An Introduction to the Electronic Structure of Atoms
and Molecules
Dr. Richard F.W. Bader
Professor of Chemistry / McMaster University / Hamilton,
Ontario
|
The Magnetic Properties of the Electron
So far, the only motion we have considered for the
electron is a motion in three-dimensional space. Since this motion is ultimately
described in terms of an orbital wave function, we term this the orbital
motion of the electron. However, the electron may possess an internal motion
of some kind, one which is independent of its motion through space. Since
the electron bears a charge, such an internal motion, if it does exist,
might be expected to generate a magnetic moment. We have previously pointed
out that when an electron is in an atomic orbital for which l is
not equal to zero, the resultant angular motion of the electron gives rise
to a magnetic moment. We would anticipate then that an electron in an s
orbital (l = 0) should not exhibit any magnetic effects as its angular
momentum is zero. If an electron in these circumstances did exhibit a magnetic
effect, it would indicate that another type of motion was possible, presumably
an internal one. Whether or not an electron in an s orbital does
possess a magnetic moment may be determined by means of an atomic beam
experiment similar to the one previously described (Fig.
3-12).
In the present experiment a beam of hydrogen atoms is passed
through the apparatus. All of the hydrogen atoms in the beam will be in
their ground state with l = 0 and hence they will not possess an
orbital magnetic moment. However, when the magnetic field is applied, something
does happen to the beam of atoms. It is split into two distinct beams,
one of which is deflected to the N pole of the magnet and the other to
the S pole. Thus even when atoms possess no magnetic moment because of
the orbital motion of the electrons, they may still exhibit magnetic effects!
As striking as the behaviour of the atoms as small magnets is the splitting
of the beam into two distinct components. Let us consider first the origin
of the magnetic effect, and second, the splitting of the beam into two
distinct beams.
The observed magnetism of the hydrogen atoms must
be due to some motion of the electrons. The nucleus of a hydrogen atom
does possess a magnetic moment but its magnitude is too small, by a factor
of roughly a thousand, to account for the deflections observed in this
experiment. A magnetic moment will be observed only when the charged particle
possesses angular momentum. Since the orbital angular momentum for an electron
in the ground state of hydrogen is zero, we are forced to assume that the
electron possesses some internal motion which has associated with it an
angular momentum. A classical analogue of the internal angular momentum
would be a spinning motion of the electron about its own axis. For this
reason it is referred to as a spin angular momentum and the
associated magnetic effect as a spin magnetic moment. These
effects are separate from, and in addition to, the orbital angular momentum
of the electron (classically, the rotation of the electron around the nucleus)
and its associated magnetic effects.
We are familiar enough with the predictions of quantum
mechanics to anticipate that the spin angular momentum and its component
along some axis will be quantized. As in the case of orbital angular momentum,
the effect of the quantization will be to limit the number of values which
the component of the spin magnetic moment may have along any given axis.
The magnitude of the spin angular momentum will determine the number of
possible values its component may have along a given axis. Each of the
possible values will in turn cause some fraction of the total spin magnetic
moment to be aligned along the same axis. In the case of the electron's
orbital motion, we found that as l and hence the orbital angular
momentum was increased, the number of possible values for the component
of the orbital magnetic moment along a given axis was increased, the number
being equal to (2l + 1).
We can use a magnetic field to inquire into the nature
of the spin angular momentum as well. In fact, we have already discussed
the pertinent experiment. The beam of hydrogen atoms was split into just
two components in the atomic beam experiment. This means that the component
of the electron's spin magnetic moment (and spin angular momentum) along
a given axis may have only one of two possible values; the component may
be aligned with the field and hence be attracted, or it may be opposed
to the field and be repelled. The electron's spin magnetic moment has been
detected in many different kinds of experiments and the results are remarkable
in that only two components of constant magnitude are ever observed. The
electron is always either repelled by the field or attracted to it. This
implies that the magnitude of the spin angular momentum for a single electron
may have only one possible value. Since the number of possible
values for the component of a given amount of angular momentum of any type
in quantum mechanics is (2l + 1), l must equal ½ and
only ½ for the spin angular momentum, and the values of m
for the electron spin, which assume values from a maximum of l to
a minimum of -l in steps of unity, must equal +½ and -½.
In this respect the spin angular momentum of the electron is quite different
from its orbital angular momentum, which may have many possible values,
as the value of l for the orbital motion is restricted only in that
it must equal zero or an integer.
It should be stressed that the splitting of the beam of
hydrogen atoms into only two components is again evidence of quantization.
If the atomic magnets (the hydrogen atoms) behaved according to classical
mechanics, then the effect of the magnetic field would be simply to broaden
the beam. The orientations of the atomic magnets would be random when they
first entered the field of the magnet and classically the individual atomic
magnets could be aligned at any and all angles with respect to the field,
giving all possible components of the spin magnetic moment along the direction
of the field. The inhomogeneous field would then exert a force proportional
to the magnitude of the component, and the beam would broaden but not split.
Since the spin magnetic moment is an intrinsic property
of the electron, even a beam of free electrons should be split into two
components in a magnetic field. However, the charge possessed by the free
electron also interacts with the magnetic field and the much smaller magnetic-magnetic
interaction is masked by the usual deflection of a charge species in a
magnetic field. By employing a neutral atom, the complications of the electronic
charge may be avoided. The original experiment was performed on a beam
of silver atoms by Stern and Gerlach in 1921. (We shall see shortly that
the electrons in a silver atom do not possess any orbital angular momentum.)
