Dr Hariree,
The
wave mechanics
of
an atom, etc.
89
The Wave
Mechanics
of
an Atom tvith
a
Non-Voulomb Ventral
Field.
Part
I.
Theory
and
Methods.
By D. R.
HARTREE, Ph.D.,
St John's College.
[Received
19 November, read 21 November, 1927.]
§
1.
Introduction.
On
the
theory
of
atomic structure proposed
by
Bohr,
in
which
the electrons
are
considered
as
point charges revolving
in
orbits
about
the
nucleus,
the
Orbits being specified
by
quantum
con-
ditions,
it
is well known that both
a
qualitative and an approximate
quantitative explanation
of
many features
of the
simpler optical
spectra
and of
X-ray spectra
of
atoms with many electrons (e.g.
Rydberg sequences
in
optical spectra, term magnitudes
in
both
X-ray
and
optical spectra)
can be
given,
if
the assumption
is
made
that
the
effects
of
the electrons
on one
another can be represented
by supposing each to move in
a
central non-Coulomb field
of
force*;
further,
the
additional concept
of a
spinning electron provides
a
similar explanation
of
other features
of
these specbraf
(e.g.
doublet struoture
of
terms
and
magnitude
of
doublet separation,
anomalous Zeeman effect). This assumption
of a
central field
was
admittedly
a
rough approximation made
in the
absence
of any
detailed ideas about the interaction between the different electrons
in an atom, but in view
of
its success as
a
first approximation
for the
orbital atom model,
the
question arises whether, the same simple
approximations may
not
give useful results when applied
to the
new formulation
of
the quantum theory which
has
been developed
in
the
last two years.
The wave mechanics
of
Schrodinger} appears
to be the
most
suitable form
of the
new quantum theory
to use for
this purpose,
and will
be
adopted throughout. Further,
if i|r
is
a
solution
of
the
wave equation (suitably
normalised),
the suggestion has been made
by Schrodinger,
and
developed
by
Klein§, that |i/r|
2
gives
the
volume density
of
charge
in the
state described
by
this
i/r;
whether this interpretation
is
always applicable
may be
doubtful,
but
for the
wave functions corresponding
to
closed orbits
of
elec-
trons
in an
atom, with which alone this paper will
be
concerned,
it
has the
advantage that
it
gives something
of a
model both
of
the stationary states
(if i/r
only contains
one of the
characteristic
functions)
and of the
process
of
radiation
(if i/r is the sum of
* See,
for
example,
M. Bom,
Vorlesungen Uber
Atommechanik
(or the
English
translation, The
Mechanics
of
tlie
Atom),
Ch. in.
t
For a
general review,
see R. H.
Fowler, Nature, Vol. cxix,
p. 90
(1927) •
for
a more detailed treatment,
F.
Hund, Liniewpektren,
Ch. m. '
%
E. Schrodinger, Ann. der Phys., Vol.
LXXIX,
pp.
361,
489; Vol.
LXXX
D 437-
Vol.
LXXXI,
p. 109
(1926); Phys.
Rev.,
Vol. xxrai,
p.
1049 (1926)
§
F.
Klein, Zeit.
f.
Phys., Vol.
su, p. 432
(1927).
90 Dr Hartree, The wave mechanics of an atom
a number of characteristic functions), and gives a simple interpre-
tation of the formula of the perturbation theory, namely, thdt the
change in the energy is the perturbing potential averaged over the
distribution of charge. Also in considering the scattering of radia-
tion of wave length not large compared to atomic dimensions, the
coherent radiation scattered by a hydrogen atom is given by
treating the scattering as classical scattering by the distribution
of charge given by Schrodinger's suggestion (if the wave length
is not too short)*, and this is probably true for any atom. For
the purpose of this paper the suggestion will be adopted literally;
the charge distribution for an atom in a stationary state is then
static (it does not necessarily follow that the charge itself is static).
Further, the distribution of charge for a closed n
k
group of
electrons is centrally symmetricalf, which suggests that on the
wave mechanics the assumption of a central field may give results
more satisfactory in detail than could be expected on the older
form of quantum theory.
