Role of electronic kinetic energy and resultant gradient information in chemical reactivity
Abstract
The role of resultant gradientinformation concept, reflecting the kinetic energy of electrons, in shaping the molecular electronic structure and reactivity preferences of open reactants is examined. This quantuminformation descriptor combines contributions due to both the modulus (probability) and phase (current) components of electronic wavefunctions. The importance of resultant entropy/information concepts for distinguishing the bonded (entangled) and nonbonded (disentangled) states of molecular fragments is emphasized and variational principle for the minimum of ensembleaverage electronic energy is interpreted as a physically equivalent rule for the minimum of resultant gradientinformation, and the information descriptors of chargetransfer (CT) phenomena are introduced. The in situ reactivity criteria, represented by the populational CT derivatives of the ensembleaverage values of electronic energy or resultant information, are mutually related, giving rise to identical predictions of electron flows in the acid(A) — base(B), reactive systems. The virial theorem decomposition of electronic energy is used to reveal changes in the resultant information content due to the chemical bond formation, and to rationalize the Hammond postulate of reactivity theory. The complementarity principle of structural chemistry is confronted with the regional hard (soft) acid and bases (HSAB) rule by examining the polarizational and relaxational flows in such acceptor–donor reactive systems, responses to the external potential and CT displacements, respectively. The frontierelectron basis of the HSAB principle is reexamined and the intra and interreactant communications in A—B systems are explored.
Keywords
Chemical reactivity Complementarity principle Hammond postulate HSAB rule Information theory Virial theoremIntroduction
Thermodynamic principles for the minimum electronic energy in molecules can be interpreted as the variational rule for the minimum of the ensembleaverage resultant gradientinformation [1, 2], related to average kinetic energy of electrons in such (mixed) quantum states. In the grandensemble representation of the externally open molecular systems, they both determine the same set of the optimum probabilities of the system (pure) stationary states. This equivalence resembles identical predictions resulting from the minimumenergy and maximumentropy principles in ordinary thermodynamics [3]. The energy and resultant gradientinformation rules thus represent physically equivalent sources of reactivity criteria. Such an information transcription of the familiar energy principle allows one to reinterpret criteria for the charge transfer (CT) in reactive systems, the populational derivatives of electronic energy, as the associated derivatives of the overall measure of the quantuminformation in molecular states, which combines the “classical” (modulus, probability) and “nonclassical” (phase, current) aspects of molecular wavefunctions. The proportionality between the resultant gradientinformation and the system kinetic energy then allows one to use the molecular virial theorem [4] in general reactivity considerations [1, 2].
The classical information theory (IT) of Fisher and Shannon [5, 6, 7, 8, 9, 10, 11, 12] has been successfully applied to interpret in chemical terms the molecular probability distributions, e.g., [13, 14, 15, 16]. Information principles have been explored [17, 18, 19, 20, 21, 22] and density pieces attributed to atoms in molecules (AIM) have been approached [13, 17, 21, 22, 23, 24, 25], providing the information basis for the intuitive (stockholder) division of Hirshfeld [26]. Patterns of chemical bonds have been extracted from molecular electronic communications [13, 14, 15, 16, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37] and entropy/information distributions in molecules have been examined [13, 14, 15, 16, 38, 39]. The nonadditive Fisher information [13, 14, 15, 16, 40, 41] has been linked to electron localization function (ELF) [42, 43, 44] of modern density functional theory (DFT) [45, 46, 47, 48, 49, 50]. This analysis has also formulated the contragradience (CG) probe for localizing chemical bonds [13, 14, 15, 16, 51], and the orbital communication theory (OCT) of the chemical bond [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37] has identified the bridgebonds originating from the cascade propagations of information between AIM, which involve intermediate atomic orbitals [15, 16, 51, 52, 53, 54, 55, 56, 57].
