Ebook Physical chemistry Part 2
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12
12.1
12.2
12.3
12.4
12.5
12.6
12.7
12.8
12.9
12.10
12.11
12.12
12.13
12.14
Synopsis
Spin
The Helium Atom
Spin Orbitals and the
Pauli Principle
Other Atoms and the
Aufbau Principle
Perturbation Theory
Variation Theory
Linear Variation Theory
Comparison of Variation and
Perturbation Theories
Simple Molecules and the
Born-Oppenheimer
Approximation
Introduction to LCAO-MO
Theory
Properties of Molecular
Orbitals
Molecular Orbitals of Other
Diatomic Molecules
Summary
Atoms and Molecules
W
E HAVE SEEN HOW QUANTUM MECHANICS provides tools for
understanding some simple systems, up to and including the hydrogen atom itself. An understanding of the H atom is a crucial point because it
is real, not a model system. Quantum mechanics showed that it can describe
the hydrogen atom like Bohr’s theory did. It also describes other model systems that have applications in the real world. (Recall that all of the model
systems—particle-in-a-box, 2-D and 3-D rigid rotors, harmonic oscillators—
could be applied to real systems even if the real systems themselves weren’t
exactly ideal.) As such, quantum mechanics is more applicable than Bohr’s
theory and can be considered “better.” We will conclude our development of
quantum mechanics by seeing how it applies to more complicated systems:
other atoms and even molecules. What we will find is that explicit, analytic
solutions to these systems are not possible, but quantum mechanics does
supply the tools for understanding these systems nonetheless.
12.1 Synopsis
In this chapter, we will consider one more property of the electron, which is
called spin. Spin has dramatic consequences for the structure of matter, consequences that could not have been considered by the standards of classical
mechanics. We will see that an exact, analytic solution for an atom as simple
as helium is not possible, and so the Schrödinger equation cannot be solved
analytically for larger atoms or molecules. But there are two tools for studying
larger systems to any degree of accuracy: perturbation theory and variational
theory. Each tool has its advantages, and both of them are used today to study
atoms and molecules and their reactions.
Finally, we will consider in a simple way how quantum mechanics considers a molecular system. Molecules can get very complicated. However, we
can apply quantum mechanics to molecules. We will finish this chapter with
an introduction to molecular orbitals and how they are defined for a very
simple molecule, H2ϩ. Simple as this system is, it paves the way for other
molecules.
370
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12.2 Spin
371
12.2 Spin
Not long before quantum mechanics was developed, an important experimental observation was made. In 1922, Otto Stern and W. Gerlach attempted to
measure the magnetic moment of the silver atom. They passed vaporized silver atoms through a magnetic field and recorded the pattern that the beam of
atoms made after it passed through the magnetic field. Surprisingly, the beam
split into two parts. The experiment is illustrated in Figure 12.1.
Attempts to explain this in terms of the Bohr theory and quantized angular
momentum of electrons in their orbits failed. Finally, in 1925, George
Uhlenbeck and Samuel Goudsmit proposed that this result could be explained
if it was assumed that the electron had its own angular momentum. This angular momentum was an intrinsic property of the electron itself and not a consequence of any motion of the electron. In order to explain the experimental
results, Uhlenbeck and Goudsmit proposed that components of the intrinsic
angular momentum, called spin angular momentum, had quantized values of
either ϩᎏ12ᎏប or Ϫᎏ12ᎏប. (Recall that h has units of angular momentum.)
Since that proposal, it has become understood that all electrons have an intrinsic angular momentum called spin. Although commonly compared to the
spinning of a top, the spin angular momentum of an electron is not due to any
rotation about the axis of the particle. Indeed, it would be impossible for us to
determine that an electron is actually spinning. Spin is a property of a particle’s very existence. This property behaves as if it were an angular momentum,
so for all intents and purposes it is considered an angular momentum.
Like the angular momentum of an electron in its orbit, there are two measurables for spin that can be observed simultaneously: the square of the total
spin and the z component of the spin. Because spin is an angular momentum,
there are eigenvalue equations for the spin observables that are the same as for
ˆ
L 2 and ˆ
Lz , except we use the operators ˆ
S 2 and ˆ
Sz to indicate the spin observables. We also introduce the quantum numbers s and ms to represent the quantized values of the spin of the particles. (Do not confuse s, the symbol for the
Glass
plate
Beam split
into two
N
Magnet
S
Oven
Beam of
Ag atoms
Figure 12.1 A diagram of the Stern-Gerlach experiment. A beam of silver atoms passed
through a magnetic field splits into two separate beams. This finding was used to propose the existence of spin on the electron.
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372
C H A P T E R 12
Atoms and Molecules
spin angular momentum, with s, an orbital that has ᐉ ϭ 0.) The eigenvalue
equations are therefore
ˆ
S 2⌿ϭ s(s ϩ 1)ប2⌿
(12.1)
Sz⌿ϭ ms ប⌿
(12.2)
The values of the allowed quantum numbers s and ms are more restricted
than for ᐉ and mᐉ. All electrons have a value of s ϭ ᎏ12ᎏ. The value of s, it turns
out, is a characteristic of a type of subatomic particle, and all electrons have
the same value for their s quantum number. For the possible values of the z
component of the spin, there is a similar relationship to the possible values of
mᐉ and ᐉ: ms goes from Ϫs to ϩs in integral steps, so ms can equal Ϫᎏ12ᎏ or ϩᎏ12ᎏ.
Thus, there is only one possible value of s for electrons, and two possible values for ms.
Spin also has no classical counterpart. Nothing in classical mechanics predicts or explains the existence of a property we call spin. Even quantum mechanics, at first, did not provide any justification for spin. It wasn’t until 1928
when Paul A. M. Dirac incorporated relativity theory into the Schrödinger
equation that spin appeared as a natural theoretical prediction of quantum
mechanics. The incorporation of relativity into quantum mechanics was one
of the final major advances in the development of the theory of quantum mechanics. Among other things, it led to the prediction of antimatter, whose existence was verified experimentally by Carl Anderson (with the discovery of the
positron) in 1932.
Example 12.1
What is the value, in Jиs, of the spin of an electron? Compare this to the value
of the angular momentum for an electron in s and p orbitals of an H-like
atom.
Solution
The value of the spin angular momentum of an electron is determined by using equation 12.1. We must recognize that the operator is the square of the
total spin, and to find the value for spin we will have to take a square root.
We get
spin ϭ ͙s(s
ϩ 1ෆ2 ϭ
ෆ)ប
Ί
1 1
6.626 ϫ 10Ϫ34Jиs
ᎏᎏ ᎏᎏ ϩ 1 ᎏᎏ
2 2
2
2
ϭ 9.133 ϫ 10Ϫ35 Jиs
The angular momentum of an electron in an s orbital is zero, since ᐉ ϭ 0 for
an electron in an s orbital. In a p orbital, ᐉ ϭ 1, so the angular momentum is
6.626 ϫ 10Ϫ34 Jиs
͙ᐉ(ᐉ+
1ෆប ϭ ͙1ෆ
ෆ)
и 2 ᎏᎏ ϭ 1.491 ϫ 10Ϫ34 J и s
2
which is almost twice as great as the spin. The magnitude of the spin angular momentum is not much smaller than the angular momentum of an electron in its orbit. Its effects, therefore, cannot be ignored.
The existence of an intrinsic angular momentum requires some additional
specificity when referring to angular momenta of electrons. One must now
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12.2 Spin
373
differentiate between orbital angular momentum and spin angular momentum. Both observables are angular momenta, but they arise from different
properties of the electron: one from its motion about a nucleus, the other from
its very existence.
The spin angular momentum of an electron can have only certain specific
values. Spin is quantized. Like the z component of orbital angular momentum,
ms has 2s ϩ 1 possible values. In the case of the electron, s ϭ ᎏ12ᎏ, so the only
possible values of ms are Ϫᎏ12ᎏ and ϩᎏ12ᎏ. The specification of an electron’s spin
therefore represents two other quantum numbers that can be used to label the
state of that electron. In practice, however, it is convenient to not specify s,
since it is always ᎏ12ᎏ for electrons. This gives us a total of four quantum numbers: the principal quantum number n, the orbital angular momentum quantum number ᐉ, the orbital angular momentum z component mᐉ, and the spin
angular momentum (z component) ms. These are the only four quantum numbers needed to specify the complete state of an electron.
Example 12.2
List all possible combinations of all four quantum numbers for an electron
in the 2p orbital of a hydrogen atom.
Solution
In tabular form, the possible combinations are
Symbol
n
ᐉ
mᐉ
ms
Possible values
2
1
Ϫ1
ϩᎏ12ᎏ or Ϫᎏ12ᎏ
0
ϩᎏ12ᎏ or Ϫᎏ12ᎏ
1
ϩᎏ12ᎏ or Ϫᎏ12ᎏ
There are a total of six possible combinations of the four quantum numbers
for this case.
2468 cmϪ1
H
400Å
500Å
600Å
700Å
21 cmϪ1
Figure 12.2 A very high resolution spectrum
of the hydrogen atom shows a tiny splitting due
to the spin on the electron. This splitting is caused
by the electron spin interacting with the nuclear
spin of the hydrogen atom’s nucleus (a proton).
