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The theory of density functional gradient series
Solid State Communications, Printed in Great Britain.
Vol. 63, NO. 11, pp. 106771070,
003%1098/87 $3.00 + .OO 1987 Pergamon Journals Ltd.
1987. 0
THE THEORY
OF DENSITY
FUNCTIONAL
GRADIENT
SERIES
J.C. Stoddart Department
of Physics, (Received
University
of Sheffield, Sheffield S3 7RH UK
29 September
1986 by C. W. McCombie)
It is shown how a density functional correlation energy can be formulated terms.
IN A PREVIOUS PAPER the density gradient series for the exchange and correlation energy in density functional theory was investigated through the use of the effective density n(r). In this approach the exchange and correlation energy is written &c
bl = I We0 @(r))dr = j n(rkxc(r)dr,
(1)
expansion for the exchange and to include well defined dominant
B,(O,no) =
2,0
&00~2,)
;2,
2 =
etc. are easily proved, h, (4, no)
where eO(n) is the exchange and correlation energy per particle of a uniform electron gas of density n. The effective density ii was obtained in a perturbation series as,
=
=
n,
+ j W
+
1 o(r
sI, r -
-
-
s,
s2, n,Ms, s,, r -
x A(s,)A(s2)A(s~)
ds,ds,ds,
etc., then this means that fqOOn,)
=
&ooo, no)
eO(nO) + j B,(r
kW2Ws, ds2
s3, noI + . . .,
(2)
-
n(r)
=
-
sI, r -
x A(s,)A(s,)ds,ds,
no>=
i(r)
s2, no) + . . .,
=
no
x (3)
s, no)
+
i
t$~w(r)w(s). 0
(4) etc. As the sum rules
. .
+
(7)
+
sdq m
me @q,)&q2)
e'q'r3q,
n,@(q)
iq,+nz).r 4,,
+
q2, no>
. . . >
(8)
and expanding G(q, no) and &q, , q2, no) in powers of q, and comparing with (7) we get,
@(q,no> =
0
2.4, 0
1.
dq,dq2
2 -w(r, no),
B2(r, s, no>=
(6)
In accordance with the usual procedure (1) the gradient coefficients g(n) etc. can be obtained from the perturbation kernels in (2). We have,
s, n,)A(s)ds
and we obtain,
4@,
0 etc.,
n(r) + g(n(r))(Vr2n(r))
+s
+ IBzcr
=
=
The problem with using (1) in practice is that 5 is not in the right form to apply to a localized atomic density. One obvious way to remedy this is to write the associated gradient expansion for ii in the form,
with n(r) = n, + A(r). The formulation in this way corresponds to a normalized exchange and correlation hole. The kernels in this expansion are related to the ones in the usual expansion of e,,-(r), given by, =
no)
n,)A(s)ds
+ j l(r - sI, r -
e&r)
where
( drerq”B, (r,
and &(O, no) fi(r, [n]
(5) 0
1 + 02(no)q2 + . . . ,
(9)
and g(n) = -o,(n), with similar relations for other coefficients. This produces through (1) essentially a local approximation to the exchange and correlation energy, and is obviously just a re-arrangement of the usual gradient series terms, so that correction terms may be argued to be small (2).
1067
THE THEORY
1068
OF DENSITY
FUNCTIONAL
Density gradient series for the energy have been up to now the only form of local density approximation which has been available. Although useful, there are difficulties in obtaining the gradient correction terms, and attention has been focussed on obtaining non-local density functionals to represent the exchange and correlation energy. However, a local density form for the energy has considerable computational advantages, and we investigate here how the usual gradient series may be reformulated to improve considerably its usefulness. We start by recalling that a density gradient series is valid when the transform ri(q)is localised round q = 0, and as this is not usually the case large errors can enter. In fact, the coefficients in such a gradient series are obtained, under the above assumption, by expanding the perturbation kernels I?, (q,, no) etc. in (3) in power series in the q, about q, = 0. We will assume that we have an electron density n(q) which is locahsed in q around q = 0 and about another value q = K. Thus we have, n(r)
=
n, (r) + n?(r),
(10)
where n, ,n2 are derived from the two parts above. We look now for an expansion for exe(r) which uses these properties. Because of its association with a normalized exchange and correlation hole it is convenient to formulate the expansion in terms of the effective density h(r). We first write n,(r) = n, + A,(r) and expand the effective density as, ri(r. [n, + n2]) = +
-
s, r, [d
+ \&Cc
-
sI, r -
fi(r, In, + n?l
=
+ h(r, [~I~],n,(r)) V*m,(r))
+
D, (r -
s2, r, hl,
We are concerned just with the leading term in such an expansion and so will not elaborate upon the form of further terms. With the transforms b, and D”?defined by.
x
(pi3 p2,
+ 2 1 k(r + 3J”l( r -
=
s, r -
D2(r =
w(r -
G@, r, M no> =
hl, 4)
Mr,
’
c?n,
(16) etc., and from (14) u(r, [d, no> = fii(r,[n,,+ nd. From (12) and (13) we have, 1 + 2 J&
=
s2, r,
b,l, no>
k(r -
s, , r -
s2, n,)
3 J”Jr -
x
n2 @3 h-h.
We can now construct
(17)
eiP”40,
p, no)&(p)
s, r -
er(p+q)‘rkO, p, q, no)
s, no>
d2(0, 0, r, [n,], n,)
(12)
. . .>
(18)
=
3 s&
d’P”i(OOp, no)
way a gradient
.)
(19)
using the limiting values given by (6). From (2) we get a(r, [a,], rt,,) and as n, is arbitrary we