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ROBERT FIELD: Let's get started.

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00:00:26,210 --> 00:00:34,010
William Klemperer was my thesis
advisor, and he died yesterday.

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00:00:34,010 --> 00:00:37,850
It also happens that the
subject of this lecture

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00:00:37,850 --> 00:00:42,170
is really the core of
what I got from him.

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00:00:42,170 --> 00:00:46,870
He showed me how to
evaluate matrix elements

13
00:00:46,870 --> 00:00:51,130
of many-electron operators,
which is the key to being able

14
00:00:51,130 --> 00:00:53,530
to interpret--

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00:00:53,530 --> 00:00:57,820
not just tabulate--
electronic properties

16
00:00:57,820 --> 00:01:00,710
of atoms and molecules.

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00:01:00,710 --> 00:01:06,980
Our goal is to be able
to reduce the complexity

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00:01:06,980 --> 00:01:10,280
of electronic structure,
which is really complicated.

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00:01:10,280 --> 00:01:13,270
The electrons interact with
each other really strongly,

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00:01:13,270 --> 00:01:14,660
and there are a lot of them.

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00:01:14,660 --> 00:01:21,020
And it's very hard to
separate the complexity

22
00:01:21,020 --> 00:01:24,650
of the many-body
interactions into things

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00:01:24,650 --> 00:01:28,290
that we can put in our
head and interpret.

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00:01:28,290 --> 00:01:30,930
And the whole goal
of this course

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00:01:30,930 --> 00:01:33,600
is to give you the
tools to interpret

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00:01:33,600 --> 00:01:37,090
complicated phenomena.

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00:01:37,090 --> 00:01:41,290
We have the vibrational problem
as a way of understanding

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00:01:41,290 --> 00:01:45,040
internuclear interactions--

29
00:01:45,040 --> 00:01:47,230
nuclear motions.

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00:01:47,230 --> 00:01:50,770
We have electronic structure
and the hydrogen atom

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00:01:50,770 --> 00:01:54,580
as a way of understanding
what electronic structure is,

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00:01:54,580 --> 00:01:57,580
and to reduce it to,
basically, the things

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00:01:57,580 --> 00:02:01,456
we learn about hydrogen.

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00:02:01,456 --> 00:02:03,080
When we go to molecular
orbital theory,

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00:02:03,080 --> 00:02:05,330
we take what we
know about atoms,

36
00:02:05,330 --> 00:02:11,030
and build a minimally-complex
interpretive picture,

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00:02:11,030 --> 00:02:14,300
which is sort of a
framework for understanding

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00:02:14,300 --> 00:02:17,270
complicated molecular
interactions.

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00:02:17,270 --> 00:02:21,350
So one of the most important
things about understanding

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00:02:21,350 --> 00:02:23,870
electronic structure
is, how do we

41
00:02:23,870 --> 00:02:28,616
deal with many-electron
wave functions?

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00:02:28,616 --> 00:02:34,190
And one of the terrible
problems is that the electrons

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00:02:34,190 --> 00:02:36,410
are indistinguishable.

44
00:02:36,410 --> 00:02:40,970
And so we have to ensure
that the wave functions are

45
00:02:40,970 --> 00:02:43,310
anti-symmetric with
respect to permutation

46
00:02:43,310 --> 00:02:47,150
of every pair of
electrons, not just two.

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00:02:47,150 --> 00:02:52,004
In helium we just dealt with
two, and that wasn't so bad.

48
00:02:52,004 --> 00:02:57,010
But when we deal with
n electrons, what

49
00:02:57,010 --> 00:03:00,910
we are going to discover
is that in order

50
00:03:00,910 --> 00:03:03,520
to anti-symmetrize
the wave function,

51
00:03:03,520 --> 00:03:07,630
we have to write a determinantal
wave function, a determinant

52
00:03:07,630 --> 00:03:09,440
of orbitals.

53
00:03:09,440 --> 00:03:11,590
And when you expand
an n by n determinant,

54
00:03:11,590 --> 00:03:14,080
you get n factorial times.

55
00:03:14,080 --> 00:03:15,790
And when you calculate
matrix elements,

56
00:03:15,790 --> 00:03:20,090
you have n factorial
squared integrals.

57
00:03:20,090 --> 00:03:23,900
So you're not going to be
handling these one at a time,

58
00:03:23,900 --> 00:03:28,130
and looking at them lovingly.

59
00:03:28,130 --> 00:03:31,340
You're going to want to be
able to take these things

60
00:03:31,340 --> 00:03:34,100
and extract what is
the important thing

61
00:03:34,100 --> 00:03:36,320
about the electronic
structure that you're

62
00:03:36,320 --> 00:03:39,150
going to need to know.

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00:03:39,150 --> 00:03:44,970
And as a graduate student,
I was collecting numbers.

64
00:03:44,970 --> 00:03:49,020
I was collecting numbers about
spectroscopic perturbations,

65
00:03:49,020 --> 00:03:51,990
where non-degenerate
perturbation 3 breaks down,

66
00:03:51,990 --> 00:03:53,700
and interesting things happen.

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00:03:53,700 --> 00:03:55,680
But this was something
that nobody in the world

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00:03:55,680 --> 00:03:58,830
was interested in because
it was the breaking

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00:03:58,830 --> 00:04:00,990
of the usual patterns.

70
00:04:00,990 --> 00:04:05,400
And I was convinced that I
had collected some stuff that

71
00:04:05,400 --> 00:04:07,970
told an interesting story.

72
00:04:07,970 --> 00:04:13,370
And I told Klemperer about
this, and he said, well,

73
00:04:13,370 --> 00:04:17,620
have you thought about how
to evaluate these integrals--

74
00:04:17,620 --> 00:04:21,430
these numbers that you are
extracting from the spectrum,

75
00:04:21,430 --> 00:04:25,950
by doing some tricks with the
many-electron wave functions?

76
00:04:25,950 --> 00:04:29,580
And then he showed me, on a
scrap of paper, how to do it.

77
00:04:29,580 --> 00:04:31,120
And I was launched.

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00:04:31,120 --> 00:04:31,750
That was it.

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00:04:31,750 --> 00:04:35,740
That has been the
foundation of my career

80
00:04:35,740 --> 00:04:38,910
for the last 50 years.

81
00:04:38,910 --> 00:04:41,252
And I didn't think that
Klemperer knew that.

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00:04:41,252 --> 00:04:42,960
I didn't think anybody
knew it, because I

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00:04:42,960 --> 00:04:45,090
didn't think it was knowable.

84
00:04:45,090 --> 00:04:47,490
But he just gave it to
me on a silver platter.

85
00:04:47,490 --> 00:04:50,970
And so I'm going
to try to give you

86
00:04:50,970 --> 00:04:54,840
at least the rudiments of
what it is you're up against,

87
00:04:54,840 --> 00:04:57,120
and how you reduce
them to things

88
00:04:57,120 --> 00:05:00,570
that you care about,
that you can think about.

89
00:05:00,570 --> 00:05:02,940
And you can understand
the hydrogen atom

90
00:05:02,940 --> 00:05:05,160
in rather complete detail.

91
00:05:05,160 --> 00:05:08,400
Or at least you can understand
how one observable relates

92
00:05:08,400 --> 00:05:10,690
to another.

93
00:05:10,690 --> 00:05:16,480
And so the relationship
between the effective quantum

94
00:05:16,480 --> 00:05:20,080
number and the ionization
energy of a state

95
00:05:20,080 --> 00:05:24,010
then provides a
hydrogen-atom-based structural

96
00:05:24,010 --> 00:05:26,950
model for everything
you can observe.

97
00:05:26,950 --> 00:05:36,410
Now spectroscopists have
the unfortunate habit

98
00:05:36,410 --> 00:05:38,570
of saying we're
interested in structure.

99
00:05:38,570 --> 00:05:41,030
Structure is static.

100
00:05:41,030 --> 00:05:46,070
Dynamics is magical,
and special, and hard.

101
00:05:46,070 --> 00:05:52,040
But if you understand
structure in a way which is not

102
00:05:52,040 --> 00:05:55,880
the exact eigenstates, not
the exact wave functions,

103
00:05:55,880 --> 00:06:00,200
but something that the molecule
was trying to do and sort of

104
00:06:00,200 --> 00:06:00,920
missed.

105
00:06:00,920 --> 00:06:03,920
And the dynamics is
just what happens

106
00:06:03,920 --> 00:06:07,200
when this preparation
isn't in eigenstate,

107
00:06:07,200 --> 00:06:09,440
which would be boring.

108
00:06:09,440 --> 00:06:12,110
And you get dynamics,
which you can understand,

109
00:06:12,110 --> 00:06:14,180
as opposed to just
saying, I'm going

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00:06:14,180 --> 00:06:17,370
to tabulate the dynamics too.

111
00:06:17,370 --> 00:06:21,230
You don't know anything unless
you have a reductionist picture

112
00:06:21,230 --> 00:06:23,040
of what's going on.

113
00:06:23,040 --> 00:06:28,520
And since the hardest part
of dealing with molecules

114
00:06:28,520 --> 00:06:32,370
is the fact that they
have a lot of electrons,

115
00:06:32,370 --> 00:06:37,520
this is really the core of being
able to do important stuff.

116
00:06:37,520 --> 00:06:44,770
Now it's a horrendously
complicated problem,

117
00:06:44,770 --> 00:06:47,340
and notationally awkward, too.

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00:06:47,340 --> 00:06:51,500
And let me just
try to explain it.

119
00:06:51,500 --> 00:06:54,130
And I'm going to try to do
this without too much reliance

120
00:06:54,130 --> 00:06:57,085
on my notes, because
they're terrible.

121
00:07:01,750 --> 00:07:04,150
We talked about helium.

122
00:07:04,150 --> 00:07:07,390
And helium has two electrons.

123
00:07:07,390 --> 00:07:13,090
And there's this 1 over r12
interaction between electrons,

124
00:07:13,090 --> 00:07:14,530
which looks innocent enough.

125
00:07:14,530 --> 00:07:15,480
You can write it down.

126
00:07:15,480 --> 00:07:18,680
You know, it's
just a few symbols.

127
00:07:18,680 --> 00:07:26,680
And we can call it the
first order perturbation.

128
00:07:26,680 --> 00:07:29,380
But that's really a
lie, because it's as

129
00:07:29,380 --> 00:07:32,870
big as almost everything else.

130
00:07:32,870 --> 00:07:37,790
And so, yeah, we can, in fact,
do a series of approximations.

131
00:07:37,790 --> 00:07:42,370
One is, ignore it, the
non-interacting electron

132
00:07:42,370 --> 00:07:43,730
approximation.

133
00:07:43,730 --> 00:07:46,210
And that's basically
repackaging hydrogen,

134
00:07:46,210 --> 00:07:48,280
and it's not quite enough.

135
00:07:48,280 --> 00:07:53,650
And then we can say, OK, let's
calculate the first order

136
00:07:53,650 --> 00:08:01,030
energy by calculating
expectation values of h12.

137
00:08:01,030 --> 00:08:03,350
So that's E1.

138
00:08:03,350 --> 00:08:08,690
And that's almost enough to give
us a sense of what is going on.

139
00:08:18,130 --> 00:08:29,890
1 over r12, commutate with
any electron, that's not 0.

140
00:08:32,720 --> 00:08:38,620
1 over r12, commutated with
any orbital angular momentum--

141
00:08:38,620 --> 00:08:42,039
any momentum is not 0.

142
00:08:42,039 --> 00:08:53,572
So that means that l and n
are not good quantum numbers.

143
00:08:53,572 --> 00:08:54,780
What's a good quantum number?

144
00:08:54,780 --> 00:08:58,022
What's the definition of
a good quantum number?

145
00:08:58,022 --> 00:09:00,600
Come on, this is an
important question.

146
00:09:00,600 --> 00:09:01,182
Yes.

147
00:09:01,182 --> 00:09:02,890
AUDIENCE: [INAUDIBLE]
count, and then you

148
00:09:02,890 --> 00:09:04,420
put them into some
formula, and then

149
00:09:04,420 --> 00:09:08,240
you can read off eigenvalues.

150
00:09:08,240 --> 00:09:12,810
ROBERT FIELD: That's
maybe 70% of what I want.

151
00:09:12,810 --> 00:09:14,160
You can put it into a formula.

152
00:09:14,160 --> 00:09:17,440
That means it's a
rigorously good thing.

153
00:09:17,440 --> 00:09:19,720
It means it commutes
with the Hamiltonian.

154
00:09:19,720 --> 00:09:22,710
A rigorously good quantum
number corresponds

155
00:09:22,710 --> 00:09:24,990
to a eigenvalue of
an operator that

156
00:09:24,990 --> 00:09:27,000
commutes with the Hamiltonian.

157
00:09:27,000 --> 00:09:33,050
So hydrogen, we rely on n and
l to get almost everything.

158
00:09:33,050 --> 00:09:36,530
But here we find that,
in addition to this

159
00:09:36,530 --> 00:09:43,630
being that small, it destroys
the foundation of our picture.

160
00:09:43,630 --> 00:09:48,610
And so how do we
think that we can

161
00:09:48,610 --> 00:09:57,930
make any sense of many-electron
atoms and molecules?

162
00:09:57,930 --> 00:10:03,700
Well it turns out we can
hide most of the complexity.

163
00:10:03,700 --> 00:10:09,430
And most of the complexity
is just working out the rules

164
00:10:09,430 --> 00:10:12,800
for calculating these
matrix elements.

165
00:10:12,800 --> 00:10:15,770
The matrix elements of
operators that we care about,

166
00:10:15,770 --> 00:10:22,080
like transition moments, spin
orbit, Zeeman effect, things

167
00:10:22,080 --> 00:10:25,520
that correspond
to how we observe

168
00:10:25,520 --> 00:10:28,490
atomic and molecular structure.

169
00:10:28,490 --> 00:10:32,930
And so the main
obstacle to being

170
00:10:32,930 --> 00:10:35,660
able to evaluate
these matrix elements

171
00:10:35,660 --> 00:10:38,577
is the permutation requirement.

172
00:10:44,650 --> 00:10:52,250
And it turns out that there is
a really simple way of dealing

173
00:10:52,250 --> 00:10:57,080
with the requirements
for electron permutation,

174
00:10:57,080 --> 00:11:01,400
and that is to write the wave
function as a determinant

175
00:11:01,400 --> 00:11:02,465
of one-electron orbitals.

176
00:11:05,620 --> 00:11:13,150
Because a determinant has three
really important properties.

177
00:11:13,150 --> 00:11:19,270
One, it changes its sign when
you permute any two columns.

178
00:11:19,270 --> 00:11:24,340
Two, it changes its sign when
you permute any two rows.

179
00:11:24,340 --> 00:11:27,190
And three, if you have two
identical columns or rows,

180
00:11:27,190 --> 00:11:29,560
it's 0.

181
00:11:29,560 --> 00:11:33,720
That's really fantastic.

182
00:11:33,720 --> 00:11:37,460
And that is the Pauli
exclusion principle-- not what

183
00:11:37,460 --> 00:11:40,300
you learned in high school.

184
00:11:40,300 --> 00:11:44,260
What you learned is a
small consequence of that.

