Mem. S.A.It. Suppl. Vol. 15, 68
c
SAIt 2010
Memorie della
Supplementi
Interatomic potentials and applications to
spectral line broadening
G. Peach
Department of Physics and Astronomy, University College London, Gower Street, London
WC1E 6BT, United Kingdom; e-mail: g.peach@ucl.ac.uk
Abstract. The use of three-body models to construct a Hamiltonian and hence obtain in-
teratomic potentials is discussed and the accuracy of the model assessed. Extensive tests
of this approach have been carried out for the NaH molecule, since this system is particu-
larly important for applications to the solar spectrum where the broadening of alkali lines is
mainly due to perturbation by neutral hydrogen. The broadening of spectral lines by neutral
atoms is considered and calculations of the pressure broadening of alkali resonance lines by
helium are presented. This data is important for the interpretation of the observed spectra of
cool dwarf stars. Finally, the validity of the simple Van der Waals formula for the line width
is tested against detailed quantum mechanical calculations.
Key words. Interatomic potentials: Na-H system – Spectral line broadening: calculations
– Cool stars: alkali resonance doublets
1. Introduction
Accurate pressure broadened profiles of alkali
resonance doublets are needed for modelling
of the atmospheres of cool stars and for gener-
ating their synthetic spectra in the region 400
- 900 nm. When the usual impact theory of
line broadening is used, see Baranger (1958),
the profile is simply Lorentzian and the widths
and shifts of the lines can be calculated, pro-
vided that interaction potentials for the emitter-
perturber system are available. However when
the lines utterly dominate their region of the
spectrum, it becomes important to also accu-
rately represent the profile in the line wings,
where the impact theory is no longer valid.
The non-Lorentzian profiles of the strongly
broadened Na I and K I doublets have been
Send offprint requests to: G. Peach
studied by Burrows & Volobuyev (2003) and
Allard et al. (2003), with the emphasis on ap-
proximate or unified semiclassical models that
can describe the far wings of the profiles. Also
Zhu et al. (2005) have carried out quantum
mechanical calculations of the emission and
absorption spectra for the wings of the lithium
resonance line. However highly accurate calcu-
lations of the central Lorentzian cores are still
needed and in this paper we present calcula-
tions of the widths for alkali lines broadened
by helium.
Accurate calculations of line widths are im-
possible unless interatomic potentials for the
emitter-perturber system are well known. The
calculation of these potentials for both ground
and excited electronic states, valid over a wide
range of interatomic separations, represents a
big challenge in itself and in this paper, three-
body models of atom-atom systems are dis-
Peach: Interatomic potentials and line broadening 69
cussed and their accuracy assessed by compar-
ison with data obtained using approaches de-
veloped in quantum chemistry.
The original impact theory of spectral line
broadening that Baranger developed is purely
quantum mechanical, but further approxima-
tions to this theory are often made. They are
of two types, firstly the quantum-mechanical
treatment is replaced by a semi-classical one
and secondly the interatomic potential used is
replaced by the asymptotic Van der Waals in-
teraction. To test the validity of these approxi-
mations, results for Lorentzian line widths are
presented for lithium and sodium lines broad-
ened by helium for 70.0 K ≤ T ≤ 3000 K.
2. Interatomic potentials
Large quantum chemistry calculations can pro-
vide very accurate potentials for electronic
states of atom-atom systems at short and in-
termediate separations, but are limited to the
study of electronic states of relatively low exci-
tation. However spectral line broadening prob-
lems often involve low-energy scattering by in-
teraction potentials for the more highly excited
states and so methods must be developed to ob-
tain an accurate representation of these poten-
tials at medium and large interatomic separa-
tions. If a valence electron of the emitter is in
an excited state and the perturber is assumed
to be always in its ground state, a three-body
model of the system is intuitively appealing
and is also practicable.
The adiabatic molecular potentials for the
A
m+
+ B
n+
+e
−
system can obtained by us-
ing a three-body model in which the ions
A
m+
and B
n+
are represented by polarisable
atomic cores. This model is most accurate
if these two cores are tightly bound so that
the excitation of core states need only be ac-
counted for by second-order perturbation the-
ory through the inclusion of their polariz-
abilities. Model potentials which may be l-
independent or l-dependent, are used to repre-
sent the electron-atom and electron-atomic ion
interactions and the basic methods adopted for
obtaining these model potentials are discussed
by Peach (1982). For example if m , 0, poten-
tials for the system A
m+
+ e
−
are determined
by fitting the spectrum of the ion A
m+ −1
, but
if m = 0, the potentials are adjusted so as
to reproduce the elastic scattering phase shifts
for electrons scattered by the neutral atom A
0
.
