Electronic States of AtomsElectronic States of AtomsQuantum numbers for electronsQuantum numbers for electrons Quantum numbers for many-electron atomsQuantum numbers for many-electron atoms

ll: orbital angular momentum quantum: orbital angular momentum quantum LL: orbital angular momentum quantum number: orbital angular momentum quantum number number (0,1, … n-1 number (0,1, … n-1 e.g., for 2 e-: e.g., for 2 e-: LL = = ll11++ll22, , ll11++ll22 -1, -1, ll11++ll22 -2, …,| -2, …,| ll11--ll22 | | where 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 0 = S, 1 = P, 2 = D, 3 = F

mmll: orbital magnetic quantum number: orbital magnetic quantum number MMLL: orbital magnetic quantum number (: orbital magnetic quantum number (mmll)) ((ll, , ll-1, …, 0, …, --1, …, 0, …, -l l )) 2 2LL+1 possible values+1 possible values

ss: electron spin quantum number (1/2) : electron spin quantum number (1/2) SS: total spin quantum number: total spin quantum number S = sS = s11++ss22, , ss11++ss2 2 -1, …,| -1, …,| ss11--ss2 2 ||

S = 0 singlet, S = 1 doublet, S = 2 tripletS = 0 singlet, S = 1 doublet, S = 2 triplet

mmss: spin magnetic quantum number: spin magnetic quantum number MMSS: spin magnetic quantum number (: spin magnetic quantum number (mmss)) (+1/2, -1/2)(+1/2, -1/2) 2 2SS+1 possible values+1 possible values

JJ: total angular quantum number: total angular quantum number J = L+S, L+S-J = L+S, L+S-11, …, , …, | | L-SL-S||

Spectroscopic Description of Spectroscopic Description of Atomic Electronic States – Term SymbolsAtomic Electronic States – Term Symbols

Multiplicity (2S +1) describes the number of possible orientations of total Multiplicity (2S +1) describes the number of possible orientations of total spin angular momentum where S is the resultant spin quantum spin angular momentum where S is the resultant spin quantum number (1/2 x # unpaired electrons)number (1/2 x # unpaired electrons)

Resultant Angular Momentum (L) describes the coupling of the orbital Resultant Angular Momentum (L) describes the coupling of the orbital angular momenta of each electron (add the mangular momenta of each electron (add the mLL values for each values for each electron)electron)

Total Angular Momentum (J) combines orbital angular momentum and Total Angular Momentum (J) combines orbital angular momentum and intrinsic angular momentum (i.e., spin).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 if less than half of the subshell is occupied, take the minimum value J

= | L − S | ; = | L − S | ; if more than half-filled, take the maximum 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. if the subshell is half-filled, L = 0 and then J = S.

Spectroscopic Description of Spectroscopic Description of Ground Electronic States – Term SymbolsGround Electronic States – Term Symbols

Term Symbol Form: Term Symbol Form: 2S+12S+1{L}{L}JJ

2S+1 – multiplicity2S+1 – multiplicityL – resultant angular momentum quantum numberL – resultant angular momentum quantum numberJ – total angular momentum quantum numberJ – total angular momentum quantum number

Ground state has maximal S and L values. Ground state has maximal S and L values.

Example: Ground State of Sodium – 1sExample: Ground State of Sodium – 1s222s2s222p2p663s3s11

Consider only the one valence electron (3sConsider only the one valence electron (3s11))L = l = 0, L = l = 0, S = s = ½, S = s = ½, J = L + S = ½J = L + S = ½so, the term symbol is so, the term symbol is 22SS½½

Are you getting the concept?Are you getting the concept?

Write the ground state term symbol for fluorine.Write the ground state term symbol for fluorine.

C – 1sC – 1s222s2s222p2p22

Step 1:Consider two valence p electronsStep 1:Consider two valence p electrons11stst 2p electron has n = 2, l = 1, m 2p electron has n = 2, l = 1, mll = 0, ±1, m = 0, ±1, mss = ±½ → 6 possible sets of quantum = ±½ → 6 possible sets of quantum

numbersnumbers22ndnd 2p electron has 5 possible sets of quantum numbers (Pauli Exclusion 2p electron has 5 possible sets of quantum numbers (Pauli Exclusion

Principle)Principle)For both electrons, (6x5)/2 = 15 possible assignments since the electrons are For both electrons, (6x5)/2 = 15 possible assignments since the electrons are

indistinguishableindistinguishable

Spectroscopic Description of Spectroscopic Description of All Possible Electronic States – Term SymbolsAll Possible Electronic States – Term Symbols

Step 2: Draw all possible microstates. Step 2: Draw all possible microstates. Calculate MCalculate ML L and Mand MSS for each state. for each state.

Step 2: Draw all possible Step 2: Draw all possible microstates. Calculate Mmicrostates. Calculate ML L

and Mand MSS for each state. for each state.

C – 1sC – 1s222s2s222p2p22

Step 3: Count the number of microstates for each MStep 3: Count the number of microstates for each MLL—M—MSS possible combination possible combination

Spectroscopic Description of Spectroscopic Description of All Possible Electronic States – Term SymbolsAll Possible Electronic States – Term Symbols

Step 4: Extract smaller tables representing each possible termStep 4: Extract smaller tables representing each possible term

C – 1sC – 1s222s2s222p2p22

Step 5: Use Hund’s Rules to determine the relative energies of all possible states.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.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 2. For terms of the same multiplicity, the highest L value has the lowest energy (D < P < S).energy (D < P < S).3. For subshells that are less than half-filled, the minimum J-value state is of 3. For subshells that are less than half-filled, the minimum J-value state is of lower energy than higher J-value states.lower energy than higher J-value states.4. For subshells that are more than half-filled, the state of maximum J-value 4. For subshells that are more than half-filled, the state of maximum J-value is the lowest energy.is the lowest energy.

Based on these rules, the ground electronic configuration for carbon has the Based on these rules, the ground electronic configuration for carbon has the following energy order: following energy order: 33PP00 < < 33PP11 < < 33PP22 < < 11DD22 < < 11SS00

Spectroscopic Description of Spectroscopic Description of All Possible Electronic States – Term SymbolsAll Possible Electronic States – Term Symbols

Hund’s Rules