Resultant gradient information, kinetic energy and molecular virial theorem

1. Introduction

The Quantum Information Theory (QIT) [1-4]has been shown to provide a solid, unifying basis for understanding - in chemical terms - the electronic structure of molecules, and explaining general trends in their chemical behavior, [5-8]. Thermodynamic energy principle has been interpreted as physically equivalent rule for the resultant content of the overall gradient-information in electronic wavefunction, the dimensionless descriptor related to the state average kinetic energy. In the grand-ensemble both these variational principles determinetheequilibrium stateofan open molecular system. This equivalence parallels the same predictions resulting from the minimum-energyandmaximum-entropyprinciplesofthe ordinarythermodynamics [9]. It explains the proportionality relations between energetic and informational criteria of chemical reactivity, measured by the corresponding populational derivatives of the ensemble-average functionals.

The QIT transcription of the variational principle for the electronic (thermodynamic) energy thus allows one to interpret reactivity criteria as the associated populational derivatives of the state resultant gradient-information (dimensionless kinetic energy) content. The latter combines the classical (probability) and nonclassical (current) contributions,due to the modulus and (local) phase components of the molecular wavefunction, respectively. The proportionality between the resultant gradient-information and the system kinetic energy also suggests the use of molecular virial theorem [10]in general reactivity considerations [5-8].

To paraphrase Prigogine [11], the electron density alone carries the information reflecting a “static” structure of “being”, missing a “dynamic” structure of “becoming” contained in the state phase or current distributions. Both these manifestations of the electronic “organization” in molecular systems ultimately contribute to overall measures of the structural entropy or information content in generally complex wavefunctions, reflected by the resultant QIT concepts [1-4]. Their classical contributions, conceptually rooted in Density Functional Theory (DFT) [12-17], probe the entropic content of incoherent (disentangled) local “events”, while the nonclassicaltermsprovidetheinformationsupplementdue to the coherence (entanglement) of such local events. The resultant measures allow one to distinguish the information content of states generating the same electron density but differing in their phase/current distributions, e.g., the bonded (entangled) and nonbonded (disentangled) states of molecular fragments [18-27].

The classical Information Theory (IT) of Fisher and Shannon [28-35]has been successfully applied to generate the chemical interpretation of molecular probability distributions, [36-39]. Information principles have been explored [5-8,40-45] and density pieces attributed toAtoms-in-Molecules(AIM) have been approached [36-39,43-47], providing the information basis for the intuitive (stockholder) division of Hirshfeld [48]. Patterns of chemical bonds in molecules have been extracted from electronic orbital communications [1,36-59], and entropy/information densities have been explored [1,36-38,60,61]. The nonadditive Fisher information [1,36 -38,62,63] has been linked to the Electron Localization Function (ELF) [64-66] of modern DFT. This analysis has also formulated the Contragradience (CG) probe [1,36-38,67] for spatial localization of chemical bonds, and the Orbital Communication Theory (OCT) of the chemical bond has identified the bridge-bonds originating from the cascade propagations of information between AIM, which involve intermediate orbitals [1,38,68-73]. The DFT-based approaches to classical issues in reactivity theory [74-80] use the energy-centered arguments in justifying the observed reactivity preferences. It is the main purpose of this work to show that general reactivity rules can be alternatively treated using the resultant-information/kinetic-energy concepts of QIT.

We begin with a short summary of the overall gradient-information concept. The resultant QIT descriptor will be introduced and its classical and nonclassical components identified. Populational derivatives of its thermodynamic, ensemble-average value generate alternative indices of chemical reactivity, adequate in predicting both the direction and magnitude of electron flows in donor-acceptor systems [5-8]. The molecular virial theorem will be used to generate the information perspective on the bond-formation and the Hammond [81] postulate of reactivity theory. The theorem will be generalized to cover the ensemble-average energy components and the role of electronic kinetic energy or the resultant gradient-information in chemical processes will be examined. Physical equivalence of the energy and information reactivity descriptors in the grand-ensemble representation of thermodynamic-equilibria will be stressed, the relation between energetic and information reactivity indices will be examined, and the “production” of the overall structural information in chemical reactions will be addressed.

2. Resultant gradient-information and kinetic energy of electrons

Consider a general (complex) quantum state |ψ⟩ of an electron described by the associated wavefunction in position representation,

$$\psi \left( r \right){\rm{ }} = \langle r|\psi \rangle = R{(r)_{}}exp[i\phi \left( r \right)] \label{eq1}$$(1)

with R(r) and φ(r) denoting its modulus and phase components, respectively. They determine the particle probability distribution,

$$p\left( r \right){\rm{ }} = \psi {\left( r \right)^*}\psi \left( r \right){\rm{ }} = R{\left( r \right)^2} \label{eq2}$$(2)

and the current density

$$\begin{aligned} j\left( r \right){\rm{ }} &= {\rm{ }}{\left[ {\hbar /\left( {2mi} \right)} \right]_{}}[\psi {\left( r \right)^*}\nabla \psi \left( r \right) - \psi {\left( r \right)^{}}\nabla \psi {\left( r \right)^*}]{\rm{ }}\\ &= {\rm{ }}\left( {\hbar /m} \right)p\left( r \right)\nabla \phi \left( r \right) \equiv p{\left( r \right)_{}}V\left( r \right) \end{aligned} \label{eq3}$$(3)

The effective velocity V(r) of this probability “fluid” measures the current-per-particle and reflects the state phase-gradient:

$$V\left( r \right){\rm{ }} = j\left( r \right)/p\left( r \right){\rm{ }} = {\rm{ }}{\left( {\hbar /m} \right)_{}}\nabla \phi \left( r \right) \label{eq4}$$(4)

The average Fisher’s [28] measure of the classical gradient-information for locality events containedinthe electronic probabilitydensityp(r) is reminiscent of von Weizsäcker’s [82] inhomogeneity correction to density-functional for the kinetic-energy:

$$\begin{aligned} I\left[ p \right]{\rm{ }} &= \int {} p{(r)_{}}{[\nabla lnp(r)]^2}_{}dr \equiv \int {} p{(r)_{}}{I_p}{(r)_{}}dr\\ &= {\rm{ }}4\int {} {[\nabla R(r)]^2}_{}dr \equiv I\left[ R \right] \end{aligned} \label{eq5}$$(5)

Here p(r) I_p(r) denotes functional’s overall density with I_p(r) = [∇lnp(r)]² standing for the associated density-per-electron. The amplitude form I[R] reveals that this classical descriptor reflects a magnitude of the state modulus-gradient. It characterizes an effective “narrowness” of the probability distribution, i.e., a degree of determinicity in particle’s position.

