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Electronic States of Atoms
Quantum numbers for electrons Quantum numbers for many-electron atoms
l: orbital angular momentum quantum
number (0,1, … n-1
where 0=s, 1=p, 2=d, 3=f)
L: orbital angular momentum quantum number
ml: orbital magnetic quantum number
(l, l-1, …, 0, …, -l )
ML: orbital magnetic quantum number (Sml)
2L+1 possible values
s: electron spin quantum number (1/2)
S: total spin quantum number
S = s1+s2, s1+s2 -1, …,| s1-s2 |
S = 0 singlet, S = 1 doublet, S = 2 triplet
ms: spin magnetic quantum number
(+1/2, -1/2)
MS: spin magnetic quantum number (Sms)
2S+1 possible values
e.g., for 2 e-: L = l1+l2, l1+l2 -1, l1+l2 -2, …,| l1-l2 |
0 = S, 1 = P, 2 = D, 3 = F
J: total angular quantum number
J = L+S, L+S-1, …, | L-S|
Spectroscopic Description of
Atomic Electronic States – Term Symbols
Multiplicity (2S +1) describes the number of possible orientations of total
spin angular momentum where S is the resultant spin quantum
number (1/2 x # unpaired electrons)
Resultant Angular Momentum (L) describes the coupling of the orbital
angular momenta of each electron (add the mL values for each
electron)
Total Angular Momentum (J) combines orbital angular momentum and
intrinsic angular momentum (i.e., spin).
To Assign J Value:
if less than half of the subshell is occupied, take the minimum value J
=|L−S|;
if more than half-filled, take the maximum value J = L + S;
if the subshell is half-filled, L = 0 and then J = S.
Spectroscopic Description of
Ground Electronic States – Term Symbols
Term Symbol Form:
2S+1{L}
J
2S+1 – multiplicity
L – resultant angular momentum quantum number
J – total angular momentum quantum number
Ground state has maximal S and L values.
Example: Ground State of Sodium – 1s22s22p63s1
Consider only the one valence electron (3s1)
L = l = 0,
S = s = ½,
J=L+S=½
so, the term symbol is 2S½
Are you getting the concept?
Write the ground state term symbol for fluorine.
Spectroscopic Description of
All Possible Electronic States – Term Symbols
C – 1s22s22p2
Step 1:Consider two valence p electrons
1st 2p electron has n = 2, l = 1, ml = 0, ±1, ms = ±½ → 6 possible sets of quantum
numbers
2nd 2p electron has 5 possible sets of quantum numbers (Pauli Exclusion
Principle)
For both electrons, (6x5)/2 = 15 possible assignments since the electrons are
indistinguishable
Step 2: Draw all possible microstates.
Calculate ML and MS for each state.
Step 2: Draw all possible
microstates. Calculate ML
and MS for each state.
Spectroscopic Description of
All Possible Electronic States – Term Symbols
C – 1s22s22p2
Step 3: Count the number of microstates for each ML—MS possible combination
Step 4: Extract smaller tables representing each possible term
Spectroscopic Description of
All Possible Electronic States – Term Symbols
C – 1s22s22p2
Step 5: Use Hund’s Rules to determine the relative energies of all possible states.
1. The highest multiplicity term within a configuration is of lowest energy.
2. For terms of the same multiplicity, the highest L value has the lowest
energy (D < P < S).
3. For subshells that are less than half-filled, the minimum J-value state is of
lower energy than higher J-value states.
4. For subshells that are more than half-filled, the state of maximum J-value
is the lowest energy.
Based on these rules, the ground electronic configuration for carbon has the
following energy order: 3P0 < 3P1 < 3P2 < 1D2 < 1S0
Hund’s Rules