Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics
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Atomic Spectroscopy 10.16 Spectral Lines: Selection Rules, Intensities, Transition Probabilities, f Values, and Line Strengths 193
10.16.5 Relationships Between Line
and Multiplet Values
The relations between the total strength and f value of
a multiplet (M) and the corresponding quantities for the
lines of the multiplet (allowed transitions) are
S
M
=
S
line
, (10.21)
f
M
=
¯
λ
¯
g
i
−1
J
k
,J
i
g
i
λ(J
i
, J
k
) f(J
i
, J
k
). (10.22)
¯
λ is the weighted (“multiplet”) wavelength in vacuum:
¯
λ = n
¯
λ
air
= hc/ ∆E , (10.23)
Table 10.7 Relative strengths for lines of multiplets in LS coupling
Normal multiplets S–P, P–D, D–F, etc.
J
m
J
m
− 1 J
m
− 2 J
m
− 3 J
m
− 4
J
m
− 1 x
1
y
1
z
1
J
m
− 2 x
2
y
2
z
2
J
m
− 3 x
3
y
3
z
3
J
m
− 4 x
4
y
4
Multiplicity
1 2 3 4 5
S–P
S
M
= 3 6 9 12 15
x
1
3.00 4.00 5.00 6.00 7.00
y
1
2.00 3.00 4.00 5.00
z
1
1.00 2.00 3.00
P–P
S
M
= 9 18 27 36 45
x
1
9.00 10.00 11.25 12.60 14.00
x
2
4.00 2.25 1.60 1.25
x
3
1.00 2.25
y
1
2.00 3.75 5.40 7.00
y
2
3.00 5.00 6.75
P–D
S
M
= 15 30 45 60 75
x
1
15.00 18.00 21.00 24.00 27.00
x
2
10.00 11.25 12.60 14.00
x
3
5.00 5.00 5.25
y
1
2.00 3.75 5.40 7.00
y
2
3.75 6.40 8.75
y
3
5.60 6.75
z
1
0.25 0.60 1.00
z
2
1.00 2.25
z
3
3.00
Symmetrical multiplets P–P, D–D etc.
J
m
J
m
− 1 J
m
− 2 J
m
− 3
J
m
x
1
y
1
J
m
− 1 y
1
x
2
y
2
J
m
− 2 y
2
x
3
y
3
J
m
− 3 y
3
x
4
Multiplicity
12345
D–D
S
M
= 25 50 75 100 125
x
1
25.00 28.00 31.11 34.29 37.50
x
2
18.00 17.36 17.29 17.50
x
3
11.25 8.00 6.25
x
4
5.00 1.25
y
1
2.00 3.89 5.71 7.50
y
2
3.75 7.00 10.00
y
3
5.00 8.75
y
4
5.00
D–F
S
M
= 35 70 105 140 175
x
1
35.00 40.00 45.00 50.00 55.00
x
2
28.00 31.11 34.29 37.50
x
3
21.00 22.40 24.00
x
4
14.00 14.00
x
5
7.00
y
1
2.00 3.89 5.71 7.50
y
2
3.89 7.31 10.50
y
3
5.60 10.00
y
4
7.00
z
1
0.11 0.29 0.50
z
2
0.40 1.00
z
3
1.00
where
∆E = E
k
− E
i
= (
¯
g
k
)
−1
J
k
g
k
E
k
− (
¯
g
i
)
−1
J
i
g
i
E
i
(10.24)
and n is the refractive index of standard air.
10.16.6 Relative Strengths for Lines
of Multiplets in LS Coupling
Table 10.7 lists relative line strengths for frequently
encountered symmetrical (P → P, D → D) and nor-
mal (S → P, P → D) multiplets in LS coupling. The
strongest, or principal, lines are situated along the main
diagonal of the table and are called x
1
, x
2
, etc. Their
Part B 10.16
194 Part B Atoms
strengths normally diminish along the diagonal. The
satellite lines y
n
and z
n
are usually weaker and devi-
ate more from the LS values than the stronger diagonal
lines when departures from LS coupling are encoun-
tered. The total multiplet strengths S
M
are also listed in
Table 10.7. A discussion of their normalization as well
as more extensive tables are give n in [10.35].
