Document1

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CHAPTER 1Bond Differentiation andOrbital Decoupling in theOrbital-Communication Theoryof the Chemical Bond

Roman F. Nalewajskia, Dariusz Szczepanikb,and Janusz Mrozekb

Contents 1. Introduction 32. Molecular Information Channels in Orbital Resolution 63. Decoupled (Localized) Bonds in Hydrides Revisited 114. Flexible-Input Generalization 155. Populational Decoupling of Atomic Orbitals 216. Bond Differentiation in OCT 297. Localized σ Bonds in Coordination Compounds 348. Restricted Hartree–Fock Calculations 36

8.1. Orbital and condensed atom probabilities ofdiatomic fragments in molecules 37

8.2. Average entropic descriptors of diatomicchemical interactions 40

9. Conclusion 44References 45

Abstract Information-theoretic (IT) probe of molecular electronic structure, within theorbital-communication theory (OCT) of the chemical bond, uses the standardentropy/information descriptors of the Shannon theory of communication

a Department of Theoretical Chemistry, Jagiellonian University, Cracow, Polandb Department of Computational Methods in Chemistry, Jagiellonian University, Cracow, Poland

Advances in Quantum Chemistry, Volume 61 c© 2011 Elsevier Inc.ISSN 0065-3276, DOI: 10.1016/B978-0-12-386013-2.00001-2 All rights reserved.

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http://dx.doi.org/10.1016/B978-0-12-386013-2.00001-2

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2 Roman F. Nalewajski, Dariusz Szczepanik, and Janusz Mrozek

to characterize the scattering of electron probabilities and their informationcontent throughout the system network of chemical bonds generated bythe occupied molecular orbitals (MOs). Thus, the molecule is treated asinformation network, which propagates the “signals” of the electron alloca-tion to constituent atomic orbitals (AOs) or general basis functions betweenthe channel AO “inputs” and “outputs.” These orbital “communications” aredetermined by the two-orbital conditional probabilities of the output AOevents given the input AO events. It is argued, using the quantum-mechanicalsuperposition principle, that these conditional probabilities are proportionalto the squares of corresponding elements of the first-order density matrixof the AO charges and bond orders (CBO) in the standard self-consistentfield (SCF) theory using linear combinations of AO (LCAO) to represent MO.Therefore, the probability of the interorbital connections in the molecu-lar communication system is directly related to the Wiberg-type quadraticindices of the chemical bond multiplicity. Such probability propagation inmolecules exhibits the communication “noise” due to electron delocaliza-tion via the system chemical bonds, which effectively lowers the informationcontent in the output signal distribution, compared with that contained inprobabilities determining its input signal, molecular or promolecular. Theorbital information systems are used to generate the entropic measures ofthe chemical bond multiplicity and their covalent/ionic composition. Theaverage conditional-entropy (communication noise, electron delocalization)and mutual-information (information capacity, electron localization) descrip-tors of these molecular channels generate the IT covalent and IT ionic bondcomponents, respectively. A qualitative discussion of the mutually decou-pled, localized bonds in hydrides indicates the need for the flexible-inputgeneralization of the previous fixed-input approach, in order to achieve abetter agreement among the OCT predictions and the accepted chemi-cal estimates and quantum-mechanical bond orders. In this extension, theinput probability distribution for the specified AO event is determined bythe molecular conditional probabilities, given the occurrence of this event.These modified input probabilities reflect the participation of the selectedAO in all chemical bonds (AO communications) and are capable of the con-tinuous description of its decoupling limit, when this orbital does not formeffective combinations with the remaining basis functions. The occupationalaspect of the AO decoupling has been shown to be properly representedonly when the separate communication systems for each occupied MO areused, and their occupation-weighted entropy/information contributions areclassified as bonding (positive) or antibonding (negative) using the extra-neous information about the signs of the corresponding contributions tothe CBO matrix. This information is lost in the purely probabilistic modelsince the channel communications are determined by the squares of suchmatrix elements. The performance of this MO-resolved approach is thencompared with that of the previous, overall (fixed-input) formulation of OCTfor illustrative π-electron systems, in the Huckel approximation. A qualita-tive description of chemical bonds in octahedral complexes is also given. Thebond differentiation trends in OCT have been shown to agree with both the

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Bond Differentiation and Orbital Decoupling in Orbital-Communication Theory 3

chemical intuition and the quantum-mechanical description. The numericalRestricted Hartree–Fock (RHF) applications to diatomic bonds in representa-tive molecular systems are reported and discussed. The probability weightedscheme for diatomic molecular fragments is shown to provide an excellentagreement with both the Wiberg bond orders and the intuitive chemicalbond multiplicities.

1. INTRODUCTION

The techniques and concepts of information theory (IT) [1–8] have been shownto provide efficient tools for tackling diverse problems in the theory ofmolecular electronic structure [9]. For example, the IT definition of Atoms-in-Molecules (AIM) [9–13] has been reexamined and the information contentof electronic distributions in molecules and the entropic origins of the chem-ical bond has been approached anew [9–18]. Moreover, the Shannon theoryof communication [4–6] has been applied to probe the bonding patternsin molecules within the communication theory of the chemical bond (CTCB)[9, 19–28] and thermodynamic-like description of the electronic “gas” inmolecular systems has been explored [9, 29–31]. The CTCB bonding patternsin both the ground and excited electron configurations have been tackledand the valence-state promotion of atoms due to the orbital hybridizationhas been characterized [28]. This development has widely explored the useof the average communication noise (delocalization, indeterminacy) andinformation-flow (localization, determinacy) indices as novel descriptors ofthe overall IT covalency and ionicity, respectively, of all chemical bonds inthe molecular system as a whole, as well as the internal bonds present in itsconstituent subsystems and the external interfragment bonds.

The electron localization function [32] has been shown to explore thenonadditive part of the Fisher information [1–3] in the molecular orbital(MO) resolution [9, 33], whereas a similar approach in the atomic orbital(AO) representation generates the so-called contragradience (CG) descrip-tors of chemical bonds, which are related to the matrix representation of theelectronic kinetic energy [34–38]. It should be recalled that the molecularquantum mechanics and IT are related through the Fisher (locality) mea-sure of information [34–41], which represents the gradient content of thesystem wavefunction, thus being proportional to the average kinetic energyof electrons. The stationary Schrodinger equation indeed marks the optimumprobability amplitude of the associated Fisher information principle, includ-ing the additional constraint of the fixed value of the system potential energy[34, 39–41]. Several strategies for molecular subsystems have been designed[9, 22, 25, 26] and the atomic resolution of bond descriptors has been pro-posed [24]. The relation between CTCB and the Valence Bond (VB) theory has

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been examined [23, 27] and the molecular similarities have been explored[9, 42]. Moreover, the orbital resolution of the “stockholder” atoms andthe configuration-projected channels for excited states have been developed[43–45].

The key concept of CTCB is the molecular information system, whichcan be constructed at alternative levels of resolving the electron probabili-ties into the underlying elementary “events” determining the channel inputsa={ai} and outputs b={bj}, for example, of finding an electron on the basisset orbital, AIM, molecular fragment, etc. They can be generated withinboth the local and the condensed descriptions of electronic probabilities ina molecule. Such molecular information networks describe the probability/information propagation in a molecule and can be characterized by the stan-dard quantities developed in IT for real communication devices. Because ofthe electron delocalization throughout the network of chemical bonds in amolecule the transmission of “signals” from the electron assignment to theunderlying events of the resolution in question becomes randomly disturbed,thus exhibiting the communication “noise.” Indeed, an electron initiallyattributed to the given atom/orbital in the channel “input” a (molecularor promolecular) can be later found with a nonzero probability at severallocations in the molecular “output” b. This feature of the electron delocaliza-tion is embodied in the conditional probabilities of the outputs given inputs,P(b|a)={P(bj|ai)}, which define the molecular information network.

Both one- and two-electron approaches have been devised to constructthis matrix. The latter [9] have used the simultaneous probabilities of twoelectrons in a molecule, assigned to the AIM input and output, respec-tively, to determine the network conditional probabilities, whereas the for-mer [38, 46–48] constructs the orbital-pair probabilities using the projectedsuperposition-principle of quantum mechanics. The two-electron (correla-tion) treatment has been found [9] to give rise to rather poor representationof the bond differentiation in molecules, which is decisively improved in theone-electron approach in the AO resolution, called the orbital-communicationtheory (OCT) [38, 46–48]. The latter scheme complements its earlier orbitalimplementation using the effective AO-promotion channel generated fromthe sequential cascade of the elementary orbital-transformation stages[43–45, 49, 50]. Such consecutive cascades of elementary information sys-tems have been used to represent the underlying orbital transformationsand electron excitations in the resultant propagations of the electron prob-abilities, determining the orbital promotions in molecules. The informationcascade approach also provides the probability scattering perspective onatomic promotion due to the orbital hybridization [28].

In OCT the conditional probabilities determining the molecular commu-nication channel in the basis-function resolution follow from the quantum-mechanical superposition principle [51] supplemented by the “physical”projection onto the subspace of the system-occupied MOs, which determines

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the molecular network of chemical bonds. Both the molecule as a whole andits constituent subsystems can be adequately described using the OCT bondindices. The internal and external indices of molecular fragments (groupsof AO) can be efficiently generated using the appropriate reduction of themolecular channel [9, 25, 46, 48] by combining selected outputs and largerconstituent fragment(s).

In this formulation of CTCB the off-diagonal orbital communications havebeen shown to be proportional to the corresponding Wiberg [52] or relatedquadratic indices of the chemical bond [53–63]. Several illustrative modelapplications of OCT have been presented recently [38, 46–48], covering boththe localized bonds in hydrides and multiple bonds in CO and CO2, as wellas the conjugated π bonds in simple hydrocarbons (allyl, butadiene, andbenzene), for which predictions from the one- and two-electron approacheshave been compared; in these studies the IT bond descriptors have beengenerated for both the molecule as whole and its constituent fragments.

After a brief summary of the molecular and MO-communication sys-tems and their entropy/information descriptors in OCT (Section 2) themutually decoupled, localized chemical bonds in simple hydrides will bequalitatively examined in Section 3, in order to establish the input proba-bility requirements, which properly account for the nonbonding status ofthe lone-pair electrons and the mutually decoupled (noncommunicating,closed) character of these localized σ bonds. It will be argued that each suchsubsystem defines the separate (externally closed) communication channel,which requires the individual, unity-normalized probability distribution ofthe input signal. This calls for the variable-input revision of the original andfixed-input formulation of OCT, which will be presented in Section 4. Thisextension will be shown to be capable of the continuous description of theorbital(s) decoupling limit, when AO subspace does not mix with (exhibit nocommunications with) the remaining basis functions.

Additional, occupational aspect of the orbital decoupling in the OCTdescription of a diatomic molecule will be described in Section 5. It intro-duces the separate communication channels for each occupied MO andestablishes the relevant weighting factors and the crucial sign conventionof their entropic bond increments, which reflects the bonding or antibondingcharacter of the MO in question, in accordance with the signs of the asso-ciated off-diagonal matrix elements of the CBO matrix. This procedure willbe applied to determine the π -bond alternation trends in simple hydrocar-bons (Section 6) and the localized bonds in octahedral complexes (Section 7).Finally, the weighted-input approach for diatomic fragments in moleculeswill be formulated (Section 8). It will be shown that this new AO-resolveddescription using the flexible-input (bond) probabilities as weighting fac-tors generates bond descriptors exhibiting excellent agreement with both thechemical intuition and the quantum-mechanical bond orders formulated inthe standard SCF-LCAO-MO theory.

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Throughout this article, the bold symbol X represents a square or rectan-gular matrix, the bold-italic X denotes the row vector, and italic X standsfor the scalar quantity. The entropy/information descriptors are measuredin bits, which correspond to the base 2 in the logarithmic (Shannon) measureof information.

2. MOLECULAR INFORMATION CHANNELS IN ORBITALRESOLUTION

In the standard MO theory of molecular electronic structure the network ofchemical bonds is determined by the system-occupied MO in the electronconfiguration in question. For simplicity, let us first assume the closed-shell (cs) ground state of the N= 2n electronic system in the RestrictedHartree–Fock (RHF) description, involving the n lowest (real and orthonor-mal), doubly occupied MO. In the LCAO-MO approach, they are givenas linear combinations of the appropriate (orthogonalized) basis functionsχ = (χ1,χ2, . . . ,χm)={χi}, 〈χ |χ〉= {δi, j} ≡ I, for example, Lowdin’s symmetri-cally orthogonalized AO, ϕ= (ϕ1,ϕ2, . . . ,ϕn)={ϕs}=χC, where the rectangu-lar matrix C={Ci,s} groups the relevant LCAO-expansion coefficients.

