Lectures on Modern Physics
Jiunn-Ren Roan
4 Oct. 2007
Atoms and Molecules
The Hydrogen Atom
The Schrödinger Equation for the Hydrogen Atom
Quantization of Orbital Angular Momentum
Quantum Number Notation
Electron Probability Distributions
The Zeeman Effect and Electron Spin
The Zeeman Effect
The Stern-Gerlach Experiment
Electron Spin
Selection Rules
Spin-Orbit Coupling
Atoms and Molecules
Many-Electron Atoms
The Schrödinger Equation for the Helium Atom
Independent-Electron Approximation
Central-Field Approximation
Hartree Approximation
The Periodic Table
Atomic Term Symbols
Hund’s Rules for Ground-State Terms
The Hydrogen Molecule
The Schrödinger Equation for the Hydrogen Molecule
The Valence-Bond Method
The Molecular-Orbital Method
Molecular Term Symbols
Atoms and Molecules
Appendix
Solving the Schrödinger Equation for the Hydrogen
Atom
References
The Hydrogen Atom
The Schrödinger Equation for the Hydrogen Atom
For the hydrogen atom the Schrödinger equation
has a spherically symmetric potential energy:
Hence, it is most convenient to work in spherical
coordinates
and write
Solving this equation (see Appendix A), we find
the energy is quantized:
where is the reduced mass of the proton-electron
system and
is called the principal quantum number.
The Hydrogen Atom
Quantization of Orbital Angular Momentum
It can be shown that the finiteness of the wave function on the z-axis requires
that the orbital angular momentum be quantized:
where
is called the orbital angular-momentum quantum number or orbital quantum
number.
Also, it can be shown that because of angular periodicity of the wave function,
the z-component of the orbital angular momentum must be quantized as well:
where
is called the orbital magnetic quantum number or magnetic quantum
number.
The Hydrogen Atom
The Hydrogen Atom
Quantum Number Notation
Because the quantized energy En is determined solely by the principal quantum
number n, distinct quantum states with different quantum numbers may have
the same energy. These degenerate states are often labeled with letters:
l
0
1
2
3
4
5
···
Label
s
p
d
f
g
h
···
The letters s, p, d, and f are the first letters of “sharp”, “principal”, “diffuse”,
and “fundamental”, respectively, used in the early days of spectroscopy.
Another widely used notation is
n
1
2
3
4
···
Shell
K
L
M
N
···
The Hydrogen Atom
Electron Probability Distributions
The radial probability distribution function P(r) is given by integrating over
the angular variables:
The electron is most likely to be found at the maximum of P(r), which for the
states having the largest possible l for each n (such as 1s, 2p, 3d, and 4f states)
occurs at n2a0, where
is the Bohr radius, the radius of the ground state in the Bohr model. In the
atomic unit system, it is used as the length unit, called a bohr.
For states without spherical symmetry, to reveal the angular dependence the
three-dimensional probability distribution function
has to be used.
The Hydrogen Atom
The Hydrogen Atom
The Zeeman Effect and Electron Spin
The Zeeman Effect (Zeeman, 1896)
The quantization of angular momentum is confirmed experimentally by the
splitting of degenerate states and the associated spectrum lines when the atoms
are placed in a magnetic field—the Zeeman effect.
m
A charged particle orbiting an oppositely charged center generates a magnetic
dipole moment m that is proportional to the angular momentum L:
q
The ratio g is called the gyromagnetic ratio. In the Bohr model,
(m: magnetic moment; m: reduced mass). In a magnetic field B directed along
the +z-axis, the potential energy associated with m is given by
Thus, in a magnetic field the 2l+1degenerate states associated with a particular
subshell are no longer degenerate but split into distinct energy levels according
to
where mB is the Bohr magneton:
The Zeeman Effect and Electron Spin
The Zeeman Effect and Electron Spin
The Zeeman Effect and Electron Spin
From D. A. McQuarrie, Quantum Chemistry, Oxford University Press, Oxford (1983).
The Zeeman Effect and Electron Spin
Only the so-called normal Zeeman effect can be explained by quantization of
orbital angular momentum. In the anomalous Zeeman effect, the splitting does
not follow the prediction.
Heisenberg and Landé independently found that the anomalous Zeeman effect
can be explained by introducing half-integer quantum numbers. While Heisenberg
was strongly discouraged (for good reasons) by his mentor, Sommerfeld, and a
close friend, Pauli, and did not publish it, Landé published it and got his name
forever associated with the Zeeman effect.
