Chapter
Two
THOMAS
The
m a y be u s e f u l
were
moving
this
idea very
crude
effective
neral
s t r u c t u r e 1,2
Z
V = - -- +
r
challenge
trostatic
way.
_
V
V2
Ves,
due
part]
n
tool
the
us t h a t
We s h a l l
explicit
that
it
as if t h e y
n o w take
assumptions
V possesses
the
ge-
,
(2-I)
electron-electron
for achieving
this
part
aim
in a
is the
elec-
,
(2-2)
es
relates
b o t h Ves
=
making
It is c l e a r
in f i n d i n g
taught
(or ion)
potential.
however,
V.
fundamental
Chapter
in an a t o m
[electron-electron
equation
4~
which
without,
Poisson
I
preceding
electrons
potential,
consists
The
of the
MOD£L
in an e f f e c t i v e
seriously,
the
consistent
the
independently
about
and the
models
to t r e a t
- FERMI
the
to the
electron
electrons.
and n in t e r m s
density,
As
of V,
soon
n,
to the
as we
shall
Eq. (2) w i l l
electrostatic
potential,
have managed
to e x p r e s s
determine
the
effective
poten-
tial.
General
formalism.
The
independent-particle
H = lp2
The
electrons
all
states
pied,
thus
equals
the
the
energy
with
by the
number
electrons
is c o n t r o l l e d
by the
(2-3)
eigenstates
binding
those
determined
of the
operator
+ V(~)
fill
with
whereas
dynamics
Hamilton
less
larger
binding
requirement
of e l e c t r o n s
of H s u c c e s s i v e l y
N.
than
Just
a certain
energy
that
the
in such
value,
are not.
count
The
a way
{, are o c c u -
parameter
of o c c u p i e d
as in Eq. (I-27)
that
this
~ is
states
is e x p r e s s e d
as
N = tr
where
trace.
we r e m e m b e r
~ (-H-{)
that
the
,
spin mulitplicity
(2-4)
of two
is i n c l u d e d
in the
28
The
The
sum of i n d e p e n d e n t - p a r t i c l e
Eip
= tr H~(-H-~)
combination
function
~,
invites
Eip
which,
with
H+~,
that
appears
rewriting
aid of
is,
analogously,
(2-5)
in the
Eip
= tr(H+~)u(-H-~)
the
energies
argument
of H e a v i s i d e ' s
step
as
- ~tr ~ (-H-~)
,
(2-6)
(4) and the d e f i n i t i o n
E I { tr(H+~)~(-H-~)
(2-7)
,
reads
Eip
In this
E I are
ding
= E I - ~N
equation,
N is the
function(al)s
energy
(2-8)
of the
given
number
effective
of e l e c t r o n s ,
potential
and b o t h
Eip
V and the m i n i m u m
and
bin-
~.
Let us m a k e
contact
with
Eqs. (I-27)
and
(I-32),
in t h a t w e
write
co
E I (~)
= - Sd~ ' N(~')
N(~')
= tr ~(-H-~')
,
(2-9)
where
is the
count
appears
now
of states
with
(2-10)
binding
energy
exceeding
~'.
Equation
as
N = N(~)
Equation
(2-11)
(9) can be e q u i v a l e n t l y
If ~ d e v i a t e s
from
by the
8~,
amount
its
then
~E I
& ~ E I = ~--~ 8~
This
has
the
(4)
important
correct
presented
value
[which
as a d i f f e r e n t i a l
is d e t e r m i n e d
statement.
by Eq. (11)]
E I is off by
= N(~)~
implication
= N&~
that
(2-12)
E i p of Eq. (8) is s t a t i o n a r y
un-
29
der variations
of
8( Eip
~
= 8
In a d d i t i o n
of b o t h
energies
density
n:
first
this
E I - N6(
= 8V E I =
follows
b e r of e l e c t r o n s
and
second
expresses
as t h e t r a c e
derivative
trace
the
of the
of t h a t
operation,
8 H f(H)
Under
the trace
also
of the
we
change
let us
from
need
of the
8H
that
possible
the
response
electron
a formal
because
we r e g a r d
proof.
N is the
as
independent
numof
identity:
(2-15)
of a f u n c t i o n
change
in the
(15)
with
if
The
given
,
commutes
only
local
(2-14)
following
[Note t h a t
f' (H)
The
exhibits
supply
(8),
the
f' (H)
on V.
n(~')
in the t r a c e
product
= 6H
depend
potential
( is a p a r a m e t e r
function.
the
(2-13)
obvious,
= tr 6H
of course):
= o
immediately
unless
value,
f(d~')SV(~')
equality,
8 H tr f(E)
which
correct
to (, E I and E i p
is i n t u i t i v e l y
equality
V. F o r t h e
its
to v a r i a t i o n s
6 V Eip
Although
(around
of an o p e r a t o r
operator,
8H,
H
and the
is not t r u e w i t h o u t
the
~:
[6H,H]
noncommutativity
= o.
does
(2-16)
not m a t t e r . ]
In o u r
application,
f(H)
=
(H+()D(-H-()
(2-17)
f' (H)
[compare
with
= q(-H-()
Eq. (I-29)],
and 8H
= 6V.
Accordingly,
8 v E 1 = tr 8V q(-H-~)
(2-18)
= 2
W e use,
of two
again,
is,
f(d~')
primes
o n c e more,
<r' 18V(~)
<r'lSV(~)q(-H(p,r
to d i s t i n g u i s h
the
numbers
spin multiplicity.
= 8V(~')
<r' I
,
) - ~)I~'>
from o p e r a t o r s ;
Now,
the
factor
since
(2-19)
30
and,
anticipating
that
2<~' I~(-H-~)]~'>
Eq. (18) implies Eq.(14).
representation
all occupied
= n(~')
Indeed,
,
(2-20)
equation
(20) is nothing but the
of the density as the sum of squared wavefunctions
states.
Upon labelling these wavefunctions
gies E' and additional
quantum numbers,
over
by their ener-
~, the left-hand
side of
(20)
is
w
2 ~,
~E,,a(~')
~(-E'-~)~E,,~(~')
: 2~
I~E,,~(~') 12~(-E'-~I
(2-21)
,
E',~
which is recognized
as the usual definition
For consistency,
the integrated
of the density.
density must equal the number
of electrons,
N = I(d~')n(~')
This
follows
immediately
f(d~')n(~')
(2-22)
from Eq. (20):
= 2 f(d~')<~' I~(-H-~)I~'>
(2-23)
= tr n(-H-~)
Another,
and more instructive,
= N(~)
proof makes use of
n in Eq. (14);
(ii) the circumstance
individually,
but only on the sum V+~;
infinitesimal
changes
implying 6Ei=o.
= N
(i) the definition
of
that E I does not depend on V and
(iii) Equation
in ~ and V such that 6V(~)
In view of Eqs. (12) and
(12). Consider
= -5~. 3 Then 5(V+~)=o,
(14) this means
o = 5~ E I + 5 V E I = NS~ + ;(d~')(-5~)n(~')
(2-24)
= 5~ (N- f ( d ~ ) n ( 5 ) )
which is equivalent
to
,
(22). This second proof has the advantage of re-
maining valid when the trace in E 1 is evaluated
no assurance that the densities
in a certain approximation.
definition.
ter,
(We shall,
in Chapter
Four.)
derived
from
approximately.
(14) and
There is
(20) are identical
If they are not, Eq. (14) is the preferable
indeed,
be confronted with this p o s s i b i l i t y
la-
31
Equation
(14) relates
the density to the effective
so that we have taken care of the right-hand
with the problem of expressing
trons,
V
es
potential,
side of Eq. (2). We are left
the electrostatic
potential
of the ele-
, in terms of V.
We proceed
from noting that Eip is not the energy of the sy-
stem.
Just as in the preceding Chapter
65)],
the use of the effective
the electron-electron
[recall the remark after Eq.(I-
potential
interaction
causes
energy,
Vee w h i c h is the e l e c t r o n - e l e c t r o n
Eee. The interaction
part of V in Eq.
given as the response of Eee to variations
8Eee = f(d~')
&n(~')
a double counting of
potential
(I), is naturally
of the density,
Vee(~')
(2-25)
[Please do not miss the analogy to Eq. (14).] Since V and ~ are the fundamental quantities
be regarded
in our "potential-functional
formalism,"
&n(r) must
as the change in the density induced by variations
Some evidence in favor of
electrostatic interaction energy
=
Ees
I
~ ;(d~') (d~")
of V and
(25) is supplied by considering
n(~')n(~")
the
(2-26)
I~'-~" I
'
for which
8E
= /(d~')&n(~')
S(d~")
n(~")
Thus,
Eq. (25) implies
V
the familiar expression
(~') = ;(d~")
es
n(~'")
to the Poisson
The e l e c t r o n - e l e c t r o n
ly contained
,
i÷1
r -r÷,, I
which is equivalent
in Eip("double
tr V
ee
(2-27)
1~'-~"1
es
equation
interaction
(2-28)
(2).
energy,
counting of pairs"),
as it is incorrect-
is
~ (-H-~)
= 2 f(d~')
Vee(~')<~' I~(-H-~)IF,>
= ; (d~')Vee (~')n (~')
;
(2-29)
32
the last s t e p uses Eq. (20). C o n s e q u e n t l y ,
the c o r r e c t
energy expression
is
E = Eip - / ( d ~ ) V e e n
The s e c o n d t e r m r e m o v e s
interaction
contained
+ Eee
the i n c o r r e c t
in Eip,
stationary
6
account
for the e l e c t r o n - e l e c t r o n
and the last t e r m adds the c o r r e c t
The e n e r g y of Eq. (30)
of b e i n g
(2-30)
is e n d o w e d w i t h the i m p o r t a n t
under variations
of b o t h V and
~,
E = 8 v E = o.
In o r d e r to see this,
(2-31)
first
6( - / ( d ~ ) V e e n
amount.
property
appreciate
. Eee )
(2-32)
= - ;(d~)(6Veen
=
which
and
-
+ VeeSn)
+ f ( d ~ ) S n Vee
f(d~)nSVee
is an i m p l i c a t i o n
of Eq. (25). Further,
a consequence
of Eqs. (13)
(14) is
8Eip
= 8~Eip
+ 8 V Eip
= f (d~)nSV
Then,
the c h a n g e
(2-33)
in E is
5E -- S ( d ~ ) n ( ~ V - S V e e ) = f ( d ~ ) n S ( V - V e e )
In v i e w of
[Eq. (I)]
V = " Z + V
r
ee
the v a r i a t i o n
6(V-Vee)
It is u s e f u l
part,
change
Ees'
(2-34)
,
(2-35)
vanishes,
and Eq. (34) i m p l i e s
to s e p a r a t e
E
ee
of Eq. (26), and the r e m a i n d e r
interaction
and p o s s i b l y
Eee = Ees + E'ee
'
other
Eq. (31), indeed.
into the c l a s s i c a l
electrostatic
E'
w h i c h c o n s i s t s of the exee'
effects. A c c o r d i n g l y , we w r i t e
(2-36)
33
and l i k e w i s e
V
ee
= V
+ V'
ee
es
The electrostatic
(2-37)
contribution
w i t h t h e aid of the P o i s s o n
field - ~ V
es
to the e n e r g y
equation
(30) can be r e w r i t t e n ,
(2), in terms
of the e l e c t r o s t a t i c
:
f ( d ~ ) n Ves
+ E
as
=
I
- ~ f ( d ~ ) n Ves
(2-38)
I f(d~) (V2Ves)Ves
= 8--~
[The s u r f a c e
t e r m of the p a r t i a l
for l a r g e r.
] Further,
V
thereby
es
=
V
+
expressing
V'
r
es
integration
w e c o m b i n e Eqs. (35)
_Z _
V
I f(dr)
÷ (~Ves) 2
= - 8-~
is zero,
and
because
V
(37) into
,
ee
N/r
es
(2-39)
in t e r m s of V, as n e e d e d
in
(2). The e n e r g y now
reads
E = Eip -~
I
Z
f (d~) [~(V+ ~ - V e e ) ] 2
(2-40)
- f (d~)n V'ee + E'ee
This
expression
models
[Eqs. (7) and
taken
for the e n e r g y
emerge depending
(8)]
is e v a l u a t e d ,
i n t o account.
lanced treatment
The TF m o d e l .
is our b a s is
u p o n the a c c u r a c y
Of course,
for a p p r o x i m a t i o n s .
to w h i c h the t r a c e
and u p o n the e x t e n t
a consistent
Various
in Eip
to w h i c h E'
is
ee
r e q u i r e s a ba-
description
of both.
The s i m p l e s t
model based
u p o n Eq. (40)
is the TF model.
It
neglects
E'
e n t i r e l y [then V'
also d i s a p p e a r s from (40)]
and e v a l u ee
ee
'
ates t h e t r a c e of Eq. (7) in the h i g h l y s e m i c l a s s i c a l a p p r o x i m a t i o n of
Eq. (I-43).
T h e TF e n e r g y
expression
is t h e r e f o r e
I 2- v - e )
ETF = 2 f ( d ~ ) (d~) i~_p2+V÷~)~l_ yp
- ~N
(2~) 3
I
- 8-~ f (d~)[~(V+
We recognize
the last t e r m as the q u a n t i t y
Z ]2
~)
E 2 of E q . ( I - 6 7 ) ,
(2-41)
w h i c h was
34
there introduced to remove the doubly counted
(electrostatic)
inter-
action energy; the term plays the same role here. The p h a s e - s p a c e integral is the TF v e r s i o n of El, p r o p e r l y denoted by
however,
(EI)TF. We shall,
suppress the subscript TF until it will become a n e c e s s a r y
d i s t i n c t i o n from other models.
The step function cuts off the m o m e n t u m integral at the
dependent)
maximal momentum
P = -/~ (V+ ~ )
(the so-called
(r-
"Fermi momentum")
,
(2-42)
so that
P
(2~) 3
0
(2-43)
rg2
or,
square roots of negative arguments being zero,
E I = f(d~) (-
I ) [-2(V+~)] 5/2
15~ 2
(2-44)
This is the T h o m a s - F e r m i result for ~I" The entire e n e r g y functional in
the TF model is then
ETF = E I + E 2 - ~N
(2-45)
= f(d~)(-
I )[_2(v+~)15/2
a~f(d~)[~(V+
Z
~)12_~
.
15~ 2
Is there any reality to it? Yes. Look back to Chapter One, w h e r e
has been used u n c o n s c i o u s l y
situation,
E 2 equals zero,
[see Eqs.(1-26)
through
for the Coulomb potential V=-Z/r.
(45)
In this
and ETF gives the leading term of Eq.(I-22)
(I-37)]. Since V is e s s e n t i a l l y equal to the
C o u l o m b potential in a'highly ionized atom, we conclude
ET F
m
-
ZZ(~N) I/3
for
N << Z
(2-46)
We shall return to highly ionized systems in a while and find the modification of
(46) w h e n a c c o u n t i n g for the e l e c t r o n - e l e c t r o n repulsion.
Before doing so, we have to study some implications of Eq. (45).
The s t a t i o n a r y p r o p e r t y of ETF with respect to v a r i a t i o n s of
V and ~ reads
35
o = 6ETF = f(d~)6V{
+ 6~{
I 2 (V+ r)
z }
I [_2 (V+~)13/2 + ~-~V
3rL2
](d~)
I [_2(V+~)]3/2 _ N}
3~2
- --I f(d~)~.
4~
(2-47)
(~v ~(v+
Z
~))
The v a l u e of the last integral is zero, because the e q u i v a l e n t integration over a remote surface vanishes
tions of V and ~ are independent,
zero individually.
in view of 6V=o for r+~. The varia-
so that the two curly b r a c k e t s
equal
Accordingly,
I V~(V+ z
~)
=
4K
I [_2(V+~)]3/2
3~2
(2-48)
and
f(d~)
I [-2 (v+~)]3/2 = N
3~ 2
,
(2-49)
of w h i c h the first is the Poisson equation,
zation of the d e n s i t y to N. Obviously,
and the second the normali-
Eq. (49) is the TF v e r s i o n of
(11),
as we n o t i c e that Eq. (I0) is realized as
N(~ , ) = 2 2:(d~) (d~)
(2~) 3
This,
1 2_V - ~') = ;(d~)
N(- ~p
inserted into Eq. (9), reproduces
On the r i g h t - h a n d side of
of
(44),
I [-2(V+~ , )]3/2
3~2
(2-50)
as it should.
(48) as well as under the integral
(49) we have the TF d e n s i t y
I
n = --~'--[-2(V+~) ]
3K 2
3/2
In the c l a s s i c a l l y forbidden domain,
vanishes.
(2-51)
c h a r a c t e r i z e d by V>-~,
this d e n s i t y
There is a sharp b o u n d a r y assigned to atoms in the TF model.
In contrast,
in an exact q u a n t u m m e c h a n i c a l d e s c r i p t i o n the t r a n s i t i o n
from the c l a s s i c a l l y allowed to the c l a s s i c a l l y forbidden region is
smooth. We have just learned about one of the d e f i c i e n c i e s of the TF
model.
It is going to be removed later w h e n we shall i n c o r p o r a t e q u a n t u m
corrections of the sort d i s c u s s e d b r i e f l y after Eq. (I-43).
The d i f f e r e n t i a l
equation
(48) for V, known as the TF e q u a t i o n
for V, is s u p p l e m e n t e d by the c o n s t r a i n t
(49) and the short d i s t a n c e be-
36
havior
of V,
r V ÷ - Z
It s i g n i f i e s
tial
the
is m a i n l y
tron with
ness
the
for
r + o
physical
given
requirement
nucleus;
formally,
is large
ses the p o t e n t i a l
square
root
in
ty is zero,
law,
with
V =
and the
radius
--
negative,
less
region,
negative;
Z-N
r
for
r o of the
V ( r = r o)
and
= - ~
it e q u a l s
beyond
(52),
to
is
and the
Poisson
the
finite-
for s m a l l
large;
distance,
r,
as r i n c r e a at the e d g e
argument
ro,
of the
the d e n s i -
equation.
Gauss's
implies
,
=
poten-
of an e l e c -
ensure
finally,
-~,
this
then
r > rO
atom
the e f f e c t i v e
energy
situation:
less n e g a t i v e ;
is the h o m o g e n e o u s
Eqs. (49)
~
following
and the d e n s i t y
and
turns
(48)
for r÷o,
potential
is n e c e s s a r y
the
allowed
(51)
so t h a t
combined
(52)
we have
becomes
of the c l a s s i c a l l y
that
by the e l e c t r o s t a t i c
of E 2. C o n s e q u e n t l y ,
the p o t e n t i a l
(2-52)
(2-53)
is d e t e r m i n e d
by
,
(2-54)
or,
Z-N
-
r
The
electric
an atom);
field
-~V
is
in p a r t i c u l a r ,
d
Neutral
at the
o
do not h a v e
It is u s e f u l
just
x
of w h i c h
= Z1/3r/a
have
~=o,
have
learned
to m e a s u r e
charged
surfaces
in
V+~
b o t h V and d V / d r
Eq.(48),
a solution
that
requires
satisfying
neutral
TF atoms
only
"inside".
an
as as m u l t i p l e
a function
these
are
vanish
ro=~,
bounda-
infinite-
of t h e p o t e n t i a l
f(x),
,
is r e l a t e d
,
so that
for V,
an "outside!,,
by introducing
V + ~ = - Z f(x)
r
argument
N=Z,
r e it c a n n o t
ly large,
the
no
edge w e h a v e
the TF e q u a t i o n
We h a v e
of the n u c l e u s
are
r o
systems,
ry c o n d i t i o n s .
they
(there
(2-56)
r 2
o
at r=r o. C o n s e q u e n t l y ,
for a f i n i t e
continuous
Z-N
v (r) I
r
since
(2-55)
o
(2-57)
to the p h y s i c a l
I~3~%2/3
a = ~,-~,
distance
= 0.8853...
r by
(2-58)
37
The
constant
a is c h o s e n
d2
called
dary
f (x)
=
f(o)
= I ,
q,
(58).
f(x)
for
f(x),
(52),(54),
introduces
to r ° t h r o u g h
notice
and
f(x o)
that
Consequently,
tion,
q,
so that
for
f(x)
is s o l e l y
all ions
with
Z;
served
the
for
f(x),
through
the
The
the
same
in C h a p t e r
shrinking
One,
translate
N
= I - ~
ionization.
Of course,
now
as
appears
appear
The boun-
- q
,
(2-60)
x ° is r e l a t e d
(2-61)
individually
determined
same
Z/r,
x of Eq.
in Eqs. (59)
b y the d e g r e e
q possess
potential
of h e a v i e r
when
factors.
into
d
-Xo d--~ f(xo)
o
factor
of the TF v a r i a b l e
of n u m e r i c a l
x > x°
Z and N do not
depend
implies
of
(53)
and of t h e d e n s i t y .
first
(56)
the d e g r e e
Equation
potential
on
equation
(2-59)
is free
= o ,
= q ( 1 - x / x o)
(60).
Z dependence
differential
,
I/2
t h e TF e q u a t i o n
Please
the
x
2
conditions
which
that
[f (x) ] 3/2
-
dx
such
a common
V itself
but
(58).
atoms
then
The
also
Bohr
atoms
Fig.1
a sketch
with
of the
of course,
because
factor
that we have
considering
shape
does,
and
of i o n i z a -
of the
Z I/3 t h e r e
already
shielding,
obsee
Eq. (I-93).
For illustration,
for w h i c h
(60)
x ° ~ 3. T h e
geometrical
shows
significance
of
f(x)
of the
for q = I/2,
third
equation
in
is i n d i c a t e d .
1
Z ....
±
)X
0
1
I,
2
Xo
4
Fig. 2-I. Sketch of f ( x ) for q = 1 / 2 .
Neutral
T F atoms.
For the
solution
of Eqs. (59)
and
(60)
that
belongs
to
38
q=o,
we write
F(x)
and call
F(x)
=
d2
= I
slope
F(x)
additive
tely write
with
the main
additional
The
the known
=
simultaneous
has no e f f e c t
significance.
We
-
for
body
the change
energy
6ETF
x << I
,
insert
(2-64)
(64)
into
(57),
r ÷ o
is t h e i n t e r a c t i o n
charge
of that
for
_- _ Z + B Z4/3
r
a
down
(2-63)
use
at
of the nuclear
nect with
= o
B,
constant
the nucleus,
F(~)
physical
arrive
V(r)
The
,
= I - B x + ...
an important
and
(2-62)
,
to
F(o)
(58),
It o b e y s
x ~/2
and is s u b j e c t
has
function.
