1. Historical introduction
2. The Schrödinger equation for one-particle problems
3. Mathematical tools for quantum chemistry
4. The postulates of quantum mechanics
5. Atoms and the ‘periodic’ table of chemical elements
6. Diatomic molecules
7. Ten-electron systems from the second row
8. More complicated molecules
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30
1/ 48-5
The one-electron atom
H (Z = 1), He+ (Z = 2), Li2+ (Z = 3), . . ., U91+ (Z = 92), . . .
Electronic Hamilton operator (for point-like clamped nucleus)∗ :
c = Tb + Vb = −
H
e
en
el
~2 2 Z e2
∇ −
2me
κ0 r
(139)
Determination of stationary bound states, i.e. solutions h r | ψ i = ψ(r )
c
with E < 0, to the time-independent Schrödinger equation (with H
el
from above):
c − E ψ(r ) =
H
el
!
~2 2 Z e2
−
− E ψ(r ) = 0
∇ −
2me
κ0 r
(140)
Firstly, we remove the fundamental constants by switching to ‘atomic
units’. This reduces the mathematical work to pure numbers, and
eliminates quantities which have experimental uncertainties.
∗ The finite mass of the nucleus can be taken into account by switching from the electron mass m
e
to a reduced mass µ, where µ−1 = me −1 + mN −1 and mN is the nuclear mass.
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30
2/ 48-5
Atomic unitsa
Physical quantity
Symbol (name)
me
e
~
κ0
mass
charge
angular momentum, action
el. permittivity 4πε0
length κ0 ~2/(me e2 ) = ~/(αme c)
time ~/Eh = ~/(α2 me c2 )
velocity a0 Eh /~ = αc
linear momentum ~/a0 = αme c
force Eh /a0
energy e2 /(κ0 a0 ) = α2 me c2
power Eh 2 /~
charge density e/a03
el. current eEh/~
el. potential Eh /e
el. capacitance κ0 a0
el. resistance ~/e2
el. field strength (E ) Eh /(ea0)
el. displacement (D ) e/a02
el. dipole moment ea0
el. quadrupole moment ea0 2
el. polarizability (ea0)2 /Eh
magn. flux ~/e
magn. flux density (B ) ~/(ea02 )
magnetizing force (H ) eEh /(a0 ~)
magn. dipole moment e~/me = 2µB
a0 (bohr)
Eh (hartree)
Value in SI unitsb
9.1093826(16)
× 10−31
1.60217653(14) × 10−19
1.05457168(18) × 10−34
1.112650056 . . .
× 10−10
5.291772
2.418884
2.187691
1.992852
8.238723
4.359744
1.802378
1.081202
6.623618
2.721138
5.887891
4.108236
5.142206
5.721476
8.478353
4.486551
1.648777
6.582119
2.350517
1.251682
1.854802
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
10−11
10−17
106
10−24
10−8
10−18
10−1
1012
10−3
101
10−21
103
1011
101
10−30
10−40
10−41
10−16
105
108
10−23
kg
C
Js
F m−1
m
s
m s−1
kg m s−1
N
J
W
C m−3
A
V
F
Ω
V m−1
C m−2
Cm
C m2
C2 m2 J−1
Wb
T
A m−1
J T−1
a Based on CODATA recommended values 2002 (http://physics.nist.gov/constants/).
b The standard deviation uncertainty in the least significant digits is given in parentheses.
Now the Schrödinger equation reads
c − E ψ(r ) = − 1 ∇2 − Z − E ψ(r ) = 0
H
(141)
el
2
r
which is transformed from cartesian coordinates (x, y, z) to spherical
coordinates (r, θ, φ), with 0 ≤ r < ∞, 0 ≤ θ ≤ π, and 0 ≤ φ ≤ 2π.
Laplace operator ∆ and squared angular momentum operator lb2 (in
atomic units) in spherical coordinates:
lb2
1 ∂ 2∂
lb2
1 ∂2
∆ = ∇2 = 2
r
− 2=
r
−
r (
∂r ∂r r
r ∂r 2
r2 )
∂
1
∂2
1 ∂
2
b
sin θ
+
l = −
sin θ ∂θ
∂θ
sin2 θ ∂φ2
(142)
(143)
Separation ansatz for the state function:
ψ(r ) = ψ(r, θ, φ) = R(r) Y (θ, φ)
(144)
(this is always possible for ‘central fields’, i.e. V = V (r)).
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30
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The angular part is found to be a spherical harmonic, Y (θ, φ) =
Ylm(θ, φ), which is an ‘angular momentum eigenfunction’, i.e. a simultaneous eigenfunction of lb2 and lbz (in atomic units):
lb2 Ylm(θ, φ) = l(l + 1) Ylm(θ, φ)
lbz Ylm(θ, φ) = m Ylm(θ, φ)
(145)
(146)
Orbital angular momentum quantum number l: l = 0, 1, 2, 3, . . .,
Magnetic quantum number m: −l ≤ m ≤ l.
Explicit form for the spherical harmonics (with Condon-Shortley† phase
convention), −l ≤ m ≤ l:
Ylm(θ, φ) = Θlm(θ) Φm (φ) = (−1)m Nlm Plm (cos θ) ei m φ
Nlm =
s
2l + 1 (l − m)!
,
4π (l + m)!
Z
Ω
(147)
Yl,−m(θ, φ) = (−1)m Ylm∗(θ, φ)
Ylm∗(θ, φ) Yl0m0 (θ, φ) dΩ = δll0 δmm0
† E. U. Condon (1902-1974), G. H. Shortley (∗ 1910)
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30
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Notation for single-particle eigenfunctions of angular momentum:
lb2 | ψαlm i = l(l + 1) ~2 | ψαlm i
l
0
s
lbz | ψαlm i = m ~ | ψαlm i
1
p
2
d
3
f
4
g
5
h
6
i
7
k
8
l
9
m
10
n
...
...
Notation for many-particle eigenfunctions of total angular momentum:
2
b2 | Ψ
L
αLM i = L(L + 1) ~ | ΨαLM i
b =
L
z
L
0
S
b |Ψ
L
z
αLM i = M ~ | ΨαLM i
n
X
i=1
lbz,i ,
1 2
P D
3
F
b =
L
4
G
5
H
n
X
i=1
6
I
bl ,
i
7
K
8
L
b · L
b
b2 = L
L
9
M
10
N
...
...
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30
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The associated Legendre‡ functions Plm (cos θ) are related to the Legendre polynomials Pl (x) = Pl0 (x) (x = cos θ). The following relations
hold (for −l ≤ m ≤ l, where applicable):
l+m
1
2 m/2 d
2
l
(1
−
x
)
(x
−
1)
2l l!