The Zeeman Effect and Electron Spin
The Stern-Gerlach Experiment (Stern and Gerlach, 1922)
Passing a beam of silver atoms through an inhomogeneous magnetic field, Stern
and Gerlach expected to see the 2l+1 degenerate states split into an odd number,
2l+1, of components. However, they were surprised to see that the beam split
into only two components.
The Zeeman Effect and Electron Spin
Electron Spin (Uhlenbeck and Goudsmit, 1925)
Pauli postulated in 1925 that an electron can exist in two distinct states and
introduced in a rather ad hoc manner a fourth quantum number to describe the
two states. Although with this he could explain the Stern-Gerlach experiment,
no interpretation was given to the fourth quantum number.
Before long, two Dutch graduate students, Uhlenbeck and Goudsmit, proposed
that the electron might behave like a spinning sphere of charge instead of a
point particle and the spinning motion would give an additional spin angular
momentum S and spin magnetic moment ms.
According to this proposal, the spin angular momentum, like the quantized
orbital angular momentum, is also quantized:
where the spin quantum number s = ½, and what the Stern-Gerlach
experiment measures is the z-component of the spin angular momentum:
where the spin magnetic quantum number ms, the fourth quantum number
introduced by Pauli, has two values +½ and -½, corresponding to the two
orientations, up and down, respectively, of the spin angular momentum.
The Zeeman Effect and Electron Spin
Like the relation between orbital angular momentum and magnetic moment, the
spin magnetic moment is also proportional to the spin angular momentum:
where the g-factor for electron gs is needed to obtain agreement with experimental
observations. The g-factor for electron does not have classical analog. In 1928
Dirac developed a relativistic generalization of Schrödinger equation for electrons,
which gave gs = 2, exactly. The observed value, however, differs from Dirac’s
prediction by a very small amount: gs = 2.00231930436170. A theory, quantum
electrodynamics, developed from early 1930s to 1950s is able to give a value that
agrees with the experimental value to 10-13.
Therefore, adding the contributions from the orbital motion and the intrinsic spin,
which gives in a magnetic field
In the Stern-Gerlach experiment, the silver atom was in an S-state (l = 0), so m
must be 0, leading to two components corresponding to ms = ±½. Thus the
Stern-Gerlach experiment directly confirmed the existence of electron spin.
The Zeeman Effect and Electron Spin
From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).
The Zeeman Effect and Electron Spin
Selection Rules
According to the interaction energy,
inclusion of electron spin can further split the energy levels. For example, the
2p state splits into five levels (taking gs = 2) instead of only three in the absence
of electron spin; and the 1s state now splits into two levels. At first sight, it
appears that the spectrum corresponding to 2p → 1s would comprise all the
possible transitions. However, this is not the case, because the emitted photon
carries one unit (ħ) of angular momentum and therefore conservation of angular
momentum requires the selection rules
be held. The six allowed transitions for 2p → 1s (Dl = -1) are
identical to those obtained without spin (normal Zeeman effect).
The Zeeman Effect and Electron Spin
Spin-Orbit Coupling
When the magnetic field is not very strong, however, the allowed transitions
for 2p → 1s exhibit additional splitting, resulting in anomalous Zeeman effect.
This is because the magnetic moments associated with the orbital and spin
angular momenta are coupled – the magnetic field created by the orbital motion
interacts with the spin magnetic moment. If the external magnetic field is not
strong enough to render the magnetic field created by the orbital motion
completely negligible, then the interaction between L and S must be taken into
account.
The strength of this interaction is proportional to L·S. In a strong magnetic
field, the coupling shifts all the levels with mms > 0 upward slightly and those
with mms < 0 downward slightly, and also removes the remaining degeneracy
in the 2p state. This splits each of the outer lines into two closely spaced lines
(anomalous Zeeman effect), which agree with experimental observations.
Many-Electron Atoms
The Schrödinger Equation for the Helium Atom
The Schrödinger equation for the helium atom is
where the kinetic energy Ki depends only on the position of the
i-th electron ri and the potential energy contains three terms,
two electron-nucleus interactions and one electron-electron
interaction
-e
-e
r2
r1
Ze
(Z = 2 for helium). The equation can be written as
The wave function, of course, depends on both positions:
Many-Electron Atoms
Independent-Electron Approximation
The helium-atom Schrödinger equation has no known exact solution. Many
approximations have been invented to tackle this difficult three-body problem.