Schrodinger's suggestion concerning the interpretation of
yfr
affords a hope that it may be possible to consider the internal field
of the atom as being due to the distribution of charge given by the
characteristic functions for the core electrons; we may, in fact,
attempt to find a field of force such that the total distribution of
charge, given by the characteristic functions in this field (taken in
suitable multiples corresponding to the numbers of electrons in
different n
k
groups), reproduces the field. The solution of this
problem, or rather a refinement of it, is, indeed, the main object of
the quantitative work to be considered here.
One point of contrast between the old ' orbital' mechanics and
the new wave mechanics may be emphasised here at once. On the
orbital mechanics the motion of an electron in an orbit lies wholly
between two radii (the potential energy and so the total energy
depends on the field at distances greater than the maximum
radius, but this does not affect the motion); on the wave
mechanics the solution of the wave equation is different from zero
at all but a finite number of values of the radius, and depends on
the field at all distances (though certainly the solution is very
small, and depends on the field to an extent negligible in practice,
at very large radii and usually at very small radii).
It will be seen later
(§ 2)
that, for a given characteristic function,
it is possible to specify two points which may to a certain extent
be considered to mark the apses of the corresponding orbit in the
orbital atom model, but the characteristic function is not zero
* I. Waller,
Nature,
Vol. oxx, p.
155
(1927);
Phil.
Mag.
Nov.
1927
(Supplement).
•f See A. Unsold, Ann. der Phys., Vol.
LXXZII,
p.
355,
§§ 5
and 6. Unsold proves
this by considering the energy of an indefinitely small charge in the
field
of a closed
group; it also follows directly for the charge distribution from Unsdld's formulae
'67),
(69).
with a nan-Coulomb central field 91
outside them; as a consequence, it is not possible to make a strict
separation between 'penetrating' and 'non-penetrating' solutions
of the wave equation as it was possible to divide orbits into two
such classes. This suggests that it would be interesting to enquire
what occurs in the wave mechanics in cases when on the orbital
mechanics an atom may have a ' penetrating' and a ' non-pene-
trating' orbit with the same quantum numbers* «*, and also
whether the comparatively large doublet separations of some terms
corresponding to orbits classed as 'non-penetrating' can be ex-
plained by the non-zero fraction of the total charge which must, on
the wave mechanics, lie inside the core.
Associated with this question is that of the assignment of the
principal quantum number n to a solution of the wave equation in
a non-Coulomb field; in the orbital mechanics this was assigned in
a perfectly definite way, to which there is no direct analogy on the
wave mechanics. In this paper I will be written for the subsidiary
quantum number, taking integer values from zero upwards, which
is less by unity than Bohr's azimuthal quantum number k; this
follows the practice adopted by various writers f; to avoid altering
a notation which has become familiar, k will be retained as a suffix
in referring to the quantum numbers of an electron,
so
that l=k
—
1.
It seems best to define n such that
n —
k
=
n
—
l
+
l is the number
of values of the radius r for which i|r = 0, excluding r = 0 (if it is
a root) and r = oo; n
—
I is then the number of values of r for
which |^|
2
is a maximum. This agrees with Bohr's principal
quantum number n for the hydrogen atom.
Both in order to eliminate various universal constants from
the equations and also to avoid high powers of 10 in numerical
work, it is convenient to express quantities in terms of units, which
may be called ' atomic units.' defined as follows:
Unit of
length,
a
B
=
A
2
/4Tr
2
me
2
,
on the orbital mechanics the
radius of the
1-quantum
circular orbit of the H-atom with
fixed nucleus.
Unit of
charge,
e, the magnitude of the charge on the electron.
Unit of
mass,
m, the mass of the electron.
Consistent with these are:
Unit of
action,
hl2ir.
Unit of
energy,
e*/a
= potential energy of charge e at distance a
from an equal charge = 2hcR =
twice
the ionisation energy
of the hydrogen atom with fixed nucleus.
Unit of time,
1/47TC.R.
* For example, on the orbital mechanics, Eb, Cn,
Ag,
Au, have 3
3
X-ray orbits,
and for the neutral atoms of these elements the first d term corresponds to a non-
penetrating
3
3
orbit.
t See, for example, P. Hand, op. dt., passim.
92
Dr
Hartree, The wave mechanics
of
an atom
These units being consistent,
the
ordinary equation?
of
classical
and wave mechanics hold
in
them;
in
particular Schrodinger's
wave equation for the motion of
a
point electron with total energy
E,
in
a
static field
in
which its potential energy
is
V, becomes
V°-^+2(E-V)yJr=*0.
