Review_Exam_2_2011

Review_Exam_2_2011 - Chemistry 341 Molecular Structure...

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Unformatted text preview: Chemistry 341 Molecular Structure, Chemical Bonding and Dynamics Fall 201 1 EXAM 2 REVIEW SHEET [Exam 2: Friday October 28, 11:10 AM—12:00 PM, Packard Lab 101] I. ELECTRONIC STRUCTURE OF ATOMS H-Like Atomic Species 1. Coulombic interaction (— 6) (~ e) W) = — (4 mg) r 2. Bohr radius 712 (47:80) a0 = 2 me e 3. Hartree energy 2 E = =27.21 eV h (47%;) a0 4. Energy z2 En : ‘ 2 n2 Eh 5. Atomic orbital ' whim = R7110”) Ylm(6’,¢) n= 1, 2, Z =0,1,2, ...,n—1 m=0,i1,i-2, ,ifl o labels 0 Atomic orbitals are normalized and orthogonal l i=j iWindT:{0 iij} 0 Radial behavior Dn,(r) : pnl(r) : r2 11,3,(1’) is the radial distribution function r occurs Where Dn,(r) or pnl (r) is a maximum mp < r >= [:07' Dn,(r) dr = $07” 19,110) d’” r probability between r1 and r2 = P(r1 —> r2) = IganAr) dr = ffpnxr) dr 0 Volume element in spherical polar coordinates in 3-D d1 = r2 sin6 dr d6 d¢ o Angular functions We can combine degenerate imaginary functions to derive real functions, e. g. e”5 + e‘i¢ 0c cos¢ em - e‘il’ 0c sin¢ Line spectra of atoms transition ni ——>nf where ni < 11f for absorption and nf < ni for emission ~_l_Ri i V—xi_ n; 11,.2 R=iR m Rm = 109,737.31534 cm—1 0 Series in emission spectrum of H Pfund Ionization energies of H—like atomic species IE = Elmo — En:I Z2 22 IE 2 Eh = 7Eh Magnetic dipole due to orbital motion of an electron _. e -> = - — L #1 2 me #2 = _/’tB m With _fl #B*2me Em=*,LtZB=,uBBm (m=0,i-1,...,il) origin of the normal Zeeman effect 9. Magnetic dipole due to spin of an electron l/ 2 "spin up” S : h : Z ms (ms {— 1/ 2 " spin down‘} Bl: {s(s+1)}1/2 h (S=1/2) The addition of a 4th quantum number, spin, explained: (1) the fine structure of spectral lines, (2) Strem—Gerlach type-experiments where a beam of atoms is split in an inhomogeneous magnetic field, and (3) the anomalous Zeeman effect. _, gee _. = — S #5 mg #2 : _ge #8 ms Eint = ge ,uB Bms (ms 2 +1/2,— 1/2) origin of the anomalous Zeeman effect He Atom 1. Hamiltonian operator A 712 Z 2 1 1 2 H: _—{v; +V§}— e —+— +6— 2111 (47mg) 7‘1 72 (47r30)r12 2. Neglect e—e repulsion in H L1’ = 15(1) 1s(2) E = —108.8 eV (Eexpt = —79.0 eW therefore must include e-e repulsion in H 3. Variational method 8: I (pl/guess ‘H Wguess dz- 2 Egd(frue) i.e. energy 8 is an upper bound to the true ground state energy 0 Particle in 1-D box as an example with l/Igum = N x (a— x) 4. Now try ‘I’ guess = 1s(l) 1s(2) with 5. Now try WW = 1s’(1)1s'(2) with Z, 3 112 _5: 1.5" = )3 J 8 a“ 7mg 27 8(2) = {(2)2 — g2} Eh Z’ — 2 3 best _ g(z,;es,) = —77.46eV Pauli Exclusion Principle 0 Formal statement For electrons with spin s=1/2 the total wavefimction is antisymmetric with respect to interchange of both spatial and spin coordinates. 0 Working statement No two electrons in an atom may have the same 4 quantum numbers (71,1, In], my) . Ground state of He: (Is)2 1 “Pa = M) M) 724041) 15(2) — fl(1) 042)} 1E0 = 2E” +2J1 s,ls 2 with = ma) 1s(2) 15(1) 1s(2) d 1'1 d 1'2 4 71' so >132 Excited state of He: (ls) (25) “P; = :154141) 2s(2) + 2s(1) 15(2)} VIE—{041) r5(2) — [1(1) {1(2)} (singlet state / Ms: 0) 1E* = E13 + E25 + Jls,2s + K1333 with = J J1s(1)2s(2) $149242)de and KM = I [15(1) 2s(2) (e— 2s(1) 15(2)dr1 drz 4 nso)r12 * 3‘? = %{ls(l) 2s(2) — 2s(1)1s(2)}a(1)a(2) (triplet state / Ms: +1) 3%,; z %{1S(1)25(2)— 2s(l)1s(2)}:/1§{a(l)fl(2)+ [3(1) 042)} (triplet state / Ms: 0) 3L11;: fihsfl) 25(2) — 25(1)1s(2)} fl(1) m2) (triplet state / Ms: —1) 3E* = Els + Ezs + J1 ‘ Kls,2s 5,23 0 1E * — 3E* = 2Klfls and 1E * — 3E* = 0.8 eV experimentally i.e. excited state triplet state is lower in energy than the excited state singlet state made from the same electron configuration (ls) (2S). Multielectron Atom 1. Ordering of energy levels Because of the penetration effect [e.g., Zefl(28) > Zefi(2p) and as a result E2S < Ezp, etc.], the subshells corresponding to a particular value of n, i.e. a particular shell, will no longer be equal in energy as they are for a H—like atomic species. Autbau or building—up principle Here we start filling the available energy levels starting at the lowest energy level and putting 2 electrons per orbital with each electron having opposite spin so as not to violate the working