The one-electron atom
H (Z = 1), He+ (Z = 2), Li2+ (Z = 3), . . ., U91+ (Z = 92), . . .
Electronic Hamilton operator (for point-like clamped nucleus)∗:
Hel = Te + Ven = − ~2
2me∇2 − Z e
2
κ0 r(139)
Determination of stationary bound states, i.e. solutions 〈 r |ψ 〉 = ψ(r)
with E < 0, to the time-independent Schrodinger equation (with Helfrom above):
(Hel −E
)ψ(r) =
(− ~2
2me∇2 − Z e
2
κ0 r−E
)ψ(r) = 0 (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 me
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
1. Historical introduction
2. The Schrodinger 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
Now the Schrodinger equation reads
(Hel − E
)ψ(r) =
(− 1
2∇2 − Z
r− E
)ψ(r) = 0 (141)
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 l2 (in
atomic units) in spherical coordinates:
∆ = ∇2 =1
r2∂
∂rr2∂
∂r− l2
r2=
1
r
∂2
∂r2r − l2
r2(142)
l2 = −
1
sin θ
∂
∂θsin θ
∂
∂θ+
1
sin2 θ
∂2
∂φ2
(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 4/ 48-5
Atomic unitsa
Physical quantity Symbol (name) Value in SI unitsb
mass me 9.1093826(16) × 10−31 kgcharge e 1.60217653(14) × 10−19 Cangular momentum, action ~ 1.05457168(18) × 10−34 J sel. permittivity 4πε0 κ0 1.112650056 . . . × 10−10 F m−1
length κ0~2/(mee2) = ~/(αmec) a0 (bohr) 5.291772 × 10−11 m
time ~/Eh = ~/(α2mec2) 2.418884 × 10−17 s
velocity a0Eh/~ = αc 2.187691 × 106 m s−1
linear momentum ~/a0 = αmec 1.992852 × 10−24 kgm s−1
force Eh/a0 8.238723 × 10−8 Nenergy e2/(κ0a0) = α2mec
2 Eh (hartree) 4.359744 × 10−18 J
power Eh2/~ 1.802378 × 10−1 W
charge density e/a03 1.081202 × 1012 C m−3
el. current eEh/~ 6.623618 × 10−3 Ael. potential Eh/e 2.721138 × 101 Vel. capacitance κ0a0 5.887891 × 10−21 Fel. resistance ~/e2 4.108236 × 103 Ωel. field strength (E) Eh/(ea0) 5.142206 × 1011 V m−1
el. displacement (D) e/a02 5.721476 × 101 C m−2
el. dipole moment ea0 8.478353 × 10−30 C mel. quadrupole moment ea0
2 4.486551 × 10−40 C m2
el. polarizability (ea0)2/Eh 1.648777 × 10−41 C2 m2 J−1
magn. flux ~/e 6.582119 × 10−16 Wbmagn. flux density (B) ~/(ea0
2) 2.350517 × 105 Tmagnetizing force (H) eEh/(a0~) 1.251682 × 108 A m−1
magn. dipole moment e~/me = 2µB 1.854802 × 10−23 J T−1
aBased on CODATA recommended values 2002 (http://physics.nist.gov/constants/).
bThe standard deviation uncertainty in the least significant digits is given in parentheses.
Notation for single-particle eigenfunctions of angular momentum:
l2 |ψαlm 〉 = l(l+ 1) ~2 |ψαlm 〉lz |ψαlm 〉 = m ~ |ψαlm 〉
l 0 1 2 3 4 5 6 7 8 9 10 ...
s p d f g h i k l m n ...
Notation for many-particle eigenfunctions of total angular momen-
tum:
L2 |ΨαLM 〉 = L(L+ 1) ~2 |ΨαLM 〉Lz |ΨαLM 〉 = M ~ |ΨαLM 〉
Lz =n∑
i=1
lz,i , L=n∑
i=1
li , L2 = L · L
L 0 1 2 3 4 5 6 7 8 9 10 ...
S P D F G H I K L M N ...
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30 6/ 48-5
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 si-
multaneous eigenfunction of l2 and lz (in atomic units):
l2 Ylm(θ, φ) = l(l+ 1)Ylm(θ, φ) (145)
lz Ylm(θ, φ) = mYlm(θ, φ) (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† phaseconvention), −l ≤ m ≤ l:
Ylm(θ, φ) = Θlm(θ)Φm(φ) = (−1)mNlmPml (cos θ) eimφ (147)
Nlm =
√2l+ 1
4π
(l−m)!
(l+m)!, Yl,−m(θ, φ) = (−1)mYlm
∗(θ, φ)
∫
ΩYlm∗(θ, φ)Yl′m′(θ, φ) dΩ = δll′ δmm′
† E. U. Condon (1902-1974), G. H. Shortley (∗1910)
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30 5/ 48-5
The first Legendre polynomials
l Pl(x)
0 1
1 x
2 12 (3x2 − 1)
3 12 (5x3 − 3x)
4 18 (35x4 − 30x2 + 3)
5 18 (63x5 − 70x3 + 15x)
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30 8/ 48-5
The associated Legendre‡ functions Pml (cos θ) are related to the Le-
gendre polynomials Pl(x) = P0l (x) (x = cos θ). The following relations
hold (for −l ≤ m ≤ l, where applicable):
Pml (x) =1
2ll!(1− x2)m/2 dl+m
dxl+m(x2 − 1)l (148)
Pl(x) =1
2ll!
dl
dxl(x2 − 1)l = 2F1(−l, l+ 1; 1; (1− x)/2) (149)
P−ml (x) = (−1)m(l−m)!