The simplest approximation is to neglect the electron-electron interaction:
so that the two electrons do not interact and behave independently, leading to
where the one-electron wave function , called an atomic orbital, satisfies
the Schrödinger equation for a hydrogen-like atom:
Substituting the wave function into the Schrödinger equation gives
Thus, the total energy is the sum of the two one-electron energies:
where
Many-Electron Atoms
In this approximation the ground state of the helium atom is characterized by
the wave function
and energy
which is larger than the true ground-state energy, -79.0 eV. Apparently, this is
not a good approximation and there is much room for improvement.
Applying this approximation to multi-electron atoms other than the helium atom
seems rather straightforward. For example, the ground state of the three-electron
lithium atom might be
. However, this
is not true, because the Pauli exclusion principle says that no two electrons in
an atom can have the same set of quantum numbers (n, l, m, ms).
Therefore, the spin wave function a and b, corresponding to ms = ½ and -½,
respectively, must be included, forming the so-called spin orbitals:
.
Then the ground-state wave function for the helium atom should be
where 1sa and 1sb are shorthand notations; the ground state of lithium has only
two electrons in the 1s orbital and the third electron must be in an n = 2 state.
Many-Electron Atoms
The Pauli exclusion principle is a consequence of a fundamental theorem called
the spin-statistics theorem: the wave functions of a system of indistinguishable
half-integer-spin particles (fermions) are antisymmetric under interchange of any
pair of particles, whereas the wave functions of a system of indistinguishable
integer-spin particles (bosons) are symmetric under interchange of any pair of
particles. Accordingly, the ground-state wave function of the helium atom must
be the “anti-symmetrized form” of
namely,
It can be readily seen that
as required by the spin-statistics theorem.
Many-Electron Atoms
Central-Field Approximation
A less trivial and more useful approximation is to assume that each of the
atomic electrons moves independently of the others in a spherically symmetric
potential energy Vc(r) that is produced by the nucleus and all the other electrons.
For the helium atom,
Because the overall effect of the electrons is to screen the nuclear Coulomb
field, the effect becomes more appreciable at greater distances:
Apparently, Vc(r) must be non-coulombic, in which the degeneracy between
states of the same n and different l is removed. This is because the electrons
with smaller l penetrate closer to the nucleus, seeing a more negative Vc(r). So,
for a given n, the states of lowest l have the lowest energy. On the other hand,
since Vc(r) is spherically symmetric, the degeneracy in m is not affected.
In general, the same quantum numbers (n, l, m, ms) can be used to label states,
but the energy now depends on both n and l. The restrictions on values of the
quantum numbers are the same as before.
Many-Electron Atoms
Hartree Approximation (Hartree, 1928)
A method for obtaining a central field is given by Hartree:
where
is the charge density associated with the j-th electron.
To solve the approximate Schrödinger equation,
with
one must know Vc(ri), but Vc(ri) in turn is determined by the wave function to
be solved. Therefore, this equation can only be solved self-consistently.
Fock correctly included spin wave functions into the Hartree approximation
and obtained a better approximation called the Hartree-Fock approximation.
Calculations using the Hartree-Fock approximation gives results that well
agree with experimental observations.
Many-Electron Atoms
Ar
Electron-diffraction data
Hartree-Fock calculation
From I. N. Levine, Quantum Chemistry, 4th ed., Prentice Hall, Englewood Cliffs (1991).
Many-Electron Atoms
The Periodic Table
In non-coulombic fields,
for a given n, the states of
the lowest l have the lowest
energy, but the degeneracy
in m is kept intact.
In some cases the intrashell
splitting (same n, different l)
is larger than the intershell
splitting (different n), so
that an “inversion” of level
order occurs. Thus, 4s < 3d,
5s < 4d, and 6s < 5d < 4f.
The level ordering in this
figure is common for neutral
atoms.
From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).
Many-Electron Atoms
Many-Electron Atoms
Atomic Term Symbols
The ground-state configurations as given in the table do not completely specify
the state of an atom with partly filled shells (also called open shells), because
electrons with given n and l can be distributed among the different possible m
and ms values. To completely specify the state, it is necessary to have additional
information on m and ms, which is given by the so-called term symbols (“terms”
in spectroscopic language means energy levels).