185
00:11:47,080 --> 00:11:51,510
So if we can build
anti-symmetric wave functions,

186
00:11:51,510 --> 00:11:55,890
we have aufbau, we can only
put one electron in an orbital.

187
00:11:55,890 --> 00:12:00,810
We have all sorts of stuff, but
it's too complicated to tell

188
00:12:00,810 --> 00:12:07,170
a student in high
school that you can't--

189
00:12:07,170 --> 00:12:10,650
just the question of
indistinguishable electrons

190
00:12:10,650 --> 00:12:14,350
is such a subtle thing
that you can't say,

191
00:12:14,350 --> 00:12:18,010
well, they have to
be anti-symmetric.

192
00:12:18,010 --> 00:12:21,200
But it's easy to say, you can't
put more than one electron

193
00:12:21,200 --> 00:12:23,670
in a spin orbital.

194
00:12:23,670 --> 00:12:25,510
But we don't talk
about spin orbitals.

195
00:12:25,510 --> 00:12:30,350
We say, we can't put more than
two electrons in an orbital,

196
00:12:30,350 --> 00:12:35,580
because we're protecting you
from unnecessary knowledge.

197
00:12:35,580 --> 00:12:37,606
OK, well, I'm not
going to protect you.

198
00:12:37,606 --> 00:12:39,390
[LAUGHTER]

199
00:12:39,390 --> 00:12:50,430
OK, so we know that
the Hamiltonian

200
00:12:50,430 --> 00:12:53,400
has to commute with--

201
00:12:57,880 --> 00:13:01,940
these capital letters mean,
many electrons' angular momenta.

202
00:13:01,940 --> 00:13:05,500
And this is the spin, this is
the projection of the spin.

203
00:13:05,500 --> 00:13:08,470
We know this is true because
the Hamiltonian doesn't

204
00:13:08,470 --> 00:13:11,930
operate on spin.

205
00:13:11,930 --> 00:13:22,770
It's a trivial result, but
it's a very important result.

206
00:13:22,770 --> 00:13:32,980
OK, so we have to worry about
spin, and spin eigenstates,

207
00:13:32,980 --> 00:13:34,930
and other things like that.

208
00:13:34,930 --> 00:13:41,005
OK, so Slater determinants.

209
00:13:52,650 --> 00:13:56,490
J.C. Slater was an MIT
professor in physics.

210
00:13:59,070 --> 00:14:03,480
He invented these
things in 1929.

211
00:14:03,480 --> 00:14:05,170
I have a reference.

212
00:14:05,170 --> 00:14:07,620
I don't know if I've
ever read this paper,

213
00:14:07,620 --> 00:14:12,000
but it's probably beautiful.

214
00:14:21,900 --> 00:14:25,770
So basically what
Slater did is showed,

215
00:14:25,770 --> 00:14:34,190
yeah, you can do the necessary
algebra to deal with any atom,

216
00:14:34,190 --> 00:14:38,980
and to be able to reduce an atom
to a small number of integrals

217
00:14:38,980 --> 00:14:42,000
that you really care about.

218
00:14:42,000 --> 00:14:44,150
And there are two
ways of doing this.

219
00:14:44,150 --> 00:14:47,140
One is the truth, and
one is the fit model.

220
00:14:50,000 --> 00:14:54,860
Now the truth is really
boring, because you

221
00:14:54,860 --> 00:14:58,130
lose all the insights,
and the fit model

222
00:14:58,130 --> 00:15:01,670
gives you the things you have
to think about and understand.

223
00:15:01,670 --> 00:15:05,990
And a fit model also tells you
what are the import the actors.

224
00:15:05,990 --> 00:15:08,480
And maybe they're in
costume, maybe they're not,

225
00:15:08,480 --> 00:15:12,520
but we can deal with them.

226
00:15:12,520 --> 00:15:16,420
But the truth is really
very complicated.

227
00:15:16,420 --> 00:15:26,870
And as I said many times, when
you go from hydrogen to helium,

228
00:15:26,870 --> 00:15:30,360
you can't solve the
Schrodinger equation exactly.

229
00:15:30,360 --> 00:15:34,390
This was perhaps a
little bit of a surprise,

230
00:15:34,390 --> 00:15:36,480
but I think it was only
a surprise in newspapers.

231
00:15:36,480 --> 00:15:38,550
I think physicists
knew immediately,

232
00:15:38,550 --> 00:15:40,320
when you go for a
two-bodied problem

233
00:15:40,320 --> 00:15:42,600
to a three-bodied
problem, there is no way

234
00:15:42,600 --> 00:15:45,830
you can have an exact solution.

235
00:15:45,830 --> 00:15:47,550
And that's the truth.

236
00:15:47,550 --> 00:15:52,340
You can't solve helium or any
more-than-one-electron problem

237
00:15:52,340 --> 00:15:54,250
exactly.

238
00:15:54,250 --> 00:15:58,660
But you can do it
really well, and it just

239
00:15:58,660 --> 00:16:00,820
costs computer time.

240
00:16:00,820 --> 00:16:05,470
And if the computer is doing
the work, you don't really care.

241
00:16:05,470 --> 00:16:08,530
Because once you've told
the computer the rules,

242
00:16:08,530 --> 00:16:09,940
then it's off to the races.

243
00:16:09,940 --> 00:16:14,380
You can go have lives or you can
go have a life, and come back,

244
00:16:14,380 --> 00:16:18,910
and the computer will tell
you whether you made a mistake

245
00:16:18,910 --> 00:16:21,070
and you're getting
a nonsense result,

246
00:16:21,070 --> 00:16:26,690
or that you have
the correct result.

247
00:16:26,690 --> 00:16:32,690
So what we know is this
permutation operator,

248
00:16:32,690 --> 00:16:40,220
operating on any two-electron
function, has to make--

249
00:16:49,880 --> 00:16:54,510
OK, I'm skipping steps, and my
notes are really kind of stupid

250
00:16:54,510 --> 00:16:55,620
sometimes.

251
00:16:55,620 --> 00:16:59,625
P 2, 1, which has to
be equal to minus.

252
00:17:04,050 --> 00:17:07,410
And now if you apply the
permutation operator twice,

253
00:17:07,410 --> 00:17:08,849
you get back to the same thing.

254
00:17:08,849 --> 00:17:11,490
So there's only two
possible eigenvalues.

255
00:17:11,490 --> 00:17:15,960
You can have minus 1 or plus 1.

256
00:17:15,960 --> 00:17:18,690
And the minus 1
corresponds to fermions,

257
00:17:18,690 --> 00:17:22,950
things that have half-integer
spin, like electrons.

258
00:17:22,950 --> 00:17:25,890
And the plus 1 corresponds to
things that have integer spin,

259
00:17:25,890 --> 00:17:31,300
like photons, and
vibrons, and other things.

260
00:17:31,300 --> 00:17:36,810
And actually, it's harder to
construct a symmetric function

261
00:17:36,810 --> 00:17:39,060
than an anti-symmetric function.

262
00:17:39,060 --> 00:17:42,580
But the thing is, you've
got lots of electrons,

263
00:17:42,580 --> 00:17:49,270
and you have very few quanta
of vibrations in a single mode,

264
00:17:49,270 --> 00:17:51,760
and you have very few
photons interacting

265
00:17:51,760 --> 00:17:53,840
with a molecule at once.

266
00:17:53,840 --> 00:17:59,920
And so the boson
symmetry is less

267
00:17:59,920 --> 00:18:04,130
important in most applications.

268
00:18:04,130 --> 00:18:07,430
And so we just have
to kill this one.

269
00:18:07,430 --> 00:18:22,300
OK, so suppose we want to talk
about something like this,

270
00:18:22,300 --> 00:18:25,290
the 1s, 2s configuration.