dxl+m
l
1 d
Pl (x) = l
(x2 − 1)l = 2F1 (−l, l + 1; 1; (1 − x)/2)
l
2 l! dx
(l − m)! m
Pl−m (x) = (−1)m
P (x)
(l + m)! l
l+m
m (x) = 2l + 1 x P m (x) −
m (x)
Pl+1
Pl−1
l
l−m+1
l−m+1
2m
x Plm (x) − (l + m)(l − m + 1) Plm−1 (x)
Plm+1 (x) = q
1 − x2
Plm (x) =
(148)
(149)
(150)
(151)
(152)
‡ A.-M. Legendre (1752-1833)
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30
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The first Legendre polynomials
l
Pl (x)
0
1
1
x
2
1 (3x2 − 1)
2
1 (5x3 − 3x)
2
1 (35x4 − 30x2 + 3)
8
1 (63x5 − 70x3 + 15x)
8
3
4
5
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30
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The first associated Legendre functions
l
m
Plm (cos θ)
1
3
0
0
cos θ
3
+1
1
+1
sin θ
3
1
−1
1 sin θ
−2
3
−1
+2
1
3
2 (5 cos θ − 3 cos θ)
3 sin θ (5 cos2 θ − 1)
2
2
−1
8 sin θ (5 cos θ − 1)
15 sin2 θ cos θ
3
3 sin θ cos θ
3
−2
+3
1 sin2 θ cos θ
8
15 sin3 θ
3
−3
1 sin3 θ
− 48
l
m
0
0
1
2
0
Plm (cos θ)
1 (3 cos2 θ − 1)
2
2
+1
2
2
−1
+2
−1
2 sin θ cos θ
2
−2
1
2
8 sin θ
3 sin2 θ
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30
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The first spherical harmonics
(Condon-Shortley phase convention)
m
Ylm(θ, φ)
0
0
1
+1
1
0
1
−1
2
+2
2
+1
2
0
2
−1
2
−2
1
4π
q
3 sin θ eiφ
− 8π
q
3 cos θ
4π
q
3 sin θ e−iφ
8π
q
5 sin2 θ e2iφ
3 96π
q
5 sin θ cos θ eiφ
−3 24π
q
5 (3 cos2 θ − 1)
1
2
4π
q
5 sin θ cos θ e−iφ
3 24π
q
5 sin2 θ e−2iφ
3 96π
l
q
l
m
3
+3
3
+2
3
+1
3
0
3
−1
3
−2
3
−3
Ylm(θ, φ)
q
7
3
3iφ
2880π sin θ e
7 sin2 θ cos θ e2iφ
15 480π
q
7 sin θ (5 cos2 θ − 1) eiφ
−3
2 48π
q
7 (5 cos3 θ − 3 cos θ)
1
2
4π
q
3
7 sin θ (5 cos2 θ − 1) e−iφ
2 48π q
7 sin2 θ cos θ e−2iφ
15 480π
q
7
sin3 θ e−3iφ
15 2880π
−15
q
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30
10/ 48-5
Two-point boundary value problem for P (r) = r R(r) (as resulting
from the separation ansatz):
d2
+ 2 [ E − Vl (r) ]
dr 2
!
(153)
P (r) = 0
l(l + 1) Z
− ,
P (0) = 0 ,
lim P (r) = 0
r→∞
2 r2
r
Physically acceptable (i.e. normalizable) solutions exist only for a
discrete set of energies: Quantization due to the boundary conditions.
Vl (r) =
Radial functions (eigenfunctions):
(2l+1)
Pnl (r) = r Rnl (r) = Nnl xl+1 Lnr
x=
2Z
r,
n
Z ∞
0
Nnl =
1
n
s
2
Z
nr !
,
(n + l)!
Rnl (r) Rn0l (r) r dr =
Z ∞
0
(x) e−x/2
(154)
nr = n − l − 1 ≥ 0
Pnl (r) Pn0l (r) dr = δnn0
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30
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Energy eigenvalues (in atomic units):
1 Z2
2 n2
These are degenerate for n > 1 (further details below).
Principal quantum number n: n = 1, 2, 3, . . .
En = Enlm = −
(155)
This result for En is also an important hint for the understanding of
the stability of matter: En > − ∞, i.e. the electron does not collapse
into the nucleus, despite the singular attractive potential, due to a
balance between kinetic and potential energy.
Generalized Laguerre§ polynomials:
(α)
Lk (x) =
(α)
Lk (x) =
(α + 1)k
1F1 (−k; α + 1; x)
k!
2k + α − 1 − x (α)
k + α − 1 (α)
Lk−1 (x) −
Lk−2 (x)
k
k
(156)
(k ≥ 2)
(157)
§ E. N. Laguerre (1834-1886)
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30
12/ 48-5
The first generalized Laguerre polynomials
(α)
Lk (x)
k
0
1
1
(α + 1) − x
2
3
4
5
h
i
1 (α + 1)(α + 2) − 2(α + 2)x + x2
2
h
i
1 (α + 1)(α + 2)(α + 3) − 3(α + 2)(α + 3)x + 3(α + 3)x2 − x3
6
h
1
24 (α + 1)(α + 2)(α + 3)(α + 4) − 4(α + 2)(α + 3)(α + 4)x
i
2
3
4
+ 6(α + 3)(α + 4)x − 4(α + 4)x + x
h
1
120 (α + 1)(α + 2)(α + 3)(α + 4)(α + 5)
− 5(α + 2)(α + 3)(α + 4)(α + 5)x + 10(α + 3)(α + 4)(α + 5)x2
− 10(α + 4)(α + 5)x3 + 5(α + 5)x4 − x5
i
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30
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The first radial functions of one-electron atoms
n
l
nr
1s
1
0
0
2s
2
0
1
2p
2
1
0
3s
3
0
2
3p
3
1
1
3d
3
2
0
4s
4
0
3
4p
4
1
2
4d
4
2
1
4f
4
3
0
Pnl (r) (x = 2Zr/n)
√
Z x e−x/2
1
2
q
Z x (2 − x) e−x/2
2
q
1 Z x2 e−x/2
2 6
q
1 Z x (3 − 3x + x2/2) e−x/2
3 3
q
Z 2 (4 − x) e−x/2
1
3 24 x q
1
Z
3 −x/2
3 120 x e
q
1 Z x (4 − 6x + 2x2 − x3/6) e−x/2
4 4q
1
Z 2
− 5x + x2/2) e−x/2
4 60 x (10
q
1
Z
3
−x/2
4 720 xq (6 − x) e
Z
1
4 −x/2
4 5040 x e
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30
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Radial functions Pnl (r) of the one-electron atom (for n = 1, 2, 3, 4)
0.8
0.6
Pnl (r)
Z 1/2 0.4
0.2
0.0
n = 1 (1s)
0.4
Pnl (r)
Z 1/2 0.2
.
......
... ....
... ...
.... .....
.. .
.. ...
.. ....
..
...
...
....
...
..
...
...
...
....
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
....
....
.....
...
.........
...
.................................
............................
.
0
0.0
-0.2
n = 3 (3s, 3p, 3d)
.............................................................................................
...........
........
... ...............
..........
......
.........
.....
.........
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.
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........................................................................ .... .... .... .... .... .... .... .......
-0.4
10
0
Zr
10
20
30 Zr
n = 2 (2s, 2p)
0.4
Pnl (r)
Z 1/2 0.2
0.0
-0.2
-0.4
............
..... .........
....
...
....
...
....
....
....
..
....
....
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....
.
.
....
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.....
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.
.
...
..
....
....
.....................
0.4
Pnl (r)
Z 1/2 0.2
.... .... .... .... .... .... ................................................
0.0
-0.2
n = 4 (4s, 4p, 4d, 4f)
..............
........................... ..............................................................................................................................
.................
..............
.... ....................
...........
................
................
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................
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............................................................................................
.................................................................................................................... .... .... .... .... .... ....