10.17 Atomic Lifetimes
The radiative lifetime τ
k
of an atomic level k is related
to the sum of transition probabilities to all levels i lower
in energy than k:
τ
k
=
i
A
ki
−1
. (10.25)
The branching fraction of a particular transition, say
to state i
,isdefinedas
A
ki
/
i
A
ki
= A
ki
τ
k
. (10.26)
If only one branch (i
) exists (or if all other branches
may be neglected), one obtains A
ki
τ
k
= 1, and
τ
k
= 1/A
ki
. (10.27)
Precision lifetime measurement techniques are dis-
cussed in Chapts. 17 and 18.
10.18 Regularities and Scaling
10.18.1 Transitions in Hydrogenic
(One-Electron) Species
The nonrelativistic energy of a hydrogenic transition
(10.1, 10)is
(∆E)
Z
= (E
k
− E
i
)
Z
= R
Z
hc Z
2
1/n
2
i
− 1/n
2
k
.
(10.28)
Hydrogenic Z Scaling
The spectroscopic quantities for a hydrogenic ion of
nuclear charge Z are related to the equivalent quanti-
ties in hydrogen (Z = 1) as follows (neglecting small
differences in the values of R
Z
):
(∆E)
Z
= Z
2
(∆E)
H
, (10.29)
(λ
vac
)
Z
= Z
−2
(λ
vac
)
H
, (10.30)
S
Z
= Z
−2
S
H
, (10.31)
f
Z
= f
H
, (10.32)
A
Z
= Z
4
A
H
. (10.33)
For large values of Z, roughly Z > 20, relativistic
corrections become noticeable and must be taken into
account.
f -value Trends
f values for high series members (large n
values) of
hydrogenic ions decrease according to
f
n, l → n
, l ± 1
∝
n
−3
. (10.34)
Data for some lines of the main spectral series of hydro-
gen are given in Table 10.8.
10.18.2 Systematic Trends and Regularities
in Atoms and Ions with Two
or More Electrons
Atomic quantities for a given state or transition in an
isoelectronic sequence may be expressed as power series
expansions in Z
−1
:
Z
−2
E = E
0
+ E
1
Z
−1
+ E
2
Z
−2
+ ... , (10.35)
Z
2
S = S
0
+ S
1
Z
−1
+ S
2
Z
−2
+ ... , (10.36)
f = f
0
+ f
1
Z
−1
+ f
2
Z
−2
+ ... , (10.37)
where E
0
, f
0
,andS
0
are hydrogenic quantities. For
transitions in which n does not change (n
i
= n
k
), f
0
= 0,
since states i and k are degenerate.
For equivalent transitions of homologous atoms,
f values vary gradually. Transitions to be compared in
the case of the “alkalis” are [10.36]
nl − n
l
Li
→
(n + 1)l −
n
+ 1
l
Na
→
(n + 2)l −
n
+ 2
l
Cu
→ ... .
Complex atomic structures, as well as cases involv-
ing strong cancellation in the integrand of the transition
integral, generally do not adhere to this regular behavior.
Part B 10.18
Atomic Spectroscopy 10.19 Spectral Line Shapes, Widths, and Shifts 195
Table 10.8 Some transitions of the main spectral series of hydrogen
Cus- Cus-
Tran- tomary λ
b
A
ki
Tran- tomary λ
b
A
ki
sition name
a
(Å) g
c
i
g
k
(10
8
s
−1
) sition name
a
(Å) g
c
i
g
k
(10
8
s
−1
)
1–2 (L
α
) 1215.67 2 8 4.699 2–6 (H
δ
) 4101.73 8 72 9.732 (−3)
1–3 (L
β
) 1025.73 2 18 5.575 (−1)
d
2–7 (H
) 3970.07 8 98 4.389 (−3)
1–4 (L
γ
) 972.537 2 32 1.278 (−1) 3–4 (P
α
) 18 751.0 18 32 8.986 (−2)
1–5 (L
δ
) 949.743 2 50 4.125 (−2) 3–5 (P
β
) 12 818.1 18 50 2.201 (−2)
1–6 (L
) 937.803 2 72 1.644 (−2) 3–6 (P
γ
) 10 938.1 18 72 7.783 (−3)
2–3 (H
α
) 6562.80 8 18 4.410 (−1) 3–7 (P
δ
) 10 049.4 18 98 3.358 (−3)
2–4 (H
β
) 4861.32 8 32 8.419 (−2) 3–8 (P
) 9545.97 18 128 1.651 (−3)
2–5 (H
γ
) 4340.46 8 50 2.530 (−2)
a
L
α
is often called Lyman α,H
α
= Balmer α,P
α
= Paschen α,etc.
b
Wavelengths below 2000 Å are in vacuum; values above 2000 Å are in air
c
For transitions in hydrogen, g
i(k)
= 2(n
i(k)
)
2
,wheren
i(k)
, is the principal quantum number of the lower (upper) electron shell
d
The number in parentheses indicates the power of 10 by which the value has to be multiplied
10.19 Spectral Line Shapes, Widths, and Shifts
Observed spectral lines are always broadened, partly due
to the finite resolution of the spectrometer and partly
due to intrinsic physical causes. The principal physi-
cal causes of spectral line broadening are Doppler and
pressure broadening. The theoretical foundations of line
broadening are discussed in Chapts. 19 and 59.