Although not considered until now, the ms of the electron in a hydrogen
atom is either ϩᎏ12ᎏ or Ϫᎏ12ᎏ. A fascinating astronomical consequence of spin is the
fact that an electron in hydrogen has a slightly different energy depending on
the relative spin orientations of the electron and the proton in the nucleus. (A
proton also has a characteristic spin quantum number of ᎏ12ᎏ.) If an electron in
a hydrogen atom changes its spin, there is a concurrent energy change that is
equivalent to light having a frequency of 1420.40575 MHz, or a wavenumber
of about 21 cmϪ1, as shown in Figure 12.2. Because of the pervasiveness of hydrogen in space, this “21-cmϪ1 radiation” is important for radio astronomers
who are studying the structure of the universe.
Finally, since spin is part of the properties of an electron, its observable values should be determined from the electron’s wavefunction. That is, there
should be a spin wavefunction part of the overall ⌿. A discussion of the exact
form of the spin part of a wavefunction is beyond our scope here. However,
since there is only one possible observable value of the total spin (s ϭ ᎏ12ᎏ) and
only two possible values of the z component of the spin (ms ϭ ϩᎏ21ᎏ or Ϫᎏ21ᎏ), it is
typical to represent the spin part of the wavefunction by the Greek letters ␣ and
, depending on whether the quantum number ms is ϩᎏ21ᎏ or Ϫᎏ21ᎏ, respectively.
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374
C H A P T E R 12
Atoms and Molecules
Spin is unaffected by any other property or observable of the electron, and the
spin component of a one-electron wavefunction is separable from the spatial
part of the wavefunction. Like the three parts of the hydrogen atom’s electronic
wavefunction, the spin function multiplies the rest of ⌿. So for example, the
complete wavefunctions for an electron in a hydrogen atom are
⌿ ϭ Rn,ᐉ и ⍜ᐉ,mᐉ и mᐉ и ␣
for an electron having ms of ϩᎏ12ᎏ. A similar wavefunction, in terms of , can be
written for an electron having ms ϭ Ϫᎏ12ᎏ.
12.3 The Helium Atom
e1
r12
e2
r1
2+
r2
Figure 12.3 Definitions of the radial coordi-
nates for the helium atom.
In the previous chapter, it was shown how quantum mechanics provides an exact, analytic solution to the Schrödinger equation when applied to the hydrogen atom. Even the existence of spin, discussed in the last section, does not alter this solution (it only adds a little more complexity to the solution, a
complexity we will not consider further here). The next largest atom is the helium atom, He. It has a nuclear charge of 2ϩ, and it has two electrons about
the nucleus. The helium atom is illustrated in Figure 12.3, along with some of
the coordinates used to describe the positions of the subatomic particles.
Implicit in the following discussion is the idea that both electrons of helium
will occupy the lowest possible energy state.
In order to properly write the complete form of the Schrödinger equation
for helium, it is important to understand the sources of the kinetic and potential energy in the atom. Assuming only electronic motion with respect to a
motionless nucleus, kinetic energy comes from the motion of the two electrons. It is assumed that the kinetic energy part of the Hamiltonian operator
is the same for the two electrons and that the total kinetic energy is the sum
of the two individual parts. To simplify the Hamiltonian, we will use the symbol ᭞2, called del-squared, to indicate the three-dimensional second derivative
operator:
Ѩ2
Ѩ2
Ѩ2
᭞2 ϵ ᎏᎏ2 ϩ ᎏᎏ2 ϩ ᎏᎏ2
Ѩx
Ѩy
Ѩz
(12.3)
This definition makes the Schrödinger equation look less complicated. ᭞2 is
also called the Laplacian operator. It is important to remember, however, that
del-squared represents a sum of three separate derivatives. The kinetic energy
part of the Hamiltonian can be written as
ប2
ប2
Ϫᎏᎏ᭞21 Ϫ ᎏᎏ᭞22
2
2
where ᭞21 is the three-dimensional second derivative for electron 1, and ᭞22 is
the three-dimensional second derivative for electron 2.
The potential energy of the helium atom has three parts, all coulombic in
nature: there is an attraction between electron 1 and the nucleus, an attraction
between electron 2 and the nucleus, and a repulsion between electron 1 and
electron 2 (since they are both negatively charged). Each part depends on the
distance between the particles involved; the distances are labeled r1, r2, and r12
as illustrated in Figure 12.3. Respectively, the potential energy part of the
Hamiltonian is thus
2e 2
2e 2
e2
ˆ ϭ Ϫᎏ
V
ᎏ Ϫ ᎏᎏ ϩ ᎏᎏ
4⑀0r1
4⑀0r2
4⑀0r12
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12.3 The Helium Atom
375
where the other variables have been defined in the previous chapter. The 2 in
the numerator of each of the first two terms is due to the 2ϩ charge on the
helium nucleus. The first two terms are negative, indicating an attraction, and
the final term is positive, indicating a repulsion. The complete Hamiltonian
operator for the helium atom is
2
2
2e 2
2e 2
e2
ˆ ϭ Ϫᎏបᎏ᭞21 Ϫ ᎏបᎏ᭞22 Ϫ ᎏ
H
ᎏ Ϫ ᎏᎏ ϩ ᎏᎏ
2
2
4⑀0r1
4⑀0r2
4⑀0r12
(12.4)
This means that for the helium atom, the Schrödinger equation to be solved is
ប2
ប2
2e 2
2e 2
e2
Ϫᎏᎏ᭞21 Ϫ ᎏᎏ᭞22 Ϫ ᎏᎏ Ϫ ᎏᎏ ϩ ᎏᎏ ⌿ ϭ Etot⌿ (12.5)
2
2
4⑀0r1
4⑀0r2
4⑀0r12
where Etot represents the total electronic energy of a helium atom.
The Hamiltonian (and thus the Schrödinger equation) can be rearranged by
grouping together the two terms (one kinetic, one potential) that deal with
electron 1 only and also grouping together the two terms that deal with electron 2 only:
2
ប2
2e 2
2e 2
e2
ˆ ϭ Ϫᎏបᎏ᭞21 Ϫ ᎏ
H
ᎏ ϩ Ϫᎏᎏ᭞22 Ϫ ᎏᎏ ϩ ᎏᎏ (12.6)
2
4⑀0r1
2
4⑀0r2
4⑀0r12
This way, the Hamiltonian resembles two separate one-electron Hamiltonians
added together. This suggests that perhaps the helium atom wavefunction is
simply a combination of two hydrogen-like wavefunctions. Perhaps a sort of
“separation of electrons” approach will allow us to solve the Schrödinger equation for helium.
The problem is with the last term: e 2/4⑀0r12. It contains a term, r12, that
depends on the positions of both of the electrons. It does not belong only with
the terms for just electron 1, nor does it belong only with the terms for just
electron 2. Because this last term cannot be separated into parts involving only
one electron at a time, the complete Hamiltonian operator is not separable and
it cannot be solved by separation into smaller, one-electron pieces. In order for
the Schrödinger equation for the helium atom to be solved analytically, it
either must be solved completely or not at all.
To date, there is no known analytic solution to the second-order differential Schrödinger equation for the helium atom. This does not mean that there
is no solution, or that wavefunctions do not exist. It simply means that we
know of no mathematical function that satisfies the differential equation. In
fact, for atoms and molecules that have more than one electron, the lack of
separability leads directly to the fact that there are no known analytical solutions
to any atom larger than hydrogen. Again, this does not mean that the wavefunctions do not exist. It simply means that we must use other methods to understand the behavior of the electrons in such systems. (It has been proven
mathematically that there is no analytic solution to the so-called three-body
problem, as the He atom can be described. Therefore, we must approach multielectron systems differently.)
Nor should this lack be taken as a failure of quantum mechanics. In this text,
we can only scratch the surface of the tools that quantum mechanics provides.
Quantum mechanics does provide tools to understand such systems. Atoms
and molecules having more than one electron can be studied and understood
by applying such tools to more and more exacting detail. The level of detail depends on the time, resources, and patience of the person applying the tools. In
theory, one can determine energies and momenta and other observables to the
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376
C H A P T E R 12
Atoms and Molecules
same level that one can know such observables for the hydrogen atom—if one
has the tools.
Example 12.3
Assume that the helium wavefunction is a product of two hydrogen-like
wavefunctions (that is, neglect the term for the repulsion between the electrons) in the n ϭ 1 principal quantum shell. Determine the electronic energy
of the helium atom and compare it to the experimentally determined energy
of Ϫ1.265 ϫ 10Ϫ17 J. (Total energies are determined experimentally by measuring how much energy it takes to remove all of the electrons from an atom.)