In entropic theories of molecular electronic structure, one ultimately requires such quantum extensions of the complementary classical measures of Fisher [5, 6] and Shannon [7, 8], of the information/entropy content in probability distributions, which are appropriate for complex probability amplitudes (wavefunctions) of quantum mechanics (QM). The IT distinction between the bonded (entangled) and nonbonded (disentangled) states of molecular subsystems also calls for their generalized (resultant) information descriptors [16, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70], which combine the classical (probability) and nonclassical (current) contributions. Probability distributions generate the classical entropy/information descriptors of electronic states. These contributions reflect only the wavefunction modulus, while the wavefunction phase, or its gradient determining the current density, give rise to the corresponding nonclassical supplements in the resultant measure the overall information content in molecular electronic states [16, 58, 59, 60]. The variational principles of such generalized entropy concepts have been used to determine the phaseequilibria in molecules and their constituent fragments [16, 61, 62, 63, 64, 65].
Paraphrasing Prigogine [71], one could regard the molecular probability distribution as determining an instantaneous structure of “being”, while the system’s current pattern generates the associated structure of “becoming”. Both of these levels of the system electronic organization contribute to the state overall entropy/information content. In quantum information theory (QIT), the classical information term, conceptually rooted in DFT, probes the entropic content of the incoherent (disentangled) local “events”, while it is the nonclassical supplement that provides the information contribution due to coherence (entanglement) of such local events. For example, resultant measures combining the probability and phase/current contributions allow one to distinguish the information content of states generating the same electron density but differing in their phase/current distributions [47, 72, 73].
The resultant Fishertype gradientinformation in specified electronic state is proportional to the state average kinetic energy [1, 2, 13, 16, 18, 40]. This allows one to interpret the variational principle for electronic energy as equivalent quantuminformation rule. The latter forms a basis for the novel, informationtreatment of reactivity phenomena [1, 2]. Various DFTbased approaches to classical issues in reactivity theory [74, 75, 76, 77, 78, 79, 80] use the energycentered arguments in justifying the observed reaction paths and relative yields of their products. Qualitative considerations on preferences in chemical reactions usually emphasize changes in energies of both reactants and of the whole reactive system, which are induced by displacements (perturbations) in parameters describing the relevant (real or hypothetical) electronic states. In such treatments, usually covering also the linear responses to these primary shifts, one explores reactivity implications of the electronic equilibrium and stability criteria [13, 15, 74, 75, 79]. For example, in charge sensitivity analysis (CSA) [74, 75], the energy derivatives with respect to the system external potential (v) and its overall number of electrons (N) and the associated charge responses of both the whole reactive systems and their constituent subsystems have been explored as potential reactivity descriptors. In R = acid(A) ← base(B) ≡ AB complexes, consisting of the coordinated electronacceptor and electrondonor reactants, respectively, such responses can be subsequently combined into the corresponding in situ indices characterizing the B → A CT [74, 75]. These difference characteristics of polarized subsystems can be expressed in terms of elementary (principal) charge sensitivities of reactants [74, 75, 78, 79]. The nonclassical IT descriptors of polarized subsystems can be similarly combined into the corresponding in situ properties describing the whole reactive system.
In this work, the role of resultant gradientinformation/kineticenergy in shaping the chemical reactivity preferences will be explored and variations of the kinetic energy of electrons in the bondforming/bondbreaking processes will be examined. The continuities of the principal physical degreesoffreedom of electronic states, the modulus/probability and phase/current distributions, respectively, will be summarized and the virial theorem will be used to interpret, in information terms, the bondformation process. The theorem implications for the Hammond [81] postulate of reactivity theory will also be explored. The frontierelectron approximation to molecular interactions will be adopted to extract the information perspective on Pearson’s [82] hard (soft) acids and bases (HSAB) principle of structural chemistry (see also [83]), and physical equivalence of the energy and information reactivity descriptors in the grandensemble representation of molecular thermodynamic equilibria will be stressed. The populational derivatives of resultant gradientinformation will be examined and advocated as alternative indices of chemical reactivity, adequate in predicting both the direction and magnitude of electron flows in reactive systems. The phasedescription of hypothetical stages of reactants in chemical reactions will be explored, the activation (“promotion”) of molecular substrates will be examined, and the in situ populational derivatives of resultantinformation will be applied to determine the optimum amount of CT in donor–acceptor reactive systems.
Classical and nonclassical sources of the structureinformation in molecular states
The physical descriptors p(r, t) and j(r, t) of a complex quantum state constitute independent sources of an overall information content of the molecular electronic structure: the probability distribution alone generates its classical contribution while the current (velocity) density determines its nonclassical complement in the resultant measure [16, 58, 59, 60].