(l+m)!Pml (x) (150)
Pml+1(x) =2l+ 1
l −m+ 1xPml (x)− l+m
l−m+ 1Pml−1(x) (151)
Pm+1l (x) =
2m√1− x2
xPml (x)− (l+m)(l −m+ 1)Pm−1l (x) (152)
‡ A.-M. Legendre (1752-1833)
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30 7/ 48-5
The first spherical harmonics
(Condon-Shortley phase convention)
l m Ylm(θ, φ) l m Ylm(θ, φ)
0 0√
14π 3 +3 −15
√7
2880π sin3 θ e3iφ
1 +1 −√
38π sin θ eiφ 3 +2 15
√7
480π sin2 θ cos θ e2iφ
1 0√
34π cos θ 3 +1 −3
2
√7
48π sin θ (5 cos2 θ − 1) eiφ
1 −1√
38π sin θ e−iφ 3 0 1
2
√74π(5 cos3 θ − 3cos θ)
2 +2 3√
596π sin2 θ e2iφ 3 −1 3
2
√7
48π sin θ (5 cos2 θ − 1) e−iφ
2 +1 −3√
524π sin θ cos θ eiφ 3 −2 15
√7
480π sin2 θ cos θ e−2iφ
2 0 12
√54π(3 cos2 θ − 1) 3 −3 15
√7
2880π sin3 θ e−3iφ
2 −1 3√
524π sin θ cos θ e−iφ
2 −2 3√
596π sin2 θ e−2iφ
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30 10/ 48-5
The first associated Legendre functions
l m Pml (cos θ) l m Pml (cos θ)
0 0 1 3 0 12 (5 cos3 θ − 3cos θ)
1 0 cos θ 3 +1 32 sin θ (5 cos2 θ − 1)
1 +1 sin θ 3 −1 −18 sin θ (5 cos2 θ − 1)
1 −1 −12 sin θ 3 +2 15 sin2 θ cos θ
2 0 12 (3 cos2 θ − 1) 3 −2 1
8 sin2 θ cos θ
2 +1 3 sin θ cos θ 3 +3 15 sin3 θ
2 −1 −12 sin θ cos θ 3 −3 − 1
48 sin3 θ
2 +2 3 sin2 θ
2 −2 18 sin2 θ
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30 9/ 48-5
Energy eigenvalues (in atomic units):
En = Enlm = − 1
2
Z2
n2(155)
These are degenerate for n > 1 (further details below).
Principal quantum number n: n = 1,2,3, . . .
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:
L(α)k (x) =
(α+ 1)kk! 1F1(−k;α+ 1;x) (156)
L(α)k (x) =
2k+ α− 1− xk
L(α)k−1(x)−
k+ α− 1
kL
(α)k−2(x) (k ≥ 2)
(157)
§ E. N. Laguerre (1834-1886)
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30 12/ 48-5
Two-point boundary value problem for P (r) = r R(r) (as resulting
from the separation ansatz):(
d2
dr2+ 2 [ E − Vl(r) ]
)P (r) = 0 (153)
Vl(r) =l(l+ 1)
2 r2− Zr, P (0) = 0 , lim
r→∞P (r) = 0
Physically acceptable (i.e. normalizable) solutions exist only for a
discrete set of energies: Quantization due to the boundary condi-
tions.
Radial functions (eigenfunctions):
Pnl(r) = r Rnl(r) = Nnl xl+1L
(2l+1)nr (x) e−x/2 (154)
x =2Z
nr , Nnl =
1
n
√Z
nr!
(n+ l)!, nr = n− l− 1 ≥ 0
∫ ∞0
Rnl(r)Rn′l(r) r2dr =
∫ ∞0
Pnl(r)Pn′l(r) dr = δnn′
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30 11/ 48-5
The first radial functions of one-electron atoms
n l nr Pnl(r) (x = 2Zr/n)
1s 1 0 0√Z x e−x/2
2s 2 0 1 12
√Z2 x (2− x) e−x/2
2p 2 1 0 12
√Z6 x
2 e−x/2
3s 3 0 2 13
√Z3 x (3− 3x+ x2/2) e−x/2
3p 3 1 1 13
√Z24 x
2 (4− x) e−x/2
3d 3 2 0 13
√Z
120 x3 e−x/2
4s 4 0 3 14
√Z4 x (4− 6x+ 2x2 − x3/6) e−x/2
4p 4 1 2 14
√Z60 x
2 (10− 5x+ x2/2) e−x/2
4d 4 2 1 14
√Z
720 x3 (6− x) e−x/2
4f 4 3 0 14
√Z
5040 x4 e−x/2
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30 14/ 48-5
The first generalized Laguerre polynomials
k L(α)k (x)
0 1
1 (α+ 1)− x2 1
2
[(α+ 1)(α+ 2)− 2(α+ 2)x+ x2
]
3 16
[(α+ 1)(α+ 2)(α+ 3)− 3(α+ 2)(α+ 3)x+ 3(α+ 3)x2 − x3
]
4 124
[(α+ 1)(α+ 2)(α+ 3)(α+ 4)− 4(α+ 2)(α+ 3)(α+ 4)x
+ 6(α+ 3)(α+ 4)x2 − 4(α+ 4)x3 + x4]
5 1120
[(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]
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30 13/ 48-5
Eigenfunctions and energy eigenvalues (in atomic units) of the one-
electron atom:
ψnlm(r) =1
rPnl(r)Ylm(θ, φ) , Enlm = En = − 1
2
Z2
n2(158)
n = 1,2,3,4, . . . , l = 0,1, . . . , n− 1 , m = −l,−l+ 1, . . . , l
〈nlm |n′l′m′ 〉 =∫ψnlm(r)ψn′l′m′(r) dr = δnn′ δll′ δmm′
Ground state and corresponding energy:
ψ100(r) =
√Z3
πe−Zr =
1
r
√Z 2Zr e−Zr
1√4π
, E1 = − Z2
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. Kloffler, H.-U. Raab, Chemie in unserer Zeit 12
(1978) 23-26.
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30 16/ 48-5
Radial functions Pnl(r) of the one-electron atom (for n = 1, 2, 3, 4)
n = 1 (1s)
0 10 Zr
0.0
0.2
0.4
0.6
0.8
Pnl(r)
Z1/2
...........
