PHYSICAL REVIEW A
VOLUME 33, NUMBER
JANUARY 1986
1
Atomic potentials, polarizabilities, and nonadiabatic corrections
in high-angular-momentum
Rydberg states
Department
S. H. Patil
of Physics, Indian Institute of Technology, Bombay 400-076, India
(Received 5 August 1985)
Analytic expressions are obtained for direct and exchange penetration energies, corrections to polarization energies„and nonadiabatic corrections in high-angular-momentum
Rydberg states. It is
found that exchange energies and corrections to polarization energies are of comparable magnitude
to that of the direct term and must be included in the analysis of these states.
I. INTRODUCTION
The high-angular-momentum
states of a Rydberg atom
can be described' in terms of the properties of the core
such as its polarizabilities.
Alternatively,
one may
analyze the energy levels of nonpenetrating
orbits to
deduce the polarizabilities of the core. A good example of
this is the work of Freeman and Kleppneri who analyzed
the high-angular-momentum
energy levels of Na and obtained the dipolar and quadrupolar polarizabilities of Na+
(in atomic units),
a~
—0.998,
a2 —1.91+0.15 .
lowing.
(1) We have expressed electron densities and potentials
in terms of ionization energies and susceptibilities.
(2) Closed expressions are given for direct and exchange
penetration energies. It is found that the exchange term is
very important and must be included in the analysis of
Rydberg levels.
(3) A surprising result is that while the dominant potentials from second-order Born terms are due to polarizabilities, there are corrections to these potentials.
These
bl
ri
j2
e
(2g '
)&/&
a —1.00,
~
This value of quadrupolar
is somewhat
polarizability
6
larger than the theoretical estimations
which give a
value of about 1.63. However, in obtaining the values in
(1.1), one needs to apply nonadiabatic corrections which
are not as yet rehably known. Similar analysis has been
carried out for other alkah metals
but with much less
success. The reason for this is the absence of accurate
data for sufficiently high-angular-momentum
states of
these atoms.
We present here a discussion of the energy levels of
high-angular-momentum
states of alkali atoms and their
isoelectronic sequences. In particular, we investigate the
importance of exchange interaction and corrections to polarization potentials in the analysis of Rydberg states. We
develop simple expressions for electron densities and potentials which allow us to obtain analytic expressions for
the energies. The main results of our analysis are the fol-
%(ri, rz, . . . )
corrections give energies which are of the same order of
magnitude as the penetration energies, and should be included in the description of Rydberg states. We call these
terms distortion terms, to distinguish them from the usual
polarization terms.
(4) We also obtain expressions for the nonadiabatic effects which allow us to obtain the polarizabilities from the
observed effective polarizabilities.
A reanalysis of Freeman-Kleppner results, incorporating the above corrections gives
b /2
'r~'
e
(2g ' )&/2„
(1.2)
whose ratio is in agreement with that from theoretical calculations.
II. ELECTRON DENSITIES IN
ATOMS
The electron densities in an atom are important for
deducing various atomic properties. They may be obtained from the Thomas-Fermi model, Hartree equation,
or Hartree-Fock equation; however, this would involve a
The long-range
great deal of numerical computations.
behavior of the electron densities is of major importance
in the calculation of susceptibility, penetration energy, etc.
For the electron density p(r) in an atom or an ion, we
note the following well-known'
properties:
f p(r)d r =N,
f p(r)r d —r =6+,
p( r)
-r
b)
'e
2(2E ')
) 1/2p
(2. 1)
(2.2)
for r
~ oo,
(2.3)
with
bi
—(Z N+ 1)(2/Ei )' —2, —
(2.4)
where N is the number of electrons, Z is the charge of the
'
nucleus, X= —, (r ) is the diamagnetic susceptibility, and
Ei is the ionization energy of the atom or ion. Indeed,
more generally one has for the wave function'o
. . . for ri
33
a2 —1.64+0. 15,
))r2))
90
'
'
))1
(2.5)
Q~1986 The American Physical Society
ATOMIC POTENTIALS, POLARIZASILITIES, AND. . .
b
=q
(2'. )'"—2,
g T~a;
(2.6)
g T~e
Ti
—Gi
1
T~
—G;
for
(2.7)
—
b, —
bi
p(r)=2+
i&1,
i
(2. 12)
fgj(r)
(2. 13)
f
where the factor of 2 is due to spin. Now, for the inert
gases and their isoelectronic sequences, the outer-shell
electrons are in the 1=1 state, so that for long-range
properties, we may write
with G; being constants. In cases of interest to us (see
Table I), the value of bi is between 0 and 1, so that the Ti
operator simulates the power behavior in Eq. (2.3), the
simulation being exact when bi —
0 or 1. For a;, we take
the values
a;=2(2E;)'~,
=X/16m. .
of the type given in Eq. (2.7)
Many parameterizations
have been used by others, e.g. , the Moliere representation.
