Analysis of Bond Orders

In MO theory, the Mulliken population analysis [149], where the population of the MOs is decomposed into atomic populations and diatomic overlap populations, considerably influenced the formulation of bonding concepts, although it does not describe a bond order per se. On the basis of the Mulliken population analysis, Mayer introduced a popular definition of the number of bonds between two centers A and B originally formulated for one-determinant wave functions [142-144, 150], 

I 8 Chemical Bonding in Open-Shell Transition-Metal Complexes 

In order to obtain a bond order formula for open-shell systems that can be applied for both the indep)endent-partide model and correlated wave functions and which simultaneously yields unique bond orders for all spin multiplet components (in the absence of a magnetic field), Alcoba et al. [151, 152] derived a general expression (in the Hilbert space partitioning scheme) from a second-order reduced density matrix. Furthermore, as the first- and second-order reduced density matrices are invariant with respect to the spin projection, they are only a function of the total spin or similarly of the maximum projection S = and the bond order can be evaluated for the highest spin-projected state = S. They arrived at the following expression for the bond order 

For a reliable estimate of the bond multiplicity, the active space should contain (at least partially) the bonding and antibonding MOs that describe, for example, the metal-metal bond of interest In the case of strong bonds, for which rjj, 2, while ri 0, the corresponding EBO contribution is equal to 1. For weak bonds, ri strongly deviates from 2 and vice versa from 0 which reduces the overall EBO, while in the limit rjf, 1, the contribution to the total EBO vanishes and the 

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A theoretical study of the thermal decomposition kinetics of ethyl fluoride 1,1-difluoroethane 1,1,1-trifluoroethane and 1,1,2,2-tetrafluoroethane has been carried out at the B3LYP/6-31-H-G, B3PW91/6-31-H-G, and MP2/6-31-h-hG levels of theory. The calculated data demonstrate that in the HF elimination reaction of the compounds studied, the polarization of the C(l)-F(3) bond is rate determining. Analysis of bond order, charges, bond indexes, and synchronicity parameters suggests that HF elimination occurs through a concerted and asynchronous four-membered cyclic TS type of mechanism. 

In this chapter, we use the definitions of bond order and valence indices provided by Mayer [4-6] (for a historical account, see Ref. [6a] and for other types of bond indices, see Ref. [6b]). In terms of electronic structure theory, they represent an extension to Mulliken s population analysis. The bond order is defined as... 

One way of getting rid of distortions and basis set dependence could be that one switches to the formalism developed by Bader [12] according to which the three-dimensional physical space can be partitioned into domains belonging to individual atoms (called atomic basins). In the definition of bond order and valence indices according to this scheme, the summation over atomic orbitals will be replaced by integration over atomic domains [13]. This topological scheme can be called physical space analysis. Table 22.3 shows some examples of bond order indices obtained with this method. Experience shows that the bond order indices obtained via Hilbert space and physical space analysis are reasonably close, and also that the basis set dependence is not removed by the physical space analysis. 

Lee etal. [129] have studied adsorption configuration and local ordering of sdicotungstate anions (STA) on Ag(lOO) electrode surfaces. Voltammetric studies have shown that STA passivates the Ag surface and thus slows down the electron transfer the dissolved redox species participate in. STA species is oriented with its fourfold axis perpendicular to the Ag(lOO) surface and the center of the STA molecule is located 4.90 A above the top layer of the Ag substrate. From the analysis of bond lengths, it has been found that four terminal O atoms are located near the hollow sites and that an Ag—O bond length is... 

Let us now evaluate exo-endo selectivity in cyclopropene addition to benzo[c]heterocycles. On the basis of bond order analysis, the exo addition of cyclopropene to benzo[c]furan was selected over the endo addition while in the case of benzo[c]pyrrole and benzo[c]thiophene, it was suggested that SOI might be responsible for formation of an exo cycloadduct product. The computed activation energies for cyclopropene addition to the benzo[c]-fused heterocycle clearly favors formation of the exo cycloadduct. The activation barrier with benzo[c]furan was... 

Carbon dioxide is a molecule with two atoms attached (SN = 2) to the central atom via double bonds. The electrons in each double bond must be between C and O, and the repulsion between these electron groups forces a linear structure on the molecule. Sulfur trioxide has three atoms bound to the sulfur (SN = 3), with equivalent partial doublebond character between sulfur and each oxygen, a conclusion rendered by analysis of its resonance forms. The best positions for the oxygens to minimize electron-electron repulsions in this molecule are at the corners of an equilateral triangle, with O—S—O bond angles of 120°. The multiple bonding does not affect the geometry, because aU three bonds are equivalent in terms of bond order. 

The main requirement in the determination of bond orders is to derive rules on how to measure the number of electrons shared between two atoms. For this purpose, a definition of an atom in a molecule is required, which, however, cannot be formulated in a unique and unambiguous way [169]. Quantum chemical calculations are typically performed in the Hilhert-space analysis, where atoms are defined by their basis orbitals. Such an analysis, however, strongly depends on both the atomic basis set chosen and the type of wave function used. The position-space representation, on the other hand, where atoms are defined as basins in three-dimensional physical space does not suffer from these insufficiencies. In this chapter, we present one option for a three-dimensional atomic decomposition scheme and the reader is referred to Refs. [170-173] for further examples. 

Photoinduced infrared absorption studies show that the photoinduced infixired modes are much weaker in intensity, than the photoinduced electronic transition [23], in contrast to the behavior of polyacetylene [51,52] and polythiophene [6]. Analysis, within the amplitude mode formalism [53], indicates that the polarons are massive, > or -60 m, while use of bond order [54] or Holstein [55] polaron formalisms leads to an even larger estimate of the mass of the polarons. Photoexcitation, into either the exciton peak or the 7C-to-7t peak of emeraldine base, produces essentially identical long-lived photoinduced infrared modes [29]. The long-lived photoinduced spectra of the leucoemeraldine base are much weaker. Similarly, the photoinduced infrared absorptions in the pemigraniline base are much weaker, than the photoinduced electronic transitions, again indicating massive photoinduced defects [30]. 

While seminal works intended to reveal SOC effects on the bonding schemes were discussed in term of spinors [9-13], canonical molecular spinors are not suited for the bonding analysis in complex systems, as opposed to small and/or symmetric model systems. Some have promoted the use of localized spinors [14], and in order to recover some chemical significance in terms of bonding, lone pairs and core orbitals, natural spinors similar to natural orbitals in the non-relativistic firameworks have been derived and implemented [15, 16]. It is worth noting that the concept of bond order in the context of multiconfigurational wave functions have been extended recently to two-step spin-orbit coupling approaches [17]. 

The simple reason for this is now well established quantum mechanics, like relativity, is the nonclassical theory of motion in four-dimensional space-time. All theories, formulated in three-dimensional space, which include Newtonian and wave mechanics, are to be considered classical by this criterion. Wave mechanics largely interprets elementary matter, such as electrons, as point particles, forgetting that the motion of particulate matter needs to be described by particle (Newtonian) dynamics. TF and HF simulations attempt to perform a wavelike analysis and end up with an intractable probability function. On assuming an electronic wave structure, the problem is simplified by orders of magnitude, using elementary wave mechanics. Calculations of this type are weU within the ability of any chemist without expertise in higher mathematics. It has already been shown that the results reported here define a covalence function that predicts, without further assumption, interatomic distances, bond dissociation energies, and harmonic force constants of all purely covalent interactions, irrespective of bond order. In line with the philosophy that... 

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