...........
..........
..........
..........
.........
.........
.........
.........
.........
........
........
........
........
........
........
........
........
........
.........
.........
.........
.........
..........
..........
........
......
.......
.......
........
.........
..........
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
n = 3 (3s, 3p, 3d)
0 10 20 30 Zr
-0.4
-0.2
0.0
0.2
0.4Pnl(r)
Z1/2
..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...............................................................................................................................................................................................................................................................
...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..........................................
.....................................
.....................................
.........................................
................................................
...................................................................
...............................................................................................
........................................................................................................................................................................................................................
.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
n = 2 (2s, 2p)
0 10 20 Zr
-0.4
-0.2
0.0
0.2
0.4Pnl(r)
Z1/2
.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.............................................................................................................................................................................................................
.................................................
.......................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
n = 4 (4s, 4p, 4d, 4f)
0 10 20 30 40 Zr
-0.4
-0.2
0.0
0.2
0.4Pnl(r)
Z1/2
.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
....................................................................
.............................................................
..................................................................
.........................................................................
...............................................................................
.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......................................................
...........................................................................................................................................................................................................................................................................................................................................................................................................
....................................................................................................................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..............................................................
..............................................................
........................................................................
...................................................................................................
.................................................................................................
.............................................
....................................
....................................................
......................................
...................................................
......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30 15/ 48-5
The ‘periodic’ table of the chemical elements
(2004)
1
2
3
4
5
6
7
1
1
H
3
Li
11
Na
19
K
37
Rb
55
Cs
87
Fr
2
4
Be12
Mg
20
Ca
38
Sr
56
Ba
88
Ra
3
21
Sc
39
Y
∗
∗∗
∗ 57
La58
Ce59
Pr60
Nd61
Pm62
Sm63
Eu64
Gd65
Tb66
Dy67
Ho68
Er69
Tm70
Yb71
Lu∗∗ 89
Ac90
Th91
Pa92
U93
Np94
Pu95
Am96
Cm97
Bk98
Cf99
Es100
Fm101
Md102
No103
Lr
4
22
Ti
40
Zr
72
Hf
104
Rf
5
23
V
41
Nb
73
Ta
105
Db
6
24
Cr
42
Mo
74
W106
Sg
7
25
Mn
43
Tc
75
Re
107
Bh
8
26
Fe
44
Ru
76
Os
108
Hs
9
27
Co
45
Rh
77
Ir
109
Mt
10
28
Ni
46
Pd
78
Pt
110
Ds
11
29
Cu47
Ag
79
Au
111
X
12
30
Zn
48
Cd80
Hg112
(X)
13
5
B
13
Al
31
Ga
49
In
81
Tl
113
14
6
C
14
Si
32
Ge
50
Sn
82
Pb114
(X)
15
7
N
15
P
33
As
51
Sb
83
Bi
115
16
8
O
16
S
34
Se
52
Te
84
Po116
(X)
17
9
F
17
Cl
35
Br
53
I
85
At
117
18
2
He
10
Ne
18
Ar
36
Kr
54
Xe
86
Rn118
(?)
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30 18/ 48-5
Degree of degeneracy¶:
m degeneracy: gl =l∑
m=−l1 = 2l+ 1 (160)
l degeneracy (without spin): gn =n−1∑
l=0
gl = n2 (161)
l degeneracy (spin included): gsn = 2 gn = 2n2 (162)
The value gsn essentially determines the length of the rows (‘periods’)
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 (withpoint-like nucleus), and is not present in many-electron atoms. For example, the 2s and 2p statesof 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