[F (x) ] 3/2
dx 2
Its i n i t i a l
it t h e T F
energy
of electrons.
in energy
Z to Z + 6Z.
charge,
energy,
(2-65)
which
caused
o f an e l e c t r o n ,
We
c a n u s e it t o
where
a minus
is t h a t
immedia-
by an infinitesimal
It is t h e a n a l o g o u s
sign
near
change
electrostatic
is n e e d e d
to c o n -
of an e l e c t r o n :
a Z 4/3 6Z
~
increase
(2-66)
of t h e n u m b e r
on the energy
since
of electrons
~E/~N
from N=Z
= - ~ = o f o r N=Z,
to N = Z + 6 Z
see E q . ( 5 5 ) .
Consequently,
-ETF
This
is t h e T F
The
= 3 B Z7/3
7 a
formula
constant
B is w e l l
the results
of a numerical
to
approximations
the
crude
result
binding
known
integration
that
led to t h e T F m o d e l ,
obtained
of k n o w i n g
is g i v e n
energy
of Eqs. (62)
an e s t i m a t e
estimate
(2-67)
numerically.
find
of t h e d e v e l o p m e n t ,
first
N = Z
for the total
our
insight
for
f o r B. I n d e e d ,
B better
there
in t h e m o d e l
of Bohr
atoms.
But before
and
(63),
quoting
let us u s e
in v i e w o f t h e p h y s i c a l
is no need,
than within
by the comparison
of n e u t r a l
of
atoms with
at t h i s
a few percent.
(67) w i t h
shielding:
stage
A
(I-51),
39
B ~ 7a 9 3 I/3
3 14(2 )
We have
no w a y
see
later
des
a tool
The
for a t r i a l
error
stationary
potential
has
V gives
B
where
the
Maximum
denoted
simal
by
good
V and
As we
an u p p e r
< -7a
property
from
for the
only
correct
A~,
be,
correct
functional
but
shall
value.
(45)
If w e e v a l u a t e
followina
for
section,
potential.
constant
,
provi-
ETF(V,~)
in
energy
any
,
(2-69)
B:
~=o and V = - ( Z / r ) F ( x ) .
functional.
V and
Let
the
us
correct
as d i s t i n g u i s h e d
Whereas
the
N=Z),
Consequently,
(for N = Z)
potential
respectively,
6V and 6~.
energy
for B.
correct
ETF(V, ~)
holds
may
the
(this m u c h w e k n o w for sure w h e n
3 B Z7/3
-7 a
w i l l be of s e c o n d o r d e r
of the TF p o t e n t i a l
the
by AV and
variations
bound
number
5% from
of the
see in the
for the
Z -7/3
sign
from
(2-68)
this
than
estimates
~ = o
shall
= 1.52
accurate
less
property
a maximum
equal
deviations
how
of E T F ( V , ~ = o )
of V.
functional
trial
judging,
it d e v i a t e s
for o b t a i n i n g
the d e v i a t i o n
the
of
that
= 9 ~
8 (-)2/3
A~ is q u i t e
consider
value
from t h e
arbitrary,
finite
for
~,
infinite-
AV is s u b j e c t
to
r AV ÷ o
for
r ÷ o
,
AV + o
for
r + ~
,
(2-70)
which
are
viations
consequences
of the
three
of
terms
AE I = S(d~ ) (_
(52)
and the
of ETF
in
normalization
(45)
I ) ([-2(V+AV+
15~ 2
V(r ÷~)=o.
T h e de-
are t h e n
~+A~)] 5/2
- [ - 2 ( V + ~ ) ] 5/2)
, (2-71)
and
AE2
1
÷
= - 8--~ f(d~) ([V(V+AV+
Z
2
~) ] - [ 7 ( V +
Z
~)]
2
)
(2-72)
1
z
8~
S(d~)[~(Av)] 2- -~1 /(d~)~IAV)-~iv+ ~)
which
after
a partial
integration
and the
use
of Eq. (48)
reads
40
AE2 =
as w e l l
8~ ;(d~) [~(AV)] 2_ S(d~)A V
(2-73)
as
. . (A~)N
. .
A (- ~N)
where
I [-2(V+~)]3/2
3rL2
S (dr)
÷ A~
Eq. (49) has b e e n e m p l o y e d .
I [-2 (V+~) ] 3/2
3~ 2
(2-74)
Accordingly,
AETF = AE I + AE 2 + A(-~N)
= S(d~ ) (_ 3
15~ 2
){[_2(V+~)_2(AV+A~)]5/2_[_2(V+~)]5/2
+ 5(AVeA~) [ - 2 ( V + ~ ) ] 3 / 2 }
(2-75)
I
- 8--~ S (d~) [~(AV)]2
The contents
of the c u r l y b r a c k e t s
[U+V]5/2 - u5/2 - ~ 5v
is of t he s t r u c t u r e
u3/2
(2-76)
4
dv' (v-v') [u+v' ] I/2 > 0
,
O
where
u = -2(V+~)
if v = o, or,
which
and v = - 2 ( A V + A ~ ) .
if u+v'
circumstance
AETF
In words:
~ o
;
a minimum.
same sense,
property
the e l e c t r o n s
that we compare;
functional
True,
it is c o m m o n
might
to a r r a n g e
implies
and
A~ = o
AV = o
of Eq.(45)
only
(under
(2-77)
has an a b s o l u t e
come as a surprise,
themselves
maximum
it is n a t u r a l
electron
to m a x i m i z e
(and c l o s e l y
distributions
potentials.
for the right d e n s i t y
for the r i g h t p o t e n t i a l
as one n a i v e -
such that the e n e r g y
is a m o n g d i f f e r e n t
s t r a t e this p o i n t by the a n a l o g o u s
electrostatics.
This
but it is not d i f f e r e n t
the c o m p e t i t i o n
in w h i c h
(76) holds
range of i n t e g r a t i o n
root v a n i s h e s ) .
o n l y for
sign in
~.
This m a x i m u m
ly e x p e c t s
energy,
= o
the TF p o t e n t i a l
at the c o r r e c t V and
achieves
o v e r the w h o l e
the s q u a r e
S o
The e qual
related)
In the
to m i n i m i z e
the
it. Let us illusituation
in
41
An e l e c t r o s t a t i c
static
analogy.
potential,
are related
Consider
~, to a given
the p r o b l e m
of finding
charge density,
to each o t h e r by the Poisson
the electro-
p, in the vacuum. 4
_ I
V2 ~ = P
4~
The e l e c t r o s t a t i c
I
=
(2-78)
energy
J' IdOl p{
can be e x p r e s s e d
in various
ways:
f IdOl I {I 2
=
1
+
= [(d~)[p~ - ~-{(q~,) 2]
If we insert
the
~ that obeys
(2-79)
(78) into
any of these
all give the same answer.
Suppose,
however,
rect
to using
an a p p r o x i m a t e
{ and have to resort
it is a d v i s a b l e
energy,
to employ
because,
at the correct
unlike
deviation
close,
Here
I
In this
situation,
(79) in c a l c u l a t i n g
expression
the
is s t a t i o n a r y
electrostatic
potential
results
for the right
(2-81)
~. The analogy
since the same t e r m occurs
application
"potential
the e n e r g y
in
is given by
l(dr) IV(A~)I 2 < 0
is a little
the e l e c t r o s t a t i c
we e v a l u a t e
they
(2-80)
A# from the right
is m a x i m a l
indeed,
of
this
one.
(~) 2]
error in E that
AE = -
is,
expressions,
that we do not know the cor-
{:
the second o r d e r
the energy
the third v e r s i o n
the o t h e r ones,
1
A finite
They
equation
also
to the TF f u n c t i o n a l
in
(75).
of the s t a t i o n a r y
functional. "5 Instead
property
of i n s e r t i n g
of
%(~),
for ~(I~) :
(2-82)
For
I=I,
it is the correct
energy.
Consequently,
42
d E(1)
d--~
= o
for
I=I
(2-83)
w h i c h implies
I
8-{ f Ida) I~) ~ -- ; Ida) p~. I-~®)
(2-84)
We have thus found an unusual e x p r e s s i o n for the e l e c t r o s t a t i c energy:
the integral of the scalar product of the dipole d e n s i t y
electric field - ~ .
Note,
in particular,
Since a t r a n s l a t e d charge d i s t r i b u t i o n
pr w i t h the
that there is no factor of I/2.
p(r+R) has the same e l e c t r o s t a -
tic energy,
;(d~) p(r+R)+
÷ +r • ( - ~ ( ~ + R ) )
+
+
= f(d~)p(~)~. [-~#(~)]
,
(2-85)
on the left hand side,
that the self
+
we find, after s u b s t i t u t i n g r÷r-R
force of any charge density vanishes:
/(d~)p(-~#)
(The stresses,
=o
of course,
(2-86)
do not.)
A d i f f e r e n t p r o b l e m is that of finding the correct charge density on the surface,
In this situation,
/dS'
S, of a c o n d u c t o r carrying a given total charge,
Q.
the relevant equations are
a (~')
_
const,
for
~ on S
,
(2-87)
i~-~' I
and
;dS a(~)
= Q
,
(2-88)
w h e r e a denotes the surface charge density.
Here the s t a t i o n a r y energy
e x p r e s s i o n is
E = ~
~/dSdS'
~d
o(r)a(r')
fir_r,
I+
+
#o(Q-;dSa(~))
The last term incorporates the constraint
of both a and ~o
the
(constant)
imply Eqs. (87) and
(88).
(2-89)
Infinitesimal variations
(88), t h e r e b y i d e n t i f y i n g ~o as
e l e c t r o s t a t i c potential on S. This energy is a m i n i m u m
if only a's o b e y i n g
(88) are allowed in the competition,
ous d i s t r i b u t i o n s of the same, given,
i.e.,
if vari-
amount of charge are compared.
43
W e get
As = 1j'dSdS'
where
Ao
Ao
(~)
is t h e d e v i a t i o n
the e l e c t r o s t a t i c
Ao ( ~ ' )
2-90)
from the o p t i m a l
density
o. Since this
e n e r g y of some c h a r g e d i s t r i b u t i o n ,
it is,
is
indeed,
positive.
TF d e n s i t y
raises
functional.
the q u e s t i o n
d e n s i t y,
It r e q u i r e s
is r e p l a c e d
electrostatic
into the r e a l m of e l e c t r o s t a t i c s
to w r i t e
to the p o t e n t i a l
on the energy,
indeed.
potential
if it is p o s s i b l e
in a d d i t i o n
upper bounds
done,
This d i g r e s s i o n
down a functional
functional
of
l o w e r ones on the c o n s t a n t
appropriate
in t e r m s
rewriting
of the d e n s ity.
of
of the
(45), thus g e t t i n g
B. This
can be
(45), w h e r e b y
the
B o t h Eq. (51) and the
relation
÷
V(r)
Z
] ÷
= - ~ +
(dr')
n(r')
I~-~' I
(2-91)
can and m u s t be u s e d in this p r o c e s s .
We s t a r t by u n d o i n g
E I is split
i n t o the k i n e t i c
E 1 = J" ( d ~ )
=
t
( d r+)
the s t e p
energy,
from Eq. (43) to Eq. (44),
Eki n, and a p o t e n t i a l
I [-2(V+~) ]5/2 _ ;(d~)
1 01t 2
E 2 is r e w r i t t e n
b y first p e r f o r m i n g
use of the P o i s s o n
1 [-2 (V+~)]I+3/2
6rc 2
I (3T~2n)5/3 + /(d~) (V+6)n
I 0T~2
÷
Z
÷
= Eki n + ; (dr) (V+ r ) n - [ (dr)
equation,
(2-92)
Zn
÷
r + {; (dr) n
a partial
followed
integration,
by e m p l o y i n g
Eq.
I
Z
I
Z)
I
E 2 = - ~-~ ;(d~) [~(V+ 3) ] 2 = 2;(d~) (Ve r ~ V
I
= - ~ f (d~)(V+
Z
~
~)n(r)
I
= - ~f(d~)(d~'
the two
last v e r s i o n s
into
then making
(91):
2
Z
(V+ 3)
)n(~)n(~')
+ +,
Ir-~ I
Combining
so that
e n e r g y part:
(2-93)
44
I
n(~)n(~')
E 2 = ~ f ( d ~ ) (d~')
;(d~) (V+~) n
l ~-~' I
(2-94)
makes the p o t e n t i a l d i s a p p e a r from the sum of E I and E 2. The r e s u l t i n g
TF d e n s i t y functional is
E = E I + E 2 - cN
1 -(3~2n)5/3
1 o~ ~
f(d~)
- ~(N-
-
;(d~)Zn
r
+
~-i(d~)(d'~')
n(~)n(~')
[~-~' t
;(d~)n)
(2-95)
All we know at this stage is that Eq. (95) gives the correct value of
the energy, p r o v i d e d we insert the correct density.
To be useful this
functional has to be s t a t i o n a r y about the right density. Not surprisingly, it is:
I
Z + ;(d~')
n(~')
+
:]
rr-r' I
(2-96)
-
w h i c h uses Eqs.(51)
V(7)
.
and the c o n s t r a i n t
and
6~
(N
-
f(d~)n)
=
o
(91) in the c o m b i n a t i o n 6
.I
.2 ÷. 2/3
.
~ ( 3 ~ n(r))
.
~
Z + f (dr)
÷
r
n(r')
Ir-r' I
(2-97)
(49), now reading
(2-98)
f (dr)
÷ n = N
The s u c c e s s i v e terms in Eq. (95) have the p h y s i c a l significance of the k i n e t i c energy•
the electrons•
the p o t e n t i a l energy b e t w e e n the nucleus and
and the e l e c t r o n - e l e c t r o n potential energy. The last term
incorporates the constraint
(98), thereby i d e n t i f y i n g ~ as the corres-
p o n d i n g L a g r a n g i a n multiplier.
In contrast,
the p o t e n t i a l functional of
Eq. (45) consists of the sun% of independent p a r t i c l e energies•
EI-~N,
plus the removal of the d o u b l y counted e l e c t r o n - e l e c t r o n - i n t e r a c t i o n
energy, E 2. It is important to a p p r e c i a t e this d i f f e r e n c e in structure.
Let us now check if the density
functional does have the ex-
pected p r o p e r t y of being m i n i m a l for the correct n and ~.
45
Minimum property of the TF density
vious d i s c u s s i o n
we consider
functional.
In analogy to the pre-
of the m a x i m u m property of the TF potential
finite deviations
A( is quite arbitrary,
functional,
An and A( from the correct n and ~. Again,
whereas
An is restricted by the requirement
that
the density be non-negative,
n + An
> o
The d e r i v a t i o n of
been implicitly
for all r
(2-99)
(95) made use of
(51) so that negative densities
had
excluded.
The various
contributions
AEki n = f(d~)
to AE are then
(3~2)5/3 [ (n+An) 5/3 - n5/3]
I 0~ 2
,
(2-I 00)
and
A(- ;(d~) Zn + 1 ; ( d ~ ) (d~')
r
2
n(~)n(~'))
Ir-r' J
Z
= ;(d~)An(~) [- r + f (d~')
+ ~I [ Cd~) Ida')
= ;(d~)
]
An (7) An(~')
(3~2)5/3[ - 5 2/3.
~n
n(~')
,
An] - (;(d~)An
I 0~ 2
(2-101)
I
An(~)
An(~')
[r-r' I
which uses Eq. (97), as well as
A[- ( ( N -
;(d~)n)]
= (( + A() ;(d~)An
(2-102)
Consequently,
AE = fld~)
(3~2) 5/3
[ (n+An)5/3
1 0~ 2
+ ~ [ (dr) (d~')
-
n5/3 - ~n
5_2/3•AnJ
An(~)÷ +An(~')
lr-r'f
+
46
+ AC [(d~-)An
(2-103)
The first term here is p o s i t i v e definite,
we write it
[compare Eq. (76)]
w h i c h becomes obvious w h e n
in the form
An
S(d~ ) {3E2) 5/3 10 Sd~(An_v ) (n+v) -I/3
9
10~2
0
~o
;
= O
The second term in
sity £n(~),
only if
(103)
£n(~)=o
(2-I04)
for all
is the e l e c t r o s t a t i c energy of the charge den-
thus it is also positive,
unless
An=o
everywhere.
The third
term does not have a definite sign. T h e r e f o r e we restrict the class of
trial densities n and trial
~'s such that
A~ S(d~)An = o
(2-105)
Then
AE
> o
;
= o
only for
An(~)=o
for all
~
;
(2-106)
the TF d e n s i t y functional of Eq. (95) has an absolute m i n i m u m at the correct density,
p r o v i d e d Eq. (I05) holds.
In general,
trial densities
satisfying
(105) will mean to c o n s i d e r only such
that obey the constraint
(98),
since then
S(d~)An = o
(2-107)
The m a i n e x c e p t i o n are neutral atoms,
sequently,
Eq. (I05)
trial values
about w h i c h we k n o w that ~=o. Con-
for ~ need not be chosen,
so that A~=o. Then
is satisfied without r e s t r i c t i n g the density according to
(107).
This o b s e r v a t i o n will prove useful, when seeking lower bounds on the
constant B.
Upper bounds on B. We pick up the story at Eq. (69). The c a l c u l a t i o n is
c o n s i d e r a b l y simplified by employing the TF variables x, x o, and f(x),
which have been introduced in Eqs. (57) through
potential functional appears as
E = -
2
(Z7/3/a) {~
[f' (x) E~xf(X)]
co
;dx If(x)]512
I/2
0
(60). In these,
X
co
+
I Sdx[f'(xl÷
0
J2+
the TF
47
(2-108)
+ q !!-q)_ }
Xo
where,
replacing
Whereas
V and
arbitrary
~, it is n o w
variations
f(x)
of x o m a y
and x o t h a t
have
be c o n s i d e r e d ,
to be
f(x)
found. 7
is s u b j e c t
to
f(o)
= 1
(2-I 09)
and
f' (x)
The
first
of t h e s e
~, in Eq. (57),
ensure
the
the trial
= - -~xo
[see(60)].
into
functions
the
of t h e
do not
of
second
have
,
(2-110)
second
the definition
= o
This,
x ÷~
is Eq. (52),
finiteness
f(x o)
for
f(x).
In
integral.
from the
(108)
Note,
inclusion
it is n e e d e d
in p a r t i c u l a r ,
-Xof' (Xo)
= q
to
that
(2-111)
equation
[f(x)]3/2
I/2
=
of
to o b e y
and t h e d i f f e r e n t i a l
f" (x)
comes
(2-112)
X
[see (59)]
is h o w
are i m p l i c a t i o n s
it works:
of the
infinitesimal
stationary
variations
of
property
f(x)
of
cause
(108).
a change
Here
in E,
o = 6[-(a/Z7/3)E]
8fix){EfIX)f312
I/2
-
co
0
co
f" Ix) } +
X
;dx
£ {6f(x)
[f' ( x ) +
1}
0
(2-113)
oo
=
;dx 5f(x) {
o
where
the
first
uses
(109)
and
differential.
clude
that
equality
(110)
Thus
beyond
[f (x) ] 3/2
i/2
is the
in f i n d i n g
(112)
-
f" (x) }
x
stationary
the
is implied.
a certain
property
null value
We combine
(yet u n s p e c i f i e d )
and the
of the
~,
it w i t h
f(x)
last
integrated
(110)
one
total
to con-
is n e g a t i v e
and
linear:
f(x)
X--X
= q Xo
for
x >_- ~
(2-114)
48
Next,
we
consider
variations
of x o.
They
produce
o = 8[-(a/Z7/3)E]
co
,
(2-115)
o
implying
of
the vanishing
(114),
the
of the
integration
contents
stops
o = fdx[f' (x)+ ~q]
o
x°
+
of the
curly
brackets.
In v i e w
at x:
(l-q)
(2-116)
= f(~)
the
last
that
- f(o)
step makes
(114)
+ q~-~ + I -
use of
becomes
(61)
Let us n o w t u r n
of the
functional
binding
energy
(109)
and
(108),
and
where
the
together
of X o = ~
is a m o n g
equal
with
for
to n e u t r a l
trial
f(o)
and o b t a i n
the
right
functions
= I
q=o.
The m a x i m u m
property
form of the
neutral
to
that
f(x)=F(x).
need
(109)
subject
we can
(~>o)
function.
to use
and
f' (x ÷ ~)
,
trial
for
are
f(x),
÷ f(~x)
,
(2-117)
(110),
Note
that
explicitly
0
[f~x)]5/2
1/2
x
x o disappeared
our knowledge
the c o m p e t i t i o n
in
(117)
to
= o.
always
(2-118)
change
the
scale,
,
(2-119)
The
optimal
choice
of
= g
atom
o
w e do not
,
3 B < 2 fdx
7
only
According
side
so
the k n o w n
I fdx[f' (X)] 2
I/2 +3
holds
so that
another
hand
x = x o,
reads
x
For any trial
f(x)
see t h a t
co
sign
q,
atoms,
with
[f(x)]5/2
o
q=o.
N o w we
(111).
co
2 fdx
7B--<~
(114).
implies
combined
, Eq. (67),
3
q = q (~-~ -I)
I
+ y ;dx[
0
f(~x)l 2 :
for
~ minimizes
40
oo
-I/2 2 Sd x
5
o
[f(x)] 5/2 + ~ g I Sd x[f, (x)]2
112
o
x
(2-120)
It is given by
3/2
= 50
We insert it into
2 -
(210)
(2-121)
dx[f' (x)] 2
and arrive at the scale invariant v e r s i o n of
(117):
7
fax [f (x)15/212/3
;dx[f' (x) l ~
o
o
x I/2
w h e r e now the equal sign holds
]
for f(x)=F(~x)
(2-122)
w i t h a r b i t r a r y ~>o.
We are now ready to invent trial functions and produce upper
bounds on B. Before doing so, let us make a little observation.
equals F, the optimal ~ in
holds only for f(x)=F(x).
in
(121)
is
(3/7)B. We c o n c l u d e
(120)
Consequently,
are equal for f=F.
2 Sd x
o
is unity,
(117)
the n u m e r a t o r and d e n o m i n a t o r
In this s i t u a t i o n the related sum in
[F(x)]5/2 _ 2 B
x I/2
-
If f
since the equal sign in
,
(117)
(2-123)
and
co
I fdx[F' (x)]2 = "~
BI
(2-124)
o
An i n d e p e n d e n t
(and rather clumsy)
the d i f f e r e n t i a l
partial integrations.
