schrödinger equation for hydrogen molecule ion · lecture 13: molecular structure ... schrödinger...
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Lecture 13: Molecular structureLecture 13: Molecular structure
o Hydrogen molecule ion (H2+)
o Overlap and exchange integrals
o Bonding/Anti-bonding orbitals
o Molecular orbitals
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Schrödinger equation for hydrogen molecule ionSchrödinger equation for hydrogen molecule ion
o Simplest example of a chemical bond is the
hydrogen molecule ion (H2+).
o Consists of two protons and a single electron.
o If nuclei are far from each another, electron is
localised on one nucleus. Wavefunctions are then
those of atomic hydrogen.
rab
rbra
++
-
ab
o !a is the hydrogen atom wavefunction of electron belonging to nucleus a. Must therefore
satisfy
and correspondingly for other wavefunction, !b. The energies are therefore
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Schrödinger equation for hydrogen molecule ionSchrödinger equation for hydrogen molecule ion
o If atoms are brought into close proximity, electron localised on b will now experience and
attractive Coulomb force of nucleus a.
o Must therefore modify Schrödinger equation to include Coulomb potentials of both nuclei:
where
o To find the coefficients ca and cb, substitute into Eqn. 2:
(2)
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Solving the Schrödinger equationSolving the Schrödinger equation
o Can simplify last equation using Eqn. 1 and the corresponding equation for Ha and Hb.
o By writing Ea0 !a in place of Ha !a gives
o Rearranging,
o As Ea0 = Eb
0 by symmetry, we can set Ea0 - E = Eb
0 - E = -!E,
(3)
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Overlap andOverlap and exchange integralsexchange integrals
o !a and !b depend on position, while ca and cb do not. Now we know that for orthogonal
wavefunctions
o But !a and !b are not orthoganal, so
o If we now multiply Eqn. 3 by !b and integrate the results, we obtain
o As -e|!a|2 is the charge density of the electron => Eqn. 4 is the Coulomb interaction energy
between electron charge density and nuclear charge e of nucleus b.
o The -e!a !b in Eqn. 5 means that electron is partly in state a and partly in state b => an
exchange between states occurs. Eqn. 5 therefore called an exchange integral.
Normalisation integral
Overlap integral
(4)
(5)
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Overlap andOverlap and exchange integralsexchange integrals
o Eqn. 4 can be visualised via figure at right.
o Represents the Coulomb interaction energy of an
electron density cloud in the Coulomb field on the
nucleus.
o Eqn. 5 can be visualised via figure at right.
o Non-vanishing contributions are only possible
when the wavefunctions overlap.
o See Chapter 24 of Haken & Wolf for further
details.
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Orbital energiesOrbital energies
o If we mutiply Eqn 3 by !a and integrate we get
o Collecting terms gives,
o Similarly,
o Eqns. 6 and 7 can be solved for c1 and c2 via the matrix equation:
o Non-trivial solutions exist when determinant vanishes:
(6)
(7)
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Orbital energiesOrbital energies
o Therefore
o This has two solutions: (8)
and (9)
o From Eqn. 8,
o Substituting this into Eqn. 6 => cb = -ca = -c
o The total wavefunction is thus
o Similarly solving for !E in Eqn. 9 and substituting into Eqn. 6 => ca = cb = c, giving
Anti-bonding orbital
Bonding orbital
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Bonding and anti-bonding Bonding and anti-bonding orbitalsorbitals
o If two nuclei approach each other, the
electronic energy E slits depending on whether
dealing with a bonding or an anti-bonding
wavefunction.
o For symmetric case (top right), occupation
probability for " is positive.
o Not the case for a asymmetric wavefunctions
(bottom right).
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Bonding and anti-bonding Bonding and anti-bonding orbitalsorbitals
o The energy E of the hydrogen molecular ion can finally be written
o ‘+’ correspond to anti-bonding orbital energies, ‘-’ to bonding.
o Energy curves below are plotted to show their dependence on rab
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Molecular Molecular orbitalsorbitals
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Molecular Molecular orbitalsorbitals
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Molecular Molecular orbitalsorbitals
o Energies of bonding and anti-bonding
molecular orbitals for first row diatomic
molecules.
o Two electrons in H2 occupy bonding
molecular orbital, with anti-parallel spins.
If irradiated by UV light, molecule may
absorb energy and promote one electron
into its anti-bonding orbital.
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Molecular Molecular orbitalsorbitals