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Solution of eigenvalue problems by the complemented vector method
BRIEF COMMUNICATIONS SOLUTION OF EIGENVALUE PROBLEMS BY TBE COMPLE%lENTEDVECTOR METHOD*f N. N.
KALITKIN MOSCOW
(Received
A method duction linear by Berg
of of
solving the
and
equations.
of
states
[II the of
for
is
(whether It is finding
1964)
September
problems
problem
Wilets
An example electron
eigenvalue
given
algebraic
7
proposed,
linear
or
a generalization the eigenfunctions.
application
of
a compressed
iron
the
method
atom
is
based
not)
in
to of
on the
a system
the
method
finding
the
reof
non-
proposed
one-
given.
1. Statement of the method We take
the
eigenvalue
problem F(u(s),
x E G.
A) = 0,
h the U(X) is the eigenfunction, Here, (possibly non-linear) fully describing
eigenvalue
ential
or
conditions,
linear
problem,
a norw
vychisl.
.Uat.
integral
equations, lization
the
boundary completing
(1) and
problem; the
it and
definition
F an
operator
may include in
the of
case the
differof eigen-
function.
*
Zh.
t
Presented in
to
the
mat. Fiz. Al l-Union
5,
6,
Conference
1965. 176
1107
- 1115,
1965.
on Computationa
1 Mat hemat its
a
Solution
We introduce vector
into
y with
the
region
G a mesh which
N + 1 components,
y,, = u(x,,) Replacing
the
operator
of
Fh(y)
a system
of
is
also
N + 1 non-linear*
We solve
system
(3)
!--I
the
the
consider
a
vector (2)
Fh of
algebraic
some
order
of
problem
,V + 1 components, so that with the same number of
iterational
-
and
coaulemented
= 0.
a vector with equations
&y(“) =
(1 < R
operator
(1)
by Newton’s
aFh (3)
177
lens
yo = h.
difference
instead
F,,(y) Obviously,
x,
we term
1 < n &N.
for
F by the
we obtain
approximation,
prob
eigenvalue
of
F’h”,,
(3) is unknowns.
method
ycs+u =
y(s)+
6yW.
dY
Here,
aFh jay
linear
in
Sufficient the
conditions
Newton
process
criterion
is
bourhood
are
which
der ivat ives, neighbourhood
and
Det(
of
this
the
rate
the
tortion The
(3).
not
e.g.
root
# 0, and
the
and
in
of
continuous
3Fh/ay)
method
of
N + 1 equations,
[21.
uniqueness In
of
Practice,
equation
(3)
exists,
together
with
its
it
is
unique
iterations
can
function is
method,
one
in
(4)
over
all
method
is
the
root
the
of
fOllOWing
in first
the
neigh-
and
second
a sufficiently
are
small
convergent
problems
difficult
to
apply.
*
if (1) were linear, of the form Au(x)
system -
in
general
components
yeyh.
as
to
it
the
convenient of
quantum
satisfy
(3)
will
the large
finding
mechanics,
be non-linear
error
concerned. one
any
computation of
for
The
boundary
of y and that
a differential
conditions.
of numerical than
an iterational
iteration
boundary to
the
therefore
each
close
not
no greater
in
are
the
error
is
methods
of
does
regarded at
the
functions
wave
be
large
~(‘1
As a result, values
also
satisfies
therefore
datnped
Even terms
a system
existence
a root
satisfying
u(x)
small
present
is
the
distributed of
the
if
used
which
present
equations formly
is
(q < 1).
widely in
while equation, in determining In
(4)
discussed
Fh(y)
w qzs
Another process,
for
convenient:
of
at
while
a matrix,
iS
y.
is
relative
of uni-
dis-
values. exponentially where
other
because
of
the
178
N.A’.
2. We shall tions
are
cients
assume
The case that
linear.
the
complex)
of
a linear
differential
We replace
(possibly
Kalitkin
them
operator
ankyk
+
and
by difference
independent
L,, =
operator
of
yobnyn
=
boundary
equations
the
condi-
with
coeffi-
solution*:
l
0,
(5)
k=i
We complement weight
is
system
chosen
(5)
Prom
by a normalization
orthogonality
condition
with
a weight;
the
considerations
N Lo =
2
bkgkyk -
C=
0,
c #
0.
Pa)
k=l
System
(5)
responding
to
is
known
to
different
have
8 solutions
eigenvalues
are
yr;
bk_i&‘Yk” =
i
the
eigenfunctions
orthogonal
with
the
cor-
weight
b,
0.
9=1
If
the
b,
yo
are
real
are
real (if
and
all
the
the
matrix
a,k
are
{a,k) real,
is
Ifermitian,
the
eigenfunctions
real
eigeavalues can
be
ob-
tained). Theorem If then
1 a,k
Det
= ikn, (dL,
yk
/8yk)
(1 &
k <
0 (0 <
that
N)
Det
y3
is
k, 72 < (a&
On the
is
3 simple
N)
(not
on solving
/C?yk) =
an eigenfunction
$, j&n-$ =
k=i
racy
and
0, in
multiple) system
which
eigenvalue,
(5)
and
case
its
rows
to
yo.
then
(53). will
be
dependent
o= *
#
Suppose
Proof. linearly
If
b.. = 5,
use [31,
ao
of
non-linear the
bk.?kYk
+
k=,
?I=0
see
i
corresponding
note
schemes, to
the
&.h
i
n=,
k=,
giving
present
an
section.