10.19.1 Doppler Broadening
Doppler broadening is due to the thermal motion of
the emitting atoms or ions. For a Maxwellian velocity
distribution, the line shape is Gaussian; the full width at
half maximum intensity (FWHM)is,inÅ,
∆λ
D
1/2
=
7.16×10
−7
λ(T/M )
1/2
. (10.38)
T is the temperature of the emitters in K, and M the
atomic weight in atomic mass units (amu).
10.19.2 Pressure Broadening
Pressure broadening is due to collisions of the emitters
with neighboring particles (see also Chapts. 19 and 59).
Shapes are often approximately Lorentzian, i. e., I(λ) ∝
{1 +[(λ − λ
0
)/∆λ
1/2
]
2
}
−1
. In the following formulas,
all FWHMs and wavelengths are expressed in Å, particle
densities N in cm
−3
, temperatures T in K, and energies
E or I in cm
−1
.
Resonance broadening (self-broadening) occurs
only between identical species and is confined to lines
with the upper or lower level having an electric dipole
transition (resonance line) to the ground state. The
FWHM may be estimated as
∆λ
R
1/2
8.6×10
−30
(g
i
/g
k
)
1/2
λ
2
λ
r
f
r
N
i
, (10.39)
where λ is the wavelength of the observed line; f
r
and λ
r
are the oscillator strength and wavelength of the reso-
nance line; g
k
and g
i
are the statistical weights of its
upper and lower levels. N
i
is the ground state number
density.
For the 1s2p
1
P
◦
1
− 1s3d
1
D
2
transition in He i
[λ = 6678.15 Å; λ
r
(1s
21
S
0
− 1s2p
1
P
◦
1
) = 584.334 Å;
g
i
= 1; g
k
= 3; f
r
= 0.2762] at N
i
= 1×10
18
cm
−3
:
∆ λ
R
1/2
= 0.036 Å.
Van der Waals broadening arises from the dipole
interaction of an excited atom with the induced dipole
of a ground state atom. (In the case of foreign gas broad-
ening, both the perturber and the radiator may be in their
respective ground states.) An approximate formula for
the FWHM, strictly applicable to hydrogen and similar
atomic structures only, is
∆λ
W
1/2
3.0×10
16
λ
2
C
2/5
6
(T/µ)
3/10
N , (10.40)
where µ is the atom-perturber reduced mass in units
of u, N the perturber density, and C
6
the inter-
Part B 10.19
196 Part B Atoms
action constant. C
6
may be roughly estimated as
follows: C
6
= C
k
− C
i
, with C
i(k)
= (9.8×10
10
)α
d
R
2
i(k)
(α
d
in cm
3
, R
2
in a
2
0
). Mean atomic polarizabil-
ity α
d
≈ (6.7×10
−25
)(3I
H
/4E
∗
)
2
cm
3
,whereI
H
is
the ionization energy of hydrogen and E
∗
the en-
ergy of the first excited level of the perturber atom.
R
2
i(k)
≈ 2.5 [I
H
/(I − E
i(k)
)]
2
,whereI is the ioniza-
tion energy of the radiator. Van der Waals broadened
lines are red shifted by about one-third the size of the
FWHM.
For the 1s2p
1
P
◦
1
− 1s3d
1
D
2
transition in He i,and
with He as perturber: λ = 6678.15 Å; I = 198 311 cm
−1
;
E
∗
= E
i
= 171 135 cm
−1
; E
k
= 186 105 cm
−1
; µ = 2.
At T = 15 000 K and N = 1×10
18
cm
−3
: ∆λ
W
1/2
=
0.044 Å.