Trial functions (unnormalized)
for the ground state of the one-electron atom
0
1
f(x)
0 1 2 3x
S: exp(−x), x = ζr
G: exp(−x2), x =√αr
L: 1/(1 + x2), x = ar
P: 1/(1 + x)2, x = cr
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30 20/ 48-5
The variation principle at work (I)
The ground state 1s1 2S of the one-electron atom
Definition of the system under consideration:
H = T + V , T = − 1
2∆ , V = Vnuc(r) = − Z
r
A set of trial functions‖:
1. Slater function (ζ > 0):
ψS = NS exp (−ζr)
2. Gauß function (α > 0):
ψG = NG exp (−αr2)
3. Lorentz function (a > 0):
ψL = NL
[1 + (ar)2
]−1
4. Preuß function (c > 0):
ψP = NP
[1 + cr
]−2
‖Refs.: 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
1. The Slater function ψS = NS exp (−ζr), ζopt =?:
〈rk〉 = N2S 4π
∫ ∞0
dr rk+2e−2ζr = N2S 4π
Γ(k+ 3)
(2ζ)k+3(k > −3)
〈1〉 = 〈r0〉 = N2Sπ
ζ3≡ 1 ⇒ NS =
√ζ3/π
〈rk〉 = Γ(k+ 3)
2k+1
1
ζk
〈T 〉 = +1
2N2
S4π
ζ
∫ ∞0
dx
(d
dxx e−x
)2
=1
2N2
S4π
ζ
∫ ∞0
dx (1− x)2e−2x =1
2ζ2
〈V 〉 = − Z ζ
〈E 〉 = 1
2ζ2 − Z ζ ⇒ d〈E 〉
dζ= ζ − Z = 0 ⇒ ζopt = Z
〈E 〉Z2
= x
(1
2x− 1
)(x = ζ/Z) ⇒ Emin = − 1
2Z2
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30 22/ 48-5
Estimate for the ground state energy:
〈E 〉 = 〈H 〉 = 〈ψ | H |ψ 〉 =∫ψ(r) H ψ(r) dr = 〈T 〉+ 〈V 〉
〈T 〉 = − 1
2〈ψ |∆ |ψ 〉 = +
1
2〈∇ψ |∇ψ 〉 , 〈V 〉 = − Z 〈r−1〉
Mathematical preliminaries:
• The beta function:
B(a, b) =Γ(a) Γ(b)
Γ(a+ b)= B(b, a)
B(a, b) =∫ 1
0ta−1(1− t)b−1 dt =
∫ ∞0
ta−1
(1 + t)a+bdt (a > 0, b > 0)
• Useful integral formulas:∫ ∞0
xs−1 exp (−p xn) dx =1
n
Γ(s/n)
ps/n(p > 0, s/n > 0)
∫ ∞0
xs−1
(1 + xn)pdx =
1
nB(p− s/n, s/n) (s/n > 0, np− s > 0)
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30 21/ 48-5
3. The Lorentz function ψL = NL
[1 + (ar)2
]−1, aopt =?:
〈rk〉 = N2L 4π
∫ ∞0
rk+2 dr
(1 + (ar)2)2= N2
L4π
ak+3
1
2B(
1− k2,k+ 3
2)
〈1〉 = 〈r0〉 = N2Lπ2
a3≡ 1 ⇒ NL =
√a3/π
〈rk〉 = 2
π
1
akB(
1− k2,k+ 3
2) (−3 < k < 1)
〈T 〉 = +1
2N2
L4π
a
∫ ∞0
dx
(d
dx
x
1 + x2
)2
=1
2N2
L4π
a
∫ ∞0
dx(1− x2)2(1 + x2)4
=1
4a2
〈V 〉 = − 2Z a/π
〈E 〉 = 1
4a2 − 2Z
a
π⇒ d〈E 〉
da=
1
2a− Z 2
π= 0 ⇒ aopt =
4Z
π〈E 〉Z2
= x
(1
4x− 2
π
)(x = a/Z) ⇒ Emin = − 4
π2Z2
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30 24/ 48-5
2. The Gauß function ψG = NG exp (−αr2), αopt =?:
〈rk〉 = N2G 4π
∫ ∞0
dr rk+2e−2αr2 = N2G 4π
1
2
Γ(k+32 )
(2α)k+32
(k > −3)
〈1〉 = 〈r0〉 = N2G
(π
2α
)3/2≡ 1 ⇒ NG =
(2α
π
)3/4
〈rk〉 = 2√π
Γ(k+32 )
(2α)k/2
〈T 〉 = +1
2N2
G4π√α
∫ ∞0
dx
(d
dxx e−x
2)2
=1
2N2
G4π√α
∫ ∞0
dx (1− 2x2)2e−2x2 =3
2α
〈V 〉 = − 2Z√
2α/π
〈E 〉 = 3
2α− 2Z
√2α
π⇒ d〈E 〉
dα=
3
2− Z
√2
πα= 0 ⇒ αopt =
8Z2
9π
〈E 〉Z2
= x
(3
2x− 2
√2
π
)(x =
√α/Z) ⇒ Emin = − 4
3πZ2
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30 23/ 48-5
Variation of the ground state energy 〈E 〉 of the one-electron atom
with the trial function parameter
〈E 〉Z2
= f(x) = a2 x2 + a1 x
−0.5
0
f(x)
1 2x
S: x2/2− x, x = ζ/Z
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, 〈V 〉 / 〈T 〉 = −2, is fulfilled, and
thus 〈E 〉 = 〈V 〉 /2 = − 〈T 〉 in all four cases.
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30 26/ 48-5
4. The Preuß function ψP = NP
[1 + cr
]−2, copt =?:
〈rk〉 = N2P 4π
∫ ∞0
rk+2 dr
(1 + cr)4= N2
P4π
ck+3B(1− k, k+ 3)
〈1〉 = 〈r0〉 = N2P
4π
3c3≡ 1 ⇒ NP =
√3c3
4π
〈rk〉 = 3
ckB(1− k, k+ 3) (−3 < k < 1)
〈T 〉 = +1
2N2
P4π
c
∫ ∞0
dx
(d
dx
x
(1 + x)2
)2
=1
2N2
P4π
c
∫ ∞0
dx(1− x)2(1 + x)6
=1
5c2
〈V 〉 = − Z c/2
〈E 〉 = 1
5c2 − Z c
2⇒ d〈E 〉
dc=
2
5c− Z
2= 0 ⇒ copt =
5Z
4〈E 〉Z2
= x
(1
5x− 1
2
)(x = c/Z) ⇒ Emin = − 5
16Z2
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-06-30 25/ 48-5
into eq. (163):
0 =
H
(0) − E(0) +∞∑
k=1
(H(k) − ε(k)
)λk
ψ(0) +
∞∑
l=1
χ(l) λl
=(H(0) − E(0)
)ψ(0) +
∞∑
m=1
λm(
H(0) −E(0))χ(m)
+m−1∑
k=1
(H(k) − ε(k)
)χ(m−k) +
(H(m) − ε(m)
)ψ(0)
Thus we obtain from the coefficient of λm:
m = 0 :(H(0) −E(0)
)ψ(0) = 0
m = 1 :(H(0) −E(0)
)χ(1) +
(H(1) − E(1)
)ψ(0) = 0
m = 2 :(H(0) −E(0)
)χ(2) +
(H(1) − E(1)
)χ(1)
+(H(2) −E(2)
)ψ(0) = 0
...
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14 28/ 48-5
Perturbation theory
Time-independent Rayleigh-Schrodinger perturbation theory for non-
degenerate states
Find a solution for the Schrodinger equation
Hλψλ = Eλψλ ⇔(Hλ −Eλ
)ψλ = 0 (163)
with the Hamilton operator
Hλ = H(0) +∞∑
k=1
λk H(k) , (164)
where λ is a (natural or artificial) perturbation parameter (|λ| < λmax),
and assume that the solutions of the unperturbed problem
H(0)ψ(0) = E(0)ψ(0) (165)
are completely known (with all E(0) non-degenerate). Then put
Eλ = E(0) +∞∑
k=1
ε(k) λk and ψλ = ψ(0) +∞∑
l=1
χ(l) λl (166)
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14 27/ 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∑
k=1
ε(k) λk
ψλ ≈ ψ(n) = ψ(0) +n∑
l=1
χ(l) λl
• 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
∂λn
∣∣∣∣∣λ=0
= n! ε(n)
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 ‘intermediate normalization’ condition
〈ψ(0) |ψλ 〉 = 1 with 〈ψ(0) |ψ(0) 〉 = 1
implies orthogonality between ψ(0) and all χ(l):
〈ψ(0) |χ(l) 〉 = 0 (for l = 1,2, . . .)