For a group of k electrons, the total angular momenta and their z-components
are given by
where L, S, M, and MS are the corresponding quantum numbers. Addition of
angular momenta in quantum mechanics is a complicated business. Fortunately,
for the addition of two angular momenta L1 and L2, the rule is simple:
This can be understood as lining up L1 and L2 parallel to obtain the greatest value
of L and in the opposite direction to obtain the least value.
Many-Electron Atoms
The possible values of the total orbital angular momentum quantum number L
for the k-electron system, therefore, can be obtained by repeatedly applying the
addition rule for two angular momenta. The result is
If all the quantum numbers li are equal, Lmin is zero; if one of the li is larger than
others, Lmin is given by orienting the other angular momenta to oppose it.
The possible values of the total spin angular momentum quantum number S can
be obtained similarly:
If k is even, Smin = 0; if k is odd, Smin = ½.
In addition to L and S, the total angular momentum
is used to further distinguish states that have the same L and S values (there are
totally (2L+1)(2S+1) such states). The possible total angular momentum quantum
numbers are
Many-Electron Atoms
The term symbol is written as
and capital letters are used for L:
l
0
1
2
3
4
5
6
7
8
···
Label
S
P
D
F
G
H
I
K
L
···
and the electron spin superscirpt 2S+1 is read as follows:
2S+1
1
2
3
4
5
6
···
singlet
doublet
triplet
quartet
quintet
sextet
···
corresponding to the fact that for L > S, the number of possible J levels is equal
to 2S+1 (called the multiplicity of the term). Thus, the term symbol for an atom
with L = 3, S = 3/2, J = 5/2 is 4F5/2 and is read “quartet F five halves”.
For closed shells and subshells, all orbitals with the same n and l are doubly
occupied, so L = 0 and S = 0, giving 1S0. Thus, the contributions from completely
filled shells or subshells are always 1S0 and can be ignored.
For open shells, consider a carbon atom in the excited state 1s22s22p3p as an
example. The possible values are L = 2, 1, 0 and S = 1, 0, so the possible terms
are 3D3,2,1 (L = 2, S = 1), 1D2 (L = 2, S = 0), 3P2,1,0 (L = 1, S = 1), 1P1 (L = 1, S = 0),
3S (L = 0, S = 1), and 1S (L = 0, S = 0).
1
0
Many-Electron Atoms
For the ground-state configuration 1s22s22p2, l1 = 1, l2 = 1, s1 = ½, s2 = ½, Pauli
exclusion principle limits the possible m and ms values, so that there are totally
15 possible combinations (remember that electrons are indistinguishable):
From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).
Since the largest value of M is 2, and it occurs only with MS = 0, there must be
a state with L = 2 and S = 0, i.e., a 1D2 , which corresponds to (2L+1)(2S+1) = 5
combinations of M and MS.
The remaining combinations has Mmax = 1, so L = 1 and M = 0, ±1. Each of these
M values occurs with a value of MS = 0, ±1, so S = 1. Thus, the term is 3P2,1,0 and
it corresponds to 9 combinations of M and MS.
The remain only one combination: M = 0 and MS = 0, corresponding to L = 0 and
S = 0, i.e., 1S0.
Many-Electron Atoms
For electrons in different subshells, called non-equivalent electrons, there is no
restriction from the Pauli exclusion principle. Electrons in the same subshell
(equivalent electron), on the other hand, must face the restriction imposed by
the exclusion principle and, therefore, some terms derived for nonequivalent
electrons are not possible.
From I. N. Levine, Quantum Chemistry, 4th ed., Prentice Hall, Englewood Cliffs (1991).
Many-Electron Atoms
Hund’s Rules for Ground-State Terms
Knowing the terms, we can find which term characterizes the ground state by
three empirical rules, called Hund’s rules, due to Hund:
(a) The stability decreases with decreasing S, so the ground state has
maximum spin multiplicity.
(b) For a given value of S (or spin multiplicity), the state with maximum
L is most stable.
(c) For given S and L, the minimum J value is most stable if there is a
single open subshell that is less than half-filled and the maximum J
is most stable if the subshell is more than half-filled.
Thus,
From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).
Many-Electron Atoms
From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).
Many-Electron Atoms
From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).
The Hydrogen Molecule
The Schrödinger Equation for the Hydrogen Molecule
e1
r12
r1A
e2
r2B
r2A r1B
HA
R
HB
The hydrogen molecule has two protons and two electrons, so the
total energy contains the kinetic energy of relative motion of the
nuclei with reduced mass mp/2 (mp = proton mass), kinetic energy of
the two electrons, and Coulombic energy of the six particle-particle
pairs:
Because mp is much greater than the electron mass, in many cases the massive
nuclei can be assumed to be stationary and the associated kinetic energy be
neglected. This is called the Born-Oppenheimer approximation. With this
approximation, the Schrödinger equation to be solved becomes
For large R, all potential energies but
and
are small, so the
r12
e1
equation becomes that of two non-interacting hydrogen atoms:
r1A
whose ground-state solution can be easily found to be
e2
r2B
r2A r1B
HA
R
HB
The Hydrogen Molecule
The Valence-Bond Method (Heitler and London, 1927)
A natural starting point for finding the solution of the Schrödinger equation is
the large-R solution
. Since the electrons are indistinguishable,
there is no way to find out which is associated with which nucleus. Therefore,
two equally valid solutions for large-R are
e1
r12
r1A
e2
e2
r2B
r2A
r2A r1B
HA
R
r12
e1
r1B
r1A r2B
HB
HA
R
HB
They are the ground-state solution for each of the two widely-separated,
non-interacting hydrogen atoms:
and therefore are the large-R solution to the Schrödinger equation for the
hydrogen molecule:
In the atomic unit system the magnitude of 2E1s (27.2 eV) is used as the energy
unit, called hartree.
The Hydrogen Molecule
The valence-bond (or Heitler-London) method uses linear combination of the
large-R solution
as a trial function and requires it satisfy the Schrödinger equation
This gives
which can be expanded:
where
are called matrix elements. The coefficients c1 and c2 that minimize the energy
E will give the best approximation to the true ground-state solution.
The Hydrogen Molecule
The coefficients c1 and c2 that minimize the energy can be found from
which gives
A nontrivial solution exists if and only if
The matrix elements Sij are computed as follows (remember the 1s orbital is real):
The Hydrogen Molecule
where
r12
e1
r1A
e2
r2B
r
r
2A
1B
integrates over all the space where the two 1s orbitals,
HA
HB
R
one centered on nucleus A and the other on nucleus B,
are simultaneously nonzero. In other words, S computes how much 1sA and 1sB
overlap and thus is called the overlap integral.
Computation of the matrix elements Hij utilizes the fact that
and
large-R solution of the Schrödinger equation with energy 2E1s:
and
are the
The Hydrogen Molecule
The integral Q defined in H11 is
e1
r12
r1A
e2
r2B
r1A
R
e2
e1
r2B
HB
HA
R
r12
r1A
r2A r1B
r2A r1B
HA
r12
e1
e2
e1
r2B
r1A
r2A r1B
HB
HA
R
r12
e2
r2B
r2A r1B
HB
HA
R
It represents the classical Coulombic interaction of the charge clouds [1sA(1)]2
with nucleus B, of the charge cloud [1sB(2)]2 with nucleus A, of the charge cloud
[1sA(1)]2 and [1sB(2)]2, and of the nuclei with one another, so Q is called the
Coulomb integral.
The integral J defined in H12 is
Since 1sA(1)1sB(1) is not an electron density in the ordinary sense, J cannot be
interpreted as a classical electrostatic interaction of two charge clouds.
HB
The Hydrogen Molecule
The strictly quantum-mechanical quantity J can be written as
This indicates that J arises as a result of exchanging electrons between the two
nuclei, so J is called the exchange integral.
With these matrix elements the condition for nontrivial coefficients becomes
The minimized energy is
or, relative to the energy of two isolated hydrogen atoms,
Since DE+ has a minimum at a finite R, the two nuclei are in a bound state,
forming a stable diatomic molecule. The corresponding bonding and antibonding
wave functions are
The Hydrogen Molecule
From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).
From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).
The Hydrogen Molecule
The electron density distributions are given by
From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).
The Hydrogen Molecule
The Molecular-Orbital Method
In the atomic-orbital approach to the electronic structure of many-electron
atoms, one-electron wave functions satisfying the Schrödinger equation with
an approximate potential energy such as
are used to build up, under the constraint imposed by the exclusion principle,
a many-electron atom’s configurations
corresponding to the ground state, the first excited state, and so on.
rA
HA
For the electronic structure of many-electron molecules, the molecular-orbital
(MO) method developed in the early 1930s by Hund, Mulliken, and others is
a generalization of the atomic-orbital method. To construct the electron
configurations of the hydrogen molecule, the MO theory first considers the
e
corresponding one-electron molecule: H2+, the simplest molecule.
rB
The Schrödinger equation for H2+ is
R
HB
To find the ground-state configuration of H2, the large-R ground-state wave
functions for H2+, 1sA (the 1s orbital centered on nucleus A) and 1sB (the 1s
orbit centered on nucleus B), are an appropriate starting point.
