NALEWAJSKI Roman F.

?

Chemical Reactivity Description in Density-Functional and Information Theories

NALEWAJSKI Roman F.*

()

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 ([]) and Fisher ([]) information terms probe the entropic content of incoherent local events of the particle localization, embodied in the probability distribution, while their nonclassical phase-companions,[] and[], provide relevantinformation 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.

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 thegeneralization of the original () information concepts of Classical Information Theory (CIT) pioneered by Fisher1and Shannon2, which are appropriate for the complex amplitudes (wavefunctions) of Quantum Mechanics (QM). The electron distribution alone generates the state averageinformation, i.e., the information received from outcomes of thelocal events of the particle-position measurements. It has been argued elsewhere3–8that in Quantum IT (QIT)3both 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 thepart of the overall information content, while the wavefunction phase or its gradient (probability-current) generate itssupplement.

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(“disentangled”) local events, the outcomes of the repeated measurements of the particle position, while their nonclassical companions provide the information supplement due to the(“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”-transformation of the Schr?dinger 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-variable. Within the electron-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(promotion) of the mutually-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)14–16or Conceptual DFT13,17,18. The chemical indices of electronic13,19–23,24,25,(or)26, andconcepts18,27help 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-description and communication treatment can be also applied to reactive systems3,39–41. The CIT descriptors generate theof 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 bond3,36–38examines the entropic bond descriptors generated by communication networks of the probability propagation between basis functions of quantum-mechanical calculations. It has identified the new38,45–51of 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, suchbonds 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)20allow 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-electron wave functions, loses the(“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-electron case will often be assumed, the modulus (density) and phase (current) aspects of general-electron states can be similarly separated using the Harriman-Zumbach-Maschke(HZM)53,54construction 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 (= 1) electron described by the complex wavefunction specified by its (real)() and() parts:() =()exp[i()]. In accordance with SP of QM52the wavefunction() =á|?measures the amplitude of-probability() =|()|2=()2, i.e., theprobability(|) of observing in|?the localized-electron state|?,

(|) =á|?á|?oá||?

=á|?á|?oá||?, (1)

(|)o() = (/)()?()()(), (2)

with the current-per-particle() =()/() = (/)?() defining the effective velocity of the probability flux.

The dynamics of a general wavefunction(,) =á|()?is described by the Schr?dinger equation (SE),

where the Hamiltonian

d(;)/d≡(;) = ?(;)/?+?×(;) = 0 or

(;)/?= ??×(;), (5)

and the phase-dynamics equation:

(;)/?= (/2){(;)-1?2(;) – [?(;)]2}

?()/. (6)

It should be observed that the elementary particle-localization states {|?} are regarded in CIT as being incoherent in character, i.e., having no prior, common molecular “ancestor”. Indeed, they represent the “disentangled” (-unrelated) measurements of the particle position. However, in Eq.(1) the outcomes of the localization experiments areon the quantum state|?, which thus determines the common molecular “ancestor” of the underlying (“entangled”,) conditional events {(|)=(|,)}.Onefurther notices that(square moduli of wavefunctions) loose the information about phases/currents of such elementary-states, which is preserved in the corresponding probability amplitudes themselves. Therefore, the amplitude (wavefunction) data should be used to determine the(“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 Fisher1measure of the classical-information content in the probability density() is reminiscent of von Weizs?cker's55inhomogeneity correction to the kinetic energy functional,

[] =ò?()]2/()d=ò()[?ln()]2doò()I()d

= 4ò[?()]2do[] (7)

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

The complementary classical descriptor of the Shannon2entropy,

[] = ?ò()ln()doò()S()d

= ?2ò2()ln()do[] (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:S[]o[]. The densities-per-electron of these probability/modulus functionals satisfy the classical relation

I() = [?S()]2(9)

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

[] = ?4á|?2|?oá||?= 4ò|?()|2doò()()d

=[] + 4ò()[?()]2doò()[I() +()]d

o[] +[]

=[] + (2/)2ò2()/()d=ò()[I() +()]d

o[] +[] or (10)

[] = ?á|ln+2|?oá||?=ò()*()d

=ò()[S() ? 2()]doò()[S() +()]d

oò()()do[] +[]. (12)

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

() = [?ln()]2+ 4[?()]2, (13)

and of theentropy[]:

() = ?[ln() + 2()]. (15)

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

is proportional to the average kinetic energy[] =á||?corresponding to the operator,

[]oá||?= (8/2)[],

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

() = [?()]2, (16)

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

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

() =S() + i() = ?[ln() + 2i()]. (18)

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

ln= ln||+ i. (19)

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

oH() +(), (20)

[]oá||?=á|-2ln|?=[] + i[]

=[] + i[]o[] +[].(21)

The complex entropy3,9thus provides a natural-amplitudegeneralization of the familiar Shannon measure2of the entropy content in the probability distribution.