(ank
+
increased
yobk&,k)Y!+
order
of
accu-
Solution
The last
sum vanishes
calling
that
{a,k)
eigenualue
of
It
by (5).
now follows
is Hermitian,
179
problems
from
(5a)
we can now write
that
(91
a0 = 0.
Re-
as
N
2 an (zkn
+
Yobkbk)
0,
=
(74
i
II = *
c
atlbnyn = 0.
The root
ye is
simple,
so that,
from
But
(7b)
is now impossible,
(‘7a),
_
on =
(7b)
i
yn,
since
it
contradicts
The theorem
(Sa).
is
proved. if
Thus, are
the
conditions
convergent
to
the
of
Theorem
solution
1 are
the
satisfied,
in a sufficiently
iterations
neighbourhood
small
of
it. If
yo is
y,.,’
is
(61
is
a multiple
another
root,
we take
eigenfunction
a0 = 0, an = ;a’
corresponding
to
the
(1 < n < .V), where
given
ye.
and Det( a&, /ayk) = 0 on the solution of satisfied, The question of the convergence of the iterations (5a).
and
investigated It
is
for essential
be normalized This
is
For duced
known to
the
the
proof
sense
of
Hermitian
iterations
(4)
are
all
the
all
the
scheme,
scheme
convergent
> 0 in
b,
with
if
any other
close in the
has not
been
(5)
and a,k
weight LOS
to
the
matrix
the real
the of
{a,k)
can constant.
can be re-
rows by suitably coefficients,
solution
chosen
therefore,
provided
the
eigen-
(5a).
= Ok,,, the
p,, (for
eigenfunction
normalization
h, > 0.
the
with
sense
1 that
a non-zero
form by multiplying
can be normalized
normalized
Theorem with
difference
For any three-point
function
of
(5a)
be possible
any three-point to
case, (5)
case.
for the
in
well
factors.
If
this
In this system
which
eigenfunction
can be
c # 0). (5b)
&i,,,Y,+C=0. k=i
l
If
only
in
[lo].
one of
the i?~k is
non-zero,
the
method
is
the
same as employed
180
In
this
case
solution
a theorem
similar
Note. proved of
of
accuracy,
first
a,k
system
(5)
and,
of
ank
the
are
the
iterations
close
to
the
of
using in
three-point
a scheme
of
im-
dependent
(with
schemes
of
the
the
fourth
second
order
order
of
ys.
and
The
of
formulated.
instance,
weakly
(5a)
first
firstly,
twice:
after
secondly,
and
accuracy,
are
be
a means For
eigenvalue
calculation.
values cable.
convergence 1 can
indicated
the
c,k(yO)
on the Theorem
accuracy.
on the
We solve ence
to
Samarskii order
smallness)
of
Kalitkin
N.N.
calculation
second
to
employed
neglecting
substituting the
the
gives
fourth
during
each
us
order.
the
depend-
yo obtained y to
the
Here,
calculation,
from
second
since
the
order
constant
Theorem
1 is
appli-
3. Finding the required eigenfunction If, values,
as
in
the
we shall
process
(4)
is
accuracy)
In us
not
given If
case
derive in
ye
becomes
in
there
operator,
advance
to
eigenvalue
which
is
of
a spectrum
them
the
of
eigen-
iterational
choice of zero aparoximation. problem is important: given
yo of
a linear
operator,
to
find
In (to
a
the
eigenfunction.
this
Let
the
a linear
for a given the following
therefore,
corresponding
the
of know
convergent
applications, certain
case not
is
the the
method
of
sufficient
Berg
and
Wilets
conditions
for
is
immediately
its
convenient.
applicability,
which
are
Lll. an eigenvalue,
a linear
one
combination
of
of
the
the
equations,
number
remainder.
no
Eliminating
of
this,
system
(5)
we get
system N
Lo =
2
b@,yk
-
c # 0,
c = 0,
A=i
anr,,yk
+
(8) yob,,yn,
=
0,
1 <
n’
<
N,
n’
Z
no.
A=1
This
is
a non-linear
of N equations with iterational process. Theorem
2
(as
a result
N unknowns A similar
of
the
normalization
yk (1 < k
We
of
condition) solve
Theorem
system
it by Newton’s 1 holds for
Solution
If (8)
the
is
conditions
non-zero
of
1. Assuming arrive
at
Theorem
on solving
4. Calculation the
field
for
1 are
the
lue
181
lems
prob
satisfied,
the
Jacobian
of
system
system.
of the one-electron levels of an atom of
by substitution
an equation
eigenrla
of
the
the
atom to be spherically
in the
Schrodinger
radial
component
symmetric,
we
equation
of
the
wave function
(see
e.g
[41)
(9)
The condition of
the
on the
following
outer
forms
boundary
of
the
$(ro, 8, cp) = 0 or
w
--I Br
An isolated large
ro with
=o
the
dRvl __dr
We employ racy.