Stark broadening due to charged perturbers, i. e.,
ions and electrons, usually dominates resonance and van
der Waals broadening in discharges and plasmas. The
FWHM for hydrogen lines is
∆λ
S,H
1/2
=
2.50×10
−9
α
1/2
N
2/3
e
, (10.41)
Table 10.9 Values of Stark-broadening parameter α
1/2
of
the H
β
line of hydrogen (4861 Å) for various temperatures
and electron densities
T(K) N
e
(cm
−3
)
10
15
10
16
10
17
10
18
5000 0.0787 0.0808 0.0765 ...
10 000 0.0803 0.0840 0.0851 0.0781
20 000 0.0815 0.0860 0.0902 0.0896
30 000 0.0814 0.0860 0.0919 0.0946
where N
e
is the electron density. The half-width param-
eter α
1/2
for the H
β
line at 4861 Å, widely used for
plasma diagnostics, is tabulated in Table 10.9 for some
typical temperatures and electron densities [10.35]. This
reference also contains α
1/2
parameters for other hy-
drogen lines, as well as Stark width and shift data for
numerous lines of other elements, i. e., neutral atoms and
singly charged ions (in the latter, Stark widths and shifts
depend linearly on N
e
). Other tabulations of complete
hydrogen Stark profiles exist.
10.20 Spectral Continuum Radiation
10.20.1 Hydrogenic Species
Precise quantum-mechanical calculations exist only for
hydrogenic species. The total power
cont
radiated (per
unit source volume and per unit solid angle, and ex-
pressed in SI units) in the wavelength interval ∆λ is
the sum of radiation due to the recombination of a free
electron with a bare ion (free–bound transitions) and
bremsstrahlung (free–free transitions):
cont
=
e
6
2π
3
0
(6πm
e
)
3/2
N
e
N
Z
Z
2
×
1
(kT )
1/2
exp
−
hc
λkT
∆λ
λ
2
×
2 Z
2
I
H
kT
n
n≥(Z
2
I
H
λ/hc)
1/2
γ
fb
n
3
exp
Z
2
I
H
n
2
kT
+
¯
γ
fb
exp
Z
2
I
H
(n
+ 1)
2
kT
− 1
+ γ
ff
(10.42)
where N
e
is the electron density, N
Z
the number dens-
ity of hydrogenic (bare) ions of nuclear charge Z, I
H
the
ionization energy of hydrogen, n
the principal quantum
number of the lowest level for which adjacent levels
are so close that they approach a continuum and sum-
mation over n may be replaced by an integral. (The
choice of n
is rather arbitrary; n
as low as 6 is found
in the literature.) γ
fb
and γ
ff
are the Gaunt factors,
which are generally close to unity. (For the higher free-
bound continua, starting with n
+ 1, an average Gaunt
factor
¯
γ
fb
is used.) For neutral hydrogen, the recom-
bination continuum forming H
−
becomes important,
too [10.37].
In the equation above, the value of the constant
factor is 6.065 × 10
−55
Wm
4
J
1/2
sr
−1
. [Numerical ex-
ample: For atomic hydrogen (Z = 1), the quantity
cont
has the value 2.9Wm
−3
sr
−1
under the fol-
lowing conditions: λ = 3×10
−7
m; ∆λ = 1×10
−10
m;
N
e
(= N
Z=1
) = 1×10
21
m
−3
; T = 12 000 K. The lower
limit of the summation index n is 2; the upper limit n
has been taken to be 10. All Gaunt factors γ
fb
,
¯
γ
fb
,and
γ
ff
have been assumed to be unity.]
10.20.2 Many-Electron Systems
For many-electron systems, only approximate theo-
retical treatments exist, based on the quantum-defect
Part B 10.20
Atomic Spectroscopy References 197
method (for results of calculations for noble gases,
see, e.g., [10.38]). Experimental work is centered on
the noble gases [10.39]. Modifications of the con-
tinuum by autoionization processes must also be
considered.
Near the ionization limit, the f values for bound-
bound transitions of a spectral series (n
→∞)
make a smooth connection to the differential os-
cillator strength distribution d f/d in the con-
tinuum [10.40].
10.21 Sources of Spectroscopic Data
Access to most of the atomic spectroscopic databases
currently online is given by links at the Plasma
Gate server [10.41]. Extensive data from NIST com-
pilations of atomic wavelengths, energy levels, and
transition probabilities are available from the Atomic
Spectra Database (ASD)attheNIST site [10.13]. Sec-
tion 10.15 includes additional references for wavelength
tables.