() = [?()]2= [i?()]2=-[?()]2. (22)

3 Equilibrium states

The-equilibrium state of an electron corresponds to the maximum of resultant measures of the- or-entropy content in|?:

{á||??á|?} = 0 or

{á||??á|?} = 0, (23)

whereanddenote the relevant Lagrange multipliers associated with the (“geometric”) constraint of the wavefunction normalization3–8. These information principles have been applied to the Schr?dinger states() giving the specified probability distribution(),?, to determine the optimum (“thermodynamic”) phase

[;] = ?(1/2) ln()o(), (24)

which defines the “thermodynamic” (-transformed) equilibrium state:

[;] =()exp[i()]o().(25)

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

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

()o|()|2p()o|()|2, (26)

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

[] =á||?

=á||?+ [2/(2)]ò()[?()]2d

o[] +[]

=[] + (/2)ò()[()/()]2d

o[] +[], (27)

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

[] = (/)?[] = ?[/(2)]?. (28)

In the ground-stateY0() of-electron systemthe electronic energy is determined by the functional of electronic density0() =0(),

E[Y0()] =E[0()]oò0()()d+[0] =[,],(29)

where the universal functional[0] generates the sum of electron kinetic and repulsion energies. This electron distribution also marks the equalized local chemical potential0() at thelevel[,],

=[,]o(?[,]/?). (30)

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

=[,] + [2/(8)]{[0()-1?0()]2?20()-1D0()}

o[,] +[0,]. (31)

Therefore, the-equilibrium of the electron configurationY0() corresponding to the Slater determinant constructed fromlowest (occupied)Kohn-Sham (KS)11MO of Eq.(25), exhibiting a common thermodynamic phase of Eq.(24) for the ground-state electron distribution0=0, differs from the original-equilibrium ground-state configurationY0() =Y0[0] constructed from the original KS orbitals. Indeed, the latter corresponds to the vanishing spatial phase,[0] = 0, and hence to the zero electronic current,[0] =, while the-displaced ground-state exhibits a finite equilibrium current[0] =Nj[0] = ?[/(2)]?0, and hence also the nonvanishing density-source term() =[0;],

() = d()/d|=?()/?|+?×() =?×()

=-[/(2)]D0,

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

?()/?|= ??×() +() =?()/?|0= ??×() = 0.(32)

The current(), “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” way56of combining the energetic and entropic aspects of molecular electronic structure calls for the maximum-entropy principle-coupled to the electronic energy. For example, in the single-electron system such variational rules read:

{á||??-1á||??á|?} = 0 or

{á||??-1á||??á|?} = 0, (33)

These (energy-constrained) variational principles, with respect to the (complex) trial state|?for the adopted measure of the resultant entropy and specified probability distribution0, give the associated Euler equations determining the equilibrium state|[0]?o|?:

Indeed, by the familiar variational principle of QM any attempt to conserve energy, while modifying the system (exact) wavefunction0() of the nondegenerate ground-state,

where the state local energy

e.g., as a result of the “thermodynamic”-transformation of Eq.(25), must fail since such manipulations always raise the energy above0:[]>0. This implies that the global variational principles of Eq.(33) can be satisfied only for the exactly vanishing “thermodynamic” phase() = 0 which follows directly from the stationary SE (36).

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

However, in the variational principle (38) the fixing of0() at each point also implies an effective total energy constraint in the “thermodynamic”-transformation,

ò0()0()d=ò0()0()d

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[0;]o() of the electronic wavefunction()=0()exp[i()] for the prescribed ground-state distribution0(),

0()= ?ln0() ? 2(),

= {?[2/(2)][D0+ i(0D+ 2?0×?)

0(?)2] +0} exp(i)

Since the given particle distribution0=0fixes the classical, DFT partE[0] of the state electronic energy, one finally observes that the entropic optimization determining the equilibrium phase[0;] 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-energyE[0]. 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 stagesin 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 ofR=A+Belectrons, in several intermediate stages involving the mutually(nonbonded, disentangled) or(bonded, entangled) reactants14,15,36,38,40,41:

0() =0()/0,ò0()d= 1,(46)

or the associated Rnormalized densities of subsystems in R￥,

0() =0()/R= (0/R) [0()/0]o00(),

?ò0() d=?0= 1.(47)

Here0=0/Rstands for the condensed probability of subsystem0in R￥. In this “bonded” equilibrium system each fragment exhibits the “molecular” thermodynamic phase,

R￥=R￥[R￥],

R￥= (A0+B0)/RoR0/R=?0(), (48)

and equalized chemical potentials:

A*(Ψ,￥) =B*(Ψ,￥) =R[R0].

