PHYSICAL REVIEW A 76, 032706 共2007兲
Long-range dispersion interactions. II. Alkali-metal and rare-gas atoms
J. Mitroy and J.-Y. Zhang
Faculty of Technology, Charles Darwin University, Darwin NT 0909, Australia
共Received 4 June 2007; revised manuscript received 17 July 2007; published 11 September 2007兲
The dispersion coefficients for the van der Waals interactions between the rare gases Ne, Ar, Kr, and Xe and
the low-lying states of Li, Na, K, and Rb are estimated using a combination of ab initio and semiempirical
methods. The rare-gas oscillator strength distributions for the quadrupole and octupole transitions were derived
by using high-quality calculations of rare-gas polarizabilities and dispersion coefficients to tune Hartree-Fock
single-particle energies and expectation values.
DOI: 10.1103/PhysRevA.76.032706
PACS number共s兲: 34.20.Cf, 31.25.Jf, 31.15.Pf, 32.10.Dk
II. GENERATION OF THE TRANSITION
MOMENT ARRAYS
I. INTRODUCTION
Recently a formalism 关1兴 was presented that allows the
van der Waals interaction between two atoms 关2,3兴, e.g.,
V共R兲 = −
C6 C8 C10
−
−
−¯
R6 R8 R10
共1兲
to be computed in a relatively straightforward manner. The
Cn parameters in Eq. 共1兲 are the dispersion coefficients while
R is the distance between the two nuclei.
In this previous work, the dispersion coefficients for
ground and excited states of alkali-metal atoms interacting
with helium and atomic hydrogen were presented 关1兴. The
present work extends the calculations to encompass the
heavier rare gases, namely, neon, argon, krypton, and xenon,
to give improved descriptions of the interaction potential at
large distances.
One of the main applications of this work lies in the area
of pressure broadening 关4–6兴. The theory of spectral line
broadening relies on an accurate description of the potential
curves between the two states undergoing the transition and
the perturbing atom. The size of the dispersion parameters
has a major influence upon the broadening since many of the
collisions occur at large impact parameters. While there have
been a number of investigations of the dispersion interaction
between ground state alkali-metal atoms and the rare gases
关2,7–12兴, this is not true for the alkali-metal-atom
excited states which have received only the most cursory
attention 关12兴.
This is surprising since the alkali-metal–rare-gas combination represents an almost ideal laboratory in which to investigate the general theory of pressure broadening of radiation by a perturbing atom or molecule. There have been a
number of laboratory investigations of broadening for
lithium 关13–15兴, sodium 关16–21兴, potassium 关22兴, and rubidium 关23,24兴. In addition, there have been a number of
theoretical investigations of alkali metal spectral line broadening caused by the rare gases 关15,25–30兴. Knowledge of the
van der Waals interaction is also important for the determination of the refractive index of alkali-metal-atom matter
waves traveling through rare gases 关31–33兴. The present paper gives the dispersion coefficients between the two lowest
alkali-metal s states and the rare gases. In addition, dispersion coefficients between lowest lying p and d states and the
rare gases are also given. All dispersion coefficients are
reported in atomic units 共a.u.兲.
1050-2947/2007/76共3兲/032706共8兲
A. Theoretical overview
The approach used to generate the dispersion coefficients
is based on the work of Dalgarno who originally derived
expressions in terms of oscillator strength sum rules 关2,3兴.
This reduced the calculation of the Cn parameters for two
spherically symmetric atoms to sums over the products of the
absorption oscillator strengths 共originating in the ground
state兲 divided by an energy denominator. The sums should
include contributions from all discrete and continuum excitations. In practice a pseudostate representation is used
which gives a discrete representation of the continuum
关1,34兴. The sum over oscillator strengths needs to be rewritten in terms of a sum over the reduced matrix elements of the
electric multipole operator in cases where one 共or both兲 of
the atoms is in a state with L ⬎ 0 关1兴.
The major part of any calculation involves the generation
of lists of reduced transition matrix elements for the two
atomic states. This involves quite lengthy calculations to
generate the excitation spectrum of the pseudostate representation. It is then a relatively straightforward calculation to
use the formalism outlined in Zhang and Mitroy 关1兴 to process the lists of matrix elements and generate the dispersion
coefficients.
B. The alkali-metal atoms
The transition arrays for the alkali-metal atoms are those
which were used in calculations of the dispersion interactions between these atoms and the ground states of hydrogen
and helium 关1兴. These are computed by diagonalizing the
fixed core Hamiltonian in a large basis of Laguerre-type orbitals. The core Hamiltonian is based upon an HF description
of the core with a semiempirical core polarization potential
tuned to reproduce the energies of low-lying spectrum.
Core excitations are included in the dispersion parameter
calculation. Oscillator strength distributions were constructed by using independent estimates of the core polarizabilities to constrain the sum rules 关34–37兴. The methodology of using constrained sum rules is also used for the rare
gases and is discussed more fully in the next section.