Resulting expressions for ε(1) and ε(2):
ε(1)= 〈ψ(0) | H(1) |ψ(0) 〉 (167)
ε(2)= 〈ψ(0) | H(1) |χ(1) 〉+ 〈ψ(0) | H(2) |ψ(0) 〉 (168)
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14 29/ 48-5
For the spatial part f(r1, r2) 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) −Eopt/a.u.
e−ζr1e−ζr2 = e−ζs 2.8477
ϕ(r1)ϕ(r2) 2.8617 b
e−ζr1e−ηr2 + e−ηr1e−ζr2 2.8757 c
e−ζs+cu 2.8896
e−ζs (1 + cu) 2.8911 d
e−ζs+cu cosh (at) 2.8994e−ζs
(c0 + c1u+ c2t2 + c3s+ c4s2 + c5u2
)2.9032
exact 2.9037 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 two-electron atom
H− (Z = 1), He (Z = 2), Li+ (Z = 3), . . ., U90+ (Z = 92), . . .
Hamilton operator (in atomic units):
Hel = − 1
2∇2
1 −1
2∇2
2 −Z
r1− Z
r2+
1
r12= h1 + h2 +
1
r12(169)
General structure of state functions for two-electron systems∗∗:
Φ(x1,x2) = f(r1, r2)Θ(σ1, σ2)
Spin part Θ = |SMS 〉: Singlet (S = 0, para-He) or triplet (S = 1,
ortho-He)
|00 〉 = (αβ − βα)/√
2
|11 〉 = αα |10 〉 = (αβ + βα)/√
2 |1− 1 〉 = 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
Electron configuration of neutral atoms in the ground state (designated as 2S+1LJ)
1 H 1s12S1/2
2 He 1s21S0
3 Li [He] 2s1 2S1/2
4 Be [He] 2s2 1S0
5 B [He] 2s2 2p1 2P1/2
6 C [He] 2s2 2p2 3P0
7 N [He] 2s2 2p3 4S3/2
8 O [He] 2s2 2p4 3P2
9 F [He] 2s2 2p5 2P3/2
10 Ne [He] 2s2 2p6 1S0
11 Na [Ne] 3s1 2S1/2
12 Mg [Ne] 3s2 1S0
13 Al [Ne] 3s2 3p1 2P1/2
14 Si [Ne] 3s2 3p2 3P0
15 P [Ne] 3s2 3p3 4S3/2
16 S [Ne] 3s2 3p4 3P2
17 Cl [Ne] 3s2 3p5 2P3/2
18 Ar [Ne] 3s2 3p6 1S0
19 K [Ar] 4s1 2S1/2
20 Ca [Ar] 4s2 1S0
21 Sc [Ar] 3d1 4s2 2D3/2
22 Ti [Ar] 3d2 4s2 3F2
23 V [Ar] 3d3 4s2 4F3/2
24 Cr [Ar] 3d5 4s1 7S3
25 Mn [Ar] 3d5 4s2 6S5/2
26 Fe [Ar] 3d6 4s2 5D4
27 Co [Ar] 3d7 4s2 4F9/2
28 Ni [Ar] 3d8 4s2 3F4
29 Cu [Ar] 3d10 4s1 2S1/2
30 Zn [Ar] 3d10 4s2 1S0
31 Ga [Ar] 3d10 4s2 4p1 2P1/2
32 Ge [Ar] 3d10 4s2 4p2 3P0
33 As [Ar] 3d10 4s2 4p3 4S3/2
34 Se [Ar] 3d10 4s2 4p4 3P2
35 Br [Ar] 3d10 4s2 4p5 2P3/2
36 Kr [Ar] 3d10 4s2 4p6 1S0
37 Rb [Kr] 5s1 2S1/2
38 Sr [Kr] 5s2 1S0
39 Y [Kr] 4d1 5s2 2D3/2
40 Zr [Kr] 4d2 5s2 3F2
41 Nb [Kr] 4d4 5s1 6D1/2
42 Mo [Kr] 4d5 5s1 7S3
43 Tc [Kr] 4d5 5s2 6S5/2
44 Ru [Kr] 4d7 5s1 5F5
45 Rh [Kr] 4d8 5s1 4F9/2
46 Pd [Kr] 4d10 1S0
47 Ag [Kr] 4d10 5s1 2S1/2
48 Cd [Kr] 4d10 5s2 1S0
49 In [Kr] 4d10 5s2 5p1 2P1/2
50 Sn [Kr] 4d10 5s2 5p2 3P0
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14 34/ 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 de-
termined 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 (anantisymmetrized product of spin orbitals), or a linear combination thereof.