Solution
Using equation 12.6 and neglecting the electron-repulsion term by assuming
that the wavefunction is the product of two hydrogen-like wavefunctions:
⌿He ϭ ⌿H,1 ϫ ⌿H,2
the Schrödinger equation for the helium atom can be approximated as
΄
ប2
ប2
2e 2
2e 2
Ϫᎏᎏ᭞21 Ϫ ᎏᎏ ϩ Ϫᎏᎏ᭞22 Ϫ ᎏᎏ ⌿H,1⌿H,2 Ϸ EHe⌿H,1⌿H,2
2
4⑀0r1
2
4⑀0r2
΅
where EHe is the energy of the helium atom. Because the first term in brackets is a function of only electron 1 and the second term in the brackets is a
function of only electron 2, this Schrödinger equation can be separated just
like a two-dimensional particle-in-a-box can be separated. Understanding
this, we can separate the Schrödinger equation above into two parts:
ប2
2e 2
Ϫᎏᎏ᭞21 Ϫ ᎏᎏ ⌿H,1 ϭ E1⌿H,1
2
4⑀0r1
ប2
2e 2
Ϫᎏᎏ᭞22 Ϫ ᎏᎏ ⌿H,2 ϭ E2⌿H,2
2
4⑀0r2
where EHe ϭ E1 ϩ E2. These expressions are simply the one-electron
Schrödinger equations for a hydrogen-like atom where the nuclear charge
equals 2. An expression for the energy eigenvalue for such a system is known.
From the previous chapter, it is
Z 2e4
E ϭ Ϫᎏ
ᎏ
8⑀20h2n2
for each hydrogen-like energy. For this approximation, we are assuming that
helium is the sum of two hydrogen-like energies. Therefore,
EHe ϭ EH,1 ϩ EH,2
e 4
22e4
22e4
ϭ Ϫᎏ
2 ᎏ
2ᎏ
2 2 Ϫ ᎏ
2ᎏ
2 2 ϭ Ϫᎏ
8⑀0h n
8⑀0h n
⑀0h2n2
where we get the final term by combining the two terms to the left. Keep in
mind that is the reduced mass for an electron about a helium nucleus, and
that the principal quantum number is 1 for both terms. Substituting the values of the various constants, along with the value for the reduced mass of the
electron-helium nucleus system (9.108 ϫ 10Ϫ31 kg), we get
EHe ϭ Ϫ1.743 ϫ 10Ϫ17 J
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12.4 Spin Orbitals and the Pauli Principle
377
which is low by ϳ37.8% compared to experiment. Ignoring the repulsion
between the electrons leads to a significant error in the total energy of the
system, so a good model of the He atom should not ignore electron-electron
repulsion.
The example above shows that assuming that the electrons in helium—and
any other multielectron atom—are simple combinations of hydrogen-like electrons is naive assumption, and predicts quantized energies that are far from the
experimentally measured values. We need other ways to better estimate the
energies of such systems.
12.4 Spin Orbitals and the Pauli Principle
Example 12.3 for the helium atom assumed that both electrons have a principal quantum number of 1. If the hydrogen-like wavefunction analogy were
taken further, we might say that both electrons are in the s subshell of the first
shell—that they are in 1s orbitals. Indeed, there is experimental evidence
(mostly spectra) for this assumption. What about the next element, Li? It has
a third electron. Would this third electron also go into an approximate 1s
hydrogen-like orbital? Experimental evidence (spectra) shows that it doesn’t.
Instead, it occupies what is approximately the s subshell of the second principal quantum shell: it is considered a 2s electron. Why doesn’t it occupy the 1s
shell?
We begin with the assumption that the electrons in a multielectron atom
can in fact be assigned to approximate hydrogen-like orbitals, and that the
wavefunction of the complete atom is the product of the wavefunctions of each
occupied orbital. These orbitals can be labeled with the nᐉ quantum number
labels: 1s, 2s, 2p, 3s, 3p, and so on. Each s, p, d, f, . . . subshell can also be labeled
by an mᐉ quantum number, where mᐉ ranges from Ϫᐉ to ᐉ (2ᐉ ϩ 1 possible
values). But it can also be labeled with a spin quantum number ms, either ϩ12
or Ϫ12. The spin part of the wavefunction is labeled with either ␣ or , depending on the value of ms for each electron. Therefore, there are several simple possibilities for the approximate wavefunction for, say, the lowest-energy
state (the ground state) of the helium atom:
⌿He ϭ (1s1␣)(1s2␣)
⌿He ϭ (1s1␣)(1s2)
⌿He ϭ (1s1)(1s2␣)
⌿He ϭ (1s1)(1s2)
where the subscript on 1s refers to the individual electron. We will assume that
each individual ⌿He is normalized. Because each ⌿He is a combination of a
spin wavefunction and an orbital wavefunction, ⌿He’s are more properly called
spin orbitals.
Because spin is a vector and because vectors can add and subtract from
each other, one can easily determine a total spin for each possible helium spin
orbital. (It is actually a total z component of the spin.) For the first spin orbital equation above, both spins are ␣, so the total spin is (ϩᎏ12ᎏ) ϩ (ϩᎏ12ᎏ) ϭ 1.
Similarly, for the last spin orbital, the total spin is (Ϫᎏ12ᎏ) ϩ (Ϫᎏ12ᎏ) ϭ Ϫ1. For
the middle two spin orbitals, the total (z-component) spin is exactly zero. To
summarize:
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378
C H A P T E R 12
Atoms and Molecules
Approximate wavefunction
Total z-component spin
⌿He ϭ (1s1␣)(1s2␣)
⌿He ϭ (1s1␣)(1s2)
⌿He ϭ (1s1)(1s2␣)
⌿He ϭ (1s1)(1s2)
ϩ1
0
0
Ϫ1
At this point, experimental evidence can be introduced. (The necessity of
comparing the predictions of theory with experiment should not be forgotten.) Angular momenta of charged particles can be differentiated by magnetic
fields, so there is a way to experimentally determine whether or not atoms have
an overall angular momentum. Since spin is a form of angular momentum, it
should not be surprising that magnetic fields can be used to determine the
overall spin in an atom. Experiments show that ground-state helium atoms
have zero z-component spin. This means that of the four approximate wavefunctions listed above, the first and last are not acceptable because they do not
agree with experimentally determined facts. Only the middle two, (1s1␣)(1s2)
and (1s1)(1s2␣), can be considered for helium.
Which wavefunction of the two is acceptable, or are they both? One can suggest that both wavefunctions are acceptable and that the helium atom is doubly degenerate. This turns out to be an unacceptable statement because, in
part, it implies that an experimenter can determine without doubt that electron 1 has a certain spin wavefunction and that electron 2 has the other spin
wavefunction. Unfortunately, we cannot tell one electron from another. They
are indistinguishable.
This indistinguishability suggests that the best way to describe the electronic
wavefunction of helium is not by each wavefunction individually, but by a
combination of the possible wavefunctions. Such combinations are usually
considered as sums and/or differences. Given n wavefunctions, one can mathematically determine n different combinations that are linearly independent.
So, for the two “acceptable” wavefunctions of He, two possible combinations
can be constructed to account for the fact that electrons are indistinguishable.
These two combinations are the sum and the difference of the two individual
spin orbitals:
1
⌿He,1 ϭ ᎏᎏ[(1s1␣)(1s2) ϩ (1s1)(1s2␣)]
͙2ෆ
1
⌿He,2 ϭ ᎏᎏ[(1s1␣)(1s2) Ϫ (1s1)(1s2␣)]
͙2ෆ
The term 1/͙2ෆ is a renormalization factor, taking into account the combination of two normalized wavefunctions. These combinations have the proper
form for possible wavefunctions of the helium atom.
Are both acceptable, or only one of the two? At this point we rely on a postulate proposed by Wolfgang Pauli in 1925, which was based on the study of
atomic spectra and the increasing understanding of the necessity of quantum
numbers. Since electrons are indistinguishable, one particular electron in helium can be either electron 1 or 2. We can’t say for certain which. But because
the electron has a spin of ᎏ21ᎏ, it has certain properties that affect its wavefunction
(the details of which cannot be considered here). If electron 1 were exchanged
with electron 2, Pauli postulated, the complete wavefunction must change sign.
Mathematically, this is written as
⌿(1, 2) ϭ Ϫ⌿(2, 1)
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12.4 Spin Orbitals and the Pauli Principle
379
The switch in order of writing the labels 1 and 2 implies that the two electrons
are exchanged. Electron 1 now has the coordinates of electron 2, and vice versa.
A wavefunction having this property is called antisymmetric. (By contrast, if
⌿(1, 2) ϭ ⌿(2, 1), the wavefunction is labeled symmetric.) Particles having
half-integer spin (ᎏ21ᎏ, ᎏ23ᎏ, ᎏ25ᎏ, . . .) are collectively called fermions. The Pauli principle
states that fermions must have antisymmetric wavefunctions with respect to
exchange of particles. Particles having integer spins, called bosons, are restricted
to having symmetric wavefunctions with respect to exchange.
Electrons are fermions (having spin ϭ ᎏ12ᎏ) and so according to the Pauli principle must have antisymmetric wavefunctions. Consider, then, the two possible approximate wavefunctions for helium. They are
1
⌿He,1 ϭ ᎏᎏ[(1s1␣)(1s2) ϩ (1s1)(1s2␣)]
͙2ෆ
(12.7)
1
⌿He,2 ϭ ᎏᎏ[(1s1␣)(1s2) Ϫ (1s1)(1s2␣)]
͙2ෆ
(12.8)
Are either of these antisymmetric? We can check by interchanging electrons 1
and 2 in the first wavefunction, equation 12.7, and get
1
⌿(2, 1) ϭ ᎏᎏ[(1s2␣)(1s1) ϩ (1s2)(1s1␣)]
͙2ෆ
(Note the change in the subscripts 1 and 2.) This should be recognized as the
original wavefunction ⌿(1, 2), only algebraically rearranged. (Show this.)