Equations
(123)
i m m e d i a t e check of the e q u a l i t y in
7
B = ~
d e r i v a t i o n of these equations uses
e q u a t i o n obeyed by F(x)
(~-
I/2 2
2/3
7 B)
More about relations
like
on the scaling p r o p e r t i e s
(123)
[Eq.(62)],
and
can be employed
for an
(122) for f(x)=F(~x) :
2
1/3
(~ 7 B)
and
(124)
combined w i t h some
(2-125)
(124) will be said in the section
of the TF model.
A v e r y simple trial f u n c t i o n is
f(x)
I
= i;x
"
(2-126)
50
for w h i c h
oo
2
co
of
dx
[the i n t e g r a l ,
[f(x)]5/2
xi/2
in t e r m s
= 2 fdx x - I / 2 ( 1 + x ) - 5 / 2
5 o
of E u l e r ' s
co
Beta
function,
15
:-8
(2-127)
I
is B(~,2}
4
= ~],
j-
and
co
;dx[f' (x)] 2 = f
o
o
dx
(1+x) 4
=
I
3
--
(2-128)
Accordingly,
B < ~
A better
value
5 -2/3
= 1.596
is o b t a i n e d
(2-129)
for
r
f (x) =
(~]__)4/3
,
(2-130)
I ~ X
when
B < 2-19/9
This
number
is no p o i n t
Lower
bounds
in t e r m s
is,
(--737--)I/3
as w e
shall
in c o n s i d e r i n g
on B.
=
see,
more
In o r d e r
of TF v a r i a b l e s ,
n(~)
-
[~ (21 :]2/3 [(1) :]-4/3
close
to e x p r e s s
to the
trial
the d e n s i t y
1.5909
actual
(2-131)
one;
so t h e r e
functions.
functional
of Eq. (95)
we w r i t e
I
V2 (V+ ~)
Z
I
I d2
~-~
very
complicated
=
Z
[r(V+ ~)]
4~ r dr 2
I
I
d2
- -
(2-132)
(zf(x))
4~ r dr 2
=
or,
more
I Z 2 f" (x)
4n a 3
x
conveniently
here,
'
51
I
n
The
function
constant.
g(x),
constant
= f' (x) + -qxo
in v i e w of
1110)
g(x ÷~)
With
-/(d~)
and
to r e q u i r i n g
(135) we h a v e
for the i n t e r a c t i o n
Z7/3 ~
Z7/3
a
Idx g' (x) = a
g(o)
o
~r n
equals
(108), g i v e s
(2-134)
e n e r g y bet-
and the e l e c t r o n s
energy
n(~)n(~')
÷ ÷,
Ir-r I
ty f u n c t i o n a l
by a
(2-135)
I f(d~) (d~')
of
from f' (x), at most,
,
is e q u i v a l e n t
the electron-electron
[This q u a n t i t y
differs
to be q / x o,
= o
(133)
w e e n the n u c l e u s
whereas
(2-133)
thus i n t r o d u c e d ,
We choose this
g(x)
which
Z 2 g' (x)
a
-E2,
this result.]
(2-136)
is
_ z 7/3 I fdx[gix)] 2
a
2
(2-137)
o
so t h a t Eq. (134),
c a n be e x p r e s s e d
,
The remaining
in t e r m s
used
in the s e c o n d
contributions
of g(x)
integral
to the d e n s i -
immediately.
We a r r i v e
at
E -
Z7/3{ 3 ;dx x I/3
5/3
a
5
[g' (x)]
+ g(o)
o
+
I ;dx[g(x)] 2
o
(2-138)
co
- --q-It-q- fdx x g' (x)]}
xo
o
Again
arbitrary
variations
ted by the r e q u i r e m e n t
g' (x) > o
of x o m a y be c o n s i d e r e d ,
of n o n - n e g a t i v e
,
and b y Eq. (135). T o g e t h e r ,
whereas
g is r e s t r i c -
densities,
(2-139)
they imply
52
g(x)< o
(2-140)
A c c o r d i n g to the discussion
upper bounds on the energy,
of Eq.(103),
Eq.(138)
provided that Eq. (I05)
supplies
is obeyed.
Expressed
in terms of x o and g, it reads
A(~)
"'O
fdx x ~g' (x) = o
o
We did notice already
[see the remark following Eq.(107)],
situation of neutral atoms,
so that
(141)
(2-141)
our knowledge
is satisfied without
of Xo=~ results
further ado.
that in the
in ~ ( ~ ) = o ,
In particular,
g~x)
need not be subject to
; dx x ~g' (x) = o
o
or
,
(2-142)
[this is Eq. (98)], more precisely,
oo
fdx x g' (x) = 1-q ; = I
for
q=o
(2-143)
o
The m i n i m u m property of the functional
the known form of the neutral
atom
(138), together with
(q=o) binding energy,
Eq.(67),
im-
plies
3 B ~ - {~
3
;dx x I/3 [g' (X)] 5/3 + g(o) + ~I fdx[g(x)]
o
2}
,
(2-144)
o
where the equal sign holds only for g(x) = F' (x). For this g(x), the
3
I
value of the two integrals is ~B and ~B, respectively, as follows from
Eqs. (123) and
(124), and the differential
equation
(62) obeyed by F(x).
Consequently,
F' (o) = -B
,
(2-145)
which is nothing more than the original definition
As in the p r e c e d i n g
scale.
Here the possible
g(X) ÷ ~ig(~2 x)
section,
scalings
of B in
(64).
we can consider changes of the
are even more general,
, (~i,~2 > o)
,
(2-146)
53
because there is no analog to the restriction
watch before.
f(o)=1,
that we had to
The optimal choices for ~I and ~2 maximize the right hand
side of
oo
3
{3 ;dx x I/3
d
5/3
+ #Ig(o)
7B > - 5
[~Id--xg (#2 x) ]
o
+ I ;dx[#ig(#2x) ] 2 }
0
oo
5/3
I/3 3 [dx x ~/3
5/3
t~1
~2
~
[g' (x)]
+ ~ig(o)
o
2
~I I ;dx[g(x)] 2}
co
+
- -
~2~o
(2-147)
They are
co
(1) 413
~1
=
[-g [o) ] 4/3
I ;dx[g(x) ] 2)-I/3
oo
(y
~-fdx I / 3 [g, ix) ]5/3
o
2-148)
o
and
co
~2
(1)4/3
=
[-g(o) ] 4/3
I ;dx[g(x) ] 2)2/3
I~
~
1;d x xl/3[g, (x)15/3
2-149)
o
0
Inserted into
(147) they produce the scale invariant version of
B > (1) 4/3 .
[-g(o)]7/3
=
co
eo
(l/dx xl/3[g ' (x)]5/3)(l[dx[g(x )]2)I/3
o
o
144):
(2-150_)
,
where the equal sign holds only for g(x)=~iF' (~2x) with arbitrary
~i,~2>o.
Indeed,
for such a g(x), we get
B = (1) 4/3
[~I B]7/3
.....
.513 113 1B)(~I2
(IJ,~I
I~ 2
,u,2
1B)113
(2-151)
The main contribution to the energy of an atom comes from the
vicinity of the nucleus. Now, Eqs. (62) and (63) imply,
54
F" (x)
Consequently,
-'- ~ x
for
a good
trial
1
An example
It t u r n s
(2-152)
x ÷ o
g(x)
for
has
to be
such
that
x ÷ o
(2-153)
is
g(x)
=-
out,
that
that w e m a y
(1+/x) -~
the
right
immediately
helps
in this
limit,
stead
of g(x)
for ~+~.
,
~>o
hand
side
consider
since
the
it a l l o w s
The l i m i t i n g
(2-154)
of
(150)
limit
increases
~÷~.
The
to e v a l u a t e
trial
with
scaling
~,
so
invariance
g ( x / ~ 2) for ~÷~,
function
is a s i m p l e
in-
exponen-
tial:
g(x)
For this
= lim[-(1+
[Z÷oo
g(x),
B
>
we h a v e
I
-a ] -K/~)
in
-
e
-/x
(2-155)
(I 50)
[-(-i)] 7/3
25 (.~) 4/3
--y
(~5 2"2/3) (1) I/3 =
(1) 4/3
(2-156)
= 1.5682
Binding
in
(131)
energy
and
of n e u t r a l
a lower
1.5682
one
TF atoms.
in
(156).
< B < 1.5909
We have
N o w we
found
combine
an u p p e r
the
two
bound
on B
and s t a t e
,
(2-157)
or
B = 1.580
The margin
know
cal
B with
effort
in
(158)
is about
a precision
was
needed
physical
picture
entirely
sufficient.
(2-158)
-+ 0.012
that
of
1.5%
of the
average
0.75%.
Please
appreciate
in o b t a i n i n g
this
we
using,
are s t i l l
A higher
precision
result.
value,
little
In v i e w
of the
the v a l u e
is not
so that
how
for B in
called
we
numericrude
(158)
for at t h i s
is
stage
55
of the development.
I n s e r t e d into
(67), this B v a l u e produces
-ETF = 0.765 Z 7/3
w h i c h is the TF p r e d i c t i o n
8
I
,
(2-159)
for the total b i n d i n g energy of neutral
I
I
I
1
TF
6
I-IF
/
.,.+
÷
÷
OL
0
i
J
j
J
25
50
75
100
_I
125
Z
Fig. 2-2. Compar~on of the TF prediction (160) with HF binding £n~gi~ (crosses).
56
atoms.
In Fig.2
-ETF
12
the q u a n t i t y
= 1.53
Z I/3
(2-160)
yz
is com p a r e d
of Z. This
atomic
to the c o r r e s p o n d i n g
plot
binding
shows
is n e v e r t h e l e s s
crudeness
Z values
viation
and
than
fractional
is 29,
120,
(160)
remarkable
TF curve
21,
duction.
Z~20
energies?
they were
available
be m e a s u r e d
particle
misled
against
Hamilton
This
that
17,
is,
15,
and
13 percent
with
energies
operator
by r e l a t i v i s t i c
for Z=I0,
of
based
upon,
are the more
de-
60,
90,
experimen-
in the Introonly up to
from the HF crosses).
of Z, the TF result
which
30,
and not w i t h
(I-7) ; this w a y we are
effects,
since it is
relative
20,
are k n o w n e x p e r i m e n t a l l y
predictions
at small
Z.
are the ones m e n t i o n e d
for large values
different
of this
it
the
In Fig.2
HF crosses
increasing
indistinguishable
trend of the
despite
a deception
The amount
values
is clear,
it represents.
w i t h HF predictions,
The reasons
are
that
however,
counts.
It d e c r e a s e s
Total b i n d i n g
(in Fig. 2 they
the TF model works
approximation
W h y do we compare
tal b i n d i n g
the general
for r e f i n e m e n t s
is closer to the integer-Z
difference
24,
reproduce
the need
how well
at large ones.
respectively.
does
Altough
of the p h y s i c a l
the continuous
the
that
energies.
HF p r e d i c t i o n s 8 for integer
Even
should
e.g.,
if
still
the many-
sure to not be
significant
the
lar-
ger Z.
TF function
F(x).
the TF function.
shall
We have
do so in this
(63). F(x)
d2
dx 2 F(x)
and the b o u n d a r y
F(o)
Upon u s i n g
appears
as
= I
there
a lot about
is m u c h more
the initial
slope B of
to say about
F(x).
We
section.
Let us p r o c e e d
(62) and
learned
Naturally,
from r e c a l l i n g
obeys
the d e f i n i n g
the d i f f e r e n t i a l
= F" (x)
properties
of Eqs.
equation
[F(x) ] 3/2
I/2
x
(2-161)
conditions
,
F(~)
= o
/x as the m a i n variable,
(2-162)
the d i f f e r e n t i a l
equation
(161)
57
d
F(x)
= 2/x F' ix)
,
(2-163)
d F' (x) : 2[F(x)] 3"2
/
d6x
Whereas
(161)
is w e l l
behaved
panded
is s i n g u l a r
at /x=o
in p o w e r s
F (x) =
which
F(x)
, F(x)
equations
can be ex-
,
(2-164)
as B a k e r ' s
s e r i e s 9 The c o m p a r i s o n
with
,
We g a i n
sI = o
,
s 2 = -B
calculation
(2-166)
of the Sk'S
it by i n s e r t i n g
left h a n d
F"(x)
(2-165)
shows
For the s u c c e s s i v e
The
s y s t e m of d i f f e r e n t i a l
t h at a r o u n d x=o
= I - Bx + 0(x 3/2)
sO = I
(161).
s k x k/2
known
[this is Eq. (64)]
relation.
, this
of /x :
~
k=o
has b e c o m e
at x=o
. We c o n c l u d e
(164)
for k>2, we need a r e c u r r e n c e
into the d i f f e r e n t i a l
equation
side is simple:
-
k k
= ~
sk ~ ( ~
k=o
I
)x k-2-2/
(2-167)
= ~3 s3 x -I/2
where
s1=o
has b e e n used,
right
hand
side of Eq. (161)
ries b e c o m e s
+
x (/-I)/2
a n d the s u m m a t i o n
is n o n l i n e a r
more complicated.
[F (x) ] 3 / 2 / x 1 / 2
~
Z=I
(£+I) (£+3)s/+3
4
index s h i f t e d
in F(x),
(k=/+3).
The
so that the p o w e r
se-
We h a v e
= x-I/211
+ ~
k=2
s k xk/2] 3/2
(2-168)
= x-I/211 + ~(
3/2
~
j ) (7k=2 sk
xk/2) j
]
r
58
where
the b i n o m i a l
theorem
is employed.
of the sum over k e x p l i c i t
k I , k 2 , ....
81,k
kj
=
is used to c o l l e c t
by w r i t i n g
Next,
1
for
k = 1
for
k ~ I
all terms
[F(x)]3/2/x1/2
the
it as the p r o d u c t
; then the K r o n e c k e r
o
we make
Delta
symbol
j-th p o w e r
of j sums over
,
(2-169)
'
x -I/2 x 1/2
of o r d e r
x (/-I)/2
=
= x-I1211+~(3<2)~-~
~---.. ~"
j=1
~ ki=2 k2=2
"kj=2
...
Skl
Sk2
×..Sk. x(k1+k2+'''+kj )/2]
3
= x I/2 +
s,
x
J
k1=2"''k,=2
J
Ski''
3
× 6/,k1+k2+...+kj
(2-170)
Since
each k is at least
two,
we have
1 = k I + k 2 + ... + kj => 2j >= 2
so that
the s u m m a t i o n
does not e x c e e d
tion
for the
tions
stop,
which
are j-1
over
1 starts
1/2 : j ~ [//2]
largest
integer
at the latest,
in number,
[F(x)]3/2/x1/2
really
at /=2
, w h i c h makes
contained
at /-2(j-I)
(2-171)
,
, and the
largest
use of the G a u s s i a n
in 1/2. The i n d i v i d u a l
, since,
again,
are not less than two each.
j
nota-
k-summa-
the o t h e r
k's,
Accordingly,
= x-I/2
/=2
j=l
×'''/k
3
k I =2
=2 S k 1 " ' ' S k j 6 / , k 1 + . . . + k .
3
3
(2-172)
This must
equal
(167),
implying
59
s4 = o
s 3 = 4/3
,
(2-173)
and
[//2]
~--(3/2)
j=1
3
(/+1) (l+3)
sl+ 3 =
~
/-2j+2
..>
.
k1= 2
k.=2 Ski "SkjS/'kl +''+k"
3
3
(2-174)
for
The
sk with
the l a r g e s t
it is s 1. Thus
that
this
index occurs
s/+ 3 is here
is a r e c u r r e n c e
der /=2,3,4,
£ = 2,3, ....
expressed
relation,
and 5. T h e r e
on the right
in terms
indeed.
43/2
43(_B)
=
side for j=1;
of Sk'S w i t h
k~/,
For i l l u s t r a t i o n ,
is only the j=1 term
s5 = ]-5( I )s2 - 1 5 2
hand
for /=2
so
consi-
and /=3:
2
-~B
(2-175)
If3/2%
134
s6 = 6' I 's3 = 6 2 3
For /=4
and /=5
, there
I
- 3
are both the j=1
and the j=2 c o n t r i b u t i o n :
4
s7 = ~ [ (
3/2,
I ;s4 +
3/2
2] = 4 3
( 2 )s2
~[~0
I
s8 = ~ [ (
3/2,
3/2
I Js5 +( 2 ) (s2s3+s3s2)]
= I--21
[3(_ 2B)+ 3~2 (-B) 4]
It is not d i f f i c u l t
(only boring)
tent w i t h w h a t we have
F(X)
3(-B) 2]
~0
+ ~
=
B2
, (2-176)
= - ~5 B
to c o m p u t e
more
Sk'S.
Let us be con-
3+
B2x712
so far:
= I - Bx + ~4 x 3z2 - ~2X
5/2
1
+ ~x
BX 4
(2-177)
+ 0 (x 9/2)
The B d e p e n d e n c e
relation
(174)
of the c o e f f i c i e n t s
prohibit
pansion
(164). We can,
to F(x)
obtained
done by i n s e r t i n g
asking
however,
by t e r m i n a t i n g
the t r u n c a t e d
and t h e i r c o m p l i c a t e d
recurrence
for the r a n g e of c o n v e r g e n c e
test the q u a l i t y
the s u m m a t i o n
series
at,
of the ex-
of the a p p r o x i m a t i o n
say,
k=8.
into the d i f f e r e n t i a l
This
is
equation
60
obeyed by F(x), and c o m p a r i n g both sides:
~k(k-2)Skx(k-4)/2
k= 3 4
For B=1.580
~
3/2/xi/2
~ [ = skxk/2]
, our estimate in
(2-178)
(158), the c o m p a r i s o n is made in Table
It shows us, that this a p p r o x i m a t i o n to F(x)
I.
solves the d i f f e r e n t i a l
0.25
; of
% for /x
i~
<
;
. This kind of analysis can be r e p e a t e d for sums
equation w i t h an a c c u r a c y of I% for /x ~ 0.4
of i ~ 0 % for /x ~ 0.20
t r u n c a t e d at a value of k m u c h larger.
One observes that a h i g h l y accu-
rate solution to the d i f f e r e n t i a l e q u a t i o n
for ~
< 0.4
only.
This is, therefore,
(161)
the
is given by these sums
(numerical)
gence of the series in Eq. (164); as a consequence,
powers of /x
this e x p a n s i o n in
is u t t e r l y useless. I0
Table 2-I. Left hand side
(LHS) and right hand side
Eq. (178), and their relative d e v i a t i o n
..., 0.50
/x
range of conver-
(DEV)
(RHS) of
for /x = 0.05,
. For B the value of Eq. (158) is used.
LHS
RHS
DEV
19.8866012
19 8866017
2.3 × 10 -8
0.10
9.783683
9 783698
1.5 x 10 -6
0.15
6.35805
6 35816
1.7 × 10 -5
0.20
4.60944
4 60991
1.0 x 10 -4
0.25
3.5373
3 5387
4.0 x 10 -4
0.30
2.8071
2 8106
1.2 x 10 -3
0.35
2.275
2 282
3.3 x 10 -3
0.40
1.867
I 882
7.8 × 10 -3
0.45
1.542
1 568
1.7 × 10 -2
0.50
1.27
1 32
3.4 x 10 -2
0.05
For a precise k n o w l e d g e of F(x), we cannot rely upon
(164) be-
cause of its small range of convergence.
The d i f f e r e n t i a l e q u a t i o n
itself has to be integrated numerically.
It is not advisable to attempt
doing this by starting from x=o
with
F(o)=1
and
(161)
F' (o)=-B , using
a suitable guess for B [as in Eq. (158)]. If the chosen value for B is
too large,
the n u m e r i c a l F(X) will turn n e g a t i v e eventually;
if B is
too small, it will start growing instead of d e c r e a s i n g steadily.
One
could imagine p i n n i n g down the correct value of B by an i t e r a t i o n based on this d i s t i n c t i o n between trial B's that are too large or too
61
small.
This
errors
cause
one w o u l d
is not
a wrong
start
circumvented,
stead
going
properties
large-x
with
the
however,
of o u t w a r d s .
start
equation
it s a t i s f i e s
the
one
because
numerical
of B. This
inwards
therefore,
turn
approaches
invites
the
that
from x=~
our
rounding-off
F(x),
difficulty
even
if
can be
towards
attention
1 4 4 / x 3 is a p a r t i c u l a r
(161)! I Of the
two
at x =~, w h e r e a s
144/x 3 for x ÷ ~
F (x) ~ 1 x3
44
This
of the
value
integrating
us,
by n o t i n g
differential
F(x)
behaviour
correct
by
Let
unfortunately,
x=o
to the
in-
large-x
of F(x).
We
that
to work,
for
boundary
conditions
it is i n f i n i t e
from
solution
at x=o.
in
of the
(162)
It is clear,
"below":
x ÷ ~
(2-179)
ansatz
F(x)
- 144 G(y(x))
x3
(2-180)
y(x)
÷ o
for
x
G(y)
= 1
for
y = o
with
÷ ~
(2-I 81)
and
The
best
into
the
choice
for the
differential
function
equation
I [xy' (x) ] 2G" (y(x))
I--2
=
which
takes
portional
on a s c a l e
(2-182)
y(x)
obeyed
must
be
found
by F(x),
from
Eq. (161).
inserting
This
leads
to
+ 1 ~ [ x 2 y " (x) - 6xy' (x)]G' (y (x) ) +C- (y (x) )
[G(y(x))] 3/2
invariant
form
,
(2-183)
if w e c h o o s e
xy' (x) to be p r o -
to y(x):
(2-184)
xy' (x) = - yy(x)
The optimal
immediate
(180)
value
for y>o has
to be d e t e r m i n e d .
Equation
(184)
and
its
consequence
x2y"(x)
=
(X d - I )
xy' (x)
= ¥(¥+l)y(x)
,
(2-185)
62
used in
(183), p r o d u c e
¥2
I--2 Y2G"(Y)
+ y(¥+7)
12
y G ' (y) + G(y)
(2-186)
= [G(y)] 3/2
The a f o r e m e n t i o n e d scale i n v a r i a n c e is obvious here: with G(y)
G(~y)
is a solution to
also
(186), for a r b i t r a r y ~. A unique value for y is
now implied by the requirement that G(y) be regular at y=o,
G(y)
= I - y + 0(y 2)
for
y ÷ o
N o t e that because of the scale i n v a r i a n c e of
(2-187)
(186), the c o e f f i c i e n t in
front of the term linear in y can be chosen to be minus one
be n e g a t i v e to not be in conflict w i t h Eq. (179)]. W i t h
[it has to
(187), Eq. (186)
reads
I -
[I+
~ ( ¥ + 7 ) 1 y + 0(y 2) = I - ~ y + 0(y 2)
,
(2-188)
whence
y(y+7)
= 6
,
(2-189)
or,
1
y = ~(-7+ 7 ~ )
The second solution to
(2-I 90)
= 0.77200...
