Faculty of Chemistry, Gronostajowa 2, 30-387 Krakow, Poland

C/ Maria Aurèlia Capmany, 69, 17003 Girona, Catalonia, Spain

*Electron delocalization in planar metallacycles: Hückel or Möbius aromatic?*In this work the relationship between the formal number of π-electrons, d-orbital conjugation topology, π-electron delocalization and aromaticity in d-block metallacycles is investigated in the context of recent findings concerning the correlation of π-HOMO topology and the magnetic aromaticity indices in these species. It is demonstrated that for π-electron rich d-metallacycles the direct link between aromaticity, the number of π-electrons and the frontier π-orbital topology does not strictly hold and for such systems it is very difficult to unambiguously associate their aromaticity with the "4n+2" (Hückel) and "4n" (Möbius) rules. It is also shown that the recently proposed electron density of delocalized bonds (EDDB) method can successfully be used not only to quantify and visualize aromaticity in such difficult cases, but also - in contrast to magnetic aromaticity descriptors - to provide a great deal of information on the real role of d-orbitals in metallacycles without the ambiguity of bookkeeping of electrons in the π-subsystem of the molecular ring. Interestingly, some of the metallacycles studied cannot be classified exclusively as Hückel or Möbius because they have a hybrid Hückel-Möbius or even quasi-aromatic nature.

*Effect of solvent on the structural diversity of quasi-aromatic Möbius cadmium(II) complexes fabricated from the bulky N6 tetradentate helical ligand.*M.P. Mitoraj, G. Mahmoudi, F. Afkhami, A. Castineiras, G. Giester, I. Konyaeva, A.A. Khandar, F. Qu, A. Gupta, F. Sagan,__D.W. Szczepanik__, D.A. Safin,19 (2019), 1649−1659. DOI: 10.1021/acs.cgd.8b01569. URL**Cryst. Growth Des.**We present the synthesis and structural and computational characterization of a heteroleptic dinuclear discrete complex [Cd2(μ1,3-NCS)2(NCS)2(LI)2]·4MeOH (1·4MeOH), where LI is the product of hydrolysis of one of the 2-PyC(Ph) groups of the parent ligand 1,2-diphenyl-1,2-bis((phenyl(pyridin-2-yl)methylene)hydrazono)ethane (L), fabricated from a mixture of Cd(NO3)2·4H2O and NH4NCS in methanol. An analogous procedure but in EtOH and iPrOH yielded crystals of the heteroleptic dinuclear discrete complex [Cd2(μ1,3-NCS)2(NCS)2(L)2] (2) and the polymeric complex of the composition [Cd3(NCS)6(L)]n (3). The bulky helical ligand L in the constituting Cd(II) based synthons has been found to exhibit a quasi-aromatic Möbius features as revealed by the electron density of delocalized bonds method as well as the extended transition state coupled with natural orbitals for chemical valence calculations. It means that a transition metal induces π-electron delocalization solely in the ligand's part; however, the d orbitals of Cd(II) do not overlap with the π-system of L. It is further found that the donor/acceptor character of the substituents attached to L can tune the Möbius quasi-aromaticity. Finally, formation of the crystals is driven not only by the typical ionic/dative Cd−N bonds, but also by a number of noncovalent interactions starting from classic hydrogen bonds (N−H···N, N−H···O) going through π···π, C−H···π, to end up with nonintuitive and recently topical Lp···Lp interactions (Lp - lone pair): e.g., C···C, N···N, N···S, S···S as well as London dispersion dominated homopolar dihydrogen interactions C−H···H−C.

*A simple alternative for the pseudo-π method.*In this work, we introduce an approximate method for the multicenter index calculation that is very simple in implementation and has the same computational cost as the pseudo-π approach. In contrast to the latter, however, the newly proposed method does not require additional single-point calculations and is capable of quantifying multicenter electron sharing in aromatic rings containing heteroatoms and transition metals.

*Aromaticity of acenes: the model of migrating circuits.*__D.W. Szczepanik__(), M. Solà, T.M. Krygowski, H. Szatylowicz, M. Andrzejak, B. Pawelek, J. Dominikowska, M. Kukulka, K. Dyduch20 (2018) 13430−13436. DOI: 10.1039/c8cp01108g. URL**Phys. Chem. Chem. Phys.**In this work we extend the concept of migrating Clar's sextets to explain local aromaticity trends in linear acenes predicted by theoretical calculations and experimental data. To assess the link between resonance and reactivity and to rationalize the constant-height AFM image of pentacene we used the electron density of delocalized bonds and other functions of the one-electron density from conceptual density functional theory. The presented results provide evidence for migration of Clar's p-sextets and larger circuits in these systems, and clearly show that the link between the theoretical concept of aromaticity and the real electronic structure entails the separation of intra- and inter-ring resonance effects, which in the case of [n]acenes (n = 3, 4, 5) comes down to solving a system of simple linear equations.

*Quasi-aromatic Möbius metal chelates.*G. Mahmoudi, F. Afkhami, A. Castineiras, I. Garcia-Santos, A. Gurbanov, F.I. Zubkov, M.P. Mitoraj, M. Kukulka, F. Sagan,__D.W. Szczepanik__, I.A. Konyaeva, D.A. Safin,57 (2018) 4395−4408. DOI: 10.1021/acs.inorgchem.8b00064. URL**Inorg. Chem.**We report the design as well as structural and spectroscopic characterizations of two new coordination compounds obtained from Cd(NO3)2·4H2O and polydentate ligands, benzilbis(pyridin-2-yl)methylidenehydrazone (LI) and benzilbis(acetylpyridin-2-yl)methylidenehydrazone (LII), in a mixture with two equivalents of NH4NCS in MeOH, namely [Cd(SCN)(NCS)(LI)(MeOH)] (1) and [Cd(NCS)2(LII)(MeOH)] (2). Both LI and LII are bound via two pyridyl-imine units yielding a tetradentate coordination mode giving rise to the 12 π electron chelate ring. It has been determined for the first time (qualitatively and quantitatively), using the EDDB electron population-based method, the HOMA index, and the ETS-NOCV charge and energy decomposition scheme, that the chelate ring containing CdII can be classified as a quasi-aromatic Möbius motif. Notably, using the methyl-containing ligand LII controls the exclusive presence of the NCS− connected with the CdII atom (structure 2), while applying LI allows us to simultaneously coordinate NCS− and SCN− ligands (structure 1). Both systems are stabilized mostly by hydrogen bonding, C−H···π interactions, aromatic π···π stacking, and dihydrogen C−H···H−C bonds. The optical properties have been investigated by diffused reflectance spectroscopy as well as molecular and periodic DFT/TD-DFT calculations. The DFT-based ETS-NOCV analysis as well as periodic calculations led us to conclude that the monomers which constitute the obtained chelates are extremely strongly bonded to each other, and the calculated interaction energies are found to be in the regime of strong covalent connections. Intramolecular van der Waals dispersion forces, due to the large size of LI and LII, appeared to significantly stabilize these systems as well as amplify the aromaticity phenomenon.

*The electron density of delocalized bonds (EDDB) applied for quantifying aromaticity.*__D.W. Szczepanik__(), M. Andrzejak, J. Dominikowska, B. Pawełek, T.M. Krygowski, H. Szatylowicz, M. Solà

19 (2017) 28970−28981. DOI: 10.1039/c7cp06114e. URL**Phys. Chem. Chem. Phys.**In this study the recently developed electron density of delocalized bonds (EDDB) is used to define a new measure of aromaticity in molecular rings. The relationships between bond-length alternation, electron delocalization and diatropicity of the induced ring current are investigated for a test set of representative molecular rings by means of correlation and principal component analyses involving the most popular aromaticity descriptors based on structural, electronic, and magnetic criteria. Additionally, a qualitative comparison is made between EDDB and the magnetically induced ring-current density maps from the ipsocentric approach for a series of linear acenes. Special emphasis is given to the comparative study of the description of cyclic delocalization of electrons in a wide range of organic aromatics in terms of the kekulean multicenter index KMCI and the newly proposed EDDBk index.

