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1AMQ, Part V Molecules
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The Molecular Hydrogen Ion
Hydrogen Molecule and Covalent
Bonding
•K.Krane, Modern Physics, Chapter 9
• Eisberg and Resnick, Quantum Physics,
Chapter 12.
1AMQ P.H. Regan &
W.N.Catford
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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 twocentred 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.
1AMQ P.H. Regan &
W.N.Catford
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1AMQ P.H. Regan &
W.N.Catford
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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
y tot  y 1 y 2
Recalling that for hydrogen, in the 1s state,
2
r / a
Hydrogen
y 100
( r , ,  ) 
e
4 a02 / 3
0
and thus, | y 100 | is W.N.Catford
proportion al to e
21AMQ P.H. Regan &
 2 r / a0
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(b) For smaller separations the y1 and y2
wavefunctions overlap and then, y1+y2 and
y1y2 have different probability distributions.
1AMQ P.H. Regan &
W.N.Catford
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Since the wavefunctions have different
probability density distributions, the
energies E+total and E-total are also different.
For (y1+y2) 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
(y1+y2) and take it to infinite distance
costs energy. It also follows that
E+total < E-total
The total energy is given by E+total = E+ + Up,
where E+ is the energy of the electron in
(y1+y2) 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)
1AMQ P.H. Regan &
W.N.Catford
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1AMQ P.H. Regan &
W.N.Catford
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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, Energy=2 x (-13.6eV)=27.2 eV
(b) At small separations, suppose we add one more
electron to the (y1+y2) energy level in H2+ (labelled
Up+E+ 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 (-> 31.7eV at 0.07nm separation).
The extra electron binds the two protons closer
together . The COVALENT BONDING is produced
by the symmetric wavefunction, (y1+y2).
1AMQ P.H. Regan &
W.N.Catford
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1AMQ P.H. Regan &
W.N.Catford
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W.N.Catford
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