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18  1
5.73
Lecture
#18
modified 10/9/02 10:21 AM
Variational Method
(See CTDL 11481155, [Variational Method]
252263, 295307[Density Matrices])
Last time:
QuasiDegeneracy
→
Diagonalize a part of infinite
H
* submatrix :
H
(0)
+
H
(1)
* corrections for effects of outofblock elements:
H
(2)
(the Van Vleck transformation)
*diagonalize
H
eff
=
H
(0)
+
H
(1)
+
H
(2)
coupled HO’s 2 : 1 (
ω
1
≈
2
ω
2
) Fermi resonance example: polyads
1. Perturbation Theory vs. Variational Method
2. Variational Theorem
3. Stupid nonlinear variation
4. Linear Variation
→
new kind of secular Equation
5. Linear combined with nonlinear variation
6. Strategies for criteria of goodness — various kinds of variational
calculations
1. Perturbation Theory vs. Variational Method
Perturbation Theory in effect uses
∞
basis set
goals:
parametrically parsimonious fit model,
H
eff
fit parameters (molecular constants)
↔
parameters that define V(x)
order  sorting
H
nk
(
1
)
E
n
(
0
)
−
E
k
(
0
)
<1
— errors less than this “mixing
angle” times the
previous order
non–zero correction term
(n is inblock, k is outof block) because diagonalization is
∞
order
(within block).
Variational
Method
best possible estimate for lowest few E
n
,
ψ
n
(and properties derivable from
these) using finite basis set and exact form of
H
.
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View Full Document 18  2
5.73
Lecture
#18
modified 10/9/02 10:21 AM
Vast majority of computer time in Chemistry is spent in variational calculations
Goal is numbers.
Insight is secondary.
“
Ab Initio
” vs. “semiempirical” or
“fitting”
[intentionally bad basis set:
Hückel, tight binding –
qualitative behavior obtained by a fit to a few microscopic–
like
control parameters]
2. Variational Theorem
not necessarily
normalized
any observable
α≡
φ
A
φφ
≥
a
0
If
is approximation to eigenfunction of
ˆ
A
belonging to lowest eigenvalue a
0
, then
PROOF:
eigenbasis (which we do not know – but know it must exist)
A
n
=
a
n
n
=
n
n
n
∑
A
=
n
n,
′
n
∑
n
A
′
n
′
n
=
n
2
n
∑
a
n
=
n
n
n
∑
=
n
2
n
∑
completeness
a
n
δ
nn
′
A
=
a
n
n
2
n
∑
′
n
2
′
n
∑
subtract a
0
from both sides
α−
a
0
=
a
n
−
a
0
()
n
2
n
∑
′
n
2
′
n
∑
≥0
expand
φ
in eigenbasis of
A
, exploiting completeness
the variational Theorem
all terms in both sums are
≥
0
again, all terms in
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This note was uploaded on 11/28/2011 for the course CHEM 5.74 taught by Professor Robertfield during the Spring '04 term at MIT.
 Spring '04
 RobertField

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