A Brief Review of Elementary Quantum Chemistry

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Contents

1 The Motivation for Quantum Mechanics                                                               4
  1.1   The Ultraviolet Catastrophe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       4
  1.2   The Photoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      5
  1.3   Quantization of Electronic Angular Momentum . . . . . . . . . . . . . . . . . . .             6
  1.4   Wave-Particle Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       6

2 The Schr
          odinger Equation                                                                            8
  2.1   The Time-Independent Schr
                                 odinger Equation          . . . . . . . . . . . . . . . . . . . .    8
  2.2   The Time-Dependent Schr
                               odinger Equation . . . . . . . . . . . . . . . . . . . . .            10

3 Mathematical Background                                                                            12
  3.1   Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    12
        3.1.1   Operators and Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . .          12
        3.1.2   Basic Properties of Operators . . . . . . . . . . . . . . . . . . . . . . . . .      13
        3.1.3   Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     14
        3.1.4   Eigenfunctions and Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . .       15
        3.1.5   Hermitian Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      16
        3.1.6   Unitary Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      18
  3.2   Commutators in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . .           18
  3.3   Linear Vector Spaces in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . .          20

4 Postulates of Quantum Mechanics                                                                    26

5 Some Analytically Soluble Problems                                                                 29
  5.1   The Particle in a Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    29



                                                  2
  5.2   The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       29
  5.3   The Rigid Rotor     . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   30
  5.4   The Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .         31

6 Approximate Methods                                                                                 33
  6.1   Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       33
  6.2   The Variational Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        35

7 Molecular Quantum Mechanics                                                                         39
  7.1   The Molecular Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .         39
  7.2   The Born-Oppenheimer Approximation . . . . . . . . . . . . . . . . . . . . . . . .            40
  7.3   Separation of the Nuclear Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . .         43

8 Solving the Electronic Eigenvalue Problem                                                           45
  8.1   The Nature of Many-Electron Wavefunctions . . . . . . . . . . . . . . . . . . . . .           45
  8.2   Matrix Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      48




                                                  3
1     The Motivation for Quantum Mechanics

Physicists at the end of the nineteenth century believed that most of the funda-
mental physical laws had been worked out. They expected only minor refinements
to get "an extra decimal place" of accuracy. As it turns out, the field of physics
was transformed profoundly in the early twentieth century by Einstein's discovery
of relativity and by the development of quantum mechanics. While relativity has
had fairly little impact on chemistry, all of theoretical chemistry is founded upon
quantum mechanics.
   The development of quantum mechanics was initially motivated by two ob-
servations which demonstrated the inadeqacy of classical physics. These are the
"ultraviolet catastrophe" and the photoelectric effect.


1.1   The Ultraviolet Catastrophe

A blackbody is an idealized object which absorbs and emits all frequencies. Clas-
sical physics can be used to derive an equation which describes the intensity of
blackbody radiation as a function of frequency for a fixed temperature--the result
is known as the Rayleigh-Jeans law. Although the Rayleigh-Jeans law works for
low frequencies, it diverges as  2 ; this divergence for high frequencies is called the
ultraviolet catastrophe.
   Max Planck explained the blackbody radiation in 1900 by assuming that the
energies of the oscillations of electrons which gave rise to the radiation must be
proportional to integral multiples of the frequency, i.e.,

                                      E = nh                                       (1)

Using statistical mechanics, Planck derived an equation similar to the Rayleigh-
Jeans equation, but with the adjustable parameter h. Planck found that for h =
6.626  10-34 J s, the experimental data could be reproduced. Nevertheless, Planck
could not offer a good justification for his assumption of energy quantization.

                                           4
Physicicsts did not take this energy quantization idea seriously until Einstein
invoked a similar assumption to explain the photoelectric effect.


1.2   The Photoelectric Effect

In 1886 and 1887, Heinrich Hertz discovered that ultraviolet light can cause elec-
trons to be ejected from a metal surface. According to the classical wave theory
of light, the intensity of the light determines the amplitude of the wave, and so a
greater light intensity should cause the electrons on the metal to oscillate more vi-
olently and to be ejected with a greater kinetic energy. In contrast, the experiment
showed that the kinetic energy of the ejected electrons depends on the frequency
of the light. The light intensity affects only the number of ejected electrons and
not their kinetic energies.
    Einstein tackled the problem of the photoelectric effect in 1905. Instead of
assuming that the electronic oscillators had energies given by Planck's formula (1),
Einstein assumed that the radiation itself consisted of packets of energy E = h ,
which are now called photons. Einstein successfully explained the photoelectric
effect using this assumption, and he calculated a value of h close to that obtained
by Planck.
   Two years later, Einstein showed that not only is light quantized, but so are
atomic vibrations. Classical physics predicts that the molar heat capacity at
constant volume (Cv ) of a crystal is 3R, where R is the molar gas constant. This
works well for high temperatures, but for low temperatures Cv actually falls to
zero. Einstein was able to explain this result by assuming that the oscillations
of atoms about their equilibrium positions are quantized according to E = nh ,
Planck's quantization condition for electronic oscillators. This demonstrated that
the energy quantization concept was important even for a system of atoms in a
crystal, which should be well-modeled by a system of masses and springs (i.e., by
classical mechanics).



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