Overlapping between two atoms

A covalent bond arises from the sharing of electrons between atoms. This results in an increase in electron density between the two atoms. Thus, covalent bonds are represented as the overlap of atomic orbitals. The overlap between two atomic orbitals, i2, on atoms 1 and 2 is represented in terms of the overlap integral, S, which is defined as... 

The net atomic population neglects the electrons associated with the overlap between two atoms. Mulliken arbitrarily divided the overlap population equally between the two atoms, producing the gross atomic population... 

A feature common to the semi-empirical methods is that the overlap matrix, S (in Equation (2.225)), is set equal to the identity matrix I. Thus all diagonal elements of the overlap matrix are equal to 1 and all off-diagonal elements are zero. Some of the off-diagonal elements would naturally be zero due to the use of orthogonal basis sets on each atom, but in addition the elements that correspond to the overlap between two atomic orbitals on different atoms are also set to zero. The main implication of this is that the Roothaan-Hall equations are simplified FC = SCE becomes FC = CE and so is immediately in standard matrix form. It is important to note that setting S equal to the identity matrix does not mean that aU overlap integrals are set to zero in the calculation of Fock matrix elements. Indeed, it is important specifically to include some of the overlaps in even the simplest of the semi-empirical models. 

There are two mechanisms by which a phase change on the ground-state surface can take place. One, the orbital overlap mechanism, was extensively discussed by both MO [55] and VB [47] formulations, and involves the creation of a negative overlap between two adjacent atomic orbitals during the reaction (or an odd number of negative overlaps). This case was temied a phase dislocation by other workers [43,45,46]. A reaction in which this happens is... 

The space filling model developed by Corey, Pauling, and Koltun is also known as the CPK model, or scale model [197], It shows the relative volume (size) of different elements or of different parts of a molecule (Figure 2-123d). The model is based on spheres that represent the "electron cloud . These atomic spheres can be determined from the van der Waals radii (see Section 2.10.1), which indicate the most stable distance between two atoms (non-bonded nuclei). Since the spheres are all drawn to the same scale, the relative size of the overlapping electron clouds of the atoms becomes evident. The connectivities between atoms, the bonds, are not visualized because they are located beneath the atom spheres and are not visible in a non-transparent display (see Section 2.10). In contrast to other models, the CPK model makes it possible to visualize a first impression of the extent of a molecule. 

Valence bond and molecular orbital theory both incorporate the wave description of an atom s electrons into this picture of H2 but m somewhat different ways Both assume that electron waves behave like more familiar waves such as sound and light waves One important property of waves is called interference m physics Constructive interference occurs when two waves combine so as to reinforce each other (m phase) destructive interference occurs when they oppose each other (out of phase) (Figure 2 2) Recall from Section 1 1 that electron waves m atoms are characterized by their wave function which is the same as an orbital For an electron m the most stable state of a hydrogen atom for example this state is defined by the Is wave function and is often called the Is orbital The valence bond model bases the connection between two atoms on the overlap between half filled orbifals of fhe fwo afoms The molecular orbital model assembles a sef of molecular orbifals by combining fhe afomic orbifals of all of fhe atoms m fhe molecule... 

The characteristic feature of valence bond theory is that it pictures a covalent bond between two atoms in terms of an m phase overlap of a half filled orbital of one atom with a half filled orbital of the other illustrated for the case of H2 m Figure 2 3 Two hydrogen atoms each containing an electron m a Is orbital combine so that their orbitals overlap to give a new orbital associated with both of them In phase orbital overlap (con structive interference) increases the probability of finding an electron m the region between the two nuclei where it feels the attractive force of both of them... 

Valence bond theory (Section 2 3) Theory of chemical bond mg based on overlap of half filled atomic orbitals between two atoms Orbital hybridization is an important element of valence bond theory... 

A covalent bond is formed when an electron pair is shared between atoms. According to valence bond theory, electron sharing occurs by overlap of two atomic orbitals. According to molecular orbital (MO) theory, bonds result from the mathematical combination of atomic orbitals to give molecular orbitals, which belong to the entire molecule. Bonds that have a circular cross-section and are formed by head-on interaction are called sigma (cr) bonds bonds formed by sideways interaction ot p orbitals are called pi (77-) bonds. 