Our point is that the conditions in Eqs. (2.2), (2.3), and
(2.5) are very important for Rydberg atoms and should be
incorporated in the densities. The predictions of Eq. (2.7)
will be shown to be of comparable accuracy to that of
Hartree-Fock calculations for evaluating penetration and
other long-range effects.
In the approximation that the wave function is made up
of single-particle wave functions P~ (r), one has
where q& is the charge scen by the ith electron, and E; is
the ionization energy of the ith electron. We therefore
propose a density of the form
p(r)=
91
f (r, r') = g QJ'(r)PJ(r')
=1,2, 3
(2.9)
a, =2(2S. )'",
where E, is the average energy of the electrons in the
atom or ion, and the i =4 term in the density describes
(2. 14)
where 8&i is the angle between r and r'. This equation has
the correct long-range hmit for r=r'. The expressions in
Eqs. (2.7) and (2. 14) will be used to obtain analytic expresRydsions for the energies of high-angular-momentum
berg levels of alkali atoms.
the average behavior of the electrons. One also expects
that since the i =1,2, 3 terms roughly correspond to the
three successive asymptotic terms, their contributions to
E in Eq. (2. 1) are approximately equal. We, therefore,
take
III. PENETRATION AND EXCHANGE
(2. 10)
ENERGIES
These relations determine two of the coefficients G;. The
remaining two coefficients are determined by Eqs. (2. 1)
and (2.2), which lead to
The first-order penetration energy is given by
— p(r)
E~„—
f
(2. 1 1)
TABLE I. Summary of relevant information
for some quantum
1
1
——
—
R
fR rf
f
%(R) d r d
f
defects and nonadiabatic
t:orrec-
tions.
Cs+
G2
1.74
2.63
3.64
15.9
0.98
6.439
11.64
63
64
18.95
52.0
x
GI
3
SI
$2
&p
5,„
3 21
3.07 x 10-'
3.95 x 10-'
0.99x 10
2.04x 10-'
—1.39
1.169
1.691
2.239
32.6
3.1
6.358
10.92
16.64
249.3
3
9.25 x 10-'
6.76 x 10-'
1.18 x 10-4
2. 18 x 10-'
—2.06
1.011
1.471
1.910
80.5
5.0
6.351
11.64
17.23
9.26x10'
8.91 x 10-'
4.83 x 10-'
2.74x 10-'
4.96x 10-'
—2.38
0.923
1.36
1.84
138.8
7.9
7.485
14.73
23. 15
3.40x 10'
4
1.227 x 10-'
5.90 x 10-'
6.23 x 10-'
1.10x 10-'
—2.61
R,
(3.1)
33
S. H. PATIL
92
where V(R) is the hydrogenic wave function of the outer
electron. Using the expression for p(r) given in Eq. (2.7),
we get
V(R) =
Carrying out the R integrations,
q(n +I)!(2q/a;)
E~„=—+4mTa; ' 2 —a; dag
electron, n'=n
for the quantum
a;+ —e
+—4~T;a;
(3.3)
l
E~„= VR %R ' 'R,
of the core,
where q is the charge
V(R} being
with the potential
2 ~+
F( n'
(2l+1}!n'!
hypergeometric
n' —
—
2l+2
4q /n a
(3 4)
)
numb r, I is the orbital quantum numb r of the valence
which is the case of interest to us, we get
I
n
principa quantum
n is the
—I —1, and F is the
'+
one obtains
»
For
function.
»1,
defect 5,
-3
. .
5~„=g 4m T~a;
2—
a;.
q(2q/a;) '+
d
(3.5)
(21+1)!