The(disentangled, nonbonded) free reactants of R￥= A0+ B0are uniquely characterized by their separate (-reactant equalized) levels of chemical potentials and thermodynamic phases determined by the isolated subsystem densities:

0=0[0,] =0[0] and

[0] =0[0,].(49)

R(RA―B) =A(RA―B) +B(RA―B).(50)

The(nonbonded, disentangled) reference R0=(A0|B0) corresponds to the electronically and geometrically “frozen” (mutually-) free sybsystems brought to RA―B, and separately exhibiting the same distributions as in R￥. The reactant distributions then determine the promolecular electron density:

R0(RA―B) =A0(RA―B) +B0(RA―B)oR0R0(RA―B),

òR0()=R0=A0+B0=R. (51)

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

The nonbondedstage R+= (A+|B+) for the same nuclear positions RA―Bconsists of the mutually-(disentangled) reactants,internally relaxed electronicallybutgeometrically rigid, characterized by their promoted densities {+=+[A0,B0,R]o0+}, the equilibrium distributions for the initial (integer) numbers of electrons in subsystems and the overall external potential of R+as a whole, which combine into the overall distribution of the polarized reactive system:

R+(RA―B) =A+(RA―B) +B+(RA―B)oRR+(RA―B),

òR+()d=Ror

R+=A0A++B0B+,òR+()d= 1. (52)

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

+[A0,B0,R] =+[A+,B+],A+1B+. (53)

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

[+;] =[+]=0[0,R],= A, B. (54)

Finally, the-reactant equilibrium state, of the geometrically “frozen” but the mutually-(bonded, electronically relaxed), entangled reactants in R(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

*=*[A*,B*,R]o**oRX*,

ò*()d=*,?*=R, (55)

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

A*+B*=R[R,R]oRR[R,R]. (56)

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

CT=A*?A0=B0?B*> 0, (57)

which equalizes the chemical potentials of both subsystems:

A*[A*,B*,R] =B*[A*,B*,R] =R[R,R]. (58)

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

[R(Ψ)] =[R,R;] = ?(?)lnR() =[R;],(59)

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

(*)o*[R(Ψ)] =*[{*},R] =[{*}]

=[R] =[R(Ψ)],= A, B. (60)

In DFT the optimum wavefunctions for the specified electron density ofelectrons,=, are constructed using the HZM scheme3,4,53,54. Such functions also appear in Levy's12constrained-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 thewave-type:

{() =()exp[i(;)]}. (61)

They adopt equal, density-dependent modulus part() =()1/2and the spatial phase function composed of the “orthogonality” [F()] and “thermodynamic” [(;)] parts,

(;) =[]×[;] +[,;]oF() +(;),(62)

with the density-dependent vector function[;]o() common to all orbitals and linked to the Jacobian of the?() transformation, and the equilibrium phase from the maximum-entropy principle3–8:

(;) =? (?)ln() ?(k[])×[;]

o()–(k)×(),(k) =-1?. (63)

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

(;) =×() ? (?)ln(),

=?(k),?= 0. (64)

This spatial phase generates the resultant current ofelectrons

(;)= ?[/(2)]?(). (65)

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

The EO-vector (reduced momentum)=[] and the density-dependent vector field() 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(), 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)= (1,2, …,). The AO states|?then identify the associated varieties of the mutually exclusive, elementary events in the molecular-system determined byoccupied MO:= (1,2, …,). In what follows we consider the simplest,-configuration approximation of HF or KS SCF calculations, with the occupied MO expanded in the AO basis:=C; here the unitary matrix C, C?C = I, groups coefficients of expanding the spatial (MO) partsof the-occupied-orbitals.

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

P('|) = {(|)o(|)o(,)/po|(|)|2}, (66)

or the associated amplitudes A('|) = {(|)} defining the nonclassical network of scattering the(input) states|?= {|?} among the(output) states|'?= {|?}, result from the bond-projected SP of QM; here(,)o(,) stands for the joint probability of simultaneously observingandin the molecular-system. In AO representation this bond subspace defines the idempotent Charge and Bond-Order (CBO) matrix:

γ = {,j} =á|?á|’?= CC?, γ2= γ. (67)

In OCT the molecular joint probabilities of AO, of the simultaneousandAO events (,)o(,) in the bond-system of a molecule, are thus proportional to the square of the corresponding CBO matrix element:

(,) =,j,i/=|,j|2/,

?(,) =-1?,j,i=,i/=p. (68)

The conditional probabilities between AO,

P('|) = {(|) =(,)/p},

(|) =|,|2/,o|(|)|2,?(|) = 1, (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-input signal()ogenerates the same distribution in the output of the AO probability network,

(') ==P('|) = {?pP(|)o?(,) =p} =, (70)

thus identifyingas thevector 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-measured by the average-entropy of the system molecular outputs given molecular inputs2,3:

(|)o=-??(,)log(|). (71)