It
(9b),
the
uniform
the
This
dr
iterations
R
A
Vl
=: 0.
(9b J
.
(&
T=To
scheme with close
for
it
the
on it
1 =O
2 that,
convergent
I=T”
of
<
the
second
boundary to
(9c)
0).
order
of
conditions
a simple
root
accu-
(9a)
and
on any non-
mesh.
mesh is
enough
close
functions,
on* x, = (n/N)‘ro.
convenient to
the
for
nucleus,
and at the
With this
calculating in the
same time
gives
mesh*+ all
atoms, region
since
of
fairly
sharp simple
formulae.
** This
in one
(94
We can introduce
condition
three-point
are
I
radius.
asymptotic
from Section
The mesh was based
.
dR r-----R
or
WnI-hI
linear
follows
be written
R(m) = 0,
T=ru
atom has an infinite
radius
atom will
151:
is
obviously
also
true
of
a uniform
mesh.
it
the is
variation
b, > 0.
fine of
computational
the
182
N.N.
In order
for
successive of
the
linearized
system
substitutions,
to
p,
Let
us find
the
difference
scheme
we denote
the
eigenvalue
by A
of
by the (5b)
method
was used,
of
only
one
non-zero. for
boundary
2mVro2 V==------, fi*
2mEro2
A=-, in which
be solvable
normalization
the
(2),
being
the
Ka itkin
iv
condition
(9a);
as in
Y, = &(5n),
case
i
yo = Once the
iterations
accordance
with
2.
Linear
more than general become
have
0,
Y&-i
the
converged,
c.
=
eigenfunction
is
accuracy
the
renormalized
in
(5s).
schemes
three
YN =
of higher
points;
reducible
to
the
order
matrix
the Hermitian
of of
their
form.
than
coefficients
Also,
the
second is
not
computational
contain in formulae
much more complicated. TABLE 1
-
N
-211.34 -203.71 -201.93
16 32 64
In the such
present
the
paper,
computations
meshes,
by successive
the
order
-201.17
1.78
-201.34
the
therefore,
elimination
accuracy of
accuracy
of
of
errors
in A
7.63
on a sequence of
nent
-
-
gating
tmprove-
AA
A
was improved
condensing
the
result
of higher
meshes.
can
by investiIf
be improved
and higher
we have
orders.
For practical reasons, three meshes The order of accuracy was only improved
(N = 64, 128, 256) were employed. as far as the third order, while
the
accuracy
“extra” 3.
Taking
three-point
mesh was used the linear
problem scheme,
to of
check
the
Section the
minor
3. of
it
will
rank
of
the
result.
be observed N -
p
by p-l
1 obtained
that,
for
by striking
the
Solution
out
the
top
Since
all
from
(10)
row and last
the
“-1
# 0 in
by striking
Using stead of
the the
Note.
fact first
Let
eigenvalue Clearly,
the
the
ratio of
The process
terminates In the
respect
variable
to the
[~jl and
matter
how high
the
process
the
of
equal
type
(8)
to can be got
out
the
We obtain
the
order
Table
last
of
in-
accuracy
1 for
the
6 = 64.77). differences
iVe take
when the
error
case
= 4:
as the
this
x = ST and with solution (9)
final
1, in accordance result h = -201.3
can be expanded is
exact
their
the
improving
meshes.
first
present
of
is
b 0 we can strike
an_l,n
the
lank)
row (no = 1).
the
1s (with of
matrix
a system
first
accuracy.
mesh interval. potential
the
condensing
level
order
of
(lo),
the
that all row.
successively of
second
out
us illustrate
by means of
the
a,
column
183
problems
of eigenvalue
true:
in powers
the mesh is
this
choice
have
continuous
of
with * 0.2. of
uniform
variable
the with
the
derivatives,
no
order.
5. Sehaviour of electrons when an atom is compressed Calculations iron
were performed
atom in the
compression
lute temperature. model [sI. Both the
outer
boundary
of
The mathematical
1.
Table of
indices (9b),
l
with
In fact,
that
2 represents
in the
energies
the
level
V(ro)
and depends
of the
atom on the
relative the
atom,
accuracy
than
an iron
and the
the
one-electron
from 0.8
to
states
notation
is
the
was assumed*
calculation
physical
energy
(in
correspond
the
that
work function
= 0.
- O.OOl,%, i.e.
of
the
different
electron-volts).
to boundary
of
l’(rg)
was 0.01
the
electrons
electron The
conditions
6 = ~/PO. The strong may be seen.
compression.
abso-
hypotheses.
dependence
compression
a compressed
at zero
on the Thomas - Fermi boundary conditions on
compression
compression
on the
of the
and it
of
130 times,
The atomic potential was based (9a) and (9b) were taken for the
was much higher
levels
of range
(9a)
and
increase
in the
from the
material
184
N.N.
Kalitkin
TABLE
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11.9.51
Solution
of
eigenvalue
FIfi.
2.
problems
185
N.N.
186
Figure
1 shows
in Angstroms curious
effect
3p-electrons 2s-
(for
the
mean distsnces
the
notation
remote
and Ip-electrons,
from the
but
is
of
the
only
i?
nucleus
(this
observable
At fairly the
high
placed
inside
inner
electrons
mentioned
the
effect.
pressures
compression
At moderate
electrons.
atom by the
from the
from the
nucleus
effect at large
also
holds
for
the
compressions).
