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THE UNIVERSITY OF BIRMINGHAM. CHM2S1A Introduction to Quantum Mechanics Dr R. L. Johnston. 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.
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OF BIRMINGHAM
CHM2S1AIntroduction to Quantum MechanicsDr R. L. JohnstonII: 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 Manyelectron atoms.
6.1The BornOppenheimer 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.
5.1 The Hydrogen Atom and Hydrogenic Ions
all have: 1 electron (charge –e)
nucleus (charge+Ze)
me
+Ze
mN
Hamiltonian Operator
Kinetic Energy where
Potential Energy where
(electrostatic attraction between electron and nucleus).
r
wavefunction
(spherical harmonic)
radial
wavefunction
(r,,)=R(r).Y(,)
n,,m(r,,)=Rn,(r).Y,m(,)
n,,m(r,,)=Rn,(r).Y,m(,)
1. Principal Quantum Number (n)
(this is the same result as from the Bohr model).
= 0, 1, 2 … (n1)
e.g. n = 1 = 0
n = 2 = 0, 1
n = 3 = 0, 1, 2
3. Orbital Magnetic Quantum Number (m)
m = 0, 1, 2 …
e.g. = 0m = 0
= 1m = 0, 1
Jz = m
angular momentum vector J
has magnitude J = 6and
5 allowed orientations:
Jz = 0, , 2.
3d orbitals
l=2
ml=1
ml=2
ml=2
ml=1
ml=0
n=3
3p orbitals
l=1
ml=1
ml=1
ml=0
3s orbital
l=0
ml=0
shell
subshells
orbitals
principal quantum number n.
() and orbitals (m) have the same energy
– i.e. they are degenerate.
s orbitals have l = 0 and ml= 0
pz
px
py
d x2 y2
dz2
dxy
dxz
dyz
Shapes of Atomic Orbitals1. The orbital (n,,m)
2. The spin state of the electron
– spin angular momentum q. no.s ( =½ for all electrons)
– spin magnetic q. no.ms ( = ½).
Sz = ms = ½
ms = +½ “spin up”
ms = ½ “spin down”
5.6 Properties of Atomic Orbitals
1. Shape – determined by the angular wavefunction Y,m(,)
1s orbital (n = 1, = 0, J = 0)
(spherically symmetric, depends only on r).
2s orbital ns orbital

+


+
2p orbitals (n = 2, = 1, J = 2)
a) Angular Nodes
b) Radial Nodes
2s 1 2p 0
3s 2 3p 1 3d 0
Example: H(1s)
a0
r
r1
r12
Z = +2e
e(2)
r2
5.7 ManyElectron Atoms
2 terms in
3 terms in
S.E.
where
Approximations must be made.
The Orbital Approximation
N = (r1,r2,r3 … rN) =(r1)(r2)(r3)…(rN)
Zeff < Z
(Z = actual charge on nucleus = atomic number)
Zeff = Z
( = shielding constant).
A
B
A
B
+
R
Properties of
wavefunctions (AOs) AandB depletion
of electron density between the nuclei
(decrease of and 2).
the nuclei leads to decreased en attraction:
E() > E(A,B)
is an antibonding MO.
labelled as a MO (or sometimes *, as it is antibonding).
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 …
degenerate
e.g. E(2px) = E(2py) = E(2pz)
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).
H1s1
He1s2
Li 1s22s1
X
1s < 2s < 2p < 3s < 3p < 4s ~ 3d < 4p < 5s < 4d …
Cr[Ar]4s13d5
Cu[Ar]4s13d10
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 subshells of the inner
(lower n) orbitals become approx.
degenerate
e.g. N[He]2s22p3 = [He]2s22px12py12pz1
e.g. C[He]2s22p2 = [He]2s22px12py1
(equivalent to … 2px12pz1 and 2py12pz1).
Parallel Spins Antiparallel Spins
rA
rB
RAB
HA
HB
PE
KE
nuclear K.E.
electron K.E.
en attraction
nn repulsion
6. Bonding in MoleculesExact solution of the Schrödinger Equation is not possible for any molecule – even the simplest molecule H2+.
Full Hamiltonian operator for H2+:
they move much more slowly.
=e.n
rA
rB
R
HA
HB
electron K.E.
en attraction
nn repulsion
(constant)
Example 1: H2+
motion:
r1A
r1B
r12
R
HA
HB
r2A
r2B
e2
Example 2: H2
(atomic no. Z):
electron K.E.
en attraction
nn repulsion
ee repulsion
Re
R
0
De
R1
R2
6.2 Potential Energy Curves
6.3 Molecular Orbital Theory
P(r) =  (r)2d
A
B
A
B
Example: H2+ and H2
+ = N+(A+B) inphase (bonding)
= N(AB) outofphase (antibonding)
+
B
A
A
B
R
Properties of +
wavefunctions (AOs) AandB buildup
of electron density between the nuclei
(increase of and 2).
+
+
side view
cross section
E(+) < E(A,B)
+ is a bonding MO.
Covalent bonding – due to sharing of electrons.
labelled as a MO:
node
B
A
A
B
R
Properties of
wavefunctions (AOs) AandB deplete
of electron density between the nuclei
(decrease of and 2).
+
E() > E(A,B)
is an antibonding MO.
labelled as a * MO (* denotes antibonding character):
+ = N+(A+B)
= N(AB)
where = + or
atomic orbitals are normalized:
define overlap integral between orbitals AandB:
normalization constant:
Energy
A=H(1sA)
B=H(1sB)
+ = 1
V(R)
R
Re
0
+
De
6.4 Molecular Orbital Diagrams
Example 1. H2+
1e in bonding orbital bound state.
1e in antibonding orbital unbound state.
H2+ H + H+
Energy
A=H(1sA)
B=H(1sB)
+ = 1
Example 2. H2
2e in bonding orbital bound state.
Energy
A=He(1sA)
B=He(1sB)
+ = 1
Example 3. He2
No net covalent bonding (bonding and ab contributions cancel out).
Only weak dispersion forces hold He atoms together (see Intermolecular Forces lectures).
+
+
+
A(2s)+B (2px)
A(2px)+B (2pz)
6.5 MOs for 2nd Row Diatomic Molecules
= N(A(2s)B (2s)) 1(2s) and 2 *(2s) (as for H2)
4*
4*
2*
2p
2p
1
3
3
1
2pz2pz overlap > overlap
34* splitting > 12*.
Due to 2s2p mixing
(hybridization) which
raises 3 and lowers
2*.
As Z, the 2s2p separation
increases, so sp mixing
is weaker.
3 > 1(B2, C2, N2)
3 < 1(O2, F2)
NB = number of electrons in bonding MOs
N* = number of electrons in antibonding MOs
2*
2p
2p
1
3
6.8 Paramagnetic Molecules*(1s2p)
(1s+2p)
nonbonding
F(2px,2py)
AOs.
6.9 Heteronuclear Diatomic MoleculesAO has more character
(greater LCAO coefficient) of
that AO.
usually have opposite characters.
= 0.19H(1s)+0.98F(2pz)
* = 0.98H(1s)0.19F(2pz)