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quista: Atlas of the Spectrum of a Platinum/Neon
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edn. (American Inst. Physics Press, Springer, New
York 2000)
10.36 A. W. Weiss: J. Quant. Spectrosc. Radiat. Transfer
18, 481 (1977)
10.37 H. R. Griem: Spectral Line Broadening by Plasmas
(Academic, New York 1974)
10.38 J. R. Roberts, P. A. Voigt: J. Res. Natl. Bur. Stand.
75, 291 (1971)
10.39 I. I. Sobelman: Atomic Spectra and Radiative Tran-
sitions, 2nd edn. (Springer, Berlin, Heidelberg
1992)
10.40 A. T. M. Wilbers, G. M. W. Kroesen, C. J. Timmer-
mans, D. C. Schram: J. Quant. Spectrosc. Radiat.
Transfer 45, 1 (1991)
10.41 Y. Ralchenko: Databases for Atomic and Plasma
Physics; at site
http://plasma-gate.weizmann.ac.il/DBfAPP.html
Part B 10
199
High Precision
11. High Precision Calculations for Helium
Exact analytic solutions to the Schrödinger
equation are known only for atomic hydrogen, and
other equivalent two-body systems (see Chapt. 9).
However, very high precision approximations are
now available for helium, which are essentially
exact for all practical purposes. This chapter
summarizes the computational methods and
tabulates numerical results for the ground
state and several singly excited states. Similar
methods can be applied to other three-body
problems.
11.1 The Three-Body
Schrödinger Equation .......................... 199
11.1.1 Formal Mathematical
Properties ................................ 200
11.2 Computational Methods ....................... 200
11.2.1 Variational Methods .................. 200
11.2.2 Construction of Basis Sets ........... 201
11.2.3 Calculation of Matrix Elements.... 202
11.2.4 Other Computational Methods .... 205
11.3 Variational Eigenvalues........................ 205
11.3.1 Expectation Values of Operators
and Sum Rules.......................... 205
11.4 Total Energies ..................................... 208
11.4.1 Quantum Defect Extrapolations... 211
11.4.2 Asymptotic Expansions .............. 213
11.5 Radiative Transitions ........................... 215
11.5.1 Basic Formulation ..................... 215
11.5.2 Oscillator Strength Table............. 216
11.6 Future Perspectives.............................. 218
References .................................................. 218
11.1 The Three-Body Schrödinger Equation
The Schrödinger equation for a three-body system con-
sisting of a nucleus of charge Ze,andmassM,andtwo
electrons of charge −e and mass m
e
is
1
2M
P
2
N
+
1
2m
e
2
i=1
P
2
i
+ V(R
N
, R
i
)
Ψ = EΨ,
(11.1)
where P
i
= ( /i)∇
i
and
V(R
N
, R
i
) =−
Ze
2
|R
N
−R
1
|
−
Ze
2
|R
N
−R
2
|
+
e
2
|R
1
−R
2
|
(11.2)
depends only on the relative particle separations. Since
the center of mass (c.m.) is then an ignorable coordinate,
it can be eliminated by defining the relative particle
coordinates
r
i
= R
i
− R
N
to obtain
1
2µ
2
i=1
p
2
i
+
1
M
p
1
· p
2
+ V(r
1
, r
2
)
Ψ = EΨ,
(11.3)
where µ = m
e
M/(m
e
+ M) is the electron reduced mass
and the term H
mp
= p
1
· p
2
/M is called the mass polar-
ization operator. For computational purposes, it is usual
to measure distance in units of a
µ
= (m
e
/µ)a
0
and en-
ergies in units of e
2
/a
µ
=2(µ/m
e
)R
∞
so that (11.3)
assumes the dimensionless form
−
1
2
2
i=1
∇
2
ρ
i
−
µ
M
∇
ρ
1
·∇
ρ
2
+ V(ρ
1
, ρ
2
)
Ψ = εΨ ,
(11.4)
where ρ
i
= r
i
/a
µ
, ε = E/(e
2
/a
µ
),and
V(ρ
1
, ρ
2
) =−
Z
ρ
1
−
Z
ρ
2
+
1
|ρ
1
− ρ
2
|
.
(11.5)
The limit µ/M → 0 defines the infinite nuclear mass
problem with eigenvalue ε
0
and eigenfunction Ψ
0
.Ifthe
mass polarization term is treated as a small perturbation,
then the total energy assumes the form
E =
ε
0
+
µ
M
ε
1
+
µ
M
2
ε
2
+···
µ
m
e
e
2
a
0
,
(11.6)
Part B 11
200 Part B Atoms
where ε
1
=−Ψ
0
|∇
ρ
1
· ∇
ρ
2
|Ψ
0
determines the first-
order specific mass shift and ε
2
is the second-order
coefficient. The common (µ/m
e
)ε
0
mass scaling of all
eigenvalues determines the normal mass shift (isotope
shift). Since µ/m = 1 − µ/M,theshiftis−(µ/M)ε
0
.