271
00:18:25,290 --> 00:18:29,640
A configuration is a list
of the occupied orbitals--

272
00:18:29,640 --> 00:18:32,430
not the occupied
spin orbitals, which

273
00:18:32,430 --> 00:18:36,540
is a spin associated
with an orbital.

274
00:18:36,540 --> 00:18:39,240
The world of spin
orbitals is where I live,

275
00:18:39,240 --> 00:18:42,450
but we do that for a reason.

276
00:18:42,450 --> 00:18:52,990
And so this two-electron thing
can be expressed as a space

277
00:18:52,990 --> 00:18:53,490
part--

278
00:19:08,660 --> 00:19:10,510
there are various
conventions that--

279
00:19:16,320 --> 00:19:17,450
times the spin part.

280
00:19:21,960 --> 00:19:34,890
And alpha 1, beta 2, and then
we have minus or plus beta 1,

281
00:19:34,890 --> 00:19:36,280
alpha 2.

282
00:19:36,280 --> 00:19:39,910
I'm looking at my notes
because some people always

283
00:19:39,910 --> 00:19:43,030
keep the electron in
the first position,

284
00:19:43,030 --> 00:19:47,710
and some people keep always the
orbital in the first position.

285
00:19:47,710 --> 00:19:50,080
And it doesn't
matter, because you

286
00:19:50,080 --> 00:19:52,180
can permute rows or columns.

287
00:19:52,180 --> 00:19:55,500
But I just want to write
what is in my notes.

288
00:19:55,500 --> 00:20:00,970
OK, so this thing, this
two-electron function,

289
00:20:00,970 --> 00:20:05,780
has two anti-symmetrized
possibilities.

290
00:20:05,780 --> 00:20:10,530
And one is a singlet,
and one is a triplet.

291
00:20:10,530 --> 00:20:15,140
So s equals zero, s equals 1.

292
00:20:15,140 --> 00:20:19,850
We recognize this alpha
beta minus beta alpha

293
00:20:19,850 --> 00:20:26,570
as the singlet spin state,
and alpha beta plus beta alpha

294
00:20:26,570 --> 00:20:29,610
as the triplet spin state.

295
00:20:29,610 --> 00:20:34,610
So we have alpha beta plus
beta alpha, and alpha alpha

296
00:20:34,610 --> 00:20:40,800
and beta beta, and we have
alpha beta minus beta alpha.

297
00:20:40,800 --> 00:20:45,810
So we call s = 0 a singlet,
and this a triplet,

298
00:20:45,810 --> 00:20:47,280
because of the number of states.

299
00:20:49,930 --> 00:20:55,900
And this wave function has
the necessary spin symmetry

300
00:20:55,900 --> 00:20:57,775
and the necessary
permutation symmetry.

301
00:21:01,180 --> 00:21:10,270
OK, so if, instead of
two electrons, we have 1,

302
00:21:10,270 --> 00:21:22,220
2 dot dot dot N, then Mr.
Slater says we do this--

303
00:21:22,220 --> 00:21:22,720
whoops.

304
00:21:32,030 --> 00:21:46,960
OK, N, 1, and then K1,
and KN N. So that's

305
00:21:46,960 --> 00:21:49,780
a determinant-- an
N by N determinant.

306
00:21:49,780 --> 00:21:54,220
And the rows correspond
to the electrons,

307
00:21:54,220 --> 00:21:59,010
and the columns corresponding
to the orbitals.

308
00:21:59,010 --> 00:22:04,080
Now this, because of the
properties of the determinant,

309
00:22:04,080 --> 00:22:08,130
is anti-symmetric with respect
to permutation of any two

310
00:22:08,130 --> 00:22:12,974
electrons or any two orbitals.

311
00:22:12,974 --> 00:22:14,890
But we don't really care
about the permutation

312
00:22:14,890 --> 00:22:16,760
of the orbitals, because
it's really the same thing

313
00:22:16,760 --> 00:22:18,140
is permuting the electrons.

314
00:22:18,140 --> 00:22:24,340
And so this N factorial is a
consequence of normalization,

315
00:22:24,340 --> 00:22:27,490
because when you expand
an N by N determinant,

316
00:22:27,490 --> 00:22:33,126
you get N factorial, additive
products of N functions.

317
00:22:33,126 --> 00:22:34,250
It looks horrible.

318
00:22:37,540 --> 00:22:40,630
And because we're
normalizing, we

319
00:22:40,630 --> 00:22:44,950
need this 1 over the
square root of N factorial

320
00:22:44,950 --> 00:22:46,570
in order to have
this thing come out

321
00:22:46,570 --> 00:22:49,940
to be 1 when you calculate
the normalization integral.

322
00:22:54,450 --> 00:22:58,540
Now this notation is
horrible because you've

323
00:22:58,540 --> 00:23:00,740
got too many symbols.

324
00:23:00,740 --> 00:23:04,180
And so depending on what
you're trying to convey,

325
00:23:04,180 --> 00:23:08,620
you reduce the symbols, and
you can reduce it simply

326
00:23:08,620 --> 00:23:15,630
by, instead of writing psi every
time, just writing the state.

327
00:23:15,630 --> 00:23:18,360
Or you can-- since
you don't need psi,

328
00:23:18,360 --> 00:23:20,940
you don't need the state
letter, you can just

329
00:23:20,940 --> 00:23:22,110
have the state number.

330
00:23:25,000 --> 00:23:28,360
But the best way to do
this is simply to say--

331
00:23:40,200 --> 00:23:44,350
this is just the main
diagonal of the determinant.

332
00:23:44,350 --> 00:23:48,090
It conveys everything you need.

333
00:23:48,090 --> 00:23:52,820
Again, if you permute any two of
these guys, any adjacent pair,

334
00:23:52,820 --> 00:23:54,720
the sign changes.

335
00:23:54,720 --> 00:23:56,470
And it contains
everything you need,

336
00:23:56,470 --> 00:23:59,910
and it doesn't require
you to look at stuff

337
00:23:59,910 --> 00:24:02,650
you're not going to use.

338
00:24:02,650 --> 00:24:05,740
And your goal is going to
be to take these things,

339
00:24:05,740 --> 00:24:08,410
and calculate matrix of them.

340
00:24:08,410 --> 00:24:11,680
And so you'll be dealing
with the orbitals one or two

341
00:24:11,680 --> 00:24:13,240
at a time.

342
00:24:13,240 --> 00:24:15,880
And this is very convenient.

343
00:24:15,880 --> 00:24:20,170
And soon, you start to
take this for granted.

344
00:24:20,170 --> 00:24:22,210
And it's a very
simple thing, but it

345
00:24:22,210 --> 00:24:27,090
isn't, because you're doing
a huge number of tricks.

346
00:24:27,090 --> 00:24:29,950
OK, I'm going to skip
over what's in my notes.

347
00:24:29,950 --> 00:24:33,490
Demonstrating that for a two
by two, that what I asserted

348
00:24:33,490 --> 00:24:36,115
is correct, you can
do that very easily.

349
00:24:47,480 --> 00:24:53,780
OK, so we can count,
and we have an atom,

350
00:24:53,780 --> 00:24:57,020
and we know how many
electrons it has.

351
00:24:57,020 --> 00:25:00,550
And so we immediately know
what our job is going to be.

352
00:25:00,550 --> 00:25:04,300
We're going to be having to
write some Slater determinant

353
00:25:04,300 --> 00:25:07,000
of those number of electrons.

354
00:25:07,000 --> 00:25:11,980
And the goal is to be able
to do the algebra in a way

355
00:25:11,980 --> 00:25:14,050
that maybe you can't
describe to your friends

356
00:25:14,050 --> 00:25:16,010
because it's too complicated.

