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CHE-30042
Inorganic, Physical & Solid State Chemistry
Advanced Quantum Chemistry: lecture 1
Rob Jackson
LJ1.16, 01782 733042
[email protected]
www.facebook.com/robjteaching
@robajackson
Background reading
Recommended
Atkins’ Physical Chemistry, 9th edition
Peter Atkins, Julio de Paula
Supplementary (more detailed)
Quantum Mechanics for Chemists
David O Hayward
(RSC Tutorial Chemistry Text no. 14)
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Lecture 1 contents
1. Recap from CHE-20028
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Schrödinger Equation: solution for model systems
(particle in box, harmonic oscillator)
Hydrogen atom – energies and orbitals
Hydrogen-like orbitals
2. Wavefunctions for ‘many-electron’ atoms
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–
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Determinant notation introduced
Wavefunctions for He – C
The Self-Consistent Field Method
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Recap. of material from CHE-20028– (i)
• Schrödinger equation: H = E
– Note – operator algebra - the Hamiltonian, H acts on
the wavefunction,  to give the energy, E.
• Particle in a box: permitted energies are:
En = n2h2/8mL2 (with n = 1,2,3 ...)
• Harmonic oscillator:
H=
– ‘x’ is the particle displacement
2 2 1 2

 2 kx
2
2m x
– En = (n+½)  (n = 0, 1, 2 ,3 …) (giving zero point energy)
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Recap. of material from CHE-20028– (ii)
• The hydrogen atom
– Hamiltonian is H = Te + Vne, or written in full:
H = (-ħ2/2m) 2 -e2/40r
– (check che-20028 slides for definitions).
– Wavefunction is written as:
(r,,) = R(r) Y(, )
– ‘s’ orbitals only depend on r, while ‘p’ orbitals
(onwards) also depend on (, ).
– ‘hydrogen-like’ orbitals assume no electron repulsion.
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Hydrogen-like orbitals
• The orbitals widely used in Chemistry (1s,
2s, 2p, 3s, 3p, 3d etc.) are known as
hydrogen-like orbitals because in their
‘ideal’ form they only apply to hydrogen
(with no electron repulsion), or to other 1electron ‘atoms’ like He+, Li2+ etc.
• A link to a good diagram will be given (also
see teaching pages). Also see next slide.
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Screenshot from http://csi.chemie.tu-darmstadt.de
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Atoms with more than 1 electron (i)
• For He, we might write the wavefunction as
1s2, shorthand for 1s(1)1s(2)
• This means electron 1 in 1s with spin up
and electron 2 in 1s with spin down.
• Wavefunctions have to satisfy the Pauli
Exclusion Principle, which states that the
wavefunction must change sign if the
electrons are interchanged.
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Atoms with more than 1 electron (ii)
• So the wavefunction for He is written as:

  (1/ 2) 1s(1)1s(2)  1s(2)1s(1)

– Note that the (1/√2) term is a normalising factor.
• The expression can be simplified by using
determinant notation (will be written and
explained in lecture).
– Look up determinants and how to solve them. Links
will be provided to useful web resources.
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Atoms with more than 1 electron (iii)
• The determinant notation can be used to simplify
the expressions for wavefunctions of atoms with
more electrons.
– Li will be done in the lecture
– Self-test: try Be, B & C
• Using this procedure we can (in principle) write
out the wavefunction of any atom in terms of
orbitals.
– Note: expressions for the orbitals are still needed.
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Correcting hydrogen like orbitals: the SelfConsistent Field method
• We can write the wavefunction of an atom in
terms of orbitals, e.g. for Li:
 = 1s2 2s1 (or in determinant notation to take into
account the Pauli Exclusion Principle).
• Each term is a 1-electron orbital, so how is
electron repulsion taken into account?
• This is done using the Self-Consistent Field
method.
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The Self-Consistent Field (SCF) method – (i)
• The s, p, d, f … orbitals that we are familiar with
are 1-electron orbitals – i.e. they don’t take into
account the presence of other electrons.
• The SCF method provides a way of correcting
wavefunctions for the presence of more than one
electron.
• This can explain, e.g., why electrons in 4s orbitals
have lower energy than those in 3d orbitals in K
and Ca
• How has this been explained up to now?
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The Self-Consistent Field (SCF) method – (ii)
• What is the Hamiltonian for a 1-electron atom?
• It will include 2 terms: electron kinetic energy (Te)
and electron-nucleus potential energy (Vne).
H = Te + Vne
• If we have more than one electron, there will be
an additional term due to electron-electron
repulsion (Vee):
H = Te + Vne + Vee
• (Note there will generally be more than one Te
and Vne term).
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The Self-Consistent Field (SCF) method – (iii)
• There is a problem in calculating the electron
repulsion energy because we are using 1-electron
orbital wave functions – i.e. each orbital only
contains one electron, so how can we explain
how they interact?
• The SCF method provides a way of correcting
orbitals for the effect of other electrons.
• It starts by calculating the average potential
energy of interaction between the first electron
and the others (i.e. Vee in the previous equation).
This is done assuming 1-electron orbitals.
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Application of the SCF method – (i)
•
If we now consider a particular example, Be,
which has electronic structure 1s22s2 (ignoring
spin and antisymmetry for now), we follow this
procedure:
– (i) calculate the Vee term assuming 1-electron
orbitals.
– (ii) solve the Schrödinger equation for the 2s
orbital using this Vee term.
– (iii) This gives a new updated 2s orbital.
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Application of the SCF method – (ii)
–
–
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(iv) using the new 2s orbital obtained, the Vee
term is recalculated, and the Schrödinger
equation solved for the 1s orbital.
(v) An updated 1s orbital is obtained, and this
is then used to amend the value of Vee, which
is then used to recalculate the 2s orbital.
(vi) The procedure is then repeated until
there is no change in the orbitals and their
energies in successive calculations.
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Application of the SCF method – (iii)
• For Be, the ordering of the orbitals will not be
affected, but in the case of K and Ca, the energy
of the 4s orbital will be found to be lower than 3d
at the end of the calculation, explaining the
orbital occupancies.
• A problem with the SCF method is that it does not
treat electron correlation (i.e. electrostatic
repulsion) properly, and this has to be corrected
for, using Configuration Interaction calculations.
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Summary of Lecture
• Material from che-20028 has been revisited.
• Wavefunctions that obey the Pauli Exclusion
Principle have been introduced.
• The determinant notation has been introduced
for atoms beyond hydrogen.
• The SCF method for atoms has been introduced
and explained.
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