*The role of the long-range exchange corrections in the description of electron delocalization in aromatic species.*__D.W. Szczepanik__(), M. Solà, M. Andrzejak, B. Pawełek, J. Dominikowska, M. Kukułka, K. Dyduch, T.M. Krygowski, H. Szatylowicz

38 (2017) 1640−1654. DOI: 10.1002/jcc.24805. URL**J. Comput. Chem.**In this article, we address the role of the long−range exchange corrections in description of the cyclic delocalization of electrons in aromatic systems at the density functional theory level. A test set of diversified monocyclic and polycyclic aromatics is used in benchmark calculations involving various exchange−correlation functionals. A special emphasis is given to the problem of local aromaticity in acenes, which has been a subject of long−standing debate in the literature. The presented results indicate that the noncorrected exchange−correlation functionals significantly overestimate cyclic delocalization of electrons in heteroaromatics and aromatic systems with fused rings, which in the case of acenes leads to conflicting local aromaticity predictions from different criteria.

*From quantum superposition to orbital communication.*__D.W. Szczepanik__(), E.J. Zak, J. Mrozek

1115 (2017) 80−87. DOI: 10.1016/j.comptc.2017.05.041. URL**Comput. Theor. Chem.**The orbital communication theory (OCT) by Nalewajski is derived step by step from first principles of quantum mechanics. It is shown that the entropy representation within the molecular orbital theory arises as a natural consequence of the probabilistic interpretation of quantum superposition. The algebra of selected types of molecular information channels is reinvestigated within the framework of the theory of Markov chains and several representative models of molecular communication systems in atomic-orbital resolution are discussed. The presented results show that the Shannon entropy alone, i.e. with no insight into its components - mutual information and conditional entropy, does not allow one to correctly identify the source of uncertainty connected with the electron probability distribution, which in some cases leads to wrong conclusions about the electron delocalization effects in a molecule.

*On the three-center orbital projection formalism within the electron density of delocalized bonds method.*A new development of the Electron Density of Delocalized Bonds formalism (EDDB) is proposed that provides marked improvement in the description of electron delocalization in aromatic rings. Special attention is paid to charged aromatic hydrocarbons of different size, for which the total population of electrons delocalized between adjacent bonds from the original formulation of the EDDB method significantly overestimates the multicenter π-electron sharing effects. The revised bond-orbital projecting scheme gives rise to systematic improvement of the results of the EDDB analysis, which now supports findings by other researchers.

*A new perspective on quantifying electron localization and delocalization in molecular systems.*The original method of electron density partitioning is introduced that allows one to probe electron localization and delocalization within one theoretical paradigm. The newly proposed method makes use of the age-old concept of bond-order orbitals as well as the recently developed bond-orbital projection formalism to decompose the one-electron density into density layers representing electrons localized on atoms (inner shells, lone pairs), shared between atoms (chemical bonds) and delocalized between adjacent bonds (multi-center bonding). The details of the current implementation are briefly discussed and several illustrative examples are provided.

*The lowest triplet states of bridged cis-2,2'-bithiophenes - theory vs experiment.*M. Andrzejak,__D.W. Szczepanik__, Ł. Orzeł17 (2015) 5328−5337. DOI: 10.1039/c4cp03327b. URL**Phys. Chem. Chem. Phys.**Theoretical methods that were previously used to give a good quantitative description of the 31Bu state of trans-2,2'-bithiophene are applied to characterize the lowest triplet states of three bridged cis−2,2'−bithiophenes: 3,3'−cyclopentadithiophene (CPDT), 3,3'−dithienylpyrrole (DTP), and 3,3'−dithienylthiophene (DTT). By obtaining highly accurate reproductions of the phosphorescence spectra of all three compounds, we rationalize the observed vibronic activity, further explore the performance of the applied theoretical methods, and address the quality of the reported experimental spectra. Over the course of this study we have, first, characterized the changes in the electronic structures between the ground state and the lowest triplet state and, second, expressed the related geometrical differences in terms of the Huang−Rhys factors. The Huang−Rhys factors have then been used to generate theoretical emission spectra with vibronic resolution. The applied procedure has yielded quantitative reproductions of the previously reported experimental phosphorescence spectra of DTT and DTP. The experimental spectrum of CPDT, on the other hand, turned out to be considerably narrower and intensity-deficient in its low energy region when compared with the theoretical results. Our experimental reinvestigation of the CPDT phosphorescence has given a refined spectrum that is significantly wider than the previously reported one, and is in nearly quantitative agreement with the theoretical prediction. This enabled us to attribute the observed discrepancy to an experimental artifact associated with the sensitivity characteristics of the commonly used photomultipliers.