-0.4
0
0
10
10
20
30
40 Zr
20 Zr
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30
15/ 48-5
Eigenfunctions and energy eigenvalues (in atomic units) of the oneelectron atom:
1
ψnlm(r ) = Pnl (r) Ylm(θ, φ) ,
r
n = 1, 2, 3, 4, . . . ,
0 0
1 Z2
Enlm = En = −
2 n2
l = 0, 1, . . . , n − 1 ,
0
h nlm | n l m i =
Z
(158)
m = −l, −l + 1, . . . , l
ψnlm(r ) ψn0l0m0 (r ) dr = δnn0 δll0 δmm0
Ground state and corresponding energy:
ψ100(r ) =
s
Z 3 −Zr
1√
1
e
=
Z 2Zr e−Zr √
,
π
r
4π
Z2
E1 = −
2
(159)
For isovalue plots — i.e. representations of all points r with ψnlm(r ) =
|c| for chosen c ∈ R — of the eigenfunctions of the one-electron atom,
see J. Brickmann, M. Klöffler, H.-U. Raab, Chemie in unserer Zeit 12
(1978) 23-26.
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30
16/ 48-5
Degree of degeneracy¶:
m degeneracy:
l degeneracy (without spin):
l degeneracy (spin included):
gl =
l
X
1 = 2l + 1
(160)
m=−l
n−1
X
gl = n 2
(161)
l=0
s
g n = 2 gn = 2 n2
(162)
gn =
s essentially determines the length of the rows (‘periods’)
The value gn
in the table of chemical elements (2, 8, 18, 32), whereas the value
2 gl = 2(2l + 1) determines the block structure of the ‘periodic’ table
(s-, p-, d-, and f-block for l = 0, 1, 2, 3, respectively).
¶ The degeneracy with respect to l, eq. (161), is a special property of the one-electron atom (with
point-like nucleus), and is not present in many-electron atoms. For example, the 2s and 2p states
of a one-electron atom are degenerate, i.e. they have the same energy, but the ‘orbital energies’
for the 2s and 2p orbitals in any state of a many-electron atom are always different.
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30
17/ 48-5
The ‘periodic’ table of the chemical elements
(2004)
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18
1
1
2
H
He
2
3
4
5
6
7
8
9
10
Li
Be
B
C
N
O
F
Ne
3
11
12
Na Mg
4
19
K
37
Ar
31
32
33
34
35
36
24
V
Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr
41
42
47
30
18
Cl
23
5
38
39
40
Rb Sr
Y
Zr Nb Mo Tc Ru Rh Pd Ag Cd
6
∗
77
78
Cs Ba
Hf Ta W Re Os
Ir
Pt Au Hg Tl Pb Bi Po At Rn
7
106
109
Fr Ra
55
87
56
88
∗
∗∗
72
73
74
75
76
46
29
17
S
22
45
28
16
P
Ti
44
27
15
21
43
26
14
Si
Ca Sc
20
25
13
Al
79
48
80
∗∗
104
111
112
Rf Db Sg Bh Hs Mt Ds
X (X)
57
58
65
66
105
59
60
107
61
108
62
63
110
64
49
50
In
Sn Sb Te
81
82
113
67
114
( X)
68
51
83
115
69
52
84
116
(X)
70
53
54
I
Xe
85
86
117
118
(?)
71
La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu
89
90
91
Ac Th Pa
92
U
93
94
95
96
97
98
Np Pu Am Cm Bk Cf
99
100
101
102
103
Es Fm Md No Lr
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30
18/ 48-5
The variation principle at work (I)
The ground state 1s1 2 S of the one-electron atom
Definition of the system under consideration:
c = Tb + Vb ,
H
A set of trial functionsk :
Tb = −
1. Slater function (ζ > 0):
1
∆,
2
Vb = Vnuc(r ) = −
3. Lorentz function (a > 0):
h
ψS = NS exp (−ζr)
ψL = NL 1 + (ar)
i
2 −1
4. Preuß function (c > 0):
2. Gauß function (α > 0):
ψ G = NG
Z
r
h
exp (−αr 2)
ψP = NP 1 + cr
k Refs.:
i−2
C. Zener, Phys. Rev. 36 (1930) 51, J. C. Slater, Phys. Rev. 36 (1930)
57 (Slater function) — S. F. Boys, Proc. R. Soc. London A 200 (1950) 542,
H. Preuß, Z. Naturforsch. A 11 (1956) 823 (Gauß function) — H. Preuß, Z.
Naturforsch. A 13 (1958) 439 (Preuß function).
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30
19/ 48-5
Trial functions (unnormalized)
for the ground state of the one-electron atom
1 ......
f (x)
0
.......................
.........
...
.......
...
......
...
.....
.....
...
.....
...
.......
...
........
...
........
...
.........
...
........
...
........
...
........
........
...
........
...
... ...
...
... .....
...
... ....
...
... ....
...
... ....
.... ....
...
...
... .....
...
... ....
... .....
...
... ....
...
... .....
...
... ....
...
... .....
...
...
.
...
... ........
...
...
.....
...
...
....
...
...
.....
...
...
.....
....
....
.....
....
....
.....
....
.....
....
...
....
.....
.
...
....
.....
.
...
....
.....
.....
...
.....
...
.....
......
...
.....
.....
...
......
.....
....
......
.....
....
.......
.....
....
.......
......
........
....
......
.........
....
......
.........
....
.......
..........
....
........
...........
.........
....
...........
..........
....
............
...........
....
.............
............ .....
..............
............. ....
...............
................
................
................
..................
..... ..................
....................
..... ....................
......................
.....................
......
........................
.
.
.
........................
......
............................
............................
.......
.................................
.................................
........
......................................
.
.
.
.
.
.........................................
.........
.............
....................................................
..........
...................................................................
............
....................
..............
..................
..........................
........................................................
...........................................................................................................................................................
...
...
...
S: exp(−x), x = ζr
...
...
2), x = √αr
...
G:
exp(−x
...
....
....
....
L: 1/(1 + x2), x = ar
....
....
....
....
P: 1/(1 + x)2 , x = cr
....
.....
.....
......
......
......
......
......
.......
.......
........
..........
...........
............
.............
...............
..................
......................
............................
.........................................
....