C. The rare gases
The rare-gas transition arrays that contribute to the dispersion parameter calculation are obtained from constrained
032706-1
©2007 The American Physical Society
PHYSICAL REVIEW A 76, 032706 共2007兲
J. MITROY AND J.-Y. ZHANG
TABLE I. Multipole oscillator strength 共f 共l兲兲 distributions and
corresponding excitation energies for the rare gases. The numbers in
the square brackets denote powers of 10.
l=2
⌬E
␣共l兲 = 兺
l=3
f 共2兲
⌬E
f 共3兲
i
0.133878
3.86864
12.2909
2.45818
33.6045
2.76243
1.68244
0.772737
1.75422关−2兴
11.2038
58.9965
11.7993
119.162
12.8742
10.1235
1.82935
1.14302
0.475217
1.53543关−3兴
4.31731关−1兴
1.14016
56.1225
314.699
62.9399
9.64829关−3兴
1.65122关−1兴
3.84045关−1兴
1.32692
4.12244
7.42956
12.1612
44.5482
8.90964
520.610
70.3471
63.4538
11.2936
8.77559
4.26933
1.59703
0.968281
0.401311
8.88348关−5兴
1.76317关−2兴
3.68588关−2兴
1.03610
3.53790
8.61329
90.0166
528.064
105.613
4.23665关−3兴
6.86382关−2兴
1.54981关−1兴
4.63478关−1兴
1.32979
1.85270
2.50232
8.22655
17.5948
17.7446
62.4884
12.4977
Xe
1224.61
189.582
178.024
40.4172
35.4631
26.3603
8.09781
6.24985
3.01936
1.18636
0.699226
0.372196
1.70417关−5兴
3.00160关−3兴
5.88693关−3兴
1.21755关−1兴
3.49418关−1兴
4.64015关−1兴
3.42416
12.7347
38.5874
1.81736关2兴
9.51970关2兴
1.90394关2兴
Ar
119.209
12.9212
10.1705
1.87635
1.19002
0.497717
3.98387关−2兴
8.04905关−1兴
2.09210
9.39598
33.1090
6.62181
Kr
520.604
70.3421
63.44877
11.28845
8.77049
4.26423
1.59193
0.963181
0.445401
1224.66
189.625
178.067
40.4602
35.5061
26.4033
8.14081
6.29285
3.06236
1.22936
0.742226
0.380326
共l兲
f 0i
2
⑀0i
共2兲
and
Ne
33.9925
3.15043
2.07044
0.759537
the polarizabilities and the dispersion coefficients between
two identical rare-gas atoms 关34,37兴. We use the sum rule for
the multipole polarizability ␣共l兲
sum rules. The data for the dipole transitions come from a set
of previously published pseudo-excitation-energies and dipole oscillator strengths 关38兴. The pairs of 共f 0i , E0i兲 were constrained to give f-value sum rules in agreement with empirical estimates of these sum rules 关39兴. For all practical
purposes, these sets of empirical estimates 共f 0i , E0i兲 are expected to give estimates of dispersion coefficients that are
accurate to about 1% 关38,39兴.
The data for the quadrupole and octupole transitions were
derived by tuning an f-value distribution obtained from
Hartree-Fock calculations to high-accuracy calculations of
共l兲
lN具r2l−2典 = 兺 f 共l兲
i = S 共0兲
共3兲
i
关40兴 to help estimate an f 共l兲-value distribution function of
reasonable accuracy. Equation 共3兲 reduces to the well known
Thomas-Reiche-Kuhn sum rule
共1兲
N = 兺 f 共1兲
i = S 共0兲
共4兲
i
for l = 1. In these equations, N is the total number of electrons, and 具r2l−2典 is a radial expectation value of the groundstate wave function.
The other sum rules used to constrain the 共f 共l兲
i , ⑀i兲 distributions are those for the dispersion coefficients C8 and C10.
These are
C8 =
共1兲 共2兲
共1兲
f 0i
f 0j
f 共2兲
15
15
0i f 0j
+ 兺
兺
4 ij ⑀0i⑀0j共⑀0j + ⑀0i兲 4 ij ⑀0i⑀0j共⑀0j + ⑀0i兲
共5兲
and
C10 = 7 兺
ij
+
共1兲 共3兲
共3兲 共1兲
f 0i
f 0j
f 0i
f 0j
+ 7兺
⑀0i⑀0j共⑀0j + ⑀0i兲
ij ⑀0i⑀0j共⑀0j + ⑀0i兲
共2兲 共2兲
f 0i
f 0j
35
.
兺
2 ij ⑀0i⑀0j共⑀0j + ⑀0i兲
共6兲
First, we assume that the contribution from each closed subshell is equal to the number of electrons in the subshell 共Ni兲
multiplied by the mean value of r2l−2 for the subshell. The
具r2l−2典 expectation value for each subshell is computed by
using the Hartree-Fock 共HF兲 wave function. These expectation values are expected to be accurate at the level of 1–2 %
共ignoring relativistic effects兲 for the systems under consideration.
Next, the excitation energy for each subshell is set to the
Koopmans energy ⑀i 共i.e., the single-particle energy coming
from a HF calculation兲 plus an energy shift ⌬共l兲
1 . The motivation for this is the expectation that the HF single-particle
energies will give a reasonable initial approximation to the
oscillator strength distribution originating from each shell
关34,37兴. The overall purpose of the energy shift is to refine
the distribution so that it is consistent with the results of
some sum rules for quantities such as the quadrupole polarizability for which reasonably accurate values are known.
For the valence np shell there are two energy shifts. Five
of the six electrons are given energy shifts of ⌬共l兲
1 , while one
electron is given a different energy shift of ⌬共l兲
2 . The multipole polarizability computed with this oscillator strength distribution is
032706-2
LONG-RANGE … . II. ALKALI-METAL AND …
PHYSICAL REVIEW A 76, 032706 共2007兲
TABLE II. The polarizabilities 共in a.u.兲 and homonuclear dispersion coefficients 共in a.u.兲 computed from
the f 共ᐉ兲-value distributions of Table I and Ref. 关38兴 are in the row labeled present. The reference values used
to tune the f 共2兲 and f 共3兲 oscillator strength distributions are discussed in the text. Values from nonrelativistic
MBPT calculations 关41兴 and relativistic MP2 calculations 共TDMP2兲 关42兴 are also given. The notation a关b兴
represents a ⫻ 10b.
