University of Windsor University of Windsor
Scholarship at UWindsor Scholarship at UWindsor
Physics Publications Department of Physics
1981
Application of discrete-basis-set methods to the Dirac equation Application of discrete-basis-set methods to the Dirac equation
Gordon W. F. Drake
University of Windsor
S. P. Goldman
Follow this and additional works at: https://scholar.uwindsor.ca/physicspub
Part of the Physics Commons
Recommended Citation Recommended Citation
Drake, Gordon W. F. and Goldman, S. P.. (1981). Application of discrete-basis-set methods to the Dirac
equation.
Physical Review A
, 23 (5), 2093-2098.
https://scholar.uwindsor.ca/physicspub/97
This Article is brought to you for free and open access by the Department of Physics at Scholarship at UWindsor. It
has been accepted for inclusion in Physics Publications by an authorized administrator of Scholarship at UWindsor.
For more information, please contact scholarship@uwindsor.ca.
PHYSICAL
RKVIK%
A
VOLUME
23,
NUMBER
5
MAY
1981
Application
of discrete-basis-set
methods
to
the
Dirac
equation
G.
W. F.
Drake and
S.P.
Goldman
Department
of
Physics, University
of
8'indsor,
Windsor,
Ontario, Canada
N98 3P4
IReceived
21
November
1980)
Variational
solutions
to the Dirac
equation
in
a
discrete I.
'
basis
set
are
investigated. Numerical
calculations
indicate
that
for a
Coulomb
potential,
the
basis
set
can
be chosen in
such
a
way
that the
variational
eigenvalues
satisfy
a
generalized
Hylleraas-Undheim
theorem.
A number
of
relativistic sum
rules
are calculated
to
demonstrate
that
the variational
solutions form
a
discrete
representation
of
the
complete
Dirac
spectrum
including both
positive-
and
negative-energy states. The
results
suggest
that
widely
used
methods for
constructing I.
'
representations of
the
nonrelativistic
electron
Green
s function
can
be
extended
to the
Dirac
equation.
As
an
example,
the
relativistic
basis
sets
are used to
calculate
electric
dipole
oscillator
strength sums from
the
ground
state,
and
dipole
polarizabilities.
I. INTRODUCTION
In
nonrelativistic
quantum
mechanics,
varia-
tional methods
provide
a
powerful
and
widely
used
technique
for
the construction
of
approxi-
mate
eigenvalues
and
eigenfunctions of
the
Schro-
dinger
equation.
To briefly
review,
if
$„
is
any
normalizable trial
function
and
H
is the
Schro-
dinger Hamiltonian,
then
the
expression
is
an
upper
bound
to
the
ground-state
energy.
If
g„
is expanded
in an
orthonormal basis
set
4,
with linear variational
coefficients
a&,
Then
E„
is
optimized
with
respect
to the
a,
by
solving
the set of
N
homogeneous equations
BE
tr
BQ)
for the
a,
.
This is
equivalent
to
diagonalizing
the
NxN
matrix
H
with matrix
elements
If„=
&C.
,
(H)e,
&
.
(1.4)
The
jth
eigenvector
~C
&)
gives
the
optimum
values
of
the
a,
and,
by
the
Hylleraas-Undheim
theorem,
'
the
X
eigenvalues
are
upper
bounds to
the
true
eigenvalues of H.
The bounds
progressively
im-
prove
as
Ã
.is
increased since the matrix
eigen-
values
necessarily
move
downward
(or
remain
fixed).
The bounds
discussed above
cannot
in
general
be extended to the
Dirac Hamiltonian
H~
because,
unlike
H,
F~
is
not bounded from below.
Any
positive-energy
eigenvalue
of the
matrix
H~
cor-
responding
to
(1.
4)
can
collapse
without limit
to
a negative-
energy
eigenvalue
as the basis
set
is
enlarged.
Variational methods have
previously
been used
to solve the
Dirac-Hartree-Fock equations"'
and
the
stability
of
these solutions
against collapse
to
negative-energy
states
has
recently
been
dis-
cussed.
~
In this
context,
stability
is
ensured
by
projecting
out the
negative-energy states.
In
the
present
work,
we
obtain
instead a
variational
representation of the
complete
Dirac
spectrum
without the
explicit. use of
projection operators.
We
suggest
on the basis
of numerical
evidence
that
for
the
special case
of
the
Dirac
equation
with a
Coulomb
potential,
the basis
set
can
be
chosen in
such
a
way
that the
eigenvalues
do
yield
upper
bounds
to
the discrete
positive
energies,
while
the
negative eigenvalues lie
in
the negative-
energy
continuum.
