# Electronic States of Atoms Quantum numbers for electrons

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 (ml) 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 (ms) 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 =|LS|; 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: {L}J 2S+1 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 = , J=L+S= so, the term symbol is 2S 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 = 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 M LMS 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 Hunds 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 Hunds Rules

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