## Electronic Structure

The most mysterious part of the atom is the electrons. The nucleus is relatively easy to comprehend because we think of it as a core of positive charge. But the electrons have a lot more space (remember the analogy with the football field) and it is natural to inquire about things like--are the electrons moving or are they stationary? If the electrons are moving, how do they move, and finally, what are electrons? These were questions that occupied the minds of the physicists at the beginning of the 20th century, a period that was probably one of the most interesting and influential in the last 100 years of science. This period was alive with other important discoveries and theories. One of the most important set of observations concerns the nature of light. At the time of the great physicist Isaac Newton, the light that allows us to see was thought to consist of tiny particles. It was said that when these particles hit the retina of the eye they induce sight.

Around the turn of the 20th century an experiment produced a result that could not be explained by the particle theory. This experiment is illustrated in Figure 10 where a the light from a light bulb is shown passing through two slits and then striking a screen. If light consists of particles, the expected result was that two images of the slits would appear on the screen--the particles would pass through the slits and strike the screen and cause two bands of light. In fact, the experiment showed that around the two bands of light there were less intense bands that appear to be images of the "original" bands. This result was explained by the suggestion that light actually was a wave, or, at least, has some of the properties of a wave.

Figure 10. The double slit experiment.

Because waves are so important in science we should take a minute to define some of their characteristics. Imagine standing alongside a very still pond. In the tree above you a bird drops a seed that strikes the water in the pond and sends out a set of ripples, or waves.

These waves actually have the shape of what in mathematics is called a sine (or cosine) wave (see Figure 11). The speed at which the waves move outward is their velocity. The distance between the waves is the wavelength, usually designated by the Greek symbol lambda (λ or l). Another characteristic of waves is their frequency, which is the number of wave crests, or troughs, that pass a particular point in the water each second. The frequency (n, Greek ν) is designated as cycles per second, one cycle being one complete motion from the top of a wave to the bottom of the trough and up again to the top of the wave. The unit cycles per second is also called Hertz (abbreviated Hz) in honor of the physicist Heinrich Hertz who did much of the early experimentation on wave motion.

Figure 11. A sine wave.

Now let us return to the double slit experiment. If light has a wave motion, we should be able to use the pebble in the water analogy (scientists frequently use analogy to produce explanations of observations). When a pebble hits a body of water, it will produce a set of waves. Imagine what will happen if these waves encounter a barrier that has two openings. These openings will initiate a new set of waves. Of course, these waves are just the off-spring of the waves created by the pebble so they have the same wavelength and frequency and speed. The waves in turn may "interfere" with one another.

Figure 12 shows that some waves will be "in phase" and will add to produce a larger wave; others will be exactly "out of phase" and crests will cancel troughs to produce no wave at all. These are analogous to the bright spots on the screen of the double-slit experiment and, thus, the experiment has been explained by constructive and destructive interference of the waves.

Figure 12. The Double Slit Experiment - Video

Of course, you will now be thinking, "Okay, I can buy this stuff about the waves, but aren't we talking about light? Light is not like water, so what is it that is moving up and down?" This is a question that baffled scientists for many years and produced models like the one that proposed that light travels through a universal medium called the "ether". Eventually, however, physicists agreed that light is perpendicular oscillating electric and magnetic fields. Light has a constant speed, but it can have a variety of wavelengths and frequencies. Indeed, these three characteristics of light are related by a simple equation:

c = λn

which tells that if c is a constant and if λ increases n must decrease. In mathematical terms we say that λ and n are inversely proportional.

In any event, the term light is now used to mean any electromagnetic radiation--from radio waves, which have long wavelengths of several meters, to visible light, with its wavelengths of about 10^{-5} meters to x-rays which have wavelengths of 10^{-8} meters. The electromagnetic "spectrum" is shown in Figure 13.

Figure 13. The electromagnetic spectrum.

Just when everyone seemed to agree about the nature of light, Einstein explained the photoelectric effect (which we won't describe except to say that it involves the velocity with which electrons leave the surface of metals when the metal is exposed to light) by proposing that light consists of little bundles of energy called **photons**. In essence, this is a bit of a return to a particle theory for light because we now (according to the photon theory) think of light as little packages of electromagnetic radiation. Einstein, moreover, made use of some work that Max Planck had done earlier and showed that these packages of light have an energy given by another simple relationship:

E = hn

where E is the energy of a photon, n is the frequency of the wave, and h is Planck's constant ( a number, 6.627 x 10^{-34} J-s).