The molecular channel probed by the-input signal0{p0}, 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-, given by the-information in the channel promolecular-input (0) and molecular-output () signals2,3:

(0:) =??(,)log[(,)/(p0q)]

=??(,)log[pP(,)/(pqp0)]

=??(,)[?logq+ log(|) + log(p/p0)]

=(') ?+D(|0)o+D(|0)o0. (72)

Above,D(|0) denotes the molecular entropy-deficiency of Kullback and Leibler57,58, relative to the promolecular distribution0,

D(|0) =?plog(p/p0).(73)

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

(0) = ??p0logp0,(74)

which has not been dissipated into communication “noise” in the molecular channel. In particular, for molecular input signal0=, and henceD(|) = 0,

(:) =() ?o. (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,

(0;) =+0=() +D(|0)o0. (76)

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

(;) =+=()o.(77)

These IT-descriptors can be further decomposed into theandcontributions in the adopted AO resolution. The communication-(covalency) represents a difference between theandterms of the information contained in the molecular CBO matrix γ.These contributions also define the associatedcomponentS(γ) of the overall (total) entropy content in γ,S(γ)oS(γ) +S(γ),

(|) = ???(,) log[(,)/p]

=-1[???,j,ilog(,j,i)+?,ilog,i]

o-1{S(γ) ?S(γ)}o-1S(γ) (78)

S(γ) =-??,j,ilog(,j,i)

S(γ) =?,ilog,i, (79)

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

(:) =??(,)log[(,)/(pq)]

=-1{2S(γ) ?S(γ) + log}

=-1{S(γ) ?S(γ) + log}. (80)

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

(;) =(|) +(:) =-1{S[γ] +log}. (81)

To summarize, in the-determinant approximation of the molecular state thepart of the Shannon entropy in molecular AO-communications (CBO matrix) represents theentropic bond-multiplicity, thepart reflects the IT-(indeterministic, noise) descriptor, while their difference measures the complementary IT-(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= (A,B) determine the relevant AO events on AIM in each reactant and their molecular() = (A,B) =and promolecular0() = (A0,B0) =0probabilities. 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('|) = {P(|),,?(A, B)}, which produce the associated output signals:

(')o= (A,B) =P('|) =and

0(')o0= (A0,B0)=0P('|). (82)

Alternatively, the optimum MO of the separated subsystems,R= (A,B)o, identifying-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()= (A,B) =and promolecular0()= (A0,B0) =0. The conditional probabilities P('|)={P(|),,?(A,B)},theclassicalcommunicationsin the MO-resolved R, ultimately determine the associated output signals:

(') == (A,B) =P('|) =and

0(') =0= (A0,B0) =0P('|). (83)

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

γ =á|?á|?= CC?

= {á|?á|?oCC?oγ,,,?(A, B)}; (84)

herecombines the occupied MO of R and the reactant parts {C=á|?} of the LCAO MO matrix C =á|?= {C} group the expansion coefficients ofin terms of the reactant AO. The corresponding MO channel is similarly related to the transformed CBO matrix in MO representation,

γMO=áR|?á|R?oUU?

= {á|?á|?oUU?,,?(A, B)}, (85)

where {U=á|?} in U =áR|?determine the expansion ofin terms of the reactant MOR= {}. At these two resolution levels the molecular or promolecular input signals then generate the corresponding overall descriptors of the OCT bond multiplicities (,0) or (MO,0,MO), and their covalent/ionic components: thenetwork conditional-entropies ()andmutual-informations(): (,or0) or (MO,MOor0,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-statewithin 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,mutual arrangements Rof 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-orbital events. The amplitudes of their conditional probabilities are given by products of the direct stage-amplitudes and generate the IT descriptors of the(bridge)in the reactive system45–50.

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-bridge communications between the acidic () and basic () parts of reactants in R, which are suggested by the dominatingflowsof the-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 R: the most important displacementCT(1)is expected between the basic partBof B and the acidic fragmentAof A, with the predicted complementary (reverse) flowCT(2), from the basic partAof A to the acidic fragmentBof B. As shown in Fig.2, these (primary) partial CT between the mutually-reactants, giving rise to the net B?A electron transfer,

CT=CT(1)?CT(2)=A*?A0=B0?B*> 0,(86)

should be accompanied by the(relaxational) adjustments (broken arrows) in electron populations of these fragments, reflected by the CT-induced-fragment flows:A,fromAtoA, andB, fromBtoB. In this flow pattern the primary and induced flowseach other giving rise to the least-activation displacement of the electronic structure in R59.

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 R. Among the cascade communications the most important are the-bridge propagations shown in Panel II of Fig.2. They can be classified as representing either the P-scatterings, when they combine two-reactant (broken arrow) scatterings, the CT-inducedpolarizations involving single-reactant (P) and-reactant (CT) (solid arrow) communications, or the pure CT bridges, when they involve two-reactant (CT) links.

