Ground and Excited States

We know that an electron ina hydrogen atom in a stationary state will be described by one of the atomic orbital functions ?_1s, ?_2s,?_2px and so forth. We can make this statement in a more abbriviated form by saying that the electron is in one of the orbitals ?_1s, ?_2s,?_2px,…, and we use this more economical kind of statement.

The orbital that has associated with the lowest energy is ?_1s if the electron is in this orbital, it has the lowest total energy possible, and we say that the atom is in its electronic ground state. If we were to give more energy, say enough to put it to ?_2px orbital, the atom would be in an electronic excited state. In general, for any atom or molecule, the state in which all the electrons are in teh possible energy orbitals is the electronic ground state. Any higher energy state is called an electronic excited state.

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.

The rule is based on teh electron pair repulsion model, which postulates that because electron pairs repel each other, they will try to stay as far apart as possible. In trigonal and tetrahedral geometries, the shape will be exactly trigonal (120° bond angles), or exactly tetrahedral (109.5° bond angles) if the electron groups are all equivalent, as for example in BH₃, or CH₃⁺ (trigonal), or in CH₄, or NH₄⁺ (tetrahedral).

If the groups are not all equivalent, the angles will deviate from the ideal values. Thus in NH, (four electron groups, three in N-H bonds, one an unshared pair), the unshared pair, being attracted only by the nitrogen nucleus, will be closer to the nitrogen on the average than will the bonding pairs, which are also attracted by a hydrogen nucleus. Therefore the repulsion between the unshared pair and a bonding pair is greater than between two bonding pairs, and the bonding pairs will be pushed closer to each other. The H-N-H angle should therefore be less than 109.5°. It is found experimentally to be 107°. Similarly, in H₂O (four electron groups, two unshared pairs, and two 0-H bonds), the angle is 104.5°.

Ambiguity may arise when more than one structure contributes. Then unshared pairs in one structure may become multiple bonds in another, so that the number of electron groups around a given atom is not the same in both structures. An example is methyl azide (19). The central nitrogen is clearly linear (two electron groups), but the nitrogen bonded to CH₃, has three electron groups in

19a and four in 19b. In such a situation, the number of electron groups is determined from the structure with the larger number of honds. Thus the nitrogen in question in 19 is trigonal, not tetrahedral.

and so forth, are to a first approximation constant from molecule to molecule, and one can therefore tell immediately from the Lewis structure of a substance that one has never encountered before roughly what the chemical properties will be.

There is a class of structures, however, for which the properties are not those expected from the Lewis structure. A familiar example is benzene, for which the heat of hydrogenation (Equation 1.1) is less exothermic by about 37 kcal/mole than one would have expected from Lewis structure 1 on the basis of the measured

heat of hydrogenation of ethylene. The thermochemical properties of various
types of bonds are in most instances transferable with good accuracy from molecule to molecule; a discrepancy of this magnitude therefore requires a fundamental modification of the bonding model. The difficulty with model 1 for benzene is that there is another Lewis structure, 2, which is identical to 1 except for the placement of the double bonds.

Whenever there are 2 alternative structures for a single compound, and any one of the strucutre becomes an inaccurate representation for the molecular structure. The actual structure of the molecule is actually a hybrid of these 2 strucutres. It is like a “superposition” of these 2 strucutres. The superposition of two more more Lewis structures into a composite picture of the compound is called resonance.

This terminology is well established, but unfortunate, because the term resonance when applied to a pair of pictures tends to convey the idea of a changing back and forth with time. It is therefore difficult to avoid the pitfall of thinking of the benzene molecule as a structure with three conventional double bonds, of the ethylene type, jumping rapidly back and forth from one location to another. This idea is incorrect. The electrons in the molecule move in a field of force created by the six carbon and six hydrogen nuclei arranged around a regular hexagon.