In either case, these potentials generate wave
functions that contain the correct number of
nodes and this means that the model poten-
tials also support unphysical bound states cor-
responding to the presence of closed shells in
the A
m+
and B
n+
ions. This effect is taken into
account in the calculation of the molecular po-
tentials by including the unphysical states in
the atomic basis used in the diagonalization of
the Hamiltonian for the three-body model.
Molecular potentials for the Li–He, Na–
He and K–He systems have been obtained us-
ing the three-body model and are discussed
by Leo et al. (2000). Subsequently they have
been used in line broadening calculations by
Mullamphy et al. (2007).
2.1. Modelling of atom-atom system
The electron-core interactions are modelled us-
ing the following form:
V
a,b
(r) = −
Z
r
(1 + δ + δ
0
r) exp(−αr)
−
z
r
−
α
d
2r
4
F
1
(r) −
α
q
2r
6
F
2
(r) + 3
β
d
r
6
F
3
(r)
+small energy term (optional) , (1)
where Z + z=nuclear charge, z = m, n and
α
d
and α
q
are the static dipole and quadrupole
polarizabilities. The coefficient β
d
is a dynami-
cal correction to the dipole polarizability and
the functions F
1
(r), F
2
(r) and F
3
(r) are cut-
off factors that ensure that the long-range con-
tributions to the potential vanish at the origin.
Parameters α, δ and δ
0
are then varied so as
to reproduce as closely as possible the known
atomic data.
The core-core interaction is specified by
V
c
(R) ' −z
2
a
α
b
d
2R
4
− z
2
b
α
a
q
2R
6
+short − range terms , (2)
70 Peach: Interatomic potentials and line broadening
where R is the interatomic separation. The pos-
sible options for the choice of the short-range
term are:
(a) use the three-body model itself to generate
the potential;
(b) use a simple analytic form based on pertur-
bation theory.
Choices (a) and (b) differ only for separations
of the order of R ' R
A
+ R
B
where R
A
and R
B
are the mean radii of the cores A
m+
and B
n+
.
Finally the three-body interaction is given
by
V
3
(r, R) '
α
d
r
2
R
2
P
1
(ˆr ·
ˆ
R) +
α
q
r
3
R
3
P
2
(ˆr ·
ˆ
R)
+small energy term (optional) , (3)
for large values of R where P
1
(ˆr·
ˆ
R) and P
2
(ˆr·
ˆ
R)
are Legendre polynomials.
The model Hamiltonian is then given by
H = −
1
2
∇
2
+ V
a
(r
a
) + V
b
(r
b
) + V
c
(R)
+V
3
(r
a
, R) + V
3
(r
b
, R) , (4)
where r
a
and r
b
are the position vectors of the
electron relative to cores A
m+
and B
n+
. A set
of atomic basis states on one or both centres is
used and the Hamiltonian matrix is then diag-
onalized to obtain the electronic energies.
2.2. Principles and problems
In equation (1) the form of the long-range in-
teractions produced by polarization are based
on well-known second-order perturbation the-
ory. The three-body term in equation (3) must
be included in order to ensure that the cor-
rect behaviour of the adiabatic potentials is
obtained for large separations. For example if
m = 1 and n = 0, potentials for the system
A
0
+B
0
tend to the Van der Waals form, −C
6
/R
6
as R → ∞. No circular procedures are adopted
so that the variable parameters that appear in
the model Hamiltonian (4) are determined us-
ing data that is totally independent of any ex-
isting data for the molecule concerned.
Some problems arise in constructing the
model Hamiltonian. The positions of virtual
states in electron-core model potentials are
sensitive to the precise fit to the atomic data.
This applies particularly in the case of the
electron-neutral atom potentials. Also, a differ-
ent potential may have to be used for ground
states; for example it is not possible to find
a central potential that reproduces the posi-
tions of all the excited singlet states of helium
and predicts the binding energy of He(1s
2
) cor-
rectly. This is not surprising, since in this case
the core electron and the valence electron are
in fact equivalent. Therefore it is necessary to
use a separate model potential which is chosen
to reproduce the energy of the state and to give
its correct polarizability.
2.3. The Na
∗
–H system
Extensive tests have been carried out to see
how accurate a three-body model proves to be
when applied to the system Na
∗
–H. This has
been chosen as a test case because H(1s) is
not a very tightly bound atom and in addition,
H
−
(1s
2
) is bound and so a state which asymp-
totically separates to the configuration Na
+
+
H
−
interacts strongly with the other states of
the type Na(nl) + H(1s). The core-core po-
tential is generated by initially using a sim-
ple analytic form for the Na
2+
(2p
5
)–H
+
inter-
action which is fed into the three-body model
to obtain a potential for Na
+
(2p
6
)–H
+
. This in
turn is then fed back into the model to obtain
a potential for Na
+
(2p
6
)–H. Finally this pro-
vides the input core-core potential for the Na
∗
–
H system.