The exchange energy to leading order is given by
rr' 4 r%'r'
E =—
r'.
r
(3 6)
Using Eq. (2. 14) and carrying out the angular integrations,
2
E« ——g AT;
i=1
f
'
R(r)e
r dr
one gets
1+1
f
r l+1
(2l+1)(21+3) r''
r 1 —1
I
+z
(21+1)(2I —1) r''
R(, } —,'/2,
gd,
(3.7)
» I»1,
where R(r} is the radial wave function of the outer electron. One notes that for n
the confluent hypergeometric function F( n', 2I +—
+2, 2r'q/n) about r'=r and
2, 2r'q/n) in R (r') is slowly varying. We expand F( n', 2I—
retain the first two terms. For n
I
1, one also has
»»
F'(
%e then
n', 2I +—
2, 2r'q/n)
=—
1+1
F( —n', 2l +2, 2r'q/n) .
get
(21 —1)(a;+2q/n)
(21 + 1)(21 + 1)!n'!n~ +
+
X
(2q/a;) '+
(1+2qlna;) z "
4l (1 —
2h;
81(1 —3h;)
8mT~q(n+1)!
h;=
(3.8)
F(
I+1
+ 21+3
2I —1
1
I
n', 2I+2,—
/n a;
4q—
n',
(3.10)
(I +1}(a;+2qln)
IV. DISTORTION POTENTIAL
The leading potential for nonpenetrating orbits is due to
core polarization coming from the second Born term
—"~
ED
Ep
—
rk
)R —
k
(
x
/R
—r
For estimating the corrections to the polarization
(4. 1)
poten-
d'
(3.9)
)
tial, we approximate (E0 Ez) ' by a consta—
nt and use
the closure property. We also assume that k&k' terms
are similar to k =k' terms and incorporate their effect in
an overall constant factor. On carrying out an expansion
of the potential in terms of Legendre polynomials and retaining only the L =1 and 2 terms, we get after angular
integration,
V(R) =
i
da;
(a;+2q/n) da;2
r
1
1
)
(2I —1)(a;+2q/n)~
2
g
)
r2L
2L+1 f p(r)
~ +'z 8(R —r)
+
L
z8(r
—R)
d
r,
(4.2}
ATOMIC POTENTIALS, POLARIZABILITIES, AND. . .
+x;E3L(x; )
where the constants Dl ~ill be chosen so as to give the
correct polarization potentials for R
ao. Using p(r) in
Eq. (2.4}, we carry out the r integrations to get
where x;=a;R and E3L(x) is the exponential integral
function.
The first term inside the brackets gives the
usual polarization potentials, whereas the remaining terms
are corrections which are generally taken to be small. We
find that these corrections to energies are comparable in
magnitude to that of the contribution of the penetration
~
In the limit n ~&l &&I, the expression for Io(a;
simple,
"
Io(o) =
and
Ii(a;)
-a R
+83+T,a; '
+B, QTa
4e
J
R
21. +1
L~, I'(2L+3)
82 ——8),
(4.5)
2
3
C
—I
g
P(2L
g(T a
)
L
3)—
where T; and a; are defined in Sec. II and aL are themultipolar polarizabilities. The main contribution to the distortion corrections is from the Bi term, and a smaller
contribution comes from the 83 and 83 terms.
For obtaining the contribution to energy, we take the
expectation value of V(R} between the hydrogenic states.
Apart from the dipolar and quadrupolar contributions,
one gets an additional contribution to energy from the
82, and 83 terms:
8„
Eg;„-Bi g T;o; Io(a;)+83 g T;a; Ii(o;)
q(n +1)!(2q/a;) '+
n + (21+1)!n'!
Io(at)=
)&F(
I
Ii(a;}=
I,(a, )=
J
n',
—n', 21+2,4q
Io(a;)da;,
I, (a, )da,
.
(4 8)
and I&(o;) also can be evaluated
(4.6}
—28
We are now in a position to determine the core polarizabilities from the energies of high-angular-momentuin
states of alkali metals or their isoelectronic sequences.
The procedure is illustrated by considering the Na atom in
detail.
To start with, the ionization energies of Na+, Na +,
and Na + are'
E —1.74, E2 —2. 63, and E3 —3.64,
respectively, whereas the average energy of Na+ is approximately 15.9 (it may be obtained by assuming suitable
screening for different levels). The value of X for Na+ is
0.98, as indicated by experiments'3 and
approximately
As this value is used to
most theoretical calculations. '
calculate small corrections to the energies, the results are
not sensitive to its precise value. Using Eqs. (2.8), (2. 10),
(2. 11), and (2. 12), we obtain the values of 6; given in
Table I.
Since the quantum defect varies slowly for large n, we
evaluate it for n
ao. From Eq. (3.5), we get for I =3,
'
~
(5. 1)
which is close to the value of 1.0X10 obtained by Freeman and Kleppner from Hartree-Fock wave functions.
