Electronic Structure and Angular Momentum of Transition ...

Electronic Structure and

Angular Momentum of

Transition Metal Complexes

? K. S. Suslick, 2013

Angular Momentum vs. Number of Electron Spins

To understand the energy of paramagnetic systems (i.e., ones with unpaired electrons), we must describe them in terms of their angular momentum.

Total Angular Momentum: J = Ms + ML

Total Multiplicity: (2S+1)(2L+1) Each "microstate" has the same

L and S, but different J.

S = 0 Singlet S = 1/2 Doublet S = 1 Triplet S = 3/2 Quartet S = 2 Quintet

L = 0

L = 1

L = 2

L = 3 L = 4

"S term" singly orb. deg.

"P term" triply

"D term" pentuply

"F term" "G term"

? K. S. Suslick, 2013

1

Spin

For the Total Spin of an atom or molecule the rules apply: 1. Doubly occupied orbitals do NOT contribute to the total Spin 2. Singly occupied orbitals can be occupied with either spin-up or spin-down e3. Unpaired e- can be coupled parallel or antiparallel, giving a total spin S 4. For a state with total spin S there are 2S+1 "components"

with M = S,S-1,...,-S. Hence terms singlet, doublet, triplet, ... 5. The MS quantum number is always the sum of all individual ms QNs.

? K. S. Suslick, 2013

Spin Names

"Russell-Saunders" Term Symbols

for atoms; Irr Rep Mulliken for molecules

Examples for dn configurations:

? K. S. Suslick, 2013

doublet sextet triplet

Atoms

Molecules

2H

6A

3l=1 or 3

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L?S Coupling

L-S Coupling = Russell-Saunders Coupling

If coupling of the spin angular momentum and orbital angular momentum is relatively weak (and it usually is), then

L and S remain "good" Quantum Numbers and can be treated independently of each other.

Each electronic state has its own term symbol

spin multiplicity

L 2S+1

L = 0 L = 1 L = 2 L = 3 L = 4

"S term" singly orb. deg. "P term" triply "D term" pentuplicately "F term" "G term"

(Within each term, there can be several degenerate microstates with different ML and MS.)

? K. S. Suslick, 2013

L?S Coupling

L = 0, 1, 2...total orbital angular momentum ("term")

ML = 0, 1, 2, L components of L (ML = ml for each e).

For example, for L = 1, there are three ML values: 1, 0, -1. (analogous to l = 1 and its three ml values: 1, 0, -1) # of ML states is 2L+1 = orbital degeneracy S = total spin angular momentum

Ms = S, S-1, ....-S components of S (MS = ms).

For example, for S = 1, there are three Ms values, 1, 0, -1.

Each electronic state has its own term symbol

spin multiplicity

L 2S+1

L = 0 L = 1 L = 2 L = 3 L = 4

"S term" singly orb. deg. "P term" triply "D term" pentuplicately "F term" "G term"

(Within each term, there can be several degenerate microstates with different ML and MS.)

? K. S. Suslick, 2013

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Hund's Rules

1. The ground state (GS `term') has the highest spin multiplicity (S). 2. If two or more terms have the same spin multiplicity,

then the GS will have the highest value of L. 3. For subshells less than half-filled (e.g., p2), lowest J is preferred;

for subshells more than half-filled, highest J is preferred.

Of all the states possible from degenerate orbitals, the lowest energy one will have the highest spin multiplicity (i.e., most unpaired spins).

For states with the same spin multiplicity, the highest orbital degeneracy will be lowest in energy.

? K. S. Suslick, 2013

The Problem: Electron-Electron Repulsion. d2

Consider as an example, 2 d electrons, one in z2

z2 x2-y2

eg

z2 x2-y2

eg

t2g xy xz yz

xz + z2 z

y

x

t2g xy xz yz

xy + z2 z

y x

overlapping lobes, large inter-electron repulsion

lobes far apart, small inter-electron repulsion

These two electron configurations differ in energy.

? K. S. Suslick, 2013

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Microstates and Spin Orbit Coupling

For a given L, the allowed values of ML and Ms are called microstates.

# of microstates =

(2No)! (2No ? Ne)! Ne!

2No from spin up vs. down

Where No = degeneracy of orbitals in set of subshell and Ne = number of electrons

e.g., for free atoms/ions, No for d orbitals = 5

? K. S. Suslick, 2013

Microstates and Spin Orbit Coupling

(2No)! (2No ? Ne)! Ne!

2No from spin up vs. down

? K. S. Suslick, 2013

etc. for 25

Pauli X

etc. for 20 more

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