Transcript LCAO principles

Introductory concepts:
Atomic and molecular orbitals
Jon Goss
http://aimpro.ncl.ac.uk
MMG Skills Lecture Series
Outline
Atomic orbitals (AOs)
Linear combinations (LCAO):
Hybrids
Molecular orbitals (MOs)
One-electron vs. many-body
Charge density and spin density
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Atomic Orbitals: founding principles
Electrons are Fermions:
The are indistinguishable
spin-half particles
Anti-symmetric wave functions
Obey the Pauli exclusion principle
(no two electrons can exist in the same quantum state)
The have mass and charge
They move in the potential arising from the (point)
nucleus and the other electrons in the atom.
For the hydrogen atom the solutions may be obtained
analytically.
For other atoms, in general this is not (yet) possible.
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Atomic orbitals
Electrons in atoms may be characterised by four
quantum numbers
n: principal quantum number
l: orbital angular momentum
ms: spin magnetic angular momentum
ml: orbital magnetic quantum number
[See for example, Atomic Spectra and Atomic Structure, Herzberg (Dover Press)]
In this lecture we are chiefly concerned with
properties implied by the different values of n
and l.
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Atomic orbitals: l
The orbital angular momentum can take positive integer values,
but they are commonly expressed using letters:
l
0
1
2
3
4
…
Term
s
p
d
f
g
…
-1, 0, 1
-2, -1, 0,
1, 2
-3, -2, -1,
0, 1, 2, 3
-4, -3, -2,
-1, 0, 1,
2, 3, 4
…
ml
0
We can interpret the increase in l in terms of an increase in
angular nodality.
For a given value of l, ml can take any value from –l to l
E.g. l=1, ml can be -1, 0 and +1.
These are equal in energy: orbital degeneracy!
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Atomic orbitals: l, ml
There is a radial
node, not shown
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Atomic orbitals: n
The possible values of l are restricted by the
principal quantum number, n.
l<n
Thus, for n=1, only l=0 (s) is allowed.
For n=2, l can have values 0 and 1 (s and p).
…and so on…
Increasing n implies increasing radial
nodality…
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Atomic orbitals: n, l and ml
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Atomic orbitals: mS
The final quantum number is the spin magnetic
quantum number, which can take two values:
ms=+½ and ms=-½
“up” and “down” spins
There is no real physical spin in the classical sense
involved.
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Pauli exclusion and the build up principles
The Pauli exclusion principle states that no two
electrons may have the same set of quantum
numbers…
For atoms, therefore, we have definite groups of
states (shells) that are incrementally occupied
with increasing energy:
1s up, 1s down (there is only one value of ml)
2s up, 2s down
(2p, ml=-1, ms=+½), (2p, ml=0, ms=+½), (2p, ml=1,
ms=+½), (2p, ml=-1, ms=-½), (2p, ml=0, ms=-½),
(2p, ml=1, ms=-½)
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Pauli exclusion and the build up principles
H: 1s1
He: 1s2
Li: 1s22s1
Be: 1s22s2
B: 1s22s22p1
C: 1s22s22p2
N: 1s22s22p3
O: 1s22s22p4
F: 1s22s22p5
Ne: 1s22s22p6
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Na: (Ne)3s1
Mg: (Ne)3s2
Al: (Ne)3s23p1
Si: (Ne)3s23p2
P: (Ne)3s23p3
S: (Ne)3s23p4
Cl: (Ne)3s23p5
Ar: (Ne)3s23p6
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Pauli exclusion and the build up principles
Nothing has been said about which ml states are
involved, although we’ll touch on this in terms
of the many-body effects.
It gets slightly more complicated with we move
beyond Ar as we begin filling the 4s orbitals
before the 3d…
You are referred to any reasonable inorganic
chemistry text book 
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Linear combinations: Hybrids
In the presence of an applied field (typically as a
consequence of nearby atoms) the atomic
orbitals combined together to form “hybrids”.
Some of the best known examples relate to
carbon.
Graphite: sp2.
Diamond: sp3.
In contrast the atomic orbitals, these are formed
in weighted combinations…
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Linear combinations: Hybrids
sp2:
s+px+py
s+px-py
s-px-py
pz
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sp3:
s+px+py+pz
s-px-py-pz
s+px-py-pz
s-px+py-pz
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Linear combinations: Molecular Orbitals
Both atomic orbitals and hybrids
centred on different atoms
combine to form covalent bonds.
σ-bonds (sigma-bonds) are made
up from overlapping orbitals
directed along the bond direction.
π-bonds (pi-bonds) are made up
from overlapping orbitals at an
angle to the inter-nuclear
direction:
πp bonds are combinations of porbitals perpendicular to the
bond direction
πp-d bonds are combinations of pand d-orbitals but not precisely
perpendicular to the bonddirection
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Linear combinations: Molecular Orbitals
σp-p anti-bonding combination
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Linear combinations: Molecular Orbitals
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Linear combinations: Molecular Orbitals
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Linear combinations: Molecular Orbitals
πp-bonding combination
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Linear combinations: Molecular Orbitals
πp* anti-bonding combination
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Linear combinations: Molecular Orbitals
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Linear combinations: Molecular Orbitals
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Linear combinations: Molecular Orbitals
πp-d bonding combination
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Linear combinations: Molecular Orbitals
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Linear combinations: Molecular Orbitals
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Linear combinations: Molecular Orbitals
πd-d bonding combination
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Linear combinations: Molecular Orbitals
πd-d anti-bonding combination
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Linear combinations: Molecular Orbitals
The bonds can be modelled by considering
linear combinations of the atomic orbitals or
atomic hybrids
This is only a simplification, as we shall see when we
consider simple many-body concepts.