This classical functional of the gradient-information in probability distribution generalizes naturally into the of the quantum state |ψ⟩,which combinesthe modulus (probability) and phase (current) contributions [1,18-22,62]. It is defined by quantum expectation value of the Hermitian operator ${\rm{\hat I}}(r)$ of the overall gradient information [62],related to electronic kinetic-energy operator T̂(r),

$$\begin{aligned} &\hat {I}(r) = - 4{\Delta ^\;} = {\rm{ }}{(2i\nabla )^2} = \;(8m/{\hbar ^2})\hat T(r) \equiv \sigma \hat T(r) \\&\hat T(r) = - {\left[ {{\hbar ^2}/\left( {2m} \right)} \right]_{}}{\nabla ^2} \end{aligned} \label{eq6}$$(6)

The integration by parts then gives the following expression for the average (resultant) gradient-information contained in quantum state |ψ⟩:

$$\begin{aligned} I[\psi ]{\rm{ }} &= \langle \psi |\hat I|\psi \rangle = - {4_{}}\int {} \psi {\left( r \right)^*}_{}\Delta \psi {\left( r \right)_{}}d{r^\;} \\&= {\rm{ }}{4_{}}\int {} |\nabla \psi \left( r \right){|^2}_{}dr \equiv \int {} p{\left( r \right)_{}}{I_\psi }{\left( r \right)_{}}d{\bf{r}}\\ &= I\left[ p \right]{\rm{ }} + {\rm{ }}4\int {} p{\left( r \right)_{}}{[\nabla \phi \left( r \right)]^2}_{}dr \\&\equiv \int {} p{\left( r \right)_{}}{[{I_p}\left( r \right){\rm{ }} + {I_\phi }\left( r \right)]_{}}d{\bf{r}}\\ &\equiv I\left[ p \right]{\rm{ }} + I[\phi ] \equiv I[p{,_{}}\phi ]\\ &= I\left[ p \right]{\rm{ }} + {\rm{ }}{\left( {2m/h} \right)^2}\int {} p{\left( r \right)^{ - 1}}_{}j{\left( r \right)^2}_{}dr \\&\equiv I\left[ p \right]{\rm{ }} + I\left[ j \right] \equiv I\left[ {p{,_{}}j} \right] \end{aligned} \label{eq7}$$(7)

This quantum gradient-information concept I[ψ] = I[p,φ] = I[p,j] is seen to combine the classical (probability) contribution I[p] of Fisher and the corresponding nonclassical (phase/current) supplement I[φ] = I[j] . It also reflects the particle average (dimensionless) kinetic energy T[ψ]:

$$I[\psi ]{\rm{ }} = \sigma \langle \psi |{\rm{\hat T}}|\psi \rangle T[\psi ]{\rm{ }} = \sigma T[\psi ] \label{eq8}$$(8)

The above one-electron development can be straightforwardly generalized into general N-electron state |Ψ(N)⟩, exhibiting electron density ρ(r)= Np(r), where p(r) stands for its probability (shape) factor. The corresponding information operator then combines terms due to each electron,

$$\begin{aligned} \hat I(N) = \sum\limits_{i = 1}^N {\hat I({r_i})} = \sigma \sum\limits_{i = 1}^N{\hat T({r_i})} \equiv \sigma \hat T(N) \end{aligned} \label{eq9}$$(9)

$$\begin{aligned} I(N){\rm{ }} &= \left\langle {\psi \left( N \right)|\hat I\left( N \right)} \right.\left. {|\psi \left( N \right)} \right\rangle \\ &= \sigma \left\langle {\psi \left( N \right)} \right.|\hat T\left( N \right)|\left. {\psi \left( N \right)} \right\rangle = \sigma T(N) \end{aligned} \label{eq10}$$(10)

In the given electron (orbital) configuration specified by a single Slater determinant Ψ(N) = |ψ₁ψ₂ …ψ_N|, e.g., in the familiar Hartree-Fock of Kohn-Sham theories, these N-electron descriptors combine the additive contributions due to all (singly occupied: n_s = 1), molecular orbitals (MO) ψ = (ψ₁, ψ₂, …, ψ_N) = {ψ_s}:

$$\begin{aligned} T(N) &= \sum\nolimits_{_{\text{s}}} {{n_s}\langle {\psi _s}|\hat T|{\psi _s}\rangle } \equiv \sum\nolimits_{_{\text{s}}} {{n_s}{T_S}} \\&= {\sigma ^{ - 1}}\sum\nolimits_{_{\text{s}}} {{n_s}\langle {\psi _s}|\hat I|{\psi _s}\rangle } \equiv {\sigma ^{ - 1}}\sum\nolimits_{_{\text{s}}} {{n_s}{I_s}} \end{aligned} \label{eq11}$$(11)

In the analytical LCAO MO representation, when these occupied MO are expressed as linear combinations of the (orthogonalized) atomic orbital (AO) basis χ = (χ₁, χ₂, …, χ_k, …),

$$|\psi \rangle = |\chi {\rangle _{}}C,C = \langle \chi |\psi \rangle = \{ {C_k}_{,s} = \langle {\chi _\kappa }|{\psi _s}\rangle \}$$(12)

$$\begin{aligned} I(N){\rm{ }} &= {\sum\nolimits_s}^{}{n_s}\langle {\psi _s}|\hat I|{\psi _s}\rangle \\ &= {\sum\nolimits _\kappa }{\sum\nolimits_l}\{ {\sum\nolimits _s}{C_k}_{,s}{n_s}{C_s}{_{,l}^ * }\} \langle {\chi _l}|\hat I|{\chi _\kappa }\rangle \\&\equiv {\sum\nolimits _k}{\sum\nolimits _l}{\gamma _k}_{,l}{{\rm I}_l}_{,k} = {\rm{ }}tr(\gamma {\rm I}) \end{aligned} \label{eq13}$$(13)

Here, the AO representation of the resultant gradient-information operator,

$${\rm I} = \{ {I_k}_{,l} = \langle {\chi _k}|\hat I|{\chi _l}\rangle = \sigma \langle {\chi _k}|\hat T|{\chi _l}\rangle \equiv \sigma {T_k}_{,l}\} \label{eq14}$$(14)

$$\gamma = Cn{C^{†} } = \langle \chi |\psi \rangle n\langle \psi |\chi \rangle \equiv \langle \chi |{\hat P_\psi }|\chi \rangle$$(15)

$$\begin{aligned} {\hat P_\psi } &= N{\rm{[}}{\sum\nolimits _s}|{\psi _s}\rangle {\left( {{n_s}/N} \right)_{}}\langle {\psi _s}|{\rm{]}} \\&\equiv N{\rm{[}}{\sum\nolimits _s}|{\psi _s}\rangle {P_s}\langle {\psi _s}|{\rm{]}} \equiv N{\rm{\hat d}} \end{aligned} \label{eq16}$$(16)

This expression for the average overall gradient-information assumes thermodynamic-like form, as trace of the product of CBO matrix, the AO representation of the (occupation-weighted) MO projector, which establishes the configuration density operator ${\rm{\hat d}}$, 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 quantity-matrix, while the CBO (density) matrix γ provides the “geometrical” weighting-factors reflecting the system electronic state. It has been argued elsewhere that elements of the CBO matrix also generate amplitudes of electronic communications between molecular AO “events” [1,36-38,49-59].. This observation adds a new angle to interpreting the average-information expression as the communication-weighted (dimentionless) kinetic energy of the system electrons [83].