Each of the six sides of the hexagon is entirely equivalent to the otehr side, which is why electrons should, even momentarily, seek out three sides and make them different from the other three, as the two alternative pictures 3 seem to imply that they do. The symmetry of the ring of nuclei (4) is called a sixfold symmetry because rotating the picture by one-sixth of a circle will give the identical picture again. This sixfold symmetry must be reflected in the electron distribution.

A less misleading picture would be the one above, in which the circle in the middle of the ring implies a distribution of the six double bond electrons of the same symmetry as the arrangement of nuclei. We shall nevertheless usually continue to use the notation 3, as it has certain advantages for thinking about reactions.

The most important features of the structures for which the resonance is needed are, first, that the molecule is more stable (of lower energy) than one would expect from looking at one of the individual strucutres. and second that the actual distribution of the electrons in the molecule is different form what one would expect from either of the resonance strucutres. Since the composite picture shows that certain electrons are free to move to a larger area, of the molecule than a single one of the  structure implies, resonance is often referred to as delocalization. We shall have more to say about delocalization later in connection with molecular orbitals.

While the benezene ring is the most fimiliar example of the necessity for modifying the Lewis strucutre language by addition of resonance concept, there are many others. The carboxylic acids, for example are much stronger acids than the alcohols; this difference must be largely due to greater stability of the carboxylate ion over the alkoxide iiion. It is the possiblity of writing two equivalent Lewis structures for the carboxylate ions that alertts us to this difference.

Another example is the allylic system. The ally1 cation (8), anion (9), and

radical (10), are all more stable than their saturated counterparts. Again, there is for each an alternativestructure :

In all the examples we have considered so far, the alternative structures have been equivalent. This will not always be the case, as the following examples illustrate :

Whenever there are nonequivalent strucutres, each will contribute to the composite picture to a different extent. The structure would represent the most stable (lowest-energy) molecule, where such a molecule actually to exist, contributes the most to composite, and the others successively less as they represent the higher energy molecules.

It is because the lowest-energy structures are most important that we specifiedin the rules for writing Lewis structures that the number of bonds should be maximum and the valence-shell occupancy not less than 8 whenever possible. Structures that violate these stipulations, such as 11 and 12, represent high-energy forms and hence do not contribute significantly to the structural pictures, which

The followinp; rules are useful in using resonance notatinn:

1. All nuclei must be in the same location in every structure. Structures with nuclei in different locations, for example 15 and 16, are chemically distinct substances, and interconversions between them are actual chemical changes, always designated by .

   

2. Structures with fewer bonds or with greater seperation of formal charge are less stable than those with more bonds or less charge seperation. thus 11 and 12 are higer energy respectively than 12 and 14.

3. When 2 structures with formal charge have the same number of bonds and approximately same charge seperation, the structure with charge on the more electronegative atom will usually be somewhat in the lower energy state, but the difference will ordinarily be small enough that both structures can be included in the composite picture.



4. All four groups attached to a pair of atoms joined by a double bond in any structure must lie in the same plane. For example, structure 18b cannot contribute because the bridged ring prevents the carbon 6 and 7 from lying in the same place as carbon 3, and the hydrogen on carbon 2. The impossiblity of strucures with double bonds at bridgeheads of small bridged rings is known as Bredt’s rule. Double bonds can occur at bridgehead if the rings are sufficiently large.
   

MODELS OF CHEMICAL BONDING

Understanding and progress in natural science rest largely on models. A little reflection will make it clear that much of chemical thinking is in terms of models, and that the models useful in chemistry are of many kinds. Although we cannot see atoms, we have many excellent reasons for believing in them, and when we think about them we think in terms of models. For some purposes a very simple
model suffices. Understanding stoichiometry, for example, requires only the idea of atoms as small lumps of matter that combine with each other in definite proportions and that have definite weights. The mechanism by which the atoms are held together in compounds is not of central importance for this purpose. When thinking about stereochemistry, we are likely to use an actual physical model consisting of small balls of wood or plastic held together by springs or sticks. Now the relative weights of atoms are immaterial, and we do not bother to reproduce them in the model; instead we try to have the holes drilled carefully so that the model will show the geometrical properties of the molecules. Still other models are entirely mathematical. We think of chemical rate processes in terms of sets of differential equations, and the details of chemical bonding require still more abstract mathematical manipulations. The point to understand is that there may be many ways of building a model for a given phenomenon, none of which is complete but each of which serves its special purpose in helping us understand some
aspect of the physical reality.

