'
rHE
NOh.J
i:l.
AL
S'l'
ATE
O:F
THE
HYDE.OGEN
l\
:LO
L:
L!J
CULE
In
partial
for
the
Th
esis
by
Sidney
Weinbaum
fulfillment
of
the
degree
of
Doctor
of
requirements
Philosophy
Californi
a
Institute
of
Technology
Pas
adena,
California
19
33
THE NOPJ;IAL
ST
AT
I
~
OF THE HYDHOGEN
MO
LEC ULE
.
\bs
tract
A
simple
w
ave
functi
on
for
the
normal
state
of
the
hydrogen
molecule,
taking
into
account
both
the
atomic
an
d
ionic
configurations,
was
set
up
and
treated
by
a
variational
method.
The
d
issociation
energy
was
found
to
be
4.00
v.e.
as
compared
to
the
experi
men
tal
value
of
4.68
v.e.
and
R
osen's
value
of
4.02
v.e.
obtained
by
use
of
a
function
in
v
olving
complicated
inte
g
rals.
It
was
found
that
the
atomic
function
occurs
with
a
factor
3.9
times
that
of
the
i o
nic
function.
A
simil
ar
functi
on w
ith
different
screening
c
on-
stants
f or
the
atomic
and
ionic
parts
was
also
tried.
It
was
found
t h
at
the
bes
t
results
are
obt
ai
ned
v1hen
these
screening
constants
are
equ
a
l.
T
he
addition
of
R
osen's
term
to
the
at
omi
c-i
o
nic
function
resulted
in
a
value
of
4.10
v.e.
for
the
dis
-
sociation
energy.
Attem
p
ts
to
obtain
some
of
the
properties
of
the
normal
hydrogen
m
olecule
by
\'\!ave-mechanical
methods
date
to
the
early
days
of
wave
m
echanics.
Bei
t
ler
and.
1
London
applied
a
first-order
perturbation
method,
and
Sugiura
2
,
by
evaluating
an
integral
whose
value
Heitler
and
London
had
only
esti
m
ate
d ,
obtained
results
qualitatively
comparable
with
known
experimental
data.
Eisenschitz
and
London
3
applied
a
second-order
per-
turbation
tr
e
atment
and
obtained
results
in
poorer
agreement
w
ith
experimental
values
than
the
results
of
previous
calculations.
F
or
example,
Beitler-London-
Su
g
iura's
value
for
the
dissociation
energy
is
3.2
v.e.
and
the
experimental
v
alue
corrected
for
the
zero-point
energy
is
4.68
v.e.,
while
Eisenschitz
an
d
London
ob-
tained
9.5
v.e
••
Thus
it
seems
that
the
perturbation
me
t
hod
is
not
very
satisfactory
for
the
treatment
of
the
h
ydrogen
m
olecule.
fhe
variational
method,
by
approaching
the
value
of
ener
gy
from
one
side,
is
safe
from
the
possibility
of
overshooting
the
n1ark.
W
ang
4
,
using
;:i,
variational
metho
d
involving
the
introduction
of
a
shielding
constant
as
a
parameter,
obtained
3.7
v.e.
for
the
dissociation
energy.
Rosen
5
,
by
using
the
three
-
parameter
function
3
w
here
~
0
is
the
hydrogenic
wave
function
for
the
lo
w
est
st
at
e
with
a
shieldin
g
const
an
t
Z,
<j//
is
a
function
symrnetrical
about
the
axis
but
not
about
a
plane
through
the
nucleus
per
p
endicular
to
it,
and
C
is
a
parci
..
m
eter,
has
obt
a
ined
4.02
v.e.
for
the
dissociation
energy.
The
improvement
on
th
e
previous
value
is
considerable,
but
the
calcul
a
ti
ans
are
rather
labori
o
us.
All
these
calculations
were
based
on
the
assu
n~ti
on
that
each
of
the
nuclei
always
has
one
electron
attached
to
it,
these
electrons
so
me
times
interchanging
their
positions,
which
leads
to
the
interchange
energy
.
It
was
su
gg
ested
by
Hund
and.
Nl
ulliken
6
that
a
truer
p
ic-
ture
would
be
given
by
a
wave
function
(Y:+</?)/'lf
+
</l)
which
takes
a
ccount
not
only
of
the
atomic
config
ura
tion
but
also
of
the
ionic
con
f i g
uration,
w
hen
both
elec
t
rons
are
on
the
sam.e
nucleus,
the
other
being
c omp
letely
stripped
of
electrons.
However,
a
function
of
t
he
type
sug
g
ested
by
Hund
and
l'vl
ulliken
would
give
the
hydrogen
molecule
in
the
normal
state
as
much
ionic
character
as
at
omic.
There
s e ems
to
be
no
r
eason
to
assume
t h
is,
and
a
..
lo
g
ic
a l
wa
ve
function
to
take
c
ar
e
of
the
a
to
m
ic-ionic
ch
a
racter
of
the
hydrogen
molecule
appears
to
be
where
c
is
a
parameter,
It
has
been
shown
7
that
the
integral
£ =
Jf/*llf/&
/tf?
*
</d
~
4
where
H
is
the
lfa,
miltonian
ope
rator
an
d
tf7
is
a.
function
which
satisfies
certain
boundary
conditions
but
is
other-
wise
arbitrary,
cont2,ining,
say,
some
variable
p
arameters,
has
the
property
that
the
l
owest
value
W
obta
ine
d f
rom
va
ryin
g
the
numeric
a l
pareJneters
is
the
best
approximation
to
the
value
of
E,
and
that
3 -
VT
is
always
positive
or
zero.
Henc
e
the
variational
integral
presents,
as
already
ment
ion
ed
, a
satisfactory
means
for
evaluating
the
energy
of
the
normal
state
of
the
hydrogen
mo
lecule.
The
first
test
for
the
w
ave
function
~=
t:(<tf'tf
-1-
<J;
'!(
)
-r
+ (
'/:
'/; +
'lf$f,}
would
be
to
consider
it
a
two-parameter
fun
c
tion.
The
results
obtained
by
varying
c
would
then
be
comparable
with
the
Heit
ler-London-~)ugi
ura
results.
It
is,
however,
mo
re
convenient
to
treat
f'~
c('tfof
+f
lf
)+
as
a
three-param.eter
function,
and
then,
a t a
cert
a
in
point
in
the
algebra,
to
reduce
it
to
a
two-pararneter
function
by
let
t
in
g Z =
1.
It
is
useful
to
set
up
the
following
scheme,
due
to
Sl
a
ter: