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Lectures on Modern Physics

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- 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

- 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

- Appendix
- Solving the Schrödinger Equation for the Hydrogen
- Atom

- References

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.

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.

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:

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

Electron Probability Distributions

The radial probability distribution functionP(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.

q

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.

A charged particle orbiting an oppositely charged center generates a magnetic

dipole moment m that is proportional to the angular momentum L:

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:

From D. A. McQuarrie, Quantum Chemistry, Oxford University Press, Oxford (1983).

The Zeeman Effect and Electron Spin

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

momentumS and spin magnetic momentms.

According to this proposal, the spin angular momentum, like the quantized

orbital angular momentum, is also quantized:

where the spin quantum numbers = 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 electrongs 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.

From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).

The Zeeman Effect and Electron Spin

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.

-e

r2

r1

Ze

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

(Z = 2 for helium). The equation can be written as

The wave function, of course, depends on both positions:

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

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.

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.

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.

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.

Electron-diffraction data

Hartree-Fock calculation

From I. N. Levine, Quantum Chemistry, 4th ed., Prentice Hall, Englewood Cliffs (1991).

Many-Electron Atoms

From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).

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.

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.

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

The term symbol is written as

and capital letters are used for L:

and the electron spin superscirpt 2S+1 is read as follows:

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 1S0and 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),

3S1 (L = 0, S = 1), and 1S0 (L = 0, S = 0).

From M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).

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):

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.

From I. N. Levine, Quantum Chemistry, 4th ed., Prentice Hall, Englewood Cliffs (1991).

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 M. Karplus and R. N. Porter, Atoms and Molecules, Benjamin/Cummings, Menlo Park (1970).

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:
- The stability decreases with decreasing S, so the ground state has
- maximum spin multiplicity.
- For a given value of S (or spin multiplicity), the state with maximum
- L is most stable.
- 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

e2

e2

r12

r12

e1

e1

r1A

r1A

r2B

r2B

r2A

r2A

r1B

r1B

HA

HA

HB

HB

R

R

The Hydrogen Molecule

The Schrödinger Equation for the Hydrogen Molecule

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

equation becomes that of two non-interacting hydrogen atoms:

whose ground-state solution can be easily found to be

e2

e1

r12

r12

e1

e2

r1A

r2A

r2B

r1B

r1A

r2A

r2B

r1B

HA

HA

HB

HB

R

R

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

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 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 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):

e2

r12

e1

r1A

r2B

r2A

r1B

HA

HB

R

The Hydrogen Molecule

where

integrates over all the space where the two 1s orbitals,

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 are the

large-R solution of the Schrödinger equation with energy 2E1s:

and

e2

e2

e2

e2

r12

r12

r12

r12

e1

e1

e1

e1

r1A

r1A

r1A

r1A

r2B

r2B

r2B

r2B

r2A

r2A

r2A

r2A

r1B

r1B

r1B

r1B

HA

HA

HA

HA

HB

HB

HB

HB

R

R

R

R

The Hydrogen Molecule

The integral Q defined in H11 is

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.

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

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

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

rB

rA

HA

HB

R

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.

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

corresponding one-electron molecule: H2+, the simplest molecule.

The Schrödinger equation for H2+ is

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 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.

From D. A. McQuarrie, Quantum Chemistry, Oxford University Press, Oxford (1983).

The Hydrogen Molecule

From D. A. McQuarrie, Quantum Chemistry, Oxford University Press, Oxford (1983).

The Hydrogen Molecule

s1s

s1s*

B

From A. L. Companion, Chemical Bonding, 2nd ed. McGraw-Hill, New York (1979).

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.

e2

e2

r12

r12

e1

e1

r1A

r1A

r2B

r2B

r2A

r2A

r1B

r1B

HA

HA

HB

HB

R

R

The Hydrogen Molecule

The ground state obtained by valence-bond (VB) method is

Therefore, up to normalization constants, the relation between them is

where

This suggests that the MO theory overemphasizes ionic feature, whereas VB

theory ignores it.

From I. N. Levine, Quantum Chemistry, 4th ed., Prentice Hall, Englewood Cliffs (1991).

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).

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.

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:

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 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.

From I. N. Levine, Quantum Chemistry, 4th ed., Prentice Hall, Englewood Cliffs (1991).

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).

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

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:

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

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.

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

polynomialsPl(w).

Putting things together, we obtain the solution (up to a normalization factor Nq)

where is called an associated Legendre function.

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

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

Solutions to the equation

are called Laguerre polynomialsLn+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.

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.

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