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Multielectron Atoms
Introduction
We have seen that the Schrödinger equation can be used to successfully explain the observed
properties of one-electron atoms. We shall now extend the Schrödinger formulism to
multielectron atoms.
Independent Particle Approximation (The Hartree Theory)
The presence of two or more electrons in an atom presents a significant complication in finding
solutions to the time-independent Schrödinger equation (TISE) because the mutual interactions
among several bodies have to be accounted for. In fact, an exact solution to the TISE is not
possible. Thus, approximations have to be made and a numerical solution effected. The starting
point for such solutions is the Independent Particle Approximation (IPA). Within the IPA, each
of the Z electrons in an atom moves independently with a spherically symmetric net potential
energy U(r) due to its attraction to the nucleus and repulsion from the other Z–1 electrons. The
great benefit of the IPA is that the TISE for the Z-electron atom can be separated into Z TISEs
that have the same form, each describing the motion of one electron. To solve the TISE for each
electron, one first guesses the form of U(r) and then solves the TISE numerically to obtain the
corresponding wave function. The wave function is then used to calculate the probability density
distribution from which the charge density distribution can be obtained. This can then be used to
calculate an improved approximation for U(r). The entire procedure is then repeated until the
difference between successive results for U(r) is smaller than a chosen “tolerance” value. This
iterative procedure is called the Hartree-Fock method.
How does one guess U(r)? Well, the innermost electron feels almost the full nuclear charge +Ze.
(Recall that by Gauss's law, the electric field depends on the total charge enclosed by a Gaussian
surface. The outer electrons are not enclosed by the Gaussian surface at the position of the
innermost electron.) Thus,
U (r ) ≈ −k
Ze 2
.
r
(innermost electron; r → 0)
(13.1)
The outermost electron is screened or shielded from the full positive charge of the nucleus by the
repulsion of the intervening Z–1 electrons. Thus, this electron feels a net charge that is very
nearly equal to + Ze + [−( Z − 1)e] = +e. (Gauss's law again: the electric field depends on the total
charge enclosed by a Gaussian surface at the position of the outermost electron.) Therefore,
U (r ) ≈ −k
e2
.
r
(outermost electron; r → ∞)
(13.2)
To encompass both cases given in Eqs. (13.1) and (13.2), one can write
1 U (r ) ≈ −k
Z eff e 2
r
,
(13.3)
where
Z eff ≈ Z
(innermost electron)
Z eff ≈ 1
(outermost electron)
A reasonable interpolating function is then used to describe Zeff for all the other electrons. [Show
pictures from Taylor et al. or Rohlf.]
Shell Structure and Energy Levels
Since the IPA potential energy is spherically symmetric, the angular equations for the θ and φ
coordinates of the separated TISE for each electron are the same as in the case of the hydrogen
atom. Hence, the quantum states of a multielectron atom will be specified by the quantum
numbers l and ml. The solution of the radial equation also gives rise to a principal quantum
number n. Finally, each electron has two
possible spin states (spin-up and spindown), specified by the quantum number
ms, which has the value ± 12 . Thus, in
multielectron atoms, the states of definite
energy (energy eigenfunctions) are
specified uniquely by the values of the
four quantum numbers; n, l, ml, and ms.
The picture (taken from Eisberg &
Resnick’s Quantum Physics) shows plots
of the radial probability density for the
electrons in the argon atom. We see that
for electrons with the same value of n, the
probability densities are significant only
in the same limited range of values of r.
Thus, the shell structure seen in the case
of the hydrogen atom is preserved in
multielectron atoms, with a spatial shell
being identified with each value of n.
Unlike the hydrogen atom and other one-electron atoms, the quantum states corresponding to a
given value of n are not all degenerate. In other words, the energy now depends on the value of l
in general. To understand the l-dependence of the energy, we look at the radial probability
densities for n =1, 2, and 3 for the hydrogen atom. [Show pictures.] These probability density
2 functions are qualitatively similar for multielectron atoms, as seen for the Ar atom. For n = 1, l =
0 so there is obviously no l-dependence for the 1s states. In a multielectron atom, the electrons in
the 1s states are closest to the nucleus, which is also true of the electron in a 1s state of hydrogen
or other one-electron atoms. Since there is very little screening for 1s electrons, they feel almost
the entire nuclear charge +Ze. Hence, using a one-electron atom approximation, a reasonable
estimate for the energy of 1s states is
E1s ≈ − Z eff2 (13.6 eV),
with Z eff ≈ Z .
(13.4)
(For larger multielectron atoms, a better estimate is Z eff ≈ Z − 2 .)
For n = 2, we have the 2s and 2p states. Note that the radial probability density for the 2s states
has a small maximum at small values of r, i.e., close to the nucleus, whereas the radial
probability density for the 2p states has no maximum close to the nucleus. Hence, it is more
likely for 2s electrons to be close to the nucleus because the 2s wave function is more penetrating
in the small-r region. This is precisely where Zeff is greater, i.e., where there is less screening. It
follows that, on average, 2s electrons are more strongly attracted to the nucleus than 2p electrons
and therefore have lower energy. A similar dependence of the energy on l occurs for n = 3 and
higher values of n.