The Electron Pair Bond-Lewis Structures

The familiar Lewis structure is the simplest bonding model in common use in organic chemistry. It is based on the idea that, at the simplest level, the ionic bonding force arises from the electrostatic attraction between ions of opposite charge, and the covalent bonding force arises from sharing of electron pairs between atoms. The starting point for the Lewis structure is a notation for an atom and its valence electrons. The element symbol represents the core, that is, the nucleus and all the inner-shell electrons. The core carries a number of positive charge equal to the number of valence electrons. This positive charge is called corecharge. Valence electrons are shown explicitly. For elements in the third and later rows ofthe periodic table, the d electrons in atoms of Main Groups 111, IV, V, VI, and VII are counted as part of the core. Thus :

Ions are obtained by adding or removing electrons. The charge on an ion is given by

charge = core charge – number of electrons shown exvlicidy

An ionic compound is indicated by writing the Lewis structures for the two ions. A covalent bond model is constructed by allowing atoms to share pairs of electrons. Ordinarily, a shared pair is designated by a line:

H-H

All valence electrons of all atoms in the structure must be shown explicitly. Those electrons not in shared covalent bonds are indicated as dots, for example:

If an ion contains two or more atoms covalently bonded to each other, the total charge on the ion must equal the total core charge less the total number of electrons, shared and unshared:

In order to write-correct Lewis structures, two more concepts are needed. First, consider the total number of electrons in the immediate neighborhood of each atom. This number is called the valence-shell occupancy of the atom, and to find it, all unshared electrons around the atom and all electrons in bonds leading to the atom must be counted. The valence-shell occupancy must not exceed 2 for hydrogen and must not exceed 8 for atoms of the first row of the periodic table. For elements of the second and later rows, the valence-shell occupancy may exceed 8. The structures

are acceptable.
The second idea is that of formal charge. For purposes of determining formal charge, partition all the electrons into groups as follows: Assign to each atom all of its unshared pair elec_tronsa nd half of all electrons in bonds leading it. Call the number of electrons assigned to the atom by this process its electron ownership. The formal charge of each atom is given by

formal charge = core charge – electron ownership

To illustrate formal charge, consider the hydroxide ion, OH-. The electron ownership of H is 1, its core charge is + 1, and its formal charge is therefore zero. The electron ownership of oxygen is 7, and the core charge is +6; therefore the formal charge is – 1. All nonzero formal charges must be shown explicitly in the
structure. The reader should verify the formal charges shown in the following examples

The algebraic sum of all formal charges in a structure is equal to the total charge. Formal charge is primarily useful as a bookkeeping device for electrons, but it also gives a rough guide to the charge distribution within a molecule. In writing Lewis structures, the following procedure is to be followed:

1. Count the total number of valence electrons contributed by the electrically neutral atoms. If the species being considered is an ion, add one electron to the total for each negative charge; subtract one for each positive charge.
2. Write the core symbols for the atoms and fill in the number of electrons determined in Step 1. The electrons should be added so as to make the valence shell occupancy of hydrogen 2 and the valence-shell occupancy of other atoms not less than 8 wherever possible.
3. Valence-shell occupancy must not exceed 2 for hydrogen and 8 for a first-row atom; for a second-row atom it may be 10 or 12.
4. Maximize the number of bonds, and minimize the number of unpaired erectrons, always taking care not to violate Rule 3.
5. Find the formal charge on each atom.

We shall illustrate the procedure with two examples.