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14 33/ 48-5
LS terms for electron configurations p2 and p3
p2: 3P (9), 1D (5), 1S (1)(62
)= 15 = 9 + 5 + 1
p3: 4S (4), 2D (10), 2P (6)(63
)= 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 = 1eV ≈ 8065.55 cm−1 (Eh = 2h cR∞)
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14 36/ 48-5
51 Sb [Kr] 4d10 5s2 5p3 4S3/2
52 Te [Kr] 4d10 5s2 5p4 3P2
53 I [Kr] 4d10 5s2 5p5 2P3/2
54 Xe [Kr] 4d10 5s2 5p6 1S0
55 Cs [Xe] 6s1 2S1/2
56 Ba [Xe] 6s2 1S0
57 La [Xe] 5d1 6s2 2D3/2
58 Ce [Xe] 4f1 5d1 6s2 1G4
59 Pr [Xe] 4f3 6s2 4I9/260 Nd [Xe] 4f4 6s2 5I461 Pm [Xe] 4f5 6s2 6H5/2
62 Sm [Xe] 4f6 6s2 7F0
63 Eu [Xe] 4f7 6s2 8S7/2
64 Gd [Xe] 4f7 5d1 6s2 9D2
65 Tb [Xe] 4f9 6s2 6H15/2
66 Dy [Xe] 4f10 6s2 5I867 Ho [Xe] 4f11 6s2 4I15/2
68 Er [Xe] 4f12 6s2 3H6
69 Tm [Xe] 4f13 6s2 2F7/2
70 Yb [Xe] 4f14 6s2 1S0
71 Lu [Xe] 4f14 5d1 6s2 2D3/2
72 Hf [Xe] 4f14 5d2 6s2 3F2
73 Ta [Xe] 4f14 5d3 6s2 4F3/2
74 W [Xe] 4f14 5d4 6s2 5D0
75 Re [Xe] 4f14 5d5 6s2 6S5/2
76 Os [Xe] 4f14 5d6 6s2 5D4
77 Ir [Xe] 4f14 5d7 6s2 4F9/2
78 Pt [Xe] 4f14 5d9 6s1 3D3
79 Au [Xe] 4f14 5d10 6s1 2S1/2
80 Hg [Xe] 4f14 5d10 6s2 1S0
81 Tl [Xe] 4f14 5d10 6s2 6p1 2P1/2
82 Pb [Xe] 4f14 5d10 6s2 6p2 3P0
83 Bi [Xe] 4f14 5d10 6s2 6p3 4S3/2
84 Po [Xe] 4f14 5d10 6s2 6p4 3P2
85 At [Xe] 4f14 5d10 6s2 6p5 2P3/2
86 Rn [Xe] 4f14 5d10 6s2 6p6 1S0
87 Fr [Rn] 7s1 2S1/2
88 Ra [Rn] 7s2 1S0
89 Ac [Rn] 6d1 7s2 2D3/2
90 Th [Rn] 6d2 7s2 3F2
91 Pa [Rn] 5f2 6d1 7s2 4K11/2
92 U [Rn] 5f3 6d1 7s2 5L6
93 Np [Rn] 5f4 6d1 7s2 6L11/2
94 Pu [Rn] 5f6 7s2 7F0
95 Am [Rn] 5f7 7s2 8S7/2
96 Cm [Rn] 5f7 6d1 7s2 9D2
97 Bk [Rn] 5f8 6d1 7s2 8H17/2
98 Cf [Rn] 5f10 7s2 5I899 Es [Rn] 5f11 7s2 4I15/2
100 Fm [Rn] 5f12 7s2 3H6
101 Md [Rn] 5f13 7s2 2F7/2
102 No [Rn] 5f14 7s2 1S0
103 Lr [Rn] 5f14 6d1 7s2
104 Rf [Rn] 5f14 6d2 7s2
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14 35/ 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 2 3d3.(2D2).4s 1D
| | | | |
3d2.4s2 | a 3P | 0 | 8436.618 | | 92 7 3d3.(2P).4s 3P
| | 1 | 8492.421 | 1.50 | 92 7 3d3.(2P).4s 3P
| | 2 | 8602.340 | 1.49 | 90 7 3d3.(2P).4s 3P| | | | |
3d3(4F)4s | b 3F | 2 | 11531.760 | 0.67 | 100
| | 3 | 11639.804 | 1.08 | 100
| | 4 | 11776.806 | 1.26 | 98 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 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
| | |
......
...
......
...
| | |
2s2.2p(2P*)26d | 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]
‡‡ 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 (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 Ti I (contd.)
| | | | |
3d2.4s2 | a 1G | 4 | 12118.394 | 0.98 | 90 8 3d3.(2G).4s 1G
| | | | |
......
...
| | | | |3d4 | a 5D | 0 | 28772.86 | | 100
| | 1 | 28791.62 | | 100
| | 2 | 28828.51 | | 100
| | 3 | 28882.44 | | 100
| | 4 | 28952.10 | | 100
| | | | |
......
...
| | | | |3d2.4p2 | e 3D | 2 | 48724.34 | |
| | 1 | 48724.83 | |
| | 3 | 48839.74 | |
| | | | |
3d2.4p2 | h 5D | 0 | 48802.32 | |
| | 1 | 48859.51 | || | 2 | 48915.07 | |
| | 3 | 49024.43 | |
| | 4 | 49036.46 | |
| | | | |
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14 39/ 48-5
Energy levels of Ce I (contd.)