To summarize, the classical continuity relation of QM expresses a sourceless character of the electron probability distribution, while its nonclassical companion introduces a nonvanishing phasesource combining both the classical (modulus) and nonclassical (phase) inputs.
Resultant information and kinetic energy
This average overall information thus assumes thermodynamiclike form, as the trace of the product of CBO matrix, the AO representation of the (occupationweighted) MO projector, which establishes the configuration density operator, and the corresponding AO matrix of the Hermitian operator for the resultant gradient information, related to the system electronic kinetic energy. In this MO “ensemble”averaging, the AO information matrix I constitutes the quantitymatrix, while the CBO (density) matrix γ provides the “geometrical” weighting factors in this MO “ensemble”, reflecting the system electronic state. It has been argued elsewhere [16, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37] that elements of the CBO matrix generate amplitudes of electronic communications between molecular AO “events”. This observation thus adds a new angle to interpreting this averageinformation expression: it is seen to represent the communicationweighted (dimensionless) kinetic energy of the system electrons.
The relevant separation of the modulus and phasecomponents of general Nelectron states calls for wavefunctions yielding the specified electron density [47]. It can effected using the Harriman–Zumbach–Maschke (HZM) [83, 84] construction of DFT. It uses N (complex) equidensity orbitals, each generating the molecular probability distribution p(r) and exhibiting the densitydependent spatial phases, which safeguard the MO orthogonality.
Bonded (entangled) and nonbonded (disentangled) states of reactants
To summarize, the fragment identity can be retained only for the mutually closed (nonbonded) status of the acidic and basic reactants, e.g., in the polarized reactive system R_{n}^{+} or in the equilibrium composite system Open image in new window . The subsystem electron densities {ρ_{α} = N_{α} p_{α}} can be either “frozen”, e.g., in the promolecular reference R^{0} = (A^{0}B^{0}) ≡ R_{n}^{0} consisting of the isolatedreactant distributions shifted to their actual positions in the molecular reactive system R, or “polarized” in R^{+} or R, i.e., relaxed in presence of the reaction partner. The final equilibrium in R as a whole, combining the bonded subsystems {α^{*}} after CT, accounts for the extra CTinduced polarization of reactants compared to R^{+}. As we have argued above, descriptors of this state, of the mutually bonded (formally open) reactants, can be inferred only indirectly, by examining the chemical potential equalization in the composite system Open image in new window . Similar external reservoirs are involved, when one examines the independent population displacements on reactants, e.g., in defining the fragment chemical potentials and their hardness tensor.
In this chain of hypothetical reaction “events”, the polarized system R^{+} appears as the launching stage for the subsequent CT and the accompanying induced polarization, after the hypothetical barrier for the flow of electrons between the two subsystems has been effectively lifted. This density polarization is also accompanied by the subsystem currentpromotion, reflected by the modified electron flow patterns in both substrates, compared to promolecule R^{0}, in accordance with their current equilibriumphase distributions [16, 66, 67, 68, 69]. This nonclassical (current) activation of both subsystems complements the classical (probability) polarization of reactants in presence of their reaction partners. The phase aspect is thus vital for accounting for the mutual coherence (entanglement) of reactants in the reactive system as a whole.
Grandensemble principle for thermodynamic equilibrium
Information descriptors of chemical reactivity
The positive signs of these “diagonal” populational derivatives assure the external stability of 〈M(v)〉_{ens.} with respect to hypothetical electron flows between the molecular system and its reservoir [74, 75]. Indeed, they imply an increase (a decrease) of the global energetic and information “intensities” (μ and ξ), which are coupled to Open image in new window , in response to the perturbation created by the primary electron inflow (outflow). This is in accordance with the familiar Le Châtelier and Le ChâtelierBraun principles of thermodynamics [3] that spontaneous responses in system intensities to the initial population displacements diminish effects of such primary perturbations.
