1. CHM2S1-A Introduction to Quantum Mechanics Dr R. L. Johnston
THE UNIVERSITY
OF BIRMINGHAM
II: Quantum Mechanics of Atoms and Molecules
5. Electronic Structures of Atoms
5.1 The hydrogen atom and hydrogenic ions.
5.2 Quantum numbers.
5.3 Orbital angular momentum.
5.4 Atomic orbitals.
5.5 Electron spin.
5.6 Properties of atomic orbitals.
5.7 Many-electron atoms.

2. 6. Bonding in Molecules
6.1 The Born-Oppenheimer approximation.
6.2 Potential energy curves.
6.3 Molecular orbital (MO) theory.
6.4 MO diagrams.
6.5 MOs for 2nd row diatomic molecules.
6.6 Molecular electronic configurations.
6.7 Bond order.
6.8 Paramagnetic molecules.
6.9 Heteronuclear diatomic molecules.

3. 5. Electronic Structures of Atoms
5.1 The Hydrogen Atom and Hydrogenic Ions
The series of atoms/ions H, He+, Li2+, Be3+ …
all have: 1 electron (charge –e)
nucleus (charge +Ze)
To determine the electronic wavefunction ( = e) and allowed electronic energy levels (E), we must set up and solve the Schrödinger Equation for a single electron in an atom.

4. Consider 3-D motion of the electron (-e) relative to the nucleus (+Ze):
Hamiltonian Operator
Kinetic Energy where
Potential Energy where
(electrostatic attraction between electron and nucleus).
+Ze mN -e me r

5. We must solve the 2nd order differential equation:
Because of the spherical symmetry of the atom, it is convenient to describe the position of the electron in spherical polar coordinates (r,,), rather than Cartesians (x,y,z).
Due to this spherical symmetry, the wavefunction can be separated into a product of radial and angular components:
(r,,) = R(r).Y(,)
Imposing boundary conditions  3 quantum numbers (n,,m)
 n,,m(r,,) = Rn,(r).Y,m(,)
radial
wavefunction
angular
(spherical harmonic)

6. n,,m(r,,) = Rn,(r).Y,m(,)
5.2 Quantum Numbers
n,,m(r,,) = Rn,(r).Y,m(,)
 depends on 3 quantum numbers (n,,m).
1. Principal Quantum Number (n)
Positive integer n = 1, 2, 3 …
For H and 1-e ions, the electron energy depends only on n
(this is the same result as from the Bohr model).
For many-electron atoms E depends on n and .

7. 2. Orbital Angular Momentum Quantum Number ()
For a given value of n,  can take the integer values:
 = 0, 1, 2 … (n-1)
e.g. n = 1  = 0
n = 2  = 0, 1
n = 3  = 0, 1, 2
3. Orbital Magnetic Quantum Number (m)
For a given value of , m can take the integer values:
m = 0, 1, 2 …  
e.g.  = 0 m = 0
 = 1 m = 0, 1
There are (2+1) allowed values of m.

8. 1 2 3 2 6 12 5.3 Orbital Angular Momentum  |J|
Q.Nos.  and m arise from the angular part of the wavefunction and they relate to the angular momentum of the electron due to its motion around the nucleus (orbital angular momentum).
 – determines magnitude of angular momentum vector (J)
Because  is quantized, so is |J|:  1 2 3
|J|
2
6
12

9. angular momentum vector J has magnitude |J| = 6 and
The orientation of the angular momentum vector (J) is also quantized – as it depends on m.
The component of angular momentum along a reference direction (e.g. the z-axis):
Jz = m
e.g. for  = 2 (m = 0, 1, 2)
angular momentum vector J
has magnitude |J| = 6 and
5 allowed orientations:
Jz = 0, , 2. m

10. 5.4 Atomic Orbitals
The wavefunctions (n,,m) describing an electron in an atom are known as atomic orbitals.
Each atomic orbital (1-e wavefunction) is uniquely defined by 3 quantum numbers (n,,m).
The orbital () gives the spatial distribution of the electron (via the Born interpretation of ||2).
Orbitals are often drawn as 3-D surfaces which enclose approx. 90% of the probability of finding the electron.