The system electron density ρ(r) and hence the one-electron probabil-ity distribution p(r)= ρ(r)/N, that is, the density per electron or the shapefactor of ρ, are determined by the first-order density matrix γ in the AOrepresentation, also called the charge and bond order (CBO) matrix,

ρ(r)= 2ϕ(r)ϕ†(r) = χ(r)[2CC†]χ †(r) ≡ χ(r)γχ †(r) = Np(r). (1)

The latter represents the projection operator Pϕ = |ϕ〉〈ϕ| =∑

s |ϕs〉〈ϕs| ≡∑s Ps onto the subspace of all doubly occupied MO,

γ= 2〈χ |ϕ〉〈ϕ|χ〉 = 2CC†≡ 2〈χ |Pϕ|χ〉 = {γi, j = 2〈χi|Pϕ|χj〉 ≡ 2〈i|Pϕ|j〉}, (2a)

thus, satisfying the appropriate idempotency relation

(γ)2= 4〈χ |Pϕ|χ〉〈χ |Pϕ|χ〉 = 4〈χ |P2

ϕ|χ〉 = 4〈χ |Pϕ|χ〉 = 2γ. (3)

The CBO matrix reflects the promoted, valence state of AO in the molecule,with the diagonal elements measuring the effective electron occupations ofthe basis functions, {γi,i=Ni=Npi}. The AO-probability vector in this state,p={pi=Ni/N}, groups the probabilities of the basis functions being occupiedin the molecule.

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The information system in the (condensed) orbital resolution involvesthe AO events χ in its input a={χi} and output b={χj}. It representsthe effective promotion of these basis functions in the molecule via theprobability/information scattering described by the conditional probabil-ities of AO outputs given AO inputs, with the input (row) and output(column) indices, respectively. In the one-electron approach [46–48], theseAO-communication connections {P(χj|χi) ≡ P(j|i)} result from the appropri-ately generalized superposition principle of quantum mechanics [51],

P(b|a) ={P(j|i) = Ni|〈i|Pϕ|j〉|2 = (2γi,i)

−1γi, jγj,i

},

∑j

P(j|i) = 1, (4)

where the closed-shell normalization constant Ni = (2γi,i)−1 follows directly

Eq. (3). These (physical) one-electron probabilities explore the dependen-cies between AOs resulting from their participation in the system-occupiedMO, that is, their involvement in the entire network of chemical bonds in amolecule. This molecular channel can be probed using both the promolecu-lar (p0

={p0i }) and molecular (p) input probabilities, in order to extract the IT

multiplicities of the ionic and covalent bond components, respectively.In the open-shell (os) case [48] one partitions the CBO matrix into contri-

butions originating from the closed-shell (doubly occupied) MO ϕcs and theopen-shell (singly occupied) MO ϕos, ϕ= (ϕcs,ϕos):

γ=〈χ |ϕos〉〈ϕos|χ〉 + 2〈χ |ϕcs

〉〈ϕcs|χ〉 ≡ 〈χ |Pos

ϕ |χ〉 + 2〈χ |Pcsϕ |χ〉 ≡ γos

+ γcs. (5)

They satisfy separate idempotency relations,

(γos)2=⟨χ∣∣Pos

ϕ

∣∣χ ⟩⟨χ ∣∣Posϕ

∣∣χ ⟩ = ⟨χ ∣∣(Posϕ )

2⟩χ⟩=⟨χ∣∣Pos

ϕ

∣∣χ ⟩ = γos, (6)

and (γcs)2= 2γcs (Eq. [3]). Hence,

P(j|i) = Ni|〈i|Pϕ|j〉|2 = (γ osi,i + 2γ cs

i,i )−1γi,jγj,i, (7a)

The conditional probabilities of Eqs. (4 and 7a) define the probability scat-tering in the AO-promotion channel of the molecule, in which the “signals”of the molecular (or promolecular) electron allocations to basis functions aretransmitted between the AO inputs and outputs. Such information systemconstitutes the basis of OCT [46–48]. The off-diagonal conditional probabil-ity of jth AO output given ith AO input is thus proportional to the squaredelement of the CBO matrix linking the two AOs, γj,i= γi, j. Therefore, it is alsoproportional to the corresponding AO contribution Mi, j= γ

2i, j to the Wiberg

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index [52] of the chemical bond covalency between two atoms A and B,M(A,B)=

∑i∈A

∑j∈B Mi, j, or to generalized quadratic descriptors of molecular

bond multiplicities [53–63].By separating the CBO contributions due to each occupied MO one

similarly defines the information system for each orbital. For example, inthe closed-shell system, each doubly occupied MO ϕ={ϕs} generates thecorresponding contributions to the CBO matrix of Eq. (2):

γ = 2∑

s

〈χ |ϕs〉〈ϕs|χ〉 ≡∑

s

γcss , γcs

s ={γ cs

i, j (s) = 2〈i|Ps|j〉}, (2b)

In the open-shell configuration, one separately partitions the contri-bution of γcs

=∑cs

s γcss , due to the doubly occupied MO ϕcs, and the

remaining part γos=∑os

s γoss , γos

s ={γos

i, j (s)=〈i|Ps|j〉}, generated by the singlyoccupied MO ϕos. They satisfy the corresponding idempotency relations(see Eqs. [3 and 6]): (γcs

s )2= 2γcs

s and (γoss )

2=γos

s . One then determines thecorresponding communication connections for each occupied MO,

Poss (b|a)=

{Pos

s (j|i)=γ os

i, j (s)γos

j,i (s)

γ osi,i (s)

}and Pcs

s (b|a)={

Pcss (j |i )=

γ csi, j (s)γ

csj,i (s)

2γ csi,i (s)

},

(7b)

were obtained using Eqs. (4a and 7a) with the normalization constantsappropriately modified to satisfy the normalization condition for the con-ditional probabilities: ∑

j

Pcss (j|i) =

∑j

Poss (j|i) = 1. (7c)

In OCT, the entropy/information indices of the covalent/ionic compo-nents of all chemical bonds in a molecule represent the complementarydescriptors of the average communication noise and the amount of informa-tion flow in the molecular information channel. The molecular input p(a) ≡ pgenerates the same distribution in the output of the molecular channel,

pP(b|a) =

{∑i

piP(j|i) ≡∑

i

P(i, j) = pj

}= p (8)

and thus identifying p as the stationary probability vector for the molecu-lar state in question. In the preceding equation we have used the partialnormalization of the molecular joint, two-orbital probabilities P(a, b)={P(i, j)= piP(j|i)} to the corresponding one-orbital probabilities. It should beobserved at this point that the promolecular input p(a0) ≡ p0 in generalproduces different output probability p0P(b|a)=p∗(a0)={p∗j }=p∗ 6= p.

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The purely molecular communication channel [9, 38, 46–48], with p defin-ing its input signal, is devoid of any reference (history) of the chemicalbond formation and generates the average noise index of the molecular ITbond covalency, measured by the conditional entropy S(b|a) ≡ S of the systemoutputs given inputs:

S(b|a) = −∑

i

∑j

P(i, j)log2[P(i, j)/pi]

= −

∑i

pi

∑j

P(j|i)log2P(j|i) ≡ S[p|p] ≡ S[P(b|a)] ≡ S. (9a)

Thus, this average noise descriptor expresses the difference between theShannon entropies of the molecular one- and two-orbital probabilities,

S = H[P(a, b)]−H[p];

H[p] = −∑

i

pilog2pi,

H[P(a, b)] = −∑

i

∑j

P(i, j)log2P(i, j). (9b)

For the independent input and output events, when Pind.(a, b)={pipj},

H[Pind.(a, b)]= 2H[p] and hence Sind.

=H[p].The molecular channel with p0 determining its input signal refers to the

initial state in the bond formation process, for example, the atomic pro-molecule—a collection of nonbonded free atoms in their respective positionsin a molecule or the AO basis functions with the atomic ground-state occu-pations, before their mixing into MO [9, 38, 46–48]. The AO occupationsin this reference state are fractional in general. However, in view of theexploratory character of the present analysis, we shall often refer to thesimplest description of the promolecular reference by a single (ground-state) electron configuration, which exhibits the integral occupations of AO.It gives rise to the average information-flow descriptor of the system ITbond ionicity, given by the mutual information in the channel inputs andoutputs:

I(a0 :b) =∑

i

∑j

P(i, j)log2[P(i, j)/(pjp0i )] = H[p]+H[p0]−H[P(a, b)]

= H[p0]− S = I[p0 :p] ≡ I[P(b|a)] = I, (10)

This quantity reflects the fraction of the initial (promolecular) informationcontent H[p0], which has not been dissipated as noise in the molecularcommunication system. In particular, for the molecular input, when p0

=p,

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I(a :b)=∑

i

∑j P(i, j)log2[P(i, j)/(pjpi)]=H[p]− S= I[p :p]. Thus, for the ind-

ependent input and output events, Iind.(a :b)= 0.Finally, the sum of these two bond components,

N (a0; b) = S+ I ≡ N [p0; p] ≡ N [P(b|a)] ≡ N = H[p0], (11)

measures the overall IT bond multiplicity of all bonds in the molecularsystem under consideration. It is seen to be conserved at the promolecular-entropy level, which marks the initial information content of orbital proba-bilities. Again, for the molecular input, when p0

=p, this quantity preservesthe Shannon entropy of the molecular input probabilities:N (a; b)=S(b|a)+I(a :b)=H[p].

It should be emphasized that these entropy/information descriptors andthe underlying probabilities depend on the selected basis set, for exam-ple, the canonical AO of the isolated atoms or the hybrid orbitals (HOs) oftheir promoted (valence) states, the localized MO (LMO), etc. In what fol-lows we shall examine these IT descriptors of chemical bonds in illustrativemodel systems. The emphasis will be placed on the orbital decoupling in themolecular communication channels and the need for appropriate changesin their input probabilities, which weigh the contributions to the averageinformation descriptors from each input.

There are two aspects of the orbital decoupling in chemical bonds. On oneside, the two chemically interacting AOs becomes decoupled, when they donot mix into MO, for example, in the extreme MO-polarization limit of theelectronic lone pair, when two bonding electrons occupy a single AO. On theother side, the two AOs are also effectively decoupled, no matter how strongis their mutual mixing, when the bonding and antibonding MO combina-tions are completely occupied by electrons, since such MO configuration isphysically equivalent to the Slater determinant of the doubly occupied (orig-inal) AO. We shall call these two facets the mixing (shape) and occupation(population) decouplings, respectively.

It is of vital interest for a wider applicability of CTCB to examine how thesetwo mechanisms can be accommodated in OCT. In Section 3, we shall arguethat the mutual decoupling status of several subsets of basis functions, mani-festing itself by the absence of any external communications (bond orders) inthe whole system, calls for the separate unit normalization of its input prob-abilities since such fragments constitute the mutually nonbonded (closed)building blocks of the molecular electronic structure. It will be demonstrated,using simple hydrides as an illustrative example, that the fulfillment of thisrequirement dramatically improves the agreement with the accepted chemi-cal intuition and the alternative bond multiplicity concepts formulated in theMO theory.

To conclude this section, we observe that by propagating the AO prob-abilities through the information channels of the separate MO, defined bythe conditional probabilities of Eq. (7b), one could similarly determine the

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IT bond increments for each occupied MO. We shall implement this idea inSection 5, when tackling the populational decoupling of atomic orbitals, as aresult of an increased occupation of the antibonding MO.

3. DECOUPLED (LOCALIZED) BONDS IN HYDRIDES REVISITED

In the ground-state the chemical interaction between two (singly occupied)orthonormal AOs χ = (a, b) originating from atoms A and B, respectively,gives rise to the doubly occupied, bonding MO ϕbond. and the unoccupiedantibonding MO ϕanti.,

ϕbond.=√

Pa+√

Qb, ϕanti. = −

√Qa+

√Pb, P+Q = 1. (12)

Their shapes are determined by the complementary (conditional) proba-bilities: P(a|ϕbond.)=P(b|ϕanti.)=P and P(b|ϕbond.)=P(a|ϕanti.)=Q, which con-trol the bond polarization, covering the symmetrical bond combinationfor P=Q= 1/2 and the limiting lone-pair (nonbonding) configurations forP= (0, 1). The associated model CBO matrix,

γ= 2

[P

√PQ√

PQ Q

], (13)

then generates the information system for such a two-AO model, shown inScheme 1.1a.

In this diagram one adopts the molecular input p= (P, Q= 1− P), toextract the bond IT covalency index measuring the channel average com-munication noise, and the promolecular input p0

= (1/2, 1/2), to calculate theIT ionicity relative to this covalent promolecule, in which each basis func-tion contributes a single electron to form the chemical bond. The bond ITcovalency S(P) is then determined by the binary entropy function H(P)= −Plog2P−Qlog2Q=H[p]. It reaches the maximum value H(P= 1/2)= 1 forthe symmetric bond P=Q= 1/2 and vanishes for the lone-pair configura-tions, when P= (0, 1), H(P= 0)=H(P= 1)= 0, marking the alternative ion-pair configurations A+B− and A−B+, respectively, relative to the initial AOoccupations N0

= (1, 1) in the assumed promolecular reference. The comple-mentary descriptor of the IT ionicity, determining the channel (mutual) infor-mation capacity I(P)=H[p0]−H(P)= 1−H(P), reaches the highest value forthese two limiting electron-transfer configurations P= (0, 1) : I(P= 0, 1)= 1.Thus, this ionicity descriptor is seen to identically vanish for the purelycovalent, symmetric bond, I(P= 1/2)= 0.

