Structural Chemistry

Like most books dedicated to structural chemistry, we start with a short recall of the long history leading to current theories of the atomic constitution of matter and the structure of atoms, mentioning Bohr’s crude model and, subsequently, the quantum mechanics frame. A brief introduction to quantum theory, operator and matrix techniques is provided, with annotations explaining the intriguing puzzle of the facts and non-intuitive reasons that contoured this paradigm. The story of Schrödinger’s cat is retold, where the animal is not hurt, replacing the dead or alive states with a sleep versus awake scenario, making the probabilistic paradox quite clear, as a legitimate mixing of wave functions symbolizing the state of the system. The quantum structure of the atom is presented in an original way, i.e. by putting a special emphasis on the effective role in chemistry of Spherical Harmonics functions, grasping the concepts in an intuitive manner, with the help of heuristic symmetry reasons. Taking variable transformations as artifices in the Schrödinger equation of the hydrogen atom, the spectrum of orbital energies unfolds without following through to a complete solution. The chapter offers picturesque descriptions and explanatory artifices which are original, not met in other textbooks. An incursion into the even more mysterious world of relativistic quantum mechanics is made, bringing electron spin into sight, along with related consequences, important for understanding further topics, such as atomic and molecular magnetism. The complexity of the theorization is increased by incorporating the Feynman path integral method, bringing pictures from a territory less often visited by chemists, for the sake of a complete cross-border perspective. Finally, while introducing specific particle and wave representations, as well as their ratio, in quantifying the wave-to-particle quantum information, the basic Heisenberg Uncertainty Relationship (HUR) is recovered for a large range of observable particle-wave Copenhagen duality, although with the dominant wave manifestation, while registering its progressive modification with the factor \(\sqrt {1 - n^{2} }\), in terms of magnitude \(n \in \left[ {0,1} \right]\) of the quantum fluctuation, for the free quantum evolution around the exact wave-particle equivalence.

The chapter continues the discussion of the atomic structure, at the level of clarifying concreteness and details, presenting also the basic methods of electronic structure theories, constituting the keystone of the developments debated in the next chapters. The Slater determinants are introduced as primitives for the construction of many-electrons wave functions, using the one-electron orbital functions as basic ingredients. The orbitals, known from the introductory part of atom theory, are generalized at the molecular level. The so-called Slater rules for handling the Hamiltonian matrix elements of the Slater determinants constructed from orthogonal orbitals are presented as the basic algorithm for the practical approach of quantum chemistry. Besides, the rules are generalizable to other operators working with one- and two-electron terms. The Slater rules are generalized, at the end of the chapter, for the case of non-orthogonal orbitals, implying non-orthogonal Slater determinants bases. With the help of Slater rules and symmetry dichotomy of the two-electron integrals in the atomic shells (the Slater–Condon parameters), several poly-electronic atoms are analyzed in quite advanced detail, writing down analytical formulas for the energies of their states and relating to the experimental data on spectral terms. This exercise is important, beyond the domain of free atoms, since in large classes of compounds and materials with technical applicability one deals with embedded ions, whose properties are approachable by an atom-alike phenomenology. An exemplification on lanthanide emission spectra, used in domestic lighting devices, occasions a quick excursus in the open challenges of current materials sciences and the rational design of properties. Going from atoms to molecules, the use of atomic basis sets as the background of quantum chemistry is debated in detail, with hands-on illustration of the various options: Slater-type orbitals, Gaussian-type bases, plane waves, and numerical bases. A critical eye is turned upon Gaussian-type orbitals, signaling certain significant failures of bases rated as rich and accurate. The underperformance is determined by the lack of appropriate polynomial factors in the definition of atomic orbitals assignable to high quantum numbers, an intrinsic design deficiency in the customary implementation and use of Gaussians. The common belief is that the limitation of Gaussians consists in the assumed use of  \( \text{exp}(-\alpha r^2)\) –type functions instead of more physical  \( \text{exp}(-\alpha r)\) ones, while the lack of proper polynomial factors can cause more severe drawbacks. In turn, numeric basis sets are observed as surprisingly good performers and possible alternative technical options. The final part of the chapter presents the fundamental electronic structure methods based on wave functions and first principles operators: the Hartree–Fock technique, introducing self-consistency, brought to a higher level by the multi-configurational approach, and the Valence Bond theory. The so-called Complete Active Space Self-Consistent Field methods are near the top of powerfulness among actually available methods, which, with broad conceptual scope and flexible technical leverages, allow the approach to a large number of problems, with the right picture of their mechanisms and manifestations. Certain other methodological varieties, such as second-order perturbation corrections to self-consistent Hartree–Fock or multi-configurational techniques, or the Coupled Cluster expansion are discarded from the actual synopsis. From our perspective, such procedures can bring only incremental changes to the physical picture, sometimes not in a well-tempered manner, while their non-variational nature is a hidden drawback, at least in conceptual respects, and a heavy burden to the computation routines. Somewhat greater attention is paid to the Valence Bond frame, acknowledging its merit as a foundational model of the chemical bond and also its potential in a modern methodological reshaping.