For the exchange term we get from Eq. (3.9)
5,„=2.04 X 10
(5.2)
Though this result may appear to be surprisingly large
compared with 5~„, it may be pointed out that a similarly
large ratio for exchange and penetration energies was obtained for Cs by Sansonetti et al. within the framework
of the Hartree-Fock approach. Indeed, our values of 5~„
and 5,„ for Cs given in Table I are close to the values of
6.07X10 and 9.71)&10 given by Sansonetti et al.
for the n = 11, 1 =4 state.
The evaluation of the quantum defect due to distortion
co. One may
energy in Eq. (4.6} is quite simple for
write 5„„as
n~
~di t
~1+1 ~2~2
(5.3)
The values of si and s2 are obtained from Eqs. (4.5) and
(4.6), and are given in Table I. Using the values ai —1.00
and az —
l. 63 to be justified by the final results, we get
na;
/n a;
in a closed
V. CORE POLARIZABILITIES
—0. 99 X 10
5~„—
2L —
+83 g T;a; I3(a; }
is quite
~
-a R
(4.4)
+L
q(2q/a;) '+
e
3
n '(21 + 1)!
)
orm.
terms.
For analyzing the effect of the distortion corrections,
we use the large-R expansion" for the exponential integral function, and consider the first few terms. One
then gets
(4.3)
)—
,
(4.7)
5d;st-
—0.95 & 10
(5.4)
which is comparable to 6~„ in magnitude but is opposite
in sign. The total quantum defect due to penetration, ex-
S. H. PATIL
intermmiiate states.
dipole term is
change, and distortion is
5(1)=5~„+5,„+5g;„
=2.08g10-',
I =3
value for
The corresponding
33
.
00
(5.5)
1=4 is
much smaller, and
(5.6)
We are now in a position to reanalyze the results of
Their Eq. (13) for the difference
Freeman and Kleppner.
between the energies of two levels with the same n value
can be rewritten as
b. E,„p, (l, I') = a'i
3
(
pj
(6. 1)
n
~E
~
(5.7)
3500.9 = 3396.06a1+ 146.91a2+ 61.97,
(5.8)
+ 10. 11a2+0.30 .
Solving these equations, one gets
—1.0025,
a2
0. 24+0. 15 .
(5.9)
——
While the effective dipolar moment is close to the value
1.0015 obtained by Fro~an and Kleppner, the quadrupolar moment is significantly smaller than their value of
0.48.
For comparing the experimental effo:tive polarizabilities with the static polarizabilities, one must include the
dynamic effects of the valence electron. Such an analysis
has been carried out for 4f states of Na by Eissa and
Opik, ' who write
3 11a1
I
a2
3 21a1+V22a2
.
(5. 10)
They find y» ~y2z-1. 0 and y2i ——1.41. While there is
no simple way of extrapolating to higher values of n, the
changes with n appear to be small. Taking these values of
y, we find
a1- 1.00, a2- 1.65+0. 15 .
~
~
where a', and a2 are the effective dipolar and quadrupolar
polarizabilities of the core. Using their values of b, E,„~„
Ai and pIi for n =13, 1=3,4, 5, and our values of 5(1),
we have (in MHz)
812. 1 = 807.44a'1
j
where p are the core states with energies EP, and are the
outer electron states with energies EJ. %e expand the
denominator in powers of Eo EJ fo—
r Eo&E& (contribu—
tions from Eo Ez t—
erms are small). After some simplification, one obtains for the first two terms (the higherorder terms give negligibly small contributions)
5E' '= —[S( —2)(0 1/8
(1, 1')+a2& 2(1, 1')
&(1')
2&(1) —
I
R
PJ
5(l)=1.0X10 ', 1=4.
a1
Rr;
Born term for the
gE(2)
we find
a'1
The second-order
(5. 11)
It is particularly heartening that the ratio of 1.65 for
az/ai is close to the theoretical value of 1.61 obtained by
McEachran et al. , and 1.65 obtained by Mahan.
It is clear that the dynamical effects are of great importance in deducing the value of ai in Eq. (5.11). In the following section, we give an alternate derivation of the
dynamical effects which supports our choice of the y
0) —3S( —3)(0 1/8
~
0) j,
(6.2)
where the first term is the usual carpi(coc9)p)
dipolar contribution,
S(c)=g(E~ —Zc)" +'
(0
P
'
and
.,
l
(6.3)
A similar expression was derived by Mahan' in the context of van der Waais constants for alkali ineials and inert
gases.