0
1
x
2
3
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30
20/ 48-5
Estimate for the ground state energy:
c|ψi =
hE i = hH i = hψ|H
Z
c ψ(r ) dr = h T i + h V i
ψ(r ) H
1
1
h ψ | ∆ | ψ i = + h ∇ψ | ∇ψ i ,
2
2
Mathematical preliminaries:
• The beta function:
h V i = − Z hr −1 i
hT i = −
B(a, b) =
B(a, b) =
Z 1
0
Γ(a) Γ(b)
= B(b, a)
Γ(a + b)
ta−1(1 − t)b−1 dt =
• Useful integral formulas:
Z ∞
0
Z ∞
0
xs−1 exp (−p xn) dx =
Z ∞
0
ta−1
dt
(1 + t)a+b
1 Γ(s/n)
n ps/n
xs−1
1
dx
=
B(p − s/n, s/n)
(1 + xn)p
n
(a > 0, b > 0)
(p > 0, s/n > 0)
(s/n > 0, np − s > 0)
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30
21/ 48-5
1. The Slater function ψS = NS exp (−ζr), ζopt =?:
hr i = NS2 4π
k
Z ∞
0
dr r k+2e−2ζr = NS2 4π
π
h1i = hr i = NS2 3 ≡ 1
ζ
0
⇒
NS =
Γ(k + 3)
(2ζ)k+3
q
(k > −3)
ζ 3 /π
Γ(k + 3) 1
2k+1 ζ k
2
Z
1 2 4π ∞
d
−x
h T i = + NS
dx
xe
2
ζ 0
dx
Z
1
1 2 4π ∞
dx (1 − x)2 e−2x = ζ 2
= NS
2
ζ 0
2
hV i = −Z ζ
1
dh E i
h E i = ζ2 − Z ζ ⇒
= ζ − Z = 0 ⇒ ζopt = Z
2
dζ
1
hE i
1 2
x
−
1
Z
=
x
(x
=
ζ/Z)
⇒
E
=
−
min
Z2
2
2
hr k i =
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30
22/ 48-5
2. The Gauß function ψG = NG exp (−αr 2), αopt =?:
2 4π
hr k i = NG
Z ∞
0
2
h1i = hr 0 i = NG
2
1 Γ( k+3
k+2
−2αr
2
2 )
dr r
e
= NG 4π
k+3
π 3/2
≡1
2α
⇒
2 (2α) 2
2α 3/4
NG =
π
(k > −3)
2 Γ( k+3
2 )
hr i = √
π (2α)k/2
2
Z
d
1 2 4π ∞
−x2
dx
xe
h T i = + NG √
2
α 0
dx
Z
2
1 2 4π ∞
3
= NG √
dx (1 − 2x2)2 e−2x = α
2
α 0
2
k
hV i = − 2Z
q
2α/π
s
s
3
2α
2
3
dh E i
8Z 2
= −Z
= 0 ⇒ αopt =
hEi = α −2Z
⇒
2
π
dα
2
πα
9π
s √
hEi
2
4 2
3
(x
=
α/Z)
⇒
E
=
−
x
−
2
Z
=
x
min
Z2
2
π
3π
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30
h
3. The Lorentz function ψL = NL 1 + (ar)2
Z ∞
i−1
23/ 48-5
, aopt =?:
r k+2 dr
1−k k+3
2 4π 1
=
N
L k+3 B( 2 , 2 )
2
2
a
2
0 (1 + (ar) )
q
2
0
2π
NL = a3 /π
⇒
h1i = hr i = NL 3 ≡ 1
a
2
1
1−k k+3
,
)
(−3 < k < 1)
hr k i =
B(
2
2
π ak
2
Z
1 2 4π ∞
d
x
h T i = + NL
dx
2
a 0
dx 1 + x2
Z ∞
1 2 4π
(1 − x2)2
1 2
= NL
dx
a
=
2
a 0
(1 + x2)4
4
h V i = − 2 Z a/π
1
dh E i
4Z
a
1
2
h E i = a2 − 2 Z
⇒
= a − Z = 0 ⇒ aopt =
4
π
da
2
π
π
2
4 2
1
hE i
x
−
(x
=
a/Z)
⇒
E
=
−
=
x
Z
min
Z2
4
π
π2
hr i = NL2 4π
k
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30
24/ 48-5
h
4. The Preuß function ψP = NP 1 + cr
i−2
, copt =?:
Z ∞ k+2
r
dr
2 4π
=
N
B(1 − k, k + 3)
P
ck+3
0 (1 + cr)4
s
3c3
4π
0
2
h1i = hr i = NP 3 ≡ 1
NP =
⇒
3c
4π
2
hr i = NP
4π
k
3
hr k i = k B(1 − k, k + 3)
(−3 < k < 1)
c
!2
Z
1 2 4π ∞
d
x
h T i = + NP
dx
2
c 0
dx (1 + x)2
Z
1 2
1 2 4π ∞
(1 − x)2
=
c
= NP
dx
2
c 0
(1 + x)6
5
h V i = − Z c/2
1
dh E i
5Z
c
2
Z
h E i = c2 − Z
⇒
= c − = 0 ⇒ copt =
5
2
dc
5
2
4
hE i
5 2
1
1
(x
=
c/Z)
⇒
E
=
−
=
x
x
−
Z
min
Z2
5
2
16
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30
25/ 48-5
Variation of the ground state energy h E i of the one-electron atom
with the trial function parameter
hE i
= f (x) = a2 x2 + a1 x
2
Z
x
1
..
...
...
....
..
......
...
...........
...
... ..........
... ...........
....
... ...........
..
... ..... .....
...
... ...... .......
...
.... ......
...
..
..... ......
...
.
.
..... ......
...
...
..... .......
...
...
..... .......
...
..
..... ........
...
..
........
.....
.
..
...
.
.........
.....
........
..
...
..........
.....
.........
...
...
...........
......
...........
...
............
...
......
............
.
.
.
.
.
.
............. .....
.
.
.
.....
...
.
.
.................
........
......
...
.....
.
................
.......
......
...
... .........................................................................................................................
........
.......
...
........ ......
........
...
...........
.........
....
.
.
.
.
.
.
.
.
.
.
.
.
.
....
.. .....
.....
.....
............
.... .........................
.................
.............
.....
....
....................................... .......................................................
......
.....
...
...........
......
..........................
f (x)
−0.5
2
..
...
...
.
.
...
...
...
...
.
...
.
...
...
...
...
...
.
.
....
...
....
....
.
....
.
.
.
....
....
.....
....
......
.
.
.
.
......
.....
.......
......
.
.
.
..........
.
.
.
.
........................................
0 ......
S: x2 /2 − x, x = ζ/Z
p
√
G: 3x2 /2 − 2 2/π x, x = α/Z
L: x2 /4 − 2x/π, x = a/Z
P: x2/5 − x/2, x = c/Z
For optimal choice of the parameter x (i.e. at the minima), the
quantum mechanical virial theorem, h V i / h T i = −2, is fulfilled, and
thus h E i = h V i /2 = − h T i in all four cases.
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30
26/ 48-5
Perturbation theory
Time-independent Rayleigh-Schrödinger perturbation theory for nondegenerate states
Find a solution for the Schrödinger equation
c ψ =E ψ
H
λ λ
λ λ
⇔
with the Hamilton operator
c =H
c(0) +
H
λ
∞
X
k=1
c −E
H
λ
λ
ψλ = 0
c(k) ,
λk H
(163)
(164)
where λ is a (natural or artificial) perturbation parameter (|λ| < λmax),
and assume that the solutions of the unperturbed problem
c(0)ψ (0) = E (0)ψ (0)
H
(165)
are completely known (with all E (0) non-degenerate). Then put
Eλ = E
(0)
+
∞
X
ε
(k) k
λ
and
ψλ = ψ
k=1
(0)
+
∞
X
χ(l) λl
(166)
l=1
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14
27/ 48-5
into eq. (163):




∞

X
c(0) − E (0) +
c(k) − ε(k) λk ψ (0) +
0= H
χ(l) λl 
H


l=1
k=1
(
∞
X
c(0) − E (0) ψ (0) +
c(0) − E (0) χ(m)
= H
λm H
∞ X
m=1
)
m−1
X c(k) − ε(k) χ(m−k) + H
c(m) − ε(m) ψ (0)
+
H
k=1
Thus we obtain from the coefficient of λm :
(0)
(0)
c(0)
m=0:
m=1:
m=2:
..
H
−E
ψ
=0
c(0) − E (0) χ(1) + H
c(1) − E (1) ψ (0) = 0
H
(0)
(0)
(1)
(1)
(2)
c
c
H
−E
−E
χ
+ H
χ(1)
c(2) − E (2) ψ (0) = 0
+ H
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14
28/ 48-5
The ‘intermediate normalization’ condition
h ψ (0) | ψλ i = 1
h ψ (0) | ψ (0) i = 1
with
implies orthogonality between ψ (0) and all χ(l):
h ψ (0) | χ(l) i = 0
(for l = 1, 2, . . .)
