Introduction CHEM2915 to the Electronic Structure of Atoms and Free Ions
A/Prof Adam Bridgeman
Room: 222 Email: [e-mail] www.chem.usyd.edu.au/~bridge_a/chem2915
Slide 1/21
Where Are We Going?
Week 6: Orbitals and Terms
Russell-Saunders coupling of orbital and spin angular momenta Free-ion terms for p2
Week 7: Terms and ionization energies
Free-ion terms for d2 Ionization energies for 2p and 3d elements
Week 8: Terms and levels
Spin-orbit coupling Total angular momentum
Week 9: Levels and ionization energies
j-j coupling Ionization energies for 6p elements
Slide 2/21
Revision Atomic Orbitals
For any 1 e- atom or ion, the Schrdinger equation can be solved The solutions are atomic orbitals and are characterized by n, l and ml quantum numbers
Hy = Ey
E is energy of the orbital y H is the Hamiltonian - describing the forces operating: Kinetic energy due to motion Potential energy due attraction to nucleus Total Hydrogen-like Hamiltonian
mv2 Ze - r HH-like
Slide 3/21
1
Atomic Orbitals - Quantum Numbers
For any 1-e- atom or ion, the Schrdinger equation can be solved The solutions are atomic orbitals and are characterized by n, l and ml quantum numbers Principal quantum number, n = 1, 2, 3, 4, 5, 6, Orbital quantum number, l = n-1, n-2, n-3, 0 = number of nodal planes Magnetic quantum number, ml = l, l -1, l 2, , -l
l = 0: s
l = 1: p
l = 2: d
Slide 4/21
Orbital Quantum Number
Orbital quantum number, l = n-1, n-2, n-3, 0 = number of nodal planes Magnetic quantum number, ml = l, l -1, l 2, , -l = orientation of orbital e.g. l = 2 gives ml = 2, 1, 0, -1, -2: so 2l + 1 = five d-orbitals
y z z
x
x
y
y
z
x x
y
Slide 5/21
Magnetic Quantum Number
Orbital quantum number, l = n-1, n-2, n-3, 0 = number of nodal planes
related to magnitude of orbital angular momentum
Magnetic quantum number, ml = l, l -1, l 2, , -l = orientation of orbital
related to direction of orbital angular momentum ml = 2 ml = 1 ml = 0 ml = -1 ml = -2
Slide 6/21
l=2
2
Spin Quantum Number
All electrons have spin quantum number, s = + Magnetic spin quantum number, ms = s, s -1 = + or : 2s+1 = 2 values
Slide 7/21
Many Electron Atoms
For any 2-e- atom or ion, the Schrdinger equation cannot be solved The H-like approach is taken for every electron i
HH-like = S mvi2 +
i
Treatment leads to configurations for example: He 1s2, C 1s2 2s2 2p2 Neglects interaction between electrons e- / e- repulsion is of the same order of magnitude as HH-like
Si
Ze ri
Sij
e2 rij
Slide 8/21
Many Electron Atoms p1 Configuration
A configuration like p1 represents 6 electron arrangements with the same energy there are three p-orbitals to choose from as l =1 electron may have up or down spin
ml
1 1 0
ms
+ - + - + -
microstate
+ (1)
1
0 ml
-1
0 -1 -1
Slide 9/21
3
Many Electron Atoms p2 Configuration
A configuration like p2 represents even more electron arrangements Because of e- / e- repulsion, they do not all have the same energy:
electrons with parallel spins repel one another less than electrons with opposite spins electrons orbiting in the same direction repel one another less than electrons with orbiting in opposite directions lower in energy than
lower in energy than
1
0 ml
-1
1
0 ml
-1
Slide 10/21
Many Electron Atoms L
For a p2 configuration, both electrons have l = 1 but may have ml = 1, 0, -1 L is the total orbital angular momentum
Lmax = l1 + l2 = 2 Lmin = l1 - l2 = 0 L = l1 + l2, l1 + l2 1, l1 l2 = 2, 1 and 0 For each L, ML = L, L-1, -L
ml1 = 1
ml1 = 1
ml2 = 1
L: 0, 1, 2, 3, 4, 5, 6 code: S, P, D, F, G, H, I
ml = -1
Slide 11/21
Many Electron Atoms S
Electrons have s = but may have ms = + or - S is the total spin angular momentum
Smax s1 = + s2 = 1 Smin = s1 - s2 = 0 S = s1 + s2, s1 + s2 1, s1 s2 = 1 and 0 For each S, MS = S, S-1, -S
Slide 12/21
4
Many Electron Atoms p2
L = 2, 1, 0 for each L: ML = L, L -1, L 2, , -L for each L, there are 2L+1 functions S = 1, 0 for each S: MS = S, S-1, S-2, . S for each S, there are 2S+1 functions Wavefunctions for many electron atoms are characterized by L and S and are called terms with symbol:
2S+1
L: 0, 1, 2, 3, 4, 5, 6 code: S, P, D, F, G, H, I
L
singlets: 1D, 1P, 1S triplets: 3D, 3P, 3S
Slide 13/21
Microstates p2
For example, 3D has L = 2 and S = 1 so: ML = 2, 1, 0, -1, -2 and MS = 1, 0, -1 five ML values and three MS values: 5 3 = 15 wavefunctions with the same energy
+/- +/( ml1, ml2 ) ML = ml1 + ml2 MS = ms1 + ms2
Slide 14/21
MS p2 2 1 ML 0 -1 -2
1
(1, 1) (1, 0) (0, 1) (0, 0)
0
(1, 1) (1, 0) (1, 0)
- - -
-1
(1, 1) (0, 1) (0, 1)
- - -
(1, 1) (1, 0) (0, 1)
- - -
- -
(0, 0) (1, -1) (-1, 1)
(1, -1) (-1, 1) (-1, 0) (0, -1) (-1, -1)
(0, 0) (1, -1) (-1, 1) (-1, 0) (0, -1) (-1, 0) (0, -1) (-1, -1) (-1, -1)
(1, -1) (-1, 1) (-1, 0) (0, -1) (-1, -1)
-
-
-
(0, 0)
-
-
5
Pauli Principle and Indistinguishabity
The Pauli principle forbids two electrons having the same set of quantum numbers. Thus for p2 - Microstates such as (1, 1) and (-1, -1) are not allowed Electrons are indistinguishable Microstates such as (1, -1) and BUT Microstates such as For example, for p2 6 ways of placing 1st electron, 5 ways of placing 2nd electron (Pauli) Divide by two because of indistinguishabillty: 6 5 15
(-1, 1) are the same
-
(1, -1) and (1, -1) are different
-
2
Slide 16/21
Pauli forbidden Indistinguishable
MS 1
(1, 1) (1, 0) (0, 1) (0, 0)
p2 2 1 ML 0 -1 -2
0
(1, 1) (1, 0) (1, 0)
- - -
-1
(1, 1) (0, 1) (0, 1)
- - -
(1, 1) (1, 0) (0, 1)
- - -
- -
(0, 0) (1, -1) (-1, 1)
(1, -1) (-1, 1) (-1, 0) (0, -1) (-1, -1)
(0, 0) (1, -1) (-1, 1) (-1, 0) (0, -1) (-1, 0) (0, -1) (-1, -1) (-1, -1)
(1, -1) (-1, 1) (-1, 0) (0, -1) (-1, -1)
-
-
-
(0, 0)
-
-
Working Out Allowed Terms
1. 2. 3. Pick highest available ML: there is a term with L equal to this ML 4. a. b. c. d. e. Highest ML = 2 L = 2 D t...