11.1.1 Formal Mathematical Properties
Two-Particle Coalescences
The exact nonrelativistic wave function for any many-
body system contains discontinuities or cusps in the
spherically averaged radial derivative with respect to r
ij
as r
ij
→ 0, where r
ij
=|r
i
−r
j
| is any interparticle co-
ordinate. If the masses and charges are m
i
and q
i
respectively, then the discontinuities are given by the
Kato cusp condition [11.1]
2
∂
¯
Ψ
∂r
ij
r
ij
=0
= µ
ij
q
i
q
j
Ψ(r
ij
= 0), (11.7)
where µ
ij
= m
i
m
j
/(m
i
+ m
j
) and
¯
Ψ denotes the wave
function averaged over a sphere centered at r
ij
= 0. If
Ψ vanishes at r
ij
= 0, then its leading dependence on r
ij
is of the form r
l
ij
Y
lm
(r
ij
) for some integer l > 0 [11.2].
Equation (11.7) applies to any Coulombic system. The
electron–nucleus cusp in the wave functions for hydro-
gen provides a simple example.
Three-Particle Coalescences
Three-particle coalescences are described by the Fock
expansion [11.3–6], as recently discussed by Myers
et al. [11.7]. For the S-states of He-like ions, the ex-
pansion has the form
Ψ(r
1
, r
2
) =
∞
j=0
[ j/2]
k=0
R
j
(ln R)
k
φ
j,k
, (11.8)
where [ ] denotes “greatest integer in”, and R = (r
2
1
+
r
2
2
)
1/2
is the hyperradius. The leading coefficients are
φ
0,0
= 1 ,
φ
1,0
=−
Zr
1
+ Zr
2
−
1
2
r
12
R ,
φ
2,1
=−2Z
π − 2
3π
r
1
· r
2
R
2
. (11.9)
The next term φ
2,0
is known in terms of a lengthy ex-
pression [11.7–9], but higher terms have not yet been
obtained in closed form. The Fock expansion has been
proved convergent for all R <
1
2
[11.10], and extended
to pointwise convergence for all R [11.11, 12].
Asymptotic Form
The long range behavior of many-electron wave
functions has been studied from several points of
view [11.13–15]. The basic result of [11.16]isthatat
large distances, the one-electron density behaves as
ρ
1/2
(r) ≈ r
Z
∗
/t−1
e
−tr
, (11.10)
where t = (2I
1
)
1/2
, I
1
is the first ionization potential (in
a.u.), and Z
∗
= Z − N + 1 is the screened nuclear charge
seen by the outer most electron. For hydrogenic systems
with principal quantum number n, I
1
= (Z
∗
)
2
/2n
2
.
11.2 Computational Methods
11.2.1 Variational Methods
Most high precision calculations for the bound states
of three-body systems such as helium are based on the
Rayleigh–Ritz variational principle. For any normaliz-
able trial function Ψ
tr
, the quantity
E
tr
=
Ψ
tr
|H|Ψ
tr
Ψ
tr
|Ψ
tr
(11.11)
satisfies the inequality E
tr
≥ E
1
,whereE
1
is the true
ground state energy. Thus E
tr
is an upper bound to E
1
.
The inequality is easily proved by expanding Ψ
tr
in the
complete basis set of eigenfunctions Ψ
1
, Ψ
2
, Ψ
3
, ···
of H with eigenvalues E
1
< E
2
< E
3
< ···,sothat
Ψ
tr
=
∞
i=1
c
i
Ψ
i
, (11.12)
where the c
i
are expansion coefficients. This can always
be done in principle, even though the exact Ψ
i
are not
actually known. If Ψ
tr
is normalized so that Ψ
tr
|Ψ
tr
=1,
then
∞
i=1
|c
i
|
2
= 1and
E
tr
=|c
1
|
2
E
1
+|c
2
|
2
E
2
+|c
3
|
2
E
3
+···
= E
1
+|c
2
|
2
(E
2
− E
1
) +|c
3
|
2
(E
3
− E
1
) +···
≥ E
1
, (11.13)
which proves the theorem.
