Chemistry 103A; Sections 5, 6, 7, 8; Lecture 19, 6 Oct 00

Recall:

Schrödinger found the wave equation which described the wave nature of particles.

Schrödinger's wave equation ascribed a wave nature to particles. As a result, Schrödinger's theory was called "wave mechanics" for a long time. Now we simply refer to it is "quantum mechanics."

 Schrödinger's equation provides two things. It provides a so-called "wave function," usually symbolized by the Greek letter psi (y , or Y ) and it provides an energy, E, associated with that wave function.

The wave function, Y , is a function of the spatial coordinates and time. That is, Y = Y (x, y, z, t).

The wave function is interpreted as giving a description of the probability of finding a particle in a particular position.

The square of the magnitude of y tells us where the particle is.

At a point in space where y (or its absolute square) is large the particle is likely to be there.

At a point in space where y is small the particle is not likely to be there.

Wherever in space y is zero the particle isn't there.

The Schrödinger equation for a system will, in general, have many solutions (usually in infinite number of solutions). Each solution has its own unique wave function, y , and we distinguish them from each other by adding one or more subscripts, writing, for example, y i.

Each y i corresponds to an allowed energy of the system which we indicate by Ei. These energies are constant, they do not change in time.

Very often we use energy level diagrams to show graphically the allowed energies.

There is one mechanical system where the energy levels are equally spaced. There is another where the energy levels get further apart at higher energies. You have already seen Bohr and de Broglie's energies for the hydrogen atom where the energy levels get closer together as the energy gets higher.
 
 

The Hydrogen Atom

All of our discussion of the structure of atoms is going to be based on the structure and quantum mechanics of the hydrogen atom.

We need to know two things about Schrödinger's solution to the hydrogen atom:

1. Schrödinger's theory confirmed the formula for the allowed energies of an electron in the H atom first given by Bohr and de Broglie. Namely,   .


(Whenever the energies and wave function of a system are indexed by a number we call that number a quantum number.

The n in the equation is a quantum number and it can take the values 1, 2, 3, 4, …. ¥ .

Since this is the number that determines the energy of the electron it is called the "principal quantum number."

We will see that Schrödinger's theory gives two additional quantum numbers for the energy states of the hydrogen atom.  [There is a fourth quantum number which Schrödinger's theory didn't provide.])
 

 2. Schrödinger's theory denied the existence of orbits. The concept of an orbit was replaced by the concept of an orbital. Quantum mechanics says that you can never trace out the exact trajectory (path or orbit) traced by a moving electron. (The reason for this is best explained by Heisenberg's "uncertainty principle," see p. 315. The uncertainty principle, which is a consequence of quantum mechanics, says that you can't know the position and the velocity (really, the momentum) of a particle simultaneously to arbitrary accuracy.) The best we can do is know the probability that an electron is at a certain spot in space. As we have already said, this probability is given by the square of the absolute value of the wave function, Y (x, y, z). (Under certain conditions the wave function may also be a function of t. That is, the wave function, and the probabilities calculated from it, change in time. But the energies, Ei do not change in time.) Although the Bohr theory gave the correct energies of the hydrogen atom it was not complete and missed some of the details later given by Schrödinger's theory.

According to Schrödinger's theory each state of the hydrogen atom requires three quantum numbers to describe it. There is the principal quantum number, n, which we have already described.

In addition to n there is a quantum number, l, which gives information about the angular momentum of the electron. Then there is a so-called "magnetic" quantum number, ml, that gives the component of l on the spatial z axis.

The allowed values of these quantum numbers are:

n = 1, 2, 3, 4, … ¥ ,

l = 0, 1, 3, … n - 1

- l £ ml £ l.

For n = 1 we have l = 0 with ml = 0

For n = 2 we have l = 0 with ml = 0
                   AND l = 1 with ml = - 1, 0, +1

For n = 3 we have l = 0 with ml = 0
                   AND l = 1 with ml = - 1, 0, +1
                   AND l = 2 with ml = - 2, - 1, 0, +1, +2
etc.

It is easiest to visualize the organization of these energy states of hydrogen using an energy level diagram. (Not to scale)
 
 

5__    __ __ __    __ __ __ __ __    __ __ __ __ __ __ __    _ _ _ _ _ _ _ _ _

4__    __ __ __    __ __ __ __ __    __ __ __ __ __ __ __

3__   __ __ __    __ __ __ __ __
 
 

2__    __ __ __
 
 
 
 
 
 
 
 
 
 
 
 

1__

Notice that there is one "state," or orbital, of the hydrogen atom at n = 1. But there are four states (orbitals) at n = 2, nine states (orbitals) at n = 3, sixteen states (orbitals) at n = 4, and so on.

We have grouped the orbitals according to their value of l. This turns out also to be a grouping according to the shape of the orbitals.

Orbitals have a shape and different orbitals have a different shape. Think of the orbital as a "cloud" of probability density.

The density of the cloud is given by the absolute value of the square of the wave function.

So, where the cloud is dense the electron is likely to be there.

Where the cloud is sparse the electron is not likely to be there.

Where there is no cloud the electron is not there.

It is customary to label the orbitals by letters. (This notation comes out of old spectroscopy notation.)

Orbitals with l = 0 are called "s" orbitals. That is, as we go up in energy there is the 1s, 2s, 3s, 4s, and so on.

s orbitals have spherical symmetry.

Orbitals with l = 1 are called "p" orbitals. Thus we have 2p, 3p, 4p, and etc. orbitals.

Recall that there are three p orbitals at each allowed value of the principal quantum number n. We could call them p-1, po, and p+1,

but it is more common and sometimes more useful to refer to them by coordinate axes as

px, py, and pz.

The px, py, and pz orbitals have cylindrical symmetry.

 Orbitals with l = 2 are called "d" orbitals. Thus we have 3d, 4d, 5d, etc orbitals. There are five possible d orbitals at each value of n. We could label them - 2, - 1, 0, +1, +2, but it is more common for chemists to label them xy, yz, zx, x2 - y2, and z2. (We will see that this labeling indicates where in space most of the electron density resides.) There is no simple description of the symmetry of the d orbitals, except that dz2 has cylindrical symmetry. The other four do not. Orbitals with l = 3 are called "f " orbitals. There are seven of them with ml ranging from - 3 to +3.

In summary, at each level there is one "s" orbital which is spherically symmetric. At each level above n = 1 there are three "p" orbitals. At each level above n = 3 there are five "d" orbitals. Next up there are seven "f " orbitals and then nine "g" orbitals, and so on. (After g the labeling goes alphabetically, except that they skip j.)

The s-orbitals all have spherical symmetry, but they differ in their internal structure.

The p-orbitals all have cylindrical symmetry, but they, too, differ from each other as n increases.

There is no easy way to describe in words the symmetries of the f, g, … orbitals.

There are graphics of s, p, d, and f orbitals on our course web site. Go back to the home page.