11. Electronic Shells, Sub-shells and Orbitals
All orbitals with the same value of n together constitute an electronic shell.
For each n, orbitals with the same value of  together constitute an electronic sub-shell.
Each sub-shell consists of (2+1) orbitals, each with a different m value. n 1 2 3 4
Shell K L M N  1 2 3
Sub-shell s p d f
No. orbitals 5 7

12. Possible combinations of shells and sub-shells.
Total number of orbitals in shell n is n2.
  n 1 2 3 … 1s 2s 2p 3s 3p 3d 4 4s 4p 4d 4f

13. Electronic Shells, Sub-shells and Orbitals
ml =2
ml =1
ml =0
ml =1
ml =2
n=3
3p orbitals
l=1
ml =1
ml =0
ml =1
3s orbital
l=0
ml =0
shell
sub-shells
orbitals

14. Atomic Orbital Energy Diagram for H
Orbital energies depend only on the
principal quantum number n.
For a given n value (shell), all sub-shells
() and orbitals (m) have the same energy
– i.e. they are degenerate.

15. Quantum Numbers and Atomic Orbitals
Quantum theory postulates that electrons in atoms are fixed in regions of space corresponding to atomic orbitals. The probability of finding an electron at a particular point in space (r) is proportional to |(r)|2 (Born Interpretation).
Orbitals differ from each other in size, shape and orientation:
the orbital size is defined by the principle quantum number n
the type of orbital (it’s shape) is defined by the angular momentum quantum number 
the orientation of the orbital is defined by the orbital magnetic quantum number m

16. Shapes of Atomic Orbitals
s orbitals have l = 0 and ml = 0
p orbitals have l = 1 and ml = 1 , 0, 1 pz px py
d x2 y2
dz2
dxy
dxz
dyz
d orbitals have l = 2 and ml = 2, 1, 0 ,1, 2

17. 5.5 Electron Spin
Stern and Gerlach (1921) observed that a beam of Ag atoms is split into 2 beams by an inhomogeneous magnetic field.
Dirac (1930) introduced relativistic effects into Quantum Mechanics and showed that to completely describe the state of an electron we must specify:
1. The orbital (n,,m)
2. The spin state of the electron
Electron spin is characterised by 2 quantum numbers:
– spin angular momentum q. no. s ( = ½ for all electrons)
– spin magnetic q. no. ms ( = ½).

18. Spin angular momentum (S) has magnitude:
The projection of S on the z-axis:
Sz = ms = ½ 
There are two possible electron spin states:
ms = +½ “spin up” 
ms = -½ “spin down” 
Spin is an intrinsic property of the electron and is not connected with orbital motion.
Complete specification of an electron in an atom requires 4 quantum numbers (n,,m,ms) – as s is fixed.

19. + 5.6 Properties of Atomic Orbitals
1. Shape – determined by the angular wavefunction Y,m(,)
1s orbital (n = 1,  = 0, |J| = 0)
Normalized wavefunction:
Has no angular dependence
(spherically symmetric, depends only on r).
All s orbitals are spherically symmetrical.
2s orbital ns orbital
As n, orbitals expand – average radius r +

20. + - 2p orbitals (n = 2,  = 1, |J| = 2)
There are 3 degenerate orbitals, with 3 different m values (0,1).
e.g. m = 0  Jz = m  = 0  2pz orbital (pointing along z-axis)
m = 1  Jz =    2px, 2py orbitals + -

21. 2. Nodal Properties
a) Angular Nodes
s orbitals – have no angular nodes (spherically symmetric)
p orbitals – have one angular node (a nodal plane, where  = 0)
No. of angular nodes =  (s = 0, p = 1, d = 2 …)
b) Radial Nodes
s orbitals – have a maximum in  at nucleus (r = 0)
For other orbitals (p, d, f …),  = 0 at nucleus due to angular nodes.
All orbitals decay exponentially at large r values.
Radial nodes – nodal surfaces where  = 0, with  changing sign either side of the node.
No. of radial nodes = n1 1s  0
2s  p  0
3s  p  d  0
Total number of nodes (angular + radial) = n1

22. Angular and Radial Nodes in 
Angular Nodes
Radial Nodes + -
3s orbital
 = 0 - +
 = 0
2pz orbital

23. 3. Radial Distribution Functions
Electron density at point (r) in space:
Probability of finding electron at point r:
Radial Distribution Function – probability density of finding electron at distance r from nucleus:
P(r)dr – probability of finding electron in a shell of thickness dr at distance r from the nucleus.
For non-spherical orbitals (p, d, f …), use:

24. Example: H(1s)
Peak in RDF at r = a0 (the Bohr radius for n = 1).
The delocalized electron is represented by a wave which has maximum probability at r = a0.
4r22 a0 r

25. Hydrogenic Ions 1e ions with nuclear charge Z. Max. in 1s RDF at
As Z, orbitals contract (rmax).
Orbital Energies
As Z, orbitals more tightly bound (En – more negative).