| | | | |
4f(2F*) 5d2(3F)6s (4F) | 5I* | 4 | 3196.607 | 0.66612 | 67 11 4f.5d.6s2 3F*
| | 5 | 3764.008 | 0.90691 | 90 3 4f.5d.6s2 3H*| | 6 | 4455.756 | 1.11714 | 64 26 (2F*) (3F)(4F) 5H*
| | 7 | 5315.803 | 1.21625 | 56 33 (2F*) (3F)(4F) 5H*
| | 8 | 6809.128 | 1.250 | 100
| | | | |
4f(2F*) 5d2(3F)6s (4F) | * | 5 | 3210.583 | 1.16277 | 41 3G* 37 (2F*) (3F)(4F) 5H*
| | | | |4f.5d 6s2 | * | 4 | 3312.240* | 1.08582 | 29 3F* 26 3G*
| | | | |
4f.5d 6s2 | 3D* | 1 | 3710.513 | 0.61549 | 64 10 4f(2F*).5d2(1D).6s(2D) 3D*
| | 2 | 4766.323 | 1.14945 | 67 12 4f(2F*).5d2(1D).6s(2D) 3D*
| | 3 | 5006.719 | 1.23674 | 58 20 1F*
| | | | |4f(2F*) 5d2(3F)6s (4F) | * | 0 | 3974.503 | | 29 5D* 23 (2F*) (1D)(2D) 3P*
| | | | |
4f(2F*) 5d2(3F)6s (4F) | * | 1 | 4020.954 | 1.49404 | 17 3S* 15 (2F*) (1D)(2D) 3P*
| | | | |
4f(2F*) 5d2(3F)6s (4F) | 3G* | 3 | 4160.283 | 0.72933 | 47 23 (2F*) (3F)(4F) 5H*
| | | | |4f(2F*) 5d2(3F)6s (4F) | * | 4 | 4173.494 | 1.02948 | 41 3G* 30 (2F*) (3F)(4F) 5H*
| | | | |
4f(2F*) 5d2(3F)6s (4F) | * | 5 | 4417.618 | 1.17790 | 29 3G* 28 4f.5d.6s2 3G*
| | | | |
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14 42/ 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 3H*
| | | | |
4f.5d 6s2 | 3F* | 2 | 228.849 | 0.76515 | 66 24 1D*| | 3 | 1663.120 | 1.07736 | 85 7 4f(2F*).5d2(1D).6s(2D) 3F*
| | 4 | 3100.151* | 1.07703 | 34 27 4f(2F*).5d2(3F).6s(4F) 5I*
| | | | |
4f.5d 6s2 | 3H* | 4 | 1279.424 | 0.88979 | 54 20 1G*
| | 5 | 2208.657 | 1.03212 | 80 12 4f(2F*).5d2(1D).6s(2D) 3H*
| | 6 | 3976.104 | 1.16032 | 78 13 4f(2F*).5d2(1D).6s(2D) 3H*| | | | |
4f.5d 6s2 | 3G* | 3 | 1388.941 | 0.73494 | 66 11 4f(2F*).5d2(1D).6s(2D) 3G*
| | 5 | 4199.367 | 1.15021 | 48 34 4f(2F*).5d2(3F).6s(4F) 5H*
| | | | |
4f(2F*) 5d2(3F)6s (4F) | 5H* | 3 | 2369.068 | 0.59978 | 66 16 4f.5d.6s2 3G*
| | 4 | 2437.629 | 0.98592 | 38 25 (2F*) (3F)(4F) 3G*| | 6 | 4746.627 | 1.16593 | 67 24 (2F*) (3F)(4F) 5I*
| | 7 | 5802.108 | 1.237 | 62 38 (2F*) (3F)(4F) 5I*
| | | | |
4f.5d 6s2 | 1D* | 2 | 2378.827 | 0.93654 | 54 23 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.5d 6s2 | * | 2 | 6303.984 | 1.419 | 36 3P* 16 4f(2F*).5d2(1D).6s(2D) 3P*
| | | | |4f(2F*) 5d2(3F)6s (2F) | * | 4 | 6475.540 | 0.897 | 43 3H* 17 (2F*)(3F)(2F) 1G*
| | | | |
4f.5d 6s2 | * | 3 | 6621.892 | 1.147 | 35 1F* 17 3D*
| | | | |
4f(2F*) 5d2(3F)6s (4F) | * | 5 | 6663.226 | 0.953 | 19 3I* 18 (2F*)(3F)(2F) 3I*
| | | | |4f(2F*) 5d2(3F)6s (4F) | * | 2 | 6836.628 | 0.68078 | 33 3F* 18 (2F*)(3F)(4F) 5G*
| | | | |
4f(2F*) 5d2(3F)6s (4F) | * | 3 | 7169.751 | 1.13 | 28 3F* 26 (2F*)(3F)(4F) 5F*
| | | | |
4f(2F*) 5d2(3F)6s (2F) | * | 4 | 7348.299 | 0.964 | 22 1G* 21 (2F*)(3F)(4F) 3H*
| | | | |4f(2F*) 5d2(3F)6s (4F) | * | 6 | 7696.210 | 1.076 | 39 3I* 19 (2F*)(1G)(2G) 3I*
| | | | |
4f(2F*) 5d2(3F)6s (4F)? | * | 5 | 7715.236 | 0.934 | 22 3I* 14 (2F*)(3F)(4F) 5G*
| | | | |
4f(2F*) 5d2(3F)6s (2F)? | * | 5 | 7841.955 | 1.063 | 24 3H* 21 (2F*)(3F)(2F) 3I*
| | | | |4f(2F*) 5d2(1D)6s (2D) | * | 1 | 7853.119 | 0.983 | 27 1P* 16 (2F*)(3F)(2F) 3D*
| | | | |
4f(2F*) 5d2(3F)6s (4F) | * | 4 | 7890.429 | 1.242 | 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.)
| | | | |
4f2.6s2 | 3H | 4 | 4762.718 | 0.80508 | 89 4 4f.5d(3G*).6s.6p(1P*) 3H
| | 5 | 6238.934 | 1.035 | 92 4 4f.5d(3G*).6s.6p(1P*) 3H
| | 6 | 7780.202 | 1.169 | 92 4 4f.5d(3G*).6s.6p(1P*) 3H
| | | | |
4f(2F*) 5d2(3F)6s (4F) | 3S* | 1 | 5097.777 | 1.88257 | 49 25 (2F*) (3F)(4F) 5D*| | | | |
4f(2F*) 5d2(3F)6s (4F) | * | 2 | 5210.906 | 1.22694 | 28 5D* 27 (2F*) (3F)(4F) 5G*
| | | | |
4f(2F*) 5d2(3F)6s (4F) | 5G* | 2 | 5409.236 | 0.773 | 35 24 (2F*) (3F)(4F) 5D*
| | 3 | 6234.792 | 1.04965 | 69 13 (2F*) (3F)(4F) 5D*
| | 4 | 6856.559 | 1.150 | 72 6 (2F*) (3F)(2F) 1G*| | 5 | 7467.160 | 1.179 | 69 7 (2F*) (3F)(4F) 3H*
| | 6 | 8055.526 | 1.207 | 50 18 (2F*) (3F)(4F) 3H*
| | | | |
4f(2F*) 5d2(3F)6s (4F) | * | 3 | 5519.751 | 1.24530 | 32 5D* 21 (2F*) (3F)(4F) 3F*
| | | | |
4f(2F*) 5d2(3F)6s (4F) | * | 0 | 5571.156 | | 44 5D* 35 4f.5d.6s2 3P*| | | | |
4f(2F*) 5d2(3F)6s (4F) | * | 4 | 5572.074 | 1.31658 | 29 5D* 27 (2F*) (3F)(4F) 3F*
| | | | |
4f.5d 6s2 | * | 1 | 5637.233 | 1.389 | 30 3P* 26 4f(2F*).5d2(3F).6s(4F) 5D*
| | | | |
4f(2F*) 5d2(3F)6s (4F) | 5F* | 1 | 5674.829 | 0.140 | 73 6 (2F*) (3F)(4F) 5D*
| | 2 | 5904.006 | 0.905 | 66 13 (2F*) (3F)(4F) 5G*| | 3 | 6337.061 | 1.232 | 42 16 (2F*) (3F)(4F) 5D*
| | 4 | 7174.156 | 1.373 | 67 19 (2F*) (3F)(4F) 5D*
| | 5 | 7933.558 | 1.345 | 81 5 (2F*) (3P)(4P) 5F*
| | | | |
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14 43/ 48-5
Energy levels of Ce I (contd.)