System
Method
␣共1兲
␣共2兲
␣共3兲
C6
C8
C10
Ne
Present
MBPT
TDMP2
Present
MBPT
TDMP2
Present
MBPT
TDMP2
Present
MBPT
TDMP2
2.669
2.656
7.518
7.328
42.07
42.07
11.08
11.06
11.15
16.79
17.21
16.85
27.16
28.22
27.17
52.80
51.86
52.78
98.20
99.30
96.75
213.7
223.3
202.8
536.4
536.4
553.1
1.255关3兴
1.273关3兴
1.241关3兴
3455
3640.6
6.383
6.553
6.543
64.30
64.30
64.80
129.6
135.1
130.1
285.9
302.3
288.4
90.27
90.34
91.79
1.621关3兴
1.623关3兴
1.644关3兴
4.040关3兴
4.187关3兴
3.981关3兴
1.200关4兴
1.281关4兴
1.139关4兴
1.533关3兴
1.536关3兴
1.589关3兴
4.903关4兴
4.906关4兴
5.024关4兴
1.501关5兴
1.555关5兴
1.474关5兴
5.882关5兴
6.198关5兴
Ar
Kr
Xe
␣共l兲 =
N lr2l−2
5lr2l−2
lr2l−2
i i
np
np
+
.
兺
共l兲 2 +
共⑀np + ⌬共l兲兲2 共⑀np + ⌬共l兲兲2
i⫽np 共⑀i + ⌬ 兲
1
1
共7兲
2
共2兲
共2兲
and C8
The parameters ⌬共2兲
1 and ⌬2 are adjusted until ␣
agree with the reference values of these quantities. The C8
dispersion parameter also requires knowledge of the dipole
oscillator strength distribution which is taken from the empirical distributions of Kumar and Meath 关38兴. Once the
quadrupole oscillator strength distribution has been fixed, the
process can be repeated using ␣共3兲 and C10 to fix the values
共3兲
of ⌬共3兲
1 and ⌬2 .
A tabulation of the f 共l兲 distributions derived from this process are given in Table I. Table II gives the polarizabilities
and Cn values 共for homo-nuclear dimers兲 computed with
these f 共l兲 distributions. Once the oscillator strengths have
been determined, the reduced matrix elements needed for the
evaluation of the dispersion parameters are obtained from
共l兲
=
f 0n
2兩具␺0 ;L0兩兩rlCl共r̂兲兩兩␺n ;Ln典兩2⌬En0
,
共2l + 1兲共2L0 + 1兲
共8兲
where ⌬En0 = En − E0. This expression can be inverted and the
positive square root taken without any loss of generality
since the rare-gas ground states have an angular momentum
of zero.
The usefulness of this approach does depend on using
accurate reference values of the quadrupole and octupole polarizabilities, and accurate values of C8 and C10 to fix ⌬共l兲
i .
These are now discussed.
impact of triple excitations 关CCSD共T兲兴 关43兴. This calculation
gave ␣共1兲 = 2.68a30. An independent CCSD共T兲 calculation
gave ␣共2兲 = 7.525a50 关44兴.
The reference value for the octupole polarizability
42.07a50 was taken from a many-body perturbation theory
共MBPT兲 calculation 关41兴. This MBPT calculation gave ␣共1兲
= 2.656a30. The reference values for the Ne-Ne dispersion coefficients C8 = 90.344 a.u. and C10 = 1535.6 a.u. were also
taken from an MBPT calculation 关41兴.
2. Argon
Using the oscillator strength distributions of Kumar and
Meath 关38兴 gave ␣共1兲 = 11.08a30 and C6 = 64.30 a.u. The value
of 52.8a50 was adopted for ␣共2兲. This was derived from a
coupled cluster calculation with full allowance for single and
double excitations and with an estimate of the impact of
triple excitations 关43兴. This calculation also gave ␣共1兲
= 11.12a30. This is a light atom and relativistic effects have a
small impact upon the polarizabilities 关a CCSD共T兲 calculation using a pseudopotential for the core gave a relativistic
correction of only −0.1a50 to ␣共2兲 关45兴兴.
The reference value for the octupole polarizability,
536.4a70 was taken from a MBPT calculation 关41兴. The
MBPT calculation gave ␣共1兲 = 11.06a30 and ␣共2兲 = 51.86a50. The
reference Ar-Ar dispersion coefficients C8 = 1623.2 a.u. and
C10 = 40963 a.u. were also taken from this MBPT calculation 关41兴.
3. Inclusion of relativistic effects
1. Neon
Using the oscillator strength distributions of Kumar and
Meath 关38兴 gave ␣共1兲 = 2.669a30 and C6 = 6.383 a.u. A reference value of 7.52a50 was adopted for ␣共2兲. This was taken
from a coupled cluster calculation with full allowance for
single and double excitations and with an estimate of the
Relativistic effects have a larger impact upon the structure
and excitation spectrum for krypton and xenon than they do
for neon and argon. Preferably, one would choose reference
values for krypton and xenon directly from relativistic calculations, but the problem with this is that the best nonrelativistic calculations usually have a better treatment of electron
032706-3
PHYSICAL REVIEW A 76, 032706 共2007兲
J. MITROY AND J.-Y. ZHANG
TABLE III. The dispersion coefficients 共in a.u.兲 between two
ground-state rare-gas atoms. Values from nonrelativistic MBPT calculations 关41兴 and relativistic MP2 calculations 共TDMP2兲 关42兴 are
also listed. The numbers in the square brackets denote powers of
ten.