/ FIG.
all
electrons
see Table 2, ro is the atom radius). A for moderate compressions the 3s- and
may be observed: are
Kalitkin
pressures compression
nucleus
occurs,
4.
of
the
the
orbits
external
predominates electrons
and a strong which
leads
for are
dis-
of
the
screening to
the
above-
Solution
of
eigenvalue
problems
187
The size of this effect is quite small: it may therefore be due to the model approximations (use of a Thomas - Fermi potential, making the electron density close to the nucleus
conceivably atomic to high).
Figs 2 and 3 respectively To illustrate the influence of compression, give the radial wave functions for an uncompressed (6 = 9.7465) and strongly compressed (6 = 128.4) substance (with boundary condition (9a); for the notation see Table 2; the numbers of the Points of the computa; tional mesh are marked along the axis of abscissae). 2. Suppose that boundary conditions (9a) and (9b) define respectively the upper and lower parts of the Bloch bands of the allowed energies (see [5]), and that the level density in the band is constant (this approximation cannot be used for wide bands; our further discussion is therefore qualitative*).
FIQ.
5. Mean square of orbital momentum: 1 - present paper; 2 - from [91.
Table 2 gives the dependence. calculated Fermi level EF on the compression (regarding see the previous footnote but one. tive.
*
An accurate in [71.
calculation
of the
band
on this assumption, of the the fact that EF is POST-
structure
for
iron
may be found
188
N.A’. Kalitkin
It the
is
clear
Fermi
higher
are
liberated
This not
the
171to
is
band with
increases
atom
It
of
clear
Zs,
2p are
the
bands
electrons
of
the
to
the
orbital
below and
5s
partially
filled
electrons
band with
v = 3.
momentum
on compression and Sp-bands
3d-
in the
always 4d,
on compression,
v = 4 and pass
mean square
(the
Is,
while
that,
monotonically 3.23
bands
filled,
from
filled
and
1.38
for
the
the
Ss-band
filled). Figure
also
5 also
performed
where there gives
higher
thus
the
for
atom,
The
calculation
of
In the high-compression
the quasiclassical < i2 > than the
for the
quasiclassical
[S].
figures
with
sufficiently
Glasko
the of for
of
an iron
not
and V.B.
Zadykhailo
of
figure
coincidence
Ack~o~ledgenents.
Gal’ din
data
potential
more justification
Even for is
the
the
times (the
random). picture
gives
with
is
a 1.5-2
calculation
ing
4.
from the
quantity
the
The numbers
in Fig,
5 represents
isolated
that
completely
filled.
given
Figure
table
and are
are never
bands are
from the
level
for
approach, it quantum-mechanical
compressions
26 electrons,
[91,
region,
the
*
1 is
quasiclassical
accurate.
author sincerely thanks 4.A. Samarskii, V.Ya. for discussion and valuable remarks; also 1.3.
programming
the
computer
and I.A.
Govorukhin
for
perform-
computations. Translated
by D.E.
Brown
REFERENCES 1.
BERG,
ity 2.
R.
and WILETS, L. On the
correction
KANTOROVICH. A.V. spaces
1950.
order
of
validity
Phys.
Sco.,
and AKILOV, R.P.
Moscow,
TIKHONOV, A.N. high
Proc.
(Funktsional’nyi
Fizmatgiz, 3.
term,
1959.
analiz
Functional
(Translation
Dokl.
the
A68,
Weizsacker 3,
published
Uniform
Akad.
Nauk
inhomogene-
229 - 239,
analysis
v normirovannykh
and SAMARSKII, A.A. accuracy.
of
in
1955.
normed
prostranstvakh), by Pergamon
difference
schemes
SSSR,
3, 514
131,
Press). of - 51'7,
Solution
4.
LANDAU, L.D. nika),
5.
GOMBAS, P.
7.
LATTER,
- 292,
- Dirac Uber
phys.
171,
matrice 1955.
- Fermi 1955.
dksp.
energy
Quantum
die
hung.
3,
Une methode quelconque,
(Kvantovaya
mekha-
Theorie
komprimierter
Atom,
Acta
~~by.5.
1958. model
of
compressed
teor.
levels
315
for
43, the
Rev.
- 319,
mecanique Bull.
atoms.
calculation Fiz.
Druckabhlngigkeit 8,
mechanics
1963.
Phys.
potential,
la9
problems
Quantum-mechanical Zh.
iron.
DERWIDUE. L. d’une
- 358,
R. Atomic
GAMBAS, P. Acta
10.
of
E.M.
statistischen 321
GANDEL’MAN, G.M.
Fermi 9.
280
eigenvalue
Moscow,
R. Thomas
2,
state a.
Zur
a, 3.
LATTER, 24.
LIFSHITS,
Fizmatgiz,
hung. 6.
and
of
7,
of
them.
.I.
the
equation
131 - 145,
Thomas 2,
des
Drehimpulses
510
of
1952.
- Fermi
99,
Phys.
and
- 519.