Density Functional Theory (DFT), the most productive and attractive method of computational chemistry in the last decades, a beacon expected to endure for a long time, is discussed in this chapter in terms of its conceptual and practical sides. After introducing the historical roots of DFT, the proofs of the founding Hohenberg–Kohn theorems are exposed. The issue of the exchange-correlation hole is introduced and illustrated by a strategy of heuristic artifices, describing the promises of DFT and its current limitations and compromises. Several technical issues are approached, making clear, for instance, the analytic operations leading to the celebrated ρ ^4/3 pattern for the approximated local density exchange functional. The various flavors of DFT calculations (species and acronyms of consecrated functionals) are treated briefly. A special emphasis is put on features specific to DFT, not attainable in the techniques pertaining to the wave function theories (WFT). These are: the conceptual possibility of fractional occupation numbers and an orbital formalism (Kohn–Sham) with eigenvalues substantiated as derivatives of the total energy with respect of level populations. The first derivative of energy as a function of occupation numbers (taken with changed sign) is invested with an important meaning: absolute electronegativity. The second derivatives are interpreted as measures for the strength of acids and bases in the Lewis definition (accepting or donating electronic density), namely the so-called chemical hardness. The chapter proposes the term of electrorigidity, instead of chemical hardness, underlining the meaning of this second derivative as a “force constant” resistant to the deformation of the electronic cloud. Although not extensively, the chemical significance of the lemmas enabled by DFT is emphasized, mentioning the principles of electronegativity equalization, maximum hardness, and the mutual affinity trends in the hard and soft acids and bases (HSAB) taxonomy. One issue is the so-called DFT+U technique, using plane wave methods to alleviate non-physical trends in the account of metal ions in compounds. The explanation brings to the level of chemists’ intuition technicalities from physicists’ language. A very suggestive illustration of exposed issues is done with the help of an original development: the energies of atomic bodies as continuous functions of shell populations. With the ad hoc proposed model, one concretizes several aspects of conceptual DFT, discussing comparisons between calculation and experiment, verifying theorems and approximations related to absolute electronegativity and chemical hardness (electrorigidity). After giving a general overview of the DFT realm and a compressed briefing of its constitutional rules, several technical aspects are revisited in more detail in the final part of the chapter. This gives an analytical survey of the main workable kinetic, exchange, and correlation density functionals, within local and gradient density approximations. They generally fulfill the N-contingency, assure the total energy minimization, influence the different levels of approximation, i.e. local density or gradient density frameworks, control the bonding through electronic localization functions, and decide upon reactivity through the electronic exchange relating the electronegativity and chemical hardness indices.

This chapter plunges into applied quantum chemistry, with various examples, ranging from elementary notions, up to rather advanced tricks of know-how and non-routine procedures of control and analysis.

In the first section, the first-principles power of the ab initio techniques is illustrated by a simple example of geometry optimization, starting from random atoms, ending with a structure close to the experimental data, within various computational settings (HF, MP2, CCSD, DFT with different functionals). Besides assessing the performances of the different methods, in mutual respects and facing the experiment, we emphasize the fact that the experimental data are affected themselves by limitations, which should be judged with critical caution. The ab initio outputs offer inner consistency of datasets, sometimes superior to the available experimental information, in areas affected by instrumental margins. In general, the calculations can retrieve the experimental data only with semi-quantitative or qualitative accuracy, but this is yet sufficient for meaningful insight in underlying mechanisms, guidelines to the interpretation of experiment, and even predictive prospection in the quest of properties design.

