Electronic States of Atoms

2. Electronic States of Atoms Quantum numbers for electronsQuantum numbers for many-electron atoms l: orbital angular momentum quantum L: orbital angular momentum quantum number number (0,1, … n-1 e.g., for 2 e-: L = l1+l2, l1+l2 -1, l1+l2 -2, …,| l1-l2 | where 0=s, 1=p, 2=d, 3=f) 0 = S, 1 = P, 2 = D, 3 = F ml: orbital magnetic quantum number ML: orbital magnetic quantum number (Sml) (l, l-1, …, 0, …, -l ) 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 MS: spin magnetic quantum number (Sms) (+1/2, -1/2) 2S+1 possible values J: total angular quantum number J = L+S, L+S-1, …, | L-S|

3. 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.

4. 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

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

6. 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.

7. Step 2: Draw all possible microstates. Calculate ML and MS for each state.

8. 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

9. 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

12. Hund’s Rules