A separation of the modulus- and phase-components of general N-electron states calls for wavefunctions yielding the specified electron density [14]. It can effected using the Harriman-Zumbach-Maschke (HZM) [84 85]construction of DFT, which uses N (complex) equidensity orbitals, each separately generating the molecular probability distribution p(r) and exhibiting the density-dependent spatial phase which safeguards the MO orthogonality.

3. Grand-ensemble description of molecular equilibria

In an open molecule M(v), identified by the external potential v(Q) of the Born-Oppenheimer (BO) approximation for the molecular geometry Q specified by coordinates of the fixed nuclei of the system constituent atoms, the populational derivatives of the average electronic energy or the resultant gradient-information call for the grand-ensemble representation of thermodynamic equilibria [5-8,15,86,87]. A molecule is then coupled to a hypothetical (macroscopic) electron reservoir R(μ) exhibiting the chemical potential μ, and the heat bath B(T) identified by its absolute temperature T in the composite (macroscopic) system

$$M(\mu ,T;v){\rm{ }} = {\rm{ }}[R(\mu )M\left( v \right)B\left( T \right)] \label{eq17}$$(17)

$$\begin{aligned} &{\langle {\rm N}\rangle _{ens.}} \equiv {N_\;} = {\text{ }}tr(\hat D\hat N)\; = {\sum\nolimits _i}{P_i}{N_i} \hfill \\ &\;\;{\sum\nolimits _i}{P_i} = {\text{ }}1,\;\;\;{P_i } \geqslant 0 \hfill \end{aligned} \label{eq18}$$

$$\hat N\; = {\sum\nolimits _i}{\sum\nolimits _j}|{\psi _j}\left( {{N_i}} \right)\rangle {N_i}\langle {\psi _j}\left( {{N_i}} \right)|$$(19)

$$\hat D(\mu ,T;v)= {\sum\nolimits _i}{\sum\nolimits _j}|{\psi _j}\left( {{N_i}} \right){\rangle _{}}{P_j}^i(\mu ,T;v)\langle {\psi _j}\left( {{N_i}} \right)|$$(20)

$$\begin{aligned} &\hat H({N_i}{,_{}}v)|{\psi _j}\left[ {{N_i}{,_{}}v} \right]\rangle = {E_j}{\left( {{N_i}} \right)_{}}|{\psi _j}\left[ {{N_i}{,_{}}v} \right]\rangle \\&or\;\;\;\;\; \hat H|{\psi _j}^i\rangle = {E_j}^i|{\psi _j}^i\rangle \end{aligned} \label{eq21}$$(21)

Such electronic N-derivatives are involved in definitions of the system Charge Transfer (CT) criteria of chemical reactivity [15,74-79], e.g., the chemical potential (negative electronegativity) [15,86-90] or the hardness (softness) [91] and Fukui Function (FF) [92]descriptors of electrons. They are thus definable only for the mixed-state of the molecular (microscopic) system M(v), e.g., that corresponding to the thermodynamic equilibrium imposed by intensities (μ, T) characterizing the external (macroscopic) subsystems R(μ) and B(T) in M(μ, T; v), μ = μ_R and T=T_B, i.e., for the equilibrium grand-canonical density operator of : ${{\rm{\hat D}}_{eq.}}$$\equiv{\hat D(\textit{$\mu{}$}, $_{ }$\textit{T}; \textit{v})}$ .

The grand-canonical intensities determine the ensemble thermodynamic potential, called the grand-potential, given by the corresponding Legendre-transform [9] of the ensemble-average energy

$$\begin{aligned} {\langle E\rangle _{ens.}} &\equiv \;E[\hat D] \equiv E(N,S; v) = {\rm{ }}tr(\hat D\hat H)\; \\&= {\sum\nolimits _i}{\sum\nolimits _j}{P_j}{^i_{}}{E_j}^i \end{aligned} \label{eq22}$$(22)

$$\Omega = \varepsilon - (\partial \varepsilon /\partial N)N - (\partial \varepsilon /\partial S)S = \varepsilon [\hat{D}]-\mu N[\hat{D}] - TS[\hat{D}]$$(23)

It minimizes at the optimum state-probabilities {P_jⁱ(μ,T;v)}  ≡  P_eq.(μ,T;Q):

$$\begin{aligned} \mathop {\min }\limits_{{\rm{\hat D}}} \Omega {\rm{[\hat D]}} &= \Omega {\rm{[\hat D(}}\mu {\rm{,}}T{\rm{;}}v{\rm{)]}} \\&= E{\rm{[}}{{{\rm{\hat D}}}_{eq.}}] - \mu N{\rm{[}}{{{\rm{\hat D}}}_{eq.}}] - TS{{\rm{[}}{{{\rm{\hat D}}}_{eq.}}]_{}}\; \\&\equiv \Omega {\rm{(}}\mu {\rm{,}}T{\rm{;}}v{\rm{)}} {\rm{ }} \Rightarrow {\rm{ }}{P_{eq.}}{\rm{(}}\mu {\rm{,}}T;Q{\rm{) }} \end{aligned} \label{eq24}$$(24)

As indicated in the preceding equation, the ensemble parameters μ and T ultimately determine the associated optimum probabilities of the (pure) stationary states {|ψ_j[N_i,v]⟩}, eigenstates of Hamiltonians {${{\rm{\hat H}}_i}$},

which define the equilibrium density operator of for the specified geometrical structure Q. Here Ξ stands for the grand-ensemble partition function, k_B denotes the Boltzmann constant, and β = (k_BT)^− 1.