(189)
is -(y+7)
= -7.772...
and of no use to us
in the present context.
The d i f f e r e n t i a l e q u a t i o n
m a k i n g use of
¥--~2y2G"(y)
12
G(y)
+ lyG'
is thereby subject to
entirely.
(186) is s i m p l i f i e d a little bit by
(189):
(y) + G(y)
= [G(y)] 3/2
(2-191)
(187), w h i c h determines the solution to
This does not m e a n that we know F(x) after finding G(y),
(191)
sin-
ce the i m p l i c a t i o n of Eq.(184)
y(x)
= 8 x -¥
(2-192)
contains an u n d e t e r m i n e d constant,
B. Its value is fixed by the require-
ment F(x=o)=1.
analogous to the previous
This is, of course,
situation
when F' (o)=-B was d e t e r m i n e d by F(x÷~)=o.
Since G(y)
it in powers of y:12
is, by construction,
regular at
y=o, we can expand
63
(2-193)
G (y) = ~
Ck yk
k=o
where
co
I
,
cI
The steps that
and
-I
(2-194)
led us to
(193) with the appropriate
sides of Eq. (191)
(174)
changes.
can be repeated
Comparing
(191)
of yl on both
gives
y2
1
+ ~ l
[-~- / ( / - 1 )
+ 1]c£
Z-j+1
=
13(2) 7 - - j=1
3
(2-195)
Z-j+I
....
>
k I =I
.. c 5
kj =I Ck I " kj l,kl+k2+...+kj
3
for £~I. The j=1 term equals ~c I and has to be brought
hand side. We then arrive
cl =
here for
powers
12
(y2Z+6) (£-I)
at the recurrence
over to the left
relation
~ ( 3 ( 2 ) ~ - ~ + 1 . . . 2 C k6. .l. C'k jk 1 + ' ~
j=2
3
k1= I
k.=1
1
3
.+kj
"
'
(2-196)
for
Z = 2, 3, ...
For example,
c2
= i__3_21312)c
9
2y2+ 6
201+21
4y2+12
608
= 0.625697 . . . .
c3
-
13/2
3/2, 3 }
12
{' 2 ) (CLC2+C2CI)+( 3 ;Cl
(3y2+6) 2
3-¥2/8
=
(y2+2) (y2+3)
--
(2-197)
15377+1813 7 ~
98496
= -0.313386...
AS we did before,
in Eq. (178)
and Table
1, we can again insert trunca-
64
ted versions
of
(193)
find the numerical
the expansion
y~1
into the differential
range of convergence
(193)
, or, x ~ I/Y
represents
(191)
of this series.
a highly
Anticipating
equation
accurate
in order to
The outcome
solution
to
(191)
is:
for
that the actual value of ~ is about
this is x~30) 3 Does this mean that the expansion
(193)
13,
is as useless
as
the one of Eq. (164)?
No. The power
away from x=~,
F(x),
i.e.,
Eq. (161),
series of G(y),
y=o, when integrating
inwards.
ingredients
of a computer
: (i) find G(y)
program
calculating
(193),
truncated
(depends on the chosen Yl and the accuracy
grate numerically
the differential
(~5 is a good choice),
accuracy
F(x)
of the machine);
equation
ted for the numerical
(the standard
integration);
for ~, and use Eqs. (192) and
(180)
(iii)
(ii) inte-
and G'(Y2)
choose
scheme
is
large k
(191) up to a certain
Runge-Kutta
for
range
chosen y1(~0.3
at a sufficiently
so that we now know G(Y2)
of the computer
equation
of the essential
for the whole
and G' (y) for a suitably
by employing
is needed to get
the differential
Here is a brief description
of x, o~x<~
a good choice)
for some o<y<1,
Y2
within
the
is well sui-
a trial value,
~(~13),
to find x2=(
~
~-/y2 ) I/y together with
(iv) now integrate the differential equation for F(x),
~(~2 ) and ~' (x2);
~
in the form (163) with /x as the relevant argument• down to x=o. At
this stage•
This ~(x)
we have a solution
obeys ~(x=~)=o
have to vary ~ until ~(x=o)=1
servation
enables
that,
if ~ ( x ) o b e y s
us to simply rescale
therefore:
to
(161),
the one corresponding
, but not ~(x=o)=1 •. Fortunately•
to ~=~.
one does not
in order to find F(x).
Instead, • the ob-
(161),
for arbitrary
~(x).
so does
~3~(~x)
The last step in the procedure
~>o,
is
(v) set
F(x)
= ~(x)/~(o)
,
(2-198)
where
= x/[~(o) ] I/3
Accordingly,
B
(2-199)
we have B given by
=
- F ' (o)
= -F' (o)/[~(o)]4/3
and• as a consequence
Y2 = ~ x2Y
B is related
(2-200)
of
=
2
~ x%-Y
to ~ through
,
(2-201)
65
[3 = 13(x2/~:2 )Y = ~ [ F ( o ) 1 x / 3
(2-202)
The s e n s i t i v i t y of the n u m e r i c a l results for B, ~, and F(x), to the
r o u n d i n g - o f f errors of the c o m p u t e r can be tested by v a r y i n g the parameters YI' Y2' and ~. Ideally,
them, n u m e r i c a l l y
the o u t c o m e should be i n d e p e n d e n t of
it is not. The little d e p e n d e n c e that one o b s e r v e s
shows how m a n y decimals of the results can be trusted.
For example,
the r e a l i z a t i o n of the p r o c e d u r e just d e s c r i b e d gave
(2-203)
B = 1.58807102261
and
= 13.270973848
(2-204)
on a c o m p u t e r w i t h a 15-decimal arithmetic! 4 For illustration,
we give a plot of F(x)
1,0
I
f1
in Fig.3
for o~x~10.
|
I
I
-Bx
0.8
A
X
u_ 0./-.
0.2
0
0
I
I
I
I
2
/-.
6
8
10
X
Fig.2-3.
The TF f u n c t i o n F(x).
Now that we know the actual v a l u e of B, let us look back at
the bounds that we found earlier,
Eq. (157). The upper bound is extreme-
66
ly good:
it is too large by less than 0.18%. On the other hand, the
lower bound is s i g n i f i c a n t l y worse:
it is too small by 1.25%. This is
a first sign of the s u p e r i o r i t y of the p o t e n t i a l f u n c t i o n a l o v e r the
density functional.
With
(203) we can give more s i g n i f i c a n t d e c i m a l s
binding energy formula.
in the TF
I n s e r t i n g B into Eq. (67) gives
- ETF = 0.768745 Z 7/3
(2-205)
There is no point in d i s p l a y i n g m o r e than six decimals.
Scaling p r o p e r t i e s of the TF model.
gram" for F(x) we made,
In step
in Eqs. (198)
and
(v) of our "computer pro-
(199), use of the i n v a r i a n c e
of the TF e q u a t i o n for f(x),
f"(x)
=
[f(x)]3/2
(2-206)
xl/2
[this is Eq. (59)], under the t r a n s f o r m a t i o n
f(x) ÷ ~3f(~x)
,
(~>o)
(2-207)
The TF model itself is not invariant under such a scaling,
b o u n d a r y c o n d i t i o n f(o)=1
fixes the scale.
because the
Therefore, we have to be
somewhat more careful w h e n i n v e s t i g a t i n g the scaling p r o p e r t i e s of the
TF model.
W h e n we were looking for bounds on B, we found it a d v a n t a g e o u s
to exploit certain scaling properties of the r e s p e c t i v e functionals.
The
scaling t r a n s f o r m a t i o n s that we c o n s i d e r e d then, were, Eq. (119):
f(x) ÷
f(~x)
(2-208)
and, Eq. (146) w i t h g(x)=f' (x)+q/Xo=f' (x) for q=o:
f(x) ÷ ~q f(~2 x)
,
w h e r e ~, ~I' and ~2 w e r e a r b i t r a r y
(2-209)
(positive)
numbers.
mine the implications of t r a n s f o r m a t i o n s as general as
to the TF p o t e n t i a l functional.
We return to Eq. (45),
Let us now exa(209) a p p l i e d
67
ETF = ] (d~) ( -
I
) [_2(V+~)15/2
15~ 2
(2-210)
=
E I
+
E 2
-
{N
and consider
V(r) ÷ v
Since,
V(~r)
for the existence
rV(r)
,
(~>o)
(2-211)
of E2, we need
÷ - Z
for
such a scaling t r a n s f o r m a t i o n
[Eq. (52)]
r ÷ o
(2-212)
,
of V has to be accompanied by a scaling
of Z,
Z ÷ ~-I
For convenience,
÷ v
Z
(2-213)
we also scale ~ by
E
,
(2-214)
so that the structure V+E is conserved.
In terms of f(x),
f(x) + v - 1
which identifies
whereas
f(~x)
(207) and
(211) and
(214), without
(213), read
,
(2-215)
(208)
as the special
situations
(209) is realized by ~2=~ and ~i=~ ~. However,
just have
(208), as we should have,
ring f(o)=1;
and only
Under
(211),
since
with
v=4 and v=1,
(213), we
(212) is equivalent
to requi-
(208) is consistent with this constraint.
(213), and
(214)
the various
contributions
to ETF
scale according to
E 1 ÷ E I (~) = f(dr) (-
I ) [_2 V(V(~r)+~)]5/2
15~ 2
(2-216)
= 55v/2-3
El
•
and
E2 ÷ E2(I~)
= _ 8~f(d~ ) [~(llVV(gr)
IIVZ.
+ _~)]2
=
68
(2-217)
=
29-I
E2
,
as w e l l as
~N +
V
~N
(2-218)
Consequently
ET F ÷ ETF(~)
For ~=I, t h i s
the e n e r g y
first o r d e r
= ~5~/2-3
is just ETF;
is s t a t i o n a r y
El + 2 v - I
for ~=I+5~,
under
~ETF
~
=
O n the o t h e r hand,
6
from
( v-1
(2-219)
w e have ETF+6 ETF . Now,
infinitesimal
changes must originate
5 ETF
E2 _ ~V~N
variations
in the s c a l i n g of Z. T h a t
is
~ETF
= (~-I)Z--~--5~
Z)
since
of V and ~, all
(2-220)
(219) w e get
5 ETF = [(5v-3)E I +
(2~-I)E 2 - ~ N ] 6 ~
,
(2-221)
so t h a t w e c o n c l u d e
(-~-3)E I ÷ (2~-I)~ 2 - ~
= (v-1)z~z~T~
T h i s is a l i n e a r e q u a t i o n
in v. It has to h o l d
two i n d e p e n d e n t
among the different
relations
"virial theorems."
Besides
(215)],
choice
for any v. So we o b t a i n
energy quantities
- two
v=1, w h e n
I
- ~ E I + E 2 - ~N = o
the o t h e r n a t u r a l
(2-222)
,
(2-223)
is t h e TF s c a l i n g
v=4
[see the c o m m e n t
to Eq.
for w h i c h
7E I + 7E 2 - 4~N = 3 Z ~ z E T F
The latter combines with ETF=EI+E2-~N
8-~ ETF = - ~
(2-224)
and
(2-225)
69
to give
(2-226)
~ ETF (z ,N)
7ETF (Z ,N) = 3 (Z-~+Ny~)
We have made explicit here,
that the energy of an atom is a function
of Z and N.
For N=Z,
Eq. (226) has the simple implication
7ETF(Z,Z)
= 3Z~zETF(Z,Z)
,
(2-227)
or
ETF(Z,Z)
= - C Z 7/3
,
(2-228)
where the constant C is yet undetermined.
knowledge of
C=- ~ETF/~N=o
for
(dr) ( V + )
N=Z
It is found by combining our
with
V 2Z
r
~
(2-229)
:
-
z(v+
)1
;:
-
B Z7/3
for
N = Z
r=o
The third step here is a partial
zes -ZS(r)
valid
that
the fourth one recogni-
as the source of the Coulomb potential
for N=Z only, uses Eq. (65).
(229)
integration;
identifies
Z/r; the last one,
[The comment to that equation says
the interaction
energy between the nuclear charge
and the electrons:
ENe = Z~ZETF
which,
according to
(2-230)
,
(I-96),
TF model in its validity.]
ETF(Z,Z) -- 71Z
÷N
is a general
statement,
)~T~(Z,m[
: ~Z~ET;(Z,N) [
N=Z
3 B Z7/3
=-TK
which
identifies
not limited to the
Now,
the constant C. This is, of course,
N=Z
(2-231)
the result that
70
we had
found
earlier
The
first
in Eq. (67).
of our
"virial"
theorems,
Eq. (223),
has
the
conse-
quence
E I = 2(E2-%N)
= 2(ETF , E I
,
(2-232)
orr
2
E I = ~ ETF
,
(2-233)
and
I
E 2 - ~N = ~ E T F
For
a Coulomb
usual
system,
theorems
they
about
the
essential
+ E
to
emerge
(2-234)
like
Eki n = - ETF
Indeed,
.
the
one
kinetic
,
from
we
and
are
the
considering,
potential
one
expects
energy:
Epo t = 2ETF
the
relations
the
(2-235)
that
r e m e m b e r how E 1 and E 2 a r e
we
have
found
so
far.
It
is
c o m p o s e d o f E k i n and E p o t = ENe
:
ee
E I = Eki n + ENe
+ 2Eee
+ ~N
,
(2-236)
E 2 = _ Eee
Note
in p a r t i c u l a r
in E I. W i t h
(230)
the
double
counting
of t h e
electron-electron
energy
we have
Epo t = ENe
+ Eee
= Z~TETF~_ - E 2
(2-237)
which
makes
produce
the
use
of
second
(225).
Now
statement
Eqs. (226)
of
(235),
and
(234)
which
then
can
be
implies
employed
the
to
first
one
immediately.
The
relative
E
: Eki n
ee
sizes
: (-ENe)
are
:
(- ~--ETF-~N): (-ETF): (- 7 E T F + ~ N )
= I : 3 : 7
for
N = Z, w h e n
;
~ = o
.
(2-238)
71
In words:
the electron-electron
of the kinetic
energy of a neutral TF atom is one third
energy and one seventh of the
(negative of the)
nucleus-
electron energy.
For ions, there is less specific
merely
implies that ETF(Z,N)
introducing
tion of q=I-N/Z,
e(o)
It
in the form
(2-239)
a reduced binding energy,
e(q),
that is a func-
the degree of ionization:
Z7/3
= - ~
e(q)
ETF(Z,N)
We know e(q)
can be written
in Eq. (226).
N
= Z 7/3 x [function of ~]
ETF(Z,N)
This invites
information
for q=o,
3
= 7B
(2-240)
i.e., N=Z
:
,
(2-241)
which is simply Eq. (231). The factor m u l t i p l y i n g
same as the one in Eqs. (108) and
of the TF potential
(density)
e(q)
(138). The m a x i m u m
functional
in
(240)
(minimum)
is, therefore,
is the
property
here expressed
as
2 fax
{5 o
[f(x)]5/2
x I/2
I fdx[f' (x)+ q ] 2
+ ~ o
Xo
+ q(1-q)}
x°
(2-242)
e(q)
oo
_
{! fdx x I/3
5
'
[g ( x ) ]
5/3
+ g(o)
o
The competing
g(x)'s are hereby restricted
by
+ yI fdx[g(x) ] 2 } .
o
[Eq. (98)]
co
fdx xg' (x) = 1-q
,
(2-243)
o
whereas
f(x=o)
= 1
The equal signs in
Eqs. (59) and
(2-244)
(242) hold only for g(x)=f' (x)+q/x o, when f(x) obeys
(60), w h i c h also determine x o.
We can relate e(q)
to Xo(q)
by recognizing
that Eq. (225) says
72
Z 7/3
~
e(q))
~--~(-
-
Z 4/3 d
a
dq e(q)
_--
_
Z4/3
~
--
a
(2-245)
q
x o (q)
thus,
d
e (a) =
dq
-
q
x O (q)
(2-246)
Consequently,
q
q,
,
fdq' Xo~q')
e (q) = 73 B
- o
(2-247)
and
e(q)
of w h i c h
q'
xo(q')
whereas
q~1).
e(q=1)
has been used;
to w e a k l y
one is d e s i g n e d
the o b v i o u s
ionized
for h i g h l y
systems
ionized
(2-249)
it says:
7Z
ETF
=
no e l e c t r o n s
Eq. (229)
-
- no b i n d i n g
energy.
gives
Z
Z(V+~+ ~)I
+ ~Z
r=o
(2-250)
Z7/3
-
so
that Eq. (226)
a
[ f " (o)
q
translates
+
-~-q]
xo
into
q2
7e(q)
By w r i t i n g
atoms
statement
= o
For ions,
Z
be a p p l i e d
the second
In Eq. (248)
(2-248)
,
the first one should
(N~Z, q~o),
(N<<Z,
I
= Sdq'
q
= 3[-f'q(O)
a subscript
(2-251)
- Xo(q)]
q we e m p h a s i z e
initial slope f' (o). The c o m p a r i s o n
q
of
the q d e p e n d e n c e
of fq(X)
(251) w i t h
results
(247)
and its
in
73
q2
-f'
q
=
(o)
B
7 .q
+
-
a'
--|dq'
x O (q)
,
3 ~o
x° ( q )
(2-252)
q
q,7/3
d
I
= B ÷ ofdq'
The latter equality
]
is verified by performing
a partial
integration.
Since -f' (o) increases with increasing q, we learn here that
q
~q[q
l/3xo(q)]
We observe
tral quantity
q
(x)
(2-253)
in these relations,
is Xo(q).
cally the differential
f"
< o
It is basically
equation
that in studying
available
ions the cen-
from solving numeri-
for f (x), Eq. (59):
q
[f~(x) ] 3/2
= - - ~
(2-254)
x
with the boundary conditions
fq(O)
Nevertheless,
= I ,
ionized atoms,
fq(Xo(q))
= O
in the two limiting
ble to make precise
on 1-q(=N/Z),
(60):
statements
,
-Xo(q)fq(Xo(q))
situations
q~1
q~o
(2-255)
it fs possi-
about the analytic d e p e n d e n c e of Xo(q)
or, q, respectively.
Let us first concentrate
on highly
q~1.
Highly
ionized TF atoms.
N/Z÷o,
the interelectronic
In the limit of extremely high ionization,
interaction becomes
pared to the n u c l e u s - e l e c t r o n
ply the Coulomb potential
interaction.
-z/r,
insignificant
as com-
In this situation V is sim-
and we are dealing with Bohr atoms,
which have been studied in the first Chapter.
Eq.(46),
and
= q
We concluded already,
in
that then
ETF(Z,N)
As a statement
e(q)
= -Z2(3N) I/3
about e(q)
for
N/Z ÷ o
this reads
3
I/3
3 ~ 2/3
1/3
= a[~(1-q)]
= ~(~)
(l-q)
(2-256)
74
for q ÷ I
After
employing
Xo(q)
which
(2-257)
Eq. (246) we
= [ ~ ( 1 - q ) ] 2/3
is r e c o g n i z e d
serted
find
for
q
,
(2-258)
~=(Z4/3/a) (q/Xo) ,
to be Eq. (I-35) w h e n
etc.,
is in-
there.
If N/Z
is not t h a t
small,
Eq. (257)
tions that account
for the r e p u l s i o n
treatment
from n o t i n g
proceeds
metrization
of the p o t e n t i a l
ous to i n t r o d u c e
another
recalling
~ =
~(t)
purpose.
by m e a n s
A systematic
s u i t e d para-
It is a d v a n t a g e -
of 15
,
t = r / r o = x/x o
(259),
correc-
(2-259)
[Eq. (55)]
similarity
in
acquire
~ (r/ro)
(Z-N)/r o
of the g r e a t
and the one of ~(t)
(258)
is not the best
for the p r e s e n t
V + ~ = - ~ ~(t)
t
Because
and
a m o n g the e l e c t r o n s .
t h a t f(x)
function
Z-N
r
V (r) + ~ or,
+ I
between
the two
(2-260)
the d e f i n i t i o n
functions
of f(x)
are s i m p l y
in
related
(57)
to
e a c h other:
f(x)
= q ~ ( x / x o)
~(t)
Consequently,
obeys
# (t) = I f (tXo)
,
the d i f f e r e n t i a l
(2-261)
equation
[~(t)] 3/2
(2-262)
~" (t) =
and is s u b j e c t
q~ ( o )
where
l=l(q)
ti/2
to
=
1
q
,
-
{(1)
=
o
,
{'
(1)
=
-1
,
(2-263)
is g i v e n b y
(q) = q l / 2 [ x o ( q ) ] 3 / 2
AS a c o n s e q u e n c e
of
(258),
I is s m a l l
(2-264)
for q~1
:
75
I ~ ~-16(I-q)
- 16
~ NZ
for
ZN = 1-q
Why is it fitting to switch from f(x)
appearance
of I in
(262) offers
o
to ~(t)? The reason is that the
the possibility
powers of I, w h e r e b y the smallness
(2-265)
of expanding ~(t)
of I promises
in
a good convergence of
the expansion.