However, upon electron exchange, the second wavefunction, equation 12.8,
becomes
1
⌿(2, 1) ϭ ᎏᎏ[(1s2␣)(1s1) Ϫ (1s2)(1s1␣)]
͙2ෆ
(12.9)
which can be shown algebraically to be Ϫ⌿(1, 2). (Show this, also.)
Therefore, this wavefunction is antisymmetric with respect to exchange of
electrons and, by the Pauli principle, is a proper wavefunction for the spin
orbitals of the helium atom. Equation 12.8, but not equation 12.7, represents the correct form for a spin-orbital wavefunction of the ground state
of He.
The rigorous statement of the Pauli principle is that wavefunctions of electrons must be antisymmetric with respect to exchange of electrons. There is a
simpler statement of the Pauli principle. It comes from the recognition that
equation 12.8, the only acceptable wavefunction for helium, can be written in
terms of a matrix determinant.
Recall that the determinant of a 2 ϫ 2 matrix written as
͉ ͉
a d
c b
is simply (a ϫ b) Ϫ (c ϫ d ), which is remembered mnemonically as
ϩ
a d→ Ϫc ϫ d
→
aϫb
Ϫ c b
The proper antisymmetric wavefunction, equation 12.8, for the helium atom
can also be written in terms of a 2 ϫ 2 determinant:
͉ ͉
1
⌿He ϭ ᎏᎏ
͙2ෆ
͉
͉
1s1␣ 1s1
1s2␣ 1s2
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(12.10)
380
C H A P T E R 12
Atoms and Molecules
The term 1/͙2ෆ multiplies the entire determinant, like it multiplies the entire
wavefunction in equation 12.8. Such determinants used to represent an antisymmetric wavefunction are called Slater determinants, after J. C. Slater, who
pointed out such constructions for wavefunctions in 1929.
The use of Slater determinants to express wavefunctions that are automatically antisymmetric stems from the fact that when two rows (or two columns)
of a determinant are exchanged, the determinant of the matrix becomes
negated. In the Slater determinant shown in equation 12.10, the possible spin
orbitals for electron 1 are listed in the first row and the spin orbitals of electron 2 are listed in the second row. Switching these two rows would be the same
thing as exchanging the two electrons in the helium atom. When this happens,
the determinant changes sign, which is what the Pauli principle requires for acceptable wavefunctions of fermions. Writing a wavefunction in terms of a
proper Slater determinant guarantees an antisymmetric wavefunction.
The Slater-determinant form of a wavefunction guarantees something else,
which leads to the simplified version of the Pauli principle. Suppose both electrons in helium had the exact same spin orbital. The determinant part of the
wavefunction would have the form
͉
͉
1s1␣ 1s1␣
1s2␣ 1s2␣
which is exactly 0
(12.11)
The determinant being exactly zero is a general property of determinants. (If
any two columns or rows of a determinant are the same, the value of the determinant is zero.) Therefore the wavefunction is identically zero and this state
will not exist. The same conclusion can be reached if the spin on both electrons is . Consider, then, the lithium atom. Assuming that all three electrons
were in the 1s shell, the only two possible determinant forms of the wavefunction would be (depending on the spin function on the third electron):
͉
͉ ͉
1s1␣ 1s1 1s1␣
1s2␣ 1s2 1s2␣
1s3␣ 1s3 1s3␣
or
͉
1s1␣ 1s1 1s1
1s2␣ 1s2 1s2
1s3␣ 1s3 1s3
(12.12)
Note that in both cases, two columns of the determinant represent the same
spin orbitals for two of the three electrons (1st and 3rd columns for the first
determinant, 2nd and 3rd columns for the second determinant). The mathematics of determinants requires that if any two rows or columns are exactly the
same, the value of the determinant is exactly zero. One cannot have a wavefunction for Li having three electrons in the 1s shell. The third electron, instead, must be in a different shell. The next shell and subshell are 2s.
As we have been assigning a set of four quantum numbers to electrons in
hydrogen-like orbitals, we can do so for the spin orbitals of multielectron
atoms where we are approximating hydrogen-like orbitals. In the first row of
equation 12.11, the two spin orbitals can be represented by the set of four
quantum numbers (n, ᐉ, mᐉ, ms) as being (1, 0, 0, ᎏ12ᎏ) and (1, 0, 0, ᎏ12ᎏ): the same
four quantum numbers. (Can you see how these numbers were determined
from the expression for the spin orbital?) In the first row of equation 12.12,
the three spin orbitals in the first case have the sets (1, 0, 0, ᎏ12ᎏ), (1, 0, 0, Ϫᎏ12ᎏ),
and (1, 0, 0, ᎏ12ᎏ): the first and third spin orbitals are the same. In the second case,
for the first row, the spin orbitals can be represented by the quantum numbers
(1, 0, 0, ᎏ12ᎏ), (1, 0, 0, Ϫᎏ12ᎏ), and (1, 0, 0, Ϫᎏ12ᎏ), with the second and third spin orbitals having the same set of four quantum numbers. In all three cases, other
rows of the Slater determinant can have quantum numbers assigned to them,
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12.4 Spin Orbitals and the Pauli Principle
381
and in all cases the determinant is exactly zero, implying that the overall wavefunction does not exist.
On this basis, one consequence of the Pauli principle is that no two electrons
in any system can have the same set of four quantum numbers. (This statement
is sometimes used in place of the original statement of the Pauli principle.)
This means that each and every electron must have its own unique spin orbital, and since there are only two possible spin functions for an electron, each
orbital can be assigned only two electrons. Therefore, an s subshell can accommodate two electrons maximum; each p subshell, with three individual p
orbitals, can hold a maximum of six electrons; each d subshell, with five d orbitals, can hold ten electrons; and so on. Because this consequence of the Pauli
principle excludes spin orbitals from having more than one electron, Pauli’s
statement is commonly referred to as the Pauli exclusion principle.
Example 12.4
Show for each row of the Slater determinants for Li in equation 12.12 that
the wavefunction represented by the determinant violates the Pauli exclusion
principle.
Solution
By row, the set of four quantum numbers for each spin orbital is listed:
͉
͉
͉
͉
1s1␣ 1s1 1s1␣
1s2␣ 1s2 1s2␣
1s3␣ 1s3 1s3␣
(1, 0, 0, ᎏ12ᎏ)
(1, 0, 0, ᎏ12ᎏ)
(1, 0, 0, ᎏ12ᎏ)
(1, 0, 0, Ϫᎏ12ᎏ)
(1, 0, 0, Ϫᎏ12ᎏ)
(1, 0, 0, Ϫᎏ12ᎏ)
(1, 0, 0, ᎏ12ᎏ)
(1, 0, 0, ᎏ12ᎏ)
(1, 0, 0, ᎏ12ᎏ)
1s1␣ 1s1 1s1
1s2␣ 1s2 1s2
1s3␣ 1s3 1s3
(1, 0, 0, ᎏ12ᎏ)
(1, 0, 0, ᎏ12ᎏ)
(1, 0, 0, ᎏ12ᎏ)
(1, 0, 0, Ϫᎏ12ᎏ)
(1, 0, 0, Ϫᎏ12ᎏ)
(1, 0, 0, Ϫᎏ12ᎏ)
(1, 0, 0, Ϫᎏ12ᎏ)
(1, 0, 0, Ϫᎏ12ᎏ)
(1, 0, 0, Ϫᎏ12ᎏ)
In each case, two of the three entries in each row have the same set of four
quantum numbers and so the wavefunction is not allowed by the Pauli exclusion principle.
Wavefunctions written in terms of a Slater determinant have a normalization factor of 1/͙n!
ෆ, where n is the number of rows or columns in the determinant (and equals the number of electrons in the atom). This is because the
expanded form of the wavefunction ⌿ has n! terms. In constructing Slater determinants, we will follow the custom of writing the individual spin orbitals
going across, two spatial wavefunctions with an ␣ and a  spin wavefunction
each, and listing the electrons sequentially going down. That is:
spin orbitals
electron 1
electron 2
electron 3
.
.
.
→
1s␣ 1s 2s␣ 2s . . .
1s␣ 1s 2s␣ 2s . . .
... ... ... ... ...
͉
͉
Going across the determinant, the spin part alternates: ␣, , ␣, , . . . . You also
have to keep track of the n, ᐉ, and mᐉ quantum numbers to make sure each
shell and subshell is represented in the proper order and number. The following example illustrates the use of this idea.
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382
C H A P T E R 12
Atoms and Molecules
Example 12.5
The third electron in Li goes into the 2s orbital. Assuming a (re)normalization constant of 1/͙6ෆ, construct a proper antisymmetric wavefunction for Li
in terms of a Slater determinant.
Solution
The rows will represent electrons 1, 2, and 3; the columns will represent the
spin orbitals 1s␣, 1s, and 2s␣ (or 2s). Following the determinant setup
above, the antisymmetric wavefunction is
͉
͉
1s1␣ 1s1 2s1␣
1
⌿Li ϭ ᎏᎏ 1s2␣ 1s2 2s2␣
͙6ෆ 1s ␣ 1s  2s ␣
3
3
3
Because there are two possible wavefunctions for Li (depending on whether
the spin orbital for the last column is 2s␣ or 2s), we conclude that this
energy level is doubly degenerate.
12.5 Other Atoms and the Aufbau Principle
We have presumed, more than proved, that multielectron atoms can be conceptually approximated as combinations of hydrogen-like orbitals (even though
our helium example showed that the predicted energies are not very close).
Further, the Pauli principle restricts orbitals to having only two electrons, each
with different spin. As we consider larger and larger atoms, electrons in these
atoms will occupy orbitals described with larger and larger principal quantum
numbers.
Recall that in the hydrogen atom, the principal quantum number is the only
quantum number that affects the total energy. This is not the case with multielectron atoms, because interelectronic interactions affect the energies of the
orbitals, and now the subshells within the shells have different energies. Figure
12.4 illustrates what happens to the electronic energy levels of atoms. In the
case of hydrogen, energies of orbitals are determined by a single quantum
number. In multielectron atoms, the principal quantum number is an important factor in the energy of an orbital, but the angular momentum quantum
number is also a factor. (To a much lesser extent, the mᐉ and ms quantum numbers also affect the exact energy of a spin orbital, but their effect on the energy
is more noticeable in molecules. Their effect on the exact energies of electrons
is practically negligible for atoms outside of magnetic fields. See Figure 12.2
for an example.)
When assigning electrons to orbitals in multielectron atoms, it might be assumed that they will occupy the available shell and subshell having the lowest
energy. This is a misstatement. Electrons reside in the next available spin orbital
that yields the lowest total energy for the atom. The placement is not necessarily
determined by the individual energy of the spin orbital. Instead, the total energy of the atom must be considered. When an atom’s electrons occupy orbitals
that yield the lowest total energy, the atom is said to be in its ground state. Any
other electronic state, which by definition would have a higher total energy, is
considered an excited state. The electrons in an atom can reach excited states by
absorbing energy; this is one of the basic processes in spectroscopy.
Consider an atom of the element beryllium, which has four electrons. Two
of the electrons occupy the orbital labeled 1s. The two remaining electrons
occupy an orbital in the second shell, but which? The n ϭ 2 shell has ᐉ ϭ 0
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12.5 Other Atoms and the Aufbau Principle
383
5s
4p
4s, p, d, f
3d
4s
3p
3s, p, d
Energy
3s
2p
2s, p
2s
1s
1s
H- like
Not H- like
Figure 12.4 The effect of more than one electron on the electronic energy levels of an atom.
For hydrogen-like atoms, all of the energy levels with the same principal quantum number n are
degenerate. For atoms having more than one electron, the shells are separated by the ᐉ quantum
number as well. (Energy axis is not to scale.)
and ᐉ ϭ 1, so that the possible subshells are 2s and 2p. Because of the slightly
higher energy of the 2p subshell, the electrons occupy the 2s subshell, which
can hold two electrons if they have different spin functions. The occupation of
orbitals in an atom is listed as an electron configuration, using superscripts to
indicate the number of electrons in each individual subshell. It is assumed that
for ground states, the spins of the electrons are appropriate and satisfy the
Pauli exclusion principle. The electron configuration for Be is written as
1s2 2s2
This is an obvious electron configuration, since the 2p subshell is higher in
energy than the 2s, as shown in Figure 12.4. However, as we will see shortly, it
is not always so straightforward to assign an electron configuration.
Example 12.6
Electron configurations are rather abbreviated when compared to the more
complete Slater-determinant form of the antisymmetric wavefunction.
Compare the electron configuration of Be with the Slater-determinant form
of ⌿ for Be.
Solution
The electron configuration for Be, given above, is simply 1s2 2s2. Using the rule
from above for constructing Slater determinants, the more complete ⌿ is
͉͉
1s1␣
1 1s2␣
⌿ ϭ ᎏᎏ
͙24
ෆ 1s3␣
1s4␣
1s1
1s2
1s3
1s4
2s1␣
2s2␣
2s3␣
2s4␣
͉͉
2s1
2s2
2s3
2s4
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384
C H A P T E R 12
Atoms and Molecules
Written as a determinant, this wavefunction is indeed antisymmetric. If the
determinant were evaluated, it would expand into 24 terms. The electron
configuration, however, is a total of only six alphanumeric characters.
Although the Slater-determinant wavefunction is more complete, the electron
configuration is much more convenient.
As we consider larger and larger atoms, electrons start occupying orbitals of
the 2p subshell. It should be recognized that with three possible p orbitals,
there are several possible ways of, say, two electrons occupying the various p
orbitals. A statement known as Hund’s rule indicates that electrons occupy each
degenerate orbital singly before pairing up orbitals with two electrons of opposite spin. (The rule was enunciated by Friedrich Hund in 1925 after detailed
consideration of atomic spectra.) In the absence of any other influence, the orbitals are still degenerate, so at this point there is no preference about which p
orbitals are singly, then doubly, occupied. Therefore, one specific electron configuration for the ground state of boron can be listed as 1s2 2s2 2p1x , and a specific electron configuration for the ground state of carbon can be given as 1s2
2s2 2p1x 2p1y . If Hund’s rule is assumed, a more general electron configuration
of C can be abbreviated as 1s2 2s2 2p2.
Example 12.7
List two other acceptable ground-state electron configurations for B and C.
Give an unacceptable ground-state electron configuration for C.
Solution
Since it does not matter which p orbitals are used, the ground state of B can
also be written as 1s2 2s2 2p1y or 1s2 2s2 2p1z . For C, the other acceptable electron configurations are 1s2 2s2 2p1y 2p1z or 1s2 2s2 2p1x 2p1z . Both of these can
be abbreviated as 1s2 2s2 2p2. An unacceptable ground-state electron configuration might be 1s2 2s2 2p2x, since this has the electrons paired in a single p
orbital rather than spread out among the degenerate p orbitals, as required
by Hund’s rule.
The filling of the spin orbitals so far has taken the order 1s, 2s, 2p. As one
considers the electron configurations of larger atoms, electrons continue to occupy orbitals through 3s and 3p. But at potassium (Z ϭ 19), instead of filling
the 3d orbital, the 4s orbital is occupied first. Only after a second electron occupies the 4s orbital (for calcium) does the 3d subshell start becoming occupied with electrons.
Why? The naive answer is that the 4s orbital is lower in energy than the 3d
orbital. Since the energies of orbitals in multielectron atoms are determined by
the quantum number ᐉ as well as the quantum number n, it must be at this
point that the energy E4s becomes less than the energy E3d. Actually, this argument is misleading. The reason that the 4s orbital becomes occupied is that the
total energy of the atom is less than it would be if the electron occupied a 3d
orbital.
On the face of it, this seems peculiar. If the 3d orbital were lower in energy,
why shouldn’t it be occupied by an electron first? If it were a hydrogen-like
atom, with only a single electron, then the absolute energy of the orbital would
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12.5 Other Atoms and the Aufbau Principle
4r 2͉R ͉ 2
⌿3d
4r 2͉R ͉ 2
⌿4s
385
0
1
2
3
4
5
6
Distance from nucleus (Å)
7
8
0
1
2
3
4
5
6
Distance from nucleus (Å)
7
8
Figure 12.5 Plots of 4r2͉R͉2 for the 3d and 4s wavefunctions. Note that the plots have the
same x-axis, and that the 4s electron has some probability of being rather close to the nucleus.
For multielectron atoms, the penetration of the 4s electron combined with the shielding effect of
the other electrons serves to make the 4s orbital the next one occupied by electrons, rather than
the 3d.
be the only factor determining the order of orbital occupation. But in multielectron atoms, there is an additional factor. Not only is the absolute energy of
the orbital a factor, but the amount an electron in that orbital interacts with the
other electrons and the nucleus is also a criterion in determining the total energy of an atom.
To illustrate this point, Figure 12.5 shows the probabilities of the radial
functions in spherically symmetric shells about the nucleus (that is, 4r2͉R͉2
versus r) for the 3d and 4s hydrogen-like wavefunctions on the same scale.
Both wavefunctions show a maxima several angstroms from the nucleus.*
However, note that the 4s orbital has three relative maxima before its absolute maximum, and that several of these maxima indicate that an electron
in a 4s orbital has a considerable probability of being closer to the nucleus
than an electron in a 3d orbital. An electron in a 4s orbital is said to penetrate inward toward the nucleus. The increased penetration of the negatively
charged 4s electron toward the positively charged nucleus means an additional energy stabilization of the system as a whole, and as a result the final
electron in K occupies the 4s orbital. This allows the entire K atom to have a
lower total energy. And even though some energy of repulsion erases some
of that energy gain, the last electron of Ca, the next largest atom, also occupies a 4s orbital instead of a 3d orbital, pairing up with the first electron in
the 4s orbital. (However, there are a few exceptions, as a review of electron
configurations will show.) Only with the introduction of another electron, for
an atom of scandium, does the electron occupy a 3d orbital instead of a
4p orbital.
This building up of electron configurations by placing electrons in orbitals
is called the aufbau principle (the name of the principle comes from the
German word aufbauen, meaning “to build up”). Although it might seem at
this point that there is little regularity in building up electron configurations
of larger atoms, there is some level of consistency. For example, the periodic
table’s shape is dictated by the filling of the orbitals by electrons. The number
of valence electrons almost never exceeds 8, due to the eventual regularity in
the filling of orbitals. There are several mnemonic devices used to remember
*Actually, in a multielectron atom the maxima would be somewhat farther out because
of a shielding effect on the nucleus by the electrons occupying the inner shells.
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386
C H A P T E R 12
Atoms and Molecules
1s
2s
2p
3s
3p
3d
4s
4p
4d
5s
6s
7s
5p
6p
5d
6d
1s
2s
2p
3s
3p
4f
5f
6f
5g
6g
4s
3d
4p
5s
4d
5p
6s
*
5d
6p
7s
**
6d
7p
7p
and so on
Figure 12.6 A convenient way to
remember the order of filling of the
subshells in most atoms (1–85).
Simply follow the order of subshells
crossed by the arrows.
*
4f
**
5f
Figure 12.7 The aufbau principle rationalizes
the structure of the periodic table. Compare the
order of filling of the subshells in Figure 12.6 with
the labels in the periodic table here. (Note where
the 4f and 5f subshells are filled.)
the order in which orbitals are filled by electrons. Perhaps the most common
is shown in Figure 12.6. This ordering of orbitals, and the idea that each subshell is completely filled before electrons occupy the next orbital, is strictly applicable to the electron configurations of 85 of the first 103 elements. (There
is little or no experimental verification of the electron configurations of elements 104 or larger to date.) Figure 12.7 shows the relationship between the
aufbau principle and the structure of the periodic table. Table 12.1 lists the
electron configurations of the elements in their lowest electronic states.
12.6 Perturbation Theory
In a previous section, we presumed that the wavefunctions of multielectron
atoms can be approximated as products of hydrogen-like orbitals:
⌿Z Х ⌿H,1 и ⌿H,2 и ⌿H,3 и и и ⌿H,Z
(12.13)
where ⌿Z is the wavefunction for an atom having a nuclear charge of Z and
⌿H,1, . . . . ⌿H,Z are the hydrogen-like wavefunctions for each of the Z electrons. Generally speaking, this is a very useful qualitative description of the
electrons in larger atoms. However, we did see that in the case of He, it is not
a good system for making a quantitative prediction of the total electronic energy of the system. As noted, there is no known exact solution to the differential equation that is the Schrödinger equation for the helium atom; it has no
analytic solution. There is no known simple (or complicated, for that matter!)
expression for ⌿ that we can substitute into the Schrödinger equation as given
in equation 12.5 and have it satisfied so that an eigenvalue E is produced.
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12.6 Perturbation Theory
Table 12.1
1
387
Ground-state electron configurations of the elementsa
H 1s
He 1s2
Li 1s2 2s1
Be 1s2 2s2
B 1s2 2s2 2p1
C 1s2 2s2 2p2
N 1s2 2s2 2p3
O 1s2 2s2 2p4
F 1s2 2s2 2p5
Ne 1s2 2s2 2p6
Na 1s2 2s2 2p6 3s1
Mg 1s2 2s2 2p6 3s2
Al 1s2 2s2 2p6 3s2 3p1
Si 1s2 2s2 2p6 3s2 3p2
P 1s2 2s2 2p6 3s2 3p3
S 1s2 2s2 2p6 3s2 3p4
Cl 1s2 2s2 2p6 3s2 3p5
Ar 1s2 2s2 2p6 3s2 3p6
K 1s2 2s2 2p6 3s2 3p6 4s1
Ca 1s2 2s2 2p6 3s2 3p6 4s2
Sc 1s2 2s2 2p6 3s2 3p6 4s2 3d 1
Ti 1s2 2s2 2p6 3s2 3p6 4s2 3d 2
V 1s2 2s2 2p6 3s2 3p6 4s2 3d 3
Cr* 1s2 2s2 2p6 3s2 3p6 4s1 3d 5
Mn 1s2 2s2 2p6 3s2 3p6 4s2 3d 5
Fe 1s2 2s2 2p6 3s2 3p6 4s2 3d 6
Co 1s2 2s2 2p6 3s2 3p6 4s2 3d 7
Ni 1s2 2s2 2p6 3s2 3p6 4s2 3d 8
Cu* 1s2 2s2 2p6 3s2 3p6 4s1 3d 10
Zn 1s2 2s2 2p6 3s2 3p6 4s2 3d 10
Ga 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p1
Ge 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p2
As 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p3
Se 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p4
Br 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p5
Kr 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6
Rb 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s1
Sr 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2
Y 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 1
Zr 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 2
Nb* 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s1 4d 4
Mo* 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s1 4d 5
Tc 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 5
Ru* 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s1 4d 7
Rh* 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s1 4d 8
Pd* 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s0 4d 10
Ag* 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s1 4d 10
Cd 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10
In 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 105p1
Sn 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 105p2
Sb 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 105p3
Te 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 105p4
I 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p5
Xe 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 105p6
Cs 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 105p6 6s1
Ba 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 105p6 6s2
La* 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 5d 1
Ce* 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 1 5d 1
Pr 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 3
Nd 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 4
Pm 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 5
Sm 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 6
Eu 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 7
Gd* 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 7 5d 1
Tb 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 9
Dy 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 10
Ho 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 11
Er 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 12
Tm 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 13
Yb 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 14
Lu 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 14 5d 1
Hf 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 14 5d 2
Ta 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 14 5d 3
W 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 14 5d 4
Re 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 14 5d 5
Os 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 14 5d 6
Ir 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 14 5d 7
Pt* 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s1 4f 14 5d 9
Au* 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s1 4f 14 5d 10
Hg 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 14 5d 10
Tl 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 14 5d 10 6p1
Pb 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 14 5d 10 6p2
Bi 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 14 5d 10 6p3
Po 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 14 5d 10 6p4
At 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 14 5d 10 6p5
Rn 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 14 5d 10 6p6
Fr 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 14 5d 10 6p6 7s1
Ra 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 14 5d 10 6p6 7s2
Ac* 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 14 5d 10 6p6 7s2 6d 1
Th* 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 14 5d 10 6p6 7s2 6d 2
Pa* 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 14 5d 10 6p6 7s2 5f 2 6d 1
U* 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 14 5d 10 6p6 7s2 5f 3 6d 1
Np* 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 14 5d 10 6p6 7s2 5f 4 6d 1
Pu 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 14 5d 10 6p6 7s2 5f 6
Am 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 14 5d 10 6p6 7s2 5f 7
Cm* 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 14 5d 10 6p6 7s2 5f 7 6d 1
Bk 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 14 5d 10 6p6 7s2 5f 9
Cf 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 14 5d 10 6p6 7s2 5f 10
Es 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 14 5d 10 6p6 7s2 5f 11
Fm 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 14 5d 10 6p6 7s2 5f 12
Md 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 14 5d 10 6p6 7s2 5f 13
No 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 14 5d 10 6p6 7s2 5f 14
Lr 1s2 2s2 2p6 3s2 3p6 4s2 3d 10 4p6 5s2 4d 10 5p6 6s2 4f 14 5d 10 6p6 7s2 6d 1
a
An asterisk by the symbol indicates that the electron configuration does not exactly conform to the strict rules of the aufbau principle. However, in almost all cases, the variance is
due to a single electron. Only elements up to Z ϭ 103 are included, since the electron configurations for elements beyond that have not been experimentally verified.
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388
C H A P T E R 12
Atoms and Molecules
That does not mean that there is no understanding of, or recourse for, such
systems. Nor does it imply that quantum mechanics is a useless theory for
these systems. There are two main tools for applying quantum mechanics to
systems whose Schrödinger equations cannot be solved exactly. Use of either
depends on the type of system under study as well as what information you
want to determine.
The first of these tools is called perturbation theory. Perturbation theory assumes that a system can be approximated as a known, solvable system and that
any difference between the system of interest and the known system is a small,
additive perturbation that can be calculated separately and added on. We will
assume that all of the energy levels under discussion are singly degenerate, so
this tool is more appropriately named nondegenerate perturbation theory. Also,
perturbation theory can be taken to very complex levels. Here, we focus on the
first level of approximation, which is called first-order perturbation theory.
Perturbation theory assumes that the Hamiltonian for a real system can be
written as
ˆsystem ϭ H
ˆideal ϩ H
ˆperturb ϵ H
ˆ° ϩ H
ˆЈ
H
(12.14)
ˆsystem is the Hamiltonian of the system of interest that is being apwhere H
ˆ° is the Hamiltonian of an ideal or model system, and H
ˆЈ repproximated, H
resents the small, additive perturbation. For example, in the case of the helium
atom, the ideal part of the Hamiltonian can represent two hydrogen-like atoms.