The electronically-relaxed, equilibrium ensemble probabilities thus satisfy the following relations between the probability “gradients” for the adopted molecular geometry Q:

$$\begin{aligned} &\mathop {\left. {\frac{{\partial \Omega (P,Q)}}{{\partial P}}} \right|}\nolimits_{{P_{eq.}}} = 0\;\;\;\;\;\;{\rm{ or }}\\&\mathop {\left. {\frac{{\partial E(P,Q)}}{{\partial P}}} \right|}\nolimits_{{P_{eq.}}} {\rm{ = }}\mathop {\left. {\mu \frac{{\partial N(P)}}{{\partial P}}} \right|}\nolimits_{{P_{eq.}}} + \mathop {\left. {T \frac{{\partial S(P)}}{{\partial P}}} \right|}\nolimits_{{P_{eq.}}} \end{aligned} \label{eq25}$$(25)

The grand-potential corresponds to replacing the “extensive” state-parameters, of the average values of the particle number N = $N{\rm{[\hat D}}]$ and thermodynamic entropy [93]

$$\begin{aligned} &S[\hat D] = {\rm{ }}tr(\hat D\hat S)\; = - {k_B}{\sum\nolimits _i}{\sum\nolimits _j}{P_j}{^i_{}}ln{P_j}^i \\&\hat S = - {k_B}{\sum\nolimits _i}{\sum\nolimits _j}|{\psi _j}^i{\rangle _{}}ln{P_j}{^i_{}}\langle {\psi _j}^i| \end{aligned} \label{eq26}$$(26)

$$\begin{aligned} {\langle N\rangle _{ens.}} &= N[{\hat D_{eq.}}]\; = {\sum\nolimits _i}{[{\sum\nolimits _j}{P_j}^i(\mu {,_{}}T;v)]_{}}{N_i} \\&= {\sum\nolimits _{i\;}}{P_i}{(\mu {,_{}}T;v)_{}}{N_i}\; = {\langle N(\mu {,_{}} T; v)\rangle _{ens.\;}} \\&= {N_{}}{[\mu{,_{}} T; v]_{}} = N \end{aligned} \label{eq27}$$(27)

$$\begin{aligned} {\langle S\rangle _{ens.}} &= \;S[{\hat D_{eq.}}] \\&= - {k_B}{\sum\nolimits _i}{\sum\nolimits _j}{P_j}^i(\mu ,{\rm{ }}T;v)ln{P_j}^i(\mu {,_{}}T;v)\;\; \\&= {\langle S(\mu {,_{}}T;v)\rangle _{ens.}} = S(\mu {,_{}}T){\rm{ }} = S \end{aligned} \label{eq28}$$(28)

In equilibrium state the prescribed average extensive descriptors N and S also uniquely identify the externally-imposed state intensities, μ = μ(N,S)andT= T(N,S), and hence also the equilibrium energy function

$$\begin{aligned} E[{\hat D_{eq.}}]\; &= {\langle E(\mu ,T;v)\rangle _{ens.}} \\&= {\sum\nolimits _i}{\sum\nolimits _j}{P_j}^i{(\mu ,T;v)_{}}\langle {\psi _j}^i|{\hat H_i}|{\psi _j}^i\rangle \\&= {\sum\nolimits _i}{\sum\nolimits _j}{P_j}^i(\mu, T; v){E_j}^i \\&= E(\mu {,_{}} T) \\&\equiv E(N, S) \end{aligned} \label{eq29}$$(29)

It allows one to formally identify the intensive parameters as its partial derivatives with respect to the constrained values of the extensive state-variables:

$$\begin{aligned} &{\left. {\mu = {{\left( {\frac{{\partial E}}{{\partial N}}} \right)}_S}} \right|_{{{{\rm{\hat D}}}_{eq.}}}}{\rm{ and }}\\&T{\rm{ = }}{\left. {{{\left( {\frac{{\partial E}}{{\partial S}}} \right)}_N}} \right|_{{{{\rm{\hat D}}}_{eq.}}}} \end{aligned} \label{eq30}$$(30)

In the T → 0 limit [15,86,87]only two ground-states (j = 0), {|ψ₀ⁱ⟩, |ψ₀^i + 1⟩}, corresponding to the neighboring integers “bracketing” the given (fractional) ⟨N⟩_ens. = N, N_i ≤ ⟨N⟩_ens. ≤ N_i + 1, appear in the equilibrium statistical mixture. Their ensemble probabilities for the specified

$$\begin{aligned} {\langle N\rangle _{ens.}} = i{P_i} + {\rm{ }}\left( {i + 1} \right)(1 - {P_i}){\rm{ }} = N \end{aligned} \label{eq31}$$(31)

$$\begin{aligned} &{P_i} = {\rm{1}} + i - N \equiv 1 - \omega \;\;and\;\;\\&{P_i}_{ + 1} = N - i{\rm{ }} \equiv {\rm{ }}\omega \end{aligned} \label{eq32}$$(32)

The continuous energy function E(N,S) then consists of the straight-line segments between the neighboring integer values of N. This implies constant values of the chemical potential in all such admissible ranges of the average electron number and μ-discontinuity at N = N_i(integer) [15,86,87].

The ensemble-average value of the resultant gradient-information,

$$\begin{aligned} {\langle I\rangle _{ens.}} &\equiv I[{\hat D_{eq.}}] = {\rm{ }}tr\;[{\hat D_{eq.}}\hat I] \\&= {\sum _i}{\sum _j}{P_j}^i(\mu {,_{}}T;v)\langle {\psi _j}^i|\hat I({N_i})|{\psi _j}^i\rangle \\&\equiv {\sum _i}{\sum _j}{P_j}^i(\mu {,_{}}T;v){I_j}^i \\&{\rm{ }}{I_j}^i = {\rm{ }}\left( {8m/{\hbar ^2}} \right)\langle {\psi _j}^i|\hat T({N_i})|{\psi _j}^i\rangle \equiv \sigma {T_j}^i \end{aligned} \label{eq33}$$(33)

$$\begin{aligned} {\langle T\rangle _{ens.}} &\equiv I[{\hat D_{eq.}}] = {\rm{ }}tr({\hat D_{eq.}}\hat T) \\&= {\sum _i}{\sum _j}{P_j}^i(\mu {,_{}}T;v)\langle {\psi _j}^i|\hat T({N_i})|{\psi _j}^i\rangle \\&{\; \equiv}{\sum _i}{\sum _j}{P_j}^i(\mu {,_{}}T;v){T_j}^i = {\sigma ^{ - 1}}{\langle I\rangle _{ens.}} \end{aligned} \label{eq34}$$(34)

The proportionality constant results from relation between the associated electronic operators:

$$\begin{aligned} \;&\hat T({N_i}) = \frac{{ - {\hbar ^2}}}{{2m}}\;\sum\limits_{k = 1}^{{N_i}} {\nabla _k^2} \;\;\;\;\;\;and \\&\hat I({N_i})\;\; = - 4\sum\limits_{k = 1}^{{N_i}} {\nabla _k^2} \end{aligned} \label{eq35}$$(35)