System
Method
C6
C8
C10
Ne-Ne
Present
MBPT
TDMP2
Present
MBPT
TDMP2
Present
MBPT
TDMP2
Present
MBPT
TDMP2
Present
MBPT
TDMP2
Present
MBPT
TDMP2
Present
MBPT
TDMP2
Present
MBPT
TDMP2
Present
MBPT
TDMP2
Present
MBPT
TDMP2
6.383
6.5527
6.543
19.50
19.753
19.85
27.30
28.009
27.74
39.66
40.518
40.54
64.30
64.543
64.80
91.13
93.16
91.16
134.5
137.97
135.7
129.6
135.08
130.1
191.9
201.27
193.2
285.9
302.29
288.4
90.27
90.344
91.79
3.889关2兴
3.9012关2兴
3.965关2兴
627.38
6.3814关2兴
6.270关2兴
1.128关3兴
1.1623关3兴
1.094关3兴
1.621关3兴
1.6232关3兴
1.644关3兴
2571.1
2.6167关3兴
2.566关3兴
4.527关3兴
4.6694关3兴
4.403关3兴
4.040关3兴
4.1873关3兴
3.981关3兴
7.026关3兴
7.3891关3兴
6.774关3兴
1.200关4兴
1.2807关4兴
1.139关4兴
1.533关3兴
1.5356关3兴
1.589关3兴
9.335关3兴
9.3352关3兴
9.618关3兴
1.740关4兴
1.7658关4兴
1.768关4兴
3.857关4兴
3.8978关4兴
Ne-Ar
Ne-Kr
Ne-Xe
Ar-Ar
Ar-Kr
Ar-Xe
Kr-Kr
Kr-Xe
Xe-Xe
TABLE IV. The dispersion coefficients 共in a.u.兲 between the
ground states of alkali-metal atoms and rare-gas atoms. The data
from the Standard and Certain 共SC兲 tabulation 关8兴 are also listed;
the two numbers represent the estimates of lower and upper bounds
for the coefficients. The numbers in the square brackets denote
powers of ten.
System
Method
C6
C8
C10
Li-Ne
Present
SC
Present
SC
Present
SC
Present
SC
Present
SC
Present
SC
Present
SC
Present
SC
Present
SC
Present
SC
Present
SC
Present
SC
Present
SC
Present
SC
Present
SC
Present
SC
43.79
42.9–44.8
174.0
171–177
259.6
255–262
410.7
402–414
50.41
46–137
196.8
184–508
292.7
273–737
460.9
430–1130
77.44
73.8–83.9
299.3
292–318
444.1
432–469
697.9
680–737
88.00
78.0–94.2
336.4
310–352
498.0
460–518
780.1
725–813
2.229关3兴
2.00关3兴–2.20关3兴
9.493关3兴
8.63关3兴–9.49关3兴
1.476关4兴
1.34关4兴–1.48关4兴
2.521关4兴
2.30关4兴–2.54关4兴
2.721关3兴
2.40关3兴–3.19关3兴
1.153关3兴
1.03关4兴–1.53关4兴
1.787关4兴
1.59关4兴–2.48关4兴
3.033关4兴
2.71关4兴–4.52关4兴
5.309关3兴
4.57关3兴–5.04关3兴
2.228关4兴
1.95关4兴–2.16关4兴
3.431关4兴
3.01关4兴–3.33关4兴
5.751关4兴
5.07关4兴–5.64关4兴
6.355关3兴
5.33关3兴–5.99关3兴
2.656关4兴
2.27关4兴–2.56关4兴
4.083关4兴
3.50关4兴–3.93关4兴
6.819关4兴
5.87关4兴–6.63关4兴
1.531关5兴
1.21关5兴–1.47关5兴
6.781关5兴
5.53关5兴–6.69关5兴
1.087关6兴
8.92关5兴–1.08关6兴
1.957关6兴
1.63关6兴–1.97关6兴
1.993关5兴
1.57关5兴–1.94关5兴
8.769关5兴
7.11关5兴–9.30关5兴
1.398关6兴
1.14关6兴–1.53关6兴
2.496关6兴
2.05关6兴–2.90关6兴
4.936关5兴
3.71关5兴–4.38关5兴
2.132关6兴
1.65关6兴–1.94关6兴
3.352关6兴
2.60关6兴–3.06关6兴
5.836关6兴
4.60关6兴–5.42关6兴
6.231关5兴
4.94关5兴–5.63关5兴
2.681关6兴
2.16关6兴–2.48关6兴
4.203关6兴
3.40关6兴–3.89关6兴
7.281关6兴
5.92关6兴–6.84关6兴
Li-Ar
Li-Kr
Li-Xe
Na-Ne
Na-Ar
4.903关4兴
4.9063关4兴
5.024关4兴
8.677关4兴
8.8260关4兴
8.685关4兴
1.807关5兴
1.8425关5兴
Na-Kr
Na-Xe
K-Ne
K-Ar
1.501关5兴
1.5545关5兴
1.474关5兴
3.035关5兴
3.1603关5兴
K-Kr
5.882关5兴
6.1984关5兴
Rb-Ar
K-Xe
Rb-Ne
Rb-Kr
Rb-Xe
correlations than the best relativistic calculations. The strategy adopted is to use estimates of the relativistic corrections
to the ␣共l兲 and Cn coefficients to adjust the best nonrelativistic values.