*A uniform approach to the description of multicenter bonding.*__D.W. Szczepanik__(), M. Andrzejak, K. Dyduch, E.J. Zak, M. Makowski, G. Mazur, J. Mrozek,

16 (2014) 20514−20523. DOI: 10.1039/c4cp02932a. URL**Phys. Chem. Chem. Phys.**A novel method for investigating the multicenter bonding patterns in molecular systems by means of the so-called Electron Density of Delocalized Bonds (EDDB) is introduced and discussed. The EDDB method combines the concept of Jug's bond-order orbitals and the indirect ("through-bridge") interaction formalism and opens up new opportunities for studying the interplay between different atomic interactions as well as their impact on both local and global resonance stabilization in systems of conjugated bonds. Using several illustrative examples we demonstrate that the EDDB approach allows for a reliable quantitative description of diverse multicenter delocalization phenomena (with special regard to evaluation of the aromatic stabilization in molecular systems) within the framework of a consistent theoretical paradigm.

*Electron delocalization index based on bond order orbitals.*__D.W. Szczepanik__(), E.J. Zak, K. Dyduch, J. Mrozek593 (2014) 154−159. DOI: 10.1016/j.cplett.2014.01.006. URL**Chem. Phys. Lett.**A new index of electron delocalization in atomic rings is introduced and briefly discussed. The newly proposed delocalization descriptor is defined as an atom averaged measure of the effectiveness of forming linear combinations from two−center bond−order orbitals for a given sequence of bonded atomic triplets, and corresponds directly to electron population analysis; it allows one to get very compact and intuitive description of π−conjugation effects without additional parametrization and calibration to the reference molecular systems. The numerical results of illustrative calculations for several typical aromatic and homoaromatic compounds seem to validate the presented methodology and definitions.

*Through-space and through-bridge interactions in the correlation analysis of chemical bonds.*__D.W. Szczepanik__(), J. Mrozek1026 (2013) 72−77. DOI: 10.1016/j.comptc.2013.10.015. URL**Comput. Theor. Chem.**The formalism of the through-space and through-bridge communications proposed by Nalewajski within the framework of the Orbital Communication Theory is used to evaluate the correlation effect between electron populations of two chemical bonds. Proposed in this paper purely probabilistic formalism defines the correlation between two chemical bonds as determined by the difference between appropriately normalized fourth−order joint probabilities corresponding to direct (through−space) and indirect (through−bridge) communication within a four−state time−homogeneous Markov chain. Alternative bond correlation coefficient introduced earlier by Yamasaki and Goddard is based on the concept of the hierarchy of statistical covariance operators. The proposed correlation coefficient requires computing only the second−order probability terms when working within the condensed atomic resolution and thus they are far less computationally expensive making them suitable for correlation analysis of chemical interactions between large molecular fragments.