357
00:25:16,010 --> 00:25:19,330
I'm faced with the problem
of trying to explain

358
00:25:19,330 --> 00:25:21,190
how to do this algebra.

359
00:25:21,190 --> 00:25:25,210
But it is something
that you can learn,

360
00:25:25,210 --> 00:25:28,000
and you can ask a
computer to do it,

361
00:25:28,000 --> 00:25:30,850
and there are all sorts
of intuitive shortcuts

362
00:25:30,850 --> 00:25:32,980
where you can look at a
problem, and you could say,

363
00:25:32,980 --> 00:25:35,540
I understand.

364
00:25:35,540 --> 00:25:40,240
OK, so you're used to orbitals.

365
00:25:43,540 --> 00:25:46,950
And that's perfectly reasonable,
because for hydrogen, we

366
00:25:46,950 --> 00:25:50,260
have orbitals, and
there's only one orbital,

367
00:25:50,260 --> 00:25:51,510
and it could have either spin.

368
00:25:51,510 --> 00:25:52,920
We don't mess with that.

369
00:25:52,920 --> 00:25:55,215
But now we're going to
talk about spin orbitals.

370
00:25:58,940 --> 00:26:01,950
And that's just the
combination of the name

371
00:26:01,950 --> 00:26:05,750
of the orbital with whether
the spin is up or down.

372
00:26:05,750 --> 00:26:10,870
And the reason for this is
it's easier to do the algebra.

373
00:26:10,870 --> 00:26:15,070
And the reason the algebra is--

374
00:26:15,070 --> 00:26:18,580
it's initially harder
to do the algebra,

375
00:26:18,580 --> 00:26:21,631
because there are certain
selection rules, and stuff

376
00:26:21,631 --> 00:26:22,130
like that.

377
00:26:22,130 --> 00:26:25,640
But once you know how to
do it you do the algebra.

378
00:26:25,640 --> 00:26:27,730
And then all of a
sudden, everything

379
00:26:27,730 --> 00:26:29,640
pops out in a very useful form.

380
00:26:36,630 --> 00:26:38,910
So the stick diagrams
are very important.

381
00:26:38,910 --> 00:26:42,420
But now I'm specifying
the stick diagrams

382
00:26:42,420 --> 00:26:45,790
as spin orbitals
rather than orbitals.

383
00:26:45,790 --> 00:26:53,100
Now another point, there
are rules for how--

384
00:26:58,720 --> 00:27:01,360
the number of spin
orbitals is different

385
00:27:01,360 --> 00:27:04,270
between the left-hand side and
the right-hand side of a matrix

386
00:27:04,270 --> 00:27:05,020
element.

387
00:27:05,020 --> 00:27:07,540
There are rules that are
easily described-- and so

388
00:27:07,540 --> 00:27:14,800
for every kind of
orbital, an orbital that

389
00:27:14,800 --> 00:27:20,170
is a scalar, that doesn't
depend on quantum numbers, that

390
00:27:20,170 --> 00:27:23,750
has a selection
rule delta SO of 0;

391
00:27:23,750 --> 00:27:30,010
and for something like a
one-electron operator that

392
00:27:30,010 --> 00:27:35,290
has a selection rule delta
spin orbital of 1 and 0;

393
00:27:35,290 --> 00:27:37,780
and then we have our
friend 1 over rij,

394
00:27:37,780 --> 00:27:43,320
that has a selection rule delta
spin orbital of 2, 1, and 0.

395
00:27:43,320 --> 00:27:46,270
And the algebra for each
is something you work out,

396
00:27:46,270 --> 00:27:47,970
and then you know how to do it.

397
00:27:47,970 --> 00:27:50,260
And I'm going to try to
give you just a little bit

398
00:27:50,260 --> 00:27:51,130
of a taste of this.

399
00:27:53,800 --> 00:28:02,980
So we already looked
at something like this.

400
00:28:02,980 --> 00:28:05,930
But we use a slightly
different notation.

401
00:28:05,930 --> 00:28:07,210
So I'm going to go back.

402
00:28:07,210 --> 00:28:16,720
And we have 1s alpha, 1s
beta, 2s alpha, 2s beta.

403
00:28:16,720 --> 00:28:20,290
And so for the ground
state of helium 1s

404
00:28:20,290 --> 00:28:23,080
squared, and we would do this.

405
00:28:27,330 --> 00:28:30,540
And the stick
diagrams are great,

406
00:28:30,540 --> 00:28:36,750
because it's easier to see on
a picture, who are the actors,

407
00:28:36,750 --> 00:28:41,880
and have I included all of them,
or have I left something out?

408
00:28:41,880 --> 00:28:45,570
And so now we're interested
in the stick diagram

409
00:28:45,570 --> 00:28:52,020
for the 1s 2s configuration.

410
00:28:52,020 --> 00:28:58,090
And there are several kinds
of 1s/2s configurations,

411
00:28:58,090 --> 00:29:00,300
depending on what the
alpha and beta are.

412
00:29:00,300 --> 00:29:05,020
So we have 1s alpha, 2s alpha.

413
00:29:05,020 --> 00:29:11,645
And we have 1s alpha, 2s beta.

414
00:29:19,890 --> 00:29:29,830
1s beta, 2s alpha, 2s
alpha, 1s beta, 2s beta.

415
00:29:29,830 --> 00:29:33,370
So there's four guys, and we can
put our arrows on these things,

416
00:29:33,370 --> 00:29:40,010
and we know everything we
need to know about these guys.

417
00:29:40,010 --> 00:29:41,050
It tells us what to do.

418
00:29:46,670 --> 00:29:57,420
Well, when we do this, the
diagonal matrix elements

419
00:29:57,420 --> 00:30:02,990
of the 1 over rij
Hamiltonian can be expressed.

420
00:30:16,000 --> 00:30:24,620
And we use this notation,
J tilde minus K tilde.

421
00:30:24,620 --> 00:30:27,980
So for every two-electron thing,
we're going to get this kind

422
00:30:27,980 --> 00:30:28,730
of--

423
00:30:28,730 --> 00:30:33,470
now these are simple integrals,
and some of them are 0.

424
00:30:33,470 --> 00:30:37,805
Because this doesn't
operate on spins.

425
00:30:37,805 --> 00:30:58,150
And so if you had a 1s alpha,
2s beta, 1 over rij, 1s beta,

426
00:30:58,150 --> 00:31:09,050
2s alpha, then the 1s alpha
with the 1s beta is 0.

427
00:31:09,050 --> 00:31:14,600
The 1s alpha with the 2s
alpha is not 0, et cetera.

428
00:31:14,600 --> 00:31:16,500
There are all sorts of stuff.

429
00:31:16,500 --> 00:31:20,025
But this tilde notation
says, well, this

430
00:31:20,025 --> 00:31:22,430
is what we start with,
and we have to convert it

431
00:31:22,430 --> 00:31:25,950
into things that really matter.

432
00:31:25,950 --> 00:31:28,850
So the operation of
removing the tilde

433
00:31:28,850 --> 00:31:32,990
requires a little bit of
work, a little bit of thought.

434
00:31:32,990 --> 00:31:36,680
And that's why my
notes are crap,

435
00:31:36,680 --> 00:31:43,370
because I can't explain it well
enough to really teach this.

436
00:31:45,980 --> 00:32:02,790
So when we do the
1s squared, the J 1s

437
00:32:02,790 --> 00:32:10,240
squared tilde, is equal
to the J 1s squared,

438
00:32:10,240 --> 00:32:14,250
because the spins take
care of themselves.

439
00:32:14,250 --> 00:32:22,840
But k tilde 1s
squared is equal to 0.