In fact, in some atoms, the l dependence is so strong that the 3d states have higher energy than
the 4s states! In general, however, states with larger values of n have greater energies. This is
because the average distance from the nucleus is greater for the states with greater n. Also, the
inner electrons screen the nuclear charge, so that the effective nuclear charge Zeff felt by the outer
electrons (greater n) is smaller. One can estimate the energy of the outermost electrons (with
principal quantum number n) as
En ≈ −
(Z eff ,n )2 (13.6 eV)
n2
.
(13.5)
Pauli Exclusion Principle and Atomic Ground States
An atom will be most stable when its energy is lowest, i.e., when all the electrons are in their
lowest energy states. This is the ground state of the atom. Naively, one might expect that all the
electrons would occupy the 1s state in order to minimize the atom's energy. However, this
“condensation” into the lowest atomic energy state does not occur for electrons. We have, to a
certain extent, assumed this in our previous discussion of screening effects. The filling of atomic
energy states by electrons to produce the ground state of the atom is dictated by the Pauli
exclusion principle: No two electrons in an atom can occupy the same quantum state, i.e., have
the same set of quantum numbers n, l, ml, ms.
3 The Pauli exclusion principle extends to all quantum mechanical systems containing particles
called fermions. (Fermions have half-integral spin.) An electron is a fermion. Other examples of
fermions are neutrons, protons, and muons. Let us illustrate how the Pauli principle governs
atomic structure and the ground-state properties of atoms.
In order to minimize the energy of an atom, the quantum states will be filled by electrons, with
the states having the lowest energy being filled first. Hydrogen has only one electron, so the
electron goes into a 1s state. The spin can be either up or down, since the energy is independent
of spin (in the absence of a magnetic field). In helium (He), Z = 2. The second electron also goes
into a 1s state, since the 1s state is two-fold degenerate ( ms = ± 12 ). However, according to the
Pauli principle, n, l, ml, and ms cannot have identical values for the two electrons. Hence, one
electron is spin-up and the other spin-down. In other words, one electron has ms = + 12 and the
other ms = − 12 . The 1s states (also called the 1s subshell) cannot contain any more electrons
without violating the Pauli principle. (A subshell is associated with a particular value of l.)
Hence, the next subshell to be filled is the 2s subshell. Thus, for Z = 3 (lithium), a new shell (n =
2) is started. The 2s subshell is filled before the 2p because the former has lower energy as
discussed previously. For Z = 4 (beryllium), the second state in the 2s subshell is occupied. The
two electrons in the 2s subshell must have opposite spins. The 2s subshell is now full, so for Z =
5 (boron) the fifth electron goes into a 2p state. Now, the number of states with orbital angular
momentum quantum number l is 2(2l+1). This is also the number of electrons that can be
contained in a subshell without violating the Pauli principle. Hence, for the 2p subshell (l = 1),
the total number of electrons that can be accommodated is six. How are the p states filled? Let us
say the three p orbitals are px, py, and pz. Each can hold two electrons of opposite spin. (The
orbitals correspond to the various values of ml.) The filling proceeds as follows. The three
orbitals are first all occupied with a single electron, with the spins all aligned parallel. The
parallel alignment of spins reduces the overall energy of the atom. The reason is that due to the
Pauli principle, the electrons with parallel spins in the same subshell tend to avoid each other,
thereby reducing the total positive repulsive energy. Additional electrons in the p subshell are
then paired with electrons in each of the singly occupied orbitals. After the 2p subshell is filled,
the entire n = 2 shell is filled (8 electrons). Additional electrons go into the n = 3 shell and so
forth. The progression of the filling of atomic subshells is shown in the picture. [Show picture.]
Note that the 4s subshell is in general filled before the 3d subshell because the 4s subshell has
lower energy for reasons explained previously. While they are filling, the 4s and 3d subshells
have very similar energies. Thus, these two subshells are in the same energy shell but not in the
same spatial shell (which is dictated by the value of n).
Before we discuss the ground state properties of the elements and the periodic table, we
introduce the useful idea of electron configuration. The electron configuration of an atom is
simply the specification of the number of electrons in occupied subshells of the atom. For
4 example, the configuration of the first six atoms in their ground states are: H: 1s1; He: 1s2; Li:
1s22s1; Be: 1s22s2; B: 1s22s22p1; C: 1s22s22p2.
The Periodic Table
The periodic table is an arrangement of the chemical elements which places elements with
similar chemical and physical properties in vertical columns or groups. The periodic table was
constructed by Mendeleev long before the electronic structure of atoms was understood, and was
based largely on empirical chemical and physical data. With our knowledge of the ground state
electronic configurations of the atoms of each element, we can understand the logic of the
periodic table. The horizontal rows of the periodic table are periods, which represent the filling
of successive subshells. The first period contains only H and He. The 1s subshell is full at He.