The perturbation part of the Hamiltonian can represent the coulombic repulsion between the electrons:
2
ˆHe ϭ (H
ˆH-like ϩ H
ˆH-like) ϩ ᎏeᎏ
H
4⑀0r12
(In this case, there are two hydrogen-like Hamiltonians because there are two
electrons. Despite this rewriting, the Schrödinger equation for He has not really changed and is still analytically unsolvable.) Any number of additive perturbations can be combined with an ideal Hamiltonian. It is, of course, easier
to keep the number of terms in the Hamiltonian as small as possible. What is
usually found is that there is a trade-off between the number of terms and the
accuracy of the solution to the Schrödinger equation.
If we assume that the wavefunction ⌿ of the real system is similar to the
wavefunction of the ideal system, denoted ⌿(0), then one can say that, approximately,
ˆsystem⌿(0) Ϸ Esystem⌿(0)
H
(12.15)
where Esystem is the eigenvalue for the energy of the real system. Over the
course of many observations, one eventually determines an average value of
the observable energy, ͗E͘. By using one of the postulates of quantum mechanics ͗E͘ can be approximated by the expression
͗E͘ Ϸ
͵ (⌿
ˆsystem⌿(0) d
)*H
(0)
(12.16)
ˆsystem, one can substitute into equation 12.16 and partially
Given the form of H
evaluate:
͵ (⌿
ϭ ͵ (⌿
͗E͘ Ϸ
ˆ° ϩ H
ˆЈ)⌿(0) d
)*(H
(0)
ˆ°⌿(0) d ϩ
)*H
(0)
ϭ ͗E (0)͘ ϩ
͵ (⌿
͵ (⌿
ˆЈ⌿(0) d
)*H
(0)
ˆЈ⌿0 d ϭ ͗E (0)͘ ϩ ͗E (1)͘
)*H
(0)
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(12.17)
12.6 Perturbation Theory
389
where ͗E (0)͘ is the average energy of the ideal or model system (that is, an
eigenvalue energy, usually) and ͗E (1)͘ is the first-order correction to the energy.
Thus, the first approximation to the energy of a real system is equal to the ideal
ˆЈ⌿(0) d. If this inenergy plus some additional amount given by ͐ (⌿(0))*H
tegral can be evaluated or approximated, then a correction to the energy can
be determined. What equation 12.17 means is that when we write a Hamiltonian
as a perturbed ideal operator, the energy—the observable associated with the
Hamiltonian—is also perturbed from the ideal.
Example 12.8
What is the correction to the energy of the helium atom, assuming that the
perturbation can be approximated as a coulombic repulsion of the two
electrons?
Solution
According to equation 12.17, the perturbation is
͗E (1)͘ ϭ
͵ (⌿
e2
)*ᎏᎏ ⌿(0) d
4⑀0r12
(0)
If some way can be found to evaluate this integral, a correction to the total
energy—and thus a perturbation-theory approximation to the energy of a He
atom—can be approximated.
The integral in the above example can be approximated by mathematical
techniques and substitutions that we will not go into. (A discussion of its solution can be found in more advanced texts.) Approximations and substitutions are possible and the above integral can be estimated as
5
e2
͗E (1)͘ ϭ ᎏᎏ ᎏᎏ
4 4⑀0a0
where e is the charge on the electron, ⑀0 is the permittivity of free space, and
a0 is the first Bohr radius (0.529 Å). When we substitute the values of the constants into this expression, we get
͗E (1)͘ ϭ 5.450 ϫ 10Ϫ18 J
Combining this result with the “ideal” energy, which was determined by assuming the sum of two hydrogen electron energies (see Example 12.3), we get
for the total energy of helium
EHe ϭ Ϫ1.743 ϫ 10Ϫ17 J ϩ 5.450 ϫ 10Ϫ18 J ϭ Ϫ1.198 ϫ 10Ϫ17 J
which, when compared to the experimentally determined energy of the helium
atom (given in Example 12.3 as Ϫ1.265 ϫ 10Ϫ17 J), is found to be off by only
5.3%. Compared to the hydrogen-like approximation of helium, this is a big
improvement. It points out the usefulness of perturbation theory.
Example 12.9
In a particle-in-a-box having length a, instead of being zero the potential energy in the box is a linear function of the position. That is,
V ϭ kx
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390
C H A P T E R 12
Atoms and Molecules
a. Using perturbation theory, estimate the average energy of a particle having
mass m and whose motion is described by the lowest-energy wavefunction
(n ϭ 1).
b. The integral in part a can be solved exactly. Explain why this calculated
value is not the exact value for the energy of a particle in this system.
Solution
a. According to perturbation theory, the energy of the particle is
͗E͘ ϭ ͗E (0)͘ ϩ ͗E (1)͘
ˆ° is the Hamiltonian for the particle-in-a-box, then the
If one assumes that H
ˆ
perturbation part H Ј of the complete Hamiltonian is kx. According to equation 12.17, the energy is
n2h2
͗E͘ ϭ ᎏᎏ2 ϩ ͗E (1)͘
8ma
To evaluate ͗E (1)͘, we must evaluate the integral
͵
a
͗E
2
x
x
͘ ϭ ᎏᎏ sin ᎏᎏ * и kx и sin ᎏᎏ dx
a0
a
a
(1)
where the normalization constant has been brought outside of the integral
sign, and d and the integration limits are for the 1-D particle-in-a-box. This
integral simplifies to
͵
a
͗E
2k
x
͘ ϭ ᎏᎏ x и sin2 ᎏᎏ dx
a 0
a
(1)
This integral has a known solution (see the integral table in Appendix 1).
Evaluation of this integral specifically is left as an exercise. Substituting for
the evaluated integral, this expression becomes
a2 2k
ka
͗E (1)͘ ϭ ᎏᎏ и ᎏᎏ ϭ ᎏᎏ
4
a
2
Therefore, the energy of the n ϭ 1 level is
ka
n2h2
͗E͘ ϭ ᎏᎏ2 ϩ ᎏᎏ
2
8ma
b. This is not an exact energy for such a system because the wavefunctions
used to determine the energies were the particle-in-a-box wavefunctions, not
wavefunctions for a box having a sloped bottom. So although the integral for
the perturbation energy is solvable analytically, it does not correct the energy
to the exact value of the true energy because we are not using the eigenfunctions of the defined system. (Nor are we using the complete Hamiltonian
operator for the defined system.) Higher-order perturbation theory, not discussed in this text, may have a better chance of approaching the exact wavefunction and energy eigenvalues for this system.
As the above example shows, although we have defined a first-order energy
correction, we are still using the ideal forms of the wavefunctions. What we
also need is a correction to the wavefunctions. It is assumed that, as for the energy correction, the first-order correction to the wavefunction is some correction added to the ideal wavefunction to approximate the real wavefunction:
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12.6 Perturbation Theory
⌿real Ϸ ⌿(0) ϩ ⌿(1)
391
(12.18)
It should be understood by now that there is not just one wavefunction for
a model system. There are a large number. Many times there are an infinite
number of wavefunctions, each with their own quantum numbers. Equation
12.18 can be rewritten to recognize the fact that the many wavefunctions are
all different and should be labeled. For example, using the label n (not to be
confused with the quantum number!):
(1)
⌿n,real Ϸ ⌿(0)
n ϩ ⌿n
(12.19)
The entire group of wavefunctions for a model system is considered a complete
set of eigenfunctions. For a model system, the individual wavefunctions are
orthogonal; this fact will be important later. Such a situation is analogous to
the coordinates x, y, and z defining three-dimensional space: the set (x, y, z)
represents a complete set of “functions” used to define any point in space. Any
point in 3-D space can be described as the appropriate combination of so
many x unit vectors, so many y unit vectors, and so many z unit vectors.*
The complete set of wavefunctions is similar. Such a set can be used to define the complete “space” of a system. The true wavefunction for a real, that is,
nonmodel, system can be written in terms of the complete set of ideal wavefunctions, just like any point in space can be written in terms of x, y, and z.
Using first-order perturbation theory, any real wavefunction ⌿n,real can be
written as an ideal wavefunction plus a sum of contributions of the complete
set of ideal wavefunctions ⌿(0)
m :
(0)
⌿n,real ϭ ⌿(0)
n ϩ Α am и ⌿m
(12.20)
m
where am is the coefficient multiplying each ideal ⌿(0)
m ; they are called expansion coefficients. Each real wavefunction ⌿n,real has a different, unique set of expansion coefficients that define it in terms of the ideal eigenfunctions. A summation like equation 12.20 is called a linear combination, because it combines
the ideal wavefunctions, which are assumed to be raised to the first power
(which defines a linear type of relationship).
Although the process is lengthy, it is algebraically straightforward to determine what the expansion coefficients are for the correction to the nth real
wavefunction, ⌿n,real. Recall that each ⌿n,real is approximated initially by an
th
ideal ⌿(0)
n . The m expansion coefficient am for the perturbation to the nth real
ˆЈ,
wavefunction ⌿n,real can be defined in terms of the perturbation operator H
(0)
(0)
the nth and mth ideal wavefunctions ⌿m and ⌿n , and the energies En and
Em of the ideal wavefunctions. Specifically,
ˆЈ⌿(n0) d
͐ (⌿(m0))H
am ϭ ᎏ
(0) ᎏ
)
En Ϫ E(0
m
m
n
(12.21)
The restriction m n comes from the derivation of equation 12.21. The integration in the numerator is over the complete space of the system. The requirement that this is nondegenerate perturbation theory also eliminates the
(0)
possibility that two energies En(0) and Em
might be equal due to degenerate
wavefunctions. (The extension of perturbation theory to degenerate wavefunctions will not be discussed here.)