3. Spectral line broadening
An early version of impact theory was pub-
lished by Lindholm (1941). The relative mo-
tion of the two atoms in the collision is treated
semi-classically and is assumed to follow a
straight-line path. The half-half width w and
shift d are given by
w + id = 2πN{
Z
∞
0
v f (v)dv
×
Z
∞
0
[1 − exp(iη)]ρdρ }
Av
. (5)
where ’Av’ denotes an average over degenerate
components of the line and N is the perturber
Peach: Interatomic potentials and line broadening 71
number density. The impact parameter is de-
noted by ρ and the relative velocity is v. The
function f (v) is the Maxwell velocity distribu-
tion normalized so that
Z
∞
0
f (v)dv = 1. (6)
The phase shift η is obtained from
η(ρ, v) = −
1
~
Z
∞
−∞
V(t) dt (7)
and for Van der Waals broadening V(t) is re-
placed by −C
6
/R
6
(t) and then the integrals can
be evaluated analytically, see Peach (1981).
The quantum-mechanical impact theory of
Baranger (1958) in its simplest form can be
established simply by making the transition
(Mvρ)
2
→ ~
2
l(l + 1) (8)
and then
2ρdρ →
~
2
(Mv)
2
(2l + 1)∆l , (9)
and the integral over ρ is replaced by a sum
over l. The phase shift η(ρ, v) is replaced so that
η(ρ, v) → 2 [η
i
(l, v) − η
f
(l, v)] , (10)
where η
i
(l, v) and η
f
(l, v) are elastic scattering
phase shifts for scattering in the adiabatic po-
tentials that describe the initial and final states
of the system.
The approximation to the quantum-
mechanical theory that corresponds most
closely to the semi-classical assumption
of a straight-line path is to use first-order
perturbation theory for the phase shifts η
i
and
η
f
in equation (10), in which the scattering
wavefunctions are replaced by plane waves.
This is commonly referred to as the Born
approximation. The scattering theory required
is discussed by Geltman (1969) and its appli-
cation to the spectral line broadening problem
by Peach & Whittingham (2009).
4. Results
The X
1
Σ and A
1
Σ states have been very accu-
rately determined by Olson & Liu (1980) and
Table 1. Interaction potential energies, V(R),
for X
1
Σ states of NaH in a.u.
R(a
0
) V(R)
1
V(R)
2
2.550802 -0.0222056 0.0173076
2.576990 -0.0257018 0.0127261
2.606257 -0.0293679 0.0077912
2.639061 -0.0332010 0.0024712
2.676012 -0.0371990 -0.0032134
2.717955 -0.0413606 -0.0092307
2.766100 -0.0456853 -0.0155804
2.822284 -0.0501734 -0.0224013
2.889504 -0.0548261 -0.0297956
2.973243 -0.0596454 -0.0376331
3.085685 -0.0646341 -0.0462748
3.269457 -0.0697958 -0.0561524
3.566044 -0.0724457 -0.0637470
3.926517 -0.0697958 -0.0641757
4.239926 -0.0646341 -0.0603022
4.484732 -0.0596454 -0.0559067
4.703898 -0.0548261 -0.0514642
4.910000 -0.0501734 -0.0470963
5.108979 -0.0456853 -0.0428457
5.304375 -0.0413606 -0.0387301
5.498659 -0.0371990 -0.0347577
5.693791 -0.0332010 -0.0309332
5.891502 -0.0293679 -0.0272634
6.093461 -0.0257018 -0.0237557
6.301393 -0.0222056 -0.0204220
7.0 -0.0124200 -0.0114645
8.0 -0.0047285 -0.0043448
10.0 -0.0005987 -0.0005714
11.75 -0.0001111 -0.0001159
12.0 -0.0000889 -0.0000942
13.5 -0.0000275 -0.0000310
15.0 -0.0000116 -0.0000126
17.0 -0.0000037 -0.0000049
1
Olson & Liu (1980), Zemke et al. (1984)
2
present work
Zemke et al. (1984) using a combination of
ab initio quantum chemistry techniques and an
RKR analysis of observed spectra. Very similar
results have been obtained by Leininger et al.