440
00:32:22,840 --> 00:32:27,700
Because when we
do 1s squared, we

441
00:32:27,700 --> 00:32:31,120
have an alpha with
an alpha for the J

442
00:32:31,120 --> 00:32:35,320
term, and an alpha with
the beta for the K term.

443
00:32:35,320 --> 00:32:40,890
And alpha with beta is 0,
because the operator cannot

444
00:32:40,890 --> 00:32:42,790
change the alpha into beta.

445
00:32:50,570 --> 00:32:53,150
So this tilde notation
is a convenient thing,

446
00:32:53,150 --> 00:32:56,960
because you can use
any Slater determinant,

447
00:32:56,960 --> 00:33:00,020
and you can express it
in terms of J's and K's.

448
00:33:00,020 --> 00:33:04,310
And the sign comes
from switching

449
00:33:04,310 --> 00:33:06,410
the order of the orbitals.

450
00:33:06,410 --> 00:33:09,140
That's how the
determinants work.

451
00:33:09,140 --> 00:33:12,740
And so you're going to see
a whole bunch of stuff.

452
00:33:12,740 --> 00:33:17,210
But removing the tildes
is the tricky business.

453
00:33:17,210 --> 00:33:22,400
OK, now when you
get a problem where

454
00:33:22,400 --> 00:33:34,590
you have a configuration
where a single Slater is not

455
00:33:34,590 --> 00:33:35,668
sufficient--

456
00:33:40,250 --> 00:33:45,230
in other words, in order to
make an eigenstate of s squared

457
00:33:45,230 --> 00:33:51,230
or sz, you sometimes
need two or more Slaters,

458
00:33:51,230 --> 00:33:54,260
and you have to use a particular
linear combination of them

459
00:33:54,260 --> 00:33:59,750
to get the right
value of s and sz.

460
00:33:59,750 --> 00:34:02,030
And then what happens
is you're looking

461
00:34:02,030 --> 00:34:10,179
at matrix elements of the 1 over
rij operator, between Slaters.

462
00:34:10,179 --> 00:34:12,170
Now this is a headache.

463
00:34:12,170 --> 00:34:15,820
And I could talk until
I'm blue in the face,

464
00:34:15,820 --> 00:34:20,679
and I cannot make it
clear how to do this.

465
00:34:20,679 --> 00:34:24,090
Because it's just awful.

466
00:34:24,090 --> 00:34:28,420
But some things in life
are worth suffering for.

467
00:34:28,420 --> 00:34:36,929
And so anyway, in
the 1s, 2s situation,

468
00:34:36,929 --> 00:34:42,150
when you do everything
right, you get--

469
00:34:42,150 --> 00:34:45,179
this is just a general notation
for a two-electron wave

470
00:34:45,179 --> 00:34:46,199
function--

471
00:34:46,199 --> 00:34:52,469
1 over r12, psi 2.

472
00:34:52,469 --> 00:34:56,460
So these guys are eigenfunctions
of s squared and sz.

473
00:34:56,460 --> 00:35:05,500
And when you do that,
you get 1/2 times 2 J,

474
00:35:05,500 --> 00:35:13,960
1s, 2s, minus or
plus 2K, 1s, 2s.

475
00:35:13,960 --> 00:35:19,840
Remember, when you have
mismatched alpha and beta,

476
00:35:19,840 --> 00:35:20,920
the K's are 0.

477
00:35:20,920 --> 00:35:25,780
But when you have K, the 1 over
rij matrix element between two

478
00:35:25,780 --> 00:35:28,620
Slaters, you can fix that.

479
00:35:28,620 --> 00:35:32,550
And so this is why it's so
hard to explain, because--

480
00:35:32,550 --> 00:35:36,460
yes, I'm not even going
to apologize anymore.

481
00:35:36,460 --> 00:35:39,780
OK, so this is what you do.

482
00:35:39,780 --> 00:35:42,480
And the notes are pretty
clear about how to do them,

483
00:35:42,480 --> 00:35:43,950
and what the problems are.

484
00:35:43,950 --> 00:35:46,170
But lecturing on it would
be a little bit hard.

485
00:35:46,170 --> 00:35:48,970
OK, so now what are we
going to do with it?

486
00:35:48,970 --> 00:35:57,840
Well, we'd like qualitative
stuff and interpretive stuff.

487
00:36:06,490 --> 00:36:09,835
Qualitative is Hund's rules.

488
00:36:12,520 --> 00:36:19,352
Now if you looked at 100
textbooks, I think 95% of them

489
00:36:19,352 --> 00:36:20,560
will have Hund's rules wrong.

490
00:36:24,440 --> 00:36:26,860
You're never going
to make that mistake.

491
00:36:26,860 --> 00:36:30,200
And interpretive-- well, we want
to know the trends of things,

492
00:36:30,200 --> 00:36:33,560
and we want to be
able to do something

493
00:36:33,560 --> 00:36:36,831
like what you did in freshman
chemistry on shielding.

494
00:36:39,720 --> 00:36:43,600
Now you probably memorized some
rules about what shields what.

495
00:36:43,600 --> 00:36:45,410
But I'm going to give
you a little bit more

496
00:36:45,410 --> 00:36:46,250
insight into that.

497
00:36:46,250 --> 00:36:47,625
So we're going to
talk about this

498
00:36:47,625 --> 00:36:48,980
for the rest of the lecture.

499
00:36:52,160 --> 00:37:01,340
OK, so you specify
a configuration.

500
00:37:04,910 --> 00:37:08,180
And this configuration might
be two electrons, two spin

501
00:37:08,180 --> 00:37:11,645
orbitals, two orbitals
times e, or three, or 10.

502
00:37:14,560 --> 00:37:20,950
And often, when you specify
the occupied orbitals,

503
00:37:20,950 --> 00:37:26,120
you neglect the
field ones, which

504
00:37:26,120 --> 00:37:30,170
is nice, because you have
fewer things to worry about.

505
00:37:30,170 --> 00:37:34,750
Because field
orbitals have spin 0.

506
00:37:34,750 --> 00:37:37,140
And you don't have
to do anything.

507
00:37:37,140 --> 00:37:39,220
They're automatically
asymmetrized,

508
00:37:39,220 --> 00:37:43,510
and they basically act
as a charged distribution

509
00:37:43,510 --> 00:37:49,630
in the core that is sampled
by the electrons outside.

510
00:37:49,630 --> 00:37:52,870
And so you need some
sort of a set of rules

511
00:37:52,870 --> 00:37:55,320
for how does that work.

512
00:37:55,320 --> 00:37:57,810
And that's shielding.

513
00:37:57,810 --> 00:38:01,720
So first, we specify
a configuration.

514
00:38:01,720 --> 00:38:05,470
And you also learned-- in
high school, probably--

515
00:38:05,470 --> 00:38:10,750
how to determine the
L, S, J terms that

516
00:38:10,750 --> 00:38:14,590
result from this configuration
by some magical crossing-out

517
00:38:14,590 --> 00:38:16,170
of boxes.

518
00:38:16,170 --> 00:38:18,670
And if you didn't, I'm glad.

519
00:38:18,670 --> 00:38:21,870
Because it would have
just clouded your mind,

520
00:38:21,870 --> 00:38:26,410
and caused earlier
insanity than MIT causes.

521
00:38:26,410 --> 00:38:31,520
So anyway, so we have
orbital angular momentum.

522
00:38:31,520 --> 00:38:33,310
And we can add the
orbital angular

523
00:38:33,310 --> 00:38:37,300
momenta of the electrons
following certain rules.

524
00:38:37,300 --> 00:38:39,650
And we have spin
angular momentum.

525
00:38:39,650 --> 00:38:46,934
And J is equal to the
vector sum of L and S.

