}\left( \mathit{\boldsymbol{r}} \right)/{{\psi }_{0}}\left( \mathit{\boldsymbol{r}} \right) \right]\equiv {{S}_{0}}\left( \mathit{\boldsymbol{r}} \right)\ \ \ \ \ \ \ \ \rm{and} \\ & \left[ \rm{\hat{\tilde{I}}}\left( \mathit{\boldsymbol{r}} \right){{\psi }_{0}}\left( \mathit{\boldsymbol{r}} \right)/{{\psi }_{0}}\left( \mathit{\boldsymbol{r}} \right) \right]\equiv {{{\tilde{I}}}_{0}}\left( \mathit{\boldsymbol{r}} \right). \\ \end{align} $ (41)

    However, in the variational principle (38) the fixing of ε0(r) at each point also implies an effective total energy constraint in the "thermodynamic" phase-transformation,

    $ \begin{array}{l} \int {{p_0}\left( \mathit{\boldsymbol{r}} \right){\varepsilon _0}\left( \mathit{\boldsymbol{r}} \right){\rm{d}}\mathit{\boldsymbol{r}}} = \int {{p_0}\left( \mathit{\boldsymbol{r}} \right){\varepsilon _0}\left( \mathit{\boldsymbol{r}} \right){\rm{d}}\mathit{\boldsymbol{r}}} \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \int {{\psi _0}\left( \mathit{\boldsymbol{r}} \right){\rm{\hat H}}\left( \mathit{\boldsymbol{r}} \right){\psi _0}\left( \mathit{\boldsymbol{r}} \right){\rm{d}}\mathit{\boldsymbol{r}}} = {E_0}, \end{array} $ (42)

    which is precluded by the energy variational principle of QM.

    One thus concludes, that the coupled (global or local) variational principles combining the energetical and entropic aspects of electronic states are precluded by principles of molecular QM. To further illustrate this point, consider the Euler equations generated by the local entropic principles, which determine the unknown spatial phase ϕeq[p0; r] ≡ ϕeq(r) of the electronic wavefunction ψeq(r)=ψ0(r)exp[iϕeq(r)] for the prescribed ground-state distribution p0(r),

    $ \begin{array}{l} \left[ {{S_0}\left( \mathit{\boldsymbol{r}} \right) - {\tau _S}{{\left( \mathit{\boldsymbol{r}} \right)}^{ - 1}}{\rm{\hat H}}\left( \mathit{\boldsymbol{r}} \right)} \right]{\psi ^{eq}}\left( \mathit{\boldsymbol{r}} \right) = \lambda {\psi ^{eq}}\left( \mathit{\boldsymbol{r}} \right)\;\;\;\;\;\;\;{\rm{or}}\\ \left[ {{{\tilde I}_0}\left( \mathit{\boldsymbol{r}} \right) - {\tau _I}{{\left( \mathit{\boldsymbol{r}} \right)}^{ - 1}}{\rm{\hat H}}\left( \mathit{\boldsymbol{r}} \right)} \right]{\psi ^{eq}}\left( \mathit{\boldsymbol{r}} \right) = \kappa {\psi ^{eq}}\left( \mathit{\boldsymbol{r}} \right), \end{array} $ (43)

    $ \begin{array}{l} {S_0}\left( \mathit{\boldsymbol{r}} \right) = - \ln {p_0}\left( \mathit{\boldsymbol{r}} \right) - 2{\phi ^{eq}}\left( \mathit{\boldsymbol{r}} \right),\\ {{\tilde I}_0}\left( \mathit{\boldsymbol{r}} \right) = {\left[ {\nabla \ln {p_0}\left( \mathit{\boldsymbol{r}} \right)} \right]^2} - {\left[ {2\nabla {\phi ^{eq}}\left( \mathit{\boldsymbol{r}} \right)} \right]^2},\\ {\rm{\hat H}}{\psi ^{eq}} = \left\{ { - \left[ {{\hbar ^2}/\left( {2m} \right)} \right]\left[ {\Delta {\psi _0} + {\rm{i}}\left( {{\psi _0}\Delta {\phi ^{eq}} + 2\nabla {\psi _0} \cdot \nabla {\phi ^{eq}}} \right)} \right]} \right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {\left. { - {\psi _0}{{\left( {\nabla {\phi ^{eq}}} \right)}^2}} \right] + v{\psi _0}} \right\}\exp \left( {{\rm{i}}{\phi ^{eq}}} \right)\\ \;\;\;\;\;\;\;\;\; \equiv \left( {{\rm{\hat H}}{\psi _0}} \right)\exp \left( {{\rm{i}}{\phi ^{eq}}} \right) + {{{\rm{\hat T}}}^{nclass}}{\psi ^{eq}}. \end{array} $ (44)

    Let us further assume, for reasons of simplicity, that ψeqis by construction normalized so that the "geometrical" constraints are redundant: λ = κ = 0. It then follows that τS(r) = E0/S0(r) and τI(r) = E0/${\tilde I_0}(\boldsymbol{r})$, so that these local Euler equations indeed reduce to the energy eigenvalue problem of the stationary SE:

    $ {\rm{\hat H}}{\psi ^{eq}} = {E_0}{\psi ^{eq}} \Rightarrow {\rm{\hat H}}{\psi _0} = {E_0}{\psi _0},\;\;{\rm{i}}.{\rm{e}}.,\;\;\;{\phi ^{eq}} = 0. $ (45)

    Since the given particle distribution ρ0 = Np0 fixes the classical, DFT part Ev[ρ0] of the state electronic energy, one finally observes that the entropic optimization determining the equilibrium phase ϕeq[p0; r] resembles the thermodynamic criterion: one searches for the maximum of the resultant entropy in a trial (complex) state for the fixed value of the DFT-energy Ev[ρ0]. The strict criterion of the conserved internal energy of thermodynamics thus becomes relaxed in the local "thermodynamic" description by the physical constraint of the fixed value of the state classical (DFT) energy.

    4 Hypothetical stages in reactive systems

    It is customary to view the interaction between reactants in a bimolecular reactive system R=A-B composed of the complementary acidic (A) and basic (B) subsystems, together comprising of NR = NA + NB electrons, in several intermediate stages involving the mutually closed (nonbonded, disentangled) or open (bonded, entangled) reactants14, 15, 36, 38, 40, 41:

    The reference state of the dissociated (entangled) fragments {α[*](Ψ, ∞), α = A, B} in the Separated-Reactant Limit (SRL) Rb(Ψ) = A0(Ψ)$\underleftrightarrow \infty $B0(Ψ), represents a collection of the infinitely-separated (noninteracting) but "bonded" fragments (having a common molecular "ancestor"-stateΨ), and separately exhibiting the ground-state densities {ρα0 = ρα0[Nα0, vα]} of free subsystems {α0}, the equilibrium distributions for the subsystem (integer) number of electrons Nα0 and the external potential vα due its own nuclei. These electron densities also define the associated probability distributions, the subsystem-normalized shape factors of the isolated fragments {α0},

    $ p_\alpha ^0\left( \mathit{\boldsymbol{r}} \right) = \rho _\alpha ^0\left( r \right)/N_\alpha ^0,\;\;\;\;\;\;\int {p_\alpha ^0\left( \mathit{\boldsymbol{r}} \right){\rm{d}}\mathit{\boldsymbol{r}}} = 1, $ (46)

    or the associated R-normalized densities of subsystems in Rb,

    $ \begin{array}{l} \pi _\alpha ^0\left( \mathit{\boldsymbol{r}} \right) = \rho _\alpha ^0\left( \mathit{\boldsymbol{r}} \right)/{N_{\rm{R}}} = \left( {N_\alpha ^0/{N_{\rm{R}}}} \right)\left[ {\rho _\alpha ^0\left( \mathit{\boldsymbol{r}} \right)/N_\alpha ^0} \right] \equiv P_\alpha ^0p_\alpha ^0\left( \mathit{\boldsymbol{r}} \right),\\ {\Sigma _\alpha }\int {\pi _\alpha ^0\left( \mathit{\boldsymbol{r}} \right){\rm{d}}\mathit{\boldsymbol{r}}} = {\Sigma _\alpha }P_\alpha ^0 = 1. \end{array} $ (47)

    Here Pα0= Nα0/NR stands for the condensed probability of subsystem α0 in Rb. In this "bonded" equilibrium system each fragment exhibits the "molecular" thermodynamic phase,

    $ \begin{array}{l} \phi _{\rm{R}}^\infty = \phi _{\rm{R}}^\infty \left[ {p_{\rm{R}}^\infty } \right],\\ p_{\rm{R}}^\infty = \left( {\rho _{\rm{A}}^0 + \rho _{\bf{B}}^0} \right)/{N_{\rm{R}}} \equiv \rho _{\bf{R}}^0/{N_{\rm{R}}} = {\Sigma _\alpha }\pi _\alpha ^0\left( \mathit{\boldsymbol{r}} \right), \end{array} $ (48)

    and equalized chemical potentials:

    $ \mu _{\rm{A}}^ * \left( {\Psi ,\infty } \right) = \mu _{\bf{B}}^ * \left( {\Psi ,\infty } \right) = {\mu _{\bf{R}}}\left[ {\rho _{\bf{R}}^0} \right]. $

    The isolated (disentangled, nonbonded) free reactants of Rn = A0 + B0 are uniquely characterized by their separate (intra-reactant equalized) levels of chemical potentials and thermodynamic phases determined by the isolated subsystem densities:

    $ \begin{array}{l} \mu _\alpha ^0 = \mu _\alpha ^0\left[ {N_\alpha ^0,{v_\alpha }} \right] = \mu _\alpha ^0\left[ {\rho _\alpha ^0} \right]\;\;\;\;{\rm{and}}\\ {\phi _{eq}}\left[ {\rho _\alpha ^0} \right] = \phi _\alpha ^0\left[ {N_\alpha ^0,{v_\alpha }} \right]. \end{array} $ (49)

    The following intermediate stages of interacting species are introduced, when both subsystems already feel the presence of each other at a finite mutual separation and the given relative orientation of the geometrically "frozen" reactants in R(RA-B) = A$\underleftrightarrow {{{\boldsymbol{\text{R}}}_{{\text{A-B}}}}}$B, specified by the external potential of R as a whole for the current values of reaction coordinate RA-B:

    $ {v_{\rm{R}}}\left( {{{\bf{R}}_{{\rm{A\_B}}}}} \right) = {v_{\rm{A}}}\left( {{{\bf{R}}_{{\rm{A\_B}}}}} \right) + {v_{\rm{B}}}\left( {{{\bf{R}}_{{\rm{A\_B}}}}} \right). $ (50)

    The promolecular (nonbonded, disentangled) reference Rn0=(A0|B0) corresponds to the electronically and geometrically "frozen" (mutually-closed) free sybsystems brought to RA-B, and separately exhibiting the same distributions as in Rn. The reactant distributions then determine the promolecular electron density:

    $ \begin{array}{l} \rho _{\rm{R}}^0\left( {{{\bf{R}}_{{\rm{A\_B}}}}} \right) = \rho _{\rm{A}}^0\left( {{{\bf{R}}_{{\rm{A\_B}}}}} \right) + \rho _{\rm{B}}^0\left( {{{\bf{R}}_{{\rm{A\_B}}}}} \right) \equiv N_{\rm{R}}^0\rho _{\rm{R}}^0\left( {{{\bf{R}}_{{\rm{A\_B}}}}} \right),\\ \int {\rho _{\rm{R}}^0\left( \mathit{\boldsymbol{r}} \right)d\mathit{\boldsymbol{r}}} = N_{\rm{R}}^0 = N_{\rm{A}}^0 + N_{\rm{B}}^0 = {N_{\rm{R}}}. \end{array} $ (51)

    No equalization of the fragment chemical potential takes place at this nonequilibrium stage.

    The nonbonded polarization stage Rn+ = (A+|B+) for the same nuclear positions RA-B consists of the mutually-closed (disentangled) reactants, internally relaxed electronically but geometrically rigid, characterized by their promoted densities {ρα+ = ρα+[NA0, NB0, vR] ≡ Nα0pα+}, the equilibrium distributions for the initial (integer) numbers of electrons in subsystems and the overall external potential of Rn+as a whole, which combine into the overall distribution of the polarized reactive system:

    $ \begin{array}{l} \rho _{\rm{R}}^ + \left( {{{\bf{R}}_{{\rm{A\_B}}}}} \right) = \rho _{\rm{A}}^ + \left( {{{\bf{R}}_{{\rm{A\_B}}}}} \right) + \rho _{\rm{B}}^ + \left( {{{\bf{R}}_{{\rm{A\_B}}}}} \right) \equiv {N_{\rm{R}}}p_{\rm{R}}^ + \left( {{{\bf{R}}_{{\rm{A\_B}}}}} \right),\\ \;\;\;\;\;\;\;\int {\rho _{\rm{R}}^ + \left( \mathit{\boldsymbol{r}} \right){\rm{d}}\mathit{\boldsymbol{r}}} = {N_{\rm{R}}}\;\;\;\;\;\;\;\;\;{\rm{or}}\\ \;\;\;\;\;\;\;p_{\rm{R}}^ + = p_{\rm{A}}^0p_{\rm{A}}^ + + p_{\rm{B}}^0p_{\rm{B}}^ + ,\;\;\;\int {p_{\rm{R}}^ + \left( \mathit{\boldsymbol{r}} \right){\rm{d}}\mathit{\boldsymbol{r}}} = 1. \end{array} $ (52)

    The internal equilibrium in each subsystem of Rn+implies the equalization of chemical potential (electronegativity) within each reactant at its separate overall level:

    $ \mu _\alpha ^ + \left[ {N_{\rm{A}}^0,N_{\rm{B}}^0,{v_{\rm{R}}}} \right] = \mu _\alpha ^ + \left[ {\rho _{\rm{A}}^ + ,\rho _{\rm{B}}^ + } \right],\;\;\;\;\mu _{\rm{A}}^ + \ne \mu _{\rm{B}}^ + . $ (53)

    At this stage the thermodynamic-phase "intensities" of subsystems are determined by their own polarized probability distributions:

    $ {\phi _{eq}}\left[ {{\alpha ^ + };\mathit{\boldsymbol{r}}} \right] = {\phi _{eq}}\left[ {\rho _\alpha ^ + } \right] = \phi _\alpha ^0\left[ {N_\alpha ^0,{v_{\rm{R}}}} \right],\;\;\;\alpha = {\rm{A}},{\rm{B}}{\rm{.}} $ (54)

    Finally, the inter-reactant equilibrium state, of the geometrically "frozen" but the mutually-open (bonded, electronically relaxed), entangled reactants in Rb(Ψ) = [A*(Ψ)¦B*(Ψ)], i.e., the ground state of the reactive system as a whole for the current reaction coordinate RA-B, is characterized by the effective subsystem densities

    $ \begin{array}{l} \rho _\alpha ^ * = \rho _\alpha ^ * \left[ {N_{\rm{A}}^ * ,N_{\rm{B}}^ * ,{v_{\rm{R}}}} \right] \equiv N_\alpha ^ * p_\alpha ^ * \equiv {N_{\rm{R}}}\pi _X^ * ,\\ \int {\rho _\alpha ^ * \left( \mathit{\boldsymbol{r}} \right){\rm{d}}\mathit{\boldsymbol{r}}} = N_\alpha ^ * ,\;\;\;{\Sigma _\alpha }N_\alpha ^ * = {N_{\rm{R}}}, \end{array} $ (55)

    which sum up to the equilibrium distribution of the whole reactive system:

    $ \rho _{\rm{A}}^ * + \rho _{\rm{B}}^ * = {\rho _{\rm{R}}}\left[ {{N_{\rm{R}}},{v_{\rm{R}}}} \right] \equiv {N_{\rm{R}}}{p_{\rm{R}}}\left[ {{N_{\rm{R}}},{v_{\rm{R}}}} \right]. $ (56)

    The effective densities of such bonded subsystems exhibit effects of the net (fractional) B →A Charge Transfer (CT),

    $ {N^{{\rm{CT}}}} = N_{\rm{A}}^ * - N_{\rm{A}}^0 = N_{\rm{B}}^0 - N_{\rm{B}}^ * > 0, $ (57)

    which equalizes the chemical potentials of both subsystems:

    $ \mu _{\rm{A}}^ * \left[ {N_{\rm{A}}^ * ,N_{\rm{B}}^ * ,{v_{\rm{R}}}} \right] = \mu _{\rm{B}}^ * \left[ {N_{\rm{A}}^ * ,N_{\rm{B}}^ * ,{v_{\rm{R}}}} \right] = {\mu _{\rm{R}}}\left[ {{N_{\rm{R}}},{v_{\rm{R}}}} \right]. $ (58)

    This stationary electron density of the whole reactive system generates thermodynamic phase of Rb(Ψ) as a whole,

    $ {\phi _{eq}}\left[ {{{\rm{R}}_b}\left( \Psi \right)} \right] = {\phi _{eq}}\left[ {{N_{\rm{R}}},{v_{\rm{R}}};\mathit{\boldsymbol{r}}} \right] = - \left( {{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}} \right)\ln {p_{\rm{R}}}\left( \mathit{\boldsymbol{r}} \right) = {\phi _{eq}}\left[ {{p_{\rm{R}}};\mathit{\boldsymbol{r}}} \right], $ (59)

    which also characterizes the entangled states of each bonded reactant α*:

    $ \begin{array}{l} {\phi _{eq}}\left( {{\alpha ^ * }} \right) \equiv \phi _\alpha ^ * \left[ {{{\rm{R}}_b}\left( \Psi \right)} \right] = \phi _\alpha ^ * \left[ {\left\{ {N_\beta ^ * } \right\},{v_{\rm{R}}}} \right] = {\phi _{eq}}\left[ {\left\{ {\rho _\beta ^ * } \right\}} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\; = {\phi _{eq}}\left[ {{\rho _{\rm{R}}}} \right] = {\phi _{eq}}\left[ {{{\rm{R}}_b}\left( \Psi \right)} \right],\;\;\;\;\;\;\alpha = {\rm{A}},{\rm{B}}{\rm{.}} \end{array} $ (60)

    In DFT the optimum wavefunctions for the specified electron density of N electrons, ρ = Np, are constructed using the HZM scheme3, 4, 53, 54. Such functions also appear in Levy's12 constrained-search definition of the universal density functional for the sum of the electron kinetic and repulsion energies. This DFT framework introduces the complete set of the density-conserving Slater determinants build using EO of the plane-wave-type:

    $ \left\{ {{\varphi _\mathit{\boldsymbol{k}}}\left( \mathit{\boldsymbol{r}} \right) = R\left( \mathit{\boldsymbol{r}} \right)\exp \left[ {{\rm{i}}{\mathit{\Phi }_\mathit{\boldsymbol{k}}}\left( {N;\mathit{\boldsymbol{r}}} \right)} \right]} \right\}. $ (61)

    They adopt equal, density-dependent modulus part R(r) = p(r)1/2 and the spatial phase function composed of the "orthogonality" [Fk(r)] and "thermodynamic" [ϕeq(N; r)] parts,

    $ {\mathit{\Phi }_\mathit{\boldsymbol{k}}}\left( {N;\mathit{\boldsymbol{r}}} \right) = \mathit{\boldsymbol{k}}\left[ p \right] \cdot \mathit{\boldsymbol{f}}\left[ {p;\mathit{\boldsymbol{r}}} \right] + {\phi _{eq}}\left[ {N,p;\mathit{\boldsymbol{r}}} \right] \equiv {F_\mathit{\boldsymbol{k}}}\left( \mathit{\boldsymbol{r}} \right) + {\phi _{eq}}\left( {N;\mathit{\boldsymbol{r}}} \right), $ (62)

    with the density-dependent vector function f[p; r] ≡ f(r) common to all orbitals and linked to the Jacobian of the rf(r) transformation, and the equilibrium phase from the maximum resultant-entropy principle3-8:

    $ \begin{array}{l} {\phi _{eq}}\left( {N;\mathit{\boldsymbol{r}}} \right) = - \left( {{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}} \right)\ln p\left( \mathit{\boldsymbol{r}} \right) - \mathit{\boldsymbol{K}}\left( {{\bf{k}}\left[ p \right]} \right) \cdot \mathit{\boldsymbol{f}}\left[ {p;\mathit{\boldsymbol{r}}} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\; \equiv {\phi _{eq}}\left( \mathit{\boldsymbol{r}} \right) - \mathit{\boldsymbol{K}}\left( {\bf{k}} \right) \cdot \mathit{\boldsymbol{f}}\left( \mathit{\boldsymbol{r}} \right),\;\;\;\;\;\;\mathit{\boldsymbol{K}}\left( {\bf{k}} \right) = {N^{ - 1}}{\Sigma _i}{\mathit{\boldsymbol{k}}_i}. \end{array} $ (63)

    Here, k = {ki} groups the reduced momenta (wave numbers) of all N occupied EO determining the average vector K(k). The resultant phase of Eqs. (61) and (62) thus reads:

    $ \begin{array}{l} {\mathit{\Phi }_\mathit{\boldsymbol{k}}}\left( {N;\mathit{\boldsymbol{r}}} \right) = \delta \mathit{\boldsymbol{k}} \cdot \mathit{\boldsymbol{f}}\left( \mathit{\boldsymbol{r}} \right) - \left( {{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}} \right)\ln p\left( \mathit{\boldsymbol{r}} \right),\\ \;\;\;\;\;\;\;\;\delta \mathit{\boldsymbol{k}} = \mathit{\boldsymbol{k}} - \mathit{\boldsymbol{K}}\left( {\bf{k}} \right),\;\;\;\;{\Sigma _i}\delta {\mathit{\boldsymbol{k}}_i} = 0. \end{array} $ (64)

    This spatial phase generates the resultant current of N electrons

    $ \mathit{\boldsymbol{j}}\left( {N;\mathit{\boldsymbol{r}}} \right) = - \left[ {\hbar /\left( {2m} \right)} \right]\nabla \rho \left( \mathit{\boldsymbol{r}} \right). $ (65)

    The equilibrium phase-transformation thus gives rise to the current "promotion" of both the molecular subsystems and the reactive system as a whole.

    The EO wave-vector (reduced momentum) k = k[p] and the density-dependent vector field f(r) result from the ordinary variational principle for the system minimum electronic energy in the familiar Self-Consistent-Field (SCF) theories, e.g., the Hartree-Fock (HF) or Kohn-Sham (KS) methods. The "thermodynamic" phase ϕeq(r), common to all occupied EO, is subsequently determined using the subsidiary maximum entropy principle of QIT.

    5 Information systems and their bond descriptors

    In OCT one adopts the molecular information system reflecting electronic "communications" between basis functions of typical SCF LCAO MO calculations, e.g., the (orthogonalized) Atomic Orbitals (AO) χ = (χ1, χ2, …, χm). The AO states |χ> then identify the associated varieties of the mutually exclusive, elementary events in the molecular bond-system determined by N occupied MO: φ = (φ1, φ2, …, φN). In what follows we consider the simplest, single-configuration approximation of HF or KS SCF calculations, with the occupied MO expanded in the AO basis: φ = χC; here the unitary matrix C, CC = I, groups coefficients of expanding the spatial (MO) parts φ of the singly-occupied spin-orbitals.

    The underlying conditional probabilities of the output AO events χ' = {χj}, given the input AO events χ = {χi}, which define the classical (probability) AO channel (see also Section 6),

    $ {\bf{P}}\left( {\mathit{\boldsymbol{\chi '}}\left| \mathit{\boldsymbol{\chi }} \right.} \right) = \left\{ {P\left( {{\chi _j}\left| {{\chi _i}} \right.} \right) \equiv P\left( {j\left| i \right.} \right) \equiv P\left( {i,j} \right)/{p_i} \equiv {{\left| {A\left( {j\left| i \right.} \right)} \right|}^2}} \right\}, $ (66)

    or the associated amplitudes A(χ'|χ) = {A(j|i)} defining the nonclassical network of scattering the emitting (input) states |χ> = {|χi>} among the monitoring (output) states |χ'> = {|χj>}, result from the bond-projected SP of QM; here P(χi, χj) ≡ P(i, j) stands for the joint probability of simultaneously observing χi and χj in the molecular bond-system φ. In AO representation this bond subspace defines the idempotent Charge and Bond-Order (CBO) matrix:

    $ {\bf{ \pmb{\mathsf{ γ}} }} = \left\{ {{\gamma _{i,j}}} \right\} = \left\langle {\mathit{\boldsymbol{\chi }}\left| \mathit{\boldsymbol{\varphi }} \right.} \right\rangle \left\langle {\mathit{\boldsymbol{\varphi }}\left| {\mathit{\boldsymbol{\chi '}}} \right.} \right\rangle = {\bf{C}}{{\bf{C}}^\dagger },\;\;\;\;\;\;\;{{\bf{ \pmb{\mathsf{ γ}} }}^2} = {\bf{ \pmb{\mathsf{ γ}} }}. $ (67)

    In OCT the molecular joint probabilities of AO, of the simultaneous input and output AO events (χi, χj) ≡ (i, j) in the bond-system of a molecule, are thus proportional to the square of the corresponding CBO matrix element:

    $ \begin{array}{l} P\left( {i,j} \right) = {\gamma _{i,j}}{\gamma _{j,i}}/N = {\left| {{\gamma _{i,j}}} \right|^2}/N,\\ {\Sigma _j}P\left( {i,j} \right) = {N^{ - 1}}{\Sigma _j}{\gamma _{i,j}}{\gamma _{j,i}} = {\gamma _{i,i}}/N = {p_i}. \end{array} $ (68)

    The conditional probabilities between AO,

    $ \begin{array}{l} {\bf{P}}\left( {\mathit{\boldsymbol{\chi '}}\left| \mathit{\boldsymbol{\chi }} \right.} \right) = \left\{ {P\left( {j\left| i \right.} \right) = P\left( {i,j} \right)/{p_i}} \right\},\\ P\left( {j\left| i \right.} \right) = {\left| {{\gamma _{i,j}}} \right|^2}/{\gamma _{i,i}} \equiv {\left| {A\left( {j\left| i \right.} \right)} \right|^2},\;\;\;\;\;\;\;\;{\Sigma _j}P\left( {j\left| i \right.} \right) = 1, \end{array} $ (69)

    then reflect the electron delocalization throughout all AO used to represent MO.

    We further recall that in OCT the entropy/information indices of the covalent/ionic components of the overall IT-multiplicities of the system chemical bonds, respectively represent the complementary descriptors of the average communication-noise and the amount of information-flow in the molecular channel2, 3. The molecular-input signal P(χ) ≡ p generates the same distribution in the output of the AO probability network,

    $ \mathit{\boldsymbol{P}}\left( {\mathit{\boldsymbol{\chi '}}} \right) = \mathit{\boldsymbol{q}} = \mathit{\boldsymbol{p}}{\bf{P}}\left( {\mathit{\boldsymbol{\chi '}}\left| \mathit{\boldsymbol{\chi }} \right.} \right) = \left\{ {{\Sigma _i}{p_i}P\left( {j\left| i \right.} \right) \equiv {\Sigma _i}P\left( {i,j} \right) = {p_j}} \right\} = \mathit{\boldsymbol{p}}, $ (70)

    thus identifying p as the stationary vector of AO-probabilities in the molecular state. Such an exploration of the molecular communication channel is devoid of any reference (history) of the chemical bond formation and generates the average "noise" index of the IT bond-covalency measured by the average conditional-entropy of the system molecular outputs given molecular inputs2, 3:

    $ S\left( {\mathit{\boldsymbol{p}}\left| \mathit{\boldsymbol{p}} \right.} \right) \equiv S = - {\Sigma _i}{\Sigma _j}P\left( {i,j} \right)\log P\left( {j\left| i \right.} \right). $ (71)

    The molecular channel probed by the promolecular-input signal p0 = {pi0}, of the elementary input events in the nonbonded system of the molecularly placed free constituent atoms, refers to the initial stage in the bond-formation process. It corresponds to the ground-state occupations of AO contributed by the constituent atoms of a molecule to the system chemical bonds, before their mixing into MO. These reference probabilities give rise to the average information-flow index of the system IT bond-ionicity, given by the mutual-information in the channel promolecular-input (p0) and molecular-output (p) signals2, 3:

    $ \begin{array}{l} I\left( {{\mathit{\boldsymbol{p}}^0}:\mathit{\boldsymbol{p}}} \right) = {\Sigma _i}{\Sigma _j}P\left( {i,j} \right)\log \left[ {P\left( {i,j} \right)/\left( {p_i^0{q_j}} \right)} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\; = {\Sigma _i}{\Sigma _j}P\left( {i,j} \right)\log \left[ {{p_i}P\left( {i,j} \right)/\left( {{p_i}{q_j}p_i^0} \right)} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\; = {\Sigma _i}{\Sigma _j}P\left( {i,j} \right)\left[ { - \log {q_j} + \log P\left( {j\left| i \right.} \right) + \log \left( {{p_i}/p_i^0} \right)} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\; = S\left( {\mathit{\boldsymbol{p'}}} \right) - S + \Delta S\left( {\mathit{\boldsymbol{p}}\left| {{\mathit{\boldsymbol{p}}^0}} \right.} \right) \equiv I + \Delta S\left( {\mathit{\boldsymbol{p}}\left| {{\mathit{\boldsymbol{p}}^0}} \right.} \right) \equiv {I^0}. \end{array} $ (72)

    Above, △S(p|p0) denotes the molecular entropy-deficiency of Kullback and Leibler57, 58, relative to the promolecular distribution p0,

    $ \Delta S\left( {\mathit{\boldsymbol{p}}\left| {{\mathit{\boldsymbol{p}}^0}} \right.} \right) = {\Sigma _i}{p_i}\log \left( {{p_i}/p_i^0} \right). $ (73)

    The average mutual information reflects a fraction of the initial information content, measured by the Shannon entropy of the promolecular signal,

    $ S\left( {{\mathit{\boldsymbol{p}}^0}} \right) = - {\Sigma _i}p_i^0\log p_i^0, $ (74)

    which has not been dissipated into communication "noise" in the molecular channel. In particular, for molecular input signal p0= p, and hence △S(p|p) = 0,

    $ I\left( {\mathit{\boldsymbol{p}}:\mathit{\boldsymbol{p}}} \right) = S\left( \mathit{\boldsymbol{p}} \right) - S \equiv I. $ (75)

    The sum of the "noise" and "flow" bond-components generates the overall OCT bond-multiplicity index, of all bonds in the molecular system under consideration,

    $ M\left( {{\mathit{\boldsymbol{p}}^0};\mathit{\boldsymbol{p}}} \right) = S + {I^0} = S\left( \mathit{\boldsymbol{p}} \right) + \Delta S\left( {\mathit{\boldsymbol{p}}\left| {{\mathit{\boldsymbol{p}}^0}} \right.} \right) \equiv {M^0}. $ (76)

    For the molecular input this quantity preserves the Shannon entropy in molecular AO probabilities:

    $ M\left( {\mathit{\boldsymbol{p}};\mathit{\boldsymbol{p}}} \right) = S + I = S\left( \mathit{\boldsymbol{p}} \right) \equiv M. $ (77)

    These IT-descriptors can be further decomposed into the additive and nonadditive contributions in the adopted AO resolution. The communication-noise (covalency) represents a difference between the total and additive terms of the information contained in the molecular CBO matrix γ. These contributions also define the associated nonadditive component Snadd(γ) of the overall (total) entropy content in γ, Stotal(γ) ≡ Sadd(γ) + Snadd(γ),

    $ \begin{array}{l} S\left( {\mathit{\boldsymbol{q}}\left| \mathit{\boldsymbol{p}} \right.} \right) = - {\Sigma _i}{\Sigma _j}P\left( {i,j} \right)\log \left[ {P\left( {i,j} \right)/{p_i}} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\; = {N^{ - 1}}\left[ { - {\Sigma _i}{\Sigma _j}{\gamma _{i,j}}{\gamma _{j,i}}\log \left( {{\gamma _{i,j}}{\gamma _{j,i}}} \right) + {\Sigma _i}{\gamma _{i,i}}\log {\gamma _{i,i}}} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\; \equiv {N^{ - 1}}\left\{ {{S^{total}}\left( {\bf{ \pmb{\mathsf{ γ}} }} \right) - {S^{add}}\left( {\bf{ \pmb{\mathsf{ γ}} }} \right)} \right\} \equiv {N^{ - 1}}{S^{nadd}}\left( {\bf{ \pmb{\mathsf{ γ}} }} \right) \end{array} $ (78)

    $ \begin{array}{l} {S^{total}}\left( {\bf{ \pmb{\mathsf{ γ}} }} \right) = - {\Sigma _i}{\Sigma _j}{\gamma _{i,j}}{\gamma _{j,i}}\log \left( {{\gamma _{i,j}}{\gamma _{j,i}}} \right)\\ {S^{add}}\left( {\bf{ \pmb{\mathsf{ γ}} }} \right) = {\Sigma _i}{\gamma _{i,i}}\log {\gamma _{i,i}}, \end{array} $ (79)

    while the information-flow (iconicity) index is identified as the difference between these two contributions of Stotal(γ):

    $ \begin{array}{l} I\left( {\mathit{\boldsymbol{p}}:\mathit{\boldsymbol{q}}} \right) = {\Sigma _i}{\Sigma _j}P\left( {i,j} \right)\log \left[ {P\left( {i,j} \right)/\left( {{p_i}{q_j}} \right)} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\; = {N^{ - 1}}\left\{ {2{S^{add}}\left( {\bf{ \pmb{\mathsf{ γ}} }} \right) - {S^{total}}\left( {\bf{ \pmb{\mathsf{ γ}} }} \right) + \log N} \right\}\\ \;\;\;\;\;\;\;\;\;\;\;\; = {N^{ - 1}}\left\{ {{S^{add}}\left( {\bf{ \pmb{\mathsf{ γ}} }} \right) - {S^{nadd}}\left( {\bf{ \pmb{\mathsf{ γ}} }} \right) + \log N} \right\}. \end{array} $ (80)

    These two OCT bond-multiplicity components finally generate the overall entropic index which reflects the additive part of the molecular entropy:

    $ M\left( {\mathit{\boldsymbol{p}};\mathit{\boldsymbol{q}}} \right) = S\left( {\mathit{\boldsymbol{q}}\left| \mathit{\boldsymbol{p}} \right.} \right) + I\left( {\mathit{\boldsymbol{p}}:\mathit{\boldsymbol{q}}} \right) = {N^{ - 1}}\left\{ {{S^{add}}\left[ {\bf{ \pmb{\mathsf{ γ}} }} \right] + \log N} \right\}. $ (81)

    To summarize, in the single-determinant approximation of the molecular state the additive part of the Shannon entropy in molecular AO-communications (CBO matrix) represents the overall entropic bond-multiplicity, the nonadditive part reflects the IT-covalent (indeterministic, noise) descriptor, while their difference measures the complementary IT-ionic (deterministic, flow) index.

    As an illustration let us qualitatively examine classical communications in the reactive system R = A-B of Section 4. In orbital resolution the subsets of functions contributed by each subsystem to the overall AO basis χ = (χA, χB) determine the relevant AO events on AIM in each reactant and their molecular p(χ) = (pA, pB) = p and promolecular p0(χ) = (pA0, pB0) = p0 probabilities. They define the reactant partition of the AO communications in R as a whole (see Panel Ⅰ of Fig. 1). The global information channel is determined by the conditional probabilities P(χ'|χ) = {P(χβ|χα), α, β ∈ (A, B)}, which produce the associated output signals:

    $ \begin{array}{l} \mathit{\boldsymbol{P}}\left( {\mathit{\boldsymbol{\chi '}}} \right) \equiv \mathit{\boldsymbol{q = }}\left( {{\mathit{\boldsymbol{q}}_{\rm{A}}},{\mathit{\boldsymbol{q}}_{\rm{B}}}} \right) = \mathit{\boldsymbol{p}}{\bf{P}}\left( {\mathit{\boldsymbol{\chi '}}\left| \mathit{\boldsymbol{\chi }} \right.} \right) = \mathit{\boldsymbol{p}}\;\;\;\;\;\;\;\;\;\;{\rm{and}}\\ {\mathit{\boldsymbol{P}}^0}\left( {\mathit{\boldsymbol{\chi '}}} \right) \equiv {\mathit{\boldsymbol{q}}^0}\mathit{\boldsymbol{ = }}\left( {\mathit{\boldsymbol{q}}_{\rm{A}}^0,\mathit{\boldsymbol{q}}_{\rm{B}}^0} \right) = {\mathit{\boldsymbol{p}}^0}{\bf{P}}\left( {\mathit{\boldsymbol{\chi '}}\left| \mathit{\boldsymbol{\chi }} \right.} \right). \end{array} $ (82)

    Alternatively, the optimum MO of the separated subsystems, φR = (φA, φB) ≡ φ, identifying one-electron events of whole reactants, can be used to explore the probability propagation in the reactive system (see Panel Ⅱ of Fig. 1). These MO events generate the corresponding input signals: molecular P(φ) = (PA, PB) = P and promolecular P0(φ) = (PA0, PB0)= P0. The conditional probabilities P(φ'|φ)={P(φβ|φα), α, β∈(A, B)}, the classical communications in the MO-resolved R, ultimately determine the associated output signals:

    $ \begin{array}{l} \mathit{\boldsymbol{P}}\left( {\mathit{\boldsymbol{\varphi '}}} \right) = \mathit{\boldsymbol{Q}} = \left( {{\mathit{\boldsymbol{Q}}_{\rm{A}}},{\mathit{\boldsymbol{Q}}_{\rm{B}}}} \right) = \mathit{\boldsymbol{P}}{\bf{P}}\left( {\mathit{\boldsymbol{\varphi '}}\left| \mathit{\boldsymbol{\varphi }} \right.} \right) = \mathit{\boldsymbol{P}}\;\;\;\;\;{\rm{and}}\\ {\mathit{\boldsymbol{P}}^0}\left( {\mathit{\boldsymbol{\varphi '}}} \right) = {\mathit{\boldsymbol{Q}}^0} = \left( {\mathit{\boldsymbol{Q}}_{\rm{A}}^0,\mathit{\boldsymbol{Q}}_{\rm{B}}^0} \right) = {\mathit{\boldsymbol{P}}^0}{\bf{P}}\left( {\mathit{\boldsymbol{\varphi '}}\left| \mathit{\boldsymbol{\varphi }} \right.} \right). \end{array} $ (83)

    Fig. 1 Orbital networks of classical communications in polarized reactive system Rn+ = (A+|B+): AO-resolved (Panel Ⅰ) and MO-resolved (Panel Ⅱ)

    The reactant blocks in the AO conditional-probability matrix, of the output AO events χ', given the input AO events χ, P(χ'|χ)= {P(χβ|χα)}, are then related to the corresponding blocks of the CBO matrix in AO representation χ = (χA, χB) [Eq.(67)],

    $ \begin{array}{l} {\bf{ \pmb{\mathsf{ γ}} }} = \left\langle {\mathit{\boldsymbol{\chi }}\left| \mathit{\boldsymbol{\varphi }} \right.} \right\rangle \left\langle {\mathit{\boldsymbol{\varphi }}\left| \mathit{\boldsymbol{\chi }} \right.} \right\rangle = {\bf{C}}{{\bf{C}}^\dagger }\\ \;\;\; = \left\{ {\left\langle {{\mathit{\boldsymbol{\chi }}_\alpha }\left| \mathit{\boldsymbol{\varphi }} \right.} \right\rangle \left\langle {\mathit{\boldsymbol{\varphi }}\left| {{\mathit{\boldsymbol{\chi }}_\beta }} \right.} \right\rangle \equiv {{\bf{C}}_\alpha }{{\bf{C}}_\beta }^\dagger \equiv {{\bf{ \pmb{\mathsf{ γ}} }}_{\alpha ,\beta }},\;\;\;\alpha ,\beta \in \left( {{\rm{A}},{\rm{B}}} \right)} \right\}; \end{array} $ (84)

    here φcombines the occupied MO of R and the reactant parts {Cα= < χα|φ>} of the LCAO MO matrix C = < χ|φ> = {Cα} group the expansion coefficients of φ in terms of the reactant AO. The corresponding MO channel is similarly related to the transformed CBO matrix in MO representation,

    $ \begin{array}{l} {{\bf{ \pmb{\mathsf{ γ}} }}^{{\rm{MO}}}} = \left\langle {{\mathit{\boldsymbol{\varphi }}_{\rm{R}}}\left| \mathit{\boldsymbol{\varphi }} \right.} \right\rangle \left\langle {\mathit{\boldsymbol{\varphi }}\left| {{\mathit{\boldsymbol{\varphi }}_{\rm{R}}}} \right.} \right\rangle = {\bf{U}}{{\bf{U}}^\dagger }\\ \;\;\;\;\;\;\;{\rm{ = }}\left\{ {\left\langle {{\mathit{\boldsymbol{\varphi }}_\alpha }\left| \mathit{\boldsymbol{\varphi }} \right.} \right\rangle \left\langle {\mathit{\boldsymbol{\varphi }}\left| {{\mathit{\boldsymbol{\varphi }}_\beta }} \right.} \right\rangle \equiv {{\bf{U}}_\alpha }{{\bf{U}}_\beta }^\dagger ,\;\;\;\alpha ,\beta \in \left( {{\rm{A}},{\rm{B}}} \right)} \right\}, \end{array} $ (85)

    where {Uα = < φα|φ>} in U = < φR|φ> determine the expansion of φin terms of the reactant MO φR = {φα}. At these two resolution levels the molecular or promolecular input signals then generate the corresponding overall descriptors of the OCT bond multiplicities (M, M0) or (MMO, M0, MO), and their covalent/ionic components: thenetwork conditional-entropies (S)andmutual-informations(I): (S, I or I0) or (SMO, IMO or I0, MO).

    It is also of interest to separately examine the reactant "diagonal" (A→A, B→B) and "off-diagonal" (A→B, B→A) communications in these discrete information channels. The former are responsible for the valence-state promotions within each of the mutually nonbonded subsystems, while the latter generate descriptors of the true chemical bonds between the two reactants. Consider the energetically most favorable, complementary mutual arrangements Rc of both reactants, shown in Panel Ⅰ of Fig. 2. The direct communications of Fig. 1 also generate a series of cascade-communications (see Sections 6 and 7) involving the intermediate-orbital events. The amplitudes of their conditional probabilities are given by products of the direct stage-amplitudes and generate the IT descriptors of the intermediate (bridge) bonds in the reactive system45-50.

    Fig. 2 Concerted flows (Panel Ⅰ) in the complementary (c) arrangement of subsystems in the bimolecular reactive system ${{\bf{R}}_{\bf{c}}} \equiv \left( {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{a}}_{\bf{A}}}\mathit{\boldsymbol{ - - - }}{\mathit{\boldsymbol{b}}_{\bf{B}}}} \\ {{\mathit{\boldsymbol{b}}_{\bf{A}}}\mathit{\boldsymbol{ - - - }}{\mathit{\boldsymbol{a}}_{\bf{B}}}} \end{array}} \right);$here aα and bα denote the acidic (a) and basic (b) parts of reactant α = A, B, and most important cascade communications via single orbital intermediates (Panel Ⅱ). The latter combine either two external (inter-reactant) CT propagations (solid arrows), two internal (intra-reactant) P scatterings (broken arrows), or single external and internal communications

    Examples of such communications between fragments in R and R", through a single fragment of R', are shown in Panel Ⅱ of Fig. 2. This figure summarizes the most important single-bridge communications between the acidic (aα) and basic (bα) parts of reactants in R, which are suggested by the dominating primary flows of the inter-fragment CT (solid arrows) shown in Panel Ⅰ of the figure.

    Indeed, from the relative donor/acceptor capacities of these reactant fragments (see Fig. 3), one predicts the following primary flows of electrons in Rc: the most important displacement nCT(1) is expected between the basic part bB of B and the acidic fragment aA of A, with the predicted complemen tary (reverse) flow nCT(2), from the basic part bA of A to the acidic fragment aB of B. As shown in Fig. 2, these (primary) partial CT between the mutually-open reactants, giving rise to the net B →A electron transfer,

    $ {N_{{\rm{CT}}}} = n_{{\rm{CT}}}^{\left( 1 \right)} - n_{{\rm{CT}}}^{\left( 2 \right)} = N_{\rm{A}}^ * - N_{\rm{A}}^0 = N_{\rm{B}}^0 - N_{\rm{B}}^ * > 0, $ (86)

    Fig. 3 A qualitative diagram of the chemical-potential equalization and the Polarizational (P) or Charge-Transfer (CT) electron flows in the complementary reactive complex Rc of Fig.2I. First, the equalized levels of the chemical potential within isolated reactants Rα0 = (A0, B0) are split on their (mutually-closed) acidic (aα) and basic (bα) fragments, due to the perturbation created by the presence of the nearby bβ and aβ parts of the reaction partner Rβ0. These shifts within the initially polarized reactants {Rα+ = (aα+|bα+)} then trigger the P-flows {δNα}, which regain electronegativity equalization in {Rα+ = (aα+¦bα+)} at their internal chemical-potential levels {μX+}. The resulting chemical-potential difference △μ+ = μA+ -μB+ < 0 ultimately determines the direction B+ → A+ and amount NCT of the subsequent inter-reactant CT, which establishes the global equilibrium in Rc as a whole, with equal levels of the chemical potential of the whole bonded (mutually-open) reactants {Rα* = (aα*¦bα*)} and their constituent acidic {aα*} and basic {bα*} parts. One observes that a presence of B destabilizes A, △μA(B) > 0, while A stabilizes B, △μB(A) < 0

    should be accompanied by the induced (relaxational) adjustments (broken arrows) in electron populations of these fragments, reflected by the CT-induced intra-fragment flows: IA, from aA to bA, and IB, from aB to bB. In this flow pattern the primary and induced flows enhance each other giving rise to the least-activation displacement of the electronic structure in Rc 59.

    Thus, the concerted electronic fluxes between reactants are conditioned by the subsystem internal polarizations. This coupled-flow system emphasizes the role of the intermediate chemical interactions in the reaction Transition-State (TS) complex Rc. Among the cascade communications the most important are the single-bridge propagations shown in Panel Ⅱ of Fig. 2. They can be classified as representing either the P-scatterings, when they combine two intra-reactant (broken arrow) scatterings, the CT-induced polarizations involving single intra-reactant (P) and inter-reactant (CT) (solid arrow) communications, or the pure CT bridges, when they involve two inter-reactant (CT) links.