*Nucleophilicity index based on atomic natural orbitals.*Within the framework of the Frontier Molecular Orbital theory the effect of substituent groups is sometimes evaluated by means of the electron population analysis of the highest occupied molecular orbital (HOMO). In this paper we propose a "reverse scenario" of evaluating a semilocal (regional) nucleophilicity of ring members in aromatic species, in which we focus on the electron population of particular atom first and then we consider the highest expectation values of the Fock operator within the representation of atomic natural orbitals (ANO). Interesingly, contrariwise to the commonly used atomic indices of nucleophilicity by Franke and Fukui, the resulting "energies" calculated for each atom in the ring enable one to succesfully identify and assess the ring activation/deactivation effect of electron donating and electron withdrawing groups in the electrophilic aromatic substitution.

*Minimal set of molecule-adapted atomic orbitals from maximum overlap criterion.*__D.W. Szczepanik__(), J. Mrozek51 (2013) 2687−2698. DOI: 10.1007/s10910-013-0230-z. URL**J. Math. Chem.**The criterion of maximum overlap with the canonical free−atom orbitals is used to construct a minimal set of molecule−intrinsic orthogonal atomic orbitals that resemble the most their promolecular origins. Partial atomic charges derived from population analysis within representation of such molecule−adopted atomic orbitals are examined on example of first−row hydrides and compared with charges from other methods. The maximum overlap criterion is also utilized to approximate the exact free−atom orbitals obtained from ab initio calculations in any arbitrary basis set and the influence of the resulting fitted canonical atomic orbitals on properties of molecule−adopted atomic orbitals is briefly discussed.

*Ground-state projected covalency index of the chemical bond.*__D.W. Szczepanik__(), J. Mrozek1023 (2013) 83−87. DOI: 10.1016/j.comptc.2013.09.008. URL**Comput. Theor. Chem.**The criterion of maximum separation of non−degenerated eigenvalues in the one-electronc reduced density matrix spectrum is utilized to generalize the definition of the standard Wiberg−type bond covalency index on the case of multi−determinant state functions. The proposed definition involves factorization of natural−orbital contributions to the ground−state bond-covalency index and is essential for the Markov's stationarity condition of the one-electron reduced density matrix within representation of atomic orbitals. The ground−state projecting technique is introduced to facilitate evaluation of bond orders in excited−state molecular systems. The performance of the presented methodology is tested on selected small but informative molecules.

*On quadratic bond-order decomposition within molecular orbital space.*__D.W. Szczepanik__(), J. Mrozek51 (2013) 1619−1633. DOI: 10.1007/s10910-013-0169-0. URL**J. Math. Chem.**A simple method of analysing and localization of canonical molecular orbitals for particular chemical bond using the MO−resolved bond−order decomposition scheme is presented. An alternative definition of classical bond order orbitals is provided and links to communication theory of the chemical bond are outlined and briefly discussed. The introduced procedure of decomposition of quadratic bond orders allows one to analyse two− as well as three− center chemical bonds within the framework of the same theory.

*Stationarity of electron distribution in ground-state molecular systems.*__D.W. Szczepanik__(), J. Mrozek51 (2013) 1388−1396. DOI: 10.1007/s10910-013-0153-8. URL**J. Math. Chem.**Stationarity of electron probability distribution within the resolution of atomic orbitals is considered involving some concepts from Orbital Communication Theory and the theory of Markov Processes. A new method of evaluating electron conditional probabilities based on natural orbitals is proposed and briefly discussed.*On several alternatives for Löwdin orthogonalization.*__D.W. Szczepanik__(), J. Mrozek1008 (2013) 15−19. DOI: 10.1016/j.comptc.2012.12.013. URL**Comput. Theor. Chem.**Several alternative procedures of orthogonalization involving the metric of the linear vector space formed by columns of the matrix of LCAO MO coefficients are briefly discussed and exemplified using electron population analyzes and orbital-atom assignment descriptors. The newly proposed procedures put emphasis on the resemblance constraints between the relevant non−orthogonal and pre−orthogonalized LCAO MO column matrices representing the subspace of occupied molecular orbitals in the final orthogonalization step. Since only the subspaces of occupied MOs completely determines the electronic structure of chemical species they give rise to definitely more adequate and chemically meaningful orthogonal atomic orbitals than the original Löwdin's atomic orbitals.