Both components yield the conserved overall bond index N (P)= 1 in thewhole range of bond polarizations P ∈ [0, 1]. Therefore, this model properlyaccounts for the competition between the bond covalency and ionicity, while

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P(b|a) pp

P aa P

(a)

(b)

PQ bb Q

P

Q

Q

1 h0n h0

n1 1

S = −Plog2P−Qlog2Q = H(P)I = H[ p0]− H(P)

N = I+ S = H[ p0]

S = I = N = 0

Scheme 1.1 The molecular information system modeling the chemical bond between twobasis functions χ = (a, b) and its entropy/information descriptors. In Panel b, the corre-sponding nonbonding (deterministic) channel due to the lone-pair hybrid h0

n is shown. For themolecular input p = (P,Q), the orbital channel of Panel a gives the bond entropy-covalencyrepresented by the binary entropy function H(P). For the promolecular input p0

= (1/2, 1/2),when both basis functions contribute a single electron each to form the chemical bond,one thus predicts H[p0]= 1 and the bond information ionicity I= 1− H(P). Hence, these twobond components give rise to the conserved (P-independent) value of the single overall bondmultiplicityN = I+ S= 1.

preserving the single measure of the overall IT multiplicity of the chemi-cal bond. Similar effects transpire from the two-electron CTCB [9] and thequadratic bond indices formulated in the MO theory [53–63].

This localized bond model can be straightforwardly extended to the sys-tem of r localized bonds in simple hydrides XHr [49], for example, CH4, NH3,or H2O, for r= 4, 3, 2, respectively. Indeed, a single σ bond X–Hα, for X=C,N, O and α= 1, . . . , r, can then be approximately regarded as resulting fromthe chemical interaction of a pair of two orthonormal orbitals: the bondingsp3 hybrid hα of the central atom, directed towards the hydrogen ligand Hα,and the 1sα ≡ σα orbital of the latter. The localized bond X–Hα then originatesfrom the double occupation of the corresponding bonding MO ϕbond.(α), withthe antibonding MO ϕanti.(α) remaining empty:

ϕbond.(α) =√

Phα +√

Qσα, ϕanti.(α) = −√

Qhα +√

Pσα P+Q = 1. (14)

In the χα = (hα, σα) representation, the corresponding CBO matrix γα for asingle σ bond X–Hα{γα,β} then includes the following nonvanishing elements:

γhα ,hα = 2P, γσα ,σα = 2Q, γhα ,σα = γσα ,hα = 2√

PQ, (15)

while for each of 4− r nonbonded hybrids {hn}, describing the system lone-electronic pairs,

γhn ,hn = 2 and γhn , j = 0, j 6= hn. (16)

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The corresponding conditional probabilities (see Eq. [4]), which determinethe nonvanishing communication connections, then read:

P(hα|hα) = P, P(σα|σα) = Q, P(hα|σα) = Q, P(σα|hα) = P, P(hn|hn) = 1.(17)

Therefore, the electron probability is not scattered by the lone-pair hybrids.As a result such decoupled subchannels {hn= h0

n} representing two lonepairs of oxygen atom in H2O or a single nonbonding electron pair ofnitrogen in NH3, introduce the exactly vanishing contributions to bothbond components and hence to the overall bond index of these moleculesin OCT.

It follows from these expressions that each localized bond X–Hα in this HOrepresentation defines the separate communication system of Scheme 1.1a,consisting of inputs and outputs χα = (hα, σα), which does not exhibit anyexternal communications with AO involved in the system remaining bonds.Therefore, such orbital pairs constitute the externally closed (nonbond-ing) subsystems, determining the mutually decoupled information systemsdefined by the diagonal blocks

Pα(bα|aα)≡Pα[χα|χα]=[

P QP Q

], Pα(aα, bα)≡Pα[χα,χα]=

[P2 PQQP Q2

],

(18)

of the associated overall probability matrices in the χ ={χα} basis set:P(b|a)={Pα(bα|aα)δα,β} and P(a, b)={Pα(aα, bα)δα,β}. Such mutually closed(isolated) subchannels correspond to the separate input/output probabilitydistributions, p0

α= (1/2, 1/2) or p

α=p∗

α= (P, Q), each satisfying the unit nor-

malization [9, 26]. These separate molecular subsystems give rise to theadditive bond contributions Sα(bα|aα) ≡ Sα, Iα(a0

α:bα) ≡ Iα and Nα(a0

α: bα)=

Sα + Iα ≡ Nα to the system overall bond descriptors in OCT:

S(P)=∑α

Sα = rH(P), I(P)=∑α

Iα = r[1−H(P)], N =∑α

Nα = r. (19)

We have recognized in these expressions that each lone-pair (dou-bly occupied) hybrid hn of the central atom, which does not form anychemical bonds (communications) with the hydrogen ligands, generatesthe decoupled deterministic subchannel of Scheme 1.1b, thus exhibit-ing the unit input probability. Therefore, it does not contribute tothe resultant entropy/information index of all chemical bonds in themolecule.

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The same result follows from another, delocalized representation of chem-ical bonds in these prototype systems. Consider, as an illustration, thecanonical valence-shell MO in CH4, with four hydrogen ligands in the alter-nating corners of the cube placed in such a way, that the three axes of theCartesian coordinate system pass through the middle of its opposite walls.In such an arrangement, the four delocalized bonds are described by the four(mutually decoupled) orbital-pair interactions between the specified canon-ical AO of carbon atom and the corresponding symmetry combination offour hydrogen orbitals. Again, the net result is the four decoupled bonds inthe system giving rise to overall IT bond index N = 4, with S = 4H(P) andI = 4[1−H(P)].

One observes, however, a change in the bond covalent/ionic compositionresulting from this transformation from the localized MO description to thecanonical MO perspective [48]. As an illustration of this entropic effect, letus briefly examine the bonding pattern in the linear BeH2. In the localizedbond representation, the two bonding MOs result from the mutually decou-pled interactions between two-orbital pairs, each including one sp hybrid ofBe and 1s orbital of the corresponding hydrogen. This localized approachthus gives N = 2, with S = 2H(P) and I = 2[1−H(P)], and hence for themaximum orbital mixing (P = 1/2), the IT bond composition reads Smax.

= 2and Imax.

= 0. In the delocalized description, the two doubly occupied canon-ical MO, expressed in the basis set χ = (h1, h2, σ1, σ2) used to generate thelocalized MO of Eq. (12), read as follows:

ψ1 =√

Us+

√V2(σ1 + σ2) =

√U2(h1 + h2)+

√V2(σ1 + σ2), U + V = 1,

ψ2 =√

Tp+

√W2(σ1 − σ2) =

√T2(h1 − h2)+

√W2(σ1 − σ2), T +W = 1.

(20)

The associated CBO matrix,

γ =

U + T U − T

√UV +

√TW

√UV −

√TW

U − T U + T√

UV −√

TW√

UV +√

TW√

UV +√

TW√

UV −√

TW V +W V −W√

UV −√

TW√

UV +√

TW V −W V +W

,

(21)

indicates that all these basis orbitals in fact exhibit the nonvanishing com-munications to all outputs in this delocalized representation of the systemelectronic structure. In the maximum mixing limit of U = V = T =W = 1/2

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it is only partly decoupled,

γmax.=

1 0 1 00 1 0 11 0 1 00 1 0 1

, (22)

and so is the associated matrix P(b|a) = 1/2γmax. of molecular communica-tions (see Eq. [4]). Thus, this delocalized channel is characterized by theinput distributions p = p0

= {1/4}, which give rise to the overall unit normal-ization. The associated entropy/information indices for this channel read asfollows: Smax.

= Imax.= 1 and hence Nmax.

= 2.The variable-input norm description of the decoupled chemical bonds

gives the full agreement with the chemical intuition, of r bonds in XHr,with changing covalent/ionic composition in accordance with the actual MOpolarization and the adopted basis set representation. The more the probabil-ity parameter P deviates from the symmetrical bond (maximum covalency)value P = 1/2, due to the electronegativity difference between the centralatom and hydrogen, the lower is the covalency (the higher ionicity) of thislocalized, diatomic bond. Therefore, in this IT description the total bondmultiplicity N = r bits is conserved for changing proportions between theoverall covalency and ionicity of all chemical bonds in the system underconsideration.

In the orbital-communication theory, this “rivalry” between bond compo-nents reflects a subtle interplay between the electron delocalization (Sα =H(P)) and localization (Iα = 1−H(P)) aspects of the molecular scattering ofelectron probabilities in the information channel of a separated single chem-ical bond, decoupled from the molecular remainder. The more deterministicis this probability propagation, the higher the ionic component. Accordingly,the more delocalized is this scattering, the higher the “noise” descriptor ofthe underlying information system.

4. FLEXIBLE-INPUT GENERALIZATION

Thus, it follows from the analysis of the preceding section that a gen-eral agreement of IT descriptors with the intuitive chemical estimates fol-lows only when each externally decoupled fragment of a molecule exhibitsthe separate unit normalization of its input probabilities; this requirementexpresses its externally closed status relative to the molecular remainder.It modifies the overall norm of the molecular input to the number ofsuch mutually closed, noncommunicating fragments of the whole molecularsystem. This requirement was hitherto missing in all previous applicationsof CTCB and OCT to polyatomic systems.

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In the generalized approach the probabilities p0α={pαa } of the constituent

inputs in the given externally decoupled (noncommunicating and non-bonded) subchannel α0 of the system “promolecular” reference M0

=

(α0|β0| . . .) should thus exhibit the internal (intrasubsystem) normalization,∑

a∈α pαa = 1; we have denoted the externally closed status of each fragment inM0 by separating it with the vertical solid lines from the rest of the molecule.Therefore, these subsystem probabilities are, in fact, conditional in character;pαa = P(a|α) = pa/Pα, calculated per unit input probability Pα = 1 of the wholesubsystem α in the collection of the mutually nonbonded subsystems in thereference M0, that is, when this molecular fragment is not considered to be apart of a larger system. Indeed, the above summation over the internal orbitalevents then expresses the normalization of all such conditional probabilitiesin the separate (or isolated) subsystem α0 :

∑a∈α P(a|α) = 1.

This situation changes discontinuously in the externally coupled (commu-nicating and bonded) case, when the same subsystem exhibits non-vanishing(no matter how small) communications with the remainder of the moleculeM = (α β . . .). Such bonded fragments of the molecule are mutually open,as symbolically denoted by the vertical broken lines separating them. Theyare characterized by the fractional condensed probabilities P = {PM

α< 1},

which measure the probabilities of the constituent subsystems in M as awhole. Therefore, the input probabilities of the bonded fragment α in M,pMα= {pM

a = PMα

pαa }, are then subject to the molecular normalization:∑

a∈α pαa =PMα

∑a∈α pαa = PM

α. The need for using the molecular input probabilities then

causes a discontinuous change in the system covalent/ionic bond compo-nents compared with the corresponding decoupled (promolecular) values.Indeed, the former corresponds to the unit norm of input probabilities forall molecular subsystems, whereas in the latter, each decoupled fragmentappears as a separate system, thus alone exhibiting the unit probabilitynormalization.

In the previous, fixed-input determination of the IT bond indices thisdiscontinuity in the transition from the decoupled to the coupled descrip-tions of the molecular fragments prevents an interpretation of the formeras the limiting case of the latter, when all external communications of thesubsystem in question become infinitely small. In other words, the fixed-and flexible-input approaches generate the mutually exclusive sets of bondindices, which cannot describe this transition in a continuous (“causal”)fashion. As we have demonstrated in the decoupled approach of the pre-ceding section, only the overall input normalization equal to the number ofthe decoupled orbital subsystems gives rise to the full agreement with theaccepted chemical intuition.

Therefore, in this section we shall attempt to remove this discontinuityin the unifying, flexible-input generalization of the use of the molecularinformation systems. We shall demonstrate that in such an extension theabove limiting transition in the communication description of the subsystem

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decoupling in the molecule finds the continuous (causal) representation. Inorder to make this transition continuous, the separate input-dependent dis-tributions, tailored for each ith input event, have to explicitly depend on thestructure of its molecular communications, which is embodied in the ith rowof the system two-orbital conditional probabilities. Indeed, they have to con-tinuously increase the overall norm of the distribution for the given inputorbital with increasing localization of the molecular scattering of this inputsignal to reach the unit input norm in the limit of this orbital being totallydecoupled from the rest of the molecule.