| | | | |
4f(2F*) 5d2(3F)6s (4F) | * | 1 | 8695.201 | 1.285 | 22 3P* 12 (2F*)(3F)(4F) 5P*
| | | | |
4f(2F*) 5d2(3F)6s (2F) | * | 3 | 8902.306 | 1.128 | 20 1F* 14 4f.5d.6s2 1F*
| | | | |
4f(2F*) 5d2(3F)6s (2F) | * | 5 | 8991.451 | 1.067 | 31 3H* 17 (2F*)(3P)(4P) 5G*| | | | |
4f(2F*) 5d2(3F)6s (4F) | * | 0 | 9119.094 | | 40 3P* 32 (2F*)(3F)(2F) 1S*
| | | | |
4f(2F*) 5d2(3P)6s (4P) | 3D* | 3 | 9135.099 | 1.274 | 50 12 (2F*)(3F)(4F) 5P*
| | | | |
4f(2F*) 5d2(3F)6s (2F) | * | 2 | 9200.707 | 1.376 | 24 3D* 21 (2F*)(3P)(2P) 3D*| | | | |
4f(2F*) 5d2(3F)6s (2F) | 3I* | 6 | 9333.222 | 1.047 | 61 20 (2F*)(1G)(2G) 3I*
| | 7 | 11061.551 | 1.141 | 79 16 (2F*)(1G)(2G) 3I*
| | | | |
4f(2F*) 5d2(3F)6s (4F) | * | 1 | 9369.628 | 1.065 | 21 3P* 17 (2F*)(3F)(2F) 3D*
| | | | |
4f(2F*) 5d2(3P)6s (4P) | 3D* | 2 | 9425.529 | 1.207 | 46 8 (2F*)(3F)(4F) 5S*| | | | |
4f(2F*) 5d2(3F)6s (2F) | 3F* | 2 | 9709.012 | 0.799 | 51 13 (2F*)(3P)(2P) 3F*
| | | | |
4f(2F*) 5d2(3F)6s (2F)? | 3G* | 3 | 9787.220 | 0.868 | 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.)
| | | | |
4f(2F*) 5d2(3P)6s (4P) | 5G* | 2 | 8088.912 | 0.403 | 95 3 (2F*)(3F)(4F) 5G*
| | 3 | 8307.309 | 0.957 | 81 6 (2F*)(3P)(4P) 3G*
| | 4 | 8762.126 | 1.054 | 50 9 (2F*)(3F)(2F) 3H*
| | 5 | 9462.705 | 1.19 | 46 12 (2F*)(3P)(4P) 3G*
| | 6 | 11030.470 | 1.193 | 51 21 (2F*)(3F)(2F) 3H*| | | | |
4f(2F*) 5d2(3F)6s (4F) | 5P* | 2 | 8101.187 | 1.735 | 85 5 (2F*)(3F)(4F) 5S*
| | 3 | 8270.249 | 1.504 | 74 11 (2F*)(3F)(4F) 3D*
| | 1 | 8430.846 | 2.04 | 62 10 (2F*)(3F)(4F) 3P*
| | | | |
4f2.6s2 | 3F | 2 | 8235.605 | 0.680 | 88 4 4f.5d(3F*).6s.6p(1P*) 3F| | 3 | 9206.305 | 1.083 | 90 4 4f.5d(3F*).6s.6p(1P*) 3F
| | 4 | 9379.148 | 1.139 | 55 32 1G
| | | | |
4f(2F*) 5d2(3F)6s (2F) | 1S* | 0 | 8351.167 | | 44 17 4f.5d.6s2 3P*
| | | | |
4f(2F*) 5d2(3F)6s (4F) | * | 2 | 8366.098 | 1.525 | 33 3P* 16 (2F*)(3F)(4F) 5S*| | | | |
4f(2F*) 5d2(3F)6s (2F) | * | 5 | 8400.730 | 0.917 | 30 3I* 19 4f.5d.6s2 1H*
| | | | |
4f(2F*) 5d2(3F)6s (2F) | * | 4 | 8509.209 | 0.954 | 20 3H* 19 (2F*)(3P)(4P) 5G*
| | | | |
4f(2F*) 5d2(3F)6s (4F) | 3I* | 7 | 8587.973 | 1.155 | 59 24 (2F*)(1G)(2G) 3I*| | | | |
4f(2F*) 5d2(3F)6s (4F) | * | 6 | 8603.531 | 1.225 | 36 5G* 25 (2F*)(3F)(4F) 3H*
| | | | |
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14 45/ 48-5
H atom in a weak homogeneous electric field (Stark effect) — per-
turbation theory
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-07-14 48/ 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