The second section focuses on HF and DFT calculations on the water molecule example, revealing the relationship with ionization potentials, electronegativity, and chemical hardness (electrorigidity) and hinting at non-routine input controls, such as the fractional tuning of populations in DFT (with the ADF code) or orbital reordering trick in HF (with the GAMESS program).

Keeping the H₂O as play pool, the orbital shapes are discussed, first in the simple conjuncture of the Kohn–Sham outcome, followed by rather advanced technicalities in handling localized orbital bases, in a Valence Bond (VB) calculation, serving to extract a heuristic perspective on the hybridization scheme.

In a third section, the H₂ example forms the background for discussing the bond as spin-coupling phenomenology, constructing the Heisenberg-Dirac-van Vleck (HDvV) effective spin Hamiltonian. In continuation, other calculation procedures, such as Complete Active Space Self-Consistent Field (CASSCF) versus Broken-Symmetry (BS) approach, are illustrated, in a hands-on style, with specific input examples, interpreting the results in terms of the HDvV model parameters, mining for physical meaning in the depths of methodologies.

The final section presents the Valence Bond (VB) as a valuable paradigm, both as a calculation technique and as meaningful phenomenology. It is the right way to guide the calculations along the terms of customary chemical language, retrieving the directed bonds, hybrid orbitals, lone pairs, and Lewis structures, in standalone or resonating status. The VB calculations on the prototypic benzene example are put in clear relation with the larger frame of the CASSCF method, identifying the VB-type states in the full spectrum and equating them in an HDvV modeling. The exposition is closed with a tutorial showing nice graphic rules to write down a phenomenological VB modeling, in a given basis of resonance structures. The recall of VB concepts in the light of the modern computational scene carries both heuristic and methodological virtues, satisfying equally well the goals of didacticism or of exploratory research. A brief excursion is taken into the domain of molecular dynamics problems, emphasizing the virtues of the vibronic coupling paradigm (the account of mutual interaction of vibration modes of the nuclei with electron movement) in describing large classes of phenomena, from stereochemistry to reactivity. Particularly, the instability and metastability triggered in certain circumstances by the vibronic coupling determines phase transitions of technological interest, such as the information processing. The vibronic paradigm is a large frame including effects known as Jahn–Teller and pseudo Jahn–Teller type, determining distortion of molecules from formally higher possible symmetries. We show how the vibronic concepts can be adjusted to the actual computation methods, using the so-called Coupled Perturbed frames designed to perform derivatives of a self-consistent Hamiltonian, with respect to different parametric perturbations. The vibronic coupling can be regarded as interaction between spectral terms, e.g. ground state computed with a given method and excited states taken at the time dependent (TS) version of the chosen procedure. At the same time, the coupling can be equivalently and conveniently formulated as orbital promotions, proposing here the concept of vibronic orbitals, as tools of heuristic meaning and precise technical definition, in the course of a vibronic analysis. The vibronic perspective, performed on ab initio grounds, allows clear insight into hidden dynamic mechanisms. At the same time, the vibronic modeling can be qualitatively used to classify different phenomena, such as mixed valence. It can be proven also as a powerful model Hamiltonian strategy with the aim of accurate fitting of potential energy surfaces of different sorts, showing good interpolation and extrapolation features and a sound phenomenological meaning.

Finally, within the symmetry breaking chemical field theory, the intriguing electronegativity and chemical hardness density functional dependencies are here reversely considered by means of the anharmonic chemical field potential, so inducing the manifested density of chemical bond in the correct ontological order: from the quantum field/operators to observable/measurable chemical field.