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

Fig.2 Concerted flows (Panel I) in the complementary (c) arrangement of subsystems in the bimolecular reactive system

hereanddenote the acidic () and basic () parts of reactant= A,B, and most important cascade communicationssingle orbitalintermediates(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.

6 Amplitude and probability channels

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 Ra0 = (A0, B0) are split on their (mutually-closed) acidic (aa) and basic (ba) fragments, due to the perturbation created by the presence of the nearby bb and ab parts of the reaction partner Rb0. These shifts within the initially polarized reactants {Ra+ = (aa+|ba+)} then trigger the P-flows {dNa}, which regain electronegativity equalization in {Ra+ = (aa+|ba+)} at their internal chemical-potential levels {mX+}. The resulting chemical-potential difference Dm+ = mA+ ?mB+<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 {Ra* = (aa*|ba*)} and their constituent acidic {aa*} and basic {ba*} parts. One observes that a presence of B destabilizes A, DmA(B) > 0, while A stabilizes B, DmB(A) < 0.

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 configurationYUHF()o|1,2,…,|= det() defined by(singly-occupied) MOexhibiting lowest orbital energies {=á||?},1￡2￡…￡￡…￡. Here the () MO (SMO)() = {()()} =á|?combine the spatial (MO) components() ={()} =á|?} = [(),()] and spin functions() = {()} =á|??(,) of an electron. In typical Unrestricted Hartree-Fock (UHF) SCF calculations or the-resolved KS DFT an exploration of chemical bonds calls for the (orthonormal) AO basis set= (1,2, …,) for expanding all MO functions,and,

=C, C =á|?= [á|?,á|?] = [C, C],

á|?=á|?á|?= C?C = I or

C?C= Iand C?C= I. (87)

The ground-stateYUHF() is thus shaped by-lowest SMO, which determine the configuration bond-spacedefined by the projector onto the-occupied MO subspaceo(1,2, …,):

=|?(CC?)á|o|?γá|; (88)

here the diagonal matrix n ={n,s'} =á||?oγMOspecifies the MO occupations:n= {1,￡; 0,>}. This operator generates the CBO matrix of Eq.(84):

γ =á||?=á|?ná|?

=á|?á|?= CC?,

(γ)2=á|?[á|?á|?]á|?

= CIC?= γ. (89)

In accordance with SP of QM52the joint probability of the given pair of the input() and output () AO-events in the molecular stateYdefined by its bond systemis given by the square of the corresponding amplitude proportional to the CBO matrix element coupling the two functions [see Eq.(68)]:

(,|Y)o(,)=,j,i/o|(,)|2or

(,) =-1/2,j,

?(,) =-1?,j,i=,i/=(|Y)op. (90)

The associatedAO probabilities P('|) of Eq.(69), of the output events' = {} given the input events= {}, P('|) = {(|) =(,)/poP?}, the squared moduli of the corresponding amplitudes {P?o|A?|2}, then read:

(|) =P?= (,i)-1,j,i,?(|) = 1. (91)

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

(|) =A?= (,i)-1/2,j. (92)

and hence:

γ=á||?= 2á||?o2γ, (γ)2=γor

γ2= 4(γ)2= 4γ= 2γ. (94)

For such-states the representative conditional probability of the molecular AO-channel P('|) = {P(|)o(?)} determined by amplitudes Ac.s.('|) = {A(|)o(?) } thus reads:

P(|)o|A(|)|2= (2,i)-1,j,ior

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

????P(?|)??.(96)

It loses the memory about AO phases in the scattering amplitudes A(?|) = {(|) =A?}, i.e., the phases of elements {,j} in CBO matrix. These “coherencies” are preserved only in the associated-channel for theelectron communications in a molecule,

|?????A(?|)?|??, (97)

which is thus capable of reflecting the quantum-mechanical interference between such elementary communications, e.g., in the(cascade) propagationsAO 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-AO intermediates''in the sequential-orbital (-bridge) scatterings?''?' define the (-stage)-cascades for the- and- propagations in a molecule:

?[P(''|)?''?P(?|'')]??o?P[(?|);'']??,

|??[A(''|)?|''??A(?|'')]?|??

o|??A[(?|);'']?|??. (98)

Theconditional probabilities between AO-events and their amplitudes are then given by products of the elementary two-orbital communications in eachsubchannel:

[(|);]o?; k=??,

[(|);]o?; k=??. (99)

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

?(??; k) =??= 1. (100)

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

[(,);]o?(,)(,)=-1?,k,j=-1,j.(101)

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

[(,);]=|[(,);]|2=-1(,), (102)

and the associated AO probability

[;] =?[(,);] =-1. (103)