The d i f f e r e n t i a l
at t>1
in
(263)
~1(t)
where we wrote
equation
can be combined
(262)
into the integral
I
= I - t + I Sat' (t'-t)
t
~1(t)
and the boundary conditions
equation
[~1(t')] 3/2
(2-266)
t 'I/2
in order to emphasize the I d e p e n d e n c e of ~. After
solving this equation
for a chosen
I, the c o r r e s p o n d i n g
value of q emer-
ges from
I
I _ 91(o ) = I + I fdt tl/2[~l(t)]3/2
q
o
In the first place,
one obtains
is to be found in an additional
e(q)
supplies
the desired
by w r i t i n g
enters,
I
7/3
= fdq'
q'
q
[1(q')] 2/3
(2-267)
I/q as a function of I, from which
step.
l(q)
Then Eq. (248), here in the form
(2-268)
,
e(q). The evaluation
of this integral
is eased
it as an integration over I instead of q, since then q(1)
not 1(q). The rewriting begins with
e(q)
o
= ] d1' dq(1')
1(q)
d1'
[q(1')] 4/3
1,2/3
(2-269)
1(q)
-_3
fdl,
7
I' -2/3
o
d (i_[q(i,)]7/3 )
d1'
in view of this implication of Eq. (265)
i-2/3(I-[q(i)] 7/3)
;
:
= i-2/3(i-[1- ~61+...] 7/3)
=
76
7~
48 11/3 + 0 (14/3) ; = o
a partial
integration
An alternative
exDression
,
(2-270)
2 I5 (q)
d1' I-[q(I ,) ] 7/3
+ 7 o
1,5/3
(2-271)
makes use of (251); here it reads
3
q4/3
= 7 [i(q)']2/3
e(q)
I=o
can be performed with the outcome
3
1-q 7/3
- 7 [i(q)]2/3
e(q)
for
(-~i(o)-q)
,
(2-272)
where
I
-~(o)
is the initial
[~1(t)] 3/2
= I + I oSdt"
(2-273)
t I/2
slope of ~1(t). Note that the equivalence
(272) allows to relate ~ ( o )
to
~i(o)
= I/q(I)
which is a useful equation
(271) and
[this is Eq.(267)]:
1
-~I, (o) = [~i(o)]4/3{I
of
I - [ ~ ( o ) ] -7/3
+ 2 12/3 ofat'
f 5/3
} ,(2-274)
for checking against algebraic mistakes.
Let us now, indeed,
expand ~1(t)
in powers of I,
oo
~1(t)
This,
inserted
= I - t + ~---Ik ~k(t)
k=1
into
(266), implies
~--IZ~l(t)
£=I
The technique
(2-275)
= I 5dt' (t'-t) t '-1/211-t'+~--Ik
t
k=1
that produced
the recurrence
relation
~k(t')] 3/2. (2-276)
for the s£ in Eq.
(174) can be applied here, too. We find
1
~I (t) = 5dt' (t'-t)t '-I/2 (I-t')3/2
t
,
(2-277)
77
and
I
~l(t) = ]at'
(t'-t)t '-I/2 ~ ( 3 / 2 )
t
j=I J
(1-t') 3/2-9
(2-278)
_.
x ~ - ~ . . . i 3= ~kl (t')...~k (t')Sl_1,k1+...+k.
ki=1
9 1
9
3
for
Z=2, 3, ...
The first few ~l(t) can be expressed in terms of elementary
functions; unfortunately, the degree of algebraic complexity increases
rapidly. Let us, therefore, confine ourselves to explicitly stating only ~1(t) as it emerges from (277) :
41 (t) = ( ~ - ~t)arccos(t
3
2 _ ~t2)t1/2 (1_t) I/2
I /2 ) + (~ + ~t
(2-279)
In particular, we have
1
#I (o) = fdt t1/2(1-t)3/2
o
-~
Equation
(o)
=
I]at t -I/2
o
(1-
- ~
16
,
(2-280)
t)3/2 = -~3~
(279) is utilized in
3 ~
ti/2
I/2
(t) = 2
~2(o) = ~ d t
(l-t)
~I
15
o
3~ 2
256
,
(2-281)
and
-~(o)
3 1
I/2
I/2
I
= ~ ]at t(l-t)
~I (t) = 3
o
3~ 2
256
(2-282)
We are now prepared to employ Eq. (267) in order to
find the leading corrections to (265), (257), and (258). From
I
q
I
I- (l-q)
-
I
+
=
~i(o)
(l-q)
+
(l-q)
+
~i(o)I
2
+
...
+
~2(o)12+
(2-283)
=
I
...
78
we get
I
I (q) = ~I (o)
~2 (O)
) (l-q)
(l-q) [I+ (I
= 16(I-q) [I+ (4-
+ ...
]
[~1 (o) ] 2
512 ) (l-q)
(2-284)
+ ...
]
1 5"n;2
or
l(q)
- 16~ NZ [I+(4-
N
15~
2512) Z + 0((
)2)]
for N/Z<<I
(2-285)
Then,
using
either
e(q)
one of the Eqs. (268) , (269) , or
= ~(~)2/3(1-q)
1,311_(i_/
(271) , we find
256 ) (l-q)
45~ 2
+ ...
]
(2-286)
= ~
[I-(1-
~ + 0(
45~ 2
Also,
from
(264) or
Xo(q)
= [_~ ( i _Iq ) ]~2/311+ (3
=
The n u m e r i c a l
(246),
i024) (i_~)
45~2
N 2/3
1024)
(--- ~)
[I+(345T~ 2
versions
I = 5.093
thereof
+ ... ]
(2-287)
N
(N) 2
~ + 0(
)]
are
(I+0.5416
N
~ + ...)
,
(2-288)
e = 1.0135(
)1/3(1-0.4236
~ + ..)
x o = 2.960(
)2/3(I+0.6944
~ + ...)
Here then
is the m o d i f i c a t i o n
,
of Eq. (46) that we p r o m i s e d
at that time:
ETF(Z,N)
= -Z2(3N)I/311-(1 -
for
N<<Z
;
256)N +
]
45~2 Z
"'"
(2-289)
79
it is obtained
by inserting
(287)
into
(240).
A simple check of consistency
is provided
by
(253). This
states
d
l(q)
dq
< o
,
(2-290)
or,
d
d(N~
A quick
look at
I(q=I-N/Z)
(285)
> o
shows that this is true,
We close this section
cussion
(2-291)
of the relative
on highly
indeed.
ionized TF atoms with a dis-
sizes of Eki n, Eee,
and ENe.
In order to be
able to apply Eq. (238), we need ~N. It is given by
~N = -N~N ETF(Z,N)
(2-292)
1 2 (3 N ) I/3 [I-(4= ]Z
so that Eqs. (236) and
(234)
1024) N~ + ...]
45~ 2
produce
I
Eee = - ~ETF - ~N
(2-293)
= Z2(3N)1/3[( I-
256 ) NZ
+ ...]
45m 2
The interaction
energy of the electrons
with the nucleus
is given by
[see Eq. (230)]
ENe = Z~Z ETF (Z, N)
(2-294)
I
-2Z2 (3N) I/3 [I- (2
whereas
the kinetic
energy
pressed
in Eq. (235). Consequently,
states
zation.
N +
• ..]
is simply the negative
E
-ENeee _ (21 _ 2128)
45~
which
128)
45~2
N
Z + 0((
that Eee is negligible
) )
,
of ETF,
as is ex-
,
(2-295)
in the limit of extremely
[We have already made use of this
(physically
obvious)
high ionifact re-
80
peatedly; see, for example, Eq. (256) ] Together with the neutral,atom
statement of (238), we have, therefore,
Ee e
f
ji/7
_
for
N = Z
(2-296)
-ENe
Likewise,
~0.2118~
for
N<<Z
one obtains
~r3/7
for
N = Z
Ekin
,
(2-297)
and
Ee e
_ ~I/3
for
N = Z
(2-298)
Ekin
~0. 4236 N
Weakly ionized TF atoms.
t<1, as is evident
for
N< <Z
As I increases,
#l(t) grows larger for all
from Eq. (262), or Eq. (266). Thus q=I/~l(o)
decreases,
finally reaching q=o for the critical value
A
= l(q) I
q÷o
(2-299)
Consequently•
Xo(q)
= q-1/3 [l (q) ] 2/3
(2-300)
A2/3 q-I/3
for
q+o
•
so that Eq. (247) implies
e(q)
Accordingly,
~ 73B
- 73 ,A-2/3 q7/3
for
q÷o
we have in the limit of very weak ionization
ETF(Z'N)
~
3 Z 7/3
A-2/3 q7/3]
7 ---~[B =
(2-301)
81
= _ 3 I[B
7
for
If we insert Eq. (301)
trial functions
Z7/3
_ A-2/3(Z_N)7/3]
(2-302)
N ~ Z
into the inequalities of
f(x) and g(x)
(242), suitably chosen
supply bounds on A -2/3. Details are given
in P r o b l e m 10, from w h e r e we cite
0.0946 < A -2/3 < 0.1008
,
(2-303)
or,
A -2/3 = 0.0977±0.0031
,
(2-304)
w h i c h tells us that A -2/3 is about six percent of B. A w e a k l y ionized
TF atom has, therefore,
the n e u t r a l atom.
p r a c t i c a l l y the same binding energy that has
In other words:
the o u t e r m o s t electrons c o n t r i b u t e
very little to the total binding energy of the atom.
In the limit q÷o, the r e l a t i o n b e t w e e n f(x)
and ~(t) becomes
singular. We cannot give sense to the right hand side of
fq(X) I
= F(x)
q+o
= q ~l(q) (X/Xo (q)) I
q÷o
[Eq. (261)], b e c a u s e Xo(q÷~)
= ~ squeezes t=X/Xe(q) into an i n f i n i t e s i m a l
v i c i n i t y of t=o. T h e r e is, n e v e r t h e l e s s ,
I approaches
A.
(2-305)
We w r i t e ¢(t)
a s e n s i b l e limit to %l(t)
for this ~A
as
(t). It obeys the d i f f e r e n -
tial e q u a t i o n
~"(t)
=
A
[~(t)]3/2
tl/2
,
(2-306)
and is subject to
~(1)
= o
,
~' (I) = -I
,
(2-307)
and
#(t) ÷ ~
A l t h o u g h ~(t)
as
t ÷ o
(2-308)
is s o m e h o w c o r r e s p o n d i n g to the s i t u a t i o n of n e u t r a l TF
atoms, the t r o u b l e of Eq.(305)
signifies that it cannot be used as a
82
p a r a m e t r i z a t i o n of the p o t e n t i a l V(r).
use for Eqs. (306) t h r o u g h
Fortunately,
there is still a
(308), in as m u c h as they offer a simple and
highly precise m e t h o d for c a l c u l a t i n g
A . Here is how it goes: #(t)
p o s s e s s e s an e x p a n s i o n in powers of ~t °, w i t h a yet u n d e t e r m i n e d constant ~ and
o = 7 + y = 1(7+ 7/7~)
•
(2-309)
of the form
144/A 2
#(t)
=--[1-~ta+
t3
w h i c h is, of course,
and
9
(a~t°)z + ...]
12+ 4a 2
an i m m e d i a t e analog to Eqs. (180),
(2-310)
(193),
(192),
(190). The c o e f f i c i e n t s of the powers of ~t ° o b e y the r e c u r r e n c e
relation
(196) after r e p l a c i n g y by o. The
gence of this series is ~t°~0.6,
unity, t~0.94. On the other hand,
of
,
(numerical)
range of conver-
or, a n t i c i p a t i n g that ~ is close to
#(t)
can also be expanded in powers
(l-t) I/2,
4A;IA t ) 7/2 + -2A
~(t) = (l--t) + --~,I-- ~ (I_t)9/2
(2--311)
+ A(1-t)11/2
this series b e i n g c o n v e r g e n t for
A
is w i t h i n the bounds of
A2
+ i--~(I-t) 6 + . . . .
(l-t) 1"2
! ~ 0.35, or,
t ~ 0.88,
when
(303). There is a range of t around t=0.9
w h e r e both expansions are converging.
This allows to d e t e r m i n e
A
and
n u m e r i c a l l y by forcing the two expansions to agree w i t h i n the a c c u r a c y
to w h i c h they represent solutions to Eq. (306). Such a c a l c u l a t i o n 16
resulted in
A
= 32.729416116173
(2-312)
and
(2-313)
= 1.0401806573862
Naturally,
physics does not need this m a n y decimals;
they are r e p o r t e d
only in o r d e r to d e m o n s t r a t e the m a r v e l o u s p r e c i s i o n of this simple
method.
Please note that one cannot compute B and B in a s i m i l a r way,
because the expansions
(164)
and
(193) c o n v e r g e for x~0.15 and x~30,
83
respectively.
There
The
is no overlap.
A of
(312) yields
A -2/3 = 0.0977330
so that we obtain,
(2-314)
from Eq. (302),
-ETF(Z,N)=0.768745 Z 7/3
for
The correction
for N=Z/2
f(x)
N ~ Z
to the neutral
(2-315)
atom binding energy is rather small;
even
it is only about one percent.
Since
rapidly
_ 0.047310(Z-N) 7/3
A
(if at all)
to ~(t)
is large,
the series of Eq. (275) does not converge
for weakly
is pointless
fact that the d i f f e r e n c e
X<Xo(q) . In particular,
ionized atoms,
in this situation.
and the switching
from
Here we make use of the
between F (x) and fq(X)
is small, when q>~o and
fq' (o) does not differ s i g n i f i c a n t l y
so that fq' (o)+B is a possibly useful expansion parameter.
from -B,
We use it
in m a k i n g the ansatz
fq(x)
where the fk(x)
= F(x)
+ ~---[f' (o) + B] k fk(x)
k=1
q
,
(2-316)
are subject to
fk(o)
= o
for
k=I,2 ....
(2-317)
and
f{ (o) = I
,
f~(o)
= o
for
To first order in f '(o)+B, the d i f f e r e n t i a l
17
q
requires
d
-
3 F(x)
2[~]
7
1/21
fl
k=2,3 ....
(2-318)
equation obeyed by fq(X)
(x) = o
(2-319)
J
One solution
is
fo(X)
= F(x)
+ ~Ix F '
(x) -
I dx
d (x3F (x))
3x 2
,
(2-320)
84
because
d~
~ [--~--j
x2
3x--~ d--x x
However,
inasmuch
fo(1)
fo(X)
F(x)
(2-321 )
= o
as
= I
(319) relates
fo(X)
F(x)
- .--~--
,
f'o(1) = - ~B4
is not proportional
equation
x
to f1(x).
(2-322)
,
The Wronskian
the two functions
of the differential
to each other:
f~ (x) - f'o(X) fl (X) = I
This is equivalent
(2-323)
to
d fl (x)
-2
dx fo ~
= [fo (x)]
(2-324)
which has the consequence
f1(x)
This does,
= fo(X)
indeed,
x
dx'
f
o [fo(X')] 2
satisfy
(2-325)
the requirements
that we need not add a multiple
of fo(X)
fl (o)=o
and
f{ (o)=I
, so
on the right hand side.
For large x, we have
fo(X )
-
I d[144
3X 2
G (y (X))]
-
(12) 2 d
3X 2
[1-y(x)
+
...]
(2-326)
(12) 2 yy(x)
3
x3
which uses Eqs. (180),
or, equivalently,
(326)
(187),
= (12) 2 ¥~x-(¥+3)
3
(184),
and
(192). This
inserted
into
(323)
produces
I
I
fl (x) -- 488 y(y+o)
xO-3
for
large x
(2-327)
85
[o
= 7+y, Eq.(309)].
In deriving
3
I+~
Eq. (319)
the first order approximation
has been used for e=[f~(o)+B]fl(X)/F(x).
fq(X)
~ F(x)
must not be applied
(1+e) 3/2
Consequently,
+ [f~ (o) +B] f1(x)
to x~x o, where
(2-328)
e=-1. We can, however,
supplement
(328) with
X
fq(X)
which
~ q(1- ~O )
is valid for X~Xo~
(2-329)
[This is obviously
of Eq. (311) as it analogously
approximations
for fq(X),
three unknown quantities
the requirement
appears
Eqs. (328)
no more than the first term
in ~l(t)].
and
(329),
!
Xo(q) , x1(q) , and fq(O)+B
that fq(X)
ous at x=x I . This can be done explicitly
to employ
the large-x
and f1(x).
algebraic
equations
f,,:
q
(12) 3
s
+ [fq(O)+B]
XI
the last one uses
x
=
(o-3)(o-4)=18.
o+3/2
o+2
o
=
y+17/2
~+9
18 Xl-5
48~y(y+O)
=
37+ 7 ~
- -48
of
-q/Xo
= O
I
= 0.9488
(2-330)
,
;
These three equations
=
situation
which allows
Thus, we have the three
xI
= q(1- ~O )
432
(°-3) X1-4
- + [fq(O)+B] 488y(y+o')
Xl~
!
Xl
0-3
xI
48By(y+O)
144
+ [f~(o)+B]
3
xI
fq:
- -
forms of F(x)
are large,
from
are continu-
in the limiting
and x1(q)
x=x I. The
are determined
and its two first derivatives
very small q, since then both Xo(q)
fq:
Let us now join the two
at a certain
imply
,
(2-331)
and
Xo
as well
_
o+2
a+372
[96(o+2)]
I/3
q-
I/3
= 10.32
q-1/3
•
as
-f' (O) = B + 1(96) -(Y+I)/3
q
8y(y+O)(0+2) -O/3 qO/3 =
(2-332)
86
(2-333)
= B + 8.05 x 10 -3 q2.59
whereby
identities
these results
like 0=¥+7
and yo=6 have been used.
are based upon the simple
we should not take them too seriously.
is certainly
ql/3xo(q)
constant
right.
For instance,
approaches
a constant
is A2/3=I0.2320.
Of course,
approximations
Nevertheless,
(328)
their
and
since
(329)
structure
Eq. (332) says that the combination
as q÷o.
This much we know already
The estimate
for A 2/3 obtained
in
(332)
- the
dif-
fers from the actual value by less than one percent.
Something
new is to be learned
tion, we differentiate
~q[-fq(O)]
from Eq. (333). As a prepara L
Eq. (252) with respect
=
q7/3 d[
I
]
=
to q:
q7/3 ~q[l(cr)]-2/3
;(2-334)
dq ql/3xo (q)
the latter equality
is a consequence
(264). Now Eq. (333)
implies
of the definition
of l(q)
in Eq.
that
d
-2/3
(a-7)/3
q ~[l(q)]
% q
=
qy/3
(2-335)
We infer that
~(q)
for values
=
A [1+(powers
of q not too large.
Xo(q)
of qy/3)]
Then,
(2-336)
of course,
= A 2/3 q-I /3 [1+ (powers of qy/3)]
,
(2-337)
and
e(q)
which
= 73B -
73 A,2/3 q7/3[ 1+elqy/3
is an implication
of
challenge
consists
determine
the corresponding
cular,
Eq. (246)
(337) when
in calculating
+ e2q2y/3
it is inserted
the coefficients
coefficients
in
(336)
+ ...]
into Eq. (247). The
el, e2,
and
, (2-338)
..., which
(337).
In parti-
supplies
Xo(q ) = A2/3 q-I/311 + ~elqy/3 + ¢;7._J.Ye2q2y/3+ ...]-I
=
(2-339)
87
o q¥/3+ o
2_ o+y ,q2¥/3+
= A2/3 q-I/311_ 7ei
((7ei)
-7--e2J
...]
Then
I (q) = [ql/3xo (q) ] 3/2
(2-340)
q2~/3
3 5 (~el) 2 o+y
= A [I- ~30
e I qy/3 + ~(~
- --7--e2)
+
...]
,
and from combining (251) and (246) with (338)
-f' (o)
7
q2
7
~q
q
= ]e(q) + Xo(q) = (] - q )e(q)
(2-341)
= _qi0/3 d [ q - 7 / 3
dq
e(q)]
: qi0/3 ~q(_ 73B q-7/3 + 73 A-2/311+elqy/3+e2q2y/3+...]),
or,
-f' (o) = B + ~ A-2/3 q°/3[e I + 2e 2 qy/3 + ...]
q
(2-342
The comparison with (333) yields a first estimate for el:
e I --- 0.75
which, in
at (333),
the order
about ten
,
(2-343
view of the crudeness of the approximation used in arriving
cannot be expected to have more significance than stating
of magnitude. (We shall see below that the actual value is
percent larger.)
A systematic computation of el, e2, ... starts from the expansion (316). Comparing powers of fa(o)+B in the differential equation
obeyed by fq(X) produces
I
£
3/2-j £-j+I
fi(x I : 5-(3j2) EFIx)]
J
xlj2
j =I
£-j+I
fkl
k I =I
(xl...
kj =I
x..fk.3 (x)6£'k1+'''+kJ
(2-344)
88
The j=1 term on the right hand side is brought over to the left, so
that
[ d2
dx 2
y3 / ~ - / ~ ]
_
=
Z
fl (x,
3/2
7(j
.[F(x)]3/2-J ~-j+1
xI/2
I
9=2
k I =I
l-j+1
>
fh
(x% ...
kj=1
(2-345)
x-'-fk
3
This right hand side contains
f1(x),
(x)6Z,k1+...+k.
3
..., fl_1(x)
but not fl(x). The
solutions to the corresponding homogeneous differential
fo(X) and f1(x), given in Eqs. (320) and
equation are
(325). With their aid we can
construct Green's function G(x,x') which satisfies
d2
[-dx
3/F~-~-/x] G(x,x' ) = 8 (x-x')
-
2
(2-346)
G(x,x')
= o
and
~xG(X,X ') = o
for
x=o
It is given by
G(x,x')
Thus
fl(x)
(2-347)
= [fo(X')f I (x) - fo(X) f I (x')]O(x-x')
x
= fdx' [fo(X')fl (x)-fo(X)fl (x')]
o
x [ (d-~22-3/F(x')/x') f£(x')]
dx'
where we refrained
,
(2-348)
from explicitly inserting the right hand side of
(345).
The use of Eq. (348) does not lie primarily in explicitly
calculating f2(x),
f3(x), etc. but in studying their structure.
Recall
that F(x) can be written as
F(x)
= 144 G(y(x))=
x3
144 i] +x___c~
~--- o[y (x) k]
x3 L k=1
(2-349)
89
which is repeating Eqs. (180) and
(193). As a consequence,
fo(X) has the
form
fo(X)
Inserted into
= 14__~4
x 3 ~1y y ( x ) [ 1 + ( p o w e r s
of y(x))]
(2-350)
(325) this implies
fl (x) - 144
3
I
x o [1+(powers of y(x))]
x ~ (12)4
6y(y+o)
,
(2-351)
of which we have seen the leading term in (327). Now we employ the recurrence formula
(348) to conclude that
fz(x ) = 144[_ 3
I
x a ]Idz
x3
(12) 4 ~y(y+o)
[1+(powers of y(x))]
,
(2-352)
where the constants d I obey
d I = -I
(2-353)
and
2y
dz = (Zo+y)(Z-I)
l-j+1
l-j+1
(3 2)
~ ~
">
dk I "''dk j 61,k1+. • .+k j
j=2
k I =I
kj =I
(2-354)
This we recognize
to be the recursion for the c k of Eq. (196), after y
and o are interchanged.
Consequently,
appear in the expansion of Eq. (310).
