It does not seem possible to account for the presence of singlets below their triplets in two electron spectra simply from the ordinary exchange integrals. The exchange integrals seem to be essentially positive. It is shown that the matrix component of electrostatic energy connecting the 3s3d^1D with 3p^2^1D in Mg I is sufficiently large to account for the occurrence of 3s3d^1D below 3s3d^3D as observed experimentally. Analytic radial wave functions of the type developed by Slater are used in the calculation of the nondiagonal elements

Bacher, R. F.

Caltech Authors - Main

LETTERS TO THE EDITOR
The treatment of the second-order splitting is simplified
by the fact that there are no low lying states which combine
with the two being considered. The configuration which
gives rise to the 'P&&2 and 'P3t2 may be written S'P. Other
possible configurations are
(a) S'D or S'S',
(c) S'P',
(e) S2PD2
(b) S4F,
(d) S'DP or S'S'P,
Here S' represents a neutron or proton in an excited S
state. Of these possible intermediate states (a) and (c) are
excluded on grounds of pa, rity; (b) has a total angular
momentum of 7/2 or 5/2 and is hence noncombining. We
are therefore'restricted to such states as (d), (e), etc. , which
have odd parity and whose reduction yields components
with angular momentum 3/2 and 1/2. These configurations
involve the breaking up of the alpha-particle core and
therefore lie perhaps 15 or 20 Mev higher than the ground
state. Considering then the magnitude of the energy de-
nominators and the probable nonoverlapping of the wave
functions, it seems certain that second and higher order
perturbations will be greatly reduced.
Under these circumstances an important factor in the
splitting of the two states will be the relativistic (Thomas)
splitting described by Inglis and estimated by him for the
case of Li7. From the order of magnitude of his result, as
well as from a rough evaluation, one would put the Thomas
splitting for the 2P state of He' at one or two hundred
thousand volts, The second-order meson splitting should
not exceed this magnitude. The sign of the resultant
splitting will be more difficult to predict, since the Thomas
doublet is inverted. '
I am indebted to Dr. Gaerttner for the information, that
new work at Pasadena on the structure of the He' level
reveals it to be a doublet with a splitting of about 150—175
kev, in satisfactory agreement with theoretical expectations.
Thanks are due Professor J. R. Oppenheimer and Dr.
L. I. Schift for valuable suggestions and criticism.
S. M. DANcoFF
Department of Physics,
University of California,
Berkeley, California,
July 31, 1939.
~ W. E. Stephens and H. Staub, Phys. Rev. 54, 237 (1938).
~ D. R. Inglis, Phys. Rev. 50, 783 (1936).
Erratum: The Paschen-Back Effect.
VI. The Spectrum of Neon
{Phys. Rev. 56, 54 (1939))
In the article with the above title the upper halves of
Figs. 2C and 3B have been interchanged,
J. B. GREEN
Ohio State University,
Columbus, Ohio,
Lehigh University,
Bethlehem, Pennsylvania,
July 17, 1939.
J. A. PEOPLES, JR,
Velocity of Radio Waves in Air
(Phys. Rev. 55, 1100 (1939))
In the discussion "Velocity of Radio Waves in Air" by
G. H, Brown, which appeared in the "Letter to the Editor"
section in the Physi ca/ Review, we note that the captions of
Figs. 1 and 2 have been interchanged. Fig. 2 should read
"Intensity pattern when the velocity is 80 percen. t of that
of light. "
G. H. BRowN
RCA Manufacturing Company,
Camden, New Jersey,
July 31, 1939.
Erratum: The Interaction of Configurations: sd —P'
(Phys. Rev. 43, 264 (1933)
An error was made in the evaluation of the radial
integral R.. Using the Rydberg constant (R„=109,737)
and' correcting this error we obtain 8,=21,387 cm ' in
place of 25,620 cm '. The sd 'D is now calculated to be at
—4272 cm ' (observed —3592 cm ') from the center of sd
configuration and P 'D at 21,353 cm '. The sd 'D appears
below the 'D (as before) due to the inclusion of the inter-
configuration matrix elements, but the numerical agree-
ment between theory and experiment is markedly improved.
The sd 'D is now calculated to be 2234 cm ' below 'D
(previously 4164 cm ') whereas it is observed at 1554 cm '
below 3D.
The writer is greatly indebted to Dr. J. P. Vinti for
directing his attention to this integral.