Resulting expressions for ε(1) and ε(2) :
c(1) | ψ (0) i
ε(1) = h ψ (0) | H
(167)
c(1) | χ(1) i + h ψ (0) | H
c(2) | ψ (0) i
ε(2) = h ψ (0) | H
(168)
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14
29/ 48-5
The perturbation theory may be used advantageously
• to determine (non-variational) approximations to the solutions of
eq. (163):
Eλ ≈ E
(n)
=E
(0)
+
n
X
ε(k) λk
n
X
χ(l) λl
k=1
ψλ ≈ ψ (n) = ψ (0) +
l=1
• to obtain exact values for derivatives of the energy E or the state
function ψ to various orders in λ at λ = 0, e.g.:
∂ nE = n! ε(n)
n
∂λ λ=0
The perturbation theory presented above can be extended to include
the case of degenerate states.
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14
30/ 48-5
The two-electron atom
H− (Z = 1), He (Z = 2), Li+ (Z = 3), . . ., U90+ (Z = 92), . . .
Hamilton operator (in atomic units):
b +h
b + 1
c = − 1 ∇2 − 1 ∇2 − Z − Z + 1 = h
H
(169)
1
2
el
2 1 2 2 r1 r2
r12
r12
General structure of state functions for two-electron systems ∗∗:
Φ(x1, x2 ) = f (r 1, r 2)Θ(σ1 , σ2)
Spin part Θ = | SMS i: Singlet (S = 0, para-He) or triplet (S = 1,
ortho-He)
√
| 00 i = (αβ − βα)/ 2
√
| 11 i = αα
| 10 i = (αβ + βα)/ 2
| 1 − 1 i = ββ
∗∗
. . . as long as the Hamilton operator does not act on the spin of the particles.
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14
31/ 48-5
For the spatial part f (r 1, r 2 ) a suitable choice must be made. For S
states we can simplify further to f (r1 , r2, r12 ), or equivalently f (s, t, u)
with s = r1 + r2 , t = r1 − r2, and u = r12 .
Variational results for the helium ground state 1s2 1S a.
f (s, t, u)
e−ζr1 e−ζr2 = e−ζs
ϕ(r1 ) ϕ(r2 )
e−ζr1 e−ηr2 + e−ηr1 e−ζr2
e−ζs+cu
e−ζs (1 + cu)
e−ζs+cu cosh (at)
e−ζs c0 + c1 u + c2t2 + c3s + c4 s2 + c5 u2
exact
− Eopt /a.u.
2.8477
2.8617
2.8757
2.8896
2.8911
2.8994
2.9032
2.9037
b
c
d
e
a E. A. Hylleraas, Z. Phys. 54 (1929) 347
b C. Froese Fischer: The Hartree-Fock method for atoms. Wiley, New York, 1977
c C. Eckart, Phys. Rev. 36 (1930) 878
d W.-K. Li, J. Chem. Educ. 64 (1987) 128
e K. Frankowski, C. L. Pekeris, Phys. Rev. 146 (1966) 46
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14
32/ 48-5
The many-electron atom
A part of the knowledge of the state functions of the one-electron
atom can be transferred to the many-electron atom, if the following
assumptions are made††:
1. Central field approximation: The electrons in the many-electron
atom are assumed to move in an effective central field Veff,l (r),
so that the orbitals can be written as ψ(r ) = R(r) Y (θ, φ), with
Y (θ, φ) = Ylm(θ, φ).
2. Equivalence restriction: The radial parts are assumed to be
independent of the magnetic quantum number m: R(r) = Rnl (r).
The resulting set of radial functions Pnl (r) = r Rnl (r) has to be determined for every state of the many-electron atom, e.g.
- He ground state (singlet): 1s2 1S → P10 (r)
- He excited states (singlet or triplet): 1s1 2s1 1,3S → P10(r), P20 (r)
- Li ground state (doublet): 1s2 2s1 2S → P10 (r), P20 (r)
†† In addition to the approximation of the many-electron state function as a Slater determinant (an
antisymmetrized product of spin orbitals), or a linear combination thereof.
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14
Electron configuration of neutral atoms in the ground state (designated as
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
H
He
Li
Be
B
C
N
O
F
Ne
Na
Mg
Al
Si
P
S
Cl
Ar
K
Ca
Sc
Ti
V
Cr
Mn
1s1
1s2
[He] 2s1
[He] 2s2
[He] 2s2
[He] 2s2
[He] 2s2
[He] 2s2
[He] 2s2
[He] 2s2
[Ne] 3s1
[Ne] 3s2
[Ne] 3s2
[Ne] 3s2
[Ne] 3s2
[Ne] 3s2
[Ne] 3s2
[Ne] 3s2
[Ar] 4s1
[Ar] 4s2
[Ar] 3d1
[Ar] 3d2
[Ar] 3d3
[Ar] 3d5
[Ar] 3d5
2S
0
2S
1/2
1S
2p1
2p2
2p3
2p4
2p5
2p6
3p1
3p2
3p3
3p4
3p5
3p6
4s2
4s2
4s2
4s1
4s2
1/2
1S
0
2P
1/2
3P
0
4S
3/2
3P
2
2P
3/2
1S
0
2S
1/2
1S
0
2P
1/2
3P
0
4S
3/2
3P
2
2P
3/2
1S
0
2S
1/2
1S
0
2D
3/2
3F
2
4F
3/2
7S
3
6S
5/2
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
Fe
Co
Ni
Cu
Zn
Ga
Ge
As
Se
Br
Kr
Rb
Sr
Y
Zr
Nb
Mo
Tc
Ru
Rh
Pd
Ag
Cd
In
Sn
[Ar]
[Ar]
[Ar]
[Ar]
[Ar]
[Ar]
[Ar]
[Ar]
[Ar]
[Ar]
[Ar]
[Kr]
[Kr]
[Kr]
[Kr]
[Kr]
[Kr]
[Kr]
[Kr]
[Kr]
[Kr]
[Kr]
[Kr]
[Kr]
[Kr]
3d6 4s2
3d7 4s2
3d8 4s2
3d10 4s1
3d10 4s2
3d10 4s2
3d10 4s2
3d10 4s2
3d10 4s2
3d10 4s2
3d10 4s2
5s1
5s2
4d1 5s2
4d2 5s2
4d4 5s1
4d5 5s1
4d5 5s2
4d7 5s1
4d8 5s1
4d10
4d10 5s1
4d10 5s2
4d10 5s2
4d10 5s2
33/ 48-5
2S+1 L
J)
5D
4
4F
9/2
3F
4
2S
1/2
1S
4p1
4p2
4p3
4p4
4p5
4p6
5p1
5p2
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14
0
2P
1/2
3P
0
4S
3/2
3P
2
2P
3/2
1S
0
2S
1/2
1S
0
2D
3/2
3F
2
6D
1/2
7S
3
6S
5/2
5F
5
4F
9/2
1S
0
2S
1/2
1S
0
2P
1/2
