Bond-Orders

Single bond, BO = 1 double bond, BO = 2 triple bond, BO = 3. 

Thus the configuration represents a BO of 1, equivalent to a single bond, as in the H2 diatomic molecule, i.e. a two-electron pair single bond. In the cation, with only a single electron in the bonding MO, this represents a BO of 0.5, as does the He2 cation as the 

The interaction of two atoms results in the formation of a (3, — 1) or bond critical point in the charge density. The trajectories of the gradient vector field of p which originate and terminate at this critical point define the bond path and interatomic surface, respectively. Thus the properties of the charge density at such a critical point serve to 

For densities obtained using the 6-3IG basis, the bond order n is given by  

The MO picture gives an easy appreciation of the concept of the bond order of a chemical bond. The bond order quantifies the degree of covalent bonding between atoms. A simple estimate of bond order between two atoms, A and B, can be gained from a simple inspection of the MO diagram using the definition 

The hypothetical molecule Hc2 would have a bond order of 0. In fact, the overlap integral in the denominator of the energy-level equations. Equations (7.22) and (7.23), means that the ungerade symmetry MO destabilizes this system slightly more than the stabilization gained by filling the l(Tg+ level. Accordingly, Hc2 is unstable with respect to isolated He atoms, and so helium is a monatomic gas. 

The bond order from Equation (7.24) is a useful mle of thumb estimate of the relative bond strengths to expect from MO diagrams. However, it does not take into account several factors which are important for estimating actual bond energies. 

These are important points for any quantitative work, electron-electron interactions must be taken into account, and the theories underpinning computation of MOs do this at various levels of accuracy. Approaches such as Hartree-Fock or density functional theory adapt the Hamiltonian operator to include electron-electron terms in an averaged way so electrons see the Coulomb field of each other averaged over the calculated density associated with each MO (see the Further Reading section in this chapter). 

Four important generalizations can be made from this section  

In molecular orbital theory, the stability of a covalent bond is related to its bond order, defined as half the difference between the number of bonding electrons and the number of antibonding electrons  

Bond order = (no. of bonding electrons — no. of antibonding electrons) [9.1] 

We take half the difference because we are used to thinking of bonds as pairs of electrons. A bond order of 1 represents a single bond, a bond order of 2 represents a double bond, and a bond order of 3 represents a triple bond. Because MO theory also treats molecules containing an odd number of electrons, bond orders of 1/2,3/2, or 5/2 are possible. 

Suppose one electron in H2 is excited from the ris MO to the MO. Would you expect the H atoms to remain bonded to each other, or would the molecule fall apart  

Which electrons in this diagram contribute to the stability of the He2 ion  

Because He2 has two bonding electrons and two antibonding electrons, it has a bond order of 0. A bond order of 0 means that no bond exists. 

The strength of a bond in a molecule is the net outcome of the bonding and antibonding effects of the electrons in the orbitals. The bond order, b, in a diatomic molecule is defined as 

7) Write the ground-state electronic configuration and deduce 

With molecular orbital diagrams such as those for H2 and Hc2, we can begin to see the power of molecular orbital theory. For a diatomic molecule described using molecular orbital theory, we can calculate the bond order. The value of the bond order indicates, qualitatively, how stable a 

This molecular orbital diagram shows the two H atoms having electrons with paired spins ( and J,). H atoms whose electrons have parallel spins (t and t or J, and j.) actually repel one another and will not bond to form H2. 

CHAPTER 9 Chemical Bonding II Molecular Geometry and Bonding Theories 

Li atom 0-2S Li2 molecule Li atom Be atom Bc2 molecule Be atom 

When we draw molecular orbital diagrams, we need only show the valence orbitals and electrons. 

In the case of H2, where both electrons reside in the orbital, the bond order is [(2 - 0)/2] = 1. In the case of Hc2, where the two additional electrons reside in the cr orbital, the bond order is [(2 2)/2] = 0. Molecular orbital theory predicts that a molecule with a bond order of zero will 

As predicted by molecular orbital theory, Li2, with a bond order of 1, is a stable molecule, whereas Be2, with a bond order of 0, does not exist. 

