Introductory concepts: Atomic and molecular orbitals

1
Introductory conceptsAtomic and molecular
orbitals

• Jon Goss

2
Outline

• Atomic orbitals (AOs)
• Linear combinations (LCAO)
• Hybrids
• Molecular orbitals (MOs)
• One-electron vs. many-body
• Charge density and spin density

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.

4
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.

5
Atomic orbitals l

• The orbital angular momentum can take positive
  integer values, but they are commonly expressed
  using letters
• 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. l1, ml can be -1, 0 and 1.
• These are equal in energy orbital degeneracy!

l 0 1 2 3 4
Term s p d f g
ml 0 -1, 0, 1 -2, -1, 0, 1, 2 -3, -2, -1, 0, 1, 2, 3 -4, -3, -2, -1, 0, 1, 2, 3, 4
6
Atomic orbitals l, ml
7
Atomic orbitals n

• The possible values of l are restricted by the
  principal quantum number, n.
• lltn
• Thus, for n1, only l0 (s) is allowed.
• For n2, l can have values 0 and 1 (s and p).
• and so on
• Increasing n implies increasing radial nodality

8
Atomic orbitals n, l and ml
9
Atomic orbitals mS

• The final quantum number is the spin magnetic
  quantum number, which can take two values
• ms½ and ms-½
• up and down spins
• There is no real physical spin in the classical
  sense involved.

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 ml)
• 2s up, 2s down
• (2p, ml-1, ms½), (2p, ml0, ms½), (2p, ml1,
  ms½), (2p, ml-1, ms-½), (2p, ml0, ms-½),
  (2p, ml1, ms-½)

11
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

• Na (Ne)3s1
• Mg (Ne)3s2
• Al (Ne)3s23p1
• Si (Ne)3s23p2
• P (Ne)3s23p3
• S (Ne)3s23p4
• Cl (Ne)3s23p5
• Ar (Ne)3s23p6

12
Pauli exclusion and the build up principles

• Nothing has been said about which ml states are
  involved, although well 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 ?

13
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

14
Linear combinations Hybrids

• sp2
• spxpy
• spx-py
• s-px-py
• pz

• sp3
• spxpypz
• s-px-py-pz
• spx-py-pz
• s-pxpy-pz

15
Linear combinations Molecular Orbitals

• Both atomic orbitals and hybrids centred on
  different atoms combine to form covalent bonds.
• s-bonds (sigma-bonds) are made up from
  overlapping orbitals directed along the bond
  direction.
• p-bonds (pi-bonds) are made up from overlapping
  orbitals at an angle to the inter-nuclear
  direction
• pp bonds are combinations of p-orbitals
  perpendicular to the bond direction
• pp-d bonds are combinations of p- and d-orbitals
  but not precisely perpendicular to the
  bond-direction

16
Linear combinations Molecular Orbitals
sp-p anti-bonding combination
17
Linear combinations Molecular Orbitals
18
Linear combinations Molecular Orbitals
19
Linear combinations Molecular Orbitals
pp-bonding combination
20
Linear combinations Molecular Orbitals
pp anti-bonding combination
21
Linear combinations Molecular Orbitals
22
Linear combinations Molecular Orbitals
23
Linear combinations Molecular Orbitals
pp-d bonding combination
24
Linear combinations Molecular Orbitals
25
Linear combinations Molecular Orbitals
26
Linear combinations Molecular Orbitals
pd-d bonding combination
27
Linear combinations Molecular Orbitals
pd-d anti-bonding combination
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

29
Linear combinations Molecular Orbitals
Molecule
Energy
Atom
Atom
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
  sp3 dangling-bonds.
• Well 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
• (abcd) the bonding combination
• (abc-dab-cda-bcd) a triply degenerate
  combinations involving some anti-bonding
  character.
• Hood et al PRL 91, 076403 (2003).

31
Linear combinations Molecular Orbitals

• What happens when the originating orbitals are
  inequivalent?

32
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

33
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?

34
One-electron vs. many-body Nitrogen

• The spins in the 1s and 2s are pairs with ms½
  and ms-½, yielding no net spin from these shells
  (S0).
• We have three electrons in the n2, l1 states
  with ml1,0,-1, ms½.
• Which one combinations are involved?
• It can be shown that there are exactly three ways
  to combine the electrons
• Two have S1/2, one has S3/2.
• 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 S1/2 combinations are P and D, whereas
  S3/2 yields S.
• We write 2P, 2D and 4S, where the leading
  numerical value indicates the multiplicity of the
  state and is given by (2S1).

35
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
• Spin selection rules for optical spectra (?S0)
• Spin state (magnetism, ESR, )
• Orbital angular momentum selection rules in
  optical spectra (?L1)
• Jahn-Teller effects apply to many-body states

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?

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?

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 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 arises
  purely from the polarization of the valence
  density due to the unequal spin up and spin down
  populations,

39
Bond-centred muonium
sp3
1s
sp3
40
Bond-centred muonium
B
C
A
41
Bond-centred muonium
AB-C
Ec
A-B
Ev
ABC
Note, we are choosing to ignore most of the
electrons in the system
42
Bond-centred muonium
AB-C
Ec
A-B
Ev
ABC
Note, we are choosing to ignore most of the
electrons in the system
43
Bond-centred muonium
AB-C
Ec
A-B
Ev
ABC
Note, we are choosing to ignore most of the
electrons in the system
44

• 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 ABC
• 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.

45
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!