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Lecture 4 Electronic Microstates & Term Symbols Suggested reading: Shriver and Atkins, Chapter 20.3 or Douglas, g , 1.4-1.5 Recap from last class: Quantum Numbers F Four quantum numbers: b ““n”” , “l”, “l” “m “ l”, ” and d ““ms” Or,, equivalently: q y “n” , “l”,, “j”, j , and “mj” Orbital angular momentum: Associated with rotation off electron l cloud around nucleus Spin angular momentum: Total angular momentum: Associated with rotation/spin of electron around it’s axis Vector sum of orbital and spin angular momentum Recap from last class: Quantum Numbers F Four quantum numbers: b ““n”” , “l”, “l” “m “ l”, ” and d ““ms” Or,, equivalently: q y “n” , “l”,, “j”, j , and “mj” Orbital angular momentum: L 1 Lz m Spin angular momentum: Total angular momentum: S ss 1 J j j 1 S z ms J z m j = 0, 1, 2, ….n1 s = 1/2 j = -s,…,+s m = -..., 0, 0 … + ms = ±1/2 mj = -j,…0,…+j j 0 +j Electronic Configurations and Microstates • Electronic configurations tell us the number of electrons in each orbital, but they don’t tell us how the electrons occupy the orbitals. i.e.,, don’t always y want to draw: •“Terms”: the energy levels and configurations of atoms and ions (and molecules) • Useful in understanding ionic and coordination compound spectra General Form of Terms naTb •n: principle quantum number •a = 2S+1 “multiplicity” p y (S is the total spin angular momentum of all electrons) **a number** For example, if two electrons are in two different orbitals we might have antiparallel spins: S=s1+s2 = ½ - ½ =0 Or parallel spins: S=s1+s2 = ½ + ½ = 1 General Form of Terms naTb •n: principle quantum number of valence electrons •a = 2S+1 “multiplicity” p y (S is the total spin angular momentum of all electrons) **a number** •T=L (total orbital angular momentum of all electrons; vector sum of , the orbital momentum of individual electrons) **a letter** l L=0, 1, 2, 3, 4 S, P, D, F, G •b=J (total angular momentum) **a number** Examples of Term Symbols in the literature Free Ions/ Atoms Molecules Sundararajan Dalton Trans Sundararajan, Trans., 2009 2009, 6021-6036 6021 6036 Wang, Nature Materials 10 (2011) Co(PPH3)2Cl2 Used in palladium-catalyzed coupling reactions, 2010 Nobel Prize Hydrogen (1 electron atom) Ground state: 1s n=1; =0 L= = =0 S=s= ½ Only l one term is possible bl (i.e., only one energy level for the one microstate)) 2S ½ (read “doublet S”) J=L+S(vector sum) =0+½ In a magnetic field field, due to the Zeeman effect, effect the 12S1/2 term yields two closely-spaced energy levels (mj=½, -½) Multiplicity terms S= ½ 2S+1=a=2 doublet (electron can be spin up or spin down) S= 1 2S+1=a=3 triplet (three different spin configurations, configurations with spin wavefunction χ) (both electrons spin up) 1 ( ) (electrons are indistinguishable) 2 (both electrons spin down) Multiplicity terms S= ½ 2S+1=a=2 doublet (electron can be spin up or spin down) S= 1 2S+1=a=3 triplet (three different spin configurations, configurations with spin wavefunction χ) S=3/2 / 2S+1=a=4 quartet (four (f d ff different spin configurations) f ) 1 ( ) 3 1 ( ) 3 Hydrogen (1 electron atom) Excited state: 2p1 n=2; =1 L= = 1 S=s= ½ J=L+S = 1+½ or 1-½ = ½ or 3/2 Two terms are possible (i.e., two energy levels, depending on the orientations of the electron spin with respect to the orbital angular momentum of the electron) 2P 1/2 or 2P3/2 In a magnetic field field, due to the Zeeman effect, effect the 22P1/2 term yields two