Use of virialtheorem partitioning
The qualitative Hammond postulate emphasizes a relative resemblance of the reaction TS complex R^{‡} to its substrates (products) in the exoergic (endoergic) reactions, while for the vanishing reaction energy the position of TS complex is predicted to be located symmetrically between substrates and products. The activation barrier thus appears “early” in exoergic reactions, e.g., H_{2} + F → H + HF, with the reaction substrates being only slightly modified in TS, R^{‡} ≈ [AB], both electronically and geometrically. Accordingly, in endoergic bondbreakingbondforming process, e.g., H + HF → H_{2} + F, the barrier is “late” along the reactionprogress coordinate P and the activated complex resembles more the reaction products: R^{‡} ≈ [CD]. This qualitative statement has been subsequently given several more quantitative formulations and theoretical explanations using both the energetic and entropic arguments [20, 97, 98, 99, 100, 101, 102, 103].
This observation demonstrates that RC derivative of the resultant gradientinformation at TS complex, dI/dP_{‡}, proportional to dT/dP_{‡}, can serve as an alternative detector of the reaction energetic character: its positive/negative values respectively identify the endo/exoergic reactions exhibiting the late/early activation energy barriers, with the neutral case (ΔE_{r} = 0 or dT/dP_{‡} = 0) exhibiting an equidistant position of TS between the reaction substrates and products on a symmetrical potential energy surface, e.g., in the hydrogen exchange reaction H + H_{2} → H_{2} + H.
The reaction energy ΔE_{r} determines the corresponding change in the resultant gradientinformation, ΔI_{r} = I(P_{prod.})  I(P_{sub.}), proportional to ΔT_{r} = T(P_{prod.})  T(P_{sub.}) = ΔE_{r}. The virial theorem thus implies a net decrease of the resultant gradient information in endoergic processes, ΔI_{r}(endo) < 0, its increase in exoergic reactions, ΔI_{r}(exo) > 0, and a conservation of the overall gradientinformation in the energyneutral chemical rearrangements: ΔI_{r}(neutral) = 0. One also recalls that the classical part of this information displacement probes an average spatial inhomogeneity of the electronic density. Therefore, the endoergic processes, requiring a net supply of energy to R, give rise to relatively less compact electron distributions in the reaction products, compared with the substrates. Accordingly, the exoergic transitions, which net release the energy from R, generate a relatively more concentrated electron distributions in products, compared to substrates, and no such an average change is predicted for the energyneutral case.
Regional HSAB versus complementary coordinations
Some subtle preferences in chemical reactivity result from the induced (polarizational or relaxational) electronflows in reactive systems, reflecting responses to the primary or induced displacements in the electronic structure of the reaction complex, e.g., [16, 104]. Such flow patterns can be diagnosed, estimated, and compared by using either the energetical or information reactivity criteria defined above. One such stillproblematic issue is the best mutual arrangement of the acidic and basic parts of molecular reactants in the donor–acceptor systems [16, 104, 105].
Consider the reactive complex A—B consisting of the basic reactant B = (a_{B}…b_{B}) ≡ (a_{B}b_{B}) and the acidic substrate A = (a_{A}…b_{A}) ≡ (a_{A}b_{A}), where a_{X} and b_{X} denote the acidic and basic parts of subsystem X, respectively. The acidic (electron acceptor) part is relatively harder, i.e., less responsive to external perturbation, exhibiting lower values of the fragment FF descriptor, while the basic (electron donor) fragment is relatively softer, more polarizable, as reflected by its higher density/population responses. The acidic part a_{X} exerts an electronaccepting (stabilizing) influence on the neighboring part of the other reactant Y, while the basic fragment b_{X} produces an electrondonor (destabilizing) effect on a fragment of Y in its vicinity.
An additional rationale for this complementary preference over the regional HSAB alignment of reactants comes from examining the charge flows created by the dominating shifts in the site chemical potential due to the presence of the (“frozen”) coordinated site of the nearby part of the reaction partner. At finite separations between the two subsystems, these displacements trigger the polarizational flows {P_{X}} shown in Figs. 3 and 4, which restore the internal equilibria in both subsystems, initially displaced by the presence of the other reactant.