(11)
For
an
attractive field,
V
is always negative;
it
is
convenient
to write
V=-v,
• (1-2)
so that
the
quantity
v
calculated
in
numerical work
is
usually
positive.
For terms
of
optical
and
X-ray spectra,
E
is
always negative.
It
is
convenient
to
write
#
=
-H
(1-3)
the factor
\
being introduced
in
order that
e
shall
be
the energy
as
a
multiple
of
the ionisation energy
of
the hydrogen atom;
if a
characteristic value
of
the
solution
of
the
wave equation gives
directly
a
spectral term, of wave number v, then
e = v/R.
(1-4)
The wave equation
in
terms of v and
e is
-e)y}r
^
0.
The present paper
is
divided into
two
parts; this,
the
first,
deals with the methods used
for
solving this equation
for
a
given
non-Coulomb central field, and with the relevant theory; the second
with the question
of
the determination
of
the potential
v,
and with
an account and discussion
of
the results
for
some actual atoms.
§ 2. Theory.
When
the
field
is spherically symmetrical so that v is
a
function
of
the
radius
r
only, and spherical polar coordinates
r,
6,
<f>
are
used,
yfr
separates into
a
product
of a
function
^
(r)
of
r
only, and
a surface spherical harmonic
S
(#,
<f>);
if
I
is
the
order
of
the
spherical harmonic, the function
%
satisfies the equation
i%
+
\
2v
-
e
jj^
x
=o
(2-D
r
dr
\_
r
2
J
A
v
or, writing
'
P=r%
(2"2)
and using dashes
to
denote differentiation with respect
to
r,
P" +
\2v-e-l(l+l)jr*)P=0
(2-3)
This
is
the form
in
which the wave equation
is
used in the greater
part
of
this paper.
with a non-Coulomb central field 93
There are three advantages in working with P rather than
with
•%;
first,, the differential equation is simpler, secondly, P
2
gives the radial density of charge if P is suitably normalised
(i.e.
P*dr I1
P>dr
is the charge lying between radii r and r+dr) *,
and this is the quantity often required in applications, thirdly, as
a consequence of this, -P
2
is the weighting function for the per-
turbing potential at different radii, in the case of a perturbation
which is centrally symmetrical.
If we recall that the wave equation is derived from the classical
Hamiltonian equation of the problem by the substitution
p
x
= i^, etc.
(in atomic units), and also that on the orbital mechanics the
radial momentum p
r
in an orbit of angular momentum k (in
atomic units
hj^-w)
is given by
p
r
*
= 2v
—
e
--
equation (23) suggests that as far as we can picture an orbit
corresponding to a given solution of the wave equation, its angular
momentum in atomic units is given by
]<?=l(J,+
1) (I integral),
and that its apses are given by the roots of
2v-e-l(l+l)/r*
=
0,
(2"4)
i.e. by the points of inflexion of P other than those which occur at the
points where P =
0.
Usually this expression has two roots, between
which it is positive; between them P has an oscillatory character,
outside them it has an exponential character; it may have one
root only (in the case I = 0) or four (for I > 0 only). On the orbit
model of the atom, when four roots occur they give the apses of
an internal and an external orbit with the same energy f; what
then happens on the wave mechanics will be discussed in the
second part of this paper.
The first requirement is a method for finding values of e for
which, given tiasa function of r, the solution P of (2"3) is zero at
7=0 and
GO
(Schrodinger's condition is that i/r should be finite
* According
to
Schrodinger's interpretation
of
tf/
(§ 1)
the
charge lying
in an
element
of
volume denned
by
drd9d<f>
is
fr2sir\*6drd0d<t>l\\p
2
r
i
a\ti
i
8drd8d<l>,
the
integral being over
all
space,
so
that
the
charge lying between radii
r and
r
+ dr
is
1I
^r'dr
=
P
2
dr
/ I
"'P*dr,
the integration
of
the spherical harmonic factor cancelling
out.
t
In
general
the
internal and external orbits with the same energy will
not
both
be quantum orbits,
but
when they occur
it is
usually possible (always
if
integral
quantum numbers are used)
to
obtain
an
internal
and an
external quantum orbit
with
the
same quantum numbers.