    6 Amplitude and probability channels

    For simplicity, we again focus on molecular information channels in the electronic state described by a single Slater determinant, e.g., the ground-state electron configuration ψUHF(N) ≡ |ψ1, ψ2, …, ψN| = det(ψoccd) defined by N (singly-occupied) MO ψoccd exhibiting lowest orbital energies {εs = < φs|${\rm{\hat H}}$|φs>}, ε1ε2 ≤ … ≤ εN ≤ … ≤ εm. Here the (spin) MO (SMO) ψ(q) = {φs(r)ξs(s)} = < q|ψ> combine the spatial (MO) components φ(r) ={js(r)} = < r|φ>} = [φoccd(r), φvirt(r)] and spin functions ξ(σ) = {ξs(σ)} = < σ|ξ> ∈ (a, β) of an electron. In typical Unrestricted Hartree-Fock (UHF) SCF calculations or the spin-resolved KS DFT an exploration of chemical bonds calls for the (orthonormal) AO basis set χ = (χ1, χ2, …, χm) for expanding all MO functions, occupied and virtual,

    $ \begin{array}{l} \mathit{\boldsymbol{\varphi = \chi }}{\bf{C}},\;\;\;\;\;\;{\bf{C}} = \left\langle {\mathit{\boldsymbol{\chi }}\left| \mathit{\boldsymbol{\varphi }} \right.} \right\rangle = \left[ {\left\langle {\mathit{\boldsymbol{\chi }}\left| {{\mathit{\boldsymbol{\varphi }}^{occd}}} \right.} \right\rangle ,\left\langle {\mathit{\boldsymbol{\chi }}\left| {{\mathit{\boldsymbol{\varphi }}^{virt}}} \right.} \right\rangle } \right] = \left[ {{{\bf{C}}^{occd}},{{\bf{C}}^{virt}}} \right],\\ \;\;\;\;\;\left\langle {\mathit{\boldsymbol{\varphi }}\left| \mathit{\boldsymbol{\varphi }} \right.} \right\rangle = \left\langle {\mathit{\boldsymbol{\varphi }}\left| \mathit{\boldsymbol{\chi }} \right.} \right\rangle \left\langle {\mathit{\boldsymbol{\chi }}\left| \mathit{\boldsymbol{\varphi }} \right.} \right\rangle = {{\bf{C}}^\dagger }{\bf{C}} = {\bf{I}}\;\;\;\;\;{\rm{or}}\\ \;\;\;\;\;{{\bf{C}}^{\mathit{occd}\dagger }}{{\bf{C}}^{occd}} = {{\bf{I}}^{occd}}\;\;\;\;{\rm{and}}\;\;\;\;\;{{\bf{C}}^{\mathit{virt}\dagger }}{{\bf{C}}^{\mathit{virt}}} = {{\bf{I}}^{\mathit{virt}}}. \end{array} $ (87)

    The ground-state ψUHF(N) is thus shaped by N-lowest SMO, which determine the configuration bond-space ψoccddefined by the projector onto the singly-occupied MO subspace φoccd ≡ (φ1, φ2, …, φN):

    $ \begin{array}{l} {{\hat P}^{occd}} = {\Sigma _s}\left| {{\varphi _s}} \right\rangle {n_s}\left\langle {{\varphi _s}} \right| = \left| \mathit{\boldsymbol{\varphi }} \right\rangle {\bf{n}}\left\langle \mathit{\boldsymbol{\varphi }} \right| = \left| {{\mathit{\boldsymbol{\varphi }}^{occd}}} \right\rangle \left\langle {{\mathit{\boldsymbol{\varphi }}^{occd}}} \right|\\ \;\;\;\;\;\;\; = \left| \mathit{\boldsymbol{\chi }} \right\rangle = \left( {{{\bf{C}}^{occd}}{{\bf{C}}^{\mathit{occd}\dagger }}} \right)\left\langle \mathit{\boldsymbol{\chi }} \right| \equiv \left| \mathit{\boldsymbol{\chi }} \right\rangle {\bf{ \pmb{\mathsf{ γ}} }}\left\langle \mathit{\boldsymbol{\chi }} \right|; \end{array} $ (88)

    here the diagonal matrix n ={nsδs, s'} = < φ|${\rm{\hat P}}_\boldsymbol{\varphi} ^{{\rm{occd}}.}$|φ> ≡ γMO specifies the MO occupations: ns = {1, sN; 0, s > N}. This operator generates the CBO matrix of Eq.(84):

    $ \begin{array}{l} {\bf{ \pmb{\mathsf{ γ}} }} = \left\langle \mathit{\boldsymbol{\chi }} \right|{{{\bf{\hat P}}}^{occd}}\left| \mathit{\boldsymbol{\chi }} \right\rangle = \left\langle {\mathit{\boldsymbol{\chi }}\left| \mathit{\boldsymbol{\varphi }} \right.} \right\rangle {\bf{n}}\left\langle {\mathit{\boldsymbol{\varphi }}\left| \mathit{\boldsymbol{\chi }} \right.} \right\rangle \\ \;\;\; = \left\langle {\mathit{\boldsymbol{\chi }}\left| {{\mathit{\boldsymbol{\varphi }}^{occd}}} \right.} \right\rangle \left\langle {{\mathit{\boldsymbol{\varphi }}^{occd}}\left| \mathit{\boldsymbol{\chi }} \right.} \right\rangle = {{\bf{C}}^{occd}}{{\bf{C}}^{\mathit{occd}\dagger }},\\ \;\;\;\;{\left( {\bf{ \pmb{\mathsf{ γ}} }} \right)^2} = \left\langle {\mathit{\boldsymbol{\chi }}\left| {{\mathit{\boldsymbol{\varphi }}^{occd}}} \right.} \right\rangle \left[ {\left\langle {{\mathit{\boldsymbol{\varphi }}^{occd}}\left| \mathit{\boldsymbol{\chi }} \right.} \right\rangle \left\langle {\mathit{\boldsymbol{\chi }}\left| {{\mathit{\boldsymbol{\varphi }}^{occd}}} \right.} \right\rangle } \right]\left\langle {{\mathit{\boldsymbol{\varphi }}^{occd}}\left| \mathit{\boldsymbol{\chi }} \right.} \right\rangle \\ \;\;\;\;\;\;\;\;\;\;\; = {{\bf{C}}^{occd}}{{\bf{I}}^{occd}}{{\bf{C}}^{\mathit{occd}\dagger }} = {\bf{ \pmb{\mathsf{ γ}} }}. \end{array} $ (89)

    In accordance with SP of QM52 the joint probability of the given pair of the input (χi) and output (χj) AO-events in the molecular state ψ defined by its bond system φoccd is given by the square of the corresponding amplitude proportional to the CBO matrix element coupling the two functions [see Eq.(68)]:

    $ \begin{array}{l} P\left( {{\chi _i},{\chi _j}\left| \Psi \right.} \right) \equiv P\left( {i,j} \right) = {\gamma _{i,j}}{\gamma _{j,i}}/N \equiv {\left| {A\left( {i,j} \right)} \right|^2}\;\;\;\;\;\;{\rm{or}}\\ A\left( {i,j} \right) = {N^{ - 1/2}}{\gamma _{i,j}},\\ \;\;{\Sigma _j}P\left( {i,j} \right) = {N^{ - 1}}{\Sigma _j}{\gamma _{i,j}}{\gamma _{j,i}} = {\gamma _{i,i}}/N = P\left( {i\left| \Psi \right.} \right) \equiv {p_i}. \end{array} $ (90)

    The associated conditional AO probabilities P(χ'|χ) of Eq.(69), of the output events χ' = {χj} given the input events χ = {χi}, P(χ'|χ) = {P(j|i) = P(i, j)/piPij}, the squared moduli of the corresponding amplitudes {Pij ≡ |Aij|2}, then read:

    $ P\left( {j\left| i \right.} \right) = {P_{i \to j}} = {\left( {{\gamma _{i,i}}} \right)^{ - 1}}{\gamma _{i,j}}{\gamma _{j,i}},\;\;\;\;\;{\Sigma _j}P\left( {j\left| i \right.} \right) = 1. $ (91)

    They reflect the electron delocalization in the occupied-MO subspace and identify the scattering amplitudes A(χ'|χ) = {A(j|i) ≡ Aij} related to the corresponding elements of the CBO matrix γ:

    $ A\left( {j\left| i \right.} \right) = {A_{i \to j}} = {\left( {{\gamma _{i,i}}} \right)^{ - 1/2}}{\gamma _{i,j}}. $ (92)

    In the spin-restricted (RHF) description of the closed-shell (cs) electronic state each of the N/2 lowest (doubly-occuppied) MO, φcs ≡ (φ1, φ2, …, φN/2), accommodates two spin-paired electrons: ψRHF(N) = |φ1α, φ1β, …, φN/2α, φN/2β|, so that ${\boldsymbol{\rm{\hat P}}}_{}^{occd}$= 2|φcs> < φcs| ≡${\boldsymbol{\rm{\hat P}}}_{}^{cs}$. The idempotency of the bond-space projector ${\boldsymbol{\rm{\hat P}}}_{}^{occd}$ then reads

    $ {\left( {{{{\rm{\hat P}}}^{occd}}} \right)^2} = {\left[ {2\left| {{\mathit{\boldsymbol{\varphi }}^{cs}}} \right\rangle \left\langle {{\mathit{\boldsymbol{\varphi }}^{cs}}} \right|} \right]^2} = 4\left| {{\mathit{\boldsymbol{\varphi }}^{cs}}} \right\rangle \left\langle {{\mathit{\boldsymbol{\varphi }}^{cs}}} \right| = 2{{{\rm{\hat P}}}^{occd}} $ (93)

    and hence:

    $ \begin{array}{l} {\bf{ \pmb{\mathsf{ γ}} }} = \left\langle \mathit{\boldsymbol{\chi }} \right|{{{\rm{\hat P}}}^{occd}}\left| \mathit{\boldsymbol{\chi }} \right\rangle = 2\left\langle \mathit{\boldsymbol{\chi }} \right|{{{\rm{\hat P}}}^{cs}}\left| \mathit{\boldsymbol{\chi }} \right\rangle \equiv 2{{\bf{ \pmb{\mathsf{ γ}} }}^{cs}},\;\;\;\;{\left( {{{\bf{ \pmb{\mathsf{ γ}} }}^{cs}}} \right)^2} = {{\bf{ \pmb{\mathsf{ γ}} }}^{cs}}\;\;\;{\rm{or}}\\ {{\bf{ \pmb{\mathsf{ γ}} }}^2} = 4{\left( {{{\bf{ \pmb{\mathsf{ γ}} }}^{cs}}} \right)^2} = 4{{\bf{ \pmb{\mathsf{ γ}} }}^{cs}} = 2{\bf{ \pmb{\mathsf{ γ}} }}. \end{array} $ (94)

    For such cs-states the representative conditional probability of the molecular AO-channel Pcs(χ'|χ) = {Pcs(j|i) ≡ P(ij)} determined by amplitudes Ac.s.(χ'|χ) = {Acs(j|i) ≡ A(ij) } thus reads:

    $ \begin{array}{l} {P^{cs}}\left( {j\left| i \right.} \right) \equiv {\left| {{A^{cs}}\left( {j\left| i \right.} \right)} \right|^2} = {\left( {2{\gamma _{i,i}}} \right)^{ - 1}}{\gamma _{i,j}}{\gamma _{j,i}}\;\;\;\;\;{\rm{or}}\\ {A^{cs}}\left( {j\left| i \right.} \right) = {\left( {2{\gamma _{i,i}}} \right)^{ - 1/2}}{\gamma _{i,j}}. \end{array} $ (95)

    The classical, probability-channel, is determined by the conditional AO probabilities, e.g., P(χ'|χ) = {P(j|i) =Pij}:

    (96)

    It loses the memory about AO phases in the scattering amplitudes A(χʹ|χ) = {A(j|i) = Aij}, i.e., the phases of elements {γi, j} in CBO matrix. These "coherencies" are preserved only in the associated amplitude-channel for the direct electron communications in a molecule,

    (97)

    which is thus capable of reflecting the quantum-mechanical interference between such elementary communications, e.g., in the indirect (cascade) propagations via AO intermediates, which also represent scatterings in the AO-loop, when the outputs of the direct scattering at one stage are used as inputs in the next stage of the information propagation at the specified molecular state (see the next section).

    Such "cascades" for the indirect ("bridge") communications between atomic orbitals in molecules represent a sequential ("product") arrangement of several direct channels. For example, the single-AO intermediates χ" in the sequential three-orbital (single-bridge) scatterings χχ"→χ' define the (single-stage)-cascades for the probability-and amplitude-propagations in a molecule:

    (98)

    The indirect conditional probabilities between AO-events and their amplitudes are then given by products of the elementary two-orbital communications in each direct subchannel:

    $ \begin{array}{l} P\left[ {\left( {j\left| i \right.} \right);k} \right] \equiv {P_{i \to j;k}} = {P_{i \to k}}{P_{k \to j}},\\ A\left[ {\left( {j\left| i \right.} \right);k} \right] \equiv {A_{i \to j;k}} = {A_{i \to k}}{A_{k \to j}}. \end{array} $ (99)

    Therefore, such bridge probabilities and underlying amplitudes can be straightforwardly derived from the corresponding direct-scattering data. The general probabilities {Pij; k} of the classical cascade propagation {ikj} satisfy the bridge-normalizations:

    $ {\Sigma _k}\left( {{\Sigma _j}{P_{i \to j;k}}} \right) = {\Sigma _k}{P_{i \to k}} = 1. $ (100)

    One observes that the joint-amplitude propagation in the complete AO cascade, involving all basis functions in the bridge, χʹ=χ, generates the resultant joint-amplitude

    $ A\left[ {\left( {i,j} \right);\mathit{\boldsymbol{\chi }}} \right] \equiv {\Sigma _k}A\left( {i,k} \right)A\left( {k.j} \right) = {N^{ - 1}}{\Sigma _k}{\gamma _{i,k}}{\gamma _{k,j}} = {N^{ - 1}}{\gamma _{i,j}}. $ (101)

    It determines the two-orbital probability for such a complete quantum bridge,

    $ P\left[ {\left( {i,j} \right);\mathit{\boldsymbol{\chi }}} \right] = {\left| {A\left[ {\left( {i,j} \right);\mathit{\boldsymbol{\chi }}} \right]} \right|^2} = {N^{ - 1}}P\left( {i,j} \right), $ (102)

    and the associated AO probability

    $ P\left[ {i;\mathit{\boldsymbol{\chi }}} \right] = {\Sigma _j}P\left[ {\left( {i,j} \right);\mathit{\boldsymbol{\chi }}} \right] = {N^{ - 1}}{p_i}. $ (103)

    Therefore, the resultant conditional probability in the complete-bridge scenario of the amplitude-propagation scheme recovers the direct-scattering probability of Eq.(91):

    $ \begin{array}{l} P\left[ {\left( {j\left| i \right.} \right);\mathit{\boldsymbol{\chi }}} \right] = P\left[ {\left( {i,j} \right);\mathit{\boldsymbol{\chi }}} \right]/P\left[ {i;\mathit{\boldsymbol{\chi }}} \right] \equiv {\left| {A\left[ {\left( {j\left| i \right.} \right);\mathit{\boldsymbol{\chi }}} \right]} \right|^2}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = P\left( {i,j} \right)/{p_i} = P\left( {j\left| i \right.} \right) = {P_{i \to j}}. \end{array} $ (104)

    This property emphasizes the stationary character of the molecular electron distribution. It follows only from the resultant-amplitude propagation of Eqs. (99) and (101), with the classical (resultant-probability) cascade giving different result,

    $ \begin{array}{l} {\Sigma _k}P\left[ {i,j;k} \right] = {\Sigma _k}P\left( {i,k} \right)P\left( {k,j} \right) \ne P\left( {i,j} \right)\;\;\;\;\;\;{\rm{and}}\\ {\Sigma _k}P\left[ {\left( {j\left| i \right.} \right);k} \right] = {\Sigma _k}{P_{i \to k}}{P_{k \to j}} \ne {P_{i \to j}}, \end{array} $ (105)

    due to the interference effects, present in the amplitude-propagation and missing in the probability-cascade3.