3P
0
34/ 48-5
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
Sb
Te
I
Xe
Cs
Ba
La
Ce
Pr
Nd
Pm
Sm
Eu
Gd
Tb
Dy
Ho
Er
Tm
Yb
Lu
Hf
Ta
W
Re
Os
Ir
[Kr]
[Kr]
[Kr]
[Kr]
[Xe]
[Xe]
[Xe]
[Xe]
[Xe]
[Xe]
[Xe]
[Xe]
[Xe]
[Xe]
[Xe]
[Xe]
[Xe]
[Xe]
[Xe]
[Xe]
[Xe]
[Xe]
[Xe]
[Xe]
[Xe]
[Xe]
[Xe]
4d10 5s2 5p3
4d10 5s2 5p4
4d10 5s2 5p5
4d10 5s2 5p6
6s1
6s2
5d1 6s2
4f1 5d1 6s2
4f3 6s2
4f4 6s2
4f5 6s2
4f6 6s2
4f7 6s2
4f7 5d1 6s2
4f9 6s2
4f10 6s2
4f11 6s2
4f12 6s2
4f13 6s2
4f14 6s2
4f14 5d1 6s2
4f14 5d2 6s2
4f14 5d3 6s2
4f14 5d4 6s2
4f14 5d5 6s2
4f14 5d6 6s2
4f14 5d7 6s2
4S
3/2
3P
2
2P
3/2
1S
0
2S
1/2
1S
0
2D
3/2
1G
4
4I
9/2
5I
4
6H
5/2
7F
0
8S
7/2
9D
2
6H
15/2
5I
8
4I
15/2
3H
6
2F
7/2
1S
0
2D
3/2
3F
2
4F
3/2
5D
0
6S
5/2
5D
4
4F
9/2
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
Pt
Au
Hg
Tl
Pb
Bi
Po
At
Rn
Fr
Ra
Ac
Th
Pa
U
Np
Pu
Am
Cm
Bk
Cf
Es
Fm
Md
No
Lr
Rf
[Xe]
[Xe]
[Xe]
[Xe]
[Xe]
[Xe]
[Xe]
[Xe]
[Xe]
[Rn]
[Rn]
[Rn]
[Rn]
[Rn]
[Rn]
[Rn]
[Rn]
[Rn]
[Rn]
[Rn]
[Rn]
[Rn]
[Rn]
[Rn]
[Rn]
[Rn]
[Rn]
4f14 5d9 6s1
4f14 5d10 6s1
4f14 5d10 6s2
4f14 5d10 6s2
4f14 5d10 6s2
4f14 5d10 6s2
4f14 5d10 6s2
4f14 5d10 6s2
4f14 5d10 6s2
7s1
7s2
6d1 7s2
6d2 7s2
5f2 6d1 7s2
5f3 6d1 7s2
5f4 6d1 7s2
5f6 7s2
5f7 7s2
5f7 6d1 7s2
5f8 6d1 7s2
5f10 7s2
5f11 7s2
5f12 7s2
5f13 7s2
5f14 7s2
5f14 6d1 7s2
5f14 6d2 7s2
3D
2S
6p1
6p2
6p3
6p4
6p5
6p6
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14
3
1/2
1S
0
2P
1/2
3P
0
4S
3/2
3P
2
2P
3/2
1S
0
2S
1/2
1S
0
2D
3/2
3F
2
4K
11/2
5L
6
6L
11/2
7F
0
8S
7/2
9D
2
8H
17/2
5I
8
4I
15/2
3H
6
2F
7/2
1S
0
35/ 48-5
LS terms for electron configurations p2 and p3
p2 : 3P (9), 1D (5), 1S (1)
p3 : 4S (4), 2D (10), 2 P (6)
6
2 = 15 = 9 + 5 + 1
6
3 = 20 = 4 + 10 + 6
Energy levels of neutral tetravalent atoms from the p-, d-, and f-block
(C, Ti, Ce), within an energy range above lowest ground state level:
∆E = 1 eV ≈ 8065.55 cm−1
(Eh = 2 h c R∞)
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14
36/ 48-5
Energy levels of C I‡‡ (280 lines of data) — IP = 11.26030 eV
----------------------------------------------Configuration
| Term
| J | Level
|
|
| (cm-1)
-------------------|--------|-----|-----------2s2.2p2
| 3P
| 0 |
0.00
|
| 1 |
16.40
|
| 2 |
43.40
|
|
|
2s2.2p2
| 1D
| 2 |
10192.63
|
|
|
2s2.2p2
| 1S
| 0 |
21648.01
|
|
|
2s.2p3
| 5S*
| 2 |
33735.20
|
|
|
2s2.2p(2P*)3s
| 3P*
| 0 |
60333.43
|
| 1 |
60352.63
|
| 2 |
60393.14
|
|
|
2s2.2p(2P*)3s
| 1P*
| 1 |
61981.82
|
|
|
2s.2p3
| 3D*
| 3 |
64086.92
|
| 1 |
64089.85
|
| 2 |
64090.95
|
|
|
..
.
..
.
..
.
..
.
..
.
..
.
|
|
|
| 1F*
| 3 |
90721.0
|
|
|
2s2.2p(2P*)27d
| 1F*
| 3 |
90732.7
|
|
|
2s2.2p(2P*)28d
| 1F*
| 3 |
90742.2
|
|
|
2s2.2p(2P*)29d
| 1F*
| 3 |
90753.8
|
|
|
-------------------|--------|-----|-----------|
|
|
C II (2P*<1/2>)
| Limit |
|
90820.42
|
|
|
C II (2P*<3/2>)
| Limit |
|
90883.84
|
|
|
2s.2p2(4P)3s
| 5P
| 1 | 103541.8
|
| 2 | 103562.5
|
| 3 | 103587.3
|
|
|
2s.2p3
| 3S*
| 1 | 105798.7
|
|
|
2s.2p3
| 1P*
| 1 | [119878]
2s2.2p(2P*)26d
‡‡ Source: Atomic Spectra Database, http://physics.nist.gov/PhysRefData/contents.html
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14
37/ 48-5
Energy levels of Ti I‡‡ (380 lines of data) — IP = 6.820 eV
----------------------------------------------------------------Configuration
| Term | J | Level
| Lande | Leading
|
|
| (cm-1)
|
g
| Percentages
------------------|-------|----|------------|-------|-----------3d2.4s2
| a 3F | 2 |
0.000 | 0.66 | 100
|
| 3 |
170.132 | 1.08 | 100
|
| 4 |
386.874 | 1.25 | 100
|
|
|
|
|
3d3(4F)4s
| a 5F | 1 | 6556.828 | 0.00 | 100
|
| 2 | 6598.749 | 0.99 | 100
|
| 3 | 6661.003 | 1.25 | 100
|
| 4 | 6742.757 | 1.35 | 100
|
| 5 | 6842.964 | 1.41 | 100
|
|
|
|
|
3d2.4s2
| a 1D | 2 | 7255.369 | 1.02 | 96
|
|
|
|
|
3d2.4s2
| a 3P | 0 | 8436.618 |
| 92
|
| 1 | 8492.421 | 1.50 | 92
|
| 2 | 8602.340 | 1.49 | 90
|
|
|
|
|
3d3(4F)4s
| b 3F | 2 | 11531.760 | 0.67 | 100
|
| 3 | 11639.804 | 1.08 | 100
|
| 4 | 11776.806 | 1.26 | 98
|
|
|
|
|
2 3d3.(2D2).4s 1D
7 3d3.(2P).4s 3P
7 3d3.(2P).4s 3P
7 3d3.(2P).4s 3P
1 3d2.3s2 1G
‡‡ Source: Atomic Spectra Database, http://physics.nist.gov/PhysRefData/contents.html
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14
38/ 48-5
Energy levels of Ti I (contd.)
3d2.4s2
|
| a 1G
|
|
| 4
|
|
| 12118.394
|
|
| 0.98
|
..
.
3d4
|
| a 5D
|
|
|
|
|
|
|
|
|
|
|
|
0
1
2
3
4
|
|
|
|
|
|
|
28772.86
28791.62
28828.51
28882.44
28952.10
3d2.4p2
|
| e 3D
|
|
|
| h 5D
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
2
1
3
0
1
2
3
4
|
|
|
|
|
|
|
|
|
|
|
48724.34
48724.83
48839.74
48802.32
48859.51
48915.07
49024.43
49036.46
90
..