4. Krypton
The oscillator strength distributions of Kumar and Meath
关38兴 gave ␣共1兲 = 16.79a30 and C6 = 129.6 a.u. The value of
98.2a50 was adopted for ␣共2兲. This was derived from a
CCSD共T兲 calculation which gave 99.86 关43兴. An independent
CCSD共T兲 calculation using a pseudopotential for the core
estimated the decrease in ␣共2兲 due to relativistic effects to be
1.65a50 关45兴. The CCSD共T兲 calculation gave ␣共1兲 = 17.07a30 if
a correction of 0.01a30 is made for relativistic effects 关45兴.
The reference value adopted for the octupole polarizability was ␣共3兲 = 1254.8a70. This was determined by adding a
relativistic correction of −17.8a70 共taken from a second order
Moller-Plesset calculation 关42兴兲 to a value of 1272.6a30 taken
from a large basis MBPT calculation 关41兴 共this MBPT calculation gave ␣共1兲 = 17.21a30兲.
The reference values for the Kr-Kr dispersion coefficients
were C8 = 4040 a.u. and C10 = 1.501⫻ 105 a.u. These values
were derived from the relativistic second order MollerPlesset perturbation theory calculations 共MP2兲 of Hattig and
Hess 关42兴 共note the data that is quoted was identified as
TDMP2 in this paper兲. This calculation gave estimates of
C6 = 130.1 a.u. and ␣共1兲 = 16.85a30 which are in very good
agreement with those of Kumar. However, the MP2 values of
␣共2兲 = 96.8a50 and ␣共3兲 = 1241a70 are, respectively, 1.5 and
032706-4
LONG-RANGE … . II. ALKALI-METAL AND …
PHYSICAL REVIEW A 76, 032706 共2007兲
TABLE V. The dispersion coefficients 共in atomic units兲 for the ⌺ and ⌸ symmetries, between the resonant
np excited state of alkali-metal atoms and the ground-state atoms of rare gases. The numbers in the square
brackets denote powers of ten.
⌺
System
Li-Ne
Li-Ar
Li-Kr
Li-Xe
Na-Ne
Na-Ar
Na-Kr
Na-Xe
K-Ne
K-Ar
K-Kr
K-Xe
Rb-Ne
Rb-Ar
Rb-Kr
Rb-Xe
⌸
C6
C8
C10
C6
C8
C10
9.822关1兴
4.011关2兴
6.049关2兴
9.728关2兴
1.492关2兴
6.088关2兴
9.187关2兴
1.479关3兴
2.006关2兴
8.127关2兴
1.223关3兴
1.964关3兴
2.290关2兴
9.241关2兴
1.390关3兴
2.229关3兴
1.549关4兴
6.379关4兴
9.690关4兴
1.584关5兴
3.112关4兴
1.283关5兴
1.947关5兴
3.171关5兴
5.065关4兴
2.089关5兴
3.169关5兴
5.151关5兴
6.289关4兴
2.595关5兴
3.934关5兴
6.391关5兴
1.849关6兴
7.846关6兴
1.216关7兴
2.062关7兴
4.739关6兴
1.998关7兴
3.078关7兴
5.160关7兴
9.204关6兴
3.866关7兴
5.937关7兴
9.887关7兴
1.239关7兴
5.197关7兴
7.969关7兴
1.324关8兴
5.492关1兴
2.201关2兴
3.296关2兴
5.244关2兴
8.511关1兴
3.413关2兴
5.120关2兴
8.167关2兴
1.187关2兴
4.717关2兴
7.060关2兴
1.123关3兴
1.379关2兴
5.451关2兴
8.150关2兴
1.295关3兴
7.522关2兴
3.969关3兴
6.818关3兴
1.351关4兴
1.395关3兴
7.032关3兴
1.186关4兴
2.294关4兴
2.328关3兴
1.122关4兴
1.858关4兴
3.500关4兴
2.971关3兴
1.403关4兴
2.301关4兴
4.2797关4兴
3.586关4兴
1.949关5兴
3.529关5兴
7.578关5兴
8.680关4兴
4.354关5兴
7.562关5兴
1.539关6兴
1.670关5兴
8.022关5兴
1.358关6兴
2.664关6兴
2.256关5兴
1.066关6兴
1.785关6兴
3.448关6兴
1.1 % smaller than the reference values adopted for these
polarizabilities. Accordingly, the effective f 共l兲-value distributions of Hattig and Hess 关42兴 for l = 2 and 3 were rescaled by
1.015 and 1.011 and used in conjunction with their f 共1兲 distributions to recompute C8 and C10 resulting in the values
listed at the start of this paragraph.
5.882⫻ 105 as the reference value. This probably underestimates the size of the relativistic correction since the
quadrupole-quadrupole sum in C10 should be rescaled by
共0.949兲2 while the dipole-octupole sums are rescaled by
0.949.