The essence of the new proposition lies in a separate determination ofthe entropy/information contributions due to each AO input in the molec-ular channel specified by the conditional probabilities P(b|a). This goal canbe tackled by using the separate probability distributions tailored for eachinput. The hitherto single molecular propagation of the overall molecularinput probabilities p of the previous approach, carried out to extract the ITcovalent bond descriptor, will now be replaced by the series of m molec-ular propagations of the separate probability distributions {p(i) = {p(k; i)}for each molecular input i = 1, 2, . . . , m, which generate the associated cova-lencies: {S(i) = S[p(i)]}. The reference promolecular probabilities, also inputdependent {p(i0) = {p(k; i0)}, will be used to estimate the correspondingionic contributions due to each input: {I(i) = I[p(i0)]}. Together, these input-dependent contributions generate the corresponding total indices {N (i) =I(i)+ S(i) = N [p(i0), p(i)]}. Finally, the overall IT bond descriptor of M asa whole will be generated by the summation of all such additive con-tributions determined in the separate propagations of the input-tailoredmolecular/promolecular distributions: N =

∑iN (i). In the average molec-

ular quantities, these contributions must be weighted with the appropriateensemble probabilities of each input, for example, the molecular probabilitiesp = {pi}.

There are obvious normalization (sum) rules to be satisfied by these input-dependent probabilities. Consider first the completely coupled molecularchannel, in which all orbitals interact chemically, thus exhibiting nonvan-ishing direct and/or indirect communications with the system remainder. Inthis case all molecular inputs have to be effectively probed to the full extentof the unit condensed probability of the molecule as a whole:∑

k

p(k; i)] =∑

k

p(k; i0)] = 1. (23)

This condition recognizes a general category of these input-dependentprobabilities {p(k; i)} and {p(k; i0)} as conditional probabilities of two-orbitalevents, that is, the joint probabilities per unit probability of the specifiedinput: p(k; i) ≡ p(k|i) and p(k; i0) ≡ p(k|i0). However, it should be emphasizedthat these probabilities are also conditional on the molecule as a whole, since

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they correspond to the unit input probability in M or M0,∑

i pi =∑

i p0i = 1,

p(k|i) = p(k|i‖M) = P(k|i), p(k|i0) = p(k|i0‖M0

) = P(k|i0) ≡ P(k|i). (24)

In case of the decoupled single-orbital subsystem χ 0i , only the diagonal

probability scattering Pi(i|i) = 1 is observed in the molecule (Scheme 1.1b).The input-tailored conditional probabilities then refer to the unit inputprobability of the input i(i0) alone:

p(k|i) = p(i|i‖i) = p(i|i)δi,k = p(k|i0) = p(i|i0‖i0) = p(i0

|i0)δi,k = δi,k. (25)

In order to make the fragment decoupling continuous in this general-ized description, the input probabilities {p(i), p0(i)} have to be replaced bythe separate distributions reflecting the actual participation of ith AO inthe chemical bonds (communications) of the molecule. Therefore, they bothhave to be related explicitly to the ith row in the conditional probabilitymatrix P(b|a) = {P(j|i)}, which reflects all communications (bonds) betweenthis orbital input and all orbital outputs {j} (columns in P(b|a)). This linkmust generate the separate subsystem probabilities p0

α, when the fragment

becomes decoupled from the rest of the molecular system, α→ α0, whenP(b|aα)→ {P(bα|aα)δα,β}, where P(bα|aα) = {P(a′|a)}. Indeed, for the decou-pled subsystem α0

= (a, a′, . . .) only the internal communications of the corre-sponding block of the molecular conditional probabilities P(bα|aα) = {P(a′|a)}are allowed. They also characterize the internal conditional probabilities inα0 since

p(j|i‖α0) = p(i, j‖α0)/p(i‖α0) = P(i, j‖M)/p(i‖M) = p(j|i‖M) = P(j|i). (26)

Hence, {p(k|a‖α0)= pα(a′|a)δα,β = p(k|a0‖α0)= pα(a′|a0)δα,β =P(a′|a)δα,β}; again,

the AO inputs in α are to be probed with an overall unit condensedprobability:

∑a′∈α P(a′|a) = 1.

In the input-dependent molecular channels, all these requirements canbe shown to be automatically satisfied when one selects the input-tailoredprobabilities, we seek, as the corresponding rows of the molecular condi-tional probability matrix P(b|a) = {P(j|i)}. Consider the conditional-entropycontribution from ith channel:

S(i) = −∑

k

∑j

P(k, j; i)log2[P(j, k)/pk] = −∑

k

P(k|i)[∑

j

P(j|k)log2P(j|k)]

.

(27)

Since this entropy-covalency corresponds to the overall unit norm ofprobability distribution associated with ith input, in the average molecu-lar quantity, corresponding to all mutually open basis functions, it has tobe weighted by the actual probability pi of this input in the molecule as a

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whole. It can be directly verified that such averaging indeed reproduces themolecular index of Eq. (9):

Sav. =

∑i

piS(i) ≡∑

i

Si = −

∑i

∑k

∑j

[piP(k|i)]P(j|k)log2P(j|k)

= −

∑k

∑j

[∑i

P(k, i)

]P(j|k)log2P(j|k) = −

∑k

∑j

pkP(j|k)log2P(j|k)

= −

∑k

∑j

P(j, k)log2P(j|k) = S. (28)

A similar demonstration can be carried out for the mutual-information(ionic) contributions:

I(i) =∑

k

∑j

P(k0, j|i)log2[P(j, k)/(pjp0k)] = −S(i)−

∑k

∑j

P(j, k0)log2p0k

= −S(i)−∑

k

p0klog2p0

k = −S(i)+H[p0],

Iav. =

∑i

piI(i) ≡∑

i

Ii = −S+H[p0], (29)

Thus, it follows from these contributions that they also reproduce theoverall molecular bond index as the mean value of the partial, input AOcontributions:

N (i)=S(i)+ I(i) = −∑

k

∑j

P(k0, j)log2p0k ,

Nav.=

∑i

piN (i) ≡∑

i

Ni = N = H[p0]. (30)

To summarize, in the flexible-input extension of OCT the consistent useof the molecular channel is proposed, with only the molecular inputs beingused in probability propagation. However, the promolecular reference dis-tribution is seen to enter the final determination of the ionic (difference)components relative to the initial distribution of electrons before the bondformation.

As an illustration (see Scheme 1.2a), let us again consider the two-AOchannel of Scheme 1.1a. We first observe that the input-dependent distribu-tions in this model are identical with the molecular distribution (see Eq. [18]).The partial and average IT descriptors are also reported in this diagram, rela-tive to the reference distribution p0

= (1/2, 1/2) of the covalent promolecule,when two AOs contribute a single electron each to form the chemical bond.

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p(a), p(b)

p(a0), p(b0)

p(a), p(b)

p(a0), p(b0)

P a P a PQP

Q b Q b Q

P(b|a)

Sa = PH(P) Sb=QH(P)

Ia = P[1− H(P)] Ib = Q[1− H(P)]

Na = Sa + Ia = P Nb = Sb + Ib = Q

Sav. = Sa + Sb = H(P) = S Iav. = Ia + Ib = 1− H(P) = I Nav. = Na +Nb = 1 = N

Scheme 1.2 The flexible-input generalization of the two-AO channel of Scheme 1.1a for thepromolecular reference distribution p0

= (1/2, 1/2). The corresponding partial and averageentropy/information descriptors of the chemical bond are also reported.

The flexible-norm generalization of the previous OCT completely reproducesthe overall IT bond order and its components reported in Scheme 1.1.

It follows from the input probabilities in Scheme 1.2 that in the limit of thedecoupled (lone-pair) orbital ϕbond. = a(P = 1) or ϕbond. = b(Q = 1) its inputprobability becomes 1, while that of the other (empty) orbital identically van-ishes, as required. The unit input probability of the doubly occupied AO inthe channel input is then deterministically transmitted to the same AO inthe channel output, with the other (unoccupied) AO not participating in thechannel communications, so that both orbitals do not contribute to the resul-tant bond indices. Therefore, the flexible-input approach correctly accountsfor the MO shape decoupling in the chemical bond, which was missing in theprevious, fixed-input scheme.

It is also of interest to examine the dissociation of this model moleculeA–B into (one-electron) atoms A and B, which determine the promolecule.Such decoupled AO corresponds to the molecular configuration [ϕ1

bond. ϕ1anti.]

since the Slater determinant |ϕbond. ϕanti.| = |ab|. Indeed, using the orthogonaltransformations between χ = (a, b) and ϕ = (ϕbond.,ϕanti.),

χ = ϕ

[ √P

√Q

−√

Q√

P

]≡ ϕCT and ϕ = χC, CTC = CCT

= I,

one can directly verify that γ[ϕ1bond. ϕ

1anti.] = CCT

= I = P(b|a), so that thedecoupled AO inputs become p(a) = p(a0) = (1, 0) and p(b) = p(b0) = (0, 1),each is separately unity normalized.

Therefore, while still retaining the essence of the previous approach,the new proposition introduces in OCT of the chemical bond that is thedesired input flexibility generating the continuity in the IT description ofthe fragment decoupling process. This generalization covers in a commonframework both the completely coupled AO in the molecule and the limitingcases of its subsystems being effectively decoupled in the molecular chan-nel. In the former case, the resultant input signal corresponds to the unitnorm of the condensed probability distribution. In the case of n-mutuallyseparated fragments, this flexible normalization is automatically increased

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to n by the choice of the flexible probabilities for each input represented bythe conditional two-orbital probabilities. As we have shown in the previ-ous section, such an approach dramatically improves the agreement withthe accepted chemical intuition. It also has the conceptual and interpretativeadvantages by providing a unifying description capable of tacking both thecoupled and decoupled molecular fragments in a single theoretical frame-work and generating the continuous description of the shape-decouplinglimit, so that the decoupled subsystems appear naturally as those exhibitinginfinitely small communications with the molecular remainder.

It should be emphasized that in calculating the “ensemble” averagebond components of Eq. (28), the product

∑k piP(k|i)P(j|k) =

∑k P(i, k)

P(j|k) ≡ Pens.(i, j) represents an effective joint probability of orbitals χi and χj

in a molecule. Indeed, the amplitude interpretation of Eq. (4) gives∑

k piP(j|k)P(k|i)∝ pi

∑k〈j|ϕ〉〈ϕ|k〉〈k|ϕ〉〈ϕ|i〉 = pi〈j|PϕPχ Pϕ|i〉 = pi〈j|Pϕ|i〉 ∝ piP(j|i), since

Pχ Pϕ = Pϕ and PϕPϕ = Pϕ . Therefore, this probability product in fact mea-sures an ensemble probability of simultaneously finding an electron onorbitals χi and χj. In Section 8, we shall use such diatomic (bonding) probabil-ity weights, when χi∈A and χj∈B, in determining the effective IT descriptors ofchemical interactions in diatomic fragments of the molecule.

5. POPULATIONAL DECOUPLING OF ATOMIC ORBITALS

The previous formulation of CTCB in atomic resolution was shown to fail topredict a steady decrease in the resultant bond order with increasing occu-pation of the antibonding MO [9, 43–45]. The same shortcoming is observedin the fixed-input OCT. For example, in the N = 3, electron system describedby the two-AO model, [M(3)] = [ϕ2

bond. ϕ1anti.], one obtains S = 0.47, I = 0.48,

and N = 0.95. Therefore, despite a half occupation of ϕanti., MO the overallbond multiplicity remains almost the same as in the completely bondingconfiguration of the two-electron system [M(2)] = [ϕ2

bond.]. Moreover, thisprobabilistic approach cannot distinguish between the two bonding config-urations for N = 1, [M(1)] = [ϕ1

bond.], and N = 2, [M] = [ϕ2bond.], predicting the

same bond indices, reported in Scheme 1.1a. Similarly, for the total popu-lation decoupling in the N = 4 electron system, [M(4)] = [ϕ2

bond. ϕ2anti.], one

predicts S = 0, I = N = 1. This is because the probabilistic models loose the“memory” about the relative phases of AO in MO [43–45], which is retainedby the elements of the quantum-mechanical CBO matrix and density of thenonadditive Fisher information [34–38]. Therefore, in this approach onlythe covalent index reflects the nonbonding (noncommunicating) status ofAO in this limit. This diagnosis indicates a need for introducing into theMO-resolved scheme the information about the bonding/antibonding char-acter of specific (occupied) MO, which is not reflected by their condensedelectron probabilities in atomic resolution.

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Let us now put to the test the performance of the flexible-input chan-nels, which were shown to properly account for the MO shape decoupling.In the limiting case of the complete population decoupling in the two-AOmodel, when both doubly occupied basis functions remain effectively non-bonding in the molecule, γ = 2I and hence P(b|a) = I. Therefore, the twocompletely occupied AOs remain effectively closed (noncommunicating anddecoupled) for any level of their mixing measured by the AO-probabilityparameter P. Again, the input-dependent probabilities separately exhibitthe unit probability norm, completely localized on a single orbital: p(a) =p(a0) = (1, 0) and p(b) = p(b0) = (0, 1). Thus, this scheme correctly predictsthe nonbonding (nb) character of such a hypothetical electronic structure:Snb= Inb

= N nb= 0. Obviously, the same result follows from the flexible-

input contributions to the system average entropy/information descriptors.However, the problem of distinguishing between the two bonding cases, ahalf-bond for N = 1 and the full single bond for N = 2, still remains, andthe descriptors of the N = 3 channel also grossly contradict the chemicalintuition.