Therefore, the thermodynamic rule of , for the minimum of the constrained average value of electronic energy can be alternatively interpreted as the corresponding extremum principle for the ensemble-average (resultant) gradient-information [5-8,36-38,40]:

$$\mathop {\sigma\min}\limits_{{\rm{\hat D}}}\Omega{\rm{[\hat D] = }}\sigma\Omega{\rm{[}}{{\rm{\hat D}}_{eq.}}{\rm{] = }}I{\rm{[}}{{\rm{\hat D}}_{eq.}}] + \sigma \{ W{\rm{[}}{{\rm{\hat D}}_{eq.}}] - \mu N{\rm{[}}{{\rm{\hat D}}_{eq.}}] - TS{\rm{[}}{{\rm{\hat D}}_{eq.}}]\}$$(36)

$$\begin{aligned} W{\rm{[}}{{\rm{\hat D}}_{eq.}}]{\rm{ = }}V{\rm{[}}{{\rm{\hat D}}_{eq.}}]{\rm{ + }}U{\rm{[}}{{\rm{\hat D}}_{eq.}}] \end{aligned} \label{eq37}$$(37)

$$\begin{aligned} {\lambda _W} = - \sigma = {\left. {{{\left( {\frac{{\partial I}}{{\partial W}}} \right)}_{N,S}}} \right|_{{{{\rm{\hat D}}}_{eq.}}}} \equiv K \end{aligned} \label{eq38}$$(38)

$$\begin{aligned} \xi \equiv \sigma \mu {\rm{ = }}{\left. {{{\left( {\frac{{\partial I}}{{\partial N}}} \right)}_{W,S}}} \right|_{{{{\rm{\hat D}}}_{eq.}}}} \end{aligned} \label{eq39}$$(39)

$$\begin{aligned} \tau \equiv \sigma T = {\left. {{{\left( {\frac{{\partial I}}{{\partial S}}} \right)}_{W,N}}} \right|_{{{{\rm{\hat D}}}_{eq.}}}} \end{aligned} \label{eq40}$$(40)

The conjugate thermodynamic principles, for constrained extrema of the ensemble energy,

$$\begin{aligned} \delta {\left( {E{\rm{[\hat D]}} - \mu N{\rm{[\hat D]}} - TS{\rm{[\hat D]}}} \right)_{{\rm{\hat D}}{}_{eq.}}} = 0 \end{aligned} \label{eq41}$$(41)

$$\begin{aligned} \delta \left( {I{\rm{[\hat D]}} - \kappa \,W{\rm{[\hat D]}} - \xi \,N{\rm{[\hat D]}} - \tau \,S{\rm{[\hat D]}}} \right){\,_{{\rm{\hat D}}{}_{eq.}}} = 0 \end{aligned} \label{eq42}$$(42)

Several N-derivatives of the ensemble-average electronic energy or of the resultant gradient-information define useful and adequate CT criteria of chemical reactivity [15,74-79]. The physical equivalence of the energy and information principles indicates that such concepts are mutually related, being both capable of describing the electron-transfer phenomena in donor-acceptor systems [5-8]. The above ensemble interpretation also applies to diagonal and mixed second derivatives of the electronic energy or its kinetic-energy (information) component, which involve the population differentiation.

In energy-representation the chemical hardness [91], the “diagonal” populational second- derivative of the ensemble energy, reflects the N-derivative ofchemical potential,

$$\begin{aligned} \eta = {\left. {{{\left( {\frac{{{\partial ^2}E}}{{\partial {N^2}}}} \right)}_S}} \right|_{{{{\rm{\hat D}}}_{eq.}}}} = {\left. {{{\left( {\frac{{\partial \mu }}{{\partial N}}} \right)}_S}} \right|_{{{{\rm{\hat D}}}_{eq.}}}} > 0 \end{aligned} \label{eq43}$$(43)

$$\begin{aligned} \omega = {\left. {{{\left( {\frac{{{\partial ^2}I}}{{\partial {N^2}}}} \right)}_{W,S}}} \right|_{{{{\rm{\hat D}}}_{eq.}}}} = {\left. {{{\left( {\frac{{\partial \xi }}{{\partial N}}} \right)}_{W,S}}} \right|_{{{{\rm{\hat D}}}_{eq.}}}} = \sigma \eta > 0 \end{aligned} \label{eq44}$$(44)

The positive signs of these diagonal population derivatives assure the external stability of an open M(v), with respect to hypothetical electron flows between molecular system and its reservoir. They indeed imply an increase (a decrease) of the global energetic and information “intensities” coupled to N, μ and ξ, in response to perturbations created by the initial electron inflow (outflow). This accords with the Le Châtelier and Le Châtelier-Braun principles of thermodynamics [9], that spontaneous responses in system intensities to the initial population displacements diminish effects of the primary perturbations.

By the cross-differentiation identity the “mixed” second-derivative of the ensemble energy, measuring the system global FF [92], can be alternatively interpreted as either the response in global chemical potential per unit displacement in the external potential, or the density response per unit populational displacement.

$$\begin{aligned} f(r) &= {\left. {{{\left( {\frac{{{\partial ^2}E}}{{\partial N \partial v(r)}}} \right)}_S}} \right|_{{{{\rm{\hat D}}}_{eq.}}}} = {\left. {{{\left( {\frac{{\partial \mu }}{{\partial v(r)}}} \right)}_S}} \right|_{{{{\rm{\hat D}}}_{eq.}}}} \\&= {\left. {{{\left( {\frac{{\partial \rho (r)}}{{\partial N }}} \right)}_S}} \right|_{{{{\rm{\hat D}}}_{eq.}}}} \end{aligned} \label{eq45}$$(45)

$$\begin{aligned} \varphi r &= {\left. {{{\left( {\frac{{{\partial ^2}I}}{{\partial N \partial v(r)}}} \right)}_{W,S}}} \right|_{{{{\rm{\hat D}}}_{eq.}}}} = {\left. {{{\left( {\frac{{\partial \xi }}{{\partial v(r)}}} \right)}_{W,S}}} \right|_{{{{\rm{\hat D}}}_{eq.}}}} \\&= \,\,\,\sigma \,{\left. {{{\left( {\frac{{\partial \rho (r)}}{{\partial N}}} \right)}_{W,S}}} \right|_{{{{\rm{\hat D}}}_{eq.}}}} = \sigma \,f(r) \end{aligned} \label{eq46}$$(46)

It has been argued elsewhere [5-8], that the in situ measures of these energy and information derivatives constitute fully equivalent descriptors of electron flows between the polarized subsystems. These CT phenomena in the polarized reactive system R⁺ = (A⁺|B⁺), containing the mutually-closed and molecularly polarized acceptor (acid, A) and donor (basis, B) reactants {α⁺}, are described by populational derivatives: the substrate chemical potentials μ_R⁺ =  {μ_a⁺} and elements of the hardness matrix η_R⁺ = {η$_{$\alpha{}$}$_, β}. These descriptors again call for the grand-ensemble representation of the polarized (externally-open) reactants, in contact with their separate (macroscopic) electron reservoirs {R_a}. They represent the electron population {N_a ≡ N_a} derivatives of the ensemble-average electronic energy in R⁺, E[{N_β}, v] ≡ E_v({N_β}), the microscopic subsystem in the macroscopic (composite) system,

$$\begin{aligned} {M_R}^ + &= {\rm{ }}({M_A}^ + |{M_B}^ + ){\rm{ }} \\&= ({R_A}{A^ + }|{B^ + }{R_B}) \\&\equiv ({R_A}M{\left( v \right)^ + }{R_B}) \end{aligned} \label{eq47}$$(47)