Thomas
-
1955. von
Atomen,
1958. de Sot.
calcul r.
des Sci.
vecteurs Liege,
24,
pronres 5,
149
-
KALITKIN MOSCOW
(Received
A method duction linear by Berg
of of
solving the
and
equations.
of
states
[II the of
for
is
(whether It is finding
1964)
September
problems
problem
Wilets
An example electron
eigenvalue
given
algebraic
7
proposed,
linear
or
a generalization the eigenfunctions.
application
of
a compressed
iron
the
method
atom
is
based
not)
in
to of
on the
a system
the
method
finding
the
reof
non-
proposed
one-
given.
1. Statement of the method We take
the
eigenvalue
problem F(u(s),
x E G.
A) = 0,
h the U(X) is the eigenfunction, Here, (possibly non-linear) fully describing
eigenvalue
ential
or
conditions,
linear
problem,
a norw
vychisl.
.Uat.
integral
equations, lization
the
boundary completing
(1) and
problem; the
it and
definition
F an
operator
may include in
the of
case the
differof eigen-
function.
*
Zh.
t
Presented in
to
the
mat. Fiz. Al l-Union
5,
6,
Conference
1965. 176
1107
- 1115,
1965.
on Computationa
1 Mat hemat its
a
Solution
We introduce vector
into
y with
the
region
G a mesh which
N + 1 components,
y,, = u(x,,) Replacing
the
operator
of
Fh(y)
a system
of
is
also
N + 1 non-linear*
We solve
system
(3)
!--I
the
the
consider
a
vector (2)
Fh of
algebraic
some
order
of
problem
,V + 1 components, so that with the same number of
iterational
-
and
coaulemented
= 0.
a vector with equations
&y(“) =
(1 < R
operator
(1)
by Newton’s
aFh (3)
177
lens
yo = h.
difference
instead
F,,(y) Obviously,
x,
we term
1 < n &N.
for
F by the
we obtain
approximation,
prob
eigenvalue
of
F’h”,,
(3) is unknowns.
method
ycs+u =
y(s)+
6yW.
dY
Here,
aFh jay
linear
in
Sufficient the
conditions
Newton
process
criterion
is
bourhood
are
which
der ivat ives, neighbourhood
and
Det(
of
this
the
rate
the
tortion The
(3).
not
e.g.
root
# 0, and
the
and
in
of
continuous
3Fh/ay)
method
of
N + 1 equations,
[21.
uniqueness In
of
Practice,
equation
(3)
exists,
together
with
its
it
is
unique
iterations
can
function is
method,
one
in
(4)
over
all
method
is
the
root
the
of
fOllOWing
in first
the
neigh-
and
second
a sufficiently
are
small
convergent
problems
difficult
to
apply.
*
if (1) were linear, of the form Au(x)
system -
in
general
components
yeyh.
as
to
it
the
convenient of
quantum
satisfy
(3)
will
the large
finding
mechanics,
be non-linear
error
concerned. one
any
computation of
for
The
boundary
of y and that
a differential
conditions.
of numerical than
an iterational
iteration
boundary to
the
therefore
each
close
not
no greater
in
are
the
error
is
methods
of
does
regarded at
the
functions
wave
be
large
~(‘1
As a result, values
also
satisfies
therefore
datnped
Even terms
a system
existence
a root
satisfying
u(x)
small
present
is
the
distributed of
the
if
used
which
present
equations formly
is
(q < 1).
widely in
while equation, in determining In
(4)
discussed
Fh(y)
w qzs
Another process,
for
convenient:
of
at
while
a matrix,
iS
y.
is
relative
of uni-
dis-
values. exponentially where
other
because
of
the
178
N.A’.
2. We shall tions
are
cients
assume
The case that
linear.
the
complex)
of
a linear
differential
We replace
(possibly
Kalitkin
them
operator
ankyk
+
and
by difference
independent
L,, =
operator
of
yobnyn
=
boundary
equations
the
condi-
with
coeffi-
solution*:
l
0,
(5)
k=i
We complement weight
is
system
chosen
(5)
Prom
by a normalization
orthogonality
condition
with
a weight;
the
considerations
N Lo =
2
bkgkyk -
C=
0,
c #
0.
Pa)
k=l
System
(5)
responding
to
is
known
to
different
have
8 solutions
eigenvalues
are
yr;
bk_i&‘Yk” =
i
the
eigenfunctions
orthogonal
with
the
cor-
weight
b,
0.
9=1
If
the
b,
yo
are
real
are
real (if
and
all
the
the
matrix
a,k
are
{a,k) real,
is
Ifermitian,
the
eigenfunctions
real
eigeavalues can
be
ob-
tained). Theorem If then
1 a,k
Det
= ikn, (dL,
yk
/8yk)
(1 &
k <
0 (0 <
that
N)
Det
y3
is
k, 72 < (a&
On the
is
3 simple
N)
(not
on solving
/C?yk) =
an eigenfunction
$, j&n-$ =
k=i
racy
and
0, in
multiple) system
which
eigenvalue,
(5)
and
case
its
rows
to
yo.
then
(53). will
be
dependent
o= *
#
Suppose
Proof. linearly
If
b.. = 5,
use [31,
ao
of
non-linear the
bk.?kYk
+
k=,
?I=0
see
i
corresponding
note
schemes, to
the
&.h
i
n=,
k=,
giving
present
an
section.