The Hydrogen Molecule
The trial function
is the simplest example of the method of linear combination of atomic orbitals
(LCAO). Like the valence-bond method, MO method minimizes the energy in
and obtains
where
The corresponding bonding (s or sg, g: gerade is the German word for even)
and antibonding (s* or su, u: ungerade = odd in German) orbitals are
The energy curves are qualitatively similar to the valence-bond results, so are
the electron density distributions.
The Hydrogen Molecule
From D. A. McQuarrie, Quantum Chemistry, Oxford University Press, Oxford (1983).
The Hydrogen Molecule
s1s
s1s*
From D. A. McQuarrie, Quantum Chemistry, Oxford University Press, Oxford (1983).
The Hydrogen Molecule
Finally, placing two electrons of opposite spins into the bonding orbital gives the
ground state of the hydrogen molecule:
This method of constructing molecular wave functions is known as the LCAO-MO
method.
A
From A. L. Companion, Chemical Bonding, 2nd ed. McGraw-Hill, New York (1979).
B
The Hydrogen Molecule
The ground state obtained by valence-bond (VB) method is
Therefore, up to normalization constants, the relation between them is
where
e1
r12
r1A
e2
e1
r2B
r1A
R
e2
r2B
r2A r1B
r2A r1B
HA
r12
HB
HA
R
HB
This suggests that the MO theory overemphasizes ionic feature, whereas VB
theory ignores it.
The Hydrogen Molecule
Because the s orbitals constructed so far are made out of 1s orbitals, they are
denoted by s1s (or sg1s) and s*1s (or su1s). Additional MOs can be constructed
from other kinds of AOs in a similar way. So 2sA ± 2sB gives s2s and s*2s.
Because the 2s AO has a higher energy than the 1s AO, the energy ordering is
s1s < s*1s < s2s < s*2s.
Constructed from 2pz (or 2p0), the MO 2pz,A ± 2pz,B are symmetric about the
inter-nuclear axis and so are s orbitals. They are designated by s2pz (or sg2pz)
and s*2pz (or su2pz).
From I. N. Levine, Quantum Chemistry, 4th ed., Prentice Hall, Englewood Cliffs (1991).
The Hydrogen Molecule
The MO constructed from 2px,y (or 2p±1) has a nodal plane in both bonding and
antibonding orbitals. AOs with one nodal plane are called p orbitals, so MOs
with one nodal plane are called p (the Greek counterpart of p) orbitals. Unlike
the s orbitals, here the bonding orbital changes sign upon inversion through the
origin (i.e., it is an odd function), whereas the antibonding orbital remains
unchanged upon inversion, so for p orbitals the bonding orbital is ungerade and
the antibonding orbital is gerade. The bonding and antibonding orbitals are
denoted by p2px,y (or pu2px,y) and p*2px,y (or pg2px,y).
From I. N. Levine, Quantum Chemistry, 4th ed., Prentice Hall, Englewood Cliffs (1991).
The Hydrogen Molecule
For most homonuclear diatomic molecules built of atoms of period 2 elements,
an approximate ordering of the energy levels is, according to experiment,
s1s < s*1s < s2s < s*2s < p2px = p2py < s2pz < p*2px = p*2py < s*2pz
so
But, since the energy difference between the s2pz and p2px,y orbitals are very
small and varies with the atomic number of the nuclei and the inter-nuclear
separation, the other possible scheme is
s1s < s*1s < s2s < s*2s < s2pz < p2px = p2py < p*2px = p*2py < s*2pz
Fortunately, many of the predictions of the two schemes are the same.
The Hydrogen Molecule
Molecular Term Symbols
The molecular term symbol is written
where 2S+1 as usual is the multiplicity and
is the magnitude of the axial component (along the molecular axis) of the total
orbital angular momentum. The following Greek letters, corresponding to the
English letters (s, p, d, ...) for atomic orbitals, are used for L:
L
0
1
2
3
4
···
Label
S
P
D
F
G
···
As an example, consider H2: (sg1s)2. Both electrons have m = 0, so L = 0.