    This single-cascade development can be straightforwardly generalized to any bridge-order t. The resultant amplitude $A_{i \to j}^{(t)}$ for the complete t-cascade, consisting of t consecutive direct channels involving all AO, preserves the direct scattering probabilities,

    $ {\bf{P}}\left[ {\left( {\mathit{\boldsymbol{\chi '}}\left| \mathit{\boldsymbol{\chi }} \right.} \right);\mathit{\boldsymbol{t}} - \mathit{\boldsymbol{\chi }}} \right] = \left\{ {{{\left| {A_{i \to j}^{\left( t \right)}} \right|}^2} = {{\left| {{A_{i \to j}}} \right|}^2} = {P_{i \to j}}} \right\}{\bf{P}}\left( {\mathit{\boldsymbol{\chi '}}\left| \mathit{\boldsymbol{\chi }} \right.} \right), $ (106)

    thus satisfying the important consistency requirement of the stationary character of the molecular channel [compare Eq.(104)]. The relevant sum-rules for general bridge probability Pij; k, l, …, m, n of the classical AO cascade

    (107)

    then read [compare Eq.(100)]:

    $ \begin{array}{l} {\Sigma _k}{\Sigma _l} \cdots {\Sigma _m}{\Sigma _n}\left[ {{\Sigma _j}{P_{i \to j;k,l, \cdots ,m,n}}} \right]\\ \;\;\;\; = {\Sigma _k}{\Sigma _l} \cdots {\Sigma _m}\left[ {{\Sigma _n}{P_{i \to n;k,l, \cdots ,m}}} \right] = {\Sigma _k}{\Sigma _l} \cdots \left[ {{\Sigma _m}{P_{i \to m;k,l, \cdots }}} \right]\\ \;\;\;\; = \cdots = {\Sigma _k}\left[ {{\Sigma _l}{P_{i \to l;k}}} \right] = {\Sigma _k}{P_{i \to k}} = 1. \end{array} $ (108)

    For the specified pair of "terminal" AO, say χiχ and χjχʹ, one can similarly examine the indirect scatterings via the molecular bond system in the incomplete cascade consisting of the remaining ("bridge") functions χb = {χk(i, j)}, with the two terminal AO being then excluded from the set of admissible intermediate scatterers. The associated bridge-communications give rise to the indirect (through-bridge) components of the entropic bond multiplicities45-50, which complement the familiar direct (through-space) chemical "bond-orders" and provide a novel IT perspective on chemical interactions between more distant AIM, alternative to the fluctuational Charge-Shift mechanism60 introduced within the classical Valence-Bond (VB) theory61.

    7 Markov chains

    The cascade probabilities derived from the stochastic matrix P(χ'|χ) = {P(j|i) = Pij} ≡ P of the direct-scattering between AO can be also related to a sequence of trials in a loop, in which the outputs of the given stage determine the input of the next stage in the probability propagation. In such a chain-dependence the outcome of any trial in a sequence is conditioned by the outcome of the trial immediately preceding, but by no earlier ones. Such a stochastic chain of dependences, common in many important practical situations and well diagnosed mathematically in probability theory, is known as the Markov process62-64.

    In the molecular scenario3, 36-38 the set of AO events defines the state space of the Markov chain and the kth trial can be considered as the change of state at the given transition time tk. Since the transition probabilities are determined by the molecular electronic state, the stage transition probabilities are independent of the trial number k. Therefore, this Markov chain is homogeneous: {Pij(k) = Pij}. The output probabilities at nth trial can be thus determined from the initial input probabilities p(0) in the chain, e.g., the molecular [p(0) = p] or promolecular [p(0) = p0] signals, by the resultant transformation of the nth power of P:

    $ \mathit{\boldsymbol{q}}\left( n \right) = \mathit{\boldsymbol{p}}\left( 0 \right){{\bf{P}}^n}. $ (109)

    Consider the 2-AO model3, 36-38 of the chemical bond due to the doubly occupied bonding MO,

    $ {\varphi _b} = {P^{ - 1/2}}{\chi _{\rm{A}}} + {Q^{ - 1/2}}{\chi _{\rm{B}}},\;\;\;\;\;P + Q = 1, $ (110)

    where the complementary conditional probabilities P = P(χA|φb) and Q = P(χB|φb) and the two AO functions χ = (χA, χB) originate from atoms A and B, respectively. The corresponding CBO matrix for this closed-shell configuration ψ(2) = |φbα, jbβ|,

    $ {\bf{ \pmb{\mathsf{ γ}} }} = 2\left[ {\begin{array}{*{20}{c}} P&{\sqrt {PQ} }\\ {\sqrt {PQ} }&Q \end{array}} \right], $ (111)

    generates the idempotent stochastic probability matrix63:

    $ {\bf{P}}\left( {\mathit{\boldsymbol{\chi }}\left| \mathit{\boldsymbol{\chi }} \right.} \right) = \left[ {\begin{array}{*{20}{c}} P&Q\\ P&Q \end{array}} \right] \equiv {\bf{P}},\;\;\;\;\;\;{{\bf{P}}^2} = {{\bf{P}}^n} = {\bf{P}}. $ (112)

    Therefore, given any initial probability distribution p(0) = [pA(0), pB(0)], pA(0) + pB(0) = 1, we have in this particular case

    $ \mathit{\boldsymbol{q}}\left( n \right) = \mathit{\boldsymbol{p}}\left( 0 \right){{\bf{P}}^n} = \mathit{\boldsymbol{p}}\left( 0 \right){\bf{P}} = \left( {P,Q} \right) = \mathit{\boldsymbol{q}}\left( 1 \right). $ (113)

    Finding Pn in a general, symmetric-P case requires a reduction of the stochastic matrix to its normal form in a similarity transformation T, i.e., the diagonalization of P,

    $ \begin{array}{l} {{\bf{T}}^{ - 1}}{\bf{PT}} = {\bf{ \pmb{\mathsf{ π}} }} = \left\{ {{\lambda _\alpha }{\delta _{\alpha ,\beta }}} \right\}\;\;\;{\rm{or}}\;\;\;{\bf{P}} = {\bf{T \pmb{\mathsf{ π}} }}{{\bf{T}}^{ - 1}},\\ \;\;\;{{\bf{T}}^{ - 1}}{\bf{T}} = {\bf{T}}{{\bf{T}}^{ - 1}} = {\bf{I}}, \end{array} $ (114)

    which also marks the algebraic eigenvalue problems for determining the transformation matrices:

    $ {\bf{PT}} = {\bf{T \pmb{\mathsf{ π}} }}\;\;\;\;{\rm{and}}\;\;\;\;{{\bf{T}}^{ - 1}}{\bf{P}} = {\bf{ \pmb{\mathsf{ π}} }}{{\bf{T}}^{ - 1}}. $ (115)

    The resultant probability transformation of Eq.(109) then reads:

    $ {{\bf{P}}^n} = {\bf{T}}{{\bf{ \pmb{\mathsf{ π}} }}^n}{{\bf{T}}^{ - 1}},\;\;\;\;\;{{\bf{ \pmb{\mathsf{ π}} }}^n} = \left\{ {{\lambda _\alpha }{\delta _{\alpha ,\beta }}} \right\}. $ (116)

    As an illustrative example consider the Binary Channel3, 36-38, 64 described by the symmetric matrix of conditional probabilities:

    $ {\bf{P}} = \left[ {\begin{array}{*{20}{c}} {1 - \omega }&\omega \\ \omega &{1 - \omega } \end{array}} \right],\;\;\;\;\;\;0 < \omega < {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}. $ (117)

    It can be diagonalized in an orthogonal transformation U, UTU = UUT = I,

    $ \begin{array}{l} {\bf{U}} = \frac{1}{{\sqrt 2 }}\left[ {\begin{array}{*{20}{c}} 1&1\\ 1&{ - 1} \end{array}} \right] = {{\bf{U}}^{\rm{T}}},\\ {{\bf{U}}^{\rm{T}}}{\bf{PU}} = \left[ {\begin{array}{*{20}{c}} 1&0\\ 0&{1 - 2\omega } \end{array}} \right] = {\bf{ \pmb{\mathsf{ π}} }}. \end{array} $ (118)

    Hence, the nth stage propagation matrix of Eq.(116) reads:

    $ \begin{array}{l} {{\bf{P}}^n} = {\bf{U}}{{\bf{ \pmb{\mathsf{ π}} }}^n}{{\bf{U}}^{\rm{T}}} = \left[ {\begin{array}{*{20}{c}} a&b\\ b&a \end{array}} \right],\\ \;\;\;\;\;\;\;a = {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}\left[ {1 + {{\left( {1 - 2\omega } \right)}^n}} \right],\;\;\;\;b = {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}\left[ {1 - {{\left( {1 - 2\omega } \right)}^n}} \right]. \end{array} $ (119)

    As intuitively expected, in the limit n→∞ it gives the resultant stochastic matrix generating the maximum-noise in the underlying communication system,

    $ {{\bf{P}}^{n \to \infty }} = \frac{1}{2}\left[ {\begin{array}{*{20}{c}} 1&1\\ 1&1 \end{array}} \right], $ (120)

    and the resultant output-probabilities of this Markov chain equal to the arithmetic average of the initial input-probabilities:

    $ \begin{array}{l} \mathit{\boldsymbol{q}}\left( {n \to \infty } \right) = \mathit{\boldsymbol{p}}\left( 0 \right){{\bf{P}}^{n \to \infty }} = \left[ {\bar p\left( 0 \right),\bar p\left( 0 \right)} \right],\\ \bar p\left( 0 \right) = {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}\left[ {{p_{\rm{A}}}\left( 0 \right) + {p_{\rm{B}}}\left( 0 \right)} \right]. \end{array} $ (121)

    8 Electron flows and electronegativity equalization

    According to Sanderson's20 Electronegativity Equalization (EE) principle and the DFT10-13 variational principle for the minimum of the electronic energy Ev({Ni}) ≡ ${{\bar E}_v}\left( {\left\{ {{q_i}} \right\}} \right)$, the equilibrium electron populations {Ni*} of the mutually-open molecular fragments, e.g., Atoms-in-Molecules (AIM) {Xi*}, imply the equalization of the subsystem chemical potentials (negative electronegativities) {μi* = -χi*}13-23 at their associated global levels:

    $ \mu = - \chi :\mu _i^ * = \mu = - \chi _i^ * = - \chi . $

    The fragment quantities are defined by the corresponding partial derivatives of the grand-ensemble average21 of the system electronic energy with respect to subsystem electron population Ni or the associated net charge qi,

    $ {\mu _i} \equiv \partial {E_v}\left( {\left\{ {{N_i}} \right\}} \right)/\partial {N_i} = - \partial {{\bar E}_v}\left( {\left\{ {{q_i}} \right\}} \right)/\partial {q_i} \equiv - {\chi _i}. $ (122)

    The global properties similarly involve differentiations with respect to the resultant state-parameters, of the whole system, N = ∑iNi or Q = ∑iqi,

    $ \mu \equiv \partial {E_v}\left( N \right)/\partial N = - \partial {{\bar E}_v}\left( Q \right)/\partial Q \equiv - \chi . $ (123)

    For example, in atomic resolution N stands for the system overall number of electrons, Q = ∑iqi = ∑i(Zi -Ni) denotes (in atomic units) its net electric charge, and all derivatives are calculated for the fixed external potential v(r) due to the nuclei exhibiting charges {Zi}. The effective electron populations can be generated in the familiar schemes of the electron population analyses or via an appropriate division of the molecular electron density ρ(r),

    $ \rho \left( \mathit{\boldsymbol{r}} \right) = {\Sigma _i}{\rho _i}\left( \mathit{\boldsymbol{r}} \right),\;\;\;\;{N_i} = \int {{\rho _i}\left( \mathit{\boldsymbol{r}} \right){\rm{d}}\mathit{\boldsymbol{r}}} ,\;\;\;\;\;\int {\rho \left( \mathit{\boldsymbol{r}} \right){\rm{d}}\mathit{\boldsymbol{r}}} = N, $ (124)

    e.g., the stockholder partitioning of Hirshfeld36, 65.

    In the electronegativity-equalized reactive system, composed of the mutually-open (bonded) fragments in Rb = (A*¦B*), the local sites are distinguished by their response properties represented by the corresponding second-partials of the system energy or the relevant Legendre-transformed "thermodynamic" potentials13-18. The regional softnesses represent the population responses per unit shift in the system global chemical potential:

    $ {s_i} = \partial {N_i}/\partial \mu = \partial {q_i}/\partial \chi . $ (125)

    They sum up to the system global chemical softness,

    $ S = {\Sigma _i}{s_i} = \partial N/\partial \mu = \partial Q/\partial \chi = {\eta ^{ - 1}}, $ (126)

    the inverse of its global chemical hardness 25

    $ \eta = \partial \mu /\partial N = \partial \chi /\partial Q = {S^{ - 1}}. $ (127)

    Alternatively, the relative acidic (electron-acceptor) or basic (electron-donor) capacities of molecular fragments can be recognized using the regional Fukui Function (FF) descriptor26, which measures the fragment relative participation in the global population displacement:

    $ \begin{array}{l} {f_i} = {s_i}/S = \partial {N_i}/\partial N = \partial {q_i}/\partial Q,\\ \;\;\;\;\;{\Sigma _i}{f_i} = \partial N/\partial N = \partial Q/\partial Q = 1. \end{array} $ (128)

    When reactants are regarded as being mutually-closed (nonbonded) in Rn+ = (A+|B+), i.e., preserving their initial (integer) numbers of electrons {Nα = Nα0}, they are considered to be separated by a hypothetical "wall" preventing the inter-reactant flow of electrons, symbolized by a solid vertical line separating the two reactants. At this Polarization (P) stage of their interaction the chemical-potential/electronegativity equalizations take place only within each subsystem. This constrained, P-equilbrium of the polarized (promoted) reactants thus gives rise to generally different levels of the chemical potential in each subsystem:

    $ \begin{array}{l} \mu _{\rm{A}}^ + = \partial {E_v}\left( {{\rm{R}}_n^ + } \right)/\partial {N_{\rm{A}}} = \left\{ {\mu _i^ * \left( {{{\rm{A}}^ + }} \right)} \right\}\\ \;\;\;\;\; \ne \mu _{\rm{B}}^ + = \partial {E_v}\left( {{\rm{R}}_n^ + } \right)/\partial {N_{\rm{B}}} = \left\{ {\mu _j^ * \left( {{{\rm{B}}^ + }} \right)} \right\}. \end{array} $ (129)

    At the given (finite) inter-reactant separation the relaxed density ρα+ is displaced relative to the corresponding separated-reactant density ρα0by its equilibrium P-response △ρα+ = ρα+ -ρα0 ≡ △ρα(β) to perturbation created by the presence of the other subsystem β. These mutually polarized reactants exhibit the associated shifts in their equilibrium chemical potentials due to the reaction partner, relative to SRL values: △μα+ = μα+ -μα0 ≡ △μα(β).