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

[(|);]=[(,);]/[;]o|[(|);]|2

=(,)/=(|) =?. (104)

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

?[,;] =?(,)(,)1(,) and

?[(|);] =???1?, (105)

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

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

??[????…????]??(107)

then read [compare Eq.(100)]:

??…??[??; k,l, …, m,n]

=??…?[??; k,l, …,m] =??…[??; k,l…]

= …=?[??; k]=??=1. (108)

For the specified pair of “terminal” AO, say?and??,one can similarly examine the indirect scatteringsthe molecular bond system in thecascade consisting of the remainin(“bridge”) functions= {1(i,j)}, with the two terminal AO being then excluded from the set of admissible intermediate scatterers. The associated-communications give rise to the indirect (through-) components of the entropic bond multiplicities45–50, which complement the familiar direct (through-) chemical “bond-orders” and provide a novel IT perspective on chemical interactions between more distant AIM, alternative to the fluctuational Charge-Shift mechanism60introduced within the classical Valence-Bond (VB) theory61.

7 Markov chains

The cascade probabilities derived from the stochastic matrix P(’|) = {(|) =P?}oP of the-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-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 scenario3,36–38the set of AO events defines the state space of the Markov chain and theth trial can be considered as the change of state at the given transition timet. Since the transition probabilities are determined by the molecular electronic state, the stage transition probabilities are independent of the trial number. Therefore, this Markov chain is homogeneous: {P?() =P?}. The output probabilities atth trial can be thus determined from the initial input probabilities(0) in the chain, e.g., the molecular [(0)=]or promolecular [(0)=0] signals, by the resultant transformation of theth power of P:

() =(0)P. (109)

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

=-1/2A+-1/2B,+= 1, (110)

where the complementary conditional probabilities=(A|) and=(B|) and the two AO functions= (A,B) originate from atoms A and B, respectively. The corresponding CBO matrix for this closed-shell configurationY(2) =|,|,

generates the idempotent stochastic probability matrix63:

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

() =(0)P=(0)P = (,) =(1). (113)

Finding Pin 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,

T-1PT = π = {,} or P = TπT-1,

T-1T = TT-1= I, (114)

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

PT = Tπ and T-1P = πT-1. (115)

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

P= TπT-1, π= {,}. (116)

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

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

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

= ?[1 + (1-2)],= ?[1-(1-2)]. (119)

As intuitively expected, in the limit?￥it gives the resultant stochastic matrix generating the maximum-noise in the underlying communication system,

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

8 Electron flows and electronegativity equalization

:=.

The fragment quantities are defined by the corresponding partial derivatives of the grand-ensemble average21of the system electronic energy with respect to subsystem electron populationNor the associated net chargeq,

The global properties similarly involve differentiations with respect to the resultant state-parameters, of the whole system,=?Nor=?q,

For example, in atomic resolutionstands for the system overall number of electrons,=?q=?(Z-N) denotes (in atomic units) its net electric charge, and all derivatives are calculated for the fixed external potential() due to the nuclei exhibiting charges {Z}. The effective electron populations can be generated in the familiar schemes of the electron population analyses oran appropriate division of the molecular electron density(),

() =?(),N=ò()d,ò()d=, (124)

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

In the electronegativity-equalized reactive system, composed of the mutually-open(bonded) fragments in R= (A*|B*), the local sites are distinguished by their response properties represented by the corresponding-partials of the system energy or the relevant Legendre-transformed “thermodynamic” potentials13–18. Thesoftnesses represent the population responses per unit shift in the system global chemical potential:

s=?N/?=?q/?.(125)

They sum up to the systemchemical softness,

=?s=?/?=?/?=-1, (126)

the inverse of its global chemical hardness25

=?/?=?/?=-1. (127)

Alternatively, the relative(electron-) or(electron-) 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:

f=s/=?N/?=?q/?,

?f=?/?=?/?= 1.(128)

When reactants are regarded as being mutually-(nonbonded) in R+= (A+|B+), i.e., preserving their initial (integer) numbers of electrons {=0}, they are considered to be separated by a hypothetical “wall” preventing the-reactant flow of electrons, symbolized by avertical line separating the two reactants. At this(P) stage of their interaction the chemical-potential/electronegativity equalizations take place only within each subsystem. This constrained, Pequilbrium of the polarized (promoted) reactants thus gives rise to generally different levels of the chemical potential in each subsystem:

A+=?E(R+)/?A= {*(A+)}

1B+=?E(R+)/?B= {*(B+)}. (129)

At the given (finite)-reactant separation the relaxed density+is displaced relative to the corresponding separated-reactant density0by its equilibrium P-responseD+=+-0oD() to perturbation created by the presence of the other subsystem. These mutually polarized reactants exhibit the associated shifts in their equilibrium chemical potentials due to the reaction partner, relative to SRL values:D+=+-0oD().