The
primary
usefulness
of the
results
is that
the
N-eigenvalue
spectrum
of
FI~
forms
a
discrete
variational
representation
of the
actual
eigenvalue
spectrum of
H~,
including both
positive and nega-
tive
energies.
This
allows the
Dirac
Green's
operator
to,
be
approximated
by
the
expression
(~
)z
g
I 6)
Al
(1 5)
E
j
z
where the
sum
over
j
includes
both
positive
and
negative
variational
eigenvalues
of
H~.
The
non-
relativistic
analog
of
(1.
5)
has
been
widely
used
to
perform summations
over
complete sets
of
inter-
mediate
states,
'
and
to
extract
information
on
scattering states.
"
The
present results
suggest
that
the
same
techniques
can be
extended
to the
Dirac
Hamiltonian.
In
the.
remainder
of the
paper,
the
variational
solution
of the
Dirac
equation
is
first
discussed
in
Sec. II.
We then
show
the
variational
eigen-
values
are
bounds
on the exact
eigenvalues
in
Sec.
III,
and
suggest
on
the
basis
of
numerical
evi-
dence that
a
generalized
Hylleraas-Undheim
theorem
exists
for the
relativistic
Coulomb
pro-
blem.
The
completeness
of
the
basis
set
is
tested
by
calculating
a
number
of sum
rules
in
Sec.
IV.
The
variational
basis
set
is then
used
in
Sec.
V
2093
1981 The
American
Physical
Society
2094
G.
W. F.
DRAKE
AND S. P. GOLDMAW
to
calculate
the
electric
dipole
oscillator-strength
sum from
the
ground
state.
Z
I&'-2(s-
I
&I&(I
&I
-
y&'
]'"
(2.10)
II. VARIATIONAL
FORMULATION
%e
restrict
our
discussion
to
an
electron in a
Coulomb
potential
o(r)
=
Z—
e2r
since this has
special
properties
which
lead
to
bounds.
The
four-component
Dirac
equation
is then
,
(2.1}
with
Our
variational
procedure
consists
of
using
a
trial
function
of
the same form
as
(2.
8)
with
u,
and
b&
regarded
as
linear variational
parameters,
and
A,
a nonlinear
parameter
which can be
adjusted
arbitrarily.
Starting
with 2N basis functions
(2.
11)
Ze
2
If
=
ca
~
p
+Pmc (2.2}
(2.
12)
$2gr
'
(-fr-'
(2.3)
and
the Dirac matrices
n
and
P
have their
usual
meanings.
For
any
central
potential,
the solutions
to
(2.
1)
can
be
written
in
the
form'
(2.
13)
which
satisfy
the
computational
procedure
is
first
to
ortho-
normalize the basis
set,
and then
to diagonalize
II,
to
obtain
the linear
combinatioris
4'2=
Z
«~.
SA+&2.
if'&
/=1
where
g(r)
and
f
(r)
are
the
large
and small radial
functions and
0»&
is
a two-component
spherical
spinor
defined
as the
vector
coupled product
4]
4~dr
=
5)
~,
(2.14)
fl~w
=~
«~2&
IfM&1'P(8~m&X2
(2.4)
J
IIr@
j~+
0
(2.
15}
with
(oi E1)
All
necessary
integrals
can
easily
be evaluated
analytically.
The
e,
,
i
=
1,
.
.
.
,
2N
are
the
discrete
variational eigenvalues.
For convenience,
we
define a
real
two-compo-
nent
radial
spinor
by
@(
)
f'g(r
)
&
f(r))
Then
4
(r)
satisfies the radial
Dirac
equation
with
d
K
0'
H
=
-io
—
+o
—
+~-
rpdyx
(2.5)
(2.6}
(2.
7)
in
atomic
units
(e
=
5'
=
m
=
1).
The
&r's
are the
Pauli
spin
matrices,
and
I(.
"
is
the
Dirac
quantum
number
x
=
+
(j+
—,
')
for
j
=
/
w
—,
'.
The exact
posi-
tive-energy
solutions
to
(2.
6)
for
quantum
numbers
n
and
z
can
be
written in the
form
III.
BOUNDS
c„=
g(r)
al
I+fpI
fib
-
t'0
&oi &»
(3.
1}
In
general,
the
e,
obtained above are distributed
between
+
~
and
-~,
and move
in
an
unpredictable
way
as the
parameters
of the basis
set
are varied.