Let us summarize what we have learned about this new property
of the electron. Since an electron may exhibit a magnetic moment even when
it does not possess orbital angular momentum, it must possess some internal
motion. We call this motion the electron spin and treat it quantum mechanically
as another kind of angular momentum. Experimentally, however, all we know
is that the electron possesses an intrinsic magnetic moment. The remarkable
feature of this intrinsic magnetic moment is that its magnitude and the
number of components along a given axis are fixed. A given electron may
exhibit only one of two possible components; it may be aligned with the
field or against it. Experimentally, or theoretically, this is all we can
know about the spin magnetic moment and the spin angular momentum. Hence
only one quantum number is required to describe completely the spin properties
of a single electron. We shall denote the value of this quantum number
by or ¯,
the upwards-pointing arrow signifying that the component of the magnetic
moment is aligned with the field and the downwards-pointing arrow that
this component is opposed to the field.
A total of four quantum numbers is required to specify
completely the state of an electron when it is bound to an atom. The quantum
numbers n, l and m determine its energy, orbital angular
momentum and its component of orbital angular momentum. The fourth quantum
number, the spin quantum number, summarizes all that can be known about
the spin angular momentum of the electron. This final quantum number may
have only one of two possible values corresponding to the magnetic moment
component being (a) aligned with the field or (b) opposed to it.
The Pauli Exclusion Principle
The consequences of the spin quantum number, when applied
to the problem of the electronic structure of atoms, are not immediately
obvious. The small magnitude of the electron's magnetic moment does not
directly affect the energy of the electron to any significant degree. To
see just how the spin of the electron does influence the problem, let us
reconsider our atomic orbital model in the light of this new degree of
freedom for the electron. In particular let us reconsider those instances
in which our model failed to account for the observations.
If a beam of helium atoms is passed through
a magnetic field, no splitting and no deflection is observed. The helium
atom, unlike the hydrogen atom is not magnetic. We could account for the
absence of a magnetic moment for helium if we assumed that of the two electrons
in the helium 1s orbital, one had its magnetic moment component
up () and the other
down (¯). The two
components would then cancel and there would be no resultant magnetic effect.
Our complete description of the electronic configuration of the helium
atom would be 1s2(¯),
i.e., both electrons have n = 1, l = 0, m = 0 and
one has a spin ()
and the other a spin (¯).
You may wonder why the states of helium corresponding
to the configurations 1s2()
or 1s2(¯¯)
are not observed. These states should exhibit twice the magnetism possessed
by a hydrogen atom. They are, however not found to occur.
What about the excited states of the helium atom? An excited state results
when one electron is raised in energy to an orbital with a higher n
value. The electrons are thus in different orbitals. The spin assignments
for an excited configuration can be made in more than one way and are such
as to predict the occurrence of both magnetic and non-magnetic helium.
For example, the configuration 1s12s1
could be 1s1()2s1(¯)
and be nonmagnetic or it could equally well be 1s1()2s1()
and be magnetic. (Click
here for note.)
Both the magnetic and non-magnetic forms are indeed found
to occur for helium in an excited state. There are in fact two kinds of
excited helium atoms, those which are non-magnetic and those which are
magnetic. If the two forms of helium possess different energies even though
they have the same orbital configuration (we shall see why this should
be so later) then we have an explanation for the previously noted discrepancy
that helium exhibits twice the number of line spectra as does hydrogen.
For every set of lines in the spectrum which arises from the transition
of the electron from the configurations 1s1()np1(¯)
to the configuration 1s1()2s1(¯)
for example, there will be another set of lines due to transitions from
1s1()np1()
to 1s1()2s1().
The study of the magnetic properties of the ground and
excited states of helium is sufficient to point out a general principle.
For the ground state of helium, in which both electrons are in the same
atomic orbital, only the non-magnetic form exists. This would
imply that when two electrons are in the same atomic orbital their spins
must be paired, that is, one up ()
and one down (¯).
This is an experimental fact because helium is never found
to be magnetic when it is in its electronic ground state. When the electrons
are in different orbitals, then it is again an experimental fact that their
spins may now be either paired (¯)
or unpaired, e.g., ().
Thus when two electrons are in the same orbital (i.e., they possess the
same n, l and m values) their spins must be paired.
When they are in different orbitals (one or more of their n, l
and m values are different) then their spins may be paired or unpaired.
We could generalize these observations by stating that "no two electrons
in the same atom may have all four quantum numbers the same." Stated
in this way we see immediately that any given orbital may hold no more
than two electrons. Since two electrons in the same orbital have the same
values of n, l and m, they can differ only through
their spin quantum number. However, the spin quantum number may have only
one of two possible values, and these possibilities are given by (n,
l,
m,
)
or (n,
l,
m, ¯).
We have indeed found the principle we were seeking, one
which limits the occupation of an atomic orbital. This principle is known
as the Pauli exclusion principle. One form of it, suitable
for use within the framework of the orbital approximation, is the statement
given in quotation marks above. The Pauli principle cannot be derived from,
nor is it predicted by, quantum mechanics. It is a law of nature which
must be taken into account along with quantum mechanics if the properties
of matter are to be correctly described. The concept of atomic orbitals,
as derived from quantum mechanics, together with the Pauli exclusion principle
which limits the occupation of a given orbital, provides an understanding
of the electronic structure of many-electron atoms. We shall demonstrate
this by "predicting" the existence of the periodic table.