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

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

This complementary complex is energetically preferred66compared 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:

This preference for the complementary behaviour in Rshould 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

D() =+()-0,

in the chemical potential() of the fragment Rof the perturbed subsystem, relative to its equalized SRL value0, created by the presence of the coordinating (perturbing) fragment Rof the other reactant14,15,59. At finite separations between the two substrates these displacements trigger the polarizational flows {} of Fig.3, which restore the internal equilibria in reactants. As indicated in the figure, one predicts for Rthat the basic subsystem B destabilizes A,DA(B) > 0, while the acidic reactant A stabilizes B,DB(A) < 0. In this reaction complex the harder (acidic) site+of the polarized reactant R+= (+|+) lowers the chemical potential of the softer (basic) site+of the other polarized reactant R+= (+|+), and+rises the chemical potential level on+:

These shifts of the initially equalized chemical potentials on the two sites in0= (a*|a*) trigger the internal+?+polarizational flow, which further enhances the (external) acceptor capacity of+and the donor ability of+, thus creating more favorable conditions for the subsequent-reactant flowCT, the net effect of the partial flows of Fig.2I:

CT=DA= ?DB=CT(1)?CT(2).(131)

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

Ro?E(R)/?R=*o?E(R)/?|R

={?*(R)o?E(R)/?N?|R}=*()oE[]/()|R.

In addition to the polarizational changes of the nonbonded reactants in R+= (A+|B+), {D+=+?0}, the equlibrium densities of the bonded reactants {*} in R= (A*|B*) exhibit additional CT-induced polarization component {D*=*?+},

*=++D*=0+D++D*

and integrate to the CT-displaced (fractional) effective populations {*=ò*d}.

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

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

,i=?/?N> 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

compared to the respective (equalized) levels in A0= (A*|A*) and B0= (B*|B*) (see Fig.4).

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

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:

They generate shifts in the chemical potentials of reactant sites in

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

CT=CT(1)-CT(2)=A*-A0oDA

=B0-B*o-DB> 0. (134)

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

DR+=A+-B+oCT< 0, (135)

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

ηR+= {,=?/?;,?(A, B)}. (136)

The latter determines thehardness (CT) and softness (CT) for this process:

CT=?CT/?CT=A,A+B,B-A,B-B,A=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-reactant separation when the charge coupling between the two species is negligible, can they be approximated by the separate-reactant quantities:

CT@A0-B0oCT0andCT@A0+B0oCT0.

We further recall that the optimum amount of CT,

CT=-CTCT, (138)

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

CT=CTCT/2 =-(CT)2CT/2 < 0. (139)

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 {a+ = (aa+|ba+)}. These primary shifts subsequently trigger the polarizational flows {Ia},

which eventually generate the global electronegativity equalization in Ras a whole: R= (A*|B*) = (A*|A*|B*|B*)oRCT*.

9 Phase considerations

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

() =() =?(),

()=() =(),=ò()d.(140)

For example, the molecularly placed “free” reactants {0} in the nonbonded promolecular reference R0= (A0|B0) and their “bonded” analogs {0} in the “entangled” promolecule R0= (A0|B0) are describedby“thermodynamic”phases{,(0)=[0]} and {,(0)=[R0]}, respectively, where the promolecular densityR0and probability distributionR0are defined by relation

R0=A0+B0=A0A0+B0B0=RR0,

0=ò0()d.(141)

The mutually-(entangled) fragments in R0are thus (phase/current)-promoted compared to their mutually-(disentangled) analogs in R0. The overall promolecular density0() thus induces the equilibrium current in0,

(0) = (/)0?,(0)=-[/(2)]0?lnR0

=-[/(2)](0/0)?R0o-[/(2)]0?R0, (142)

which differs from the equilibrium current in0:

(0) = (/)0?,eq(0)=-[/(2)]0?ln0

=-[/(2)]?0. (143)

A similar-distinction is observed between the equilibrium state of an disentangledreactant*in R*= (A*|B*) and its entangled analog*in R*= (A*|B*)oR:

(*) = (/)*?,eq(*)=-[/(2)]*?ln*

=-[/(2)]?*, (144)

(*) = (/)*?,eq(*)=-[/(2)]*?lnR*

=-[/(2)](*/)?Ro-[/(2)]*?R. (145)

Consider now the disentangled polarized reactants {+} in R+= (A+|B+) and their entangled analogs {+} in the “bonded” reactive system R+= (A+|B+) exhibiting the overall electron density

R+=RR+=A++B+=A0A++B0B+,

+=ò+()d. (146)

Their equilibrium phases determine the associated subsystem currents:

(+) = (/)+?,(+)=-[/(2)]+?ln+

=-[/(2)]?+. (147)

(+) = (/)+?,(+)=-[/(2)]+?lnR+

=-[/(2)]?R+. (148)

The phase aspect of electronic communications in the nonbonded, polarized reactive system R+= (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 R+are characterized by thermodynamic phases generated by densities of the polarized. In the final (bonded) state of R*= (A*|B*)oR, their equilibrium phase is ultimately determined by theprobability distribution, of R as a whole.