closely-spaced energy levels (mj=½, -½) while the 22P3/2 yields four (mj=3/2, ½, -½, -3/2) Helium (2 electron atom) L 1 2 S s1 s2 G Ground d state: 11s2 n=1; 1=2=0 L= 1+2= 0 S=s1+s2= ½-½ (Pauli) = 0 J L S( t sum)=0 J=L+S(vector )0 1S 0 Helium (2 electron atom) L 1 2 S s1 s2 G Ground d state: 11s12p 2 1 1=0; 2=1 L= 1+2= 1 S=s1+s2= ½-½ or ½+½ = 0 or 1 J=L+S(vector sum)=0 1P 0 3P 0 “singlet P” “triplet p P” Carbon (6 electron atom) L 1 2 3 4 5 6 S s1 s2 s3 s4 s5 s6 G Ground d state: 11s22s 2 22p 2 2 1=0; 2=0; 3=0; 4=0; 5=1; 6=1 L= 2, 1, 0 D, P, S S= 0 or 1 2S+1 = 1, 3 J L S( t sum)=3, J=L+S(vector ) 3 2, 2 1 The electrons in the 1s and 2s states will be spin paired. The two electrons in p can be either in px, py, or pz orbitals, and either paired or unpaired: m=-1 m=0 m=+1 Microstates Microstates: the different ways in which electrons can occupy certain orbitals Grouping together the microstates that have the same energy when electron-electron repulsions are taken ( , the into account,, yyields the terms (i.e., spectroscopically distinguishable energy levels) Number of microstates: 2 N 0! (2 N 0 N e )! N e ! No=# degenerate orbitals (i.e., three degenerate p orbitals) Ne=# electrons Carbon, continued: [He]2p2 2 N 0! 2(3!) 15 (2 N 0 N e )! N e ! (2 3 2)!2! No=# degenerate orbitals for p=3 Ne=# # electrons in p p=22 What are the microstates, microstates and which has the lowest energy? Carbon, continued: [He]2p2 ms=+1/2 ml +1 0 ms=-1/2 ‐1 +1 0 ‐1 ML=ml MS=ms 1 1 1 2 0 1 3 ‐1 1 4 1 ‐1 5 0 ‐1 6 ‐1 ‐1 7 2 0 8 1 0 9 0 0 10 1 0 11 0 0 12 ‐1 0 13 0 0 14 ‐1 0 15 ‐2 0 Carbon, continued: [He]2p2 MS=m s ML=ml +1 0 ‐1 ‐2 0 1 0 ‐1 1 2 1 0 1 3 1 +1 1 2 1 +2 2 0 1 0 Note: the array is symmetric about the lines through ML=0 and MS=0, providing a check of the tabulation ML=22, 1, 1 0 0, -11, - 2 L L=22 a D term The values in the array occur only for MS=0 S=0 **a 1D term!** Subtracting out the 1D term: MS=m s +1 0 ‐1 ‐1 1 1 1 0 1 2 1 +1 1 1 1 ‐2 ML=ml +2 2 ML=1, 0, -1 L=1 a P term The values in the array occur MS=1, 0, -1 S=1 term!** **aa 3P term! Subtracting out the 3P term: MS=m s +1 0 ‐1 ‐1 0 0 0 0 0 1 0 +1 0 0 0 ‐2 ML=ml +2 2 ML=0 L=0 a S term MS=0 S=0 term!** **aa 1S term! The energies of the terms 1 Hund’s 1. Hund s rule: For a given configuration, configuration the term with the greatest multiplicity lies lowest in energy the triplet term of a configuration (if one is permitted) will have a lower energy than a singlet term p y, the term with 2. L_Rule: For a term of a ggiven multiplicity, the greatest value of L lies lowest in energy if L is high, the electrons can effectively avoid each other 3. J-Rule: if subshell is less than half filled, lowest J is lowest energy. If greater than half filled, highest J is lowest energy q to half-filled,, onlyy one J possible p If equal Energy Diagram for Carbon Eneergy Ground state of Carbon: 3P0 (“triplet P0”) Beyond Russell-Saunders coupling For light atoms and the 3d series (Z30), (Z30) the energies of the microstates are determined first by the electron spin (S), then their orbital angular momentum (L). i.e, as before, total momentum J is determined by g momenta,, then the summingg first the orbital angular spins, and then combining both: Russel-Saunders coupling For heavier atoms, must consider spin-orbit coupling: “jj-coupling”. l For each h electron, l find f d j=+s, then h sum j’s of each electron to find total J of atom/ion For this class, knowledge of Russell-Saunders coupling is sufficient