{(t) =
144/x°t
~ [ I+
the d~s are the coefficients
that
That is
dk(at°)k]
(2-355)
k=1
This connection between the fz(x)'s and ¢(t), which is, of course, not
accidental, is the clue to computing el, e2, ... of Eq. (338). We reveal
its significance by inserting Eqs. (349) and (352) into the ansatz (316),
fq(X)
= 14411+ ~ ( - [ f q ( 0 ) + B ] - X 3
k=1
3
(12) 4
[x° (q) ] O k
x
~ )
( _ _ ~ ,)kOdk ] +
Xolq,
90
+ ....
(2-356)
where the ellipsis indicates the terms containing "powers of y(x)."
After introducing h by
q
I
= B + 7(12) 4~8y (y+G) [hq/X O(q)]°
-fq(O)
(2-357)
Eq. (355) is employed:
co
fq(X)
= 1443 [ I+ Z dk(a[h q ~ ]X
x
k=1
O)k]
+...
(2-358)
A
2h3 #(hq ~ )
= q [i-~]
q
+ ...
where [l(q)]2=q[Xo(q)] 3 is used. What is exhibited in (358) is the part
of f (x) that goes with the zeroth power of y(x). Likewise, an arbitraq
ry power of y(x), say [y(x)] m contributes
[y(x)]m 144 [ cm + (powers of ~(hqX/Xo(q))°)
x3
]
(2-359)
~ a[~@q)]2 hq
Cm[Y(X)]m ~m(hq x--~)
x
to fq(X). The functions ~m(t) thus defined are such that
144/A 2
~m(t) -
[ I + (powers of ~t°)]
(2-360)
t3
We can now make explicit what supplements Eq. (358):
x
fq(X) = q[l--~q)] 2 h~[~(hq x--~-~)+
j
Cm[Y(x)]m~m(hq x ~ q ) ) ]
m=1
(2-361).
Whereas
(316) is an expansion that is expected to converge rapidly for
x not too close to Xo(q), the series of (361) is the faster convergent
the smaller y(x) is. This identifies large values of x (i.e., x~x o) as
91
the domain of application.
It is instructive
to make contact with the original defini-
tion of ~(t),
_ qI f q (tXo(q)) ]
#(t) : ~l(q) (t) I
q+o
q÷o
(2-362)
[see the comment to Eq. (305)]. Since
= o
y(tx o(q) ) J
: ~(tx o(q))-Y[
q÷o
q+o
the combination of Eqs. (361) and
,
(2-363)
(362) reads
(t) = h 3 ¢ (hqt) ]
q
q÷o
(2-364)
from which we learn that
hq
]
= I
(2-365)
q÷o
This tells us what e I is. Equations
el + 2e 2 qy/3 +...
=
(342) and
-[fq(O)+B]
ly
(357) together say
A2/3
q-a~3
(2-366)
7(12)4
B(y+a) A2/3
:
hq
[ql/3Xo (q)
or, after making use of ql/3xo(q)=[l(q)]2/3 ,
el + 2e 2 qy/3
7 (~) 4
+...
= ~
~B(¥+o)
A-2Y/3(hq
A
[77~71
2/3
)
(2-367)
Now the limit q÷o identifies
7
eI = ~
With ~, B, and
(_~)4 .e,13(y+a) A-2y/3
A from
(313),
rical value of e I is roughly
(204), and
(2-368)
(312), respectively,
10% larger than the estimate of
the nume(343):
92
e I = 0.825908
(2-369)
Note that Eq. (368) reveals the physical
of B and
significance
A has been clear since Eqs. (67) and
The requirement
yo(q)
fq(X=Xo(q))=o
of ~ and 6; that
(301).
relates hq to yo(q),
given by
-- Y(Xo(q) ) = 6[Xo(q)]-Y
(2-370)
= BA-2y/3 q y / 3 [ ~ ] 2 y / 3
inasmuch as x = Xo(q)
{(hq)
in Eq. (361) yields
+ f
Cm[Yo(q)] m ~m(hq)
m=1
With the aid of ~' (I)= -I
in yo(q),
or, qy/3,
(2-371)
= o.
[which is Eq. (307)], we find to first order
respectively:
h q = I + c I Yo (q) ~I (I) + 0(Yo (q))2)
(2-372)
= I - ~ A -2Y/3 ~I (1)qy/3 + 0(q2y/3)
where ci= -I has been used.
In conjunction
with Eq. (340) this has the
consequence
(hq[
a I _BA-2¥/3
]2/3)o = (i+(7e
~I (I) )qy/3 +. . .)o
(2-373)
o
= 1 + o(.Te 1
_6A-2y/3
¢1(1))_
ay/3
+
.
.
.
.
so that the order qy/3 in Eq. (367) is
o
_BA-2y/3
2e 2 = oe1( 7 e I
To proceed
~i(I))
(2-374)
further, we need to know ~i(I).
The insertion of Eq. (361) into the differential
eauation
obeyed by fq(X) produces
d2
[-dt 2
3
t-y
2 A /~-~/~]
~1 (t) = o
,
(2-375)
93
when terms linear in y(x) are identified. This is quite analogous to
Eq. (319) , so that
9o(t ) - tY[9(t)+ ~It
(2-376)
%'(t)]
is one solution of (375), the one corresponding to fo(X) of (320). The
Wronskian of 91 (t) with 9o(t) is
,
I ,12,
9o(t)~1'(t) - ~o(t)91 (t) = ~-~-;4~o(o+X)
t2
,
Y
(2-377)
which makes use of Eq. (360)~ and the small t form of 9o(t),
9o (t) = t ¥ I
d
3q 2 d--{(t3%(t))
= 13t-(2-Y) (Y) 2~t[1-c~t°+d2(°~t°)2+''']
(2-378)
I ~_2) 2~ o to+Y-3 [I-2d 2 ~t°+..]
= - 7(
Equation
(377) now implies
I ,12,
I
t '¥
~1(t) = 9°(t) [91(I)/~°(I) - ~l-A-;4 cx°(°+Y)fdt'
(t
~ )
2]
'
(2-379)
where 91(I) is determined by the t÷o form of 91 (t), statet in Eq. (360).
In connection with (378) this requires that the square brackets in (379)
possess the form
- 3t-(O+Y)
~o
[I+
(powers
o f 0~t°)]
1
1 (y)
91 (1)/90(I) - 3(°+Y)fdt't [- ~
2
c~o
t,Y
~ ]
(2-380)
I
~I (I)/9o(I) - ~ ( o + y ) f d t '
t
2
t -2(O-3) (1+4d2~t°)
~O (G+¥) fdt'{[I
~I ( ~ ) 2 ~ O ~ t1'Y2 _
t
t-2 (O-3) (I+4d20~ t°) }.
94
In the latter
t=o,
version,
since the
the
integrand
second
integrates
I
Idt'
t
is no longer
singular
at
has the s t r u c t u r e
{ ... } = a2t ' s [const.
which
integral
+ (powers
(2-381)
of o t ° )]
to
I
= fdt' { ...}
o
{ ...}
+ ~ztT[const.+(powers
of at°)]
I
The first
= ;dt'
o
{...}
integral
in
+ t -(m+x) (~tm) 2 [ c o n s t . + ( p o w e r s
of at°)]
(2-382)
(380)
_ ~(o+¥)[
is
1-t7-2°
7-20
-t 7-°
17-o
]
+ 4d20
(2-383)
_ ~(i+~d2
The c o n s e q u e n c e
91
of
o+x a) -
-V(380)
+
(I)/~o(I)
3
a-6
t-(o+x) (i+4d2 o+x at 0)
y
is t h e r e f o r e
3
o+¥
--~--o,)
~-~(I+4d 2
(2-384)
I
1,12,
tY
1+4d2at°
~o ~
t °+Y+l
- --~-3(O+¥) ] d t { [- ~-A-; 20° ~lq--T~T] 2
~0
With
o
the aid of 9o(I)
= -I/3
9
d 2 = 12+4o2
this
and,
9/2
= o(2o+y)
from
(310)
3
X
= 4 2o+y
or
} =
0
(354),
,
(2-385)
says
1 (1+3 ~ a )
91(1) = ~6
(2-386)
1
l+3~t
+ o+x [dt {
~o o
t ~+¥-I
This
expresslon
does
not
lend itself
°
"1"12"2~0 ~ - ~t -Y~ ] z }
-[3~x-~;
to further
algebraic
simplifica-
tions.
The n u m e r i c a l
value
of 91(I),
obtained
by a m e t h o d
analo-
95
gous to the one that produced
#1(I)
and or in Eqs. (312) and
= 0.3216868353717
w h i c h illustrates
Wronskian
A
,
(2-387)
once more the high precision of the algorithm.
(377), at t=1,
~1'(I)
is employed
= (4+¥)~i (I) -
the d i f f e r e n t i a l
(~)4 ~o(o+~)
~1"(1)
equation
= 2Y41'11 ) -
The
in finding
= 0.2869164052321
whereas
(313), is
(2-388)
,
(375)
supplies
yly+1)41(1 )
(2-389)
= 0.002936027410
The a l g e b r a i c
statements
41(1)
where,
because
0.15% o f t h e
= 21
of the
sum.
of
(388)
)4c~(o+y)
smallness
In conjunction
and
(389)
can be c o m b i n e d i n t o
+ g ~1"(1)
,
o f ~1"(1),
the
with
(2-390)
latter
Eq. ( 3 6 8 ) ,
this
[SA- 2 ~ / 3 41 (1) = 76 el + g1 6 A-2,~/3 ~1"(1)
w h i c h we insert into
(374)
_ 0+6
e2
here,
14
resulting
0+6
- ~ - e12
about
implies
,
;
(2-391)
(2-392)
a 0.5% correction
= 0.671015
,
to
(2-393)
in
e 2 = 0.667554
Naturally,
only
~1 ''(I)
.12,~
]
~-6 2 [-Ej ~(o+y)
the ~i"(I) term represents
e2
is
to find
I
e1211
part
the subsequent
(2-394)
coefficients
in
(338)
can be computed the
96
same way.
The results of this section are summarized
-ETF(Z,N)
= 0.768745
in
Z 7/3
-0. 047310 (Z-N) 7/3 [I+0. 825908 (I-N/Z) 0. 257334
+0. 667554 (I-N/Z) 0.514668
+ ... ]
which is the w e a k - i o n i z a t i o n
Eq. (289) [and its supplement
result of
7].
One last remark is in order.
(361)
(2-395)
analog to the h i g h - i o n i z a t i o n
of Problem
making explicit use of the requirement
to
,
How could we get around without
-Xo(q)f~(Xo(q))
= q ? As applied
it reads
OO
hq~' (hq) + >
Cm[Yo(q)]m[hq~m(hq)
- Y m ~m(hq)]
m=l
(2-396)
= - [ I(_~A) ]2 h-3
q
Indeed,
this together with
which can be converted
(371)
into %(q)
gives
%(q)
as a function of yo(q),
as a function of q, w h e r e a f t e r
Eq. (340)
identifies
el, e2, etc. Fortunately,
ween f~(o)
and hq in Eq. (356), so that we could avoid the more tedious
(though,
of course,
Arbitrarily
analytic
equivalent)
we came to know the relation bet-
procedure based upon Eq.(396).
ionized TF atoms. We have spent some time on studying the
form of such quantities
like e(q), Xo(q),
and -f~(o)
as func-
tions of q - both for q~1 and for q~o, which are the situations
ghly and weakly
ionized atoms,
respectively.
ever, did not tell us how good are few-terms approximations
(289) and
(395). Let us, therefore,
sults of numerical
integrations
fq(X)
values of q.
for various
We present
the nineteen q values
of hi-
These considerations,
make the comparison with the re-
of the differential
equation obeyed by
in Table 2 the outcome of such calculations
0.95,0.90, .... 0.05
for q=1 and q=o. The fractional
how-
as in Eqs.
, supplemented
binding energy
for
by what we know
97
e(q)/e(o)
= ETF(Z,N)/ETF(Z,Z)
is a d d i t i o n a l l y plotted,
(2-397)
as a f u n c t i o n of q, in Fig.4. We o b s e r v e that
1.0
O.8
0.6
0
S
0.&
0.2
00
I
I
0.2
I
0./-,
0.6
I
O.B
1.0
q
Fig.2-4.
The f r a c t i o n a l
binding
r e m o v i n g 30% of the e l e c t r o n s
e n e r g y e(q)/e(o) as a f u n c t i o n
from the neutral atom,
of q.
reduces the b i n d i n g
energy o n l y by I%; a r e d u c t i o n by 10% requires the removal of 65% of the
electrons.
Even when only 5% of the electrons are left, the binding ener-
gy is still m o r e than 50% of the n e u t r a l - a t o m one. Here is the quantitative v e r s i o n of the q u a l i t a t i v e remark that the innermost electrons
c o n t r i b u t e most to the b i n d i n g energy,
From Eq. (286),
e(q)/e(o)
1.489(N/Z)1/3
(241)
the o u t e r m o s t least.
and P r o b l e m 7 we find that,
N
N 2
= I - 0.4236 ~ + 0.0909(~) + ...
The s u c c e s s i v e a p p r o x i m a t i o n s that this r e p r e s e n t s
for N<<Z,
(2-398)
are c o m p a r e d to the
98
Table 2-2. TF q u a n t i t i e s Xo(q) , -fq'(O), and e(q)/e(o)
1-q = 0, 0.05,
..., I.
N/Z
x O (q)
0
0
for N/Z=
-fq'(O)
e (q)/e (o)
~
0
0.05
0.416269
3.020996
0.537084
0.10
0.685790
2.233243
0.662517
0.15
0.934348
1.952470
0.742539
0.20
1.179253
1.813524
0.800221
0.25
1.428919
1.734116
0.844082
0.30
1.689292
1.684993
0.878380
0.35
1.965691
1.653119
0.905616
0.40
2.263681
1.631819
0.927406
0.45
2.589715
1.617337
0.944875
0.50
2.951825
1.607410
0,958847
0.55
3.360561
1.600602
0.969946
0.60
3.830452
1.595965
0.978668
0.65
4.382486
1.592853
0.985410
0.70
5.048683
1.590815
0.990503
0.75
5.881272
1.589530
0.994227
0.80
6.973385
1.588763
0.996824
0.85
8.513784
1.588345
0.998508
0.90
10.92728
1.588149
0.999475
0.95
16.10273
1.588081
0.999908
I
~
B=1.588071
actual values in Fig.2-5.
I
We see that the q u a d r a t i c a p p r o x i m a t i o n repro-
duces the actual data almost p e r f e c t l y even for N~Z.
i n c l u s i o n of the next term,
[Incidentally,
the
0.0024369(N/Z) 3,18would make the d e v i a t i o n
u n r e c o g n i z a b l e in Fig.5.]
In contrast,
pansion
[Eqs. (338),
the p e r f o r m a n c e of the w e a k ionization ex-
(369), and
1-e(q)/e(o)
0.06154q7/3
is s i g n i f i c a n t l y worse;
(394)],
= I + 0.8259 qy/3 + 0.6676 q2y/3 +
see Fig.6. ObviouslY,
the coefficients
expansions do not get small as rapidly as the ones in
a few more terms in
(2-399)
....
(399)
in this
(398). One needs
for a h i g h quality a p p r o x i m a t i o n over a large
99
1.0
0.9
o,8
O"
co
0.7
0.6
0
0.2
0.4
I
I
0.6
0.8
1.0
N/Z
F i g . 2 - 5 . Comparison of t h e l i n e a r and q u a d r a t i c approximations
of Eq.(398) with t h e a c t u a l numbers ( c r o s s e s ) .
range
of q¥/3.
At this
time,
only the n u m e r i c a l
has been c a l c u l a t e d 19, w h e r e a s
value
for e 3 leads
Validity
which
to the dashed
of the TF model.
touched
upon
the p r o p e r t i e s
all its
the d e f i c i e n c i e s
us from Eq. (40)
classical
(I-43);
cept
evaluation
and
for the
(direct)
one,
because
it leads
. They
has m a d e us f a m i l i a r
the d e s c r i p t i o n
are:
one
to an i n c o r r e c t
are those w h i c h
(i) the
(highly)
semi-
to the recipy of Eq.
interactions
(in p a r t i c u l a r ,
ex-
we did not care
the first one is the m o r e
treatment
with
we m u s t
are.
the m o d e l
of e l e c t r o n - e l e c t r o n
Of the two,
This
of the TF model,
in Eip a c c o r d i n g
electrostatic
energy).
aspects,
of the m o d e l
to Eq. (41
(ii) the d i s r e g a r d
for the e x c h a n g e
discussion
that d e f i n e
of the trace
of e 3 = 0.550066
in Fig.6.
In o r d e r to improve
The a p p r o x i m a t i o n s
brought
curve
important
value
... are not k n o w n as yet.
The d e t a i l e d
of TF atoms.
now find out w h a t
e 4, e 5,
of the m o s t
serious
strongly
100
I
I
I
I
I o1
X
f
/
/
X
X
oo.
X
1
0
I
0.2
0.4
qY/3
//
/
/
///
/
!
!
0.6
0.8
1.0
F i g . 2 - 6 . Comparison of t he l i n e a r and q u a d r a t i c approximations of
Eq.(399) with t h e a c t u a l numbers ( c r o s s e s ) . The dashed curve i s
t h e corresponding cubic approximation.
bound electrons,
the energy.
the ones close to the nucleus
To make this point,
that contribute most to
let us recall that the application
Eq. (I-43) (i.e., the evaluation of traces of unordered
space integrals)
is justified when commutator
the ordering process,
are negligible.
quires that the commutator
equals
not change
significantly
operators
In the present context this re-
this oradient
be "small."
which
Small com-
is small if the potential
over the range important
for an electron.
ce the quantum standard of length,
associated with an individual
tron, is its deBroglie wave length,
l, a small gradient means
I~ ~ v l
Substantial
changes
<< Ivl
by phase
as they appear in
of the m o m e n t u m and the potential,
i times the gradient of the potential,
pared to what? Physically,
terms,
of
does
Sin-
elec-
<2-4oo)
in V occur on a scale set by the distance r, so
101
that criterion
(400)
requires
that
<< r
(2-401)
On the other hand,
I is the inverse m o m e n t u m
and pi for this kind of reasoning),
root of the potential,
as criterion
the relation
>> I
Upon introducing
(2-402)
TF variables,
Z I/3
First,
in turn is given by the square
see Eq. (42). In short, we have,
for the v a l i d i t y of the TF model,
r ~
which
(we ignore factors of two
this reads
Ix f (x) II/2 >> I
q
we learn here that,
if Z is large enough.
for a given x, the TF model is reliable only
Second,
where the a p p r o x i m a t i o n
(2-403)
there is information
about the regions
cannot be trusted.
At short distances,
left-hand
side of
(403)
I/Z. Consequently,
TF a p p r o x i m a t i o n
incorrectly
f (x) p r a c t i c a l l y equals unity, and the
q
is of the order of one, when x ~ Z -2"3,
!
or r
there is an inner region of strong binding where the
fails.
Indeed,
the innermost
Then,
near the edge of the atom at X=Xo,
ar form of Eq. (329). Now the left-hand
one, when
iX-Xol~Z-2/3/q 2, or
s i d e of
Ir-rol%I/(Zq2).
region of weak binding to be also treated
The situation
in
(403).
is not satisfied,
D(r)
fq(X) has the lineis of the order of
Thus we find the outer
inadequately
in the TF model.
for neutral
atoms,
For q=o, the TF function F(x)
Its large-x form F(x)~I/x 3 implies that the criterion
and 8 plots of the radial densities
= 4~r2n(r)
are used to illustrate
(2-404)
these observations
concerning
the v a l i d i t y of
Please note that the regions of failure shrink with in-
creasing
comes
are described
once x is of the order Z I/3," or r~1.
In Figs.7
the TF model.
(403)
is, of course not basically different
although the argument has to be modified.
appears
electrons
in the TF model.
Z. We conlcude that
exact for Z+~. 20
Nice,
(in some sense)
the TF a p p r o x i m a t i o n
but in the real world Z isn't that large,
Z I/3, which obviously
is the relevant parameter.
be-
the more so
It ranges merely
from
102
0.4
- Z -2/3
-4"
N
-,,,,.
a
~Z -2/3
O LX-~
0
Fig.2-7.
Xo
Regions of r e l a b i l i t y
an i o n i z e d
atom { q = I / 2 ) ,
a function
of t h e TF v a r i a b l e
and f a i l u r e
illustrated
4
of t h e TF model i n
by t h e r a d i a l
density
as
x.
one to roughly five over the w h o l e P e r i o d i c Table.
Clearly, m o d i f i c a -
tions aimed at i m p r o v i n g the TF m o d e l are called for. All f o l l o w i n g
C h a p t e r s are d e v o t e d to their discussion.
leading approximation,
The TF atom is thereby the
and the supplements to the TF m o d e l will all be
regarded as small corrections.
For this reason it was n e c e s s a r y to
spend so m u c h time w i t h a d e t a i l e d study of the TF model.
It is important to a p p r e c i a t e that the density, w h i c h was
used in Figs.7 and 8, is the right q u a n t i t y to plot for this purpose.
The TF p r e d i c t i o n
nTF(r)
(51)
=
I [_2(V+~)]3/2
3~ 2
I (2Z/r)
3"2-/
3~ 2
for
(2-405)
r÷o
is clearly very m u c h in error at small distances.
Also,
for an ion of
103
0.4 1"
~-213
M
N
.Z1/3
0~..~
r/////A
0
X
Fig.2-8.
Like Fig.7,
for a neutral
10
atom.
degree of i o n i z a t i o n q, we obtain
nTF(r)
~F(r)
~
I
3~ 2
= o
[2Zq(1_r/ro)]3/2for
r<r O
(2-406)
for
r>r °
for the d e n s i t y around the edge of the atom. This is a sharp b o u n d a r y
instead of the q u a n t u m - m e c h a n i c a l l y correct smooth t r a n s i t i o n into the
classically
tially.
f o r b i d d e n domain, w h e r e the real d e n s i t y decreases e x p o n e n -
In the s i t u a t i o n of n e u t r a l atoms,
the large-r b e h a v i o r of the
d e n s i t y is
nTF(r)
~
I [ 2 ~ 144 ]3/2 = 243 ~ - -I
3~2
r x3
8
r6
w h e r e the large-x form of F(x)
,
is employed. Again,
(2-407)
this is not the cor-
rect e x p o n e n t i a l d e p e n d e n c e on the distance.