This failure to properly reflect the intuitive MO-population trends by theIT bond indices calls for a thorough revision of the hitherto used overallcommunication channel in AO resolution, which combines the contributionsfrom all occupied MOs in the electron configuration in question. Instead, onecould envisage a use of the separate MO channels introduced in Section 2(Eq. [7b]). As an illustration, let us assume for simplicity the two-AO modelof the chemical bond A–B originating from the quantum-mechanical interac-tion between two AOs: χ = (a ∈ A, b ∈ B). The bond contributions betweenthis pair of AO in the information system of sth MO,

Sa,b(ϕs) = S[Ps(b |a )],

Ia,b(ϕs) = H[p0s ]− S(ϕs),

Na,b(ϕs) = Sa,b(ϕs)+ Ia,b(ϕs) = H[p0s ], (31)

would then be straightforwardly recognized as bonding (positive), whenγa,b(ϕs) > 0, or antibonding (negative), when γa,b(ϕs) < 0, and nonbonding(zero), when γa,b(ϕs) = 0. Here, p0

s denotes the input probability in the ϕs infor-mation channel. Alternatively, the purely molecular estimate of the mutualinformation Is[ps :ps] can be used to index the localized bond ionicity.

In combining such MO contributions into the corresponding resultantbond indices for the specified pair (i, j) of AO, these increments shouldbe subsequently multiplied by the MO-occupation factor f MO

={ fs = ns/2},which recognizes that the full bonding/antibonding potential of the givenMO is realized only when it is completely occupied, and by the correspond-ing MO probability PMO

={Ps=ns/N}. The resultant A–B descriptors wouldthen be obtained by summation of such occupation/probability-weightedbonding or antibonding contributions from all occupied MOs, which

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determine the system chemical bonds:

S(i, j) =∑

s

sign[γi, j(ϕs)]PsfsSi, j(ϕs), I(i, j) =∑

s

sign[γi, j(ϕs)]PsfsIi, j(ϕs),

N(i, j) =∑

s

sign[γi, j(ϕs)]PsfsNi, j(ϕs). (32)

As shown in Scheme 1.3, these resultant IT indices from the MO-resolvedOCT do indeed represent adequately the population-decoupling trends forN= 1÷ 4 electrons in the two-AO model.

Consider now another model system of the π electrons in allyl, with theconsecutive numbering of 2pz = z orbitals in the carbon chain. In the Huckelapproximation, it is described by two occupied (canonical) MOs:

ϕ1 =1√

2

[1√

2(z1 + z3)+ z2

](doubly occupied) and

ϕ2 =1√

2(z1 − z3) (singly occupied), (33)

which generate the corresponding MO and molecular CBO matrices,

γ1 =12

1√

2 1√

2 2√

21√

2 1

, γ2 =12

1 0 −10 0 0−1 0 1

,

γ = γ1 + γ2 =12

2√

2 0√

2 2√

20√

2 2

, (34)

and the molecular information system shown in Scheme 1.4. The correspond-ing MO information systems, generated by the partial CBO matrices {γs},using the MO-input probabilities of AO, ps = {p(i|s) = γij(s)/ns}, are reportedin Scheme 1.5; their normalization requires that

∑i p(i|s) = 1.

It follows from Eqs. (2b, 7a, and 7a) that there are no analytical combina-tion formulas [9] for grouping the partial MO bond indices of Scheme 1.5 intotheir overall analogs of Scheme 1.4. Indeed, the MO channels are determinedby their own CBO structure, and a variety of their nonvanishing communica-tion connections between AOs generally differ from that for the system as awhole. Moreover, the input (conditional) probabilities used in Scheme 1.5 donot reflect the two MO channels being a part of the whole molecular channel.The latter requirement is only satisfied when the two networks are paral-lely coupled [42] into the combined information system, in which the inputprobabilities are given by the corresponding products {ps = Psps}, wherethe MO probabilities PMO

= {Ps} = (2/3, 1/3). In allyl such molecular inputsgive the following IT descriptors of the two MO channels: S1 = P1S1 = 1,

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(a)γb = nb

[P

√PQ

√PQ Q

]S(ϕb) = H(P)

P a P a PQP

Q b Q b QI(ϕb)=H

( 12

)−H(P) = 1−H(P)

I(ϕb) = S(ϕb)+ I(ϕb) = 1

(b)γa = na

[Q −

√PQ

−√PQ P

]S(ϕa) = H(P)

Q a Q a QPQ

P b P b PI(ϕa)=H

( 12

)−H(P) = 1−H(P)

N(ϕa) = S(ϕa)+ I(ϕa) = 1

(c)ϕa γ =

[P

√PQ

√PQ Q

]P(b|a) =

[P QP Q

]PMO= (1, 0)

ϕb S=12 S(ϕb)=

( 12

)H(P) I=

12 I(ϕb)=

( 12

)[1− H(P)] N =

12 N(ϕb) =

12

ϕa γ = 2[

P√PQ

√PQ Q

]P(b|a) =

[P QP Q

]PMO= (1, 0)

ϕb S= S(ϕb)=H(P) I= I(ϕb)= [1− H(P)] N = N(ϕb) = 1

ja γ =

[2P+Q

√PQ

√PQ 2Q+ P

]P(b|a)=

[(P+ 1)2/(3P+ 1) PQ/(3P+ 1)QP/(3Q+ 1) (Q+ 1)2/(3Q+ 1)

]PMO=( 2

3 , 13

)jb S=

( 23

)S(ϕb)−

( 16

)S(ϕa)=

12H(P) I=

( 23

)I(ϕb)−

( 16

)I(a)=

( 12

)[1−H(P)]

N=( 2

3

)N(ϕb)−

( 16

)N(ϕa)=

12

ja γ =[

2 00 2

]P(b|a) =

[1 00 1

]PMO=( 1

2 , 12

)jb S=

( 12

)S(ϕb)−

( 12

)S(ϕa)= 0 I=

( 12

)I(ϕb)−

( 12

)I(ϕa)= 0

N=( 1

2

)N(ϕb)−

( 12

)N(ϕa) = 0

Scheme 1.3 Decoupling of atomic orbitals in the MO-resolved OCT (2-AO model) withincreasing occupation of the antibonding combination of AO. Panels a and b sum-marize the bonding and antibonding channels, while Panel c reports the associatedprobability/occupation-weighted indices.

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p, p0 p*

1/3

1/3

1/31/4

1/41/3

2/3 z1z1

1/2 z2z2

11/36

1/3 2/3 z3z3 11/36

7/18 S = 1.11 I = 0.46 N = 1.58

Scheme 1.4 The molecular information channel of π electrons in allyl and its overall ITbond indices.

j2:

1/2 z1 z11/2 1/2

1/2

1/2

1/2 1/2 1/2z3 z3

I2= 0S2= 2= 1

p2 p2P2(b|a)p1 p1

1/4 z1 z11/4 1/4

j1:

1/2

1/4

1/41/4

1/41/4

z2 z21/21/2

1/2

1/2

z3 z31/4 1/4

S1= 1= 3/2 I1= 0

P1(b|a)

Scheme 1.5 The molecular π -electron information systems for two occupied MOs in allyl(Eq. [33]). The corresponding MO bond indices (in bits) are also reported.

I1 = −P1log2P1 = 0.39; S2 = P2S2 =13 , I2 = −P2log2P2 = 0.53. Such molecular

inputs thus generate the nonvanishing IT ionicities, which sum up to thegroup entropy I = I1 + I2 = H[PMO] = −

∑s Pslog2Ps = 0.92.

One then observes that the overall index of Scheme 1.4,N = 1.58 = H[p0],predicting about 3/2 π -bond multiplicity in allyl, can be reconstructed byadding to this additive-ionicity measure, the sum of the bonding (posi-tive) entropy-covalency S1 of the first MO and the antibonding (negative)contribution (−S2) due to the second MO:

S1+ (−S2)+ I = N . (35)

One also notices that the population-weighting procedure of Scheme 1.3,with f1 = 1 and f2 = 1/2, gives a diminished bond multiplicity:

N = f1P1S1 − f2P2S2+ ( f1 I1 − f2 I2) = f1(S1 + I1)− f2(S2 + I2)

≡ f1N1 − f2N2 = 0.96, (36)

thus predicting roughly a single π bond in allyl. The latter result reflectsthe fact that only a single-bonding MO, ϕ1, is completely occupied, whereasthe antibonding combination ϕ2 of AO on peripheral carbon atoms remainspractically nonbonding.

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In the same Huckel approximation the delocalized π bonds in butadieneare determined by two doubly occupied canonical MOs with PMO

=(

12 , 1

2

)and f MO

= (1, 1),

ϕ1 = a(z1 + z4)+ b(z2 + z3), ϕ2 = b(z1 − z4)+ a(z2 − z3), 2(a2+ b2) = 1;

a =12

√1−

1√

5= 0.3717, b =

12

√1+

1√

5= 0.6015. (37)

The corresponding CBO matrices,

γ1= 2

a2 ab ab a2

ab b2 b2 abab b2 b2 aba2 ab ab a2

, γ2= 2

b2 ab −ab −b2

ab a2−a2

−ab−ab −a2 a2 ab−b2

−ab ab b2

,

γ=1√

5

√

5 2 0 −12√

5 1 00 1

√5 2

−1 0 2√

5

, (38)

generate the associated AO-information channels as shown in Schemes 1.6and 1.7.

1/4

1/4

1/4

1/4

1/4

1/4

1/4

1/4

2/5

2/5

2/5

2/5

1/10

1/10

1/10

1/101/2

1/2

1/2

1/2

z1

z2

z3

z4

z1

z2

z3

z4

S = 1.36 I = 0.64 = 2

Scheme 1.6 The overall π -electron channel in OCT for butadiene derived from the HuckelMO.

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j1: z1a2

a2

p(i|1)b2

b2zi

z2

z3

z4

p1 = {p(i|1)} = (a2 , b2 , b2 , a2)

a2= 0.1382, b2

= 0.3818

S1 = N1 = 1.85, I1 = 0

j2: z1b2

b2

p(i|2)a2

a2zi

z2

z3

z4

p2 = {p(i|2)} = (b2 , a2 , a2 , b2)

S2 = N2 = 1.85, I2 = 0

Scheme 1.7 Probability scattering in the Huckel π -MO channels of butadiene for therepresentative input orbital zi = 2pz,i and the associated MO entropies.

The overall data correctly predict the resultant double multiplicity ofall π bonds in butadiene. In the one-electron OCT treatment, they exhibitrather substantial IT ionicity [48], which indicates a high degree of deter-minism (localization) in the orbital probability scattering, compared with theprevious two-electron approach [9]. A reference to the preceding equationindicates that a half of the reported entropy for ϕ2 is associated with theantibonding interactions between AOs, as reflected by the negative values ofthe corresponding elements in the MO CBO matrix. Therefore, the bondingand antibonding components in S2 cancel each other, when one attributesdifferent signs to these AO contributions. The group ionicity I = I1 + I2 =

H[PMO] = 1 and hence Eq. (35) now reads S1 +(

12 S2 −

12 S2

)+ I = 1.925, where

Ss = PsSs, thus again predicting roughly two π bonds in the system.In the Huckel theory the three occupied MO, which determine the π bonds

in benzene, PMO=

13 1, where 1 stands for the unit row matrix, read

ϕ1 =1√

6(z1 + z2 + z3 + z4 + z5 + z6),

ϕ2 =12(z1 + z2 − z4 − z5),

ϕ3 =1√

12(z1 − z2 − 2z3 − z4 + z5 + 2z6). (39)

They give rise to the overall CBO matrix elements reflecting the π -electronpopulation on orbital χi = zi, γi,i = z1, γi,i = 1, and the chemical couplingbetween χi and its counterparts on carbon atoms in the relative ortho-, meta-,and para-positions, respectively, γi,i+1 = 2/3, γi,i+2 = 0, γi,i+3 = −1/3. The resul-tant scattering of AO probabilities for the π electrons in benzene is shown in

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zi1/22/9 zi + 1

1/6 zi 0zi + 2

1/18zi + 3

S = 1.70 I2 = 0.89 N = 2.58

Scheme 1.8 The probability scattering in benzene (Huckel theory) for the representativeinput orbital zi= 2pz,i and the associated OCT entropy/information descriptors.

Scheme 1.8. The overall bond multiplicity is somewhat lower thatN = 3 pre-dicted for the three localized π bonds in cyclohexatriene since in benzene, theπ -bond alternation is prevented by the stronger σ bonds, which assume themaximum strength in the regular hexagon structure [64–67].

