Electronic States of Atoms Quantum numbers for electronsQuantum numbers for many-electron atoms l: orbital angular momentum quantumL: orbital angular momentum quantum number number (0,1, … n-1 e.g., for 2 e-: L = l 1 +l 2, l 1 +l 2 -1, l 1 +l 2 -2, …,| l 1 -l 2 | number (0,1, … n-1 e.g., for 2 e-: L = l 1 +l 2, l 1 +l 2 -1, l 1 +l 2 -2, …,| l 1 -l 2 | where 0=s, 1=p, 2=d, 3=f) 0 = S, 1 = P, 2 = D, 3 = F where 0=s, 1=p, 2=d, 3=f) 0 = S, 1 = P, 2 = D, 3 = F m l : orbital magnetic quantum numberM L : orbital magnetic quantum number (  m l ) (l, l-1, …, 0, …, -l ) 2L+1 possible values (l, l-1, …, 0, …, -l ) 2L+1 possible values s: electron spin quantum number (1/2) S: total spin quantum number S = s 1 +s 2, s 1 +s 2 -1, …,| s 1 -s 2 | S = s 1 +s 2, s 1 +s 2 -1, …,| s 1 -s 2 | S = 0 singlet, S = 1 doublet, S = 2 triplet S = 0 singlet, S = 1 doublet, S = 2 triplet m s : spin magnetic quantum numberM S : spin magnetic quantum number (  m s ) (+1/2, -1/2) 2S+1 possible values (+1/2, -1/2) 2S+1 possible values J: total angular quantum number J = L+S, L+S-1, …, | L-S| 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 m L values for each electron) Total Angular Momentum (J) combines orbital angular momentum and intrinsic angular momentum (i.e., spin). To Assign J Value: 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 – 1s 2 2s 2 2p 6 3s 1 Consider only the one valence electron (3s 1 ) L = l = 0, S = s = ½, J = L + S = ½ so, the term symbol is 2 S ½

Are you getting the concept? Write the ground state term symbol for fluorine.

C – 1s 2 2s 2 2p 2 Step 1:Consider two valence p electrons 1 st 2p electron has n = 2, l = 1, m l = 0, ±1, m s = ±½ → 6 possible sets of quantum numbers 2 nd 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 Spectroscopic Description of All Possible Electronic States – Term Symbols Step 2: Draw all possible microstates. Calculate M L and M S for each state.

C – 1s 2 2s 2 2p 2 Step 3: Count the number of microstates for each M L —M S possible combination Spectroscopic Description of All Possible Electronic States – Term Symbols Step 4: Extract smaller tables representing each possible term

C – 1s 2 2s 2 2p 2 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: 3 P 0 < 3 P 1 < 3 P 2 < 1 D 2 < 1 S 0 Spectroscopic Description of All Possible Electronic States – Term Symbols

Hund’s Rules