*Electron population analysis using a reference minimal set of atomic orbitals.*__D.W. Szczepanik__(), J. Mrozek996 (2012) 103−109. DOI: 10.1016/j.comptc.2012.07.021. URL**Comput. Theor. Chem.**The criterion of maximum overlap with the Huzinaga's MINI basis functions as well as the "physical" orthogonalization concept are used to rationalize the standard atomic charges calculated within representation of extended basis sets and to compare charge distributions calculated using different basis sets. The generalization of "physical" orthogonalization approach on natural orbitals is introduced and briefly discussed. Numerical results fully validate the presented methodology identifying the newly proposed atomic charges as an interesting alternative to the Mulliken and Löwdin population analyses.*Symmetrical orthogonalization within linear space of molecular orbitals.*__D.W. Szczepanik__(), J. Mrozek521 (2012) 157−160. DOI: 10.1016/j.cplett.2011.11.047. URL**Chem. Phys. Lett.**An alternative procedure of orthogonalization involving a metric matrix of the linear vector space formed by the columns of LCAO matrix is introduced. It is proved that this procedure is fully equivalent the original Löwdin scheme. Also, a few modifications involving a two-step orthogonalization scheme are outlined and briefly discussed. Numerical results of test calculations of atomic charges for several representative molecules seem to fully validate the newly proposed methodology.*Basis set dependence of molecular information channels and their entropic bond descriptors.*R.F. Nalewajski,__D.W. Szczepanik__, J. Mrozek50 (2012) 1437−1457. DOI: 10.1007/s10910-012-9982-0. URL**J. Math. Chem.**Information channels from SCF MO calculations using different basis sets and their entropic bond descriptors are compared within the orbital communication theory. In this information-theoretic (IT) treatment of communications between basis functions the overall covalency and ionicity bond components reflect the average communication noise and information flow, respectively, in the resolution level specified by the adopted set of basis functions. The basis-set dependence of the orbital conditional probabilities and their entropic descriptors of the information covalency/ionicity content is explored. Compared to the minimum set of the occupied atomic orbitals of the separated constituent atoms, the extended basis sets of gaussian orbitals and/or their formal contractions generally give rise to a higher IT-covalency and lower IT-ionicity descriptors of the system chemical bonds. In the augmented set case, containing the polarization function, the use of only communications is advocated in a semi-quantitative chemical interpretation of the IT bond indices. The maximum-overlap criterion is used to transform the general (orthonormal) extended basis to its semi-augmented form, which facilitates the near minimum basis set interpretation of bond descriptors and extraction of communications involving the polarization functions. A similar transformation using the minimum information distance criterion can be also envisaged. The effect of the atomic reduction of the molecular channels, which misses the effect of the "internal" communications (bonds) on constituent atoms, is also examined. As intuitively expected, the IT descriptors of such reduced channels are found to be less sensitive to the basis set enlargement.

*Probing the interplay between multiplicity and ionicity of the chemical bond.*__D.W. Szczepanik__(), J. Mrozek