    Consider now the chemical-potential diagram shown in Fig. 3, for the complementary acidic (a) and basic (b) fragments in each reactant, e.g., the mutually-open (bonded) parts in Ab = (aA*¦bA*) and Bb = (aB*¦bB*), or the mutually-closed (nonbonded) pieces of An+ = (aA+|bA+) and Bn+ = (aB+|bB+). The donor/acceptor character of molecular fragments can be identified using regional softnesses or FF indices: the acidic (electron-acceptor) subsystem is chemically harder, exhibiting lower values of si and fi, while the basic (electron-donor) fragment is chemically softer, as reflected by its higher values of these response descriptors.

    According to the Maximum Complementarity (MC) rule14, 15, 59, 66 reactants arrange themselves in such a way that the geometrically accessible a-fragment of one reactant faces the geometrically accessible b-fragment of the other reactant (see Fig. 2I):

    $ {{\rm{R}}_c} \equiv \left[ \begin{array}{l} {a_{\rm{A}}} - - - {b_{\rm{B}}}\\ {b_{\rm{A}}} - - - {a_{\rm{B}}} \end{array} \right]. $

    This complementary complex is energetically preferred66 compared to the regional HSAB-type structure, in which the acidic (basic) fragment of one reactant faces the analogous part, of the same donor-acceptor character, in the other reactant:

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

    This preference for the complementary behaviour in Rc should be expected on purely electrostatic (ES) grounds14, 15, 66, since then the region of positive electrostatic (ES) potential around an acidic (electron deficient) site of one reactant overlaps with the region of negative potential around the basic (electron rich) part of the reaction partner, thus generating larger ES stabilization energy (or smaller ES destabilization interaction) compared to that in RHSAB. Additional rationale for this complementary preference over the regional-HSAB alignment comes from examining the particle flows created by the primary shift

    $ \Delta {\mu _\alpha }\left( \beta \right) = \mu _\alpha ^ + \left( \beta \right) - \mu _\alpha ^0, $

    in the chemical potential μα(β) of the fragment Rα of the perturbed subsystem α, relative to its equalized SRL value μα0, created by the presence of the coordinating (perturbing) fragment Rβ of the other reactant β14, 15, 59. At finite separations between the two substrates these displacements trigger the polarizational flows {δNα} of Fig. 3, which restore the internal equilibria in reactants. As indicated in the figure, one predicts for Rc that the basic subsystem B destabilizes A, △μA(B) > 0, while the acidic reactant A stabilizes B, △μB(A) < 0. In this reaction complex the harder (acidic) site aβ+ of the polarized reactant Rβ+ = (aβ+|bβ+) lowers the chemical potential of the softer (basic) site bα+of the other polarized reactant Rα+ = (aα+|bα+), and bβ+ rises the chemical potential level on aα+:

    $ \begin{array}{l} \left\{ {\mu _{{b_\alpha }}^ + \left( {{a_\beta }} \right) < 0\;{\rm{and}}\;\mu _{{a_\alpha }}^ + \left( {{b_\beta }} \right) > 0} \right\}\\ \;\;\;\; \Rightarrow \mu _{{a_\alpha }}^ + \left( {{b_\beta }} \right) - \mu _{{b_\alpha }}^ + \left( {{a_\beta }} \right) > 0. \end{array} $ (130)

    These shifts of the initially equalized chemical potentials on the two sites in a0 = (aα*¦bα*) trigger the internal aα+bα+ polarizational flowδNα, which further enhances the (external) acceptor capacity of aα+ and the donor ability of bα+, thus creating more favorable conditions for the subsequent inter-reactant flow NCT, the net effect of the partial flows of Fig. 2I:

    $ \begin{gathered} b_{\text{B}}^ + \xrightarrow{{{\text{C}}{{\text{T}}^{\left( 1 \right)}}}}a_{\text{A}}^ + \;\;{\text{and}}\;\;b_{\text{A}}^ + \xrightarrow{{{\text{C}}{{\text{T}}^{\left( 2 \right)}}}}a_{\text{B}}^ + , \hfill \\ \;\;{N_{{\text{CT}}}} = \Delta {N_{\text{A}}} = - \Delta {N_{\text{B}}} = {n_{{\text{C}}{{\text{T}}^{\left( 1 \right)}}}} - {n_{{\text{C}}{{\text{T}}^{\left( 2 \right)}}}}. \hfill \\ \end{gathered} $ (131)

    A similar analysis of the RHSAB complex predicts the following initial shifts in the site chemical potentials: $\mu _{{b_\alpha }}^ + \left( {{b_\beta }} \right) > 0$ and $\mu _{{a_\alpha }}^ + \left( {{a_\beta }} \right) > 0$. They imply the {bα+aα+} internal flows {δNα}, which lower the acceptor capacity of the acidic site aα+ and the donor capacity of the basic site bα+. These polarization flows thus create an extra electron accumulation on aα+ and the associated electron depletion on bα+, i.e., less favourable conditions for the subsequent inter-reactant CT. In other words, the interaction-induced polarization flows in Rc enhance NCT, while those in RHSAB hinder this net B →A electron transfer.

    The global CT-equilibrium in R as a whole is reached when both internally-open reactants are also mutually-open in Rb = (A*¦B*), where the hypothetical barrier for inter-subsystem flow of electrons is lifted, as symbolized by the broken vertical line. The EE in Rb then extends over reactants α = A, B and their constituent AIM {Xiα}, as well as over all local volume elements:

    $ \begin{array}{l} {\mu _{\rm{R}}} \equiv \partial {E_v}\left( {\rm{R}} \right)/\partial {N_{\rm{R}}} = \mu _\alpha ^ * \equiv \partial {E_v}\left( {\rm{R}} \right)/\partial {N_{\alpha \left| {\rm{R}} \right.}}\\ \;\;\;\;\; = \left\{ {\mu _{i \in a}^ * \left( {\rm{R}} \right) \equiv \partial {E_v}\left( {\rm{R}} \right)/\partial {N_{i \in \alpha \left| {\rm{R}} \right.}}} \right\} = {\mu ^ * }\left( \mathit{\boldsymbol{r}} \right) \equiv \delta {E_v}\left[ \rho \right]/\delta \rho {\left( \mathit{\boldsymbol{r}} \right)_{\left| {\rm{R}} \right.}}. \end{array} $

    In addition to the polarizational changes of the nonbonded reactants in Rn+ = (A+|B+), {△ρα+= ρα+ -ρα0}, the equlibrium densities of the bonded reactants {rα*} in Rb = (A*¦B*) exhibit additional CT-induced polarization component {△ρα* = ρα* -ρα+},

    $ \rho _\alpha ^ * = \rho _\alpha ^ + + \Delta \rho _\alpha ^ * = \rho _\alpha ^0 + \Delta \rho _\alpha ^ + + \Delta \rho _\alpha ^ * $

    and integrate to the CT-displaced (fractional) effective populations {Nα* = $\int {_\rho } $α*dr}.

    The complementary preference of reaction complexes also follows from the electronic stability considerations, in spirit of the familiar Le Ch telier-Braun principle of the ordinary thermodynamics56. In contrast to the P-stage analysis of Fig. 3 let us now assume the primary CT-flows of Eq.(131) in

    $ {{\rm{R}}_{{\rm{CT}}}} \equiv \left( {\frac{{{a_{\rm{A}}} \leftarrow {b_{\rm{B}}}}}{{{b_{\rm{A}}} \to {a_{\rm{B}}}}}} \right), $

    where the solid horizontal line again denotes the wall preventing the flow of electrons, and then examine the secondary (induced) intra-reactant responses to perturbations created by these primary displacements. We recall that, in accordance with the Le Châtelier stability principle, an inflow (outflow) of electrons to (from) the given site i increases (decreases) the site chemical potential, as reflected by the positive value of the site hardness descriptor:

    $ {\eta _{i,i}} = \partial {\mu _i}/\partial {N_i} > 0. $

    The partial CT-flows thus create the following shifts in chemical potentials on the four sites in the mutually and externally closed fragments of reactants in

    $ \begin{array}{l} {\rm{R}}_{{\rm{CT}}}^ + \equiv \left( {\frac{{{a_{\rm{A}}}\left( {n_{{\rm{CT}}}^{\left( 1 \right)}} \right)}}{{{b_{\rm{A}}}\left( {n_{{\rm{CT}}}^{\left( 2 \right)}} \right)}}\left| {\frac{{{b_{\rm{B}}}\left( {n_{{\rm{CT}}}^{\left( 1 \right)}} \right)}}{{{a_{\rm{B}}}\left( {n_{{\rm{CT}}}^{\left( 2 \right)}} \right)}}} \right.} \right),\\ \Delta \mu _{{a_{\rm{A}}}}^ + \left( {n_{{\rm{CT}}}^{\left( 1 \right)}} \right) > 0\;\;\;{\rm{and}}\;\;\Delta \mu _{{b_{\rm{A}}}}^ + \left( {n_{{\rm{CT}}}^{\left( 2 \right)}} \right) < 0,\\ \Delta \mu _{{a_{\rm{B}}}}^ + \left( {n_{{\rm{CT}}}^{\left( 2 \right)}} \right) > 0\;\;\;{\rm{and}}\;\;\Delta \mu _{{b_{\rm{B}}}}^ + \left( {n_{{\rm{CT}}}^{\left( 1 \right)}} \right) < 0, \end{array} $ (132)

    compared to the respective (equalized) levels in A0 = (aA*¦bA*) and B0 = (aB*¦bB*) (see Fig. 4).

    Fig. 4 Qualitative diagram of the chemical potential displacements in the complementary complex Rc+ = (A+|B+), due to the primary CT perturbations nCT(1) and nCT(2) in RCT, and subsequent induced responses IA and IB of Fig.2.I. The CT perturbations split the initially equalized levels of the chemical potential within each reactant, {a0 = (aa*¦ba*)}, with the inflow (outflow) of electron increasing (decreasing) the site chemical potential in {α+ = (aα+|bα+)}. These primary shifts subsequently trigger the polarizational flows {Ia}, which eventually generate the global electronegativity equalization in Rc as a whole: Rc = (A*¦B*) = (aA*¦bA*¦aB*¦bB*) ≡ RCT*

    These CT-induced shifts in fragment electronegativities subsequently trigger the secondary, induced flows of the figure,

    $ a_{\text{A}}^ + \xrightarrow{{{I_{\text{A}}}}}b_{\text{A}}^ + \;\;{\text{and}}\;\;a_{\text{B}}^ + \xrightarrow{{{I_{\text{B}}}}}b_{\text{B}}^ + , $ (133)

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

    Consider now the primary CT displacements in the HSAB structure:

    $ {{\rm{R}}_{{\rm{HSAB}}}} = \left( {\frac{{{a_{\rm{A}}} \leftarrow {b_{\rm{B}}}}}{{{b_{\rm{A}}} \to {a_{\rm{B}}}}}} \right) $

    They generate shifts in the chemical potentials of reactant sites in

    $ {\rm{R}}_{{\rm{HSAB}}}^ + = \left( {\frac{{{a_{\rm{A}}}}}{{{b_{\rm{A}}}}}\left| {\frac{{{b_{\rm{B}}}}}{{{a_{\rm{B}}}}}} \right.} \right), $

    that induce responses $a_{\rm{A}}^ + \leftarrow b_{\rm{A}}^ + $ and $a_{\rm{B}}^ + \leftarrow b_{\rm{B}}^ + $ which further exaggerate changes created by the primary perturbation. This is contrary to the Le Châtelier-Braun principle thus giving rise to a less stable reactive complex.

    The partial CT of Eq.(131) generate the resultant B+→A+ flow of electrons:

    $ \begin{array}{l} {N_{{\rm{CT}}}} = n_{{\rm{CT}}}^{\left( 1 \right)} - n_{{\rm{CT}}}^{\left( 2 \right)} = N_{\rm{A}}^ * - N_{\rm{A}}^0 \equiv \Delta {N_{\rm{A}}}\\ \;\;\;\;\;\;\; = N_{\rm{B}}^0 - N_{\rm{B}}^ * \equiv - \Delta {N_{\rm{B}}} > 0. \end{array} $ (134)

    Its magnitude is determined by the difference in chemical potentials of the polarized reactants,

    $ \Delta \mu _{\rm{R}}^ + = \mu _{\rm{A}}^ + - \mu _{\rm{B}}^ + \equiv {\mu _{{\rm{CT}}}} < 0, $ (135)

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

    $ {\bf{ \pmb{\mathsf{ η}} }}_{\rm{R}}^ + = \left\{ {{\eta _{\alpha ,\beta }} = \partial {\mu _\alpha }/\partial {N_\beta };\;\;\alpha ,\beta \in \left( {{\rm{A}},{\rm{B}}} \right)} \right\}. $ (136)

    The latter determines the in situ hardness (ηCT) and softness (SCT) for this process:

    $ {\eta _{{\rm{CT}}}} = \partial {\mu _{{\rm{CT}}}}/\partial {N_{{\rm{CT}}}} = {\eta _{{\rm{A,A}}}} + {\eta _{{\rm{B,B}}}} - {\eta _{{\rm{A,B}}}} - {\eta _{{\rm{B,A}}}} = S_{{\rm{CT}}}^{ - 1}. $ (137)

    It should be emphasized that both the "force" of Eq.(135) and the effective hardness tensor (electronic Hessian) include the "embedding" terms due to a presence of the other reactant at a finite distance. Only at an early stage of the reaction, at large inter-reactant separation when the charge coupling between the two species is negligible, can they be approximated by the separate-reactant quantities:

    $ {\mu _{{\rm{CT}}}} \cong \mu _{\rm{A}}^0 - \mu _{\rm{B}}^0 \equiv \mu _{{\rm{CT}}}^0\;\;\;{\rm{and}}\;\;\;{\eta _{{\rm{CT}}}} \cong \eta _{\rm{A}}^0 - \eta _{\rm{B}}^0 \equiv \eta _{{\rm{CT}}}^0. $

    We further recall that the optimum amount of CT,

    $ {N_{{\rm{CT}}}} = - {\mu _{{\rm{CT}}}}{S_{{\rm{CT}}}}, $ (138)

    generates the associated (2nd-order) CT-stabilization energy:

    $ {E_{{\rm{CT}}}} = {\mu _{{\rm{CT}}}}{N_{{\rm{CT}}}}/2 = - {\left( {{\mu _{{\rm{CT}}}}} \right)^2}{S_{{\rm{CT}}}}/2 < 0. $ (139)

    9 Phase considerations

    The phase approach to equilibria in molecules and reactive systems offers a new perspective on the promoted states of molecular fragments {α+} or {α*}, e.g., reactants, AIM, etc., determining the mutully exclusive pieces {ρα*} of the overall density ρ in the whole (molecular) system M:

    $ \begin{array}{l} \rho \left( \mathit{\boldsymbol{r}} \right) = Np\left( \mathit{\boldsymbol{r}} \right) = {\Sigma _\alpha }\rho _\alpha ^ * \left( \mathit{\boldsymbol{r}} \right),\\ \rho _\alpha ^ * \left( \mathit{\boldsymbol{r}} \right) = N_\alpha ^ * p_\alpha ^ * \left( \mathit{\boldsymbol{r}} \right) = N\pi _\alpha ^ * \left( \mathit{\boldsymbol{r}} \right),\;\;\;\;N_\alpha ^ * = \int {\rho _\alpha ^ * \left( \mathit{\boldsymbol{r}} \right){\rm{d}}\mathit{\boldsymbol{r}}} . \end{array} $ (140)