Heuristic concepts of structural chemistry, like hybridization and aromaticity, that ensure the communication with chemists specialized in experimental branches, are revisited with state-of-the-art methodologies, from an original perspective. We find that the celebrated hybrids made of s and p orbitals have not fallen into caducity, as too simple for applied structural chemistry, good only for the kindergarten of elementary chemical training. Looking beyond the sp, sp², and sp³ standard hybridization formats, exploring the meaning of s ^u p ^v differential degrees of hybridization, obtainable by means of post-computational tools of Natural Bond Orbitals (NBO) theories, meaningful lines of discussion can be drawn. Besides, the differential hybrids s ^u p ^v can be obtained in advance of calculation, on grounds of simple geometry analysis. If hybridization exists as real force (driven by the local character of electronic correlation), then the bond angles around central atoms with low site-symmetries can be interrelated. An interesting series of this sort is presented as proof of hybridization, as a non-superfluous concept. Checking the validity of hybrids made of s, p, and d functions, one finds that these cannot be invoked in Wernerian transition metal complexes (as is the case of d²sp³ octahedral hybridization), but gain relevance in organometallic systems. Here, the isolobality qualitative model, based intrinsically on the isomorphism of hybrid orbital sets from metal versus non-metal moieties, is a valuable rationalization clue for series of compounds. The concept of aromaticity is thoroughly debated, from different perspectives with various models, paying tribute to the importance of this issue and to the extremely diversified panoply of existing interpretations. With advanced multi-configuration calculations, Complete Active Space Self-Consistent Field (CASSCF) and Valence Bond (VB), followed by subsequent modeling by the Heisenberg spin Hamiltonian, that follows consistently the VB phenomenology, we dig into the causal factors of molecular geometry for the C₆H₆ and C₄H₄, taken as prototypes of aromatic and anti-aromatic behavior. It is seen then that, if only the π electrons existed, the systems would go to anti-aromatic type of bond alternating distortion, the aromaticity of benzene being secretly sustained by the strength of its σ skeleton. We present a detective story that deserves to be closely followed. Inorganic and organometallic clusters, generalizing the covering area of the aromaticity paradigm, are illustrated, with an interesting example where the theoretical prediction helps to identify specific reactivity features. The NBO frame is illustrated, by its Natural Resonance Theory (NRT) branch and specific tools of energy components analysis, as a surrogate to the VB calculations. Although the nominal meaning of resonance structures differs in the NBO versus VB computation frames (density component vs. wave function), the interpretation tempts similar heuristics. Another series of original considerations on aromaticity is constructed with reactivity criteria and on the grounds of graph theory, decorating the topological determination with meaningful parameters. It appears that aromaticity may be a tool of chemical structure and reactivity characterization while assuming for it a viable quantum definition, i.e. differently counting at molecular orbital and atoms-in-molecule chemical bonding level. Yet further insight is obtained when also the molecular topology by special adjacency in bonding is considered, within the so-called “colored” chemical reactivity by chemical topology.

Coordination compounds, alternatively called complexes, are systems where metal ions (d-type transition elements or the f-elements, lanthanides and actinides) are linked to molecules that may have standalone identity (the ligands), showing local connectivities (coordination numbers) larger than those presumable by the valence rules. The supplement of linkage capabilities is realized by weak bonding interactions, ionic and partly covalent. This situation generates special properties, the loosely bonded “nervous” electrons causing various magnetic manifestations and electronic transitions in visible or near-infrared, strongly influenced by the coordination environment and electron counts of metal ions, as well as by the long-range interactions. The specifics of this bonding regime are treated with models belonging to the Ligand Field Theory, originating from the pre-computational era, but keeping their insightful benefits also in modern times, as tools for interpreting calculations in a phenomenological way. There are several classes of ligand field (LF) models, the classical paradigm being based on the expansion of effective Hamiltonian in spherical harmonics, as operators having numeric cofactors as parameters. This construct is a perennial, possible everlasting idea, exploiting in elegant manner the symmetry factors. Other versions, such as the so-called Angular Overlap Model (AOM) are closer to the chemist’s idea about the bonding capabilities of ligands. The computation of coordination systems is often a non-trivial task, the mastering of ligand field ideas offering useful guidelines in setting the input and reading the output. The coordination bonding regime is also encountered in many solid state systems (oxides, halides), the intrinsic electronic structure features of the metal ions and their interaction with the environment being the basis of important current or future-targeted applications in the material sciences. An excursus in this problematic is drawn in this chapter. If the reader is a novice to ligand field concepts, or in the calculations serving in this domain, the presented exposition will provide helpful clues and heuristic perspectives for an illuminating initiation. For instance, for the AOM in octahedral field, we give a shortcut proof of the master formula, not demanding the full assimilation of the technique. The difficulties of multi-parametric LF in terms of spherical harmonic operators are circumvented with picturesque color maps of the LF potential on the coordination sphere. When the reader knows the principles of LF, but is longing to go to the next level, of mastering the underlying algebra, this chapter has things to offer. The computer algebra insets help very much to reach high level exercises and proofs. The same goes for people acquainted with quantum calculations, and who may be interested to know hints and tricks related with the specifics and peculiarities of the electronic structure in d- and f-based complexes, conducting numeric experiments in the spirit of the LF paradigm. Besides, we introduce, as application phenomena worth knowing, inorganic thermochromism and magnetic anisotropy. Finally, we hope that even the readers with extensive expertise in LF algebra or state-of-the-art ab initio methods, will find here original clues, interpretations, and developments. Along with basic exposition of various computational techniques (CASSCF, DFT, TD-DFT), we explain insightful handling, marking the limits of interpretations (e.g. the TD-DFT inability for certain LF problems). A special emphasis is put on the first-principles modeling of the f-type complexes, where the authors brought pioneering contributions in the methodology of multi-configuration calculations applied to such systems. The challenge to be faced is the non-aufbaunature of the f shell of the lanthanide ions in complexes and lattices, which makes problematic the routine approach. Original interpretations and methodologies are also highlighted for the issue of magnetic anisotropy, an important manifestation resulted from the imbrication of the ligand field and spin-orbit effects. The phenomenological modeling and the ab initio calculations are placed on equal footing in this chapter.