Part B 11.2
High Precision Calculations for Helium 11.2 Computational Methods 201
The basic idea of variational calculations then is to
write Ψ
tr
in some arbitrarily chosen mathematical form
with variational parameters (subject to normalizability
and boundary conditions at the origin and infinity), and
then adjust the parameters to obtain the minimum value
of E
tr
.
The minimization problem for the case of linear
variational coefficients can be solved algebraically. For
example, let
χ
p
(α, β ) = r
i
1
r
j
2
r
k
12
e
−αr
1
−βr
2
(11.14)
denote the members of a basis set, where p is an index
labeling distinct triplets of nonnegative integer values
for the powers {i, j, k},andα, β are (for the moment)
fixed constants determining the distance scale.
If Ψ
tr
is expanded in the form
Ψ
tr
=
N
p=1
c
p
χ
p
(α, β ) , (11.15)
then the solution to the system of equations
∂E
tr
/∂c
p
= 0, p = 1,... ,N, is exactly equivalent to
solving the N-dimensional generalized eigenvalue prob-
lem
Hc = λOc ,
(11.16)
where c is a column vector of coefficients c
p
;and
H and O have matrix elements H
pq
=χ
p
|H|χ
q
and
O
pq
=χ
p
|χ
q
.ThereareN eigenvalues λ
1
,λ
2
,...λ
N
,
of which the lowest is an upper bound to E
1
.
Extension to Excited States
By the Hylleraas–Undheim–MacDonald (HUM) theo-
rem [11.17, 18], the remaining eigenvalues λ
2
,λ
3
,...
are also upper bounds to the exact energies E
2
, E
3
,...,
provided that the spectrum is bounded from below. The
HUM theorem is a consequence of the matrix eigenvalue
interleaving theorem, which states that as the dimensions
of H and O are progressively increased by adding an
extra row and column, the N old eigenvalues λ
p
fall be-
tween the N + 1 new ones. Consequently, as illustrated
in Fig. 11.1, all eigenvalues numbered from the bottom
up must move inexorably downward as N is increased.
Since the exact spectrum of bound states is obtained
in the limit N →∞,noλ
p
can cross the correspond-
ing exact E
p
on its way down. Thus λ
p
≥ E
p
for every
finite N.
11.2.2 Construction of Basis Sets
Since the Schrödinger equation (11.4) is not separable
in the electron coordinates, basis sets which incorporate
E
∞
E
5
E
4
E
3
E
2
E
1
N
12345
λ
5
λ
1
λ
4
λ
3
λ
2
Fig. 11.1 Diagram illustrating the Hylleraas–Undheim–
MacDonald Theorem. The λ
p
, p = 1,... ,N are the
variational eigenvalues for an N-dimensional basis set, and
the E
i
are the exact eigenvalues of H. The highest λ
p
lie in
the continuous spectrum of H
the r
12
=|r
1
−r
2
| interelectron coordinate are most ef-
ficient. The necessity for r
12
terms also follows from
the Fock expansion (11.8). A basis set constructed from
terms of the form (11.14) is called a Hylleraas basis
set [11.19, 20]. (The basis set is often expressed in
terms of the equivalent variables s = r
1
+r
2
, t = r
1
−r
2
,
u = r
12
.)
With χ
p
(α, β ) defined as in (11.14), the general form
for a state of total angular momentum L is
Ψ
tr
=
[L/2]
l
1
=0
p
C
p,l
1
χ
p
(α, β )r
l
1
1
r
l
2
2
Y
M
l
1
L−l
1
L
(
ˆ
r
1
,
ˆ
r
2
)
± exchange ,
(11.17)
where
Y
M
l
1
l
2
L
(
ˆ
r
1
,
ˆ
r
2
) =
m
1
,m
2
Y
l
1
m
1
(
ˆ
r
1
)Y
l
2
m
2
(
ˆ
r
2
)
× l
1
l
2
m
1
m
2
|LM (11.18)
is the vector coupled product of angular momenta l
1
, l
2
for the two electrons. The sum over p in (11.17) typically
includes all terms in (11.14) with i + j + k ≤ Ω,where
Ω is an integer determining a so-called Pekeris shell of
terms, and the exchange term denotes the interchange
of r
1
and r
2
with (+) for singlet states and (−) for
triplet states. Convergence is studied by progressively
Part B 11.2
202 Part B Atoms
increasing Ω. The number of terms is
N =
1
6
(Ω + 1)(Ω + 2)(Ω + 3).