    For example, the molecularly placed "free" reactants {αn0} in the nonbonded promolecular reference Rn0 = (A0|B0) and their "bonded" analogs {αb0} in the "entangled" promolecule Rb0= (A0¦B0) are describedby"thermodynamic"phases{ϕα, eq.(αn0)=ϕeq.[pα0]} and {ϕα, eq.(αb0)=ϕeq.[pR0]}, respectively, where the promolecular density ρR0 and probability distribution pR0are defined by relation

    $ \begin{array}{l} \rho _{\rm{R}}^0 = \rho _{\rm{A}}^0 + \rho _{\rm{B}}^0 = N_{\rm{A}}^0p_{\rm{A}}^0 + N_{\rm{B}}^0p_{\rm{B}}^0 = {N_{\rm{R}}}p_{\rm{R}}^0,\\ \;\;\;\;N_\alpha ^0 = \int {\rho _\alpha ^0\left( \mathit{\boldsymbol{r}} \right){\rm{d}}\mathit{\boldsymbol{r}}} . \end{array} $ (141)

    The mutually-open (entangled) fragments in Rb0 are thus (phase/current)-promoted compared to their mutually-closed (disentangled) analogs in Rn0. The overall promolecular density ρ0(r) thus induces the equilibrium current in αb0,

    $ \begin{array}{l} \mathit{\boldsymbol{j}}\left( {\alpha _b^0} \right) = \left( {\hbar /m} \right)\rho _\alpha ^0\nabla {\phi _{\alpha ,{\rm{eq}}}}\left( {\alpha _n^0} \right) = - \left[ {\hbar /\left( {2m} \right)} \right]\rho _\alpha ^0\nabla \ln p_{\rm{R}}^0\\ \;\;\;\;\;\;\;\;\;\; = - \left[ {\hbar /\left( {2m} \right)} \right]\left( {\rho _\alpha ^0/\rho _R^0} \right)\nabla \rho _{\rm{R}}^0 \equiv - \left[ {\hbar /\left( {2m} \right)} \right]d_\alpha ^0\nabla \rho _{\rm{R}}^0, \end{array} $ (142)

    which differs from the equilibrium current in αn0:

    $ \begin{array}{l} \mathit{\boldsymbol{j}}\left( {\alpha _n^0} \right) = \left( {\hbar /m} \right)\rho _\alpha ^0\nabla {\phi _{\alpha ,{\rm{eq}}}}\left( {\alpha _n^0} \right) = - \left[ {\hbar /\left( {2m} \right)} \right]\rho _\alpha ^0\nabla \ln p_\alpha ^0\\ \;\;\;\;\;\;\;\;\;\; = - \left[ {\hbar /\left( {2m} \right)} \right]\nabla \rho _\alpha ^0. \end{array} $ (143)

    A similar current-distinction is observed between the equilibrium state of an disentangled reactant αn* in Rn* = (A*|B*) and its entangled analog αb* in Rb* = (A*¦B*) ≡ R:

    $ \begin{array}{l} \mathit{\boldsymbol{j}}\left( {\alpha _n^ * } \right) = \left( {\hbar /m} \right)\rho _\alpha ^ * \nabla {\phi _{\alpha ,{\rm{eq}}}}\left( {\alpha _n^ * } \right) = - \left[ {\hbar /\left( {2m} \right)} \right]\rho _\alpha ^ * \nabla \ln p_\alpha ^ * \\ \;\;\;\;\;\;\;\;\;\; = - \left[ {\hbar /\left( {2m} \right)} \right]\nabla \rho _\alpha ^ * , \end{array} $ (144)

    $ \begin{array}{l} \mathit{\boldsymbol{j}}\left( {\alpha _b^ * } \right) = \left( {\hbar /m} \right)\rho _\alpha ^ * \nabla {\phi _{\alpha ,{\rm{eq}}}}\left( {\alpha _b^ * } \right) = - \left[ {\hbar /\left( {2m} \right)} \right]\rho _\alpha ^ * \nabla \ln p_{\rm{R}}^ * \\ \;\;\;\;\;\;\;\;\;\; = - \left[ {\hbar /\left( {2m} \right)} \right]\left( {\rho _\alpha ^ * /{\rho _{\rm{R}}}} \right)\nabla {\rho _{\rm{R}}} \equiv - \left[ {\hbar /\left( {2m} \right)} \right]d_\alpha ^ * \nabla {\rho _{\rm{R}}}. \end{array} $ (145)

    Consider now the disentangled polarized reactants {αn+} in Rn+ = (A+|B+) and their entangled analogs {αb+} in the "bonded" reactive system Rb+ = (A+¦B+) exhibiting the overall electron density

    $ \begin{array}{l} \rho _{\rm{R}}^ + = {N_{\rm{R}}}p_{\rm{R}}^ + = \rho _{\rm{A}}^ + + \rho _{\rm{B}}^ + = N_{\rm{A}}^0p_{\rm{A}}^ + + N_{\rm{B}}^0p_{\rm{B}}^ + ,\\ \;\;\;N_\alpha ^ + = \int {\rho _\alpha ^ + \left( \mathit{\boldsymbol{r}} \right){\rm{d}}\mathit{\boldsymbol{r}}} . \end{array} $ (146)

    Their equilibrium phases determine the associated subsystem currents:

    $ \begin{array}{l} \mathit{\boldsymbol{j}}\left( {\alpha _n^ + } \right) = \left( {\hbar /m} \right)\rho _\alpha ^ + \nabla {\phi _{\alpha ,eq}}\left( {\alpha _n^ + } \right) = - \left[ {\hbar /\left( {2m} \right)} \right]\rho _\alpha ^ + \nabla \ln p_\alpha ^ + \\ \;\;\;\;\;\;\;\;\;\; = - \left[ {\hbar /\left( {2m} \right)} \right]\nabla \rho _\alpha ^ + . \end{array} $ (147)

    $ \begin{array}{l} \mathit{\boldsymbol{j}}\left( {\alpha _b^ + } \right) = \left( {\hbar /m} \right)\rho _\alpha ^ + \nabla {\phi _{\alpha ,eq}}\left( {\alpha _b^ + } \right) = - \left[ {\hbar /\left( {2m} \right)} \right]\rho _\alpha ^ + \nabla \ln p_{\rm{R}}^ + \\ \;\;\;\;\;\;\;\;\;\; = - \left[ {\hbar /\left( {2m} \right)} \right]\nabla \rho _{\rm{R}}^ + . \end{array} $ (148)

    The phase aspect of electronic communications in the nonbonded, polarized reactive system Rn+ = (A+|B+) is revealed by amplitudes of conditional probabilities determined by the corresponding elements of the CBO matrix. The conditional probabilities themselves, generated by the squared moduli of amplitudes, loose memory about such relative AO-or MO-phases of reactants. We recall that the internal equilibria in subsystems of Rn+ are characterized by thermodynamic phases generated by densities of the polarized reactants. In the final (bonded) state of Rb* = (A*¦B*) ≡ R, their equilibrium phase is ultimately determined by the molecular probability distribution, of R as a whole.

    In one-determinantal approximation of HF and KS theories the bond-subspace is spanned by N singly-occupied MO |φ> of the ground-state electron configuration. The amplitudes of AO communications are then generated by elements of the CBO matrix γ = {γi, j} [Eq.(90)],

    $ {\bf{A}}\left( {\mathit{\boldsymbol{\chi '}}\left| \mathit{\boldsymbol{\chi }} \right.} \right) = \left\{ {A\left( {j\left| i \right.} \right) = \gamma _{i,i}^{ - 1/2}{\gamma _{i,j}}} \right\}. $ (149)

    The reactant bases {|χα>}, of AO originating from the constituent atoms of subsystem α, in the overall basis |χ> = (|χA>, |χB>) arrange the electron communications and the underlying CBO matrix into the substrate-resolved blocks,

    $ \begin{array}{l} {\bf{A}}\left( {\mathit{\boldsymbol{\chi '}}\left| \mathit{\boldsymbol{\chi }} \right.} \right) = \left\{ {{\bf{A}}\left( {{\mathit{\boldsymbol{\chi }}_\beta }\left| {{\mathit{\boldsymbol{\chi }}_\alpha }} \right.} \right)} \right. = \left\{ {A\left( {b \in \beta \left| {a \in \alpha } \right.} \right)} \right\},\\ {\bf{ \pmb{\mathsf{ γ}} }} = \left\{ {{{\bf{ \pmb{\mathsf{ γ}} }}_{\alpha ,\beta }} = \left\{ {{\gamma _{a,b}}} \right\}} \right\}. \end{array} $ (150)

    The moduli {Ma, b} and phases {Φa, b} of the CBO matrix elements {γa, b = Ma, bexp(iΦa, b)} determine the corresponding parts of complex amplitudes {A(b|a)} of AO communications.The phase-content disappears in the conditional probabilities {P(b|a) = |A(b|a)|2 = |Ma, b|2/Ma, a}, which determine the classical (probability) channel.

    Let us now examine the complex CBO matrix elements between the equilibrium, phase-transformed AO states of the polarized subsystems,

    $ \begin{array}{l} \left| {{\mathit{\boldsymbol{\chi }}_{eq}}\left( {\alpha _n^ + } \right)} \right\rangle = \left\{ {\left| {{a_{eq}}\left( {\alpha _n^ + } \right)} \right\rangle = \exp \left[ {{\rm{i}}{\phi _{eq}}\left( {\alpha _n^ + } \right)} \right]\left| a \right\rangle ,} \right.\\ \;\;\;\;\left. {a \in \alpha ,\alpha = {\rm{A}},{\rm{B}}} \right\}. \end{array} $ (151)

    It should be observed that these subsystem states, corresponding to different equilibrium phases of reactants {ϕeq(αn+; r) = ϕα+(r)}, are no longer orthogonal:

    $ \begin{array}{l} \left\langle {{a_{eq}}\left( {\alpha _n^ + } \right)\left| {{b_{eq}}\left( {\beta _n^ + } \right)} \right.} \right\rangle \\ \;\;\;\;\; = \int {{a^ * }\left( \mathit{\boldsymbol{r}} \right)\exp \left\{ {{\rm{i}}\left[ {{\phi _{eq}}\left( {\beta _n^ + ;\mathit{\boldsymbol{r}}} \right) - {\phi _{eq}}\left( {\alpha _n^ + ;\mathit{\boldsymbol{r}}} \right)} \right]} \right\}b\left( \mathit{\boldsymbol{r}} \right){\rm{d}}\mathit{\boldsymbol{r}}} \\ \;\;\;\;\; \ne \int {{a^ * }\left( \mathit{\boldsymbol{r}} \right)b\left( \mathit{\boldsymbol{r}} \right){\rm{d}}\mathit{\boldsymbol{r}}} = {\delta _{a,b}}. \end{array} $ (152)

    However, the equilibrium basis of the mutually-open (bonded) fragments in Rb = (A*¦B*) = R,

    $ \left| {{\mathit{\boldsymbol{\chi }}_{eq}}\left( {\alpha _b^ * } \right)} \right\rangle = \left\{ {\left| {{a_{eq}}\left( {\alpha _b^ * } \right)} \right\rangle = \exp \left[ {{\rm{i}}{\phi _{eq}}\left( {\rm{R}} \right)} \right]\left| a \right\rangle } \right\}, $ (153)

    all exhibiting the molecular phase ϕeq(R; r) = ϕR(r), remains orthonormal:

    $ \begin{array}{l} \left\langle {{a_{eq}}\left( {\alpha _b^ * } \right)\left| {{b_{eq}}\left( {\beta _b^ * } \right)} \right.} \right\rangle = \int {{a^ * }\left( \mathit{\boldsymbol{r}} \right)\exp \left\{ {{\rm{i}}\left[ {{\phi _{eq}}\left( {{\rm{R}};\mathit{\boldsymbol{r}}} \right) - {\phi _{eq}}\left( {{\rm{R}};\mathit{\boldsymbol{r}}} \right)} \right]} \right\}b\left( \mathit{\boldsymbol{r}} \right){\rm{d}}\mathit{\boldsymbol{r}}} \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \int {{a^ * }\left( \mathit{\boldsymbol{r}} \right)b\left( \mathit{\boldsymbol{r}} \right){\rm{d}}\mathit{\boldsymbol{r}}} = {\delta _{a,b}}, \end{array} $ (154)

    and hence

    $ {\Sigma _\beta }{\Sigma _{b \in \beta }}\left| {{b_{eq}}\left( {\beta _b^ * } \right)} \right\rangle \left\langle {{b_{eq}}\left( {\alpha _b^ * } \right)} \right| = {\Sigma _\beta }{\Sigma _{b \in \beta }}\left| b \right\rangle \left\langle b \right|. $ (155)

    The relevant CBO matrix element γa, b; eq(Rn+) coupling two equilibrium AO states in Rn+,

    $ \begin{array}{l} {a_{eq}}\left( {\alpha _n^ + ,\mathit{\boldsymbol{r}}} \right) = a\left( \mathit{\boldsymbol{r}} \right)\exp \left[ {{\rm{i}}\phi _\alpha ^ + \left( \mathit{\boldsymbol{r}} \right)} \right] = {a_{eq}}\left( {{\rm{R}}_n^ + ;\mathit{\boldsymbol{r}}} \right)\;\;\;\;{\rm{and}}\\ {b_{eq}}\left( {\beta _n^ + ,\mathit{\boldsymbol{r}}} \right) = b\left( \mathit{\boldsymbol{r}} \right)\exp \left[ {{\rm{i}}\phi _\beta ^ + \left( \mathit{\boldsymbol{r}} \right)} \right] = {b_{eq}}\left( {{\rm{R}}_n^ + ;\mathit{\boldsymbol{r}}} \right), \end{array} $ (156)

    then reads:

    $ \begin{array}{l} {\gamma _{a,b;eq}}\left( {{\rm{R}}_n^ + } \right) = \left\langle {{a_{eq}}\left( {{\rm{R}}_n^ + } \right)\left| \mathit{\boldsymbol{\varphi }} \right.} \right\rangle \left\langle {\mathit{\boldsymbol{\varphi }}\left| {{b_{eq}}\left( {{\rm{R}}_n^ + } \right)} \right.} \right\rangle \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \equiv {M_{a,b;eq}}\left( {{\rm{R}}_n^ + } \right)\exp \left[ {{\rm{i}}{\mathit{\Phi }_{a,b;eq}}\left( {{\rm{R}}_n^ + } \right)} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \int {{\rm{d}}\mathit{\boldsymbol{r}}} \int {{\rm{d}}\mathit{\boldsymbol{r'}}{a_{eq}}{{\left( {a_n^ + ,\mathit{\boldsymbol{r}}} \right)}^ * }\left[ {\mathit{\boldsymbol{\varphi }}\left( \mathit{\boldsymbol{r}} \right){\mathit{\boldsymbol{\varphi }}^\dagger }\left( {\mathit{\boldsymbol{r'}}} \right)} \right]{b_{eq}}\left( {\beta _n^ + ;\mathit{\boldsymbol{r'}}} \right)} \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \equiv \int {{\rm{d}}\mathit{\boldsymbol{r}}} \int {{\rm{d}}\mathit{\boldsymbol{r'}}\left[ {{a^ * }\left( \mathit{\boldsymbol{r}} \right)b\left( {\mathit{\boldsymbol{r'}}} \right)} \right]{\gamma _{\alpha ,\beta }}\left( {\mathit{\boldsymbol{r}},\mathit{\boldsymbol{r'}}} \right)} \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \equiv \int {{\rm{d}}\mathit{\boldsymbol{r}}} \int {{\rm{d}}\mathit{\boldsymbol{r'}}{R_{a,b}}\left( {\mathit{\boldsymbol{r}},\mathit{\boldsymbol{r'}}} \right){\gamma _{\alpha ,\beta }}\left( {\mathit{\boldsymbol{r}},\mathit{\boldsymbol{r''}}} \right)} , \end{array} $ (157)