.
|
|
|
|
|
|
|
..
.
3d2.4p2
|
|
|
..
.
|
|
|
|
|
|
|
..
.
|
|
|
|
|
|
|
|
|
|
|
8 3d3.(2G).4s 1G
100
100
100
100
100
..
.
|
|
|
|
|
|
|
|
|
|
|
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14
39/ 48-5
Energy levels of Ti I (contd.)
| f 3D | 2 | 49571.69
|
|
| 3 | 49619.72
|
|
|
|
|
| f 1D | 2 | 50128.08
|
|
|
|
|
| f 1G | 4 | 52125.98
|
|
|
|
|
| e 1P | 1 | 53663.32
|
|
|
|
|
------------------|-------|----|------------|
|
|
|
|
Ti II (4F<3/2>)
| Limit |
| 55010
|
|
|
|
|
|
|
|
|
|
|
|
|
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14
40/ 48-5
Energy levels of Ce I‡‡ (953 lines of data) — IP = 5.5386 eV
------------------------------------------------------------------------------Configuration
| Term
|
J
| Level
| Lande
| Leading
|
|
| (cm-1)
|
g
| Percentages
-------------------------|---------|-------|------------|---------|-----------4f.5d 6s2
| 1G*
|
4
|
0.000 | 0.94543 | 55
29
|
|
|
|
|
4f.5d 6s2
| 3F*
|
2
|
228.849 | 0.76515 | 66
24
|
|
3
| 1663.120 | 1.07736 | 85
7
|
|
4
| 3100.151* | 1.07703 | 34
27
|
|
|
|
|
4f.5d 6s2
| 3H*
|
4
| 1279.424 | 0.88979 | 54
20
|
|
5
| 2208.657 | 1.03212 | 80
12
|
|
6
| 3976.104 | 1.16032 | 78
13
|
|
|
|
|
4f.5d 6s2
| 3G*
|
3
| 1388.941 | 0.73494 | 66
11
|
|
5
| 4199.367 | 1.15021 | 48
34
|
|
|
|
|
4f(2F*) 5d2(3F)6s (4F)
| 5H*
|
3
| 2369.068 | 0.59978 | 66
16
|
|
4
| 2437.629 | 0.98592 | 38
25
|
|
6
| 4746.627 | 1.16593 | 67
24
|
|
7
| 5802.108 | 1.237
| 62
38
|
|
|
|
|
4f.5d 6s2
| 1D*
|
2
| 2378.827 | 0.93654 | 54
23
|
|
|
|
|
3H*
1D*
4f(2F*).5d2(1D).6s(2D) 3F*
4f(2F*).5d2(3F).6s(4F) 5I*
1G*
4f(2F*).5d2(1D).6s(2D) 3H*
4f(2F*).5d2(1D).6s(2D) 3H*
4f(2F*).5d2(1D).6s(2D) 3G*
4f(2F*).5d2(3F).6s(4F) 5H*
4f.5d.6s2 3G*
(2F*) (3F)(4F) 3G*
(2F*) (3F)(4F) 5I*
(2F*) (3F)(4F) 5I*
3F*
‡‡ Source: Atomic Spectra Database, http://physics.nist.gov/PhysRefData/contents.html
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14
41/ 48-5
Energy levels of Ce I (contd.)
4f(2F*) 5d2(3F)6s (4F)
4f(2F*) 5d2(3F)6s (4F)
4f.5d 6s2
4f.5d 6s2
4f(2F*) 5d2(3F)6s (4F)
4f(2F*) 5d2(3F)6s (4F)
4f(2F*) 5d2(3F)6s (4F)
4f(2F*) 5d2(3F)6s (4F)
4f(2F*) 5d2(3F)6s (4F)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
5I*
*
*
3D*
*
*
3G*
*
*
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
4
5
6
7
8
5
4
1
2
3
0
1
3
4
5
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
3210.583 |
|
3312.240* |
|
3710.513 |
4766.323 |
5006.719 |
|
3974.503 |
|
4020.954 |
|
4160.283 |
|
4173.494 |
|
4417.618 |
|
3196.607
3764.008
4455.756
5315.803
6809.128
0.66612
0.90691
1.11714
1.21625
1.250
1.16277
1.08582
0.61549
1.14945
1.23674
1.49404
0.72933
1.02948
1.17790
|
| 67
| 90
| 64
| 56
| 100
|
| 41 3G*
|
| 29 3F*
|
| 64
| 67
| 58
|
| 29 5D*
|
| 17 3S*
|
| 47
|
| 41 3G*
|
| 29 3G*
|
11
3
26
33
4f.5d.6s2 3F*
4f.5d.6s2 3H*
(2F*) (3F)(4F) 5H*
(2F*) (3F)(4F) 5H*
37 (2F*) (3F)(4F) 5H*
26 3G*
10 4f(2F*).5d2(1D).6s(2D) 3D*
12 4f(2F*).5d2(1D).6s(2D) 3D*
20 1F*
23 (2F*) (1D)(2D) 3P*
15 (2F*) (1D)(2D) 3P*
23 (2F*) (3F)(4F) 5H*
30 (2F*) (3F)(4F) 5H*
28 4f.5d.6s2 3G*
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14
42/ 48-5
Energy levels of Ce I
|
4f2.6s2
|
|
|
|
4f(2F*) 5d2(3F)6s (4F)
|
|
4f(2F*) 5d2(3F)6s (4F)
|
|
4f(2F*) 5d2(3F)6s (4F)
|
|
|
|
|
|
4f(2F*) 5d2(3F)6s (4F)
|
|
4f(2F*) 5d2(3F)6s (4F)
|
|
4f(2F*) 5d2(3F)6s (4F)
|
|
4f.5d 6s2
|
|
4f(2F*) 5d2(3F)6s (4F)
|
|
|
|
|
|
(contd.)
3H
3S*
*
5G*
*
*
*
*
5F*
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
4
5
6
1
2
2
3
4
5
6
3
0
4
1
1
2
3
4
5
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
4762.718
6238.934
7780.202
5097.777
5210.906
5409.236
6234.792
6856.559
7467.160
8055.526
5519.751
5571.156
5572.074
5637.233
5674.829
5904.006
6337.061
7174.156
7933.558
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
0.80508 |
1.035
|
1.169
|
|
1.88257 |
|
1.22694 |
|
0.773
|
1.04965 |
1.150
|
1.179
|
1.207
|
|
1.24530 |
|
|
|
1.31658 |
|
1.389
|
|
0.140
|
0.905
|
1.232
|
1.373
|
1.345
|
|
89
92
92
4 4f.5d(3G*).6s.6p(1P*) 3H
4 4f.5d(3G*).6s.6p(1P*) 3H
4 4f.5d(3G*).6s.6p(1P*) 3H
49
25 (2F*) (3F)(4F) 5D*
28 5D*
27 (2F*) (3F)(4F) 5G*
35
69
72
69
50
24
13
6
7
18
32 5D*
21 (2F*) (3F)(4F) 3F*
44 5D*
35 4f.5d.6s2 3P*
29 5D*
27 (2F*) (3F)(4F) 3F*
30 3P*
26 4f(2F*).5d2(3F).6s(4F) 5D*
73
66
42
67
81
6
13
16
19
5
(2F*)
(2F*)
(2F*)
(2F*)
(2F*)
(2F*)
(2F*)
(2F*)
(2F*)
(2F*)
(3F)(4F)
(3F)(4F)
(3F)(2F)
(3F)(4F)
(3F)(4F)
(3F)(4F)
(3F)(4F)
(3F)(4F)
(3F)(4F)
(3P)(4P)
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14
5D*
5D*
1G*
3H*
3H*
5D*
5G*
5D*
5D*
5F*
43/ 48-5
Energy levels of Ce I (contd.)