Basis sets of this type were used by many authors,
culminating in the extensive high precision calculations
of Pekeris and coworkers [11.21] for low-lying states,
using as many as 1078 terms. Their accuracy is not easily
surpassed because of the rapid growth of N with Ω,and
because of numerical linear dependence in the basis set
for large Ω. Recently, their accuracy has been surpassed
by two principal methods. The first explicitly includes
powers of logarithmic and half-integral terms in χ
p
,as
suggested by the Fock expansion [11.22–25]. This is
particularly effective for S-states. The second focuses
directly on the multiple distance scales required for an
accurate representation of the wave function by writing
the trial function in terms of the double basis set [11.26]
Ψ
tr
=
[L/2]
l
1
=0
p
C
(1)
p,l
1
χ
p
(α
1
,β
1
) + C
(2)
p,l
1
χ
p
(α
2
,β
2
)
× r
l
1
1
r
l
2
2
Y
l
1
l
2
L
(r
1
, r
2
) ± exchange , (11.19)
where each χ
p
(α, β ) is of the form (11.14), but with dif-
ferent values for the distance scales α
1
,β
1
and α
2
,β
2
in
the two sets of terms. They are determined by a complete
minimization of E
tr
with respect to all four parameters,
producing a natural division of the basis set into an
asymptotic sector and a close-range correlation sector.
The method produces a dramatic improvement in accu-
racy for higher-lying Rydberg states (where variational
methods typically deteriorate rapidly in accuracy) and is
also effective for low-lying S-states [11.27–29]. Nonrel-
ativistic energies accurate to 1 part in 10
16
are obtainable
with modest computing resources.
Another version of the variational method is the
quasi-random (or stochastic) method in which non-
linear exponential parameters for all three of r
1
, r
2
,
and r
12
are chosen at random from certain specified in-
tervals [11.30, 31]. The method is remarkably accurate
and efficient for low-lying states, but subject to severe
roundoff error.
11.2.3 Calculation of Matrix Elements
The three-body problem has the unique advantage
that the full six-dimensional volume element (in the
c.m. frame) can be transformed to the product of
a three-dimensional angular integral (ang) and a three-
dimensional radial integral (rad) over r
1
, r
2
,andr
12
.The
transformation is
dr
1
dr
2
=
2π
0
dφ
2π
0
dϕ
1
π
0
sin θ
1
dθ
1
×
∞
0
r
1
dr
1
∞
0
r
2
dr
2
r
1
+r
2
|r
1
−r
2
|
r
12
dr
12
,
(11.20)
where θ
1
, ϕ
1
are the polar angles of r
1
and φ is the angle
of rotation of the triangle formed by r
1
, r
2
,andr
12
about the r
1
direction. The polar angles θ
2
, ϕ
2
are then
dependent variables. The basic angular integral is
Y
∗
l
1
m
1
(θ
1
,ϕ
1
)Y
l
2
m
2
(θ
2
,ϕ
2
)
ang
= 2πδ
l
1
l
2
δ
m
1
m
2
P
l
1
(cos θ) , (11.21)
where cos θ ≡
ˆ
r
1
·
ˆ
r
2
denotes the radial function
cos θ =
r
2
1
+r
2
2
−r
2
12
2r
1
r
2
, (11.22)
and P
l
(cos θ) is a Legendre polynomial. The angu-
lar integral over vector-coupled spherical harmonics
is [11.32]
Y
M
∗
l
1
l
2
L
(
ˆ
r
1
,
ˆ
r
2
) Y
M
l
1
l
2
L
(
ˆ
r
1
,
ˆ
r
2
)
ang
= δ
L,L
δ
M,M
Λ
C
Λ
P
Λ
(cos θ) , (11.23)
where
C
Λ
=
1
2
[(2l
1
+ 1)(2l
1
+ 1)(2l
2
+ 1)(2l
2
+ 1)]
1/2
× (−1)
L+Λ
(2Λ + 1)
×
l
1
l
1
Λ
000
l
2
l
2
Λ
000
Ll
1
l
2
Λ l
2
l
1
,
(11.24)
and the sum over Λ includes all nonvanishing terms. This
can be extended to general matrix elements of tensor
operators by further vector coupling [11.32].
Radial Integrals
Table 11.1 lists formulas for the radial integrals arising
from matrix elements of H,aswellasthosefromthe
Breit interaction (see Sect. 21.1). Although they can all
be written in closed form, some have been expressed
as infinite series in order to achieve good numerical
stability. The exceptions are formulas 5 and 10 in the
Table, which became unstable as α → β. More elaborate
Part B 11.2