The second period begins with the filling of the 2s subshell and then the 2p subshell. At Ne
(neon), both the 2s and 2p subshells (as well as the 1s) are full. Note that both He and Ne have
all their occupied energy shells completely filled. They are examples of the noble or inert gases
(group VIII). These gases are chemically inert or inactive because the filled or closed shell
configuration is very stable. Relatively large amounts of energy are needed to excite an electron
and to ionize an atom. Also, the total angular momentum of noble-gas atoms is zero, so they
have no magnetic moment and therefore produce negligible magnetic fields outside the atom.
Further, the charge distribution is spherically symmetric and since the total atomic charge is zero,
no significant electric field is generated outside the atom. All these attributes make the noble
gases extremely unreactive chemically.
At the beginning of each period, there is an atom with a noble-gas closed-shell configuration plus
one electron. This group of elements (group I) is called the alkali elements. They include Li,
sodium (Na), potassium (K), rubidium (Rb), cesium (Cs), and francium (Fr). The single
outermost electron occupies a new shell. Thus, its average distance from the nucleus is
significantly larger than those for the inner electrons. Also, this lone outer electron is very well
screened from the nucleus. Therefore, this electron is only weakly bound to the atom. Excitation
and ionization energies are relatively low so this electron is very readily given up to form
chemical bonds. Thus, the alkali elements are extremely chemically reactive.
Going across the periodic table to the group VII elements (halogens), we see that for each atom,
only one electron is needed for a closed-shell configuration. In fact, when an electron is added to
a halogen atom, a significant amount of energy is released. This energy is called the electron
affinity of the atom. The larger the electron affinity, the greater the likelihood (in general) that an
electron will be captured in chemical bonds. The halogens have high electron affinities and are
therefore extremely chemically active. The halogens are: fluorine (F), chlorine (Cl), bromine
(Br), iodine (I), and astatine (At).
5 The alkaline earths have a noble gas configuration plus a filled s subshell. The s electrons are
relatively loosely bound and so the alkaline earths are quite reactive also, but not as reactive as
the alkali elements.
In the middle region of the periodic table, between the completion of filling the s subshell and
starting of the filling of the p subshell, there are the transition elements. (They are all metals, so
they are called transition metals.) In the transition elements, a d subshell is filling. The transition
elements in a given period tend to have similar chemical properties because chemical properties
are in large part governed by the number of electrons in the outermost subshell, i.e., the valence
electrons. In most of the elements of a transition series, the outermost s subshell is full (with two
s electrons). For example, in the 3d transition series, most members have a filled 4s shell, i.e.,
[Ar] 4s23dn. There are two exceptions to this in chromium (Cr) and copper (Cu), which have only
one 4s electron. The reason for this is that the half-filled or completely filled 3d subshell
represents a stable configuration.
Other series of elements that are worthy of note are the lanthanides (also called the rare earths)
in which the 4f shell is filling. The rare earths have very similar chemical properties because the
4f subshell is deep within the outermost 6s subshell, which is completely full for all the rare
earths. The same picture emerges in the case of the actinides, which have the 5f subshell filling
deep within the 7s valence electrons.
It should be quite clear that the Pauli exclusion principle governs the structure and properties of
atoms. Without the exclusion principle, atoms would be very different objects indeed; the
universe would be dramatically different in its structure and contents.
Some Properties of the Elements
Ionization Energies
The ionization energy is the minimum energy needed to remove an electron from an atom in its
ground state. Since the alkali metals have a single loosely-bound valence electron, one expects
that the ionization energy for the alkali element in a period to be the lowest. As one moves to the
right across the period, Z increases. Screening for outermost electrons is enhanced, but the
increase in Z dominates. Hence, we expect the valence electrons to be more tightly bound and the
ionization energy to increase. This is what is observed. The noble gases have the highest
ionization energies of all the elements in a period.
Note that there is a slight drop in ionization for oxygen (O) compared with nitrogen (N). The
reason is that the fourth 2p electron must be paired with another electron in one of the three
hitherto singly occupied 2p orbitals. The spin of the fourth 2p electron must be opposite that of
the other electron in the orbital. Since these two electrons have distinct quantum numbers (their
spins are different), the Pauli principle cannot be violated. Thus, the electrons in the doubly
occupied orbital can be very close to one another, thereby increasing the energy of the system
6 because of increased Coulomb repulsion. The increased energy of the paired 2p electrons leads
to a smaller ionization energy. There are also drops in ionization energy when a p subshell starts
to be filled. [For example, gallium (Ga), indium (In), thallium (Tl).] This is because there is a
significant increase in energy when one goes from a filled s or d subshell to the next empty p
subshell.
Atomic Radii
We expect the alkali element in a period to have the largest atomic radius because the lone
valence electron is very loosely bound. As Z increases as one moves across the period, one
expects the atomic radius to decrease because of tighter binding to the nucleus. This is what is
observed, with few exceptions. Note the large increase in atomic radius as one moves from a
inert gas to the alkali element of the next period. This is of course due to the fact that the single
electron in the alkali element occupies a new shell, with significantly larger average distance
from the nucleus than the inner electrons.
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