526
00:38:46,934 --> 00:38:51,470
And we say we have an LS term--

527
00:38:51,470 --> 00:38:55,910
like triplet P.
And it can have J

528
00:38:55,910 --> 00:39:06,860
is equal to L plus S, L plus
S minus 1, down to L minus S

529
00:39:06,860 --> 00:39:09,050
absolute value.

530
00:39:09,050 --> 00:39:11,000
These are the possible J's.

531
00:39:11,000 --> 00:39:13,520
And so Hund's
rules is all about,

532
00:39:13,520 --> 00:39:18,650
of all of the states that belong
to a particular configuration,

533
00:39:18,650 --> 00:39:20,840
which one is the lowest?

534
00:39:20,840 --> 00:39:23,910
One-- which one, not
the second lowest.

535
00:39:23,910 --> 00:39:25,040
Which one is the lowest?

536
00:39:25,040 --> 00:39:26,330
And why do we care?

537
00:39:26,330 --> 00:39:29,510
Because in statistical
mechanics everything

538
00:39:29,510 --> 00:39:33,380
is dominated by the
lowest energy state.

539
00:39:33,380 --> 00:39:36,340
And so if you can figure out
what is the lowest energy

540
00:39:36,340 --> 00:39:39,790
state, you've basically got
as much as most people are

541
00:39:39,790 --> 00:39:42,430
going to want.

542
00:39:42,430 --> 00:39:48,280
So you want to know what are L,
S, and J for the lowest energy

543
00:39:48,280 --> 00:39:51,040
state of a configuration.

544
00:39:51,040 --> 00:39:54,830
Configurations are typically
far apart in energy.

545
00:39:54,830 --> 00:39:58,210
So if you know what the lowest
energy configuration is,

546
00:39:58,210 --> 00:40:01,210
and the lowest energy state of
it, as far as your friends--

547
00:40:01,210 --> 00:40:03,730
the statistical
machinations, you

548
00:40:03,730 --> 00:40:07,590
can tell them how to write
their partition functions.

549
00:40:07,590 --> 00:40:09,490
And the rest is details.

550
00:40:09,490 --> 00:40:11,710
And mostly, you
don't want details.

551
00:40:11,710 --> 00:40:13,779
If your friends tell
you they want details,

552
00:40:13,779 --> 00:40:15,820
well, you tell them, this
is what you have to do,

553
00:40:15,820 --> 00:40:19,420
but it's no simple
three Hund's rules.

554
00:40:19,420 --> 00:40:22,910
OK, so Hund's rules--

555
00:40:22,910 --> 00:40:25,840
you look at all of the
L, S, J states that

556
00:40:25,840 --> 00:40:29,860
are possible for a
particular configuration.

557
00:40:29,860 --> 00:40:34,530
And you can use the crossing
out of ML/MS boxes if you want.

558
00:40:34,530 --> 00:40:36,790
And I could tell you
why you would do that.

559
00:40:36,790 --> 00:40:41,170
But I don't want to cause
insanity at this stage, either.

560
00:40:41,170 --> 00:40:43,180
But I'm an expert at that.

561
00:40:43,180 --> 00:40:47,370
And you can also use
lowering operators

562
00:40:47,370 --> 00:40:50,370
to generate all the states,
once you know stuff.

563
00:40:50,370 --> 00:40:57,820
OK, so once you
know all the states,

564
00:40:57,820 --> 00:41:04,410
Hund says, which one of
these has the largest S?

565
00:41:04,410 --> 00:41:05,666
which one?

566
00:41:05,666 --> 00:41:07,860
And that's easy to know.

567
00:41:07,860 --> 00:41:14,610
And for example, if
you had 2p squared,

568
00:41:14,610 --> 00:41:26,730
you're going to get singlet
D, triplet P, and singlet S.

569
00:41:26,730 --> 00:41:30,210
And well, here's the triplet.

570
00:41:30,210 --> 00:41:33,440
That has the largest
S. So the triplet P

571
00:41:33,440 --> 00:41:37,250
is the lowest energy state.

572
00:41:37,250 --> 00:41:43,850
Now if there were multiple
triplets, as there would be,

573
00:41:43,850 --> 00:41:56,340
say, for 2p3d, then
you'd have to decide

574
00:41:56,340 --> 00:42:01,030
which of those
triplets is the lowest.

575
00:42:01,030 --> 00:42:03,610
And all you care about,
all you're allowed to say

576
00:42:03,610 --> 00:42:06,380
is which one is the lowest.

577
00:42:06,380 --> 00:42:10,900
And it's the one
with the maximum L.

578
00:42:10,900 --> 00:42:15,610
And then the last step
is, what is the lowest

579
00:42:15,610 --> 00:42:18,940
J for that LS state?

580
00:42:18,940 --> 00:42:20,740
And that's kind of cute.

581
00:42:20,740 --> 00:42:25,580
Because you have the
P shell, there's--

582
00:42:36,900 --> 00:42:46,320
for a P shell, you can have six
P orbitals to fill the shells.

583
00:42:46,320 --> 00:42:51,450
1 alpha, 1 beta, 0
alpha, 0 beta, et cetera.

584
00:42:51,450 --> 00:42:54,515
So the degeneracy
of a P orbital is 6.

585
00:42:57,650 --> 00:43:05,690
If you have p to N,
where N is less than 3,

586
00:43:05,690 --> 00:43:08,630
you have a
less-than-half-filled shell.

587
00:43:08,630 --> 00:43:19,430
And then lowest is J equal
L minus S, absolute value.

588
00:43:19,430 --> 00:43:24,680
And if N is greater than 3,
you have the lowest being--

589
00:43:24,680 --> 00:43:29,060
the highest possible value
of J is equal to L plus S--

590
00:43:29,060 --> 00:43:33,310
so for L, N greater than 3.

591
00:43:33,310 --> 00:43:35,410
And now when you have
a half-filled shell,

592
00:43:35,410 --> 00:43:48,114
the lowest state is usually
an S state with maximum spin.

593
00:43:48,114 --> 00:43:49,030
But it doesn't matter.

594
00:43:52,510 --> 00:43:55,000
When you have less-than or
more-than-half-filled shell,

595
00:43:55,000 --> 00:43:58,300
you have generally a state with
orbital angular momentum not

596
00:43:58,300 --> 00:43:59,890
equal to 0.

597
00:43:59,890 --> 00:44:02,260
And you have spin orbit
splitting of that.

598
00:44:02,260 --> 00:44:06,160
And so you do want to
care what is the lowest J.

599
00:44:06,160 --> 00:44:09,580
But when N is equal
to 3, the lowest state

600
00:44:09,580 --> 00:44:10,870
is usually an S state.

601
00:44:10,870 --> 00:44:12,520
It doesn't have a
spin orbit splitting,

602
00:44:12,520 --> 00:44:20,440
and it just has one value of J,
which is whatever the spin is.

603
00:44:20,440 --> 00:44:27,190
So Hund's rules tell
you how to identify,

604
00:44:27,190 --> 00:44:32,030
without knowing beans, what
is the lowest energy state,

605
00:44:32,030 --> 00:44:34,300
and it's never wrong.

606
00:44:34,300 --> 00:44:36,280
Well, maybe sometimes
wrong, but that's

607
00:44:36,280 --> 00:44:38,410
because of one of
my things where

608
00:44:38,410 --> 00:44:40,460
you have a perturbation
between states belonging

609
00:44:40,460 --> 00:44:41,980
to two configurations.

610
00:44:41,980 --> 00:44:44,500
But people get really
excited when they discover

611
00:44:44,500 --> 00:44:47,140
a violation of Hund's rules.

612
00:44:47,140 --> 00:44:49,730
And it's just trivial.

613
00:44:49,730 --> 00:44:53,080
So there is this.

614
00:44:53,080 --> 00:44:54,190
What time is it?