5. Xenon
III. CALCULATIONS AND RESULTS
The oscillator strength distributions of Kumar and Meath
关38兴 gave ␣共1兲 = 27.16a30 and C6 = 285.9 a.u. A value of ␣共2兲
= 213.7a50 was adopted as the reference. This was determined
by adding a relativistic correction of −9.64a50 共taken from a
second order Moller-Plesset calculation 关42兴兲 to a value of
223.29a30 taken from a large basis MBPT calculation 关41兴
共this MBPT calculation gave ␣共1兲 = 28.23a30兲. The C8 reference values for the Xe-Xe interaction were determined by a
procedure similar to that used for the Kr-Kr case. The relativistic MP2 f 共2兲-value distribution of Hattig and Hess 关42兴
was rescaled by 1.054 共=213.7/ 202.8兲 and used in conjunction with their f 共1兲 distribution to give C8 = 1.200⫻ 104.
This procedure could not be adopted for ␣共3兲 and C10 because Hattig and Hess did not report any calculations for this
case. Accordingly, the relativistic correction was estimated
by assuming it was the same relative size as the correction
for the quadrupole transition. Using the data from Hattig and
Hess, one gets a correction of 0.949= 202.8/ 213.7. Scaling
the MBPT octupole polarizability of 3640.6 by 0.949 gave a
reference value of ␣共3兲 = 3455a70. 共Note that the scale factors
for the relativistic correction for ␣共2兲 and ␣共3兲 were roughly
similar in size, e.g., for krypton they were 0.985 and 0.989,
respectively.兲 The value of C10 was estimated by simply rescaling the MBPT value of 6.198⫻ 105 by 0.949 giving
A. Dispersion parameters
The dispersion coefficients were computed using the formalism presented in Ref. 关1兴. This formalism reduces the
dispersion coefficients to sums over sets of reduced matrix
elements of the two atoms. The oscillator strengths for the
rare gases can be converted to reduced matrix elements without any error since the expressions for the dispersion relations involving one atom in an S state only involve squares
of the reduced matrix elements for the S-state atom.
B. The heteronuclear rare-gas combinations
The complete set of dispersion coefficients for all the possible rare-gas combinations are listed in Table III. The ultimate accuracy of the tabulated coefficients is largely limited
by the accuracy of the many-body calculations used to derive
the reference values. Most of the dispersion coefficients were
based on the MBPT dispersion coefficients of Thakkar et al.
关41兴. The present values of the dispersion coefficients for
combinations involving Kr or Xe should be regarded as more
accurate than the MBPT values since the impact of relativistic effects have been incorporated as corrections.
The other set of dispersion coefficients listed in Table IV
are the TDMP2 values of Hattig and Hess 关42兴. The TDMP2
032706-5
PHYSICAL REVIEW A 76, 032706 共2007兲
J. MITROY AND J.-Y. ZHANG
TABLE VI. The dispersion coefficients 共in a.u.兲 between the
共n + 1兲s excited state of an alkali-metal atom and a rare-gas atom in
its ground state. The data from the Proctor and Stwalley tabulation
关7兴 are also listed. The numbers in the square brackets denote powers of ten.
System
Li-Ne
Li-Ar
Li-Kr
Li-Xe
Na-Ne
Na-Ar
Na-Kr
Na-Xe
K-Ne
K-Ar
K-Kr
K-Xe
Rb-Ne
Rb-Ar
Rb-Kr
Rb-Xe
Method
Present
PS
Present
PS
Present
PS
Present
PS
Present
PS
Present
PS
Present
PS
Present
PS
Present
PS
Present
PS
Present
PS
Present
PS
Present
PS
Present
PS
Present
PS
Present
PS
C6
C8
C10
310.4
308.4
1.280 关3兴
1.273 关3兴
1.934 关3兴
1.921 关3兴
3.118 关3兴
3.091 关3兴
337.0
331.1
1.386 关3兴
1.368 关3兴
2.095 关3兴
2.064 关3兴
3.375 关3兴
3.322 关3兴
435.5
417.6
1.786 关3兴
1.727 关3兴
2.697 关3兴
2.607 关3兴
4.341 关3兴
4.199 关3兴
469.6
444.1
1.921 关3兴
1.838 关3兴
2.899 关3兴
2.773 关3兴
4.665 关3兴
4.468 关3兴
8.185关4兴
8.106关4兴
3.410关5兴
3.377关5兴
5.189关5兴
5.090关5兴
8.480关5兴
8.149关5兴
9.427关4兴
9.286关4兴
3.926关5兴
3.868关5兴
5.973关5兴
5.831关5兴
9.756关5兴
9.431关5兴
1.488关5兴
1.443关5兴
6.192关5兴
6.031关5兴
9.413关5兴
9.096关5兴
1.534关6兴
1.459关6兴
1.685关5兴
1.630关5兴
7.009关5兴
6.787关5兴
1.065关6兴
1.024关6兴
1.736关6兴
1.643关6兴
2.578关7兴
2.545关7兴
1.077关8兴
1.062关8兴
1.645关8兴
1.597关8兴
2.710关8兴
2.546关8兴
3.153关7兴
3.095关7兴
1.317关8兴
1.291关8兴
2.010关8兴
1.943关8兴
3.306关8兴
3.099关8兴
6.064关7兴
5.877关7兴
2.529关8兴
2.450关8兴
3.855关8兴
3.690关8兴
6.319关8兴
5.898关8兴
7.237关7兴
6.973关7兴
3.017关8兴
2.907关8兴
4.597关8兴
4.378关8兴
7.530关8兴
7.001关8兴
Cn coefficients are consistently smaller for interactions involving either krypton or xenon. This is expected since relativistic effects generally lead to contractions of the orbitals
which result in smaller polarizabilities and dispersion interactions. The present values of the dispersion coefficients
generally lie between the TDMP2 and MBPT values for interactions involving krypton or xenon.