R. F. BACHER
Cornell University,
Ithaca, New York,
July 21, 1939.


The Interaction of Configurations: sd−p^2

FEB RUARY 15, 1933 PH YSI CAL REVI EW -VOLUME 43
The Interaction of Configurations: sd —p'
R. F. BAcHER, * University of 3fichigan
(Received December 16, 1932)
It does not seem possible to account for the presence of
singlets below their triplets in two electron spectra simply
from the ordinary exchange integrals. The exchange
integrals seem to be essentially positive. It is shown that
the matrix component of electrostatic energy connecting
the 3s3d'D with 3P''D in Mg I is sufficiently large to
account for the occurrence of 3s 3d'D below 3s 3d'D as
observed experimentally. Analytic radial wave functions
of the type developed by Slater are used in the calculation
of the nondiagonal elements.
HE spectra of elements and ions which con-
tain two valence electrons outside closed P
or d shells, are in general quite regular. That is,
the energy states are for the most part those for
which one electron remains in a low s-state and
the other takes on higher values of n and 1. The
resultant states are singlet and triplet terms with
the I. determined from the second electron. For
some of the heavier elements which follow the
completion of a d-shell there are many other
terms which arise when one of the d-electrons is
excited from its closed shell. The presence of these
states causes large perturbations in the singlets
and triplets, disturbing both their multiplet
separations and their series. These perturbations
are for the greater part quite complicated. If one
considers only those cases for which only the two
electrons are excited, the theory of Houston' is
expected to hold. Condon and Shortley' have
gathered together many examples of two electron
spectra to see how they fit Houston's theory. The
radial integrals which are involved in the theory
are not calculated but are carried through as
parameters for which certain relations are ex-
pected to hold from the theory. The radial inte-
gral of type (1) which gives the singlet-triplet
separation is considered to be either positive
(triplet below singlet) or negative (triplet above
singlet). Condon and Shortley's work shows that
for the cases for which the singlet is below the
*Alfred Lloyd Fellow. The greater part of this work
was carried out while the writer was National Research
Fellow at the Massachusetts Institute of Technology.
~ W. V. Houston, Phys. Rev. 33, 297 (1929).
2 E. U. Condon and G. H. Shortley, Phys. Rev. 3S,
1342 (1930).
triplet there is very poor agreement with the
theory.
In studying these difficulties for the D terms
of Mg I which is one of the simplest cases where
the singlet lies below the triplet, it was found that
with no reasonable radial functions for magne-
sium could the singlet ever be expected to come
below the triplet. The singlet-triplet separation is
given by the following expression:
CO
—e' dr, R(3s, r,)R(3d, r~)
5 0
co p 2
dr, —R(3s, r~)R(3d, r, ), (1)
0 ~b
where R(3s) and R(3d) are one electron wave
functions multiplied by r, and r, is the smaller of
r~ and r2, and rb the larger. Fig. 1 gives a plot of
these two wave functions as determined by the
method of Slater. The negative contribution to
the first integral arises from the second maximum
of the 3s which has the opposite sign from the 3d
wave function. This part is quite small; in fact
one may get a good first approximation to the
0
0.2
l2 r
FIG. 1. R(3p), R(3s) and R(3d) of Mg I as determined
by the Slater method are plotted against r, where r is in
units of the radius of the first Bohr orbit.
264
I NTE RAC T I 0 N 0 F CO N F I GU RAT I 0 NS: sd —p' 265
integral by considering only the term in r3 which
gives the outer maximum of the 3s radial func-
tion. The same of course holds for the second
integral. In other cases the 6rst integral may be
negative but then the second will be negative too
and the product positive. This conclusion is a
consequence of the position and relative magni-
tudes of the maxima of the radial wave functions
and in general to get a negative integral would
require very large and improbable distortions of
the wave functions from hydrogen functions. It
seems then that the examples of two-electron
spectra for which the singlet lies below the
triplet should not be expected to agree with the
Houston theory and their explanation must lie
elsewhere. It will be shown that perturbations by
other configurations may be sufficiently large to
account for them.
As pointed out by Condon, ' the interaction be-
tween con6gurations often becomes so large that
it is necessary to consider both configurations
together as one in calculating the various matrix
components. Condon pointed out that this hap-
pens when two configurations of the same parity
overlap. This interaction may also be very large
under favorable circumstances even in case the
configurations are well separated.