(ank
+
increased
yobk&,k)Y!+
order
of
accu-
Solution
The last
sum vanishes
calling
that
{a,k)
eigenualue
of
It
by (5).
now follows
is Hermitian,
179
problems
from
(5a)
we can now write
that
(91
a0 = 0.
Re-
as
N
2 an (zkn
+
Yobkbk)
0,
=
(74
i
II = *
c
atlbnyn = 0.
The root
ye is
simple,
so that,
from
But
(7b)
is now impossible,
(‘7a),
_
on =
(7b)
i
yn,
since
it
contradicts
The theorem
(Sa).
is
proved. if
Thus, are
the
conditions
convergent
to
the
of
Theorem
solution
1 are
the
satisfied,
in a sufficiently
iterations
neighbourhood
small
of
it. If
yo is
y,.,’
is
(61
is
a multiple
another
root,
we take
eigenfunction
a0 = 0, an = ;a’
corresponding
to
the
(1 < n < .V), where
given
ye.
and Det( a&, /ayk) = 0 on the solution of satisfied, The question of the convergence of the iterations (5a).
and
investigated It
is
for essential
be normalized This
is
For duced
known to
the
the
proof
sense
of
Hermitian
iterations
(4)
are
all
the
all
the
scheme,
scheme
convergent
> 0 in
b,
with
if
any other
close in the
has not
been
(5)
and a,k
weight LOS
to
the
matrix
the real
the of
{a,k)
can constant.
can be re-
rows by suitably coefficients,
solution
chosen
therefore,
provided
the
eigen-
(5a).
= Ok,,, the
p,, (for
eigenfunction
normalization
h, > 0.
the
with
sense
1 that
a non-zero
form by multiplying
can be normalized
normalized
Theorem with
difference
For any three-point
function
of
(5a)
be possible
any three-point to
case, (5)
case.
for the
in
well
factors.
If
this
In this system
which
eigenfunction
can be
c # 0). (5b)
&i,,,Y,+C=0. k=i
l
If
only
in
[lo].
one of
the i?~k is
non-zero,
the
method
is
the
same as employed
180
In
this
case
solution
a theorem
similar
Note. proved of
of
accuracy,
first
a,k
system
(5)
and,
of
ank
the
are
the
iterations
close
to
the
of
using in
three-point
a scheme
of
im-
dependent
(with
schemes
of
the
the
fourth
second
order
order
of
ys.
and
The
of
formulated.
instance,
weakly
(5a)
first
firstly,
twice:
after
secondly,
and
accuracy,
are
be
a means For
eigenvalue
calculation.
values cable.
convergence 1 can
indicated
the
c,k(yO)
on the Theorem
accuracy.
on the
We solve ence
to
Samarskii order
smallness)
of
Kalitkin
N.N.
calculation
second
to
employed
neglecting
substituting the
the
gives
fourth
during
each
us
order.
the
depend-
yo obtained y to
the
Here,
calculation,
from
second
since
the
order
constant
Theorem
1 is
appli-
3. Finding the required eigenfunction If, values,
as
in
the
we shall
process
(4)
is
accuracy)
In us
not
given If
case
derive in
ye
becomes
in
there
operator,
advance
to
eigenvalue
which
is
of
a spectrum
them
the
of
eigen-
iterational
choice of zero aparoximation. problem is important: given
yo of
a linear
operator,
to
find
In (to
a
the
eigenfunction.
this
Let
the
a linear
for a given the following
therefore,
corresponding
the
of know
convergent
applications, certain
case not
is
the the
method
of
sufficient
Berg
and
Wilets
conditions
for
is
immediately
its
convenient.
applicability,
which
are
Lll. an eigenvalue,
a linear
one
combination
of
of
the
the
equations,
number
remainder.
no
Eliminating
of
this,
system
(5)
we get
system N
Lo =
2
b@,yk
-
c # 0,
c = 0,
A=i
anr,,yk
+
(8) yob,,yn,
=
0,
1 <
n’
<
N,
n’
Z
no.
A=1
This
is
a non-linear
of N equations with iterational process. Theorem
2
(as
a result
N unknowns A similar
of
the
normalization
yk (1 < k
We
of
condition) solve
Theorem
system
it by Newton’s 1 holds for
Solution
If (8)
the
is
conditions
non-zero
of
1. Assuming arrive
at
Theorem
on solving
4. Calculation the
field
for
1 are
the
lue
181
lems
prob
satisfied,
the
Jacobian
of
system
system.
of the one-electron levels of an atom of
by substitution
an equation
eigenrla
of
the
the
atom to be spherically
in the
Schrodinger
radial
component
symmetric,
we
equation
of
the
wave function
(see
e.g
[41)
(9)
The condition of
the
on the
following
outer
forms
boundary
of
the
$(ro, 8, cp) = 0 or
w
--I Br
An isolated large
ro with
=o
the
dRvl __dr
We employ racy.
It
(9b),
the
uniform
the
This
dr
iterations
R
A
Vl
=: 0.
(9b J
.