C. The alkali-metal–rare-gas coefficients
The dispersion constants between the rare gases and the
alkali-metal-atom ground states are given in Table III and
compared with the compilation of Standard and Certain 共SC兲
关8兴. The values of SC were determined by constructing
f-value distributions using a combination of experimental
and theoretical information about polarizabilities, sum rules,
and transition energies to give bounded estimates of the polarizabilities and thus the dispersion coefficients using Pade
approximates 关8,10兴. It can be seen from Table III that many
of the bounds given in the SC tabulation are quite loose, and
in the case of interactions involving sodium, the variation
between the upper and lower bounds is more than a factor of
2. Some older compilations of dispersion coefficients for the
alkali-metal–rare-gas combinations 关2,7,10–12兴 are not listed
for reasons of brevity.
The present values should be regarded as superior to the
SC compilation for three reasons. The pseudo-oscillatorstrength distributions for the higher multipoles of the rare
gases and alkali-metal cores should be better than the essentially one-term expressions for the dynamic polarizabilities
used by SC 关8,10兴. In addition, the data used to bound the
pseudo-f-value distributions is in many cases better. Finally,
the dipole f-value distributions for the rare gases 关38兴 and the
valence f-value distributions used for the alkali-metal atoms
关36兴 represent an update of the data that was available to SC
in 1985 关8兴.
The C8 and C10 dispersion coefficients were usually larger
than the upper bounds given by SC. The only alkali-metal
atom for which this does not occur is sodium, but the spread
quoted by SC for the sodium upper and lower bounds is
significantly larger than the spreads given for lithium, potassium, and cesium.
In addition to the ground state, the dispersion coefficients
between the np alkali-metal excited state and the rare gases
are given in Table V. Apart from the early work of Mahan
关12兴, these values represent the only calculations of the dispersion coefficients between the alkali-metal np states and
the rare gases from neon onward 共note that there have been
some calculations of the dispersion coefficients for helium
关1,46兴兲. Data on the excited state potentials do exist 关47–52兴
but almost nothing is known about the long-range forms of
the potentials. The present calculations rectify this omission.
The obvious trend in the table is the tendency for the dispersion coefficients to get larger as either the alkali-metal atom
or the rare-gas atom increases in size.
The dispersion coefficients for the interaction of atoms in
excited 共n + 1兲s states with the rare gases are listed in Table
VI. The results of Proctor and Stwalley 共PS兲 关7兴 are presented for comparison. The PS results were derived by using
various constraints on oscillator strength sum rules 共based on
experimental and theoretical data兲 to determine the oscillator
strength distribution as a function of excitation energy. Although PS quoted uncertainties, these are not given here
since their uncertainties in many cases seem to be overoptimistic. The present Cn are all larger than those of PS. This is
not surprising since PS did not include any contribution from
the alkali-metal-atom cores in the calculation. Therefore it
would be expected that the differences from the PS values
would get larger as the size of the alkali-metal atom increases, and this is the case. The difference between the
present and PS C6 for the Li-Ne combination was 2.0 out of
310 a.u. For the Rb-Ne case, the difference for C6 was 25.6
out of 469.6 a.u. This situation was reminiscent of the comparison with PS for the alkali-metal–He Cn coefficients of
032706-6
LONG-RANGE … . II. ALKALI-METAL AND …
PHYSICAL REVIEW A 76, 032706 共2007兲
TABLE VII. The dispersion coefficients 共in atomic units兲 for the ⌺, ⌸, and ⌬ symmetries, between the lowest nd excited state of the
alkali-metal atoms with the ground state of rare-gas atoms. The numbers in the square brackets denote powers of ten.
⌺
System
Li-Ne
Li-Ar
Li-Kr
Li-Xe
Na-Ne
Na-Ar
Na-Kr
Na-Xe
K-Ne
K-Ar
K-Kr
K-Xe
Rb-Ne
Rb-Ar
Rb-Kr
Rb-Xe
⌸
⌬
C6
C8
C10
C6
C8
C10
C6
C8
C10
428.3
1.770关3兴
2.678关3兴
4.323关3兴
422.7
1.743关3兴
2.636关3兴
4.253关3兴
310.3
1.267关3兴
1.911关3兴
3.073关3兴
258.2
1.044关3兴
1.571关3兴
2.518关3兴
3.132关5兴
1.299关6兴
1.969关6兴
3.193关6兴
3.035关5兴
1.259关6兴
1.908关6兴
3.094关6兴
1.763关5兴
7.310关5兴
1.109关6兴
1.799关6兴
1.252关5兴
5.190关5兴
7.873关5兴
1.278关6兴
1.800关8兴
7.474关8兴
1.135关9兴
1.848关9兴
1.725关8兴
7.164关8兴
1.088关9兴
1.772关9兴
8.399关7兴
3.489关8兴
5.303关8兴
8.648关8兴
5.305关7兴
2.204关8兴
3.352关8兴
5.473关8兴
380.4
1.571关3兴
2.376关3兴
3.834关3兴
375.7
1.548关3兴
2.340关3兴
3.773关3兴
276.9
1.128关3兴
1.701关3兴
2.733关3兴
231.4
932.9
1.402关3兴
2.245关3兴
1.138关5兴
4.723关5兴
7.173关5兴
1.168关6兴
1.104关5兴
4.578关5兴
6.954关5兴
1.133关6兴
6.427关4兴
2.667关5兴
4.054关5兴
6.617关5兴
4.583关4兴
1.901关5兴
2.892关5兴
4.728关5兴
1.087关7兴
4.755关7兴
7.481关7兴
1.303关8兴