The lowest of the configurations arising when
both of the valence electrons of two electron
spectra are excited is the p' configuration, but it
seemed at first that it was quite unlikely that
perturbations from this configuration could be
causing the irregularity of the ordinary singlets
and triplets, due to the fact that in some of the
examples where the irregularity of the D terms is
noted the p' levels are so much higher than this
D level that they lie above the first ionization
potential. Such an explanation seems to be the
only one remaining, however, and this seems to
be an indication that such interactions are very
much larger in many cases than they have gen-
erally been supposed to be. The examples of the
sd configuration which give the singlet below the
triplet constitute the majority of such cases for
two electron spectra, as would be expected if one
considers that the even configuration p' lies very
much lower than other con6gurations involving
two excited electrons. This configuration does not
perturb the sp and sf configurations because of
the difference in parity and affects the ss con-
figurations only in a lesser degree particularly in
view of the larger "normal" triplet-singlet separa-
tion of the latter.
THE ENERGY MATRICES
If one includes both the electrostatic interac-
tion and the spin-orbit interaction the Hamil-
tonian may be written as usual in the following
form:
Z8 1—(p'+p'+p') — +l'(r)(~ ~) +" Z—
electrons,
, 2p, psxrs ffQ
where the 6rst sum is a sum of functions for each
electron alone and the second for pairs of elec-
trons. Since there are no components of H be-
tween states with different Sf' it will be sufficient
to consider only those of a single 3fz value. In
addition since, for Mg I and other simple cases
the triplet is unaffected (the interval rule still
holds), it will be sufficient to consider M'q —2,
the largest value of MJ for which the singlet is
represented and of course the simplest to treat.
Furthermore after the transformation to (I S)—
wave functions the electrostatic energy in which
we are particularly interested will have corn-
ponents only between states having the same
I and S.
'E. U. Condon, Phys. Rev. 36, 1121 (1930).
TABLE I,
1
2
3
(-'„0)
(2, 0)
(1 0)
(—-', &)
(4, 1)
(-,', 2)
(—l, 2)
There are three states from the sd con6gura-
tion and two from the p2 included in the group
MJ =2; 'D3, 2 D2 and 382 'D~, respectively. These
6ve states have the m, and m~ values for each
electron and also resultant values for both
electrons M8 and Mg as given in Table I.
The diagorial and nondiagonal matrix compon-
266 R. F. BACHER
ents of H for these states are determined from
expressions of the sort obtained by Condon' and
by Slater, in terms of wave functions built up as
Slater4 did from one-electron wave functions.
These will be referred to as the original wave
functions. For convenience put Ze'/r—=H~ ',
U(r)(l s) =H2, and e' 1/r~2=H3. The matrix of
H3 has the following form in the original wave
TABLE II. Aviatrix of II3 in original wave functions.
1 2 3 4 5
elements both a and b are sufficiently small to be
neglected in comparison to the much larger terms
in Hl and H2.
In order to be able to assign L and S values to
the various states it is useful to have these energy
matrices in terms of (I. S) wa—ve functions. To
do this, matrices for L', 5' and L S are written
down from the work of Johnson' and the trans-
formation to make all three diagonal is found in
the same manner as his. The transformation
matrix is given in Table IV. After the transfor-
R,——,'Rb 0
5Rb
5Rb (—,'-,-) 'R,
(«2.,=) 'R,
TABLE IV. Transformation matrix.
2 3 4
(3)'
(-'=) 'R. —(«'-:) ' ~
Rd —-', R,
Rd+ —,. '—,R,
1
(«) s (1)s (1)s
(1)s (1)s
0 0
functions (see Table II) where the R's are various
radial integrals.
—Ro (3g 3g ~ 3d 3d) ~ R —R2(3g 3d, 3d 3g) mation to (I S) wave func—tions H3 and Hq are
given in Tables V and VI.
R, =R'(3s, 3p; 3d, 3p); Rd, R'(3p, 3p—; 3p, 3p)
where
R, =R'(3p 3p 3p 3p)
'D3 'D2 'D2 P2 «D2
TABLE V. 3&trix of H3 in (I—S) eave functions.
R'(a, p;7, 8)
GO CO r ka=e' dry dr2R ry R, ry Rp r2 R$ r2.
0 0 r 0+1
H& has only diagonal components which are the
same for all sd states and again for all p' states.
