Sturmian Orbitals in Quantum Chemistry
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Transcript of Sturmian Orbitals in Quantum Chemistry
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()
()
z
x
a b
(+)A B
=cost
z
x
y(+) A
P
P
(b)(a)1 2( )
1/2
(
1
2
)/2
(1
)/2
+
2
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arctan
-2 -2
0 0
2 2
4 4
(f+
2)/(2+n)
0 /2/4
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x
z
y
x
k =0.866
= /3
x
z
y
x
k =0.707
= /4
x
z
y
x
k =0.500
= /6
xx
z
y
k =0.00
= 0.00
x
z
y
x
k =1.00
= /2
z
z
z
z
z
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0 0.2 0.4 0.6 0.8 1
k2
0 0
1 1
2 2
3 3
4 4
5 5
6 6
0 0.2 0.4 0.6 0.8 1
k2
0
5
10
15
20
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,,
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,
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|l >
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1p
2p p
,P
u
u
u2
3
1
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(a)0
0 = 2
O 4 O 2 O 2
Y
n ; ; m
; ; '
O 4 O 3 O 2
Y
n ; l ; m
; ; '
x = s i n s i n c o s '
y = s i n s i n s i n '
z = s i n c o s
w = c o s
(b)x = s i n c o s '
y = s i n s i n '
z = c o s s i n
w = c o s c o s
S
m
xw y
(,,)
z
n-1
3
3l
z
y
x
y
x
z w
w z x y
m
n-1
S
(,,)
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w z x y
m
z y
Wigner rotation matrix
w
l
m
x
n-1
l
,
,
n-1
(a) (b)
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x z x y
n-1
l
w z
n-1
Z matrix
w
m
y
m
/2
(b)(a)
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0 0 0 0
0 0 0 0
0 0 0 0
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1 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 11 1 1 1
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1 1 1 1 1 1 1 1 1 1 1 1
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1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 10 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0
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1 1 1 1 1 1 1 1 1
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1 1 1 1 1 1 1 1 1
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0 0 0 0 0 0
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0 0 0 0 0 0 0 0 0
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1 1 11 1 11 1 11 1 11 1 11 1 11 1 1
1 1 1
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1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
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1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1
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1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1
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1 1 1 1 1 1
1 1 1 1 1 1
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1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
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1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 11 1 1 1 1 11 1 1 1 1 1
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1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
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1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
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1 1 1 1 1 1 1 1
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1 1 1 1 1
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0000
0
1111
1
0 0 0
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1 1 1
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1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1
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1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
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1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
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1 1 1 1 1 1 1 1
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1 1 1 1 1 1 1 1
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1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1
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1 1 1
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1 1 11 1 11 1 1
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0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1
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,
,
(c)
(k)
(a)
n-1, q(k)
(b) n-1, b(k)n-1
m
z x y w
m
x y z ww x y z
l,
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Znm
lZnm
l m m
0
k k k k
1 0 1
||Cnm
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type II type I
n1, b(k)
m
n1, q(k)
m
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k k k k
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type II type IEll. CylindricalEll. Cylindrical
m
x
n1
m
n1
m
z x y w
x y w z
lm
y w
y w
l
m
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wx y z
z x
m
m
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zwyxz
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0
1
2
3
4
0
1
2
3
4
/4 /2
Arctan k2
q b
Arctan 1/(1-k2)
n+k
2 n+1/(1-k2)
Elliptic Cylindrical Basis, Type IElliptic Cylindrical Basis, Type II
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0 2 4 6
R ( = s / p0)
-2
-1.8
-1.6
-1.4
-1.2
-1
-0.8
-0.6
E(=
-p02/2)
{1s, 2s, 2pz}
{1s}
{1s, 2s}
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0 50 100
cpu time (arb. units)
1.0010-6
1.0010-5
1.0010-4
1.0010-3
1.0010-2
1.0010-1
Err.(atomicunits)
third iteration
second iteration
first iteration
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0 2 4
R = s / k (a.u.)
-2
-1.8
-1.6
-1.4
-1.2
-1
-0.8
-0.6
E=-k
2/
2
(a.u.)
Energy of the electron in the H2
+ground state (1
g)
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1 2 3 4
R (a.u.)
-1.15
-1.1
-1.05
-1
V(R)(a.u.)
H2
ground state (1g) potential energy curve
(a)
(b)
(c)
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y
z
x
1
5
3
2
4
4
5
1
3
y
z
x
2
x
y
z
1
2
3
4
5
lax
lax
lax
lax
leqleq
leq
leq
leq
leq
lot
lot
lot
lot
D3h C4v
D3h
lpi
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v
v
C 2
v
d
d
v
C
C
2
4
C4 v
D3 h
C2 v
v
v
v
C
C
C
C S
2
2
2
3 3
h
trigonal bypiramid intermediate configuration transition state
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12
45
23
15
34
35
14
13
25
24
12
23
45
15
34
35
13
24
25
14
2
5
44
3
3
2
2
2
3
1
1
2
1
5
1
3
44
5
5
1
3
4
5
3
5
4
12
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D3h
(20)
(1)
(4)
(5)
(5)
(4)
(1)
G240
+
2
3
+ 2
+ 3
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-400 -200 0 200 400
Q (a.u.)
-840.74
-840.73
-840.72
E
nergy(Eh)
C4v
D3h
D3h
E1
E2
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