1.043关7兴
4.566关7兴
7.188关7兴
1.252关8兴
5.172关6兴
2.289关7兴
3.630关7兴
6.408关7兴
3.321关6兴
1.483关7兴
2.364关7兴
4.212关7兴
236.5
972.9
1.469关3兴
2.365关3兴
234.7
961.5
1.451关3兴
2.334关3兴
176.7
712.0
1.070关3兴
1.711关3兴
151.0
599.1
896.7
1.427关3兴
−1.609关4兴
−6.199关4兴
−8.975关4兴
−1.325关5兴
−1.552关4兴
−5.969关4兴
−8.634关4兴
−1.272关5兴
−8.498关3兴
−3.190关4兴
−4.534关4兴
−6.432关4兴
−5.600关3兴
−2.054关4兴
−2.867关4兴
−3.899关4兴
−2.874关5兴
−1.395关6兴
−2.209关6兴
−3.815关6兴
−2.770关5兴
−1.341关6兴
−2.121关6兴
−3.652关6兴
−1.423关5兴
−6.713关5兴
−1.034关6兴
−1.694关6兴
−8.989关4兴
−4.107关5兴
−6.128关5兴
−9.364关5兴
helium 关1兴. Part of the difference between the present and PS
Cn can be attributed to the core. For example, omission of
the core for the Rb共5s兲-Ne interaction results in C6 = 449.7
and C8 = 1.681⫻ 103 a.u., both of these values are still larger
than the PS values.
The PS dispersion coefficients for the Li–rare-gas systems
were recently utilized in a combined experimental and theoretical analysis of the noble gas pressure broadening of the
7
Li 2s – 3s transition by Rosenberry et al. 关15兴. Rosenberry et
al. were not able to get detailed quantitative agreement for
either the pressure broadening or pressure shift. The present
values of C6 for the Li–rare-gas case agree with those of PS
to better than 1%, and to within 5% for C8, so any inaccuracies in the PS Cn coefficients are unlikely to be a major
contributor to any error.
Table VII reports the dispersion coefficients between the
lowest nd state of the alkali-metal atoms with the rare-gas
ground states. The C6 coefficients for a given alkali-metal
atom show a tendency to get larger as the rare-gas atom gets
larger. This behavior is expected. The Cn coefficients exhibit
one pattern that is unexpected. The dispersion coefficients
for a given rare gas tend to be largest for Li and smallest for
rubidium; for example, the C6 dispersion coefficients for the
⌺, ⌸, and ⌬ states of Li-Ar decrease steadily as the Li is
substituted by Na, K, and Rb. This behavior is caused by the
nature of the downward transition from the nd state to the
resonant np excited state. The oscillator strength for this
transition is negative. As the alkali-metal atoms get larger,
the negative contribution from this transition to C6 gets
larger leading to a reduction in the value of C6. It is worth
noting that the static dipole polarizability of the lowest nd
state decreases as the alkali-metal atom gets larger 关36兴.
One of the larger sources of error in the present calculations will be the oscillator strength distributions for the
alkali-metal-atom cores since these were defined with a
single ⌬共l兲 shift. The coefficient with the largest core contribution is C6 for Rb-Ne, where 23% of C6 arises from the
core. Calculations of C6 for the positronium–rare-gas and
hydrogen–rare-gas interactions suggest the error involved in
using a single ⌬共1兲 does not exceed 5% 关37兴. So the influence
of the errors in the core f 共l兲-value distributions could result in
C6 being too large by about 1–2 %. The influence of the core
decreases as the excitation level of the alkali-metal atom increases 共e.g., the core has a 4% effect on the Rb共6s兲-Ne C6兲,
or the multipolarity of Cn increases 共e.g., the core has a 6%
effect on the Rb共5s兲-Ne C8兲, and decreases as the size of the
rare-gas atom increases 共e.g., the core has a 16% effect on
the Rb共5s兲-Xe C6兲.
IV. SUMMARY
The van der Waals coefficients between the rare gases and
some low-lying excited states of the alkali-metal atoms have
been determined using sum rules over oscillator strengths
and transition moments. The dipole oscillator strength distributions for the rare gases were taken from the empirical
tabulations of Kumar and Meath 关38兴. The oscillator strength
distributions for the higher multipoles were determined by
tuning distributions derived from HF wave functions to expectation values taken from sophisticated many-body calculations.
The most recent previous compilation of the alkali-metal–
rare-gas dispersion interactions is the one by SC 关8兴. The
032706-7
PHYSICAL REVIEW A 76, 032706 共2007兲
J. MITROY AND J.-Y. ZHANG
present values should be regarded as superseding the SC values. The underlying f-value distributions for the rare gases
and alkali-metal atoms give dipole polarizabilities and dispersion interactions that are generally accurate to about 1%
for the rare-gas–rare-gas and alkali-metal–alkali-metal cases
关1,34,53–55兴. The present C8 and C10 dispersion coefficients
are generally larger than the SC values or near the high end
of allowable range indicated by SC.
The scope of the present article has been restricted to the
ns, 共n + 1兲s, and np and lowest md states of the alkali-metal
atoms, mainly to keep the length of the present article to a
reasonable size. However, generating tables of dispersion coefficients for other low-lying excited states of alkali-metal
atoms would be relatively straightforward.
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