The in situ descriptors of CT are thus derived from the corresponding partials of the system ensemble-average energy with respect to ensemble-average electron populations on (externally-open) molecular-subsystems {α⁺} in the (mutually-closed) composite fragments {M_a⁺ =  (a⁺R_a)} of M_R⁺:

$$\begin{aligned} &{\mu _a} \equiv \partial {E_v}(\{ {N_\gamma }\} )/\partial {N_a}\\ &{\eta _\alpha }_{,\beta } = {\partial ^2}{E_v}(\{ {N_\gamma }\} )/\partial {N_\alpha }\partial {N_\beta } = \partial {\mu _\alpha }/\partial {{\rm N}_\beta }\\&\;\;\;\;\;\; = \partial {\mu _\beta }/\partial {{\rm N}_\alpha } = {\eta _\beta }_{,\alpha } \end{aligned} \label{eq48}$$(48)

$$\begin{aligned} {\mu _{CT}} = \partial {E_v}\left( {{N_{CT}}} \right)/\partial {N_{CT}} = {\mu _{\rm A}}^ + - {\mu _{\rm B}}^ + < 0 \end{aligned} \label{eq49}$$(49)

$$\begin{aligned} {\eta _{CT}} &= \partial {\mu _{CT}}/\partial {N_{CT}} \\&= ({\eta _{{\rm A},{\rm A}}} - {\eta _{{\rm A},{\rm B}}}) + ({\eta _{{\rm B},{\rm B}}} - {\eta _{{\rm B},{\rm A} + }}) \equiv {\eta _{\rm A}}^{\rm R} + {\eta _{\rm B}}^{\rm R} \\&\equiv {\Sigma _{CT}}^{ - 1} \end{aligned} \label{eq50}$$(50)

$${N_{CT}} = - {\mu _{CT}}{S_{CT}} = { - {\mu _{CT}}} / {\eta _{CT}}$$(51)

$${E_{CT}} = {\mu _{CT}}{N_{CT}}/2 = - {\mu _{CT}}^2{S_{CT}}/2 < 0 $$(52)
The corresponding CT-derivatives of the average gradient-information in AB systems similarly involve the in situ information potential,

$${\xi _{CT}} = \partial I\left( {{N_{CT}}} \right) / \partial {N_{CT}} = {\xi_A}^{+} - {\xi_B}^{+} = \sigma {\mu_{CT}}$$(53)

$${\omega _{CT}} = \partial {\xi _{CT}} / \partial {N_{CT}} \equiv \theta _{CT}^{ - 1} = \sigma {\eta _{CT}} = {\sigma _\;}{S_{CT}}^{ - 1}$$(54)

$$\begin{aligned} {N_{CT}}= - {\xi _{CT}}/ \omega _{CT} = - {\xi _{CT}}{\theta _{CT}} = - \mu _{CT} / \eta _{CT} = - {\mu _{CT}}{S_{CT}} \end{aligned} \label{eq55}$$(55)

4. Virial theorem implications

The virial theorem for the stationary electronic states |ψ_jⁱ⟩ = |ψ_j[N_i,v]⟩ in molecules reflects homogeneities of the kinetic and potential energy contributions in such pure quantum states,

$$\begin{aligned} {T_j}^i &= \langle {\psi _j}^i|\hat T({N_i})|{\psi _j}^i\rangle {\text{ and }}{W_j}^i = \langle {\psi _j}^i|\hat W({N_i},v)|{\psi _j}^i\rangle \hat W({N_i},v) \\&= \sum\limits_{k = 1}^{{N_i}} {[v(k) + \frac{1}{2}\sum\limits_{l \ne k}^{} {g(k,l)} } ] = \hat V({N_i},v) + \hat U({N_i},v) \end{aligned} \label{eq56}$$(56)

$$\begin{aligned} {E_j}^i = \langle {\psi _j}^i|{\hat H_i}|{\psi _j}^i\rangle = {T_j}^i + {W_j}^i \end{aligned} \label{eq57}$$(57)

In BO approximation both this average energy and its components are parametrically dependent upon molecular geometry specified by the fixed (Cartesian) coordinates Q of the nuclei, and so are the energy differences with respect to the adopted reference, e.g., the Separated Atoms Limit (SAL) or the separated reactants,

$$\begin{aligned} &{E_j}^i\left( Q \right){\text{ }} = {T_j}^i\left( Q \right){\text{ }} + {W_j}^i\left( Q \right)\;\;\;\;\;and\;\;\;\;\;\\&\Delta {E_j}^i\left( Q \right){\text{ }} = \Delta {T_j}^i\left( Q \right){\text{ }} + \Delta {W_j}^i\left( Q \right) \end{aligned} \label{eq58}$$(58)

The molecular virial theorem for the pure stationary state in BO approximation reads [10]:

$$\begin{aligned} 2{T_j}^i\left( Q \right) + {W_j}^i\left( Q \right) + Q \cdot [{{\partial {E_j}^i\left( Q \right)} / {\partial Q}}] \equiv 2{T_j}^i\left( Q \right) + {W_j}^i\left( Q \right) + Q \cdot {\nabla _Q}{E_j}^i\left( Q \right){\text{ }} = 0 \end{aligned} \label{eq59}$$(59)

It extracts the kinetic and potential components of the overall electronic energy for the current geometrical structure of the molecular system,

$$\begin{aligned} &{T_j}^i\left( Q \right){\text{ }} = - {E_j}^i\left( Q \right) - Q \cdot {\nabla _Q}_{}{E_j}^i\left( Q \right)\;\;\;\;\;and\;\;\;\;\;\\&{W_j}^i\left( Q \right){\text{ }} = {\text{ }}2{E_j}^i\left( Q \right) + Q \cdot {\nabla _Q}{E_j}^i\left( Q \right) \end{aligned} \label{eq60}$$(60)