*The unit vectors in the x, y, and z directions are labeled i, j, and k, respectively, so that
any point in 3-D space can be represented as xi ϩ yj ϩ zk.
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392
C H A P T E R 12
Atoms and Molecules
Using equation 12.21, the nth real wavefunction ⌿n,real is written as
ˆЈ⌿(n0) d (0)
͐ (⌿(m0))H
⌿n,real ϭ ⌿(0)
ᎏ
⌿m
n ϩ Α ᎏ
)
E(n0) Ϫ E(0
m
m
΄
΅
m
n
(12.22)
Note the ordering of the terms having m and n indices in the above equation;
it is important to keep them straight. ⌿n,real is still very similar to the nth ideal
wavefunction, but now it is corrected in terms of the other wavefunctions ⌿(0)
m
that define the complete set of wavefunctions for the model system. Real wavefunctions defined in this way are not normalized. They must be normalized independently, once the proper set of expansion coefficients has been determined.
(0)
The presence of the term E (0)
n Ϫ E m in the denominator of equation 12.22
is very useful. Although this is just a first correction to the wavefunction, in
principle equation 12.22 can add an infinite number of terms to the wavefunction. However, consider the denominator in the definition of the expansion coefficient. When the difference is small, the value of the fraction—and
(0)
therefore am—is relatively large. On the other hand, if the difference E(0)
n Ϫ Em
is large, then the fraction and therefore am are small. Negligibly small, sometimes. Consider the four-term linear expansion:
(0)
(0)
(0)
(0)
⌿0,real Ϸ ⌿(0)
0 ϩ 0.95⌿1 ϩ 0.33⌿2 ϩ 0.74⌿3 ϩ 0.01⌿4
The fourth term in the expansion, ⌿4,ideal, has a very small expansion coefficient. This suggests that either the integral in the numerator of a4 is very small
or that the denominator of a4 is very large (or both). Either way, little of the
approximation is usually lost if that term is simply neglected:
(0)
(0)
(0)
⌿0,real Ϸ ⌿(0)
0 ϩ 0.95⌿1 ϩ 0.33⌿2 ϩ 0.74⌿3
There is little way of knowing beforehand how large the integral in the numerator of the expression 12.21 will be. Although the ideal wavefunctions ⌿(0)
m
ˆ
and ⌿(0)
n are orthogonal, the presence of the operator H Ј may make the value
of the integral nonzero, perhaps even large. But the denominator is in terms
(0)
of only the energies of the model system, E (0)
m and E n . Since a model system
typically has known energy eigenvalues, a good (but not necessarily absolute)
rule of thumb is that if the eigenvalue energies of the wavefunctions are far
enough apart, the expansion coefficient will be small. What this implies is that
the most important corrections in the real wavefunction ⌿n,real will be wavefunctions whose energies are close to the ideal wavefunction ⌿(0)
n of the original wavefunction approximation. So although the complete set of wavefunctions may have an infinite number of ideal wavefunctions, only those that have
eigenvalues for energy that are close to the energy of the nth state will have a
noticeable impact on the wavefunction correction.
Example 12.10
Because of electronegativity differences, the p electron in a bond between two
different atoms—say in the CϵNϪ ion—does not act exactly like a particlein-a-(flat)-box, but like a particle in a box that has a slightly higher potential
ˆЈ ϭ kx
energy on one side than the other. Assume, then, a perturbation of H
for the ground state ⌿1 of a particle-in-a-box system.
a. Draw the perturbed system.
b. Assuming that the only correction to the real ground-state wavefunction is
the second particle-in-a-box wavefunction ⌿2, calculate the coefficient a2 and
determine the first-order-corrected wavefunction.
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12.6 Perturbation Theory
393
Solution
a. The system looks like this:
x
where the sloped line indicates the true bottom of the box.
b. In order to determine a2, we need to evaluate the expression
͵⌿
a
и kx и ⌿1,PIAB dx
0
a2 ϭ ᎏᎏᎏ
E1,PIAB Ϫ E2,PIAB
*
2,PIAB
where PIAB stands for particle-in-a-box. The wavefunctions and energies for
the particle-in-a-box system are known, so all we need do is substitute for the
wavefunctions and the energies.
͵Ίᎏ2aᎏ sin ᎏ2aᎏx и kx и Ίᎏ2aᎏ sin ᎏ1aᎏx dx
a
0
a2 ϭ ᎏᎏᎏᎏ
12h2
22h2
ᎏᎏ2 Ϫ ᎏᎏ2
8ma
8ma
Since all the functions in the integral are being multiplied together, they can
be rearranged (and the constants removed from the integral sign and the
denominator simplified) to yield
͵
a
2k
2x
1x
ᎏᎏ x и sin ᎏᎏ и sin ᎏᎏ dx
a 0
a
a
a2 ϭ ᎏᎏᎏᎏ
2
3h
Ϫᎏᎏ2
8ma
In order to integrate this, we need to substitute the trigonometric identity
sin ax и sin bx ϭ ᎏ12ᎏ[cos (a Ϫ b)x Ϫ cos(a ϩ b)x] and then use the integral
table in Appendix 1. We get:
΄ ͵
a
2k 1
x
3x
ᎏᎏ ᎏᎏ x cos ᎏᎏ Ϫ x cos ᎏᎏ dx
a 20
a
a
a2 ϭ ᎏᎏᎏᎏ
2
3h
Ϫᎏᎏ2
8ma
΅
x
ax
x
3x
3x
k a2
a2
ax
ᎏᎏ ᎏᎏ2 cos ᎏᎏ ϩ ᎏᎏ sin ᎏᎏ Ϫ ᎏᎏ2 cos ᎏᎏ Ϫ ᎏᎏ sin ᎏᎏ ͉a0
a
a
a
a
a
9
3
ϭ ᎏᎏᎏᎏᎏᎏᎏ
2
3h
Ϫᎏᎏ2
8ma
΄
΅
Evaluating this at the limits and simplifying, one finds that
128kma3
a2 ϭ ᎏᎏ
272h2
and so the approximate wavefunction is
128kma3
и ⌿2,PIAB
⌿1,real Ϸ ⌿1,PIAB ϩ ᎏᎏ
272h2
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394
C H A P T E R 12
Atoms and Molecules
For a cyanide species where the mass is me, k Ϸ 1 ϫ 10Ϫ7 kgиm/s2, and
a Ϸ 1.15 Å (that is, 1.15 ϫ 10Ϫ10 m), we can evaluate the expression above
and get
⌿1,real Ϸ ⌿1,PIAB ϩ 0.1516 и ⌿2,PIAB
A more complete treatment includes contributions from ⌿3, ⌿4, and so on,
but their contributions become less and less important as the difference between the ideal energies increases. Finally, recall that a2 for ⌿2,real (or any
other real ⌿) will be different from the a2 calculated above for ⌿1,real.
12.7 Variation Theory
The second major approximation theory used in quantum mechanics is called
variation theory. Variation theory is based on the fact that any test wavefunction for a system has an average energy that is equal to or greater than the true
ground-state energy of that system. Therefore, the general idea is that the lower
the energy, the better the approximated energy (and therefore, the better the
wavefunction). What one does is to suppose a trial wavefunction that has some
variable parameter in it, determine the expression for the energy of the system
(using the Schrödinger equation or the definition of average energy, ͗E͘), and
then determine what value the variable must have in order to yield the lowest
possible energy. Since the wavefunction should also provide average values for
other observables, those other values can be determined once a minimum energy is determined for that trial wavefunction.† One of the strengths of variation theory is that the trial wavefunctions can be any function, as long as the
function meets the standards of wavefunctions in general (that is, continuous,
integrable, single-valued, and so on) and satisfies any inherent requirement
of the system (such as approaching zero as x approaches Ϯؕ or the system
barriers).
One way of stating the basic idea behind variation theory is the following:
ˆ, true wavefunctions ⌿true, and
for a system having a Hamiltonian operator H
some lowest-energy eigenvalue E1, the variation theorem states that for any
normalized trial wavefunction :
͵ *Hˆ d Ն E
1
(12.23)
If is identically equal to ⌿true for the ground state, then equation 12.23 is an
equality. If is not exactly the ground-state wavefunction, then equation 12.23
is an inequality and the energy produced by the integral is always greater than
the true ground-state energy of the system. Therefore, the lower the predicted
energy, the closer it is to the true ground-state energy and the “better” an
energy eigenvalue it is. Proof of variation theory is left as an exercise at the
end of this chapter. For an unnormalized wavefunction, equation 12.23 is
written as
ˆ d
͐ *H
ᎏᎏ Ն E1
͐ * d
(12.24)
Usually, the trial wavefunctions have some set of adjustable parameters
(a, b, c, . . .). The energy is calculated as an expression in terms of those para†
However, there is no guarantee that this trial wavefunction will yield accurate values for
other observables.
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