Transcript LCAO principles
Introductory concepts: Atomic and molecular orbitals
http://aimpro.ncl.ac.uk
Jon Goss
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Outline
Atomic orbitals (AOs) Linear combinations (LCAO): Hybrids Molecular orbitals (MOs) One-electron vs. many-body Charge density and spin density MMG Skills Lecture Series
3 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
m s
: spin magnetic angular momentum
m l
: 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
Term
m l
0
s
0
1
p
-1, 0, 1
2
d
-2, -1, 0, 1, 2
3
f
-3, -2, -1, 0, 1, 2, 3
4
g
-4, -3, -2, -1, 0, 1, 2, 3, 4
…
…
…
We can interpret the increase in
l angular nodality
.
For a given value of l, m
l
in terms of an increase in can take any value from –l to l E.g. l=1, m
l
can be -1, 0 and +1.
These are equal in energy: orbital degeneracy!
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Atomic orbitals:
l, m
l
6
There is a radial node, not shown MMG Skills Lecture Series
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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
… MMG Skills Lecture Series
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Atomic orbitals:
n, l and m
l
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Atomic orbitals:
m S
The final quantum number is the spin magnetic quantum number, which can take two values:
m s
=+½ and m
s
=-½ “up” and “down” spins There is no real physical spin in the classical sense involved.
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10 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 m
l
) 2s up, 2s down (2p, m
l
=-1, m
s
=+½), (2p, m
l
=0, m
s
=+½), (2p, m
l
=1,
m s
=+½), (2p, m
l
=-1, m
s
=-½), (2p, m
l
=0, m
s
=-½), (2p, m
l
=1, m
s
=-½) MMG Skills Lecture Series
11 Pauli exclusion and the build up principles
H: 1s 1 He: 1s 2 Li: 1s 2 2s 1 Be: 1s 2 2s 2 B: 1s 2 2s 2 2p 1 C: 1s 2 2s 2 2p 2 N: 1s 2 2s 2 2p 3 O: 1s 2 2s 2 2p 4 F: 1s 2 2s 2 2p 5 Ne: 1s 2 2s 2 2p 6 Na: (Ne)3s Mg: (Ne)3s Al: (Ne)3s Si: (Ne)3s P: (Ne)3s 2 Ar: (Ne)3s 2 1 2 2 3p 3p 3p S: (Ne)3s 2 3p 4 2 3 Cl: (Ne)3s 2 3p 5 3p 1 2 6 MMG Skills Lecture Series
12 Pauli exclusion and the build up principles
Nothing has been said about which of the
many-body effects
.
m l
states are involved, although we’ll touch on this in terms 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 MMG Skills Lecture Series
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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: sp 2 .
Diamond: sp 3 .
In contrast the atomic orbitals, these are formed in weighted combinations… MMG Skills Lecture Series
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Linear combinations:
Hybrids
sp
2 : s+p
x
+p
y
s+p
x
-p
y
s-p
x
-p
y p z sp
3 : s+p
x
+p
y +p z
s-p
x
-p
y -p z
s+p
x
-p
y -p z
s-p
x
+p
y -p z
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15 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: π π
p p-d
bonds direction are combinations of p orbitals perpendicular to the bond direction bonds are combinations of p and d-orbitals but not precisely perpendicular to the bond MMG Skills Lecture Series
16 Linear combinations:
Molecular Orbitals
σ
p-p
anti-bonding combination MMG Skills Lecture Series
17 Linear combinations:
Molecular Orbitals
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18 Linear combinations:
Molecular Orbitals
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19 Linear combinations:
Molecular Orbitals
π p -bonding combination MMG Skills Lecture Series
20 Linear combinations:
Molecular Orbitals
π p * anti-bonding combination MMG Skills Lecture Series
21 Linear combinations:
Molecular Orbitals
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22 Linear combinations:
Molecular Orbitals
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23 Linear combinations:
Molecular Orbitals
π
p-d
bonding combination MMG Skills Lecture Series
24 Linear combinations:
Molecular Orbitals
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25 Linear combinations:
Molecular Orbitals
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26 Linear combinations:
Molecular Orbitals
π
d-d
bonding combination MMG Skills Lecture Series
27 Linear combinations:
Molecular Orbitals
π
d-d
anti-bonding combination MMG Skills Lecture Series
28 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… MMG Skills Lecture Series
29 Linear combinations:
Molecular Orbitals
Molecule
a 1u
1s a 1s b
a 1g
Atom Atom MMG Skills Lecture Series
30 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 sp
3 dangling-bonds.
We’ll label them a, b, c and d.
As in the H 2 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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31 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… MMG Skills Lecture Series
33 One-electron vs. many-body:
Nitrogen
Nitrogen atoms have the electronic configurations 1s 2 2s 2 2p 3 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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34 One-electron vs. many-body:
Nitrogen
The spins in the 1s and 2s are pairs with m
s
=+½ and m
s
=-½, yielding no net spin from these shells (S=0).
We have three electrons in the n=2, l=1 states with m
l
=1,0,-1,
m s
= ± ½.
Which one combinations are involved?
It can be shown that there are exactly
three ways to combine the electrons
G, …) term.
We write 2 P, 2 : 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, Do not confuse the total electron spin and the S orbital angular momentum The two S=1/2 combinations are P and D, whereas S=3/2 yields S.
D and 4 S, 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 MMG Skills Lecture Series
36 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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37 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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38 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 arises
purely nodal
at the muonium, so there should be zero isotropic hyperfine interaction in this form, but this is not the case – for the bond centre, the isotropic contribution to the hyperfine interaction from the polarization of the valence density due to the unequal spin up and spin down populations, MMG Skills Lecture Series
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Bond-centred muonium
sp 3 sp 3 1s MMG Skills Lecture Series
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Bond-centred muonium
B A C MMG Skills Lecture Series
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Bond-centred muonium
Note, we are choosing to ignore most of the electrons in the system
A+B-C A-B
E
c
E
v A+B+C MMG Skills Lecture Series
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Bond-centred muonium
Note, we are choosing to ignore most of the electrons in the system
A+B-C A-B
E
c
E
v A+B+C MMG Skills Lecture Series
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Bond-centred muonium
Note, we are choosing to ignore most of the electrons in the system
A+B-C A-B
E
c
E
v A+B+C MMG Skills Lecture Series
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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! MMG Skills Lecture Series