The p r i n c i p a l lesson consists
in stating that the real den-
104
sity is not of the form
n = nTF + ( a small correction)
As a consequence,
(2-408)
the TF density functional of Eq. (95) cannot be used
as the starting point when looking for corrections.
TF potential
as it behaves
tureless
potential
In contrast,
is very much like the real effective potential,
like -Z/r for r÷o and like -(Z-N)/r
in between.
The structure
for r+~, being struc-
is in the second derivative
(related to the density),
fore, we must find modifications
not in the potential
of the TF potential
(45) in order to overcome the insufficiencies
the
inasmuch
of the
itself.
functional
of the TF model.
Thereof Eq.
If this
is so, why does the vast majority of people working on TF theory use
the language of density functionals?
are historical
ones.
As far as I can see, the reasons
In the original work by Thomas and Fermi 21 the
principal variable was the density,
whereas
played the role of an auxiliary quantity.
years in basically
all presentations
textbook 22 is the most prominent
henberg-Kohn
section)
of the subject,
one. Then,
so over the
of which Gomb~s'
in 1964, the socalled Ho-
theorem 23 (of which we shall sketch a proof in the next
triggered the development
Because of this theorem,
ded theoretically,
(our introduction
certainly
nues the "general formalism"
defined as the density
functional,
theorem
lacking a "riwhich conti-
that we left after Eq. (40), we shall see
is wrong.
formulation
is in this spirit)
In the following section,
that this preconception
Hohenberg-Kohn
based upon the concept
which is widely regarded as an intuitive
foundation.
Density and potential
formalism.
appear to be well foun-
in contrast to formulations
gorous" theoretical
is the preferable
of a density functional
density functionals
of the effective potential,
approach
the effective potential
This remained
The potential
functional
is as well
and for the reasons given above it
in atomic physics.
functionals.
For a proo± of the aforementioned
(we shall state it below), we return to the ma-
ny particle Hamilton operator of Eq. (I-7), where we replace the nucleuselectron potential
~Z / r uy an arbitrary
external potential Vext(~),
and split Hmp into the kinetic energy operator Hkin,
tion energy operators
and the interac-
Hex t and Hee:
N
~ 1 ½ PJ2
Hmp = j'=
N
+ ~0=I
---Vext(rj)+
N
{~----"
I
j,k=1 rjk
=
105
(2-409)
= Hki n + Her t + Hee
Different
ground
states
I~o > will correspond
to differing
choices of
Vex t. In order to simplify the argument, we shall assume that, except
for the irrelevant possibility of a reorientation of all spins, the
ground states are unique
the external
density
(a slight,
and otherwise
potential would destroy
innocuous,
any degeneracy
anyhow).
change of
Thus,
the
in the ground state,
n(~') : ~ (d~2') (d~3')... (d~N') I<~)~½ ,r
÷'3, . .,rNi~o>I
÷'
2
+ ~(d~)(d~)...
+
(d~)
[<r~,r',r~ .... rNI~o>I 2
...
(2-410)
+ f(drl)... (drN_ I) l<r~,r 2 ..... rN_ I ,r I~o>l 2
÷, ... (dr
÷,N)
= Nf (dr2)
÷ ÷
I<~'~',
-'-2
. 4,
"
'rN[~o>12
(the latter equality makes use of the antisymmetry
of the wave function,
and a trace affecting the spin indices only is understood implicitly),
is a functional of the external potential V
ext"
Two different external potentials, Vex t and Vext, will lead
to two different ground states I~o > and I~o> , since the respective
SchrSdinger equations are different. (The situation ~ext = Vext + const.
is not interesting, since we consider only potentials that are
cally different).
by I~o > and
The expectation
I~o>, respectively,
values of H
mp
and ~
mp
are minimized
so that
<~oIHmpI~o > - <~ol~mpI~o > < <~oIHmp-~mpI~o >
(2-411)
and
<~olHmpI~o >
which are combined
- <~ol~mpl~o > > <~oIHmp-~mpI~o >
(2-412)
into
<~olHmp-~mpI~o >
Now we insert
,
< <~olHmp-~mpI~o>
(2-413)
106
N
H
mp
-3
(2-414)
mp = Hext - ~ext = ~--(Vext(~j )- ~ext(~j ))
j=l
and obtain
N
~olHmp-~mpl~o >
= ~(d~)...
j=1
(d~) (Vex t(~i)- ~ext(rj))
×
.....
2
= f(d~') [Vext(~')- ~ext(~')]n(~')
,
(2-415)
and likewise for I~o >. The implication of (413) is therefore
f(d~') [Vext(~')- ~ext(~')][n(~')- ~(~')]
< o
,
(2-416)
from which we conclude that n ~ n. Different external potentials not
only produce different ground states but also different ground-state
densities.
Consequently,
a given n corresponds to a certain Vex t which
is uniquely determined by n. In other words: Vex t is a functional of n.
And since the ground state I~o> is a functional of Vex t, it can be regarded as a functional of n as well. Then the expectation values of
Hki n and Hee in the ground states are also functionals of the density.
Here then is the Hohenberg-Kohn theorem: there exist universal (i.e.,
independent of Vex t) functionals of the density Ekin(n) and Eee(n),
so that the ground-state energy equais
E(n) = Ekin(n)
+ /(d~')Vext(~')n(~')
+ Eee(n)
,
(2-417)
where n is the ground-state density. The minimum property of <~IHmpl~ >
implies that the energy E(n) is minimized by the correct ground-state
%
density; trial densities n, which must be subject to the normalization
S(d~')n~(~ ') = N
,
(2-418)
yield larger energies E(~) than the ground-state energy E(n). It is
useful to include the constraint (418) into the energy functional by
means of
E(n,~)
= Ekin(n)
+ f(d~')Vext(~')n(~')
+ Eee(n)-
107
-~( N - f(d~)n
since this E(n,~)
)
,
(2-419)
is stationary under arbitrary variations
density n and the Lagrangian m u l t i p l i e r
Before p r o c e e d i n g
onal,
a few remarks
general one;
enter•
concerning
the related potential
The H o h e n b e r g - K o h n
the specific
The price for the generality
knowledge
~.
to construct
are in order.
in particular,
of both the
functi-
theorem is a very
forms of Hki n and Hee never
is paid in form of a total
the structure of the density
functionals
lack of
Ekin(n)
and E
(n). The theorem states no more than their existence. Obviousee
ly, the detailed form of these functionals must depend upon the specific Hki n and Hee that are investigated
sider relativistic
corrections
tions to nuclear physics,
different
[one could,
to the kinetic
for instance,
energy,
reflect upon fermion-fermion
from the Coulomb form of
(409)]. Also,
con-
or, in applicainteractions
no technical
proce-
dure is known that would enable us to perform the step from H
One must rely upon some physical
that approximate
insight,
when c o n s t r u c t i n g
to E(n).
mp
functionals
the actual E(n).
The kinetic
energy in the ground state of H
mp
of
(409) is
the e x p e c t a t i o n value
N
I
Ekin = <~o[Hkinl~o > = < ~ o I ~ P j
2
{~o >
,
(2-420)
j=1
which,
in c o n f i g u r a t i o n
space,
appears
as
Eki n = ~ N S (d~')f (dry)... (dr~)V'~o (r',r~ .... r N)
•~ ' % ( ~ ' , ~ .... ~ I
(2-421
= f(d~'l (d~"1½e (~'-~"1 ~'-~" nlll(~';r"l
Here, once more,
the a n t i s y m m e t r y
and we have introduced
of the wave function has been used,
the o n e - p a r t i c l e
density matrix
n(1)(~';P ') = Nf(dr~)
~' * +" ,r
÷,
~ ÷,2 ....÷,
+
+,2 , ..,rN)~o(r',r
• .. (drN)9 O (r
rN)
,
(2-422)
which is an immediate
itself is the diagonal
generalization
of
(410), so that the density
part of n(1)(~ , ;r")
+
,
108
n(~')
= n (I)(~' ;~')
(2- 423)
Let us now attempt to interpret n(1)(r '.÷"
,r ) as the matrix element of an
effective density operator,
n(1)(~';~ ") = 2<~' lq( _ ~I p 2 _ V ( ~
) _
~)i~,, >
(A more careful d i s c u s s i o n hereof will be presented
The effective
stage,
potential V(~)
(2-424)
in Chapter Four.)
that appears here is unspecified
except for remarking that it is a functional
at this
of the density,
÷
n(r),
because the density matrix on the left-hand
tional.
The factor of two is the spin m u l t i p l i c i t y
to make explicit
side is such a funcwhich we now choose
instead of further assuming that a trace on spin indi-
ces is left implicit.
adding a constant,
Note that V is determined without the option of
since Eq. (423)
has to hold for the given density.
The diagonal version of Eq. (424) showed up earlier,
in Eq.
(20). We are clearly back to the picture of particles m o v i n g independently in an effective potential V. The notation established
useful here,
too.
In particular,
we introduce
the independent
then is
particle
Hamilton operator
÷ ÷
H(r,p)
= ~I p2 + V(~)
(2-425)
just as in Eq. (3). The kinetic
energy of
(421)
is then rewritten
as
Eki n = f(d~') (d~") [I~,.~,, 5(~'-~")]n (I)(~' ;~")
= f(d 'l
= tr
I
~p
1½p 2
2
D(-H-~)
(2-426)
,
where we remember that the trace operation
spin factor.
2
includes m u l t i p l y i n g
E I = tr(H+~)q(-H-~)
is a functional
energy
by the
The quantity E I of Eq. (7),
of V+~,
(2-427)
thus a functional of n, as V=V(n).
(426) is contained
in
The kinetic
(427),
Eki n = E I - t r ( V + ~ ) ~ ( - H - ~ )
(2-428)
= E I - f (d~')(V(~') +~)n (~')
109
This we insert
into
E(n,~)
(419)
and arrive
= E I (V+~)
at
- ](d~') (V(~')
- Vext(~'))n(~')
(2-429)
+ Eee(n)
In the present
fore,
(429)
context,
V is still regarded
is the same functional
than reorganize
the right-hand
is stationary
under variations
just as
is stationary.
(419)
a change
- ~N
in E(n,~)
(419), we have done no more
Consequently,
of n and ~ around
An infinitesimal
= ( ~ E I (V+~)
- N)8~
= o
zero for the same reasons
sider a variation
of n. There-
the functional
their correct
variation
(429)
values,
of ~ induces
given by
8~E(n,~)
it is, indeed,
as in
side.
as a functional
;
(2-430)
that implied
Eq. (13). Now con-
of the density:
6nE(n,~)
= ~(d~')SnV(~')n(~')
- ~ (d~')6nV(~')n(~')
- / (d~') (V(~') - V e x t(~'))6n(~')
(2-431)
+ f (d~')6n(~')
Vee(~')
= ~(d~')[-V(~')+
where Eqs. (14) and
E(n,~)
thus
Vext(r
) +Vee
(25) have been employed.
(~')]Sn(~')
The stationary
property
implies
V = Vex t + Vee
In words:
the effective
tial and the effective
6nEee(n)
Note in particular,
dent) potential.
(2-432)
potential
interaction
equals
the sum of the external
potential,
Vee , defined
that V is always
a local
of the circumstance
poten-
by Eq.(25),
+
= f (dr'
÷ )Sn(r,)Vee(~,
)
So far, V has been regarded
Because
of
(2-433)
(i.e., momentum
as a functional
that the contributions
in
indepen-
of the density.
(431), that ori-
110
g i n a t e in v a r i a t i o n s of V, take c a r e of t h e m s e l v e s , we can e q u a l l y
t r e a t V as an i n d e p e n d e n t
E(V,n,()
is o b v i o u s l y
V,n,
= E I (V+~)
stationary
The e n e r g y
- f(d~')(V-Vext)n
under
independent
functional
+ Eee(n)
infinitesimal
- (N
(2-434)
variations
and (. If we do not w a n t to h a v e b o t h V and n as i n d e p e n d e n t
tities,
w e h a v e the o p t i o n of e l i m i n a t i n g
(434) b a c k to
expressing
V(n)
variable.
well
in
(419)
is d o n e by f i r s t
the p o t e n t i a l
(434).
Likewise,
one has to u s e Eq. (432),
to e x p r e s s
sity from
in t e r m s of t h e d e n s i t y ,
to o b t a i n
l e a v i n g us w i t h a p o t e n t i a l
Starting
these
The
step from
for V, t h e r e b y
and t h e n u s i n g this
of the p o t e n t i a l
is a f u n c t i o n a l
ee
of V. T h i s n(V)
Let us i l l u s t r a t e
tionals.
a functional
in w h i c h V
n as a f u n c t i o n a l
(434)
one of the two.
s o l v i n g Eq. (20)
of
quan-
alone,
of the d e n s i t y ,
then eliminates
functional
the den-
E(V,C).
ideas w i t h the r e s p e c t i v e
TF func-
from
ETF(V,n,()
= f(d~) (-
I ) [_2 (V+() ] 5/2
15~ 2
Z
f (d~) (V+ ~) n
(2-435)
÷
I
÷
÷
n(r)n(~')
+ ~ f (dr) (dr')
I~-~' I
w h e r e V e x t is n o w t h e p o t e n t i a l
-Z/r,
w e get the d e n s i t y
(N
,
e n e r g y of an e l e c t r o n w i t h the nucleus,
funcitonal
of Eq. (95), a f t e r
first
inverting
[Eq.(51)]
I
n =--[-2(V+()]
3~ 2
3/2
(2-436)
to
V =
_
which then allows
dingly.
~I ( 3 ~ 2 n ) 2/3 - (
to r e w r i t e the f i r s t
On the o t h e r hand,
v
is s o l v e d
,
(2-437)
and s e c o n d t e r m in
(435)
accor-
if
= - -z , [ (dr'
÷)
r
n(r')
÷
÷
Ir-r' I
(2-438)
for n,
n = - ~1
V 2 (V+
Z
~)
(2-439)
111
(this is, of course,
Poisson's
equation),
we can eliminate n from the
second and third term in (435) and are led to the TF potential functional of Eq. (45). Of course, within the framework of the TF model,
three functionals
E(V,n,~),
E(n,~),
and E(V,~)
lent, but I repeat:
as a basis for improvements
tion, the potential
functional
over the TF approxima-
is the preferable one.
We have investigated
TF model.
the
are perfectly equiva-
earlier the scaling properties
of the
Let us now see, what one can state about the behavior of
the exact density functionals
mations of the density,
n(~')+n
Ekin(n)
and Eee(n)
under scale transfor-
(~') : b3 n ( ~ ' )
T h e TF approximations
intuitively expect:
(&in(n))TF
(2-440)
to Eki n and Eee scale in the manner that one would
= f (d~')
1 (3~2n(~,))5/3
I 0~ 2
÷ ~ -3 f(d~' )
I ( 3 ~ob 3 n (~r')
÷ ) 5/3
I 0~ 2
= 2z (Eki n (n))TF
(2-441)
'
and likewise
I
(d~" n (~') n (~")
(Eee(n.))TF = 2 ] (d~')
) IF'-~''I
(2-442)
+ a[Eee(n)]TF
For the exact functionals,
the equations
Eki n(n ) = ~2 Ekin(n)
(2-443)
Eee(n ) = # Eee(n)
(2-444)
and
do not hold, however;
even their combination
Ekin(n ) + E e e ( n ) = ~2 Ekin(n ) + ~ Eee(n)
(2-445)
112
is only true if ~=1+e with an infinitesimal
vation has been made only recently,
that the statement
der to derive
(445)
the virial
2 Ekin(n)
from the minimum
is, indeed,
uniquely
= - Eee(n)
property
(445), we recall
a certain
ground-state
only needed
potential
Vex t and
where E ~ is the ground-state
o
I~o > according to
and a certain
I~o~>
I~>
about Eqs. (443)
÷ ,. . . , ~ 1 ~ o
<~r1'
+!
operator
U(~)
Let
to the actual
,
(2-447)
>
Inasmuch
+
<~r1,, .... ~rN[
state.
for the ones that go
for n . Clearly,
is scaled as in Eq. (440).
÷
ground
obeys
= Eo~ I~o~>
energy
+,I . .
. . . +rNl~o
,
<r
> ÷ 3~/2
U(p)
(417).
these remarks
14o> for the ones related
(Hkin + H ~ext + Hee) I~o~>
where the unitary
(2-446)
that to any given density there correspond
with the scaled density n . Thus
3N/2
note
in or-
Vext(~')
functional
proving
density n, and write Vext
~ and
then the density
for such ~ 1 ,
+ f(d~')n(~')~'.~'
of the density
external
us keep the notation
obser-
theorem
As a first step towards
through
E. This surprising
by Levy and Perdew. 24 Please
if we transform
i2-448)
,
as
÷
= <r1', .... r~IU(~ )
,
(2-449)
is given by
I N ÷ + ÷ +
= exp { i ~ - - - ( r j . p j + p j . r j )
logp }
,
(2-450)
j=1
we can read
(448)
as
(2-451)
l~o> ÷ u(~)I~o>
The point is that this scaled
from considering
immediately
I~o> is not equal to
the Schr~dinger
obtained
equation
. This
emerges
obeyed by U(~) I~o>, which
from the one satisfied
U(2) (Hki n + Hex t + Hee)
I~>
by
is
I~o >. We have
U -I (~) U(2)[~o > = EoU(~)I~o>. (2-452)
113
The action of U upon ~j and pj is simply
u(~) ~
u-I(I~) = ~ ~j
]
(2-453)
U(~) +pj U-I (~) = ~I +pj
,
so that
u-I
U(B) Hki n
=!
(~)
p2 Hkin
'
(2-454)
U(~) H
U-I (~) = I-- H
ee
~ ee
,
and
N
U(~)
(2-455)
Hex t U "I (~) = > Vext(B~ j)
j=1
Consequently,
(Hki n + ~ZU(B)HextU-I (B) + ~Hee)U(B)14o>
= B2EoU(~)I~o > ,
(2-456)
which, in view of the factor B multiplying Hee , is not of the form required for I ~ > in Eq. (447). Thus, indeed
u(~)I%> ~ I % ~>
for ~ I .
Nevertheless,
l~2>v
,
(2-457)
and U(~) I~o> are not unrelated.
cular, they give rise to the same density,
(410). This implies the equality
<~2 IH ext
~ 1%~>
In parti-
n (r), when inserted
= <~ o I u -I(~) ~ ext
~ u(~)1%>
'
into
(2-458)
and for the same reason
<~o [ Hext
I~o> = <~o~ I U(~) Hext
U-1
(~) [~o~>
(2-459)
We are now prepared to employ the minimum property of the expectation
value of the Hamilton operator of Eq. (447) in the form
<~O~ I (Hkin + H~ext + Hee) I~o~ >
=<
114
<~o I U-I(~)(Hkin+He~xt+Hee)U(~)
which,
as a consequence of Eqs. (454) and
Ekin(n
~o >
,
(2-460)
(458), says
) + E e e ( n ) =< ~2 Ekin(n)
+ ~ Eee(n)
The equal sign holds only for ~=I, in the first place.
hand side always exceeds the left hand one for ~ I ,
(2-461)
Since the right
however,
the two
sides must agree up to first order in ~=~-I, at least, so that the
equal sign actually applies to ~=I+~ with an infinitesimal
~. This is
the statement we made at Eq. (445). Another way of expressing the same
fact is
d
(n)
d-~[Ekin
+ Eee (n)] I
= 2 Ekin (n) + Eee (n)
~=I
(2-462)
We can also exploit the minimum property of the expectation value of
the Hamilton operator of Eq. (456). Here we have
~2
<~o I U-I (~) (Hkin +
U(~)Hex t U
-1
(~) + ~Hee)U(~)J~o >
(2-463)
<~o~ I (Hkin + ~2U(~)Hex t U -I (~)+~Hee) I~o~ >
or with
(454) and
(459),
~2 Ekin(n)
where,
,
+ ~2 Eee(n)
_~ Eki n(n ) + ~ Eee(n )
,
(2-464)
again, the equal sign is true for all ~'s that differ from uni-
ty at most infinitesimally.
Equations
(461) and
(464) can be combined into two state-
ments about Eki n and Eee individually,
(~-I) [Eee(n ~) - ~ Eee(n)]
namely
(2-465)
~ o
and
(~-I) [Eki n(n ) -
2
Eki n(n)]
=
< o
It seems natural to assume that the left-hand sides in (465) and
(2-466)
(466)
are of second order in ~=~-I for small g. If this were true, these equations would mean that
115
d
(n) I
- E
(n) > o
d-~ Eee
ee
~=I
(2-467)
and
ddb Ekin(n
)I
- 2 Ekin(n)
~=I
< o
(2-468)
As a matter of fact, we shall see below that equal signs have to be
written
in
left-hand
(467)
and
sides in
(468)
(465)
instead of ">" and "<". Consequently,
and
(466)
spective
square brackets of order
equality
sign holds up to order
the results of Ref.24,
and
are,
at least, of order
g3. Therefore,
g3, at least.
the
g~, the re-
also in Eq. (461) the
These remarks go beyond
where Levy and Perdew stopped at stating
(465)
(466).
For a proof of what has just been said, we have to turn to
the p o t e n t i a l
proximation
functional
EI(V+~).
to E I responds
(El (V+~) ]T F
+
In Eq. (216) we found that the TF ap-
like
~ 5v/2-3
[El (V+~.) ]TF
,
when V and ~ are scaled according to Eqs. (211) and
V(r)
÷ bY V(~r)
A l t h o u g h the exact El(V+{)
,
~ ÷
(2-469)
(214),
~
does not behave
(2-470)
like
(469)
for arbitrary
v,
it does so for v=2:
E I (V+%)
÷
÷
V(r)
V
2
El (V+{)
(2-471)
for
÷
÷ = 2
(r)
÷
V(br)
(2-472)
We d e m o n s t r a t e
this by first o b s e r v i n g that
I p2
2
÷
n(- [
- ~ V(~r)
- ~
=
=
Z
{)
(2-473)
-
U(~)
n(-
~I
p2
-
V ( r÷
)
-
~) U-I (~)
116
where
U(~) now denotes
U(~)
This i s
the one-particle
= exp {i [(r.p
+ p.r)
version
of
(450) ,
log ~}
(2-474)
used in
= tr(1 p2+ V(~)+~)~(-
El(V+{)
1 p2_ V(~)-~)
+
÷ tr(~I p2 + ~2V(~r)+~2~)q(= ~2tr U(~) (~p2+
I
= ~2 EI(V+~ )
v(~)+~)
÷
2 ~)
[I p2 - ~ 2V(~r)-~
H
I 2 -v(~)-~)
(- yp
U-1 (~)
,
(2-475)
or
EI(V ~ + ~2~)
indeed.