All matrix elements in γ1= 2〈χ |P1|χ〉= (13 )1, where χ = (z1, z2, z3, z4, z5, z6)

and all elements in the square matrix 1 are equal to 1, are positive (bonding),whereas half of them in γ2 and γ3 is negative, thus representing the antibond-ing interactions between AOs. The nonvanishing elements in γ2 are limitedto the subset χ ′ = (z1, z2, z4, z5):

γ2= 2〈χ ′|P2|χ

′〉 =

12

1 1 −1 −11 1 −1 −1−1 −1 1 1−1 −1 1 1

, (40)

while γ3 explores the whole basis set χ :

γ3 = 2 〈χ | P3|χ〉 =16

1 −1 −2 −1 1 2−1 1 2 1 −1 −2−2 2 4 2 −2 −4−1 1 2 1 −1 −2

1 −1 −2 −1 1 22 −2 −4 −2 2 4

. (41)

These CBO matrices of the occupied MO give rise to the followingcommunications and input probabilities in the associated MO channels:

P1(b|a) =16

1, p1 =16

1; P2(b|a) =14

1 1 0 1 1 01 1 0 1 1 00 0 0 0 0 01 1 0 1 1 01 1 0 1 1 00 0 0 0 0 0

,

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p2 =14(1, 1, 0, 1, 1, 0); P3(b|a) =

112

1 1 4 1 1 41 1 4 1 1 41 1 4 1 1 41 1 4 1 1 41 1 4 1 1 41 1 4 1 1 4

,

p3 =1

12(1, 1, 4, 1, 1, 4). (42)

The corresponding entropy/information descriptors then read as follows:

S1 = N1 = 2.58, I1 = 0; S2 = N2 = 2, I2 = 0; S3 = N3 = 2.25, I3 = 0. (43)

The group ionicity I = I1 + I2 + I3 = H[PMO] = 1.58 and S1 = S1/3 then alsogives rise to roughly (2.5)-bond multiplicity, with the bonding (positive) andantibonding (negative) contributions in S2 and S3 approximately cancelingeach other.

6. BOND DIFFERENTIATION IN OCT

It has been demonstrated elsewhere that the bond alternation effects arepoorly represented in both the CTCB formulated in atomic resolution [9]and in its OCT (fixed-input) extension [48]. The OCT indices from the alter-native output reduction schemes have been shown to give more realisticbut still far from satisfactory description of the bond alternation trends inthese molecular systems [48]. This is because in purely probabilistic models,the bonding and antibonding interactions are not distinguished since con-ditional probabilities (squares of the MO-CBO matrix elements) loose theinformation about the relative phases of AO in MO. However, this distinctionis retained in the off-diagonal CBO matrix elements, particularly in the sepa-rate CBO contributions {γs} from each occupied MO. Since the OCT analysisof the bonding patterns in molecules provides the supplementary, a posterioridescription to the standard MO scheme in this section we shall attempt to usethis extra information, directly available from the standard SCF MO calcula-tions, to generate more realistic “chemical” trends of the π -bond alternationpatterns in the three illustrative systems of the preceding section.

The problem can be best illustrated using the simplest allyl case. As dis-cussed elsewhere [9, 22], the entropy/information indices for the given pairof atomic orbitals can be extracted from the relevant partial channel, whichincludes all AO inputs (sources of the system chemical bonds) and the two-orbital outputs in question, defining the localized chemical interaction ofinterest. In Scheme 1.9, two examples of such partial information systems are

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1/31/3

(a)

1/3

1/3

2/3

1/2

11/36

7/18

z1

z2

z1

z2

z3

1/4

1/3

1/4

1/4

11/36

11/36

2/3

2/3

1/3

1/3

1/3

z1

z2

z3

z1

z3

(b)

S(1, 2) = 0.82 I(1, 2) = 0.24 N (1, 2) = 1.05

S(1, 3) = 0.59 I(1, 3) = 0.45 N (1, 3) = 1.05

Scheme 1.9 The molecular partial information channels and their entropy/informationdescriptors of the chemical interaction between the adjacent (Panel a) and terminal (Panel b)AO in the π -electron system of allyl.

displayed for the nearest neighbor (z1, z2) and terminal (z1, z3) chemical inter-actions. They have been obtained from the molecular channel of Scheme 1.4,by removing communications involving the third, remaining AO of this min-imum basis set of π AO. It follows from these illustrative sets of indices thatthe two partial channels give rise to identical overall index N , with onlythe IT-covalent/ionic components differentiating the two bonds: the nearestneighbor interaction exhibits a higher “noise” (covalency) component andhence the lower information-flow (ionicity) content. In the Huckel theorythe corresponding partial information systems in the butadiene π -electronsystem predict identical indices for all pairs of orbitals, S(i, j) = 0.68, I(i, j) =0.25, and N (i, j) = 0.93, thus failing completely to account for the π -bondalternation.

To remedy this shortcoming of the communication theory, one has tobring into play the known signs of interactions between the specified pair(i, j) of AO in the given MO ϕs, in order to recognize them as bonding(exhibiting a “constructive” interference), γi, j(s) > 0, or antibonding (involv-ing a “destructive” interference), γi, j(s) < 0, with γi, j(s) = 0 correspondingto the nonbonding (zero communication) case. The MO-resolved channelsare vital for the success of such an approach since the bonding interactionbetween the given pair of AOs in one MO can be accompanied by the anti-bonding interaction between these basis functions in another occupied MO.This extraneous information determines the signs of contributions in theweighted contributions of Eqs. (31 and 32) from the partial MO channels,including the two specified orbitals in their input and output, and usingthe fragment-renormalized MO probabilities [9, 26]. It should be observedthat in the flexible-input approach of Section 4 the nonbonding AOs, which

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1/3 1/4 1/4

1/2

1/4

z1 z1

2/3 1/2 1/2z2 z2

(a) z1–z2:

ϕ1: γ1, 2(ϕ1) > 0

S1,2(ϕ1) = 1 I1,2(ϕ1) = 0 N1,2(ϕ1) = 1

S(1, 2) = 2/3 I(1, 2) = 0 N(1, 2) = 2/3

1/2

1/2

1/2

1/41/4

1/41/4

1/21/2

1/2

1/4

1/4

1/4

z3 z3

z1 z1

z1 z1

1/2 1/2 1/2z3 z3

(b) z1–z3:

ϕ1: γ1, 3(ϕ1) > 0

S1,3(ϕ1) = 1 I1,3(ϕ1) = 1 N1,3(ϕ1) = 1

ϕ2: γ1, 3(ϕ2) < 0

S1,3(ϕ2) = 1 I1,3(ϕ2) = 0 N1,3(ϕ2) = 1

S(1, 3) = 2/3− 1/6 = 1/2 I(1, 3) = 0 N(1, 3) = 1/2

Scheme 1.10 The partial MO-information channels and their entropy/information descrip-tors of the chemical interactions between the nearest neighbor (Panel a) and terminal(Panel b) AO in the π -electron system of allyl.

communicate only with themselves, gives rise to the separate AO channelsof Scheme 1.1b, thus not contributing to the resultant bond descriptors.

An illustrative application of such scheme to π electrons in allyl, forwhich PMO

=(

23 , 1

3

), f MO

=(1, 1

2

), p1 =

(14 , 1

2 , 14

), and p2 =

(12 , 0, 1

2

), is reported

in Scheme 1.10. One observes that z1–z2 interaction has only the bonding con-tribution from ϕ1, while the effective z1–z3 interaction combines the bondingcontribution due to ϕ1 and the antibonding increment originating from ϕ2.This scheme is seen to generate (2/3)-bond multiplicity between the near-est neighbors and a weaker half-bond between the terminal carbon atoms.This somewhat contradicts the Wiberg’s covalency indices predicting a halfz1–z2 bond and a vanishing z1–z3 interaction. The reason for a finite value ofthis bond index in OCT is the dominating delocalization of electrons in ϕ1

throughout the whole π system.Let us similarly examine the localized π interactions in butadiene, for

which PMO=(

12 , 1

2

), f MO

= (1, 1), p1 = (a2, b2, b2, a2), and p2 = (b

2, a2, a2, b2). Areference to Eq. (38) indicates that the equivalent terminal pairs of AO,z1–z2 and z3–z4, exhibit only the bonding interactions in ϕ1 and ϕ2, whilethe remaining AO combinations involve the bonding contribution from ϕ1

and the antibonding from ϕ2. These MO increments are summarized inScheme 1.11 (see also Scheme 1.7).

These diatomic IT indices predict the strongest terminal (1–2) or (3–4) πbonds, which exhibit somewhat diminished bond multiplicity to about 92%

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of the unit value in ethylene, and the vanishing bond orders of the (1–3)and (2–4) interactions. The middle (2–3) π bond measures about 14% of theethylene reference value, while the chemical interaction between terminalcarbons (1–4) is diagnosed as being antibonding in character, in full confor-mity with the negative value of the corresponding off-diagonal element inthe overall CBO matrix (Eq. [38]). These predictions should be comparedwith the associated quadratic indices Mi, j= γ

2i, j of Wiberg, M1,2=M3,4= 0.8,

M1,3=M2,4= 0, and M1,4=M2,3= 0.2, which unrealistically equates the partialbonding (2–3) and antibonding (1–4) interactions.

As final example let us reexamine from the present perspective a differ-entiation of the localized π -bonds between the two carbon atoms in therelative ortho-, meta- and para-positions in benzene [9, 48]. This weightedMO approach makes a separate use of the diatomic parts of the canonicalMO channels, with the bonding and antibonding contributions identified bythe signs of the corresponding coupling elements in the MO density matri-ces {γs}. It should be realized that while the canonical (delocalized) MOcompletely reflect the molecular symmetry, its diatomic fragments do not.Therefore, the bond indices generated in this scheme must exhibit some dis-persions so that they have to be appropriately averaged with respect to theadmissible choices of the corresponding orbital pairs to ultimately generatethe invariant entropy/information descriptors of the ortho-, meta-, and paraπ bonds in benzene. We further observe that in this π system, PMO

=(

13 , 1

3 , 13

)and f MO

= (1, 1, 1).Scheme 1.12 summarizes the elementary entropy/information increments

of the diatomic bond indices generated by the MO channels of Eq. (42). Theygive rise to the corresponding diatomic descriptors, which are obtained fromEq. (32). For example, by selecting i= 1 of the diatomic fragment consistingadditionally the j= 2, 3, 4 carbon, one finds the following IT bond indices:

S(1, 2)=N(1, 2)= 0.42, S(1, 3)=N(1, 3)= 0.01,

S(1, 4)=N(1, 4)= −0.25.

These predictions correctly identify the bonding, a practically nonbonding,and the antibonding characters of π bonds between two carbons of the ben-zene ring in the relative ortho-, meta-, and para-positions, respectively, asindeed reflected by the overall CBO matrix elements. However, due to anonsymmetrical (fragment) use of the symmetrical MO channels, these pre-dictions exhibit some dispersions when one explores other pairs of carbonatoms in the ring, giving rise to the following average descriptors:

S(ortho)=N(ortho)= 0.52, S(meta)=N(meta)= 0.06,

S(para)=N(para)= − 0.19.

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2a2

2b2

2b2

2a2

z1

z1

z2(3)

z2(3)

z1

z1

z2(3)

z2(3)

(a) z1–z2, z1– z3:

b2

b2

b2

b2

a2

a2

a2

b2

b2

a2

a2

a2

ϕ1: γ1,2(ϕ1) > 0, γ1,3(ϕ1) > 0

S1,2(ϕ1) = N1,2(ϕ1) = 0.925 I1,2(ϕ1) = 0

S1,3(ϕ1) = N1,3(ϕ1) = 0.925 I1,3(ϕ1) = 0

ϕ2: γ1,2(ϕ2) > 0, γ1,3(ϕ2) < 0

S1,2(ϕ2) = N1,2(ϕ2) = 0.925 I1,2(ϕ2) = 0

S1,3(ϕ2) = N1,3(ϕ2) = 0.925 I1,3(ϕ2) = 0

S(1, 2)=N(1, 2)= 0.925, I(1, 2) = 0 S(1, 3) = N(1, 3) = 0, I(1, 3) = 0;

1/2

1/2

1/2

1/2

z1

z1

z4

z4

z1

z1

z4

z4

(b) z1–z4:

a2

a2

b2

b2

a2

a2

b2

a2

b2

a2

b2

b2

ϕ1: γ1, 4(ϕ1) > 0

S1,4(ϕ1) = N1,2(ϕ1) = 0.789 I1,4(ϕ1) = 0

ϕ2: γ1, 4(ϕ2) < 0

S1,4(ϕ2) = N1,4(ϕ2) = 1.061 I1,4(ϕ2) = 0

S(1, 4) = N(1, 4) = −0.136, I(1, 2) = 0

1/2

1/2

1/2

1/2

z2

z2

z3

z3

z2

z2

z3

z3

(c) z2–z3:

b2

b2

a2

a2

b2

b2

a2

b2

a2

b2

a2

a2

ϕ1: γ2, 3(ϕ1) > 0

S2,3(ϕ1) = N2,3(ϕ1) = 1.061 I2,3(ϕ1) = 0

ϕ2: γ2, 3(ϕ2) < 0

S2,3(ϕ2) = N2,3(ϕ2) = 0.789 I2,3(ϕ2) = 0

S(2, 3) = N(2, 3) = 0.136, I(2, 3) = 0

Scheme 1.11 The partial MO-information channels and their entropy/information descrip-tors for the two-orbital interactions in the π -electron system of butadiene.