10 (2011) 471−482. DOI: 10.1142/s021963361100658x. URL**J. Theor. Comput. Chem.**A new bond multiplicity measure based on the Wiberg-type bond covalency index and the atomic charge from population analysis is presented. Heuristically derived formulas allow one to evaluate the character of the chemical bond, especially its ionicity degree. Numerical results at RHF/ROHF theory level demonstrate that full multiplicities of typical chemical bonds are close to formal orders and their basis set dependence is inconsiderable, especially for highly polarized chemical bonds.*Entropic bond descriptors from separated output-reduced communication channels in AO-resolution.*Communication Theory of Chemical Bond (CTCB) in atomic orbital resolution is used to define entropic bond orders of diatomic molecular fragments. Partial communication channels for separated information flows from atomic centers and two alternative output-reducion schemes with their entropic descriptors are proposed. Also two types of information that can be transmitted through communication system are identified: information about molecular electron occupations and information about bonding shares of atomic orbitals. The former is used to evaluate an average number of electrons engaged in bond forming process while the latter provides information about electron localization in chemical bond. Calculated entropic bond orders and their IT-covalency and IT-ionicity components are in good agreement with both chemical intuition and MO theory predictions.*Bond differentiation and orbital decoupling in the orbital-communication theory of the chemical bond.*R.F. Nalewajski,__D.W. Szczepanik__, J. Mrozek61 (2011) 1−48 (Chapter 1). DOI: 10.1016/B978-0-12-386013-2.00001-2. URL**Adv. Quantum Chem.**Information-theoretic (IT) probe of molecular electronic structure, within the orbital-communication theory (OCT) of the chemical bond, uses the standard entropy/information descriptors of the Shannon theory of communication to characterize the scattering of electron probabilities and their information content throughout the system network of chemical bonds generated by the occupied molecular orbitals (MOs). Thus, the molecule is treated as information network, which propagates the "signals" of the electron allocation to constituent atomic orbitals (AOs) or general basis functions between the channel AO "inputs" and "outputs". These orbital "communications" are determined by the two-orbital conditional probabilities of the output AO events given the input AO events. It is argued, using the quantum-mechanical superposition principle, that these conditional probabilities are proportional to the squares of corresponding elements of the first-order density matrix of the AO charges and bond orders (CBO) in the standard self-consistent field (SCF) theory using linear combinations of AO (LCAO) to represent MO. Therefore, the probability of the interorbital connections in the molecular communication system is directly related to the Wiberg-type quadratic indices of the chemical bond multiplicity. Such probability propagation in molecules exhibits the communication "noise" due to electron delocalization via the system chemical bonds, which effectively lowers the information content in the output signal distribution, compared with that contained in probabilities determining its input signal, molecular or promolecular. The orbital information systems are used to generate the entropic measures of the chemical bond multiplicity and their covalent/ionic composition. The average conditional-entropy (communication noise, electron delocalization) and mutual-information (information capacity, electron localization) descriptors of these molecular channels generate the IT covalent and IT ionic bond components, respectively. A qualitative discussion of the mutually decoupled, localized bonds in hydrides indicates the need for the flexible-input generalization of the previous fixed-input approach, in order to achieve a better agreement among the OCT predictions and the accepted chemical estimates and quantum-mechanical bond orders. In this extension, the input probability distribution for the specified AO event is determined by the molecular conditional probabilities, given the occurrence of this event. These modified input probabilities reflect the participation of the selected AO in all chemical bonds (AO communications) and are capable of the continuous description of its decoupling limit, when this orbital does not form effective combinations with the remaining basis functions. The occupational aspect of the AO decoupling has been shown to be properly represented only when the separate communication systems for each occupied MO are used, and their occupation-weighted entropy/information contributions are classified as bonding (positive) or antibonding (negative) using the extraneous information about the signs of the corresponding contributions to the CBO matrix. This information is lost in the purely probabilistic model since the channel communications are determined by the squares of such matrix elements. The performance of this MO-resolved approach is then compared with that of the previous, overall (fixed-input) formulation of OCT for illustrative π-electron systems, in the Hückel approximation. A qualitative description of chemical bonds in octahedral complexes is also given. The bond differentiation trends in OCT have been shown to agree with both the chemical intuition and the quantum-mechanical description. The numerical Restricted Hartree-Fock (RHF) applications to diatomic bonds in representative molecular systems are reported and discussed. The probability weighted scheme for diatomic molecular fragments is shown to provide an excellent agreement with both the Wiberg bond orders and the intuitive chemical bond multiplicities.

*Probabilistic models of the chemical bond in the function spaces.*__D.W. Szczepanik__(supervisor: J. Mrozek)

PhD Thesis, Jagiellonian University (2013). DOI: 10.13140/RG.2.2.19414.55368. PDF*Entropic bond indices from information theory.*__D.W. Szczepanik__(supervisor: J. Mrozek)

MSc Thesis, Jagiellonian University (2008). DOI: 10.13140/RG.2.2.34514.04807. PDF

Last update: **2019-02-28**