(&
T=To
scheme with close
for
it
the
on it
1 =O
2 that,
convergent
I=T”
of
<
the
second
boundary to
(9c)
0).
order
of
conditions
a simple
root
accu-
(9a)
and
on any non-
mesh.
mesh is
enough
close
functions,
on* x, = (n/N)‘ro.
convenient to
the
for
nucleus,
and at the
With this
calculating in the
same time
gives
mesh*+ all
atoms, region
since
of
fairly
sharp simple
formulae.
** This
in one
(94
We can introduce
condition
three-point
are
I
radius.
asymptotic
from Section
The mesh was based
.
dR r-----R
or
WnI-hI
linear
follows
be written
R(m) = 0,
T=ru
atom has an infinite
radius
atom will
151:
is
obviously
also
true
of
a uniform
mesh.
it
the is
variation
b, > 0.
fine of
computational
the
182
N.N.
In order
for
successive of
the
linearized
system
substitutions,
to
p,
Let
us find
the
difference
scheme
we denote
the
eigenvalue
by A
of
by the (5b)
method
was used,
of
only
one
non-zero. for
boundary
2mVro2 V==------, fi*
2mEro2
A=-, in which
be solvable
normalization
the
(2),
being
the
Ka itkin
iv
condition
(9a);
as in
Y, = &(5n),
case
i
yo = Once the
iterations
accordance
with
2.
Linear
more than general become
have
0,
Y&-i
the
converged,
c.
=
eigenfunction
is
accuracy
the
renormalized
in
(5s).
schemes
three
YN =
of higher
points;
reducible
to
the
order
matrix
the Hermitian
of of
their
form.
than
coefficients
Also,
the
second is
not
computational
contain in formulae
much more complicated. TABLE 1
-
N
-211.34 -203.71 -201.93
16 32 64
In the such
present
the
paper,
computations
meshes,
by successive
the
order
-201.17
1.78
-201.34
the
therefore,
elimination
accuracy of
accuracy
of
of
errors
in A
7.63
on a sequence of
nent
-
-
gating
tmprove-
AA
A
was improved
condensing
the
result
of higher
meshes.
can
by investiIf
be improved
and higher
we have
orders.
For practical reasons, three meshes The order of accuracy was only improved
(N = 64, 128, 256) were employed. as far as the third order, while
the
accuracy
“extra” 3.
Taking
three-point
mesh was used the linear
problem scheme,
to of
check
the
Section the
minor
3. of
it
will
rank
of
the
result.
be observed N -
p
by p-l
1 obtained
that,
for
by striking
the
Solution
out
the
top
Since
all
from
(10)
row and last
the
“-1
# 0 in
by striking
Using stead of
the the
Note.
fact first
Let
eigenvalue Clearly,
the
the
ratio of
The process
terminates In the
respect
variable
to the
[~jl and
matter
how high
the
process
the
of
equal
type
(8)
to can be got
out
the
We obtain
the
order
Table
last
of
in-
accuracy
1 for
the
6 = 64.77). differences
iVe take
when the
error
case
= 4:
as the
this
x = ST and with solution (9)
final
1, in accordance result h = -201.3
can be expanded is
exact
their
the
improving
meshes.
first
present
of
is
b 0 we can strike
an_l,n
the
lank)
row (no = 1).
the
1s (with of
matrix
a system
first
accuracy.
mesh interval. potential
the
condensing
level
order
of
(lo),
the
that all row.
successively of
second
out
us illustrate
by means of
the
a,
column
183
problems
of eigenvalue
true:
in powers
the mesh is
this
choice
have
continuous
of
with * 0.2. of
uniform
variable
the with
the
derivatives,
no
order.
5. Sehaviour of electrons when an atom is compressed Calculations iron
were performed
atom in the
compression
lute temperature. model [sI. Both the
outer
boundary
of
The mathematical
1.
Table of
indices (9b),
l
with
In fact,
that
2 represents
in the
energies
the
level
V(ro)
and depends
of the
atom on the
relative the
atom,
accuracy
than
an iron
and the
the
one-electron
from 0.8
to
states
notation
is
the
was assumed*
calculation
physical
energy
(in
correspond
the
that
work function
= 0.
- O.OOl,%, i.e.
of
the
different
electron-volts).
to boundary
of
l’(rg)
was 0.01
the
electrons
electron The
conditions
6 = ~/PO. The strong may be seen.
compression.
abso-
hypotheses.
dependence
compression
a compressed
at zero
on the Thomas - Fermi boundary conditions on
compression
compression
on the
of the
and it
of
130 times,
The atomic potential was based (9a) and (9b) were taken for the
was much higher
levels
of range
(9a)
and
increase
in the
from the
material
184
N.N.
Kalitkin
TABLE
-. -..
-__
--
1.488
lSab 2Sa 2Pb
2Pa 3s)) 3sa 3pb %. 3db 3da r,S, Is,
‘pb +a. 4db 5sb
-
-700.6i,
-687.22
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d6.84 )) -49.42 -46.20 -21.36 -14.fx1
-57u3.38
-6i.30 -65.24 -34.71 -34.42 7.72 14.11 0.54 25.61 7.37 37.13 27.04 57.73
-;0,58 -60.28 -30.43 -29.38 10.45 20.18 3.40 40.55 12:30 54.13 37.26 84.70
11.70
16.89
ii.84
-
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-A.36 -54.31 -26.02 -23.19 13.68 27.66 8.19 6~1.22 19.78 75.73 49.70 118.65 - -___
- .-
23.48
16.43
-
31.87
3sa
3pb +a %b
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11.9.51
Solution
of
eigenvalue
FIfi.