For estimating S( —3), we note the well-known relations
S( —2)= —,'a,
(6.4)
,
S(0)= 2N .
(6.5)
Now since most of the one-electron discrete excitation energies correspond to EP —
Eo being close to separation energy of the electron, we take
S
S(u)= g E;"+'M;,
(6.6)
where Ei is the ionization energy of the core, E2 is the
ionization energy of the core from which one electron is
removed, etc. We also note that S(0) is proportional to
the number of electrons and, hence, M~ may be expected
to be proportional to E; . An additional argument in
support of the proportionality is that M; are essentially
proportional to expectation values of r; If we use. the
asymptotic form in Eq. (2.5) for the wave functions, these
expectation values again come out to be proportional to
E; '. We may therefore write
S(u)=I gE;",
(6.7)
l
where
I
ly leads
is the proportionality
constant.
This immediate-
to
(6.&)
values.
VI. NGNADIABATIC CORRECTIONS
~
Therefore, in this analysis, one gets
(6.9)
Nonadiabatic corrections to energies can be incorporated by including the summation over the intermediate
states of the valence electron in the summation over the
a2=a2 —3
gE;
gE;
ai .
(6. 10)
ATOMIC POTENTIALS, POLARIZABILITIES, AND. . .
33
In the summation
over
VII. CONCLUSIONS
i, the contribution decreases rapidFor
ly with E; so that we need only the first few terms.
5. 10 are
2. 63, E3 —3.64, and E4 —
Na, EI —1.74, E2 ——
adequate and give
(6, 11)
1.0 and y2) ——1.39. These
which means that y)] —
y22 —
values are close to the values we have used in getting Eq.
(5.11). They give
al —1.00, a2 ——1.63+0. 15,
consistent with the values
theoretical estimations. '
(6.12)
in Eq. (5.11) and with
the
B.
Edlen, Atomic Spectra, Vol. 27 of Handbuch der Physik
(Springer, Berlin, 1964).
~C. Deutsch, Phys. Rev. A 2, 43 (1970).
3R. R. Freeman and D. Kleppner, Phys. Rev. A 14, 1614 (1976).
4J. Lahiri and A. Mukherji, Phys. Rev. 153, 386 (1967).
5R. P. McEachran, A. D. Stauffer, and S. Greita, J. Phys. 8 12,
3119 (1979).
~G. D. Mahan, Phys. Rev. A 22, 1780 (1980).
7C. D. Harper, M. D. Levinsen, and S. E. Wheatley, J. Opt. Soc.
Am. 67, 579 (197")).
88. P. Stoicheff and E. %einberger, Can. J. Phys. 57, 2143
(1979).
9C. J. Sansonetti, K. L. Andrew, and J. Vergese, J. Opt. Soc.
Am. 71, 423 (1981).
' E. N. Lassettre, J. Chem. Phys. 43, 4475 (1965); J. Katriel and
It is clear that exchange and distortion terms are comparable in magnitude to that of the penetration term and
must be included in the analysis of Rydberg states. Inclusion of these terms brings the empirical value of the
quadrupolar moment of Na+ into good agreement with
the theoretical estimations.
In the absence of accurate data for high-angularmomentum states, it has not been possible ' to obtain
conclusive results for other alkali metals. We have summerized the relevant results for the iona of the alkali metals in Table I, which may be useful for future analyses.
E. R. Davidson, Proc. Natl. Acad. Sci. U. S.A. 77, 4403
(1980).
'iM. Abramowitz and I. A. Stegun, Handbook of Mathematical
Functions (Dover, New York, 1965),
I2C. E. Moore, Atomic Energy Levels, Natl. Bur. Stand. (U. S.)
Circ. No. 467 (U.S. GPO, %ashington D.C., 1949), Vol. I.
'3F. Hoare and G. Brindley, Proc. R. Soc. London, Ser. A 159,
395 (1937).
'4B.-G. Englert and J. Schwinger, Phys. Rev. A 26, 2322 (1982).
~5S. H. Patil, J. Chem. Phys. 80, 5073 (1984).
isH. Eissa and U. Opik, Proc. R. Soc. London 92, 556 (1967).
' G. D. Mahan, J. Chem. Phys. 48, 950 (1968).
'sH. A. Bethe and E. E. Salpetar, Quantum Mechanics of One
and Tue-Electron Atoms (Springer, Berlin, 1957), Secs. 62 and
61a.