The molecular magnetism is a vault full of treasures for both fundamental and applied sciences, appealing to chemists inclined for practical work, synthetic or instrumental, as well as analytic spirits, dealing with concepts and computation. It was born about two decades before the end of the past millennium and is still vivid nowadays, forty years later. Molecular magnetism, the new face of magneto-chemistry, has grown symbiotically with modern theoretical chemistry, in the age of the computer revolution, when the accessibility of quantum calculations have achieved user-friendly status. The field is dedicated to re-enacting the knowledge developed by physicists since the 1950s (only at a conceptual level and, often, in the language of solid state theory), taking case studies of molecular nature, which enable a completely new perspective. The orbital concepts have been largely embraced and enjoyed, at least at the level of qualitative models, even by people dedicated to the experimental branches, the field having an intrinsic interdisciplinary bedrock. The advanced theory and method development is necessary to the consolidation of this background, facing further more complex tasks and challenges. This chapter presents a primer in the structural chemistry of molecular magnetism and its relation with the properties, starting first with generalities on the phenomenological side. The Heisenberg-Dirac-van Vleck (HDvV) spin Hamiltonian, met previously in the frame of Valence Bond theory, is taken here in its most frequent use: the modeling of inter-center effective exchange coupling of magnetic ions. Other spin Hamiltonian components, Zeeman and Zero Field Splitting (ZFS) are introduced, aside the operational definitions of measurable quantities, magnetization and susceptibility. We call attention to the fact that, sometimes, the pragmatic use of phenomenological tools can be misleading, if not controlled by more advanced structural reasoning. The computation modeling draws guidelines for parametric dimensions unavailable by experiments. Within the current state of the art, the ab initio approach can even provide predictions, helping the goals of property engineering. We present, with application examples, the two branches of methods for calculation of exchange coupling constants: Broken Symmetry (via Density Functional Theory, BS-DFT) and multi-configurational wave function theory (e.g. Complete Active Space Self-Consistent Field, CASSCF). Different methods, or different settings within the same procedure, may give variate parametric sets, more or less close to the reproduction of experimental data, but it is important that the range and relative ratios of the values are usually stable, safe for understanding the underlying mechanisms, or for fixing parametric uncertainties of the phenomenological fit. A special area of the molecular magnetism is those based on lanthanides, in mono-nuclear or poly-nuclear complexes, where the specifics of electronic structure (partly developed in the ligand field chapter) demand special attention and strategies. The authors of this book made pioneering advances in the ab initio multi-configurational approach of realistic lanthanide complexes, analyzing first the mechanism of frequent ferromagnetic coupling in Cu–Gd complexes, recalled here briefly. The mechanism is active also in other d-f systems, but the picture becomes more complicated in the case of lanthanide sites with degenerate free ion ground states (quasi-degenerate, as ions in molecule). The quasi-degeneracy (weak ligand field splitting of multiplets) and the spin-orbit coupling are giving rise to the phenomenon of magnetic anisotropy, of crucial importance for making a molecule, and ultimately any larger system, behave as a magnet with fixed poles. Although magnetic anisotropy is a rather complex issue, we present original tools allowing a picturesque interpretation: the polar maps of state-specific magnetization functions. A detailed analysis of case studies showing the interplay of exchange coupling, ligand field, and spin-orbit effects in the magnetism of a prototypic series of d-f dinuclears illuminates the magneto-structural causalities. A section dedicated to the spin crossover effects gives new clues and perspectives to the general premises and simple modeling of the phenomena, as well as to the advanced analysis by insightful computational experiments. The Magnetism is already the basis of innumerable technical applications, its molecular and nanoscale avatars being speculated as the assets of a new future technology, called spintronics (an analogue of actual electronics, but based on spin bits). Realistically, spintronics is still a faraway desideratum, but the journey to this goal is fascinating, mustering cooperation across several borders of chemistry domains.