2.
problems
185
N.N.
186
Figure
1 shows
in Angstroms curious
effect
3p-electrons 2s-
(for
the
mean distsnces
the
notation
remote
and Ip-electrons,
from the
but
is
of
the
only
i?
nucleus
(this
observable
At fairly the
high
placed
inside
inner
electrons
mentioned
the
effect.
pressures
compression
At moderate
electrons.
atom by the
from the
from the
nucleus
effect at large
also
holds
for
the
compressions).
/ FIG.
all
electrons
see Table 2, ro is the atom radius). A for moderate compressions the 3s- and
may be observed: are
Kalitkin
pressures compression
nucleus
occurs,
4.
of
the
the
orbits
external
predominates electrons
and a strong which
leads
for are
dis-
of
the
screening to
the
above-
Solution
of
eigenvalue
problems
187
The size of this effect is quite small: it may therefore be due to the model approximations (use of a Thomas - Fermi potential, making the electron density close to the nucleus
conceivably atomic to high).
Figs 2 and 3 respectively To illustrate the influence of compression, give the radial wave functions for an uncompressed (6 = 9.7465) and strongly compressed (6 = 128.4) substance (with boundary condition (9a); for the notation see Table 2; the numbers of the Points of the computa; tional mesh are marked along the axis of abscissae). 2. Suppose that boundary conditions (9a) and (9b) define respectively the upper and lower parts of the Bloch bands of the allowed energies (see [5]), and that the level density in the band is constant (this approximation cannot be used for wide bands; our further discussion is therefore qualitative*).
FIQ.
5. Mean square of orbital momentum: 1 - present paper; 2 - from [91.
Table 2 gives the dependence. calculated Fermi level EF on the compression (regarding see the previous footnote but one. tive.
*
An accurate in [71.
calculation
of the
band
on this assumption, of the the fact that EF is POST-
structure
for
iron
may be found
188
N.A’. Kalitkin
It the
is
clear
Fermi
higher
are
liberated
This not
the
171to
is
band with
increases
atom
It
of
clear
Zs,
2p are
the
bands
electrons
of
the
to
the
orbital
below and
5s
partially
filled
electrons
band with
v = 3.
momentum
on compression and Sp-bands
3d-
in the
always 4d,
on compression,
v = 4 and pass
mean square
(the
Is,
while
that,
monotonically 3.23
bands
filled,
from
filled
and
1.38
for
the
the
Ss-band
filled). Figure
also
5 also
performed
where there gives
higher
thus
the
for
atom,
The
calculation
of
In the high-compression
the quasiclassical < i2 > than the
for the
quasiclassical
[S].
figures
with
sufficiently
Glasko
the of for
of
an iron
not
and V.B.
Zadykhailo
of
figure
coincidence
Ack~o~ledgenents.
Gal’ din
data
potential
more justification
Even for is
the
the
times (the
random). picture
gives
with
is
a 1.5-2
calculation
ing
4.
from the
quantity
the
The numbers
in Fig,
5 represents
isolated
that
completely
filled.
given
Figure
table
and are
are never
bands are
from the
level
for
approach, it quantum-mechanical
compressions
26 electrons,
[91,
region,
the
*
1 is
quasiclassical
accurate.
author sincerely thanks 4.A. Samarskii, V.Ya. for discussion and valuable remarks; also 1.3.
programming
the
computer
and I.A.
Govorukhin
for
perform-
computations. Translated
by D.E.
Brown
REFERENCES 1.
BERG,
ity 2.
R.
and WILETS, L. On the
correction
KANTOROVICH. A.V. spaces
1950.
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term,
1959.
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Uniform
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229 - 239,
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1955.
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SSSR,
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131,
Press). of - 51'7,
Solution
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LANDAU, L.D. nika),
5.
GOMBAS, P.
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dksp.
energy
Quantum
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3,
Une methode quelconque,
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mekha-
Theorie
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Acta
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1958. model
of
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teor.
levels
315
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43, the
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mecanique Bull.
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calculation Fiz.
Druckabhlngigkeit 8,
mechanics
1963.
Phys.
potential,
la9
problems
Quantum-mechanical Zh.
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R. Atomic
GAMBAS, P. Acta
10.
of
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statistischen 321
GANDEL’MAN, G.M.
Fermi 9.
280
eigenvalue
Moscow,
R. Thomas
2,
state a.
Zur
a, 3.
LATTER, 24.
LIFSHITS,
Fizmatgiz,
hung. 6.
and
of
7,
of
them.
.I.
the
equation
131 - 145,
Thomas 2,
des
Drehimpulses
510
of
1952.
- Fermi
99,
Phys.
and
- 519.
Thomas
-
1955. von
Atomen,
1958. de Sot.
calcul r.
des Sci.
vecteurs Liege,
24,
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149
-