H2, the magnetic interaction, has the form given
in Table III. a and b are radial integrals depend-
ing on the form of U(r). For the case of light
'D3 R, ——,'Rb 0
3D2
«D
3P2
«D
0 Ra ——',Rb 0
0 R,+-',Rb
0 0
0 2(-, —,.) -R,
Rd —5R,
2(-,'-.-)lR,
Rd+ (-,'—„,)R,
3D3 3D2 'D2 3P 'D2
TABLE Vl. 1lfatrix of H2 in (L—S) wave functions.
TABLE III. Matrix of H2 in original wave functi ons.
-'b
D2
1D
3P2
«D2
0
—-b2
(8) gg
0
0
(1)2a
(1)2a
—a12
—a12
' J. C. Slater, Phys. Rev. 34, 1293 (1929).
—812
H& is, of course, still diagonal. The only non-
diagonal matrix component in H3 which now ap-
b M. H. Johnson, Phys. Rev. 39, 197 (1932).
I NTERACTI ON OF CONF I GU RATIONS: sd —P' 267
pears is that between the two singlet D's. II3,
however, now contains several radial integrals
which it is necessary to evaluate before going
further. This may be simplified considerably in
any particular case by calculating only those
elements in which we are particularly interested
and taking the rest from the experimental ma-
terial. For example III and the R and R~ terms
of II3 determine only the centers of gravity of the
two electron configurations sd and p' and for the
purpose of this work this may be taken from ex-
perimental material. There remain three radial
integrals in II3 and two in the magnetic interac-
tion matrix. These last demand a knowledge of
U(r) for their evaluation a,nd this function is
known only very approximately. Fortunately for
the cases of the lighter elements which compose
many of the cases where the interaction between
configurations may be simplified to the extent
adopted here, the magnetic interactions are small
compared to the electrostatic and may be
neglected in first approximation.
THE RADIAL INTEGRALS
In order to determine the energy matrix H3
it is necessary to evaluate the radial integrals and
this involves knowledge of the radial wave func-
tions for each particular atom considered. Slater'
has recently given a method for obtaining ana-
lytic expressions to represent the Hartree wave
functions, and this may be done with great ac-
curacy. A method of interpolation provides wave
functions for those atoms for which Hartree has
not yet obtained functions. Unfortunately the
Hartree functions have not been obtained for the
optical electrons and to find wave functions for
these electrons demands an extrapolation of
Slater's curves which, while it does not give wave
functions of as great accuracy as for the inner
electrons, still gives greater accuracy than could
previously be obtained. Mg I furnishes a good
case for comparison with the experimental ma-
terial. The radial functions given by the Slater
method and normalized to unity are: (see Fig. 1)
R(3s) = (1/2. 84)(—r'e~ "'+11.96r'e '""—6.86re """)
g(Bd) (10—3/1 046)( Ps~.70(r—5.7)) p3s—o.36(~—6.71))
R(BP) = (1/2. 37)(r'e '""—10.43r'e '"")
For these radial integrals i.t is sufficient to con-
sider only the terms in r', since the other terms
have such high negative exponentials and thus
contribute only for small r. Since r enters the
integrals to a high power this makes the contri-
bution from the terms with high exponentials
small.
Experimentally the absolute energy states of
Mg I for Bs 3d and Bp' have been found to be
Bs 3d 'D = 15,268.9 'D = 13,714.7 (not separated)
and BP' 'Po, ), ' to be 3860.4, 3839.6 and 3799.0
cm ', respectively. The 'D from Bp' has been re-
ported by Sawyer' but this as pointed out by
Paschen' is located too near the triplet. Further-
more, the regularity of the Rydberg denominators
convinces one that it is a higher member of the 3s
nd series. The failure of Paschen to find 3p' 'D
may well be due to its great displacement by the
perturbation. Knowing the 'D and 'P however
' J. C. Slater, Phys. Rev. 42, 33 (1932).