There
is
no guarantee
that
any
eigenvector is
an approximate
representation
of
some particular
state,
and
the
eigentalues
are not bounds.
But
for the special
case of
a
Coulomb
potential,
it
appears
that one
does
obtain bounds
in
the
form
of
a generalized
Hylleraas-Undheim Theorem.
We
give
here
a
rigorous proof
for the
case
N
=
1,
together
with
numerical evidence
that
the theorem
can be
extended
to
arbitrary N.
For
the case
N
=
1,
consider a
trial function
of
the
form
y-l
e-
&r
~4
(15
/ON
&0) &1)
(2.8)
where
g(r)
is
an arbitrary
continuous
function
subject
to the conditions
with
y
(Z2
~2+2)1h
(2.
9)
f
g(r)'dr
=
1,
0
lim
g(r)
=
0.
r~0
(3.
2)
APPLICATION OF
DISCRETE-BASIS-SET
METHODS
TO
THE
~
~
~
2095
Then
variation
of
(8
8)
(8.
4)
where
gt'
2vr
dr
(3.5)
TV=
gt'
x
'dr
~
0
(8
5)
Clearly,
a
g(r)
can
be
chosen
to
yield
a
wide
range
of
eigenvalues if
Y
and
S"
are
independent.
But
for
the
Coulomb
potential,
V
=
-ZW
and
(3.4}
reduces to
E„=
—
ZW+
r
(1+
a'~'W')'+.
(8.
7)
The
lower
root
always
lies below
—
1/a',
and is
therefore
a
lower bound on
the
highest
negative-
energy
eigenvalue.
The
upper
root
can
be
opti-
mized
with
respect to
arbitrary
variations in
W
&
0
to yield
a
single
minimum
with
respect
to
a
and b for an
arbitrary potential
e(r)
yields
negative -energy
continuum
below
E
= —
1/n',
while
the
positive-energy eigenvalues behave
exactly
as
if
the Dirac Hamiltonian
were a
posi-
tive-definite
operator.
They
move
progressively
lower
as
N
is increased, but
never
cross the exact
energies.
The
s,
@
eigenvalues
for
Z
=
92
and
X
=
65.2
are
shown
for
progressively larger
basis
sets
in
Fig.
1.
The
spurious
root for
I(;
&0
dis-
cussed
previously
is
always
present,
but
causes
no
difficulties.
The
above
behavior
depends
on there
being
as
many
f,
[(Eq.
2.
12)]
as
g,
[(Eq.
2.
11)]
functions in
the basis set.
As
f,
functions
are
omitted,
nega-
'
tive
eigenvalues
progressively disappear
and
positive
eigenvalues
fall
progressively
below
the
exact
values.
It,
therefore,
appears
that
for
the
Coulomb-Dirac
Hamiltonian,
there
exists
a
generalized
Hylleraas-Undheim
theorem
stating
that
for
2N-dimensional
basis
sets
containing
as
many
functionally equivalent degrees
of freedom
in
the
upper
component
as
in
the
lower
component,
there are N-positive
eigenvalues
and N-negative
eigenvalues.
For
x
&0,
the
positive eigenvalues
are
upper
bounds
on the first
N
discrete
energies.
For
g
&0,
one
obtains
bounds
on the
first
N-1
discrete
energies,
together
with
a
spurious
root.
In both
cases,
the
negative
eigenvalues are
lower
bounds
on
the highest negative-energy
state
(i.
e.
,
-mc'}.
Y
tr
(8.
8)
IV.
THE DIRAC
GREEN'S
OPERATOR
AND SUM
RULES
w=z/yl~l.
(8.9)
These
are the exact
values for the lowest
posi-
tive-energy
state
having any a
&0
(i.
e.
,
j
=
l+
—,
).
For
&~0,
the
above procedure
yields
a
spurious
root
degenerate
with the
corresponding
state
with
x
&0.
For
example,
a
1p,
~,
(z
=
1)
root
is
ob-
tained
which
is degenerate
with
1s,
@(~
=
-1).
This
causes no
problem
as
the basis
set
is
en-
larged
because the
single spurious
root
can be
simply
discarded.
The
essential
point
is that
a
lowest posibve-energy
root
always
exists which
prevents
the
spectrum
from
collapsing.
For
g
&0,
the
root is
a
rigorous
upper
bound on
the
lowest
positive eigenvalue
for
any
choice
of
g(r}
subject
to
conditions
(3.