statement of the Pauli Exclusion Principle. Angular Momentum [OrbitaL Spin and Total] 132 w: {1;(L+1)}h2 w L2 1/1: ML h w (ML = +L,...,—L) s2 1/}: {S(S+1)}h2 w S2 W: MS h V : +S>---2‘S) 32 1/1:{J(J+1)}h2 1,” J2 w: MJh 1/1 (214]: +J,...,—J) Atomic Term Symbols 23+1L J name of state 5. Some examples: He: (ls)2 130 He: (ls) (2s) 150, 3S1 Li: [He] (3s) 231,2 B: [He] (2s)2 (2p) 3P3,2 and 3P1,2 ...by Hund’s empirical rule 3a 3 1,2 is the ground state 6. Tables for possible atomic term states for nonequivalent and equivalent electrons [See page on this at end of Review Material] 7. Hund’s Empirical Rules 1. The term arising from the ground configuration with the maximum multiplicity (28H) lies lowest in energy. 2.. For levels with the same multiplicity, the one with the maximum value of L lies lowest in energy. 3. For levels with same S and L, the one With the lowest energy depends on the extent to which the sublevel is filled. a. If the subshell is less than half-filled, the state with the smallest value of J is the most stable. b. If the subshell is more than half-filled, the state with the largest value of J is the most stable. 8. Spin-orbit Coupling 1 En = ghcA{J(J+1)— L(L+1)— S(S+1)} origin of fine structure of spectral lines, e. g. the Na doublet spectrum II. ELECTRONIC STRUCTURE OF MOLECULES / CHEMICAL BONDING 1. H; o For fixed nuclei 19 WW e2 {1+1}+ e2 27” e (471'80) 7C4 7‘3 (47:80)R - Born—Oppenheimer approximation 0 secular equations 0 secular determinant H AA + H AB . . . . . 0 5'1 2 —1:S-;-~ shows minimum in energy curve, 1.e. a bondlng state H AA — H AB . . . . . . 82 : —-~— shows no mimmum in energy curve, 1.6. an antlbondmg state l-S 1 1/2 . . 0 {/11 = ( 2 + 2S] {ISA + 153} bondlng molecular orbital 1 1/2 I . - WZ = {13A — lsB} antibondmg molecular orb1ta1 0 other excited states of H; H2 0 Simple molecular orbital gives equal weighting to covalent and ionic structures Correlation Diagram for A2 Diatomic Molecules [See correlation diagram Figure 11.14 ( SAB) given at end of Review Material] The correlation diagram can be used to arrive at electron configurations for homonuclear diatomic molecules. The bond order is defined as B. 0. = (N — N * ) / 2 where N is the number of bonding electrons and N * is the number of antibonding electrons. The general observations are: (1) as B.O.( T ), Re( 1 ), (2) as B.O.( T ), Dc( T ), and (3) magnetism is related to whether the molecule has any unpaired electrons if all electrons are paired the molecule is diamagnetic (slightly repelled when placed in a magnetic field) and if there are unpaired electrons the molecule is paramagnetic (attracted into the poles of a magnetic field) Molecular Term Symbols for A2 Diatomic Molecules 2S+l +0r— A goru TERM STATES Two Nonequivalent Electrons For two nonequivalent electrons (i.e. they come form subgroups with different 77 and E values), the possible term states are listed in the following table. Table —Terms of two nonequivalent electrons in s, p, d, and f orbitals. Each term occurs both as a singlet and as a triplet s p d f s S P D F p SPD PDF DFG d SPDFG PDFGH f SPDFGHI Equivalent Electrons ‘ For equivalent electrons (i.e. they come fiom the same subgroup and have the same 12 and K values), the Pauli exclusion principle restricts the term states to those listed in the following table. Table —Terms of equivalent electrons. The number of times each term occurs is shown in brackets 81; 28' p1; 2P d1; 2D 6.2; 18 P2; 1D, 3P d2; 15’, 1D, 1G, 313, 3177 p3; 2P, 2D, 45‘ d3; 2P, 2D(2)’ 217’, 2G, 2H, 4P, 4F p4;ls,1D, 3P d4;18(2),1D(2),1F,10(2),1I, 819(2), 3D, 81W2), 39, 3H, 51) ‘ 395; 2P 01“; 28, 2P, 213(3), 217(2), 20(2), 2H, 2r, 4P, 4D, 417’, 4671.68 P6; 16' ’ . Gerard W. King, Spectroscopy and Molecular Structure, Holt, Rhinehart and Winston, Inc., New York, 1964, p. 73. 15 150E United 2 Liz 02 02 N62 Separated atom R_> N2 F2 atoms Figure 1L14 Correlation diagram for homonuclear diatomic molecules The energy scale is schematic, but dashed lines show the correct sequence for filling of orbitals for the molecules indicated. (From R. S. Berry, S. A. Rice, and J. Ross, Physical Chemistry. Oxford University Press, 2000.) , ...
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Review_Exam_2_2011 - Chemistry 341 Molecular Structure...

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