We can now turn to our first model of the electronic structure of atoms. Niels Bohr, a Danish physicist, proposed that a hydrogen atom, which has only one electron and one proton, could be thought of as having the electron on one of a series of circular tracks. The electron according to Bohr must move in one circle around the nucleus and if it is undisturbed the electron must remain on that circle or track. The interesting, and very revolutionary thing about Bohr's model is that he proposed that the angular momentum, which is similar to the energy of the electron, can have only certain values. In fact, he proposed that this angular momentum can have values that are integral values of Planck's constant divided by 2^{1}. In other words,

angular momentum = nh/2^{1}

where n is an integer (1, 2, 3, 4, ...). This constituted a major break with classical physics and was the beginning of the quantum era of science. To be sure that you understand this notion, notice that the angular momentum cannot be 1.134h/2^{1}, or 34.5h/2^{1}, but it can be 1h/2^{1} or 2h/2^{1} or 55h/2^{1}. That is, the angular momentum can have only certain values and not others. The angular momentum is said to be **quantized** (from the Latin *quantus*, "how great"). Although we cannot derive the consequences here, it turns out that quantization of the angular momentum also leads to quantization of the radius of the tracks and the energy of the atom.

The first three orbits, or tracks, of the Bohr model and the energies of each are shown in Figure 14. If the electron is in the first orbit, it is characterized by the quantum number n = 1, and it has an energy of -21 x 10^{-19} joules. If it is in the second orbit, n = 2, and the energy is -5 x 10^{-19} joules. Notice again that the energy of the electron cannot be -10 x 10^{-19} joules; it can have only those values dictated by the quantization of the angular momentum.

Figure 14. The first three Bohr orbits and their energies.

The Bohr model was an immediate success because it was able to successfully predict the ionization energy of the hydrogen atom and to predict with very good precision the wavelengths of light that are given off when hydrogen is heated in an electric arc. However, the success of the model was short-lived. It could not predict the ionization energies of other atoms, and there are features of the line spectrum of even hydrogen that it could not explain. Thus, within a few years of Bohr's success, the search was on for another model of electronic structure. The model that was finally proposed by several physicists and that has remained the model used by chemists and physicists today is referred to as the wave model, **wave mechanics**, or **quantum mechanics**. This model incorporates both the wave ideas formulated for light and the quantum ideas proposed by Einstein, Planck, and Bohr.

Basically, the wave model assumes that an electron has some characteristics of electromagnetic radiation--that it can be represented, in other words, by a wave. This model is extremely mathematical in nature, and, moreover, contains a number of features that are almost totally outside of the realm of our everyday experiences. For example, it turns out that the amplitude (actually it is the square of the amplitude) of the wave that represents an electron is proportional to the probabiity of finding the electron. Suppose that a fictitious electron confined to a line has the wave shown in Figure 15(a). The square of the amplitude of the wave is shown in Figure 15(b). Notice that the negative part of the wave (on the right hand side of part (a)) becomes positive when the amplitude is squared (the square of -2 is 4).

Figure 15. A wave for an electron confined to a line.

Thus, if we want to know where we can most likely find the electron we look for that region of the line where the amplitude squared is greatest. In this case, it occurs at 1/4 d and 3/4 d. In the middle of the line the probability of finding the electron is zero. This notion that we can never be sure of the location of the electron, but can only deal in probabilities is somewhat like passing the scene of an accident involving a large truck and a small car. You cannot tell whether anyone was injured but you would be willing to bet that there is a greater probability that the person in the small car was injured than the person in the truck.

Another consequence of the wave model is that the energy of the electron is quantized, just as was true in the Bohr model. Although the wave model is superior to the Bohr model in many respects, it also cannot be solved exactly for the energies of electrons in atoms more complicated than hydrogen. However, there are approximations that can be used in the wave model that give very good estimates of the relative energies of the electrons not only in a variety of atoms, but even in complicated molecules.