To consider diatomic molecules beyond Be2, we must also consider the combination of p atomic orbitals. Like 5 orbitals, p orbitals combine both constmctively, to give bonding molecular orbitals that are lower in energy than the original atomic orbitals, and destructively, to give antibonding 

By the use of elementary number theory, to simulate uniform distribution of a valence electron over the ionization sphere, a complete set of ionization radii and electronegativities is now available for the simulation of a whole range of chemical properties as described in the papers to follow. 

On looking for a relationship between ionization radius and the chemistry of homonuclear covalent interaction, the classification into single and multiple bonds is followed as a first approximation. An immediate observation, valid for most single bonds, is a constant value of the dimensionless distance 

Divergence angles of n/% and r/16 for integer and half-integer bond orders are implied and shown in Fig. 14. This solution corresponds with the empirical values derived before, i.e.. 

The corresponding numerical solution defines bond order, b, by the equation 

The quantized variation of bond order may be rationalized by viewing overlapping charge spheres as spherical standing waves. These waves interfere constructively at specific interatomic distances that depend on wavelength. Destructive interference that occurs at intermediate distances tends to destabilize the interaction and to prevent continuous variation of bond order. Distortion of the interference pattern requires work, as measured by bond-stretching force constants. 

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Fournier R and Salahub D R 1990 Chemisorption and magnetization A bond order-rigid band model Surf. Sol. 238 330-40... 

The systematic lUPAC nomenclature of compounds tries to characterize compounds by a unique name. The names are quite often not as compact as the trivial names, which are short and simple to memorize. In fact, the lUPAC name can be quite long and cumbersome. This is one reason why trivial names are still heavily used today. The basic aim of the lUPAC nomenclature is to describe particular parts of the structure (fi agments) in a systematic manner, with special expressions from a vocabulary of terms. Therefore, the systematic nomenclature can be, and is, used in database systems such as the Chemical Abstracts Service (see Section 5.4) as index for chemical structures. However, this notation does not directly allow the extraction of additional information about the molecule, such as bond orders or molecular weight. 

With such a matrix representation, the storage space is dependent only on the number of nodc.s (atoms) and independent of the number of bonds. As Figure 2-14 dcmon.stratcs, all the e.sscntial information in an adjacency matrix can also be lound in the much smaller non-rediindant matrix. But the adjacency matrix is unsuitable for reconstructing the constitution of a molecule, because it does not provide any information about the bond orders. 

Both the adjacency and distance matrices provide information about the connections in the molceular structure, but no additional information such as atom type or bond order. One type of matrix which includes more information, the Atom Connectivity Matrix (ACM), was introduced by Spialtcr and is discussed in Ref, [38]. This approach was eventually abandoned but is listed here because it was quite a unique approach. 

The bond matrix is related to the adjacency matrix but gives information also on the bond order of the connected atoms. Elements of the matrix obtain the value of 2 if there is a double bond between the atoms, c.g, between atoms 2 and 3... 

Adjactney matrix describes connections of atoms contains only 0 and 1 (bits) no bond types and bond orders no number of free electrons... 

Distunct matrix describes geometric distances no bond types or bond orders no number of free electrons cannot be represented by bits... 

There are many ways of presenting a connection table. One is first to label each atom of a molecule arbitrarily and to arrange them in an atom list (Figure 2-20). Then the bond information is stored in a second table with indices of the atoms that are connected by a bond. Additionally, the bond order of the corresponding coimection is stored as an integer code (1 = single bond, 2 = double bond, etc.) in the third column. 

Both tables, the atom and the bond lists, are linked through the atom indices. An alternative coimection table in the form of a redundant CT is shown in Figure 2-21. There, the first two columns give the index of an atom and the corresponding element symbol. The bond list is integrated into a tabular form in which the atoms are defined. Thus, the bond list extends the table behind the first two columns of the atom list. An atom can be bonded to several other atoms the atom with index 1 is connected to the atoms 2, 4, 5, and 6. These can also be written on one line. Then, a given row contains a focused atom in the atom list, followed by the indices of all the atoms to which this atom is bonded. Additionally, the bond orders are inserted directly following the atom in-... 

