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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 HLike 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,i2, ,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 3D
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,i1,...,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 typeexperiments 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 ee 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 1D 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 halfﬁlled, the state with the smallest value of J
is the most stable. b. If the subshell is more than halfﬁlled, the state with the largest value of J
is the most stable. 8. Spinorbit 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 lS 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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