26. 5.7 Many-Electron Atoms
For atoms with 2 or more electrons, the Schrödinger Eqn. must include KE terms for all electrons and PE terms for all e-e and e-n interactions.
Example: He (2 electrons)
2 terms in
3 terms in
S.E.
where
Z = +2e
-e(1) r1 r2
r12
-e(2)

27. N = (r1,r2,r3 … rN) = (r1)(r2)(r3)…(rN)
For N electrons: N = (r1,r2,r3 … rN)
For > 1 e, S.E. cannot be solved analytically – though good numerical solutions may be obtained from computer calculations.
 Approximations must be made.
The Orbital Approximation
The total wavefunction (N) for the N-electron atom is approximated by the product of N 1-e orbitals similar to those of hydrogenic ions:
N = (r1,r2,r3 … rN) = (r1)(r2)(r3)…(rN)

28. Orbital Energies for Many-Electron Atoms
For H and hydrogenic ions, E depends only on n.
e.g. 3s, 3p and 3d orbitals are degenerate.
For many-electron atoms, E depends on n and .
In shell n, En as  i.e. E(ns) < E(np) < E(nd) …
Why ?

29. Effective Nuclear Charge
An electron at distance r from nucleus experiences an effective nuclear charge:
Zeff < Z
(Z = actual charge on nucleus = atomic number)
Electrons inside a sphere of radius r repel the electron and shield (screen) it from the nucleus:
Zeff = Z  
( = shielding constant).

30.  is cylindrically symmetrical about the internuclear (A-B) axis
Properties of 
For large R,  behaves like 2 independent H(1s) AOs.
For small R, there is significant overlap between the AOs.
Destructive interference between atomic
wavefunctions (AOs) A and B  depletion
of electron density between the nuclei
(decrease of  and 2).
The depletion of electron density between
the nuclei leads to decreased e-n attraction:
 E() > E(A,B)
  is an antibonding MO.
 is cylindrically symmetrical about the internuclear (A-B) axis
 labelled as a  MO (or sometimes *, as it is antibonding). A B R A
-B  + 

31. Electrons in orbitals with low n (core orbitals) have greater electron density (higher probability of finding the electron) close to the nucleus  experience higher Zeff  lower energy.
Electrons in orbitals with same n, but
lower , have peaks in RDF which are
closer to the nucleus  better at shielding
other electrons and better at penetrating
shielding:
Zeff ns > np > nd …
E ns < np < nd …
Orbitals with same n and  are still
degenerate
e.g. E(2px) = E(2py) = E(2pz)

32. Electronic Configurations
To determine the electronic configuration of an atom (how the electrons are distributed among the atomic orbitals), we need to know:
The available orbitals and their relative energies.
The number of electrons.
The rules for filling the orbitals.

33. X Rules for Filling Orbitals 1. The Pauli Exclusion Principle
No two electrons in a particular atom or ion can have the same values of all 4 quantum numbers (n,,m,ms).
No more than 2e can occupy a given orbital
If 2e are in the same orbital (same n,,m), they must have opposite (= paired) spins.
H 1s1
He 1s2
Li 1s22s1 X

34. 2. The Aufbau (Building-up) Principle
Add available electrons into orbitals – starting from the lowest in energy, put maximum of 2e in each orbital.
General order of orbital occupation:
1s < 2s < 2p < 3s < 3p < 4s ~ 3d < 4p < 5s < 4d …
Usually completely fill each sub-shell () before starting to fill another.
Exceptions – e.g. in the 1st row transition metals:
Cr [Ar]4s13d5
Cu [Ar]4s13d10

35. The Aufbau Principle n

36. Variation of Orbital Energy with Z
The 3d and 4s orbitals are close in
energy for the first transition metal
series.
For heavier atoms (higher atomic
number Z), E(3d) < E(4s).
As Z, the  sub-shells of the inner
(lower n) orbitals become approx.
degenerate

37. 3. Spatial Separation
Electrons occupy different orbitals (m) of a given sub-shell () before starting to pair electrons.
e.g. N [He]2s22p3 = [He]2s22px12py12pz1
Electrons in different orbitals on average are further apart  lower e-e repulsion  lower E.

38. 4. Hund’s Rule (of Maximum Multiplicity)
Provided rules 1 and 2 are satisfied, an atom in its ground state adopts the configuration with the maximum number of unpaired (= parallel) spins.
e.g. C [He]2s22p2 = [He]2s22px12py1
(equivalent to … 2px12pz1 and 2py12pz1).
Parallel Spins Antiparallel Spins
Electrons with parallel spins () tend to avoid each other better than those with antiparallel spins ()  reduced e-e repulsion  lower E.
This is known as spin correlation.