The combinations are dictated by the relative
energies of the atomic orbitals.
A prototypical example is the hydrogen
molecule…
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Linear combinations: Molecular Orbitals
Molecule
Energy
a1u
1sa
1sb
a1g
Atom
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Atom
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Linear combinations: Molecular Orbitals
The same approach can be adopted for
defects in solid solution:
The vacancy in diamond
Take out an atom and you generate four
equivalent sp3 dangling-bonds.
We’ll label them a, b, c and d.
As in the H2 molecule, we form linear
combinations of these orbitals to form the
“molecular” orbitals for the four together:
(a+b+c+d) – the bonding combination
(a+b+c-d;a+b-c+d;a-b+c+d) – a triply
degenerate combinations involving some
anti-bonding character.
Hood et al PRL 91, 076403 (2003).
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Linear combinations: Molecular Orbitals
What happens when the originating orbitals are
inequivalent?
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One-electron vs. many-body
Remember that electrons are indistinguishable
particles, so the molecular and atomic orbitals are
models for the electrons in real compound systems
A more precise description of the electronic states must
be a function of the positions of all the electrons in
the system.
This is the many-electron wave function of an atom,
molecule, defect…
As an example, lets look back at one of the atoms…
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One-electron vs. many-body: Nitrogen
Nitrogen atoms have the electronic
configurations 1s22s22p3
What does this mean in terms of the properties
of the atom?
Note, for weak spin-orbit coupling, the electron spins
combine to give the total effective spin, S.
What is the spin state of a nitrogen atom?
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One-electron vs. many-body: Nitrogen
The spins in the 1s and 2s are pairs with ms=+½ and ms=-½,
yielding no net spin from these shells (S=0).
We have three electrons in the n=2, l=1 states with ml=1,0,-1,
ms=±½.
Which one combinations are involved?
It can be shown that there are exactly three ways to combine
the electrons:
Two have S=1/2, one has S=3/2.
The combinations also yield effective orbital angular momenta, which
in the many-body sense are labeled using upper case terms (S, P, D, F,
G, …)
Do not confuse the total electron spin and the S orbital angular momentum
term.
The two S=1/2 combinations are P and D, whereas S=3/2 yields S.
We write 2P, 2D and 4S, where the leading numerical value indicates the
multiplicity of the state and is given by (2S+1).
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One-electron vs. many-body
Does this make any difference?
Obviously the answer must be yes, otherwise I
would not have tortured you with the
preceding analysis
1. Spin selection rules for optical spectra (ΔS=0)
2. Spin state (magnetism, ESR, …)
3. Orbital angular momentum selection rules in
optical spectra (|ΔL|=1)
4. Jahn-Teller effects apply to many-body states
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One-electron states vs. electron density
Experimentally observed properties may
depend more or less on the many-body effects,
with some accessible from the frontier orbitals
alone:
Optical selection rules for
orbitally non-degenerate one-electron states?
orbitally degenerate one-electron states?
ESR?
Bond strengths?
Reactive sites?
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One-electron states vs. electron density
Example of H/μ+ in diamond.
There are two main forms of this centre
“Normal muonium” – a non-bonded site
“Anomalous muonium” – residing in the centre of a carbon-carbon
bond.
In the overall neutral charge state there are an odd number of
electrons and therefore the net electron spin allows for access of
these centres in ESR-like experiments.
The interaction of the electron spin and the nuclear spin of H (or
muonium) can be determined
theoretically by analysis of
the spin-density at the nucleus.
The question is, how well is the spin density represented by the
unpaired one-electron state?
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One-electron states vs. electron density
The answer is that qualitatively the correct kind of
answer can be obtained for normal muonium, but not
for anomalous muonium.
The unpaired electron (the state in the band-gap) is centred on
the muon for the normal form, giving a large isotropic hyperfine
interaction. There will also be a contribution from the
polarisation of the valence states, but this is probably not the
dominant term.
The unpaired electron in the bond-centre is nodal at the
muonium, so there should be zero isotropic hyperfine
interaction in this form, but this is not the case – for the bondcentre, the isotropic contribution to the hyperfine interaction
arises purely from the polarization of the valence density due
to the unequal spin up and spin down populations,
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Bond-centred muonium
sp3
1s
sp3
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Bond-centred muonium
B
C
A
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Bond-centred muonium
A+B-C
Ec
A-B
Ev
A+B+C
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Note, we are choosing to ignore
most of the electrons in the system
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Bond-centred muonium
A+B-C
Ec
A-B
Ev
A+B+C
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Note, we are choosing to ignore
most of the electrons in the system
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Bond-centred muonium
A+B-C
Ec
A-B
Ev
A+B+C
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Note, we are choosing to ignore
most of the electrons in the system
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In this simplified model, the experiment can be
qualitatively explained:
the isotropic part of the hyperfine comes from the
small differences between spin up and spin down
states “A+B+C”
This is spin polarization in action!
For an accurate facsimile of the experiment,
you must include the polarisation of all the
electrons in your system.
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Summary
Atomic and molecular orbital theory is a
powerful tool for simplified, but highly
illustrative explanations of a wide range of
materials properties.
However, it must always be remembered that it
is only a simplified model!
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