Pauli exclusion principle requires that their spins must be opposite, giving S = 0.
Thus, the term symbol for H2 is 1S (a singlet sigma state). It is easy to see that
a closed subshell (each set of degenerate MO constitutes a molecular subshell)
configuration has both S = 0 and L = 0 and gives rise to only a 1S term.
For a less trivial example, consider B2: (sg1s)2(su1s)2(sg2s)2(su2s)2(pu2p)2. The
only non-trivial contribution is from the two electrons in the open subshell pu2p.
The Hydrogen Molecule
The two combinations with L = 2 have MS = 0, so S = 0 and the term is 1D (singlet
delta).
The other four combinations all have L = 0 and MS = 0, ±1, so S = 1 and the term
is 3S.
Finally, the only remaining combination has L = 0 and MS = 0, so S = 0 and the
term is 1S.
Hund’s rules apply to molecular electronic states as well, so the state with the
largest spin multiplicity will be the ground state. Thus, the ground state of B2
is a 3S state.
The Hydrogen Molecule
Superscripts + and –, and subscripts g and u can be used to indicate additional
symmetric properties of the term. If the wave function changes sign upon
reflection in a plane through the nuclei, a superscript – is supplemented; otherwise,
+ is used. Because for states with L ≠ 0, such a reflection always changes the
sign of the axial component of the total orbital angular momentum, superscripts
± are used only for S states.
Subscripts g and u are added to show the parity (symmetry under inversion through
the origin) of the term. Terms arising from an electron configuration that has an
odd number of electrons in MOs of odd parity are odd (u); all other terms are even
(g).
From I. N. Levine, Quantum Chemistry, 4th ed., Prentice Hall, Englewood Cliffs (1991).
Appendix
Solving the Schrödinger Equation for the Hydrogen Atom
To solve the equation
with spherically symmetric potential energy,
in spherical coordinates
, first write
Applying a method called separation of variables, we assume
and obtain
Appendix
The Schrödinger equation then becomes
which can be written
Collecting terms of different dependence, we obtain
The only way for terms depending on different independent variables to be equal
all the time is each term is a constant:
Appendix
The angular part can be further separated:
so
(Nf is a normalization factor) and
Note that periodicity in the azimuthal angle gives
Let
then the equation for the polar angle becomes
Appendix
Power series are often used to solve differential equations. It will be very helpful
to make the resultant recursion relation as simple as possible, i.e., involving as
few terms as possible. To achieve this goal, we substitute
into the equation. With a little algebra, it is easy to find that if
then
This form will make the power series
couple only two instead of three or more terms: the coefficient of the
term is
It can be shown that only if the series terminates at certain power can the solution
be finite at w = ±1. This requirement gives
Therefore, we conclude that
in which l is an integer.
Appendix
The differential equation now becomes
where
Because differentiating the factor 1-w2 more than twice results in zero and
if
, then the differential equation becomes
This can be simplified as
which is known as Legendre’s differential equation. Its solution is the Legendre
polynomials Pl(w).
Putting things together, we obtain the solution (up to a normalization factor Nq)
where
is called an associated Legendre function.
Appendix
The radial part now takes the form
For large r, the bound-state (E < 0) solution satisfies
so
On the other hand, for small r, the differential equation requires that
Therefore, we can try
and substitute it into the differential equation to obtain
Appendix
Again, the power series method leads us to conclude that for the series to terminate
at somewhere and result in a finite F(r) for large r, we must have
where n is an integer. In other words, the energy is quantized:
where
as before.
The differential equation to be solved thus becomes
which can be written
Appendix
Solutions to the equation
are called Laguerre polynomials Ln+l:
so solutions to the equation
are simply
where
is called the associated Laguerre polynomial.
Finally, returning to the original unknown function R(r), we get
where Nr is a normalization factor.
Appendix
The wave function therefore can be written
It can be shown that the normalization condition
gives
References
1. H. D. Young and R. A. Freedman, Sears and Zemansky’s University Physics
(Pearson, 2008) 12th ed.
2. M. Karplus and R. N. Porter, Atoms and Molecules (Benjamin, 1970).
3. D. A. McQuarrie, Quantum Chemistry (Oxford University Press, 1983).
4. L. I. Schiff, Quantum Mechanics (McGraw-Hill, 1971).
5. I. N. Levine, Quantum Chemistry (Prentice Hall, 1991) 4th ed.