2).
We
have not
yet
been
Able
to
extend
the formal
proof
of bounds to
basis sets with
N&1.
However,
we
have done
extensive
numerical
calculations
with
progressively
larger
basis sets of the
form
(2.
8)
for
g
=
+
1,
+
2,
N
up
to
16,
and
a range
of values of
In
every
case,
the
2N
eigenvalues split
into
N-positive
eigenvalues
and
N-negative
eigen-
values.
The
negative
eigenvalues
all
lie in the
To the
extent
that
the
discrete
variational spec-
trum
obtained
by
diagonalizing
H~
in
a finite
basis
set
represents the actual
spectrum of
H~,
the
Green's
operator
can
be
approximated
by
G(
)
g
l
4.
&&4.
l
6ff
—
Z
(4.
1)
w'here
the
e„are
the
variational
eigenvalues
and
the
l
p„&
the
corresponding
eigenvectors
To
tes.
t
the
validity of
(4.
1},
we have
evaluated
a number
of sum
rules of the
form
(4.
2)
where
the sum
includes
integrations
over both
positive-
and
negative-energy
continua.
By
re-
placing
factors
of
(E„-E,
)
with
[H~,
r]
as
in the
derivation
of
nonrelativistic
oscillator
strength
sum
rules,
'
the
S;(i
=
0,
.
..
,
4)
can
be calculated
exactly.
The
results
are
(4.3)
(4.
4)
(4.
5)
2096
G.
W.
F.
DRAKE
AND S.
P. GOLDMAÃ
8.
0—
4,
0-
8
g
-10
1 1
12
2.
0—
bound
t
states
1.
5-
1.
4-
1
1
1
1
1
1
1
1
1
1
1
1
1
1
ci
Q
1.
991
I I I
P
/
/
/
r
r
r
-3.
0-
-10.
0-
I
I
I
I
I
I
I
I
3
I
I
I
I
I
I
I
I
I
6
8
I
I
I
I
I
I
I
I
I
10
I
I
I
I
I
I
I
I
12
-80.
o-
I I
2
3
I I I
4
5 6
I I
I
I I I
I
I
7
8 g
10 11
12
13
14
FIG.
1.
Distribution
of the
nsf
j2
variational
eigenvalues
for
Z=
92 and
X=
65.2
as
a function of the
size of the
basis
set.
Each
basis
set
of
size
2~has
X-positive eigenvalues
in the
upper
half of the
diagram
and
N-negative
eigenvalues
in the lower half
of the
diagram.
The vertical scale is logarithmic.
8~
(40
I
&
I ko&
+
12
~
(4.
7}
(4.
8)
with
G(E,
)
approximated
by
(4.
1).
The
errors
and
S,
diverges
for
ns,
~,
and
npy/2
states. The
non-
relativistic
8,
also
diverges
even
though
the
relativ-
istic
S,
remains finite.
Equation (4.
4)
for
S,
shows
that the
contributions
from positive-
and
negative-
energy
states
to
the
nonrelativistic oscillator
strength
sum
(for
which
S,
=
1)
exactly
cancel.
This
result
was obtained
previously
by
Levinger
et
al.
'
Alternatively,
the
S;
can
be
written in
the form
2
oI~=
3$
(4.
9)
Relativistic
values
of
o'IZ'
(in
units
of
ao}
are
given
in Table II
for
a selection of hydrogenic
ions.
The
values
have converged
to
the number
of figures
quoted.
The
exact nonrelativistic
value
is
&„=
4.
5/Z'.
arising
from
the
use
of
(4.
1)
with N
=
14 are
shown
in
Table
I.
All
of
the
errors are
small,
and
extrapolate smoothly
to zero as
N
is
in-
creased. This
provides
strong
evidence
that
the
variational
representation tends to
a
com-
plete
and accurate
description of
the
Dirac
spec-
trum as the size of the
basis
set
is
enlarged.
The sum
S,
is
related
to
the
dipole
polarizability
e,
by'
![TABLE I. Comparison of sum rules [Eqs. (4.3)-(4.7)] with exact values for the 1s~g& state, using a 2x14 term basis set. M& =S&(exact) -S&(sum rule), For S&, the number tabulated is M~ instead of M~/S~. The values of y from (2.9) are 0.99997337 for Z =1 and 0.93105942 for Z =50.](/figures/table-i-comparison-of-sum-rules-eqs-4-3-4-7-with-exact-3a7aap4n.webp)