39. 6. Bonding in Molecules PE KE
Exact solution of the Schrödinger Equation is not possible for any molecule – even the simplest molecule H2+.
Full Hamiltonian operator for H2+ : e HA HB
RAB rA rB PE KE
nuclear K.E.
e-n attraction
n-n repulsion
electron K.E.

40. 6.1 The Born-Oppenheimer Approximation
Nuclei are much heavier (thousands of times) than electrons.
 they move much more slowly.
In the Born-Oppenheimer approximation, nuclei are treated as being stationary.
Consider motion of electrons relative to fixed nuclei.
Total wavefunction () is split into the product of electronic and nuclear wavefunctions:
 = e.n

41. Schrödinger Equation for electronic motion:
Example 1: H2+
Fix RAB (  R).
Schrödinger Equation for electronic
motion:
Note: although the electronic SE is solved for fixed RAB (= R), the solutions (e,Ee) depend on the value of R. e HA HB R rA rB
electron K.E.
e-n attraction
n-n repulsion
(constant)

42. Electronic Hamiltonian:HB R
r1A
r1B e2
r2B
r2A
r12
Example 2: H2
Electronic Hamiltonian:
SE for molecules containing other elements
(atomic no. Z):
e-n attraction terms  e2  Ze2
n-n repulsion terms +e2  +Z2e2.
electron K.E.
e-n attraction
n-n repulsion
e-e repulsion

43. 6.2 Potential Energy Curves
For a diatomic molecule, solving the electronic SE for different fixed positions of the nuclei (i.e. fixed inter-nuclear distances, R) gives the molecular potential energy curve V(R).
Re = equilibrium bond length
De = PE well depth = dissociation energy.
As R  , V(R)  0 (dissociation limit).
This can be extended to larger molecules:
e.g. for a general triatomic molecule we get a potential energy surface V(R1,R2,).
V(R) R Re
-De R1 R2 

44. 6.3 Molecular Orbital Theory
How can we determine  and E for an electron in a molecule?
SE can be solved exactly for (within the B-O approximation) but it is complicated and SE cannot be solved exactly for > 1e.
We need to make more approximations  Molecular Orbital theory.
6.3 Molecular Orbital Theory
Electrons in molecules have spatial distributions which are described by 1e wavefunctions called molecular orbitals (MOs) – analogous to atomic orbitals (AOs).
Let  represent a MO and  an AO.
MOs are spatially delocalised over the molecule.
Probability of finding electron at point r in space in MO :
P(r) = | (r)|2d

45. Linear Combination of Atomic Orbitals (LCAO) Approximation
Construct MOs () as linear combination of AOs ():
ci = coefficients (numbers) = contribution of ith AO to the MO.
N AOs  N MOs.
Justification
When electron is close to one nucleus (A) it experiences an electrostatic (Coulomb) attraction that is greater than that to B.
 MO wavefunction () close to A, resembles an atomic orbital centred on A (A).

46. + = N+(A + B) in-phase (bonding)
Example: H2+ and H2
MOs formed as linear combinations of H(1s) AOs.
2 AOs (A,B)  2 MOs (+,)
+ = N+(A + B) in-phase (bonding)
 = N(A  B) out-of-phase (antibonding) A A B B R

47. + A B Properties of + R
For large R, + behaves like 2 independent H(1s) AOs.
For small R, there is significant overlap between the AOs.
Constructive interference between atomic
wavefunctions (AOs) A and B  build-up
of electron density between the nuclei
(increase of  and 2). R A B R A B +

48.  Covalent bonding – due to sharing of electrons.
The accumulation of electron density between the nuclei leads to increased e-n attraction (electrons interact strongly with both nuclei):
 E(+) < E(A,B)
 + is a bonding MO.
 Covalent bonding – due to sharing of electrons.
+ is cylindrically symmetrical about the internuclear (A-B) axis
 labelled as a  MO: +
side view +
cross section

49.  -B A Properties of 
For large R,  behaves like 2 independent H(1s) AOs.
For small R, there is significant overlap between the AOs.
Destructive interference between atomic
wavefunctions (AOs) A and B  deplete
of electron density between the nuclei
(decrease of  and 2).
-B A B R A 
node

50.  is cylindrically symmetrical about the internuclear (A-B) axis
The depletion of electron density between the nuclei leads to decreased e-n attraction.
 E() > E(A,B)
  is an antibonding MO.
 is cylindrically symmetrical about the internuclear (A-B) axis
 labelled as a * MO (* denotes antibonding character): + 
nodal plane

51. Normalization of the MO Wavefunction
What are the normalization constants (N+ and N)?
+ = N+(A + B)
 = N(A  B)
Normalization condition:
where  = + or 
e.g. for +  
atomic orbitals are normalized:
define overlap integral between orbitals A and B:
normalization constant:

52. Similarly, for  we get:
Since 0 < SAB < 1, this means that N > N+ .
Similar arguments can be used to show that the antibonding orbital () is raised in energy by more than the bonding orbital (+) is lowered in energy when a bond is formed.
Note: if we ignore overlap, then N+ = N = 1/2

53. 6.4 Molecular Orbital Diagrams
Example 1. H2+
Ground state configuration: (+)1
1e in bonding orbital  bound state.
Excited state configuration: ()1
1e in antibonding orbital  unbound state.
Energies defined relative to dissociation
H2+  H + H+
A=H(1sA)
B=H(1sB)
+ = 1
 = 2*
Energy
V(R) Re
-De +  R

54. A=H(1sA) B=H(1sB) + = 1  = 2* Energy Example 2. H2
Ground state configuration: (+)2
2e in bonding orbital  bound state.
H2 has a shorter stronger bond than H2+ (more bonding electrons).
Note: De(H2) < 2De(H2+) due to e-e repulsion.
Re/pm
De/kJ mol-1
H2+
106
255 H2
74.1
430

55. A=He(1sA) B=He(1sB) + = 1  = 2* Energy Example 3. He2
Ground state configuration: (+)2()2
No net covalent bonding (bonding and a-b contributions cancel out).
Only weak dispersion forces hold He atoms together (see Intermolecular Forces lectures).
He2+ has the configuration (+)2()1 and does have net covalent bonding.

56. 6.5 MOs for 2nd Row Diatomic Molecules
Valence AOs = 2s, 2px, 2py, 2pz
Core AOs = 1s (not involved in bonding)
Linear combinations of 2s orbitals:
 = N (A(2s)  B (2s))  1(2s) and 2 *(2s) (as for H2)
Some combinations are not allowed – zero net overlap = “orthogonal orbitals”: + 
A(2s) + B (2px) + 
A(2px) + B (2pz)

57. The 2p orbitals interact to give -type (0 angular nodes with respect to the molecular axis) and -type (1 angular node) MOs, which can be bonding (,) or antibonding (*,*). 3
4* 1
2* 2p
4* 1
2* 3
2pz-2pz  overlap >  overlap
 3-4* splitting > 1-2*.

58. 6.6 Molecular Electronic Configurations
Follow same rules as for atomic electronic configurations (Aufbau principle, Hund’s rule etc.).
Note: the ordering of MOs can vary – e.g. the 3(2p) and 1(2p) MOs are sometimes reversed:
Due to 2s-2p mixing
(hybridization) which
raises 3 and lowers
2*.
As Z, the 2s-2p separation
increases, so s-p mixing
is weaker.
3 > 1 (B2, C2, N2)
3 < 1 (O2, F2)

59. 6.7 Bond Order
The strength of a covalent bond is the net outcome of occupying bonding and antibonding orbitals.
Bond Order b = ½(NB  N*)
NB = number of electrons in bonding MOs
N* = number of electrons in antibonding MOs
Examples NB N* b
H2+ 1
0.5 H2 2
He2 N2 8 3 O2 4 F2 6
F2+ 5
1.5

60. 6.8 Paramagnetic Molecules
Even with even numbers of electrons, certain molecules are paramagnetic (i.e. they have unpaired electron spins).
e.g. O2
Ground state electronic configuration: (1)2 (2*)2 (3)2 (1)4 (2*)2
There are 2 electrons in the antibonding pair of 2* orbitals.
From Hund’s rule – the lowest energy configuration has the most unpaired spins = 2.
The magnetic effects of these 2 electrons do not cancel out. 3
4* 1
2* 2p

61. 6.9 Heteronuclear Diatomic Molecules
Generally AOs of different atoms have different energies – depending on relative electronegativities of the atoms.
The MO closest in energy to an
AO has more character
(greater LCAO coefficient) of
that AO.
 bonding and antibonding orbitals
usually have opposite characters.
e.g. HF
 = 0.19H(1s)+0.98F(2pz)
* = 0.98H(1s)0.19F(2pz)
(1s+2p)
*(1s-2p) HF
non-bonding
F(2px,2py)
AOs.