Lewis structures serve admirably for many aspects of mechanistic organic chemistry. Frequently, however, we need a more accurate bonding model.
Models Based on the Quantum Theory
The description of chemical bonding must ultimately be based on an understanding of the motions of electrons. In order to improve our model, we need to appeal to the quantum theory, which summarizes the current understanding of the behavior of particles of atomic and subatomic size. The quantum theory provides the mathematical framework for describing the motions of electrons in molecules. When several electrons are present, all interacting strongly with each other through their mutual electrostatic repulsion, the complexity is so great that exact solutions cannot be found. Therefore approximate methods must be used even for simple molecules. These methodstake various forms, ranging from complex ab initio calculations, which begin from first principles and have no parameters adjusted to fit experimental data, to highly approximate methods such as the Hiickel theory, which is discussed further in Appendix 2. The more sophisticated of these methods now can give results of quite good accuracy for small molecules, but they require extensive use of computing eq~ipmentS.~uc h methods are hardly suited to day-to-day qualitative chemical thinking. Furthermore, the most generally applicable and therefore most powerful methods are frequently simple and qualitative. Our ambitions in looking at bonding from the point of view of the quantum theory are therefore modest. We want to make simple qualitative arguments that will provide a practical bonding model.
Atomic Orbitals
The quantum theory specifies the mathematical machinery required to obtain a complete description of the hydrogen atom. There are a large number of functions that are solutions to the appropriate equation; they are functions of the x, y, and z coordinates of a coordinate system centered at the nuc l eu~E.~ac h of these functions describes a possible condition, or state, of the electron in the atom, and each has associated with it an energy, which is the total energy (kinetic plus potential) of the electron when it is in the state described by the function in question. The functions we are talking about are the familiar Is, 2s, 2P, 3s,. . . atomic orbitals, which are illustrated in textbooks by diagrams like those in Figure 1.1. Each orbital function (or wave function) is a solution to the quantum mechanical equation for the hydrogen atom called the Schrodinger equation. The functions are ordinarily designated by a symbol such as ?, ?, ?, and so on. We shall call atomic orbitals ?, or ?, and designate by a subscript the orbital meant, as for example ?1s, ?2s, and so on. Later, we may abbreviate the notation by simply using the symbols Is, 2s, . . . , to indicate the corresponding orbital functions. Each function has a certain numerical value at every point in space; the value at any point can be calculated once the orbital function is known. We shall never need to know these values, and shall therefore not give the formulas; they can be found in other source.
The important things for our purposes are, first, that the numerical values are positive in certain regions of space and negative in the other regoins and second, that the value of each function approaches zero as one moves farther from the nucleus. In the figure above the positive regions are shaded and negative ones are unshaded.
Imagine walking around inside an orbital and suppose that there is some was of sensing the value – positive, negative and zero – of the orbital function as you walk from point to point. On moving from a positive region to a negative region, you must pass through some point where the value is zero. The collections of all the adjecent points at which the function is zero are called nodes. They are the surfaces in the three dimensional space ans most important ones for our purposes are like those shown in the figure above for p and d orbitals illustated. Nodes can also be spherical and of other shapes, but these are of less concern to a chemist.
The Physical Significance Of Atomic Orbital Function
The fact that an orbital function ? is of different algebraic sign in different regions has no particular physical significance for the behavior of an electron that finds itself in the state defincd by the orbital. (We shall scc shortly that the significance of the signs comes from the way in which orbitals can be combined with each other.) The quantity that has physical meaning is the value at each point of the function ?2, which is positive everywhere, since the square of a negative number is positive. the squared function, ?2, gives the probability of finding the electron at various points in the space. Diagrams like that in the figure below, with the shading of vairous regions or, more succinictly, the electron distribution or electron density, are actually pictures of ?2, and not ? itself. The general shape of ?2 will be similar to the shape of ?. the orbitals and their shapes have no edges, even though definite outlines are usually drawn in the diagrams. The values merely approach closer and closer to zero as one goes away from the nucleus.
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