615
00:44:54,190 --> 00:44:58,390
I have a few minutes to talk
about shielding, and I will.

616
00:44:58,390 --> 00:45:02,740
OK, so we have a nucleus.

617
00:45:02,740 --> 00:45:08,460
And it has a charge of z.

618
00:45:08,460 --> 00:45:10,740
Bare nucleus--
there's no electrons,

619
00:45:10,740 --> 00:45:14,150
so the atomic number
is the charge.

620
00:45:14,150 --> 00:45:20,430
And then we have a
filled shell around it.

621
00:45:20,430 --> 00:45:21,840
It's spherically symmetric.

622
00:45:24,960 --> 00:45:30,570
And so if we penetrate
inside of it, what we see--

623
00:45:30,570 --> 00:45:34,500
suppose we penetrate inside,
to this point, what we see

624
00:45:34,500 --> 00:45:40,650
is only the plus z, minus
the number of electrons

625
00:45:40,650 --> 00:45:42,510
inside this sphere.

626
00:45:42,510 --> 00:45:48,360
Now if you took Electricity and
Magnetism, you can prove this.

627
00:45:48,360 --> 00:45:50,830
If you didn't,
you can accept it.

628
00:45:50,830 --> 00:45:53,760
And so outside the
nucleus, the charge

629
00:45:53,760 --> 00:45:57,970
is plus one, because
you have a neutral atom.

630
00:45:57,970 --> 00:46:01,470
And then when you penetrate
inside this region

631
00:46:01,470 --> 00:46:06,990
of dense charge, and all of
the spins are generally paired,

632
00:46:06,990 --> 00:46:09,840
this is spherical.

633
00:46:09,840 --> 00:46:16,710
So what you end up
seeing is z effective, as

634
00:46:16,710 --> 00:46:19,000
opposed to z true.

635
00:46:19,000 --> 00:46:21,000
And let's say here here's r0--

636
00:46:21,000 --> 00:46:22,460
or this is r0.

637
00:46:25,020 --> 00:46:33,610
And so beyond r0, the charge
that you see is plus 1.

638
00:46:33,610 --> 00:46:37,830
Add 0, you see a charge of z.

639
00:46:37,830 --> 00:46:42,460
And so what ends up happening
is you get z effective,

640
00:46:42,460 --> 00:46:47,440
which is dependent on
distance from the nucleus.

641
00:46:47,440 --> 00:46:49,750
And it goes from
the integer value

642
00:46:49,750 --> 00:46:54,580
that you know, from the
atomic number, down to 1,

643
00:46:54,580 --> 00:46:59,020
because you've taken
one electron away

644
00:46:59,020 --> 00:47:02,650
from a neutral atom,
and taken it outside.

645
00:47:02,650 --> 00:47:06,550
And now we have
this wonderful thing

646
00:47:06,550 --> 00:47:08,695
called the centrifugal barrier.

647
00:47:13,690 --> 00:47:17,800
So if we have a state
that has a non-zero of l--

648
00:47:17,800 --> 00:47:19,800
well, if we have a state
with a zero value of l,

649
00:47:19,800 --> 00:47:24,710
it can penetrate all the way
into the core, to the nucleus.

650
00:47:24,710 --> 00:47:27,380
And so that means that
the shielding is less

651
00:47:27,380 --> 00:47:30,070
for s orbitals.

652
00:47:30,070 --> 00:47:34,270
And now if we have a non-zero
l, it can't get in so far.

653
00:47:34,270 --> 00:47:40,330
And the larger
the l is, the less

654
00:47:40,330 --> 00:47:43,060
it can see this extra charge.

655
00:47:43,060 --> 00:47:45,590
So high l's are
very well shielded.

656
00:47:45,590 --> 00:47:47,770
Low l's are not
so well shielded.

657
00:47:47,770 --> 00:47:54,720
And the shielding goes s least
shielded, p, less, so on.

658
00:47:54,720 --> 00:47:58,090
Now there's some other
interesting things.

659
00:47:58,090 --> 00:48:03,200
Which, you know, I
hate to say this,

660
00:48:03,200 --> 00:48:12,888
but comparing 5.111
or 5.112 to 3.091,

661
00:48:12,888 --> 00:48:16,800
there is this business of what
happens when you start to--

662
00:48:27,030 --> 00:48:32,280
you start with potassium, and so
you put an electron in the 4s,

663
00:48:32,280 --> 00:48:34,560
not in the 3d orbital.

664
00:48:34,560 --> 00:48:35,170
Right?

665
00:48:35,170 --> 00:48:35,670
Why is that?

666
00:48:38,630 --> 00:48:45,210
Well, the 4s sees the larger
charge is less shielded.

667
00:48:45,210 --> 00:48:47,080
So it goes in.

668
00:48:47,080 --> 00:48:49,770
Then when you go from
potassium to calcium,

669
00:48:49,770 --> 00:48:54,370
you put another
electron in this.

670
00:48:54,370 --> 00:48:55,850
And that's true.

671
00:48:55,850 --> 00:48:59,190
So for calcium, you
have a 4s squared.

672
00:48:59,190 --> 00:49:03,500
And 4s 3d is a
higher-lying state.

673
00:49:03,500 --> 00:49:05,570
Now you take an electron--

674
00:49:11,640 --> 00:49:13,670
I'm cooking my own goose.

675
00:49:21,550 --> 00:49:26,510
If you take one of
these electrons away--

676
00:49:26,510 --> 00:49:28,760
this is not the way I
wanted it to come out--

677
00:49:28,760 --> 00:49:36,170
you find yourself in a 3d state.

678
00:49:36,170 --> 00:49:43,450
Because 3D penetrates
a little bit under 4s.

679
00:49:47,580 --> 00:49:50,850
I can't explain it in a way
that's going to make sense.

680
00:49:50,850 --> 00:49:54,960
I really wanted to,
because I care so much

681
00:49:54,960 --> 00:49:56,970
about these simple arguments.

682
00:49:56,970 --> 00:50:01,390
But I will just be
wasting your time.

683
00:50:01,390 --> 00:50:03,570
So the order in which
you feel orbitals

684
00:50:03,570 --> 00:50:06,960
comes out, naturally, different
from the order in which you

685
00:50:06,960 --> 00:50:09,180
remove electrons from orbitals.

686
00:50:09,180 --> 00:50:14,140
And the shielding arguments
are capable of explaining that.

687
00:50:14,140 --> 00:50:18,670
OK, so this is the end of atoms.

688
00:50:18,670 --> 00:50:24,790
And I've asked you to observe
some complicated algebra which

689
00:50:24,790 --> 00:50:26,500
you're never going
to do, or at least

690
00:50:26,500 --> 00:50:27,640
never going to do much of.

691
00:50:30,830 --> 00:50:33,740
Everything you need
to know about atoms,

692
00:50:33,740 --> 00:50:37,580
you can tell a computer,
and it can do it.

693
00:50:37,580 --> 00:50:41,885
Now molecules are
much more complicated.

694
00:50:41,885 --> 00:50:43,760
And that's we're going
to start on next time.

695
00:50:43,760 --> 00:50:46,550
We're going to start with
molecular orbital theory.

696
00:50:46,550 --> 00:50:50,510
And I'm not going to be
presenting the normal textbook

697
00:50:50,510 --> 00:50:51,620
approach.

698
00:50:51,620 --> 00:50:54,500
I'm going to present an
interpretive approach, where

699
00:50:54,500 --> 00:50:56,660
you understand
why things happen,

700
00:50:56,660 --> 00:51:01,640
as opposed to memorize just
symmetries, and filling orders,

701
00:51:01,640 --> 00:51:02,630
and so on.

702
00:51:02,630 --> 00:51:05,370
OK, I'll see you on Wednesday.