4f.5d 6s2
4f(2F*) 5d2(3F)6s (2F)
4f.5d 6s2
4f(2F*) 5d2(3F)6s (4F)
4f(2F*) 5d2(3F)6s (4F)
4f(2F*) 5d2(3F)6s (4F)
4f(2F*) 5d2(3F)6s (2F)
4f(2F*) 5d2(3F)6s (4F)
4f(2F*) 5d2(3F)6s (4F)?
4f(2F*) 5d2(3F)6s (2F)?
4f(2F*) 5d2(1D)6s (2D)
4f(2F*) 5d2(3F)6s (4F)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
*
*
*
*
*
*
*
*
*
*
*
*
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
2
4
3
5
2
3
4
6
5
5
1
4
|
|
|
|
|
|
|
|
|
|
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|
|
|
6303.984
6475.540
6621.892
6663.226
6836.628
7169.751
7348.299
7696.210
7715.236
7841.955
7853.119
7890.429
|
|
|
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|
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|
|
|
|
|
|
|
|
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|
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|
|
|
0.897
|
|
1.147
|
|
0.953
|
|
0.68078 |
|
1.13
|
|
0.964
|
|
1.076
|
|
0.934
|
|
1.063
|
|
0.983
|
|
1.242
|
|
1.419
36 3P*
16 4f(2F*).5d2(1D).6s(2D) 3P*
43 3H*
17 (2F*)(3F)(2F) 1G*
35 1F*
17 3D*
19 3I*
18 (2F*)(3F)(2F) 3I*
33 3F*
18 (2F*)(3F)(4F) 5G*
28 3F*
26 (2F*)(3F)(4F) 5F*
22 1G*
21 (2F*)(3F)(4F) 3H*
39 3I*
19 (2F*)(1G)(2G) 3I*
22 3I*
14 (2F*)(3F)(4F) 5G*
24 3H*
21 (2F*)(3F)(2F) 3I*
27 1P*
16 (2F*)(3F)(2F) 3D*
25 3F*
18 (2F*)(3F)(4F) 5D*
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14
44/ 48-5
Energy levels of Ce I (contd.)
4f(2F*) 5d2(3P)6s (4P)
4f(2F*) 5d2(3F)6s (4F)
4f2.6s2
4f(2F*) 5d2(3F)6s (2F)
4f(2F*) 5d2(3F)6s (4F)
4f(2F*) 5d2(3F)6s (2F)
4f(2F*) 5d2(3F)6s (2F)
4f(2F*) 5d2(3F)6s (4F)
4f(2F*) 5d2(3F)6s (4F)
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|
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|
5G*
5P*
3F
1S*
*
*
*
3I*
*
|
|
|
|
|
|
|
|
|
|
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|
2
3
4
5
6
2
3
1
2
3
4
0
2
5
4
7
6
|
| 8088.912
| 8307.309
| 8762.126
| 9462.705
| 11030.470
|
| 8101.187
| 8270.249
| 8430.846
|
| 8235.605
| 9206.305
| 9379.148
|
| 8351.167
|
| 8366.098
|
| 8400.730
|
| 8509.209
|
| 8587.973
|
| 8603.531
|
|
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|
|
|
0.403
0.957
1.054
1.19
1.193
1.735
1.504
2.04
0.680
1.083
1.139
1.525
0.917
0.954
1.155
1.225
95
81
50
46
51
3
6
9
12
21
(2F*)(3F)(4F)
(2F*)(3P)(4P)
(2F*)(3F)(2F)
(2F*)(3P)(4P)
(2F*)(3F)(2F)
85
74
62
5 (2F*)(3F)(4F) 5S*
11 (2F*)(3F)(4F) 3D*
10 (2F*)(3F)(4F) 3P*
88
90
55
4 4f.5d(3F*).6s.6p(1P*) 3F
4 4f.5d(3F*).6s.6p(1P*) 3F
32 1G
44
17 4f.5d.6s2 3P*
33 3P*
16 (2F*)(3F)(4F) 5S*
30 3I*
19 4f.5d.6s2 1H*
20 3H*
19 (2F*)(3P)(4P) 5G*
59
24 (2F*)(1G)(2G) 3I*
36 5G*
25 (2F*)(3F)(4F) 3H*
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14
5G*
3G*
3H*
3G*
3H*
45/ 48-5
Energy levels of Ce I (contd.)
4f(2F*) 5d2(3F)6s (4F)
4f(2F*) 5d2(3F)6s (2F)
4f(2F*) 5d2(3F)6s (2F)
4f(2F*) 5d2(3F)6s (4F)
4f(2F*) 5d2(3P)6s (4P)
4f(2F*) 5d2(3F)6s (2F)
4f(2F*) 5d2(3F)6s (2F)
4f(2F*) 5d2(3F)6s (4F)
4f(2F*) 5d2(3P)6s (4P)
4f(2F*) 5d2(3F)6s (2F)
4f(2F*) 5d2(3F)6s (2F)?
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|
*
*
*
*
3D*
*
3I*
*
3D*
3F*
3G*
|
|
|
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|
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|
1
3
5
0
3
2
6
7
1
2
2
3
|
| 8695.201
|
| 8902.306
|
| 8991.451
|
| 9119.094
|
| 9135.099
|
| 9200.707
|
| 9333.222
| 11061.551
|
| 9369.628
|
| 9425.529
|
| 9709.012
|
| 9787.220
|
..
.
..
.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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|
|
|
1.285
1.128
1.067
1.274
1.376
1.047
1.141
1.065
1.207
0.799
0.868
22 3P*
12 (2F*)(3F)(4F) 5P*
20 1F*
14 4f.5d.6s2 1F*
31 3H*
17 (2F*)(3P)(4P) 5G*
40 3P*
32 (2F*)(3F)(2F) 1S*
50
12 (2F*)(3F)(4F) 5P*
24 3D*
21 (2F*)(3P)(2P) 3D*
61
79
20 (2F*)(1G)(2G) 3I*
16 (2F*)(1G)(2G) 3I*
21 3P*
17 (2F*)(3F)(2F) 3D*
46
8 (2F*)(3F)(4F) 5S*
51
13 (2F*)(3P)(2P) 3F*
42
26 (2F*)(3P)(2P) 3G*
..
.
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14
46/ 48-5
Energy levels of Ce I (contd.)
..
.
..
.
..
.
|
|
|
|
| *
|
6
| 30603.359 |
|
|
|
|
| *
| 4,3 | 30624.042 |
|
|
|
|
| *
|
2
| 30706.313 |
|
|
|
|
| *
|
4
| 30739.270 |
|
|
|
|
| *
|
3
| 30740.884 |
|
|
|
|
| *
|
5
| 30767.047 |
|
|
|
|
| *
|
4
| 30787.639 | 0.695?
|
|
|
|
| *
|
3
| 30854.052 | 0.737
|
|
|
|
| *
|
7
| 30876.234 |
|
|
|
|
| *
|
2
| 30991.841 |
|
|
|
|
-------------------------|---------|-------|------------|
|
|
|
|
Ce II (4H*<7/2>)
| Limit
|
| 44672
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14
47/ 48-5
H atom in a weak homogeneous electric field (Stark effect) — perturbation theory
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14
48/ 48-5