The wave model is also similar to the Bohr model in that it uses quantum numbers to characterize an electron. Rather than the one quantum number that was used in the Bohr model, three are required in the wave model. These numbers are given names such as n (the principal quantum number), λ (the angular momentum quantum number), and m (the magnetic quantum number). Each of the numbers can have only certain integral values, and each reveals something different about the energy and the location of the electron. Every unique set of these three quantum numbers is called an **orbital**. For example, if n = 2, λ = 1, and m = 0, this set of numbers is referred to as a 2p orbital (the 2 comes from the value of n, the p from the value of λ). Table 3 lists some of the allowed sets of n, λ, and m for values of n equal to 1 and 2.

Table 3. Allowed sets (orbitals) of n, λ, and m.

Every electron in an atom must, according to the wave model, have a set of n, λ, and m values. Moreover, only two electrons can have the same values of n, λ, and m. To make matters more complicated, two electrons that have the same values of n, λ, and m, must differ in their value of another quantum number (a result of relativity theory, not quantum theory), the spin quantum number, s. The spin quantum number is a strange one--it can have only two values, and these are half-integral. The allowed values of s are +1/2 and -1/2. Now if we include all four of the quantum numbers, the Pauli exclusion principle says that each electron in an atom must have a unique set of four quantum numbers.

If you are feeling a little confused by all of this quantum number mumbo-jumbo, consider this analogy. Think of an empy apartment house with a bunch of rooms, each of the rooms is one apartment and each has an address. The landlord for this apartment has set an unusual and probably illegal rule: no more than two people can occupy each apartment and those people must be of opposite sex. In our analogy, each apartment is an orbital and the people that rent the apartments are the electrons. The sex, male or female, of the people is analogous to the spin of the electrons. If we assume that there is only one set of stairs, and no elevator, it will even be true that each apartment (orbital) has a different location and that a different amount of energy will be required to get to each. Moreover, the floor that each apartment is on is somewhat analogous to the value of n, the principal quantum number; that is, the higher the apartment the greater the value of n. What our analogy does not convey is the probabilistic nature of the electron. In our analogy we know exactly where each person is at any particular time; we also know the exact confines of each apartment. In the strange world of wave mechanics, we know neither of these with certainty. We can only say that a particular electron is in an orbital and that it therefore has such and such a probability of being at a certain point in space (for example, at some point 2 Å from the nucleus).

In spite of the uncertainty about these orbitals, they are always represented in general chemistry texts as solid figures. Some authors are careful to indicate that these solid figures represent an area of space within which the electron can be found 95% of the time. In the orbital representations shown below (Figures 16-18) you will notice that s orbitals are spherical, p-orbitals have two lobes, and d-orbitals are even more complex. You will also notice that p and d orbitals have other designations that come from their values of m. These designations indicate (in most cases) something about on what axes the orbitals have their maximum probabilities. We have also shown several different types of representations for a p-orbital, to give you the chance to see how authors differ in their feelings about how best to represent them. Notice also that p and d-orbitals are sometimes shown with different colors for alternating lobes. These colors indicate the different signs of the equation for the wave responsible for the orbital (see Figure 15 to remind yourself that waves can have positive and negative parts).

Figure 16. Representations of s and p-orbitals.

Figure 17. Some alternative representations of p-orbitals.

The different colors and + and - signs indicate the sign of the mathematical equation that describes the orbital.

Figure 18. Photographs of models representing the d-orbitals.

Carefully examine the orientation of the orbitals relative to the x, y, and z axes.

Finally, we show in Figure 19 an energy level diagram for a fluorine atom with four orbitals completely filled and one orbital, a 2p orbital, half-filled. This is called the **electron** configuration for an atom and is usually designated as:

F 1s^{2}2s^{2}2p^{5}

Figure 19. Energy level diagram for fluorine.

This energy level diagram also suggests that electrons can be "bumped up" into higher energy levels if they somehow are subjected to the right amount of energy. For example, the electron in the half-filled p orbital could be bumped up to the 3s orbital. This would require an amount of energy exactly equal to the difference in energy between the 2p and 3s orbitals. This amount of energy can be supplied by exposing the atom to electromagnetic radiation (for example ultraviolet light). The "bumping up" process is called **excitation** and the resulting atom is said to be excited. After some time has passed the atom will try to "relax" by losing energy to go back to its ground state. When this happens, the same amount of energy will be lost, and a photon will be emitted. It is this excitation/relaxation process that is responsible for most of modern spectroscopy: the measurement of the amount or type of a substance or atom present by irradiating the sample with electromagnetic radiation.