RAMSES is usually generated from molecular structures in a VB representation. The details of the connection table (localized charges, lone pairs, and bond orders) are kept within the model and are accessible for further processes. Bond orders are stored with the n-systems, while the number of free electrons is stored with the atoms. Upon modification oF a molecule (e.g., in systems dealing with reactions), the VB representation has to be generated in an adapted Form from the RAMSES notation. 

The most well-known and at the same time the earliest computer model for a molecular structure representation is a wire frame model (Figure 2-123a). This model is also known under other names such as line model or Drciding model [199]. It shows the individual bonds and the angles formed between these bonds. The bonds of a molecule are represented by colored vector lines and the color is derived from the atom type definition. This simple method does not display atoms, but atom positions can be derived from the end and branching points of the wire frame model. In addition, the bond orders between two atoms can be expressed by the number of lines. 

The optimization of the backtracking algorithm usually consists of an application of several heuristics which reduce the number of candidate atoms for mapping from Gq to Gj. These heuristics are based on local properties of the atoms such as atom types, number of bonds, bond orders, and ring membership. According to these properties the atoms in Gq and Gj are separated into different classes. This step is known in the literature as partitioning [13]. Table 6.1 illustrates the process of partitioning. 

Figure 10.1-3. Two regioisomeric products of the training data set and their corresponding physicochemical effects used as coding vectors bo bond order difference in tr-electro-...

Covalen t radii for all th e clem cn ts are readily available an d the bond orders of all bonds arc available from the molecular graph. Prior to describing the explicit default parameter scheme, it is nee-... 

As with atomic charges, the bond order is not a quantum mechanical observable and so anuus methods have been proposed for calculating the bond orders in a molecule. 

Table 2.6 Bond order obtained from the Mayer bond order scheme [Mayer 1983].

Mayer defined the bond order between two atoms as follows [Mayer 1983] ... 

P is the total spinless density matrix (P = P + P ) and P is the spin density matrix (P = p" + P ). For a closed-shell system Mayer s definition of the bond order reduces to ... 

The bond orders obtained from Mayer s formula often seem intuitively reasonable, as illustrated in Table 2.6 for some simple molecules. The method has also been used to compute the bond orders for intermediate structures in reactions of the form H -1- XH HX -1- H and X I- XH -H H (X = F, Cl, Br). The results suggested that bond orders were a useful way to describe the similarity of the transition structure to the reactants or to the products. Moreover, the bond orders were approximately conserved along the reaction pathway. 

As with methods for allocating electron density to atoms, the Mayer method is not necessarily correct, though it appears to be a useful measure of the bond order that conforms to accepted pictures of bonding in molecules. 

The Tersoff potential [Tersoff 1988] is based on a model known as the empirical bond-order potential. This potential can be written in a form very similar to the Finnis-Sinclair potential ... 

The key term is which is the bond order between the atoms i and j. This parameter depends upon the number of bonds to the atom i the strength of the bond between i and j decreases as the number of bonds fo fhe atom i increases. The original bond-order potential [Abell 1985] is mathematically equivalent to the Finnis-Sinclair model if the bond order by is given by ... 

The Tersoff potential was designed specifically for the group 14 elements and extends the basic empirical bond-order model by including an angular term. The interaction energy between two atoms i and j using this potential is ... 

The function/c is a smoothing function with the value 1 up to some distance Yy (typically chosen to include just the first neighbour shell) and then smoothly tapers to zero at the cutoff distance, by is the bond-order term, which incorporates an angular term dependent upon the bond angle 6yk- The Tersoff pofenfial is more broadly applicable than the Stillinger-Weber potential, but does contain more parameters. 

There are a number of different ways that the molecular graph can be conununicated between the computer and the end-user. One common representation is the connection table, of which there are various flavours, but most provide information about the atoms present in the molecule and their connectivity. The most basic connection tables simply indicate the atomic number of each atom and which atoms form each bond others may include information about the atom hybridisation state and the bond order. Hydrogens may be included or they may be imphed. In addition, information about the atomic coordinates (for the standard two-dimensional chemical drawing or for the three-dimensional conformation) can be included. The connection table for acetic acid in one of the most popular formats, the Molecular Design mol format [Dalby et al. 1992], is shown in Figure 12.3. 

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