[The invariance
employed
Before
Unfortunately,
there
from considering
I-v12
[~ r
i~ +=
potential
Vext(r).
red then was the Coulomb
discussion,
we preserve
has been
one),
U
the at(473) would
(~), such that
(2-477)
÷p
except
for v=2, as emerges
of the transformed
quantities:
(2-478)
v=2.
transformation
In that earlier
[Eq. (213)],
potential
-Z/r.
the structure
properties
of the TF model
of the effective
by a corresponding
Z appropriately
to show where
The analog of
,
about the scaling
that a scaling
V(r) must be accompanied
permutations
,-v12_ ÷]
÷ -I = i ++
I
p : U [~ ,p ] U
unless
In the section
by changing
a unitary
is no such operator,
the commutator
This is a contradiction,
ternal
for v~2 fails.
(not necessarily
÷p U~I (~) = - ~ / 2
U(~)
we remarked
it is instructive
the argument
÷
(~) ~ U~I (~) = ~r
U
(2-476)
(475).]
proceeding,
an operator
,
of the trace under cyclic
in the last step of
tempt of repeating
require
= ~2 EI(V+~ )
transformation
context,
because
potential
of the ex-
this was achieved
the only Vex t conside-
In the more general
present
V-Vex t by scaling Vex t like V in
Eq.(473),
Vext(~ ) ÷ Vext,~(~ ) = ~2 V e x t ( ~ )
The density
is, of course,
scaled as in
(2-479)
(440). Under these simultane-
117
ous transformations of V,n,~, and Vex t, the potential-density functional of (434) behaves as described by
E(V,n,~) ÷ E (V,n,~) = E(V ,n ,~2~)
= ~2{EI(V+ ~) - ~(d~')(V(~') -Vext(~'))n(~')-~N}
+
Eee(n )
(2-480)
Since E(V,n,~) is stationary under infinitesimal variations of V,n, and
~, all first order changes must originate in the scaling of Vex t. [The
same argument was also applied to ETF(2) of Eq. (219).] Thus,
d-~d E (V,n,~) i
= ](d~')n(~') ~
~=I
Vext, ~(~')I
~=I
,
(2-481)
or with (479) ,
d
d-~ E2 (v'n'~) I
=
~=I
(dr')n(~') [2 Vext(r')+r'-V'Vex t (~')].
(2-482)
On the other hand, Eq. (480) implies
d
d-~ E2 (v'n'~)I
= 2 {E I (V+~)- f (dr'÷) (V(~')-Vex t(~'))
~=1
× n(~')-~N}+ ~
Eee(n ) 1
~=I
(2-483)
The equivalence of these two right-hand sides, combined with the virial
theorem (446), yields
d Ee e(n ) I
- E ee (n) = 2 Ekin (n)-2{E 1 (V+~)-f (dr')Vn-~N} .
d-~
2=I
(2-484)
The last step consists of recognizing that for the actual V,n, and ~,
the contents of the curly brackets equals the kinetic energy. This emerges from Eq. (428). Consequently, the right-hand side is zero. We arrive
at
d Eee (n)[
d-~
2 ~=I = E ee (n)
'
(2-485)
118
and as a consequence
of
(462),
d
(n) I
= 2
(n)
d-~ Ekin
Ekin
~=I
Indeed,
the statements
(2-486)
following Eq. (468) are justified.
Please be aware of the following mental trap.
is eliminated
functional
from E(V,n,~),
E(V,~),
If the density
so that we are left with the potential
one could think that the resulting kinetic
Ekin(V,~)
= E I (V+~)
- f (d~')V(~')n(~')
energy,
- ~N
(2-487)
scales according to
Ekin(V
inasmuch as
,~2~)
= ~2 Ekin(V,~)
(2-488)
[Eq. (14)]
6 V EI(V+~)
together with
= f(d~')6V(~')n(~')
,
* ~3 n ( ~ ' )
,
if V and ~ are scaled as in
(2-490)
(472). This is not so, however,
functional that is to be inserted
not the one obtained
6 n Eee(n)
[Eqs. (432) and
(2-489)
(471), implies
n(~')
potential
,
from
(433)].
into Eq. (487) for n(r')
(489), but the one that emerges
= f(d~')6n(~') [V(~')
because the
- Vex t(~')]
In the TF approximation,
is
from
(2-491)
for instance,
this is
the Poisson equation
n(~')
=
-
~
I
V,2
÷
(V(r')
-
+
Vext(r'))
in which the scaling of V and Vex t [Eqs. (472)
n(r')
different
÷
n(~r' )
from the desired
(2-492)
and
(479)] produces
,
form of
(2-493)
(490). Therefore,
Eq. (488) is not
true, not even for ~'s that differ from unity by an infinitesimal
amount.
119
The kinetic
functional
energy by itself
formalism.
is not a central
quantity
in the potential
What we have just seen is an illustration
of this
remark.
Relation
between
at the beginning
nection
between
the TF approximation
and Hartree's
of Chapter
is the promise
TF theory
This time has finally
One there
and HF theory
Somewhere
to discuss
"to some extent
the con-
in Chapter Two."
come.
Hartree's 25 basic
state wave-function
method.
idea consists
by a product
in approximating
the ground-
in which each factor refers
to just
one of the electrons:
<~I'
,r 2'
'"""
The ~j's are supposed
f(d~')
so that the wave
requirement
exchange
,rN ' i~o > ~ ~i(~i)~2(~2)
~j*(r')
~k(~')
function
(494)
= 5jk
,
(2-495)
is properly
normalized
is not satisfied
are not treated
correctly.
by
is consistent.
mation,
We are actually
not about the Hartree-Fock
This restriction
be repeated
is not essential
for a comparison
the TF model that includes
rived in Chapter
TF model
Four.
and Hartree's
(494).
TF model,
context,
neglecting
which does include
ex-
approxiexchange.
The argument
of HF theory with the proper
interaction,
The
Consequently,
about Hartree's
for the discussion.
the exchange
can
extension
of
which will be de-
At this moment we are content with the simple
ansatz
(494).
With Eq. (494) we obtain
values
talking
model,
to unity.
In the present
where we want to make contact with the original
change
(2-494)
to be orthonormal,
of antisymmetry
effects
..~N(~N)
approximations
of the three parts of the many-particle
to the expectation
Hamilton
operator
(409).
These are given by
Eki n = < % IHkin I~O>
N
I
,
.~,
~ - - - f (d~') ~ ~'~j (~')
,j
j=1
and
Eext = <~olHextl
~o >
(2-496)
120
~""f(d~') ~j*(5')
(2-497)
Vex t(~') 0j (~')
j=1
as well as
Eee = <0o I Hee I ~o >
N
N
j=1
k=1
k~j
- -. + 0 k (-~. ~ " ) ]
Ir'-r" I
x 0j (5')
(2-498)
Since the description does not pay attention to the exchange energy, we
do not have to be pedantic either when it comes to excluding the selfenergy. In other words: it is perfectly consistent to include the k=j
term in Eq. (498). The approximation to the ground-state energy is then
E = <0olHmpl~o > ~ EHartre e
N
=
~(d~'
I_+, ,(~,).V 0j (~'
(~'
(~')0j (r')
){2 V ~j
)+0j*
)Vex t
j=1
N
+ ~0jl*(~') [ ~--~(d~")0k*(~") IF'-~"II O k(~'')]0j (~')}
k=1
(2-499)
The as yet undetermined 0j's are now chosen such that EHartre e is stationary under infinitesimal variations of them. Thus
N
,
~f(d~')60j
9=I
I
,
÷,
(~') {- ~V 2+Vext(r
× ~k(~")}0j(~')
N
+
÷
) + ~ f ( d r " ) 0 k * ( r " ) - - _ r II
E~,+I
k=1
= o
(2-500)
The variations 6~j* are not arbitrary but subject to
f(d~')8~j*(~')
~k(~') = o
,
(2-501)
121
which is a consequence of the orthonormalization
(495). Therefore,
Eq.
(500) implies
N
(~') +~---~ (d~"),k* (~,, )
1 ,2 +
{- gV
Vex t
k=1
I
l~'-~"I
*k (~'') }~j (~')
¢2-502)
N
= >
ejl ~l (~')
/=I
where the constants
sjl are the Lagrange mulitpliers of the constraints
(495). The single-particle wave-functions
termined simultaneously
~j and the sjl are to be de-
from Eqs. (502) and (495).
The hermitian property of the differential operator {...} in
(502) is employed in demonstrating that the matrix (sj/) is hermitian:
Ejl = f(d~') ~/*(~'){...}
~j(~')
(2-503)
= ~(d~') ~j(~'){...}
~l*(~')
Another observation is the nonuniqueness
= e~j
of the solution to (502) and
(495). If ~j and Sjl are one solution, then
N
~j = ~-- uj/ ~£
(2-504)
/=I
and
N
=
uL
(2-505)
k,m=l
is another one, whereby
(uj/) is any unitary matrix,
N
)
(2-506)
Umk = 6jk
m=l
It is essential here that the density, that appears in (502), is invariant under such a unitary transformation:
N
(r') =
~k
~*(~')
k=1
~k (r')
122
=>
N
~k*(~')~z(~'l =
(2-5071
n(~')
k=1
[The approximate wave-function (494) is obviously not invariant under
(504). This is nothing to worry about, because as soon as (494) is antisymmetrized, the effect of (504) reduces to the mulitplication by a
phase-factor.]
Since (s41)j is hermitian, we can choose (U4/)j such that
(~j/) is diagonal,
sj/ = Ej 6j/
(2-508)
Then Eq. (502) is Schr~dinger's
{_
equation in appearance,
~ (~')
?,2 + V(~,)}~3(~, ) = ~ej ~j
where the effective single-particle
potential V is
N
V(~') = Vext(~')
(2-509)
*
+ T--f(d3")~k
k=l
I
(3")13'-3"I
~k(r")
'
(2-510)
which is equivalent to
N
>
~k* (3') ~k (3')
k=1
4rcI V'2(V(3')-Vext(3') )
(2-511)
Let us now look at the Hartree energy. It is
N
•
I
EHartree = j~=1f (d3')~j* (~') {- ~V
I (V(~')
+ ~
N
,2
-
+Vex t
Vext
(3')
(3')) }~j (~')
N
= ~](d3')~j
j =I
(3') 7-- ejl ~/(r )
/=I
(2-512)
N ¸
-lf (d3') (V(~') -Vex t (3'))Z~j*(3')~j
j=l
(3')
where both (502) and (510) have been used. With the aid of the ortho-
123
normality
of the ~j's and with Eq. (511) we obtain
N
EHartree
This will look
= ~=I
even more
N
>
~ f(d~')[~' (V(~') - V e x t (~,))]2
ejj
like the TF potential
N
ejj =
j =I
functional
. (2-513)
after we use
N
gjj =
j =I
ej
(2-514)
,
j =I
%
in conjunction
with the fact that the E. are the N smallest
3
of the single-particle Hamilton operator
H = ~I p2 + V(~)
eigenvalues
,
(2-515)
to write
N
ejj = tr H q(-H-~)
,
(2-516)
j=1
where,
of course,
~ is such that the count of occupied
states
equals
the number of electrons:
N = tr q(-H-~)
If we combine
(516) and
(2-517)
(517)
in the now familiar way,
N
ejj = tr(H+~)q(-H-~)
- ~N
3=I
= E I (V+~)
- ~N
,
(2-518)
then
EHartre e = E 1 (V+~) - 8~ f (d~') [~' (V(~') - V e x t(~'))]2- ~N.
(2-519)
It becomes
ween the TF approach
the optimal
clear now what the fundamental
and Hartree's
single-particle
method.
wave functions
difference
is bet-
The latter asks: what are
to be used in
(494)?
124
The answer is given by the Hartree equations
(502).
26
But suppose we
do not care that m u c h for the ~i's. Then we can equally w e l l put the
question:
diately:
what is the best e f f e c t i v e p o t e n t i a l in
the TF potential,
if EI(V+~)
(519)? We reply imme-
is evaluated in the semiclassi-
cal limit. Does this m e a n that the TF model is an a p p r o x i m a t i o n to
Hartree's d e s c r i p t i o n ? No,
it is rather the other way round: the Har-
tree p i c t u r e contains more detail than it should.
a p p r o x i m a t i o n s made before a r r i v i n g at
In v i e w of all the
(519), there is a b s o l u t e l y no
point in being e x t r e m e l y p r e c i s e w h e n e v a l u a t i n g EI(V+~).
S u m m i n g up: the TF model and Hartree's m e t h o d are really
two independent,
t h o u g h related,
approaches.
N o n e is a priori the bet-
ter or w o r s e one. W h e r e a s I do not want to go as far as Lieb does
TF theory is well defined.(...)
["...
- a state of affairs in m a r k e d con-
trast to that of HF theory."27],
I do have the i m p r e s s i o n that in apply-
ing TF methods one is m o r e conscious about the p h y s i c a l a p p r o x i m a t i o n s
that enter the development.
In one r e s p e c t the H a r t r e e detour over the s i n g l e - p a r t i c l e
wave functions is superior to the TF p h a s e - s p a c e integral:
ger e q u a t i o n
the Schr~din-
(509) treats the strongly bound electrons c o r r e c t l y w i t h -
out any further ado. We shall see in the next C h a p t e r how the TF m o d e l
can be modified,
in a simple way,
in order to handle these innermost
electrons properly. W i t h this i m p r o v e m e n t the TF d e s c r i p t i o n is in no
way inferior to Hartree's.
Please do not miss how n a t u r a l l y we have been led to a potential functional,
more,
Eq. (519), not to a density functional.
Here is, once
support for our v i e w that TF theory is best t h o u g h t of as formu-
lated in terms of the e f f e c t i v e potential.
Then the d e n s i t y is not a
f u n d a m e n t a l but a d e r i v e d quantity.
Problems
2-I. For the g e n e r a l i z a t i o n of the i n d e p e n d e n t - p a r t i c l e H a m i l t o n operator of Eq. (3) to
H = :I( ~ -
~(~) )2 + v(~)
,
e 2
w h e r e ~ = ~-~ = 1/137.036...
is S o m m e r f e l d ' s
and A is an e f f e c t i v e v e c t o r p o t e n t i a l
analogs of
(14) and
(20) are
fine s t r u c t u r e c o n s t a n t
(in a t o m i c units),
show that the
125
~I EIP
~I El
=
and
5(~')=2<r']
[(p-~A)~(-H-E)+~(-H-E) (p-~A)] I~'> .
Then generalize Eq. (25) to read
6Eee = f (d~') [6n(~')Vee(~')
Next conclude that, instead of
- 0.63.Aee]
(30), the stationary energy expression
is now
÷ ).j(r')
E = Eip - S (d~')Vee (~')n(~')+~S (dr'
÷ )A ee(r'
+ E ee '
since
= Aext + Aee
'
with a given external vector potential Aex t. How does the TF version
of E I depend on A?
2-2. Another application of the stationary property of the electrostatic potential functional of Eq. (78). Instead of ~(~), insert ~(~'),
where
r'
is
related
to
and an infinitesimal
r
through
an
infinitesimal
translation
b y 6~
rotation around 6~,
5 = 5' + 6~ + 6~×r'
Use
(dS) = (dS')
and
(~(5,))2
= (~,~(~,))2
to write the primed
energy as
E' = fCd~'l[pG' +6e+6exr
÷ ÷ ÷' ) ~ ( ~ ' ) _ ~ (I? ,÷~ ( ~ , ) ) 2 ]
<
= E = S(dS)[p(5)¢(5)-
1 (V¢
÷ (r))
+
~-~
21
,
where the equal sign holds to first order in 6~ and 6~. Conclude,
the self force vanishes
[Eq. (86)], and also the self torque,
f CdS) ~ (5) 5 × ( - ~ (5)) = o
2-3. Write a computer program for the TF function F(x) as outlined
around Eq.(200).
Use it to confirm
that
126
= 1.80006394
~dx F(x)
O
co
fdx[F(x)] 2 = 0 . 6 1 5 4 3 4 6 4
,
O
f d x [ - F ' (x)] 3 = 0 . 3 5 3 3 3 4 5 6
,
O
co
~dx/~'-~-/-x = 3.915933
O
2-4. W i t h the c o m p u t e r
xF(x)
2-5.
occurs
p r o g r a m of P r o b l e m
at x = 2 . 1 0 4 0 2 5 2 8 0 ,
This maximum
of xF(x)
is r e l a t i v e l y
to F (x) is t h e r e f o r e
~"(x)
=~6[~(x)]2
=
s u b j e c t to F(o)=1
,
~2
, ~(~)=o
broad,
so that
~ constant
represented
x = const.
, and
fdx x 1 / 2 [ ~ ( x ) ] 3 / 2
of
where F(x)=0.2311514708.
F" (x)/[F(x)] z = [x F(x)]-I/2
An a p p r o x i m a t i o n
3 c h e c k that the m a x i m u m
by the s o l u t i o n of
,
(to fix the v a l u e of ~)
= I
O
Find this
~ ( x ) . 28 How good i s
in c a l c u l a t i n g
2-6.
Insert
the numbers
%1(o)
~l'(o)
= -
Find
~2(t)
when i t
is employed
3 and 4?
into Eq. (274) to find ~i'(o)
and ~2' (o)
in
I + #i'(o)
C o m p a r e w i t h Eqs. (280) and
2-7.
approximation
of P r o b l e m s
of Eq. (283)
as the c o e f f i c i e n t s
this
I + ~2'(o)
12
+
...
(281).
fr o m Eq. (278); t h e n e v a l u a t e
~3(o).
Use it to show t h a t
127
in Eq. (285) :
(i)
(;)2+ 0 ((N/Z) 3)
524288)
21067
0((N/Z) z) = ( 1 2 - - +~
60~2
(ii) in Eqs. (286)
(289):
2
223
= (~+ 30~2
0((N/Z)2)
(iii)
and
262144, N z
3)
~ J
( ) +0((N/Z)
in Eq. (287) :
0((N/Z)2 ) = (22
2-8. Use the recurrence
el(t)
relation
(278) to show that
(l-t) (51+2)/2
for
t ~ I
Look back at Eq. (311) and notice
that,
indeed,
A
is in the
2-9.
(1-t) 7/2-term,
Show that,
for I>A
~(t)
AS I + % t i ÷ I
monstrate
~
1 9 0 1 9 + 2883584) (N ) 2 + 0((N/Z) ~)
90~ 2
2025~ 4
~
400
12
, SO that
and of
, el(t)
tl
(t-tl) 2
A 2 in the
has a pole,
for
the first occurence
of
(1-t) 6-term.
at
t=tl>o,
of the form
t ~ tI
¢~(t)~l[~(t)] 3/2
for
t1<t ~ I. Use this to de-
that
1-tl-~
7
d~
o /I
4 .5/2
_ ( ~ u 9 ) 2/5
for I>> A
2-10.
Upper and lower bounds
it is combined
f(x)
to A -2"3
! can be obtained
with the inequalities
of
from Eq. (301) when
(242). A suitable
trial
function
is given by
iF(x)
f (x)
X~oo(x2-x)
where q is fixed,
its derivative
for
o < x=1< x I
for
xI< x
I
x ° is arbitrary,
are continuous.
,
and x I and x 2 are such that f(x)
For q~>o, x I is sufficiently
and
large to jus-
128
tify the use of the asymptotic
implies
4
X 2 = ~X I
form
(179) for F(X~Xl).
Show that this
432
x14 = q-7~O
and
Then derive
co
2 ~dx If(x)]5/2
5
I/2
O
x
2
B
7
=
2 (12) 5 + 2(X_~o)5/2 (4)3.~13,5~-9/3,~;
35
7
xI
and
1 ~dx[f' (x) +
20
]2 =
B
56
Xo +7 ( )2 xl
x17
Putting everything together you should have
(12) s 30
x17- [7
3
> 7B-
8~ + 3 Xo
4x~I ]
3/3
3 A-2/3 q7/3
7
,
for
q÷o .
It is then useful to switch from x ° to a new independent parameter, I,
by setting XoE12/3q -I/3. Check that then xi=(432)I/4 1 I/6 q-I/3, so
that, for all I>o,
A-2/3
Optimize
7 f2/3
> ~
8
~
(56~_
i-7/6
3/3 30)
I and find the lower bound on A-2/3 of
this optimal
(303). Show that, for
I, the ratio x2/x ° does not equal unity. Consequently,
the
trial f(x) does not change its sign at X=Xo, as the actual f(x) does.
Impose x2=x ° and demonstrate that a lower bound on A -2/3 emerges, which
is worse than the previous one.
For an upper bound on A -2/3 use the trial function
g(x)
~F' (x)
for o~x~x I
+ q/x O
= ~ - q / X o ~ 1 - x / x 2)
for Xl~X~X 2
for x2~x
Make sure that g is continuous
functional of
and obeys Eq. (243). Then evaluate the g-
(242). You should get
129
A-2/3
6t
]7/3
I )I/3[ I ( ~
(5t'1)(3-t)
(I - ~ t
~ - - - 16 + 191t - 74t 2 )
< I[
_74tl/3(1 _ 1 t ) 1 / 3 (1 - t 4/3 )] ,
+
I
where the range of t=xl/x 2 is ~ < t ~ 1. Find
(numerically)
value
for t and thus the upper bound on A -2/3 of
2-11.
Insert Eq. (316)
the optimal
(303).
into
(t) = ~ f (t x o(q)) I
q
q
q÷o
and derive
(352).
2-12.
Derive Eq. (462)
2-13.
Because
dimensional
appear
directly
from Eqs. (433), (432),
of the homogeneity
space,
the density
and isotropy
functionals
finitesimally
translated
this with the stationary
(428).
of the physical
Ekin(n)
in Eq. (417), have the same numerical
and
value
and Eee(n),
for n(~')
which
and the in-
and rotated
n(r') = n(~' + 6g + 6~×r').
property
(417)
of
three-
Combine
to show that there is no net
force,
[ (dr' )n (r')
(-V Vex t IF')) --o
and no net torque,
f (d~')n(~')~'
× (-~'V ext (~')) = o
exerted
on the system by the external
Problem
2?
2-14.
Show that the density
functional
potential.
Are you reminded
of the kinetic
of
energy is given
by
Eki n(n)
if there
= f (d~)
1 (~n)Z/n
is only one electron.
no contradiction
to the general
,
This does scale
statement
like
(443). Why is there
that Eki n does not obey
(443)?