(2000) who use very large Gaussian basis sets
and carry out a full CI calculation. In tables 1
and 2 the present work based on the three-body
model is compared with that of Olson & Liu
(1980) and Zemke et al. (1984). For the X
1
Σ
state the three-body model predicts the posi-
tion of the minimum to be at R ' 3.8a
0
, which
72 Peach: Interatomic potentials and line broadening
Table 2. Interaction potential energies, V(R),
for the A
1
Σ states of NaH in a.u.
R(a
0
) V(R)
1
V(R)
2
3.18532 -0.0139258 -0.0140319
3.21537 -0.0154006 -0.0155388
3.24769 -0.0169040 -0.0170848
3.28265 -0.0184318 -0.0186636
3.32025 -0.0199810 -0.0202466
3.36126 -0.0215497 -0.0218462
3.40567 -0.0231363 -0.0234554
3.45367 -0.0247389 -0.0250770
3.50582 -0.0263556 -0.0266930
3.56289 -0.0279841 -0.0282868
3.62563 -0.0296213 -0.0298757
3.69536 -0.0312632 -0.0314701
3.77322 -0.0329057 -0.0330282
3.86185 -0.0345434 -0.0345929
3.96408 -0.0361712 -0.0361413
4.08427 -0.0377834 -0.0376901
4.22751 -0.0393748 -0.0392079
4.40363 -0.0409405 -0.0407241
4.62681 -0.0424770 -0.0422317
4.92652 -0.0439824 -0.0437476
5.39253 -0.0454570 -0.0453076
6.03466 -0.0461876 -0.0461498
6.64126 -0.0454570 -0.0454093
7.07136 -0.0439824 -0.0438514
7.37201 -0.0424770 -0.0422590
7.62429 -0.0409405 -0.0406401
7.85125 -0.0393748 -0.0389998
8.06309 -0.0377834 -0.0373436
8.26529 -0.0361712 -0.0356773
8.46163 -0.0345434 -0.0340024
8.65457 -0.0329057 -0.0323211
8.84544 -0.0312632 -0.0306394
9.03554 -0.0296213 -0.0289597
9.22565 -0.0279841 -0.0272864
9.41613 -0.0263556 -0.0256256
9.60737 -0.0247389 -0.0239820
9.79956 -0.0231363 -0.0223606
9.99307 -0.0215497 -0.0207645
10.18790 -0.0199810 -0.0191989
10.38481 -0.0184318 -0.0176630
10.58493 -0.0169040 -0.0161537
10.78959 -0.0154006 -0.0146677
11.00124 -0.0139258 -0.0131960
11.75 -0.0090740 -0.0085823
12.0 -0.0077220 -0.0072714
13.5 -0.0023670 -0.0021438
15.0 -0.0006230 -0.0005462
20.0 -0.0000047 -0.0000129
1
Olson & Liu (1980), Zemke et al. (1984)
2
present work
Table 3. Interaction potential energies, V(R),
for a
3
Σ states of NaH in a.u.
R(a
0
) V(R)
1
V(R)
2
1.5 0.457088 0.538711
1.75 0.308218 0.365211
2.0 0.209415 0.243135
2.5 0.099873 0.107539
3.0 0.051073 0.051465
3.5 0.028792 0.028141
4.0 0.017934 0.017616
4.5 0.011974 0.011953
5.0 0.008224 0.008311
6.0 0.003792 0.003822
8.0 0.000593 0.000507
10.0 0.000041 -0.000008
12.0 -0.000015 -0.000032
15.0 -0.000008 -0.000010
20.0 -0.000002 -0.000002
30.0 -0.000000 -0.000000
1
Olson & Liu (1980)
2
present work
is slightly shifted from the accurate value of
R ' 3.57a
0
, and the well depth is reduced
by about 10.6%. It is clear that the three-body
model should be more accurate for excited
electronic states, but even so for the case of the
A
1
Σ state the agreement is particularly good. In
table 3 the present results for the a
3
Σ potential
are compared with the results of Olson & Liu
(1980). The agreement is very good for R >
2.5a
0
. In all cases, the present potentials should
be the most accurate for large values of R, be-
cause atomic basis states are used which give
essentially exact results for the energies at infi-
nite separation.
In tables 4, 5 and 6 results obtained us-
ing impact theory are shown for the half-half
widths of lines of lithium and sodium broad-
ened by helium. In each case the full quantum-
mechanical result is compared with the re-
sults from the Born and Van der Waals ap-
proximations. It is clear that typically the Born
approximation predicts widths that are about
20% too small. For the lowest temperatures the
Van der Waals approximation is close to the
Born results indicating that the asymptotic part
of the potential is the dominant influence and
demonstrates the equivalence of the Born and