The chapter presents a theory with a qualitative flavor, Tensor Surface Harmonics (TSH) , that demonstrated its elegant power for explaining the bonding in clusters in the era preceding the computer revolution and the wider availability of the quantum chemical computer programs. It uses symmetry reasons, based on the parentage of delocalized molecular orbitals from spherical harmonics functions and their derivatives, when a cluster is more or less approximated with a pseudo-globular pattern. The derivative term, in the above phrase, literally meant the mathematical operation of differentiation, applied on spherical harmonics, as function of polar coordinates, diversifying the basis of elements in which the structural description can be conceived. Within TSH, quasi-spherical molecules can be regarded as giant atoms. The realization of stable occupation schemes in quasi-degenerate orbital patterns draws rationales for stereochemistry and compositional “magic numbers” in cluster chemistry. The axial symmetry, explaining the electronic structure of rings, in a manner similar to Hückel’s crude and clear approximation, can be regarded as a particularization of TSH. To be distinguished from the classical qualitative use of TSH, we revisit elements of this theory with the support of computational methods exposed in the previous chapters, proving and enlarging the illuminating power of this paradigm. One may note that we met and exploited the spherical harmonics in several instances, throughout this book, starting with the well-known encounter as the angular part of atomic orbitals, continued with the lesser known use in the parameterization of two-electron integrals (that enabled a hands-on lucrative approach of many electron atoms) and culminating with Ligand Field Theory, where the spherical harmonics are cornerstones of phenomenological Hamiltonian and effective bases. The Tensor Surface Harmonics is approached here as a welcome completion of the conceptual heuristics and phenomenological modeling leverage emerging from spherical symmetry. Finally, the aromaticity concept may be linked, in poly-aromatic hydrocarbon (PAH) molecules, with the isomers constitution. This connection appears on 16 PAHs molecules and their isomers, by combining the topo-reactivity method of specific-bond-by-adjacency procedure with Kekulé, Clar, and Fries benzenoid descriptions. The resulting approach determines a new method of classification for aromatic molecules, in general, and may be used to predict which molecules are most likely to adopt the most aromatic (Kekulé + Clar + Fries) conformation in chemical reactions.

Bondonic chemistry promotes the modeling of chemical transformations by quantum particles of the chemical field, the so-called bondons, rather than by molecular wave function. From the bondonic side, the quantum computational information, mainly regarding the bonding energy, but also with the topology of the molecular architecture, is projected on the length radii or action, bondonic mass and gravitational effects, all without eigen-equations in “classical” quantum mechanics, although being of observable nature, here discussed and compared for their realization and predictions. As a boson and responsible for chemical bonding, i.e. electronic aggregating in a stable structure (despite the inter-electronic repulsion) the gravitational side of the bondons is also manifested, and accordingly here reviewed and applied on paradigmatic chemical compounds. Being a particle of quantum (chemical) interaction, the bondon is necessarily a boson, and emerges from chemical field by a spontaneous symmetry breaking mechanism, following the Goldstone mechanism yet featuring the Higgs bosonic mass rising caring the electronic pair information by a bondon-antibondon (Feynman) coupling, eventually corresponding to the bonding-antibonding chemical realms of a given bonding. The present mechanism of bondonic mass is applied for describing the Stone-Wales topological defects on graphene, a 2D carbon material allowing electrons to unidirectionally interact in bosonic-bondonic formation; in this framework, the molecular topology as well as combined molecular topology-chemical reactivity approaches are unfolded showing that bondons fulfill quantum entangled behavior.