    $ {\gamma _{\alpha ,\beta }}\left( {\mathit{\boldsymbol{r}},\mathit{\boldsymbol{r'}}} \right) = \exp \left\{ {{\rm{i}}\left[ {\phi _\beta ^ + \left( {\mathit{\boldsymbol{r'}}} \right) - \phi _\alpha ^ + \left( \mathit{\boldsymbol{r}} \right)} \right]} \right\}\mathit{\boldsymbol{\varphi }}\left( \mathit{\boldsymbol{r}} \right){\mathit{\boldsymbol{\varphi }}^\dagger }\left( {\mathit{\boldsymbol{r'}}} \right). $ (158)

    For the complex matrix element γa, b; eq(R), coupling the two equilibrium-AO states of the mutually-bonded reactants in R,

    $ \begin{array}{l} {a_{eq}}\left( {\alpha _b^ * ;\mathit{\boldsymbol{r}}} \right) = a\left( \mathit{\boldsymbol{r}} \right)\exp \left[ {{\rm{i}}{\phi _{\rm{R}}}\left( \mathit{\boldsymbol{r}} \right)} \right] = {a_{eq}}\left( {{\rm{R}},\mathit{\boldsymbol{r}}} \right)\;\;\;\;{\rm{and}}\\ {b_{eq}}\left( {\beta _b^ * ;\mathit{\boldsymbol{r}}} \right) = b\left( \mathit{\boldsymbol{r}} \right)\exp \left[ {{\rm{i}}{\phi _{\rm{R}}}\left( \mathit{\boldsymbol{r}} \right)} \right] = {b_{eq}}\left( {{\rm{R}};\mathit{\boldsymbol{r}}} \right), \end{array} $ (159)

    one similarly finds:

    $ \begin{array}{l} {\gamma _{a,b;eq}}\left( {\rm{R}} \right) \equiv {M_{a,b;eq}}\left( {\rm{R}} \right)\exp \left[ {{\rm{i}}{\mathit{\Phi }_{a,b;eq}}\left( {\rm{R}} \right)} \right] = \left\langle {{a_{eq}}\left( {\rm{R}} \right)\left| \mathit{\boldsymbol{\varphi }} \right.} \right\rangle \left\langle {\mathit{\boldsymbol{\varphi }}\left| {{b_{eq}}\left( {\rm{R}} \right)} \right.} \right\rangle \\ \;\;\;\;\;\;\;\;\;\;\;\;\; = \int {d\mathit{\boldsymbol{r}}} \int {d\mathit{\boldsymbol{r'}}{a_{eq}}{{\left( {{\rm{R}},\mathit{\boldsymbol{r}}} \right)}^ * }\left[ {\mathit{\boldsymbol{\varphi }}\left( \mathit{\boldsymbol{r}} \right){\mathit{\boldsymbol{\varphi }}^\dagger }\left( {\mathit{\boldsymbol{r'}}} \right)} \right]{b_{eq}}\left( {{\rm{R}};\mathit{\boldsymbol{r'}}} \right)} \\ \;\;\;\;\;\;\;\;\;\;\;\;\; \equiv \int {d\mathit{\boldsymbol{r}}} \int {d\mathit{\boldsymbol{r'}}{R_{a,b}}\left( {\mathit{\boldsymbol{r}},\mathit{\boldsymbol{r'}}} \right){\gamma _{\rm{R}}}\left( {\mathit{\boldsymbol{r}},\mathit{\boldsymbol{r'}}} \right)} , \end{array} $ (160)

    $ {\gamma _{\rm{R}}}\left( {\mathit{\boldsymbol{r}},\mathit{\boldsymbol{r'}}} \right) = \exp \left\{ {{\rm{i}}\left[ {{\phi _{\rm{R}}}\left( {\mathit{\boldsymbol{r'}}} \right) - {\phi _{\rm{R}}}\left( \mathit{\boldsymbol{r}} \right)} \right]} \right\}\mathit{\boldsymbol{\varphi }}\left( \mathit{\boldsymbol{r}} \right){\mathit{\boldsymbol{\varphi }}^\dagger }\left( {\mathit{\boldsymbol{r'}}} \right). $ (161)

    This CBO matrix is idempotent [see Eqs. (154) and (155)],

    $ \begin{array}{l} {\Sigma _\beta }{\Sigma _{b \in \beta }}{\gamma _{a,b;eq}}\left( {\rm{R}} \right){\gamma _{b,a;eq}}\left( {\rm{R}} \right) = {\gamma _{a,a;eq}}\left( {\rm{R}} \right)\\ \;\;\;\; = \left\langle {{a_{eq}}\left( {\rm{R}} \right)\left| \mathit{\boldsymbol{\varphi }} \right.} \right\rangle \left\langle {\mathit{\boldsymbol{\varphi }}\left| {{a_{eq}}\left( {\rm{R}} \right)} \right.} \right\rangle {\left| {\left\langle {{a_{eq}}\left( {\rm{R}} \right)\left| \mathit{\boldsymbol{\varphi }} \right.} \right\rangle } \right|^2}\\ \;\;\;\; = {\Sigma _s}{\left| {\left\langle {{a_{eq}}\left( {\rm{R}} \right)\left| {{\varphi _s}} \right.} \right\rangle } \right|^2} = {\Sigma _s}P\left[ {{\varphi _s}\left| {{a_{eq}}\left( {\rm{R}} \right)} \right.} \right]\\ \;\;\;\; \equiv P\left[ {\mathit{\boldsymbol{\varphi }}\left| {{a_{eq}}\left( {\rm{R}} \right)} \right.} \right] \equiv P\left[ {{a_{eq}}\left( {\rm{R}} \right)\left| \mathit{\boldsymbol{\varphi }} \right.} \right], \end{array} $ (162)

    where P[φ|aeq(R)] = P[aeq(R)|φ] stands for the conditional probability of observing in the bond-system φ an electron occupying aeq(R) (P[φ|aeq(R)]), or -of finding aeq(R) in the occupied MO subspace φ(P[aeq(R)|φ]).

    The CBO matrix γeq(R) = {γa, b; eq(R)} determines the conditional probabilities determining classical communications between the equilibrium AO:

    $ \begin{array}{l} {\bf{P}}\left( {{{\mathit{\boldsymbol{\chi '}}}_{eq}}\left( {\rm{R}} \right)\left| {{\mathit{\boldsymbol{\chi }}_{eq}}\left( {\rm{R}} \right)} \right.} \right) = \left\{ {{{\bf{P}}_{eq}}\left( {{\mathit{\boldsymbol{\chi }}_\beta }\left| {{\mathit{\boldsymbol{\chi }}_\alpha }} \right.} \right) = \left\{ {P\left( {{b_{eq}}\left| {{a_{eq}}} \right.} \right),\alpha ,\beta \in \left( {{\rm{A}},{\rm{B}}} \right)} \right\}} \right\},\\ \;\;\;\;\;{\Sigma _\beta }{\Sigma _{b \in \beta }}{\rm{P}}\left( {{b_{eq}}\left| {{a_{eq}}} \right.} \right) = 1,\\ P\left( {{b_{eq}}\left| {{a_{eq}}} \right.} \right) = {\gamma _{a,b;eq}}\left( {\rm{R}} \right){\gamma _{b,a;eq}}\left( {\rm{R}} \right)/{\gamma _{a,a;eq}}\left( {\rm{R}} \right) \equiv {\left| {A\left( {{b_{eq}}\left| {{a_{eq}}} \right.} \right)} \right|^2}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. { = {M_{a,b;eq}}{{\left( {\rm{R}} \right)}^2}/P\left[ {{a_{eq}}\left( {\rm{R}} \right)\left| \mathit{\boldsymbol{\varphi }} \right.} \right]} \right\}\;\;\;\;\;\;\;\;{\rm{or}}\\ A\left( {{b_{eq}}\left| {{a_{eq}}} \right.} \right) = {\gamma _{a,a;eq}}{\left( {\rm{R}} \right)^{ - 1/2}}{\gamma _{a,b;eq}}\left( {\rm{R}} \right). \end{array} $ (163)

    Therefore, the resultant modulus Ma, b; eq(R)determines the conditional probability of the information propagation between the equilibrium AO, i.e., the classical equilibrium -AO channel, while the resultant phase Φa, b; eq(R) shapes the information descriptors of its nonclassical complement, the associated phase information network35.

    The phase/current aspect of electronic states rises a natural question of a possible favourable matching of the reactant phases in chemical reactions. Clearly, any synchronization of the equilibrium phases of reactants or their acidic/basic fragments, which is required for generating or enhancing the spontaneous flows of electrons discussed in the preceding section, is favourable for the inter-reactant bond formation processes. The concerted flows of Fig. 2I are driven by the chemical potential difference between the coordinating acidic (acceptor) and basic (donor) subsystems. Below we briefly examine the subsystem phase-requirements for effecting, in a "coherent" reactive event, the charge currents in such preferred directions.

    In the positive-phase convention the equilibrium current of an atomic fragment is directed away from its nucleus. Therefore, the in-phase process ϕRin = ϕA++ϕB+ of combining the subsystem wavefunctions in the antisymmetrized product ψRin =${{\rm{\hat A}}_{{\rm{A}} \leftrightarrow {\rm{B}}}}(\psi _{\rm{A}}^ + \psi _{\rm{B}}^ + )$ generates an accumulation of electron density between the coordinating atoms of the acidic (A) and basic (B) reactants, thus enhancing the inter-subsystem bond covalency. Accordingly, the CT flow of electrons, A+ ← B+, in the prototype coordination bond is enhanced by the equilibrium flows corresponding to the anti-phase process:

    $ \begin{array}{l} \phi _{\rm{R}}^{anti} = \phi _{\rm{B}}^ + - \phi _{\rm{A}}^ + ,\psi _{\rm{R}}^{anti} = {{{\rm{\hat A}}}_{{\rm{A}} \leftrightarrow {\rm{B}}}}\left[ {{{\left( {\psi _{\rm{A}}^ + } \right)}^ * }\psi _{\rm{B}}^ + } \right]\\ {\mathit{\boldsymbol{j}}_{eq}}\left( {{{\rm{B}}^ + }} \right) - {\mathit{\boldsymbol{j}}_{eq}}\left( {{{\rm{A}}^ + }} \right) = \left( {\hbar /m} \right)\left[ {p_{\rm{B}}^ + {\phi _{eq}}\left( {{{\rm{B}}^ + }} \right) - p_{\rm{A}}^ + {\phi _{eq}}\left( {{{\rm{A}}^ + }} \right)} \right]\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = - \left[ {\hbar /\left( {2m} \right)} \right]\nabla \left( {p_{\rm{B}}^ + - p_{\rm{A}}^ + } \right). \end{array} $ (164)

    Similar anti-phase synchronization of currents is required to enhance the resultant CT between coordinating atoms of the acidic (A) and basic (B) reactants, and the primary CT displacements nCT(1) and nCT(2)of Eq.(131), which also represent the spontaneous flows in response to the initial electronegativity differences of the relevant acidic (a) and basic (b) sites in both reactants.

    The secondary, relaxational flows {δNα} or {Ia} of Figs. 2-4, from the acidic (aα) to basic (bα) part of α, are similarly generated and enhanced by the equilibrium subsystem currents corresponding to another ϕ(aα+) -ϕ(bα+) anti-phase process:

    $ \mathit{\boldsymbol{j}}\left( {a_\alpha ^ + } \right) - \mathit{\boldsymbol{j}}\left( {b_\alpha ^ + } \right) = - \left[ {\hbar /\left( {2m} \right)} \right]\nabla \left[ {p\left( {a_\alpha ^ + } \right) - p\left( {b_\alpha ^ + } \right)} \right] $ (165)

    10 Conclusions

    To accommodate complex wavefunctions of QM the nonclassical (phase/current)-related supplements to classical (probability) descriptors of the entropy/information content in molecular electronic states are required. A generalized QIT gradient measure of the Fisher determinicity-information, related to dimensionless kinetic energy of electrons, involves a contribution due to the probability current (phase gradient), which gives rise to a nonvanishing quantum information source. The resultant global entropy, the Shannon-type descriptor of the state indeterminicity-information, similarly involves the average-phase contribution, which complements the classical Shannon functional of the electronic probability distribution. This extension satisfies the requirement that the classical link between the Shannon and Fisher information densities, for the incoherent local events, extends into the nonclassical (quantum) domain of relations between the entropy/information densities of the coherent events characterized by their phases or currents. The gradient descriptor of the state resultant entropy (indeterminicity-information) has also been introduced, including the negative current/phase contribution. The complex entropy, a natural generalization of the classical Shannon concept, generates the probability and phase contributions in the resultant measure as its real and imaginary parts.

    The information principle of the maximum resultant entropy determines the phase-equilibria in molecules and their constituent fragments. Both the global and gradient measures of the resultant entropy have been shown to give rise to the same thermodynamic-phase solution marking the system phase-equilibrium. The optimum "thermodynamic" phase of the externally-nonbonded fragment is related to the logarithm of its probability density, thus generating the subsystem nonvanishing electronic current, which modifies its nonclassical entropy/information contribution. The phase transformation of the system wavefunction also affects the probability source and the net entropy production3. This QIT perspective on molecular equilibria provides a useful theoretical framework for treating electronic states of constituent fragments in molecular or reactive systems as coherent (phase-related) events. However, as we have explicitly argued in Section 3, a truly "thermodynamic" pure-state description of the coupling between the energetic and IT-entropic aspects of molecular equilibrium states is explicitly precluded by the variational principle of QM.

    In this work we have qualitatively examined the equilibrium-phase description of typical donor-acceptor reactive systems at several hypothetical stages of a bimolecular chemical reaction, involving the mutually-closed (nonbonded, disentangled) or -open (bonded, entangled) reactants or their acidic and basic fragments. We have stressed EE processes, the equilibrium-phase aspect of the reactant polarization, and the concerted electronic flows involved. Implications of the intra-and inter-reactant electronegativity equalizations have been discussed and the current promotion of the (polarized) nonbonded (mutually-closed) and bonded (mutually-open) substrates has been explored. The electronic communications between AO, in both the molecular channel and its cascade (bridge) generalization, give rise to the classical entropic descriptors of the direct and intermediate chemical bonds between reactants. The intra-reactant communications are responsible for the valence-state promotion of the mutually nonbonded subsystems, while the inter-reactant propagations generate descriptors of the chemical bonds between the entangled reactants. In OCT both the overall bond multiplicity and its covalent/ionic components can be expressed in terms of the entropic additive and nonadditive communication contributions derived from the system CBO matrix.

    The preference for the complementary mutual arrangements of reactants in reactive complex, in which the chemically hard (soft) fragments of one reactant face the chemically soft (hard) parts of the reaction partner, has been justified by exploring the polarizational/charge-transfer flows of electrons they imply. In the complementary reaction complex, which provides a favourable matching of the ES potentials of both reactants, the electronic displacements have been shown to avoid an exaggerated electron accumulation or depletion on reactants and their constituent fragments. Such concerted displacements in the electronic structure have been shown to fulfill the familiar Le Châtelier-Braun stability requirement: the responses to the primary CT displacements act in the direction to diminish consequences of the latter, thus acting in the direction to regain the equilibrium. This behavior is in contrast to that predicted for the regional HSAB-complex, in which the hard (soft) fragments of both reactants face the like-fragments of the other reactant. The phase-synchronization effecting the desired displacements in the system electronic structure has also been commented upon.

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