7 R. A. Sawyer, J. Opt. Soc. Am. 13, 433 (1926).
from experiment, it is sufficient, with the knowl-
edge which can be obtained from the radial in-
tegral R~ and R, to determine the parts of the
electrostatic energy matrix which are the same
for all terms from the same configuration (see
Fig. 2a). Calculation of the radial integrals gives
Rq ——10,L90 cm ' R = 25,620 cm ' R, = 29,940
cm ', and the difference between the centers of
the configurations determined as above is 13,845
cm '. The electrostatic energy matrix now be-
comes if we take sd as zero energy for convenience:
(Table VII). This energy matrix leads to the fol-
lowing energy values: 'D3, 2 at —2038 cm ', 'I2
at 7857 cm ' and 'D2 at —6202 cm ' and +23,282
cm '. This places one singlet D at 4000 cm ' be-
low the triplet, a much larger displacement even
than is found in experiment. Numerically this is
not very good agreement but the difference may
easily be due to the extrapolations made in deter-
mining the coeKcients for the radial wave func-
' F. Paschen, Sitz. Ber. Preuss. Akad. 32, 709 (1931).
268 R. F. 8 ACHE R
TABLE VII. Mo,trix for II3.
3D2 'D 3P2
'D3 —2038
1D
3P
1D2
—2038 0
2038
0 13230
7857
13230
15043
30-
Ios cm
1~(
Vp~ S
25-
imp
/
/
/
/
/
/
/
'/
I
/
/
/
/
/
I
f i f'$ t T.olllzQtion Poicntia l
I
~P
tions. It is indicated, however, that even for
such a case as magnesium for which the p' con-
figuration occurs at the ionization potential, the
perturbing effect on the sd configuration may be
large enough to displace the 'D to a position
below 'D.
PERTURBATION OF THE SERIES S nd
In order to find how the higher series members
are perturbed it is necessary to know the energy
matrices of p' and s nd considered together and
also a matrix giving all the s nd members to see if
the various series members disturb each other.
This becomes very complicated when all the
series members are considered, and as an ap-
proximation only the first two series members
will be considered. The matrix of Ha of s(n+1)d
and p' is the same as that of s nd and p' in form
and differs only in the fact that R(n+1)d is used
in place of R(nd) in determining the various
radial integrals. For s nd and s(n+1)d considered
together H3 has the following form when trans-
formed to L Swave —functions (see Table VIII).
There are components only between like terms
in the two configurations. Whether the disturbing
effect is much greater between the singlets than
between the triplets depends on the size of Rgg.
A comparison between this matrix and that for
s(n+1)d and p' taken together would allow one
to determine whether perturbation of a series
takes place by the perturbing configuration dis-
turbing each configuration of the series separately
or only one member of the series is disturbed and
this disturbs the others. Since R~ is expected to
be positive, any contribution from it would tend
to increase the normal singlet triplet separation
of the higher series member. This indicates that
the perturbation of a series of terms such as those
discussed by Shenstone and Russell' is caused by
the separate perturbation of the series members
by the perturbing configuration. A calculation of
these effects for Mg I indicated that R(4d) was
not known with sufficient accuracy to give much
significance to the results.
Bs3d 'D
3s3d D sp
Frc. 2. Energy levels of 3p2 and 3s 3d of Mg I (a) without
considering interaction of configurations, (b) with inter-
action of 3s 3d and 3p2 considered, (c) from experimental
material. 3P —'D is taken from experiment in a and b.
CONCLUSION
It thus seems that it is possible to account for
the presence of singlets below their triplets in the
two electron spectrum of Mg I by considering the
perturbing effect of the low 3p' configuration. It
is believed that other examples, Al II, Zn I,
Ga II, In II, and others which are similar to Mg I
can also be accounted for in this way if the radial
'A. G. Shenstone and H. N. Russell, Phys. Rev. 39,
415 (1932).
I NTERACTION OF CONFIGURATIONS sd —P2
TABLE VIII. 2lfutrix of II3 for s nd end s (n+].)d.
269
snd
3D lD2
s(n+1)d
3/7 1D2
s nd'D2
1D2
s (n+1)d 'D2
lD
R ——,'Rb
Rg 5RQ
Rg 5Rb
RA ORB
R +-5Rb
RA+ ORB
R~ ——,'-Rgg
R,' ——',Rb'
Rg gRg
R ' —-,'Rb'
Rg+-5'R~
R~ +5Rb
functions are known. There are undoubtedly
many other cases where these large interactions
of con6gurations are important and indeed they
may account for many of the deviations which
have been found from the Slater theory. It seems
possible that these interactions may even be
large enough to account for the inversion of
doublets in alkali spectra as suggested by
White. "
It is a pleasure to the writer to thank Professor
Slater for his interest and criticism as well as for
furnishing the radial functions.
' H. E. White, Phys. Rev. 40, 316 (1932),


English

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The Interaction of Configurations: sd−p^2

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