These relations assume a particularly simple form for the energetical profiles, sections of the BO Potential Energy Surface (PES), e.g., the energy function in diatomics, for which the internuclear distance R uniquely specifies the molecular geometry, or along the reaction-coordinate (RC) R_c in chemical processes, with the trajectory arc-length P = |R_c| determining the reaction-progress variable. In diatomics the virial theorem expressed in terms of energy changes relative to SAL reads:

$$\begin{aligned} &2\Delta {T_j}^i\left( R \right)+\Delta {W_j}^i\left( R \right)+{R_{}}[{{d\Delta {E_j}^i\left( R \right)} / {dR}}]{\text{ }}= {\text{ }}0\;\;\;\;or \\&\Delta {T_j}^i\left( R \right) = - \Delta {E_j}^i\left( R \right) - {R_{}}[{{d\Delta {E_j}^i\left( R \right)} / {dR}}] = {{ - d[R\Delta {E_j}^i\left( R \right)]} / {dR}}\;\;\;\;and \\&\Delta {W_j}^i\left( R \right) = 2\Delta {E_j}^i\left( R \right) + {R_{}}[{{d\Delta {E_j}^i\left( R \right)} / {\partial R}}] = {R^{ - 1}}{{d[{R^2}\Delta {E_j}^i\left( R \right)]} / {dR}} \end{aligned} \label{eq61}$$(61)

$$\begin{aligned} &2T\left( Q \right) + W\left( Q \right) + Q \cdot {\nabla _Q}E\left( Q \right){\text{ }} = {\text{ }}0\;\;\;\;\;or\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \\&T\left( Q \right){\text{ }} = - E\left( Q \right) - Q \cdot {\nabla _Q}E\left( Q \right)\;\;\;\;\;and\;\;\;\;\; \\&W\left( Q \right){\text{ }} = {\text{ }}2E\left( Q \right) + Q \cdot {\nabla _Q}E\left( Q \right) \end{aligned} \label{eq62}$$(62)

They determine both the system thermodynamic energy,

$$\begin{aligned} E{\text{[}}{\hat D}_{eq.}]{\text{ = }}T{\text{[}}{\hat D}_{eq.}]{\text{ + }}W{\text{[}}{\hat D}_{eq.}] \end{aligned} \label{eq63}$$(63)

$$\begin{aligned} W{\text{[}}{{{\rm{\hat D}}}_{eq.}}] &= {\langle W\rangle _{ens.}} = tr\;{\text{(}}{{{\rm{\hat D}}}_{eq.}}{\rm{\hat W)}}{Q_{eq.}}\;\;\;\;\;\;\;\;\;{\;_{\;\;\;\;\;\;}} \\&= {\sum\nolimits _i}{\sum\nolimits _j}{P_j}^i(\mu ,T;v)\langle {\psi _j}^i|{\rm{\hat W(}}{N_i},v)|{\psi _j}^i\rangle \\& \equiv{\sum\nolimits _i}{\sum\nolimits _j}{P_j}^i(\mu, T; v){W_j}^i \end{aligned} \label{eq64}$$(64)

Let us briefly examine some implications of this general balance between the kinetic and potential components of the thermodynamic value of electronic energy. For the energy-minimum geometry Q_eq.(E)  = Q_eq. , determined by the vanishing gradient of thermodynamic energy,

$$\begin{aligned} {\nabla _Q}E{|_{eq.}} = 0 \end{aligned} \label{eq65}$$(65)

$$\begin{aligned} &T\left( {{Q_{eq.}}} \right){\text{ }} = - E\left( {{Q_{eq.}}} \right){\text{ }} = {\sigma ^{ - 1}}I\left( {{Q_{eq.}}} \right)\;\;\;\;\;and\;\;\;\;\;\\&W\left( {{Q_{eq.}}} \right) = 2E\left( {{Q_{eq.}}} \right) \end{aligned} \label{eq66}$$(66)

For such a geometrically-relaxed structure the minimum-energy principle of thermodynamics thus implies the thermodynamic maximum-information rule in QIT:

$$\begin{aligned} \{ mi{n_P}{[E\left( P \right)]_\mu }_{,T;Qeq.} &\Rightarrow ma{x_P}{[I\left( P \right)]_\mu }_{,T;Qeq.}\} \\&\Rightarrow P(\mu ,T;Q)\} \end{aligned} \label{eq67}$$(67)

Inother words, in thermodynamic (electronically-relaxed) equilibrium the geometrically-relaxed molecular systems assume the maximum resultant gradient-information related to its average kinetic energy. This information principle complements the familiar maximum-entropy rule of ordinary thermodynamics [9].

It should be observed that the energy-optimum structure Q_eq.(E ) of differs from that determined by the vanishing geometric gradient of the grand-potential,

$$\begin{aligned} {\left. {\frac{{\partial \Omega (P,Q)}}{{\partial Q}}} \right|_{{{\bar Q}_{eq.}}}}{\text{ = }}0{\text{ }} \Rightarrow {\text{ }}{Q_{eq.}}(\Omega ) = {\bar Q_{eq.}} \ne {Q_{eq.}} \end{aligned} \label{eq68}$$(68)

$$\begin{aligned} &{\left. {\frac{{\partial E(P,Q)}}{{\partial Q}}} \right|_{{{\bar Q}_{eq.}}}} = \mu {\left. {\frac{{\partial N(P)}}{{\partial Q}}} \right|_{{{\bar Q}_{eq.}}}}{\rm{ + }}T{\left. {\frac{{\partial S(P)}}{{\partial Q}}} \right|_{{{\bar Q}_{eq.}}}} \\&= \left( {\mu \frac{{\partial N(P)}}{{\partial P}}} \right. + \left. {T\frac{{\partial S(P)}}{{\partial P}}} \right){\left. {\left( {\frac{{\partial P}}{{\partial Q}}} \right)} \right|_{{{\bar Q}_{eq.}}}} \end{aligned} \label{eq69}$$(69)

Consider now the pure-state (micro-canonical) case summarized by the virial relations of Eqs. , which allow to extract the kinetic-energy/gradient-information differences from the corresponding energy profiles. Elsewhere [7,8] we have examined the BO energy profiles corresponding to the bond-formation process, A + B = A⎯B (see Figure 1), and the bimolecular chemical-reaction, A + B  →  R$^{‡}$  →  C + D (see Figure 2 ), where R$^{‡}$ denotes the Transition-State (TS) complex, in order to examine the accompanying changes in the resultant gradient information. Let us summarize some general conclusions of this analysis.

Figure 1 presents qualitative plots reflecting variations with internuclear distance of the ground-state bond-energy and its kinetic-energy contribution. The BO potential ΔE(R) and its kinetic-energy component ΔT(R) also reflecting variations in (resultant) gradient-information ΔI(R) = σΔT(R), relative to SAL, allow one to examine the energy/information variations with inter-nuclear distance R in the bond formation process. It follows from the figure that during a mutual approach by the constituent atoms the kinetic-energy/gradient-information is first diminished relative to the SAL reference, due to the longitudinal Cartesian component of the kinetic energy, associated with the “z” direction (along the bond axis) [94,95]. At the equilibrium distance R_e the resultant information rises above the SAL value, due to the dominating increase in transverse components of the kinetic energy, corresponding to “x” and “y” directions perpendicular to the bond axis. Therefore, at the equilibrium bond length R_e the chemical bond gives rise to a net increase in the resultant gradient-information relative to SAL, where electrons of each atom experience the external potential of only its own nucleus. This reflects a relatively more compact electron distribution in a molecule, where electrons move in the field of both nuclei.