Explain the relevance of atomic orbital overlap and of molecular orbital filling to the strength of the bond formed between two atoms. 

Because stability depends on the ability of the particles to remain at discrete distances from each other, the well-known relation described by Morse (5) can be used as a starting point for stabilization mechanisms. As shown in Figure 3, two uncharged (and nonrepelling) bodies approach each other until they have attained an equilibrium distance corresponding to the position of minimum energy. The solid line actually represents a compromise between the repulsive forces operative between two atoms when their electron clouds overlap and the attraction which always exists between two bodies. 

An example of quantum mechanical schemes is the oldest and most widely used Mulliken population analysis [1], which simply divides the part of the electron density localized between two atoms, the overlap population that identifies a bond, equally between the two atoms of a bond. Alternatively, empirical methods to allocate atomic charges to directly bonded atoms in a reasonable way use appropriate rules which combine the atomic electronegativities with experimental structural information on the bonds linking the atoms of interest. A widely used approach included in many programs is the Gasteiger-Hiickel scheme [1]. 

Both carbon atoms in ethylene molecule undergo sp2 hybridization and form three identical sp2 hybrid orbitals. One p orbital remains unhybridized. Two sp2 hybrid orbitals from each carbon atom overlap end to end with the Is orbital of a hydrogen atom and four C — Ho bonds are formed in total. Also, between the two carbon atoms, a C — Co bond is formed as a result of the overlap between two sp2 hybrid orbitals. So, in the C2H4 molecule in total there are five o bonds. Meanwhile, the unhybridized p orbitals of the two carbon atoms overlap side by side and form a rt bond. So between the two carbon atoms in the C2H4 molecule there is one o bond, formed by the overlapping of sp2 hybrid orbitals and one n bond, formed by the side by side overlapping of the unhybridized p orbitals. In total, two bonds are formed, hence a double bond exists between the two carbon atoms. 

You know that a covalent bond involves the sharing of a pair of electrons between two atoms each atom contributes one electron to the shared pair. In some cases, such as the hydronium ion, HsO", one atom contributes both of the electrons to the shared pair. The bond in these cases is called a co-ordinate covalent bond. In terms of the quantum mechanical model, a co-ordinate covalent bond forms when a filled atomic orbital overlaps with an empty atomic orbital. Once a co-ordinate bond is formed, it behaves in the same way as any other single covalent bond. The next Sample Problem involves a polyatomic ion with a co-ordinate covalent bond. 

In the overlap region between two atoms (A and B), densities are additive and the total electron density is... 

Molecular orbital theory of the covalent bond shows a direct relationship between the extent of the overlap of two atomic orbitals and the bond strength. The larger the overlap, the stronger the bond. Maximum overlapping would produce the strongest bond and the most stable system. Maximum overlap of the H and O atomic orbitals... 

A strongly stabiliztng interaction between two atoms in which certain of their atomic orbitals overlap, thereby resulting in a region of high electron density. This sharing of two electrons in the orbital by the two bonded atoms describes a molecular orbital. 

Mulliken population analysis. This treatment shows how a molecule under investigation distributes electrons according to the atomic orbital occupancy. Note that the overlap population between two atoms is divided evenly between them without consideration of possible differences in atom types, electronegativities, and so on. 

Hirshfeld (1964) pointed out that bond bending not only occurs in ring systems, but also results from steric repulsions between two atoms two bonds apart, referred to as 1-3 interactions. The effect is illustrated in Fig. 12.3. The atoms labeled A and A are displaced from the orbital axes, indicated by the broken lines, because of 1-3 repulsion. As a result, the bonds defined by the orbital axes are bent inwards relative to the internuclear vectors. When one of the substituents is a methyl group, as in methanol [Fig. 12.3(b)], the methyl-carbon-atom hybrid reorients such as to maximize overlap in the X—C bond. This results in noncolinearity of the X—C internuclear vector and the three-fold symmetry axis of the methyl group. Structural evidence for such bond bending in acyclic molecules is abundant. Similarly, in phenols such as p-nitrophenol (Hirshfeld... 

---