The above ortho result shows that the overall IT bond multiplicity betweenthe nearest neighbors N(ortho) ∼ 0.5 is indeed compromised in benzene,compared with N = 1 in ethylene, due to the effect of the prohibited bondalternation, enforced by the stronger σ bonds [64–67]. Again, the magnitudesof these IT indices generally agree with the corresponding Wiberg indices:Mortho= 0.44, Mmeta= 0, and Mpara= 0.11. Note, however, that OCT properlyrecognizes the para interactions in benzene as antibonding, whereas in theWiberg scheme, this distinction is lost.

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zi zi1/6

1/61/61/6

1/61/2

zj zj 1/61/2

(zi ≠ zj) ∈(z1, z2, z3, z4, z5, z6)

(a) j1:

γi, j(ϕ1) > 0

Si, j(ϕ1) = Ni, j(ϕ1) = 0.862 Ii, j(ϕ1) = 0

zi zi1/4

1/41/4

1/41/41/2

zj zj 1/41/2

(zi ≠ zj) ∈(z1, z2, z4, z5)

(b) j2:

γ1, 2(ϕ2) = γ4,5(ϕ2) > 0

γ1,4(ϕ2) = γ1,5(ϕ2) = γ2,4(ϕ2) = γ2,5(ϕ2) < 0

Si, j(ϕ2) = Ni, j(ϕ2) = 1 Ii, j(ϕ2) = 0

z3 z31/3

1/31/31/3

1/31/2

z6 z6 1/31/2

(zi ≠ zj) ∈(z3, z6)

(c) j3:

zi zi1/12

1/121/121/12

1/121/2

zj zj 1/121/2

(zi≠ zj) ∈(z1, z2, z3, z4, z5)

zi zi1/12

1/31/121/3

1/121/5

zj zj 1/34/5

zi ∈(z3, z6), zj ∈(z1, z2, z4, z5)

γ3,6(ϕ3) < 0

S3,6(ϕ3) = N3,6(ϕ3) = 1.057 I3,6(ϕ3) = 0

γ1,2(ϕ3) = γ1,4(ϕ3) = γ2,5(ϕ3) = γ4,5(ϕ3) < 0γ1,5(ϕ3) = γ2,4(ϕ3) > 0

Si, j(ϕ3) = Ni, j(ϕ3) = 0.597 Ii,j(ϕ3) = 0

γ1,3(ϕ3) = γ2,6(ϕ3) = γ3,5(ϕ3) = γ4,6(ϕ3) < 0γ1,6(ϕ3) = γ2,3(ϕ3) = γ3,4(ϕ3) = γ5,6(ϕ3) > 0

Si, j(ϕ3) = Ni, jϕ3) = 0.827 Ii, j(ϕ3) = 0

Scheme 1.12 The elementary entropy/information contributions to chemical interactionsbetween two different AOs in the minimum basis set {zi = 2pz,i} of the π -electron system inbenzene.

7. LOCALIZED σ BONDS IN COORDINATION COMPOUNDS

The decoupled description of hydrides (Section 3) can be naturally extendedinto the localized σ bonds between the central atom/ion X and the coordi-nated ligands {Lα}, for example, in the coordination compounds of transitionmetal ions or in SF6. Consider, for example, the octahedral complex XL6 withthe ligands placed along the axes of the Cartesian coordinate system: {L1(e),L2(e)}, e= x, y, z. The X–Lα bond then results from the chemical interaction

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between six acceptor (partially occupied) d2sp3 hybrids {Hα} of X and thecorresponding donor (doubly occupied) {σα} orbitals of ligands. The corre-sponding localized MO, which determines the communication channel ofthe separate bond M–Lα, α= 1, 2, . . . , 6, now include the (doubly occupied)bonding MO ϕb(α), with the two electrons originating from the donor σαorbital, nb=Nσ (α)= 2, and the antibonding MO ϕa(α), in general partly occu-pied with na=NX(α) electrons originating from X, which result from the twobasis functions χα = (Hα, σα):

ϕb(α)=√

PHα +

√Qσα, ϕa(α)= −

√QHα +

√Pσα, P+Q= 1. (44)

The associated CBO matrix elements and the corresponding conditionalprobabilities they generate now depend on the initial number of electrons na

on Hα, which are contributed by X to the αth σ bond (see also Scheme 1.3),

γHα ,Hα = 2P+ naQ, γσα ,σα = 2Q+ naP, γHα ,σα = γσα ,Hα = (2− na)√

PQ. (45)

Indeed, na= 0, for example, in SF6, determines the maximum value of themagnitude of the coupling CBO element γHα , σα = γσα , Hα = 2

√PQ, and na= 1

diminishes it by a factor of 2, while the double occupation of ϕa(α) givesrise to the nonbonding state corresponding to the separate, decoupledsubchannels for each orbital,

γHα ,Hα = γσα ,σα = 2 and γHα ,σα = γσα ,Hα = 0, (46)

which do not contribute to the entropy/information indices of the localizedchemical bond.

For na= 0, that is, the empty antibonding MO, when X–Lα channel isgiven by Scheme 1.1a, the IT bond indices correctly predict the overall ITmultiplicity reflecting the six decoupled bonds in this molecular system:

S(P)=∑α

Sα(bα|aα)= 6H(P),

I(P)=∑α

Iα(a0α

: bα)= 6[1−H(P)], N= 6. (47)

The highest IT covalency of the σ bond M–Lα, Smax.= 1, predicted for

the strongest mixing of orbitals P=Q= 1/2, is thus accompanied by thevanishing IT ionicity, Imax.

= 0.The corresponding conditional probabilities Pα(bα|aα) ≡ Pα[χα|χα] for the

single and double occupations of ϕa(α) are reported in the corresponding dia-grams of Scheme 1.3c. It follows from these expressions that in the latter case

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0.9 HH

(a)

(b)

0.1σσ

12 0.1

0.912

12

12

0.9 HH

0.1σσ

13 0.1

0.923

11/30

19/30

P(b|a) pp

P(b|a) p∗p0

Smax.= 0.469

Imax.= 0.479

Nmax.= 0.948

Scheme 1.13 The orbital-communication channels for the localized M–Lα bond in thefixed-input approach, for P=Q= 1/2, and the singly occupied antibonding MO: covalent(molecular input; Panel a) and ionic (promolecular input; Panel b).

the off-diagonal elements identically vanish, γHα ,σα = γσα ,Hα = 0, thus givingrise to the decoupled pair of orbitals and hence to the deterministic chan-nel of Scheme 1.1b for each of them (see the fourth diagram in Scheme 1.3c).Therefore, such separate channels do not contribute to the overall IT bonddescriptors.

For the partly bonding, open-shell configuration na= 1 (the third dia-gram in Scheme 1.3c) and the maximum covalency combination P=Q= 1/2,one obtains a strongly deterministic information system as shown inScheme 1.13. It follows from these diagrams that the fixed-input approachpredicts a practically conserved overall bond order compared with the na= 0case (the second diagram in Scheme 1.3c), with the bond weakening beingreflected only in the bond composition with now roughly equal (half-bond)covalent and ionic components.

As already discussed in Scheme 1.3, the populational decoupling trends ofAO in the coordination bond are properly reflected only in the flexible-input(MO-resolved) description, which recognizes the bonding and antibondingcontributions to the resultant bond multiplicity from the signs of the cor-responding CBO matrix elements of the system-occupied MO. It should beemphasized, however, that such treatment ceases to be purely probabilisticin character since it uses the extraneous piece of the CBO information, whichis lost in the conditional probabilities.

8. RESTRICTED HARTREE–FOCK CALCULATIONS

In typical SCF-LCAO-MO calculations the lone pairs of the valence and/orinner shell electrons can strongly affect the IT descriptors of the chemi-cal bond. Therefore, the contributions due to each AO input should be

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appropriately weighted (see Eqs. [28 and 29] in Section 4) using the joint,two-orbital probabilities that reflect the actual participation of each AO inthe system chemical bonds. In this section we describe such an approach todiatomic chemical interactions in molecules and present numerical resultsfrom standard RHF calculations for a selection of representative molecularsystems.

8.1. Orbital and condensed atom probabilities ofdiatomic fragments in molecules

The molecular probability scattering in the specified diatomic fragment(A, B), involving AO contributed by these two bonded atoms, χAB= (χA,χB),to the overall basis set χ ={χX}, is completely characterized by the corre-sponding P(χAB|χAB) block [22, 26] of the molecular conditional probabilitymatrix of Eq. (4), which determines the molecular communication system inOCT [46–48] of the chemical bond:

P(χAB|χAB) ≡ [P(χY|χX); (X, Y)∈ (A, B)]

≡ {P(j|χAB);χj ∈χAB} ≡ {P(j|i); (χi,χj)∈χAB)}. (48)

Thus, the square matrix P(χAB|χAB) contains only the intrafragment commu-nications, which miss the probability propagations originating from AO ofthe remaining constituent atoms χZ /∈χAB.

The atomic output reduction of P(χAB|χAB) [9] gives the associated con-densed conditional probabilities of the associated molecular informationsystem,

P(XAB|χAB)= [P(A|χAB), P(B|χAB)]

=

{P(X|χAB) ≡ {P(X|i)}=

∑j∈X

P(j∣∣χAB);χi ∈ χAB, X=A, B

}, (49)

where P(Y|i) measures the conditional probability that an electron on χi willbe found on atom Y in the molecule. The sum of these conditional prob-abilities over all AOs contributed by the two atoms then determines thecommunication connections {P(A, B|i)}, linking the condensed atomic output(A, B) and the given AO input χi in the associated communication system ofthe diatomic fragment:

P(A|χAB)+ P(B|χAB)=P(A, B|χAB)

=

{P(A,B|i)=P(A|i)+ P(B|i)=

∑j∈(A,B)

P( j∣∣ i) ≤ 1

}. (50)

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In other words, P(A,B|i) measures the probability that the electron occu-pying χi will be detected in the diatomic fragment AB of the molecule.The inequality in the preceding equation reflects the fact that the atomicbasis functions participate in chemical bonds with all constituent atoms,with the equality sign corresponding only to a diatomic molecule,when χAB=χ .

The fragment-normalized AO probabilities

p(AB)={pi(AB)= γi,i/NAB}, NAB=

∑i∈(A,B)

γi,i,∑

i∈(A,B)

pi(AB)= 1, (51)

where NAB stands for the number of electrons in the specified diatomic frag-ment of the molecule and pi(AB) denotes the probability that one of themoccupies χi∈(A,B), then determine the simultaneous probabilities of the jointtwo-orbital events [47]:

PAB(χAB,χAB)={PAB(i, j)= pi(AB)P(j|i)= γi,jγj,i/(2NAB)}. (52)

They generate, via relevant partial summations, the joint atom-orbital prob-abilities in AB, {PAB(X, i)}:

PAB(XAB,χAB)= [PAB(A,χAB), PAB(B,χAB)]

=

{PAB(X, i)=

∑j∈X

PAB(i, j) ≡ pi(AB)P(X|i), X=A, B}

. (53)

For the closed-shell molecular systems one thus finds

PAB(X,χAB)=

{PAB(X, i)= pi(AB)

∑j∈X

P( j∣∣ i)=

∑j∈X

γi,jγj,i

2NAB

}, X=A, B. (54)

These vectors of AO probabilities in diatomic fragment AB subsequentlydefine the condensed probabilities {PX(AB)} of both bonded atoms in sub-system AB:

PX(AB)=NX(AB)

NAB=

∑i∈(A,B)

PAB(X, i)=∑

i∈(A,B)

∑j∈X

γi,jγj,i

2NAB, X=A, B, (55)

where the effective number of electrons NX(AB) on atom X=A, B reads:

NX(AB)=∑

i∈(A,B)

∑j∈X

γi,jγj,i

2. (56)

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Therefore, in diatomic molecules, for which χAB=χ , one finds using theidempotency relations of Eq. (3),

PX(AB)=∑j∈X

(∑i

γj,iγi,j

2NAB

)=

∑j∈X

γj,j

NAB=

∑j∈X

pj(AB), X=A, B, (57)

and hence PA(AB)+PB(AB)= 1. Clearly, the last relation does not hold fordiatomic fragments in larger molecular systems, when χAB 6= χ , so that ingeneral PX(AB) 6=

∑j∈X pj and

PA(AB)+ PB(AB) 6= 1. (58)

We finally observe that the effective orbital probabilities of Eqs. (52–54)and the associated condensed probabilities of bonded atoms (Eq. 55) do notreflect the actual AO participation in all chemical bonds in AB, giving riseto comparable values for the bonding and nonbonding (lone-pair) AO in thevalence and inner shells. The relative importance of basis functions of oneatom in forming the chemical bonds with the other atom of the specifieddiatomic fragment is reflected by the (nonnormalized) joint bond probabilitiesof the two atoms, defined by the diatomic components of the simultaneousprobabilities of Eqs. (52 and 53):

Pb(A, B) ≡∑i∈B

PAB(A, i)=∑i∈A

PAB(B, i)=Pb(B,A) =∑i∈A

∑j∈B

γi,jγj,i

2NAB. (59)

The underlying joint atom-orbital probabilities, {PAB(A, i), i ∈ B} and{PAB(B, i), i ∈ A}, to be used as weighting factors in the average conditional-entropy (covalency) and mutual-information (ionicity) descriptors of the ABchemical bond(s), indeed assume appreciable magnitudes only when theelectron occupying the atomic orbital χi of one atom is simultaneously foundwith a significant probability on the other atom, thus effectively excludingthe contributions to the entropy/information bond descriptors due to thelone-pair electrons. Thus, such joint bond probabilities emphasize of AOshave both atoms are simultaneously involved in the occupied MOs.