Figure 1 Qualitative diagram of variations in the BO electronic energy ΔE(R) (solid line) with the internuclear distance R in a diatomic molecule, and of its kinetic energy component from the virial-theorem partitioning, ΔT(R) = -d/dR[RΔE(R)] (broken line), also reflecting the state resultant gradient-information ΔI(R) = σΔT(R)

Another interesting case of variations in molecular geometry is the (intrinsic) reaction coordinate R_c, or equivalently the progress-variable (arc-length) P along this trajectory, for which the virial relations assume the diatomic-like form (see Figure 2). Let us again examine the virial theorem decomposition of the corresponding energy profile along the R_c-section of PES, ΔE(R_c)  ≡  ΔE(P), in an elementary bimolecular reaction, to which the qualitative Hammond [81]postulate of reactivity theory applies. Again, the ground-state virial-theorem decomposition can be used to extract qualitative plots of the resultant gradient-information from the energy profiles corresponding either to endo- or exo-ergic reactions (upper panel), or to the energy-neutral chemical process on symmetric PES (lower panel).

The qualitative rule of Hammond is seen to be fully indexed by the sign of the geometric, P-derivative of the average resultant-information at the TS complex [5,8-10]. More specifically, this postulate emphasizes a relative resemblance of the reaction TS complex R^‡ to its substrates (products) in the exo-ergic (endo-ergic) reactions, while for the vanishing reaction energy the position of TS complex is predicted to be located symmetrically between substrates and products. In other words, the activation barrier appears “early” in the exo-ergic reactions, e.g., H₂ + F  →  H + HF, with the reaction substrates being only slightly modified in R^‡ ≈ [A −  −  − B] , both electronically and geometrically. Accordingly, in the endo-ergic bond-breaking-bond-forming process, e.g., H + HF  →  H₂ + F, the barrier is “late” along the reaction progress-variable P and the activated complex resembles more the reaction products: R^‡ ≈ [C −  −  − D] . This qualitative statement has been subsequently given several more quantitative formulations and theoretical explanations, based upon both the energetic and entropic arguments [96-103].

The energy profile along the reaction “progress” coordinate P,

$$\begin{aligned} \Delta E\left( P \right) = E\left( P \right) - E\left( {{P_{substrates}}} \right) \end{aligned} \label{eq70}$$(70)

$$\begin{aligned} \Delta T\left( P \right){\text{ }} = T\left( P \right) - T\left( {{P_{substrates}}} \right) \end{aligned} \label{eq71}$$(71)

$$\begin{aligned} \Delta I\left( P \right){\text{ }} = I\left( P \right) - I\left( {{P_{substrates}}} \right){\text{ }} = \sigma \Delta T\left( P \right) \end{aligned} \label{eq72}$$(72)

$$\begin{aligned} \Delta T\left( P \right){\text{ }} = - \Delta E\left( P \right) - P[d\Delta E\left( P \right)/dP] = -{{d[P\Delta E\left( P \right)]} / {dP}} \end{aligned} \label{eq73}$$(73)

The energy profiles ΔE(P) in the endo- or exo-directions, for the positive and negative reaction energy

$$\begin{aligned} \Delta {E_r} = E\left( {{P_{products}}} \right) - E\left( {{P_{substrates}}} \right) \end{aligned} \label{eq74}$$(74)

This demonstrates that the ground-state RC-derivative dI/dP|_‡ of the resultant gradient-information at TS complex, proportional to dT/dP|_‡, can serve as an alternative detector of the reaction energetic character: its positive/negative values identify the positive/negative reaction energy ΔE_r in endo/exo-ergic reactions, exhibiting the late/early activation barriers, respectively; the neutral case (ΔE_r = 0 or dT/dP|_‡ =0) exhibits 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 + H₂  → H₂ + H.

Figure 2 Variations of the BO total electronic energy (ΔE) and its kinetic energy component (ΔT) in the exo-ergic (ΔE_r < 0) and endo-ergic (ΔE_r > 0) reactions (upper Panel), and for the symmetrical PES (ΔE_r = 0) (lower Panel)

Since the forces acting on nuclei in the equilibrium, separated reactants or products vanish, the reaction energy ΔE_r of determines the corresponding change in the resultant gradient-information,

$$\begin{aligned} \Delta {I_r} = I\left( {{P_{products}}} \right) - I\left( {{P_{substrates}}} \right) = \sigma \Delta {T_r} \end{aligned} \label{eq76}$$(76)
proportional to the associated variation in the electronic kinetic energy:

$$\begin{aligned} \Delta {T_r} = T\left( {{P_{products}}} \right) - T\left( {{P_{substrates}}} \right){\text{ }} = - \Delta {E_r} \end{aligned} \label{eq77}$$(77)

The virial theorem thus implies a net decrease of the resultant gradient-information in endo-ergic processes, ΔI_r(endo)  ∝  -ΔE_r(endo) < 0, its increase in exo-ergic reactions, ΔI_r(exo)  ∝  -ΔE_r(exo) > 0, and a conservation of the overall gradient-information in the energy-neutral chemical rearrangements: ΔI_r(neutral)  ∝  -ΔE_r(neutral) = 0.

One recalls that the classical part of this information displacement probes an average change in the spatial inhomogeneity of electron density. Therefore, the endo-ergic processes, requiring a net supply of energy to the reactive system R, give rise to relatively less-compact electron distributions in reaction products,comparedto substrates. Accordingly, the exo-ergic transitions, with a net release of energy from R, generate on average more concentrated electron distributions in products, and no such a change is predicted in energy-neutral case.

5. Conclusion

In this overview we have explored qualitative reactivity applications of the resultant information measure in QIT. First, the concept of the overall gradient-information in specified quantum state, which combines the classical (probability) and nonclassical (phase/current) contributions, has been introduced as the expectation value of the corresponding (Hermitian) information-operator related to that of electronic kinetic energy. We have then explored the thermodynamic-average measure and its variational principle in the grand-ensemble. The electron-population derivatives, information reactivity descriptors of CT phenomena in donor-acceptor systems, have been examined, the physical equivalence of variational principles for ensemble-averages of energy and information (kinetic-energy) in thermodynamics has been emphasized, and the relation between the in situ energy and information CT criteria have been examined.

The proportionality relation between the resultant gradient-information and kinetic energy of electrons indicates that the latter plays a more important role in chemical reactivity than previously thought. The electronic energy and information/kinetic-energyrepresent alternative descriptors of molecular equilibria. They generate physically equivalent and adequate reactivity criteria for describing CT phenomena in the acid-base systems. Since for representative energy-profiles this component is readily available from the molecular virial theorem, we have briefly examined the theorem general implications for changes in the overall information content of equilibrium molecular structures, the bond-formation process, and the Hammond postulate of reactivity theory. The principle of the maximum thermodynamic information has been formulated and the dependence in chemical processes of the change in the overall gradient-information upon the reaction energy has been addressed. The Hammond postulate has been shown to be quantitatively indexed by the geometrical information derivative at TS complex.