1AMQ, Part V Molecules

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1AMQ, Part V Molecules

• The Molecular Hydrogen Ion
• Hydrogen Molecule and Covalent
• Bonding

• K.Krane, Modern Physics, Chapter 9
• Eisberg and Resnick, Quantum Physics,
• Chapter 12.

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Molecular Bonding
For a molecule, we need at least two centres of
positive charge ie, two atomic nuclei. The Time
Independent Schrodinger Equation (TISE) must be
solved for a two-centred potential, with the
electrons occupying the allowed levels.
The H2 Molecular Ion this comprises of a single
electron and two protons. It turns out that a
particular value for the separation of the two
nuclei is favoured (ie. corresponds to a minimum
energy) compared to the unbound (free infinite
separation) system.
The electron is in effect shared between the
two centres of positive charge.
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(a) Consider when the protons are very far apart,
then if y1 and y2 are the two separate hydrogen
w.functions, and the electron is equally likely
to occupy either y1 or y2, then the total
w.function is
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(b) For smaller separations the y1 and y2
wavefunctions overlap and then, y1y2 and y1-y2
have different probability distributions.
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Since the wavefunctions have different
probability density distributions, the energies
Etotal and E-total are also different. For
(y1y2) it is more likely to find the electron
between the protons, which reduces the repulsion
and hence a lower energy solution results, ie. to
take the electron from the state described by
(y1y2) and take it to infinite distance costs
energy. It also follows that
Etotal lt E-total
The total energy is given by Etotal E Up,
where E is the energy of the electron in (y1y2)
and Up is the energy of the repulsion between
the two protons.
Note, that there is a bound state which has a
minimum energy at -16.3eV (corresponding to a
separation of 0.11nm). This is 2.7eV less
binding energy than a single H atom and one
proton at infinity (-13.6 eV)
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The H2 Molecule.
This comprises of two electrons and two
protons. At infinite separation of the four
particles, the energy 0. (a) At infinite
separation of the two protons, the lowest
energy corresponds to having two hydrogen
atoms, Energy2 x (-13.6eV)27.2 eV (b) At small
separations, suppose we add one more electron to
the (y1y2) energy level in H2 (labelled UpE
in figure on next page). This is allowed by Pauli
principle (ms-1/2). This approximately doubles
the overall energy (-16.3eV x 2 -32.6 eV at
approx 0.11nm) . More accurately, the electron
interaction is more complex (-gt 31.7eV at 0.07nm
separation). The extra electron binds the two
protons closer together . The COVALENT BONDING
is produced by the symmetric wavefunction,
(y1y2).
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