The reference bonding probabilities of AO have to be normalized to thecorresponding sums P(A, B|χAB)={P(A, B|i)} of Eq. (50). Since the bondprobability concept of Eq. (59) involves symmetrically the two bondedatoms, we apply the same principle to determine the associated referencebond probabilities of AO to be used to calculate the mutual-informationbond index:

{pb(i)=P(A, B|i)/2; i ∈ (A, B)}, (60)

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where P(A,B|i) denotes the probability that an electron originating fromorbital χi will be found on atom A or B in the molecule.

8.2. Average entropic descriptors of diatomicchemical interactions

As we have already mentioned in Section 2, in OCT the complementaryquantities characterizing the average noise (conditional entropy of the chan-nel output given input) and the information flow (mutual information in thechannel output and input) in the diatomic communication system defined bythe conditional AO probabilities of Eq. (48) provide the overall descriptors ofthe fragment bond covalency and ionicity, respectively. Both molecular andpromolecular reference (input) probability distributions have been used inthe past to determine the information index characterizing the displacement(ionicity) aspect of the system chemical bonds [9, 46–48].

In the A–B fragment development we similarly define the followingaverage contributions of both constituent atoms to the diatomic covalency(delocalization) entropy:

HAB(B|χA)=∑i∈A

PAB(B, i) H(χAB|i), HAB(A|χB)=∑i∈B

PAB(A, i) H(χAB|i),

(61)

where the Shannon entropy of the conditional probabilities for the given AOinput χi ∈ χAB= (χA, χB) in the diatomic channel:

H(χAB|i)= −∑

j∈(A,B)

P(j|i)log2P(j|i). (62)

In Eq. (61) the conditional entropy SAB(Y|χX) quantifies (in bits) the delocal-ization X→Y per electron so that the total covalency in the diatomic fragmentA–B reads as follows:

SAB=NAB[HAB(B|χA)+HAB(A|χB)]. (63)

Again, it should be emphasized that the simultaneous (diatomic) proba-bilities {PAB(X, i ∈ Y), Y 6= X}, used in Eq. (61) as weighting factors of thecorresponding contributions due to the specified input AO, effectively elim-inate contributions due to the inner- and valence-shell lone pairs, sincethese weighting factors reflect the actual orbital participation in the fragmentchemical bonds.

Accordingly, the probability-weighted contributions to the averagemutual-information quantities of bonded atoms are defined in reference to

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the unbiased bond probabilities of AO (Eq. [60]):

IAB(χA : B)=∑i∈A

PAB(B, i)I(χAB : i), IAB(χB : A)=∑i∈B

PAB(A, i)I(χAB : i),

I(χAB : i)=∑

j∈(A,B)

P(j|i)log2

(P(j|i)pb(i)

). (64)

They generate the total information ionicity of all chemical bonds in thediatomic fragment:

IAB=NAB[IAB(χA : B)+ IAB(χB : A)]. (65)

Finally, the sum of the above total (diatomic) entropy-covalency andinformation-ionicity indices determines the overall information-theoreticbond multiplicity in the molecular fragment in question:

NAB=SAB + IAB. (66)

They can be compared with the diatomic (covalent) bond order of Wiberg[52] formulated in the standard SCF-LCAO-MO theory,

MAB=

∑i∈A

∑j∈B

γ 2i,j=

∑i∈A

∑j∈B

Mi,j, (67)

which has been previously shown to adequately reflect the chemical intu-ition in the ground state of typical molecular systems. Such a comparison isperformed in Tables 1.1 and 1.2, reporting the numerical RHF data of bondorders in diatomic fragments of representative molecules for their equilib-rium geometries in the minimum (STO-3G) and extended (6-31G*) basis sets,respectively.

It follows from both these tables that the applied weighting proceduregives rise to an excellent agreement with both the Wiberg bond ordersand the chemical intuition. A comparison between corresponding entriesin Table 1.1 and the upper part of Table 1.2 also reveals generally weakdependence on the adopted AO representation, with the extended basis setpredictions being slightly closer to the familiar chemical estimates of thelocalized bond multiplicities in these typical molecules. In a series of relatedcompounds, for example, in hydrides or halides, the trends exhibited by theentropic covalent and ionic components of a roughly conserved overall bondorder also agree with intuitive expectations. For example, the single chemicalbond between two “hard” atoms in HF appears predominantly covalent,

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Table 1.1 Comparison of the diatomic Wiberg and entropy/information bondmultiplicity descriptors in selected molecules: the RHF results obtained in theminimum (STO-3G) basis set

Molecule A–B MAB NAB SAB IAB

H2 H–H 1.000 1.000 1.000 0.000F2 F–F 1.000 1.000 0.947 0.053HF H–F 0.980 0.980 0.887 0.093LiH Li–H 1.000 1.000 0.997 0.003LiF Li–F 1.592 1.592 0.973 0.619CO C–O 2.605 2.605 2.094 0.511H2O O–H 0.986 1.009 0.859 0.151AlF3 Al–F 1.071 1.093 0.781 0.311CH4 C–H 0.998 1.025 0.934 0.091C2H6 C–C 1.023 1.069 0.998 0.071

C–H 0.991 1.018 0.939 0.079C2H4 C–C 2.028 2.086 1.999 0.087

C–H 0.984 1.013 0.947 0.066C2H2 C–C 3.003 3.063 2.980 0.062

C–H 0.991 1.021 0.976 0.045C6H1

6 C1–C2 1.444 1.526 1.412 0.144C1–C3 0.000 0.000 0.000 0.000C1–C4 0.116 0.119 0.084 0.035

1 For the sequential numbering of carbon atoms in the benzene ring.

although a substantial ionicity is detected for LiF, for which both Wiberg andinformation-theoretic results predict roughly (3/2)-bond in the minimumbasis set, consisting of approximately one covalent and 1/2 ionic contri-butions; in the extended basis set, both approaches give approximately asingle-bond estimate, with the information theory predicting the ionic dom-inance of the overall bond multiplicity. The significant information-ionicitycontribution is also detected for all halides in the lower part of Table 1.2. Onealso finds that all carbon–carbon interactions in the benzene ring are prop-erly differentiated. The chemical orders of the single and multiple bonds inethane, ethylene, and acetylene are also properly reproduced, and the triplebond in CO is accurately accounted. Even more subtle bond differentiationeffects are adequately reflected by the present information-theoretic results.The differentiation of the “equatorial” and “axial” S–F bonds in the irregu-lar tetrahedron of SF4 is reproduced, and the increase in the strength of thecentral bond in propellanes with increase of sizes of the bridges is correctlypredicted [9].

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Table 1.2 The same as in Table 1.1 for the extended 6-31G* basis set

Molecule A–B MAB NAB SAB IAB

F2 F–F 1.228 1.228 1.014 0.273HF H–F 0.816 0.816 0.598 0.218LiH Li–H 1.005 1.005 1.002 0.004LiF Li–F 1.121 1.121 0.494 0.627CO C–O 2.904 2.904 2.371 0.533H2O O–H 0.878 0.896 0.662 0.234AlF3 Al–F 1.147 1.154 0.748 0.406CH4 C–H 0.976 1.002 0.921 0.081C2H6 C–C 1.129 1.184 1.078 0.106

C–H 0.955 0.985 0.879 0.106C2H4 C–C 2.162 2.226 2.118 0.108

C–H 0.935 0.967 0.878 0.089C2H2 C–C 3.128 3.192 3.095 0.097

C–H 0.908 0.943 0.878 0.065C6H1

6 C1–C2 1.507 1.592 1.473 0.119C1–C3 0.061 0.059 0.035 0.024C1–C4 0.114 0.117 0.081 0.035

LiCl Li–Cl 1.391 1.391 0.729 0.662LiBr Li–Br 1.394 1.394 0.732 0.662NaF Na–F 0.906 0.906 0.429 0.476KF K–F 0.834 0.834 0.371 0.463SF2 S–F 1.060 1.085 0.681 0.404SF4 S–Fa 1.055 1.064 0.670 0.394

S–Fb 0.912 0.926 0.603 0.323

SF6 S–F 0.978 0.979 0.726 0.254B2H2

6 B–B 0.823 0.851 0.787 0.063B–Ht 0.967 0.995 0.938 0.057B–Hb 0.476 0.490 0.462 0.028

Propellanes3

[1.1.1] Cb–Cb 0.797 0.829 0.757 0.072[2.1.1] Cb–Cb 0.827 0.860 0.794 0.066[2.2.1] Cb–Cb 0.946 0.986 0.874 0.112[2.2.2] Cb–Cb 1.009 1.049 0.986 0.063

1 For the sequential numbering of carbon atoms in the benzene ring.2 Ht and Hb denote the terminal and bridge hydrogen atoms, respectively.3 Central bonds between the bridgehead carbon atoms Cb.

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Moreover, as intuitively expected, the C–H bonds are seen to slightlyincrease their information ionicity when the number of these terminal bondsincreases in a series: acetylene, ethylene, and ethane. In B2H6, the correct ≈(1/2)-bond order of the bridging B–H bond is predicted, and approximatelysingle terminal bond multiplicity is detected. For the alkali metal fluoridesthe increase in the bond entropy-covalency (decrease in information ionic-ity) with increasing size (softness) of the metal is also observed. For the fixedalkali metal in halides, for example, in a series consisting LiF, LiCl, and LiBr(Table 1.2), the overall bond order is increased for larger (softer) halogenatoms, mainly due to a higher entropy-covalency (delocalization) and noisecomponent of the molecular communication channel in AO resolution.

9. CONCLUSION

Until recently, a wider use of CTCB in probing the molecular electronicstructure has been hindered by the originally adopted two-electron con-ditional probabilities, which blur a diversity of chemical bonds. We havedemonstrated in the present work that the MO-resolved OCT using theflexible-input probabilities and recognizing the bonding/antibonding char-acter of the orbital interactions in a molecule, which is reflected by the signsof the underlying CBO matrix elements, to a large extent remedies this prob-lem. The off-diagonal conditional probabilities it generates are proportionalto the quadratic bond indices of the MO theory; hence, the strong interorbitalcommunications correspond to strong Wiberg bond multiplicities. It alsocovers the orbital decoupling limit and properly accounts for the increas-ing populational decoupling of AO when the antibonding MOs are moreoccupied. It should be also emphasized that the extra-computation effort ofthis IT analysis of the molecular bonding patterns is negligible comparedwith the standard computations of the molecular electronic structure, sinceall quantum-mechanical computations in the orbital approximation alreadydetermine the CBO data required by this generalized formulation of OCT.

We have also demonstrated that a dramatic improvement of the over-all entropy/information descriptors of chemical bonds and a differentiationof diatomic bond multiplicities is obtained when one recognizes the mutu-ally decoupled groups of orbitals as the separate information systems. Suchdecoupling process can be satisfactorily described only within the flexible-input approach, which links the specified AO-input distribution to itsinvolvement in communicating (bonding) with the remaining orbitals. Theother improvement of the IT description of the bond diversity in moleculesand their weakening with the nonzero occupation of antibonding MO hasbeen gained by applying the MO-resolved channels supplemented withthe extra sign convention of their entropy/information bond contributions,which is linked to those of the associated MO bond orders. Indeed, the

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bonding/antibonding classification, although lost in the conditional prob-abilities, is directly available from the corresponding CBO matrix elements,routinely generated in all LCAO-MO calculations and required to generatethe information channels themselves.

This orbital IT development extends our understanding of the chem-ical bond from the complementary viewpoint of the information/communication theory. The purely probabilistic models have been previ-ously shown to be unable to completely reproduce the bond differentiationpatterns observed in alternative bond order measures formulated in the stan-dard MO theory. However, as convincingly demonstrated in Section 8, thebond probability weighting of contributions due to separate AO inputs givesexcellent results, which completely reproduce the bond differentiation indiatomic fragments of the molecule implied by the quadratic criterion ofWiberg. In excited states, only the recognition of the bonding/antibondingcharacter of the orbital interactions, which is reflected by the signs of the cor-responding elements of the CBO matrix, allows one to bring the IT overalldescriptors to a semi-quantitative agreement with the alternative measuresformulated in the SCF-LCAO-MO theory.

The OCT has recently been extended to cover many orbital effects in thechemical bond and reactivity phenomena [38, 68–70]. The orbital communi-cations have also been used to study the bridge bond order components [71,72] and the multiple probability scattering phenomena in the framework ofthe probability-amplitude channel [73]. The implicit bond-dependency ori-gins of the indirect (bridge) interactions between atomic orbitals in moleculeshave also been investigated [74].

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