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

A('|)={(|) =,i-1/2,j}. (149)

The reactant bases {|?}, of AO originating from the constituent atoms of subsystem, in the overall basis|?= (|A?,|B?) arrange the electron communications and the underlying CBO matrix into the substrate-resolved blocks,

A('|) = {A(|) = {(?|?)},

γ = {γ,= {,b}}. (150)

The moduli {M,b} and phases {,b} of the CBO matrix elements {,b=M,bexp(i,b)} determine the corresponding parts of complex amplitudes {(|)} of AO communications.The-content disappears in the conditional probabilities {(|) =|(|)|2=|,b|2/M,a}, which determine the(probability).

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

|(+)?= {|a(+)?= exp[i(+)]|?,

?,= A, B}. (151)

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

áa(+)|b(+)?

=ò*()exp{i[(+;)-(+;)]}()d

1ò*()()d=,b. (152)

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

|(*)?= {|a(*)?= exp[i(R)]|?}, (153)

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

áa(*)|b(*)?=ò*() exp{i[(R;)-(R;)]}()d

=ò*()()d=,b, (154)

and hence

???|b(*)?áb(*)|=???|?á|. (155)

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

a(+,) =()exp[i+()]=a(R+;) and

b(+;)=()exp[i()]=b(R+;), (156)

then reads:

,b;eq(R+) =áa(R+)|?á|b(R+)?

oM,b;eq(R+)exp[i,b;eq(R+)]

=òdòd'a(+,)*[()?(')]b(+;')

oòdòd' [*()(')],(,')

oòdòd'R,b(,'),(,''), (157)

,(,') = exp{i[(')-+()]}()?('). (158)

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

a(*;) =()exp[iR()] =a(R,) and

b(*;) =()exp[iR()] =b(R;), (159)

one similarly finds:

,b;eq(R)oM,b;eq(R)exp[i,b;eq(R)] =áa(R)|?á|b(R)?

=òò'a(R,)*[()?(')]b(R;')

oòò'R,b(,')R(,'), (160)

R(,') = exp{i[R(')-R()]}()?('). (161)

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

???,b;eq(R),a;eq(R) =,a;eq(R)

=áa(R)|?á|a(R)?=|áa(R)|?|2

=?|áa(R)|?|2=?[|a(R)]

o[|a(R)]o[a(R)|], (162)

where[|a(R)] =[a(R)|] stands for the conditional probability of observing in the bond-systeman electron occupying a(R)([|a(R)]), or - of findinga(R) in the occupied MO subspace([a(R)|]).

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

P('(R)|(R)) = {P(|) = {(b|a),,?(A,B)}},

???P(b|a) = 1,

(b|a) =,b;eq(R),a;eq(R)/,a;eq(R)o|(b|a)|2

=M,b;eq(R)2/[a(R)|]} or

(b|a)=,a;eq(R)-1/2,b;eq(R). (163)

Therefore, the resultant modulusM,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,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-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-requirements for effecting, in a “coherent” reactive event, the charge currents in such preferred directions.

(B+)-(A+) = (/)[B+(B+)-A+(A+)]

=-[/(2)]?(B+-A+). (164)

Similarphase 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 displacementsCT(1)andCT(2)of Eq.(131), which also represent the spontaneous flows in response to the initial electronegativity differences of the relevant acidic () and basic () sites in both reactants.

The secondary, relaxational flows {} or {} of Figs.2–4, from the acidic () to basic () part of, are similarly generated and enhanced by the equilibrium subsystem currents corresponding to another(+) ?(+)-phaseprocess:

(+) ?(+) = ?[/(2)]?[(+) ?(+)] (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-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-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 thelocal events, extends into the nonclassical (quantum) domain of relations between the entropy/information densities of theevents characterized by their phases or currents. The gradient descriptor of the state resultant entropy (- 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-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 solutionmarking the system- 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-(nonbonded, disentangled) or -(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- and-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 theandchemical bonds between reactants. The-reactant communications are responsible for the valence-stateof the mutually nonbonded subsystems, while the-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 entropicandcommunication contributions derived from the system CBO matrix.

The preference for themutual arrangements of reactants in reactive complex, in which the chemically() fragments of one reactant face the chemically() 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 Chatelier-Braunrequirement: 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() fragments of both reactants face the like-fragments of the other reactant. The-synchronization effecting the desired displacements in the system electronic structure has also been commented upon.

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Corresponding author. Email: nalewajs@chemia.uj.edu.pl.