In R_{c}, the harder (acidic) site a_{Y} initially lowers the chemical potential of the softer (basic) site b_{X}, while b_{Y} rises the chemical potential level of a_{X}. These shifts trigger the internal (polariaztional) flows {a_{X} → b_{X}}, which enhance the acceptor capacity of a_{X} and donor ability of b_{X}, thus creating more favorable conditions for the subsequent interreactant CT of Fig. 3. A similar analysis of R_{HSAB} (Fig. 4) predicts the b_{X} → a_{X} polarizational flows, which lowers the acceptor capacity of a_{X} and donor ability of b_{X}, i.e., the electron accumulation on a_{X} and electron depletion on b_{X}, thus creating less favorable conditions for the subsequent interreactant CT.
The complementary preference also follows from the electronic stability considerations, in spirit of the familiar Le ChâtelierBraun principle of the ordinary thermodynamics [3]. In contrast to analysis of Figs. 3 and 4, where the CT responses follow the internal polarizations of reactants, the equilibrium responses to displacements {Δv_{X} = v_{Y}} in the external potential on subsystems, one now assumes the primary (interreactant) CT displacements {ΔCT_{1}, ΔCT_{2}} of Figs. 3 and 4, in the internally closed but externally open reactants, and then examines the induced (secondary) relaxational responses {I_{X}} to these perturbations.
Frontierelectron and communication outlooks on HSAB principle
The physical equivalence of reactivity concepts formulated in the energy and resultant gradientinformation representations has also direct implications [1, 2] for the communication theory of the chemical bond (CTCB) [13, 14, 15, 16, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37]. In OCT, the theory orbital realization, one treats a molecule as an information channel propagating signals of the AO origins of electrons in the bond system determined by the system occupied MO. It has been argued elsewhere [14, 15, 16] that elements of the CBO matrix γ = {γ_{k,l}} [Eq. (25)], the weighting factors in expression of Eq. (23) for the average resultant gradientinformation, determine amplitudes of conditional probabilities defining the direct communications between AO. Entropic descriptors of this channel then generate the information bond orders and their covalent/ionic components, which ultimately facilitate an IT understanding of molecular electronic structure in chemical terms.
The communication noise (orbital indeterminicity) in this network, measured by the channel conditional entropy, is due to the electron delocalization in the bond system of a molecule. It represents the overall bond “covalency”, while the channel information capacity (orbital determinicity), reflected by the mutual information of this communication network, measures the resultant bond “ionicity”. Therefore, the more scattering (indeterminate) the molecular information system, the higher its covalent character. Accordingly, a less noisy (more deterministic) channel represents a more ionic molecular system [13, 14, 15, 16, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37].
In chemistry the bond covalency, a common possession of electrons by interacting atoms, is indeed synonymous with the electron delocalization generating the communication noise. A classical example is provided by bonds connecting identical atoms, e.g., hydrogens in H_{2} or carbons in ethylene, when the interacting AO in the familiar MO diagrams of chemistry exhibit the same levels of AO energies. The bond ionicity accompanies large differences in atomic electronegativities generating a substantial CT. Such bonds correspond to a wide separation of the interacting AO energies in the familiar MO diagrams of chemistry. The ionic bond component introduces more determinicity (less noise) into molecular AO communications, thus representing a bond mechanism competitive with bond covalency [29, 106, 107, 108, 109, 110].
One of the celebrated (qualitative) rules of chemistry deals with stability preferences in molecular coordinations. The HSAB principle of Pearson [82] predicts that chemically hard (H) acids (A) prefer to coordinate hard bases (B) in the [HH]complexes, and soft (S) acids prefer to coordinate soft bases in [SS]complexes, whereas the “mixed” [HS] or [SH]complexes, of hard acids with soft bases or of soft acids with hard bases, respectively, are relatively unstable [83, 90]. As we have emphasized in the preceding section, this global preference is no longer valid regionally, between fragments of reactants, where the complementarity principle [104, 105] dictates the preferred arrangement between the acidic and basic sites of both reactants.

How does the [HH] or [SS] preference shape the intra and interreactant communications in the whole reactive complex?

How is the H or S character of a substrate reflected by its internal communications?

How does the HSAB preference influence the interreactant propagations of information?
In the communication perspective on reactive systems [16, 111], the H and S reactants correspond to the internally ionic (deterministic) and covalent (noisy) reactant channels, respectively. The former involves localized orbital communications between chemically bonded atoms, while the latter corresponds to strongly delocalized information scatterings between AO basis states. A natural question then arises: what is the overall character of communications responsible for the mutual interaction between reactants? Do the Ssubstrates in [SS]complex predominantly interact “covalently”, and H substrates of the [HH]complex “ionically”?
It follows from Eq. (97) that the maximum covalent component of the interreactant chemical bond is expected in interactions between soft, strongly overlapping reactants [83], since then the numerator assumes the highest value while the denominator reaches its minimum. For the same reason one predicts the smallest covalent stabilization in interactions between the hard, weakly overlapping substrates, with the mixed hardness combinations giving rise to intermediate bond covalencies.
To summarize, the [HH]complex exhibits the maximum ionicstabilization, the [SS]complex the maximum covalentstabilization, while the mixed combinations of reactant hardnesses in [HS] and [SH]coordinations exhibit a mixture of moderate covalent and ionic interactions between the acidic and basic subsystems [83]. Therefore, communications representing the interreactant bonds between the chemically soft (covalent) reactants are also expected to be predominantly “soft” (delocalized, indeterministic) in character, while those between the chemically hard (ionic) subsystems are predicted to be dominated by the “hard” (localized, deterministic) propagations in the communication system for R as a whole [1, 2].
This observation adds a communication angle to the classical HSAB principle of chemistry.
Conclusions
In this work, we have attempted the QIT description of the bimolecular donor–acceptor reactive system, including all hypothetical processes that accompany the bondbreaking/bondforming processes of chemical reactions. The present (resultant) information analysis of reactivity phenomena complements earlier (classical) DFTIT approaches, e.g., [115, 116, 117, 118, 119, 120, 121]. It should be emphasized, however, that the present resultantinformation analysis has followed the standard thermodynamic approach to open microscopic systems, which does not imply any new “thermodynamic” transcription of DFT, see, e.g., [120, 121]. The continuities of the classical (modulus/probability) and nonclassical (phase/current) state parameters have been examined and contributions, that these molecular degreesoffreedom generate in the resultant gradientinformation descriptor of a quantum state, have been identified. The need for nonclassical (phase/current) complements of the classical (probability) measures of the information content in molecular electronic states has been reemphasized. It has been argued that the electron density alone reflects only the structure of “being”, missing the structure of “becoming” contained in the current distribution. Both of these manifestations of the molecular “organization” ultimately contribute to the overall information content in generally complex electronic wavefunctions, reflected by the resultant QIT concepts. Their importance in describing the mutual bonding and nonbonding status of reactants has been stressed and the in situ populational derivatives in the energy and information representations have been examined.
The DFTbased theory of chemical reactivity distinguishes several intuitive, hypothetical stages involving either the mutually bonded (entangled) or nonbonded (disentangled) states of reactants for the same electron distribution in constituent subsystems. These two categories are discerned only by the phase aspect of the quantum entanglement between molecular fragments. The equilibrium phases and currents of reactants can be related to the relevant electron densities using the entropic principles of QIT. This generalized approach deepens our understanding of the molecular promotions of constituent fragments and provides a more precise framework for monitoring the reaction progress.
The grandensemble description of thermodynamic equilibria in externally open molecular systems has been used to demonstrate the physical equivalence of the energy and resultant gradientinformation principles. The populational derivatives of the resultant gradientinformation, related to the system average kinetic energy, have been suggested as reliable reactivity criteria. They were shown to predict both the direction and magnitude of the electron flows in reactive systems. The grandensemble description of thermodynamic equilibria in the externally open molecular systems has been outlined and the physical equivalence of variational principles for the electronic energy and resultant gradientinformation has been emphasized. The virial theorem has been used to explain the qualitative Hammond postulate of the theory of chemical reactivity, and the information production in chemical reactions has been addressed. The ionic and covalent interactions between frontier MO of the acidic and basic reactants have been examined to justify the HSAB principle of chemistry and to provide the communication perspective on interaction between reactants. It has been argued that the internally soft and hard reactants prefer to externally communicate in the like manner, consistent with their internal communications. This preference should be also reflected by the predicted character of the interreactant bonds/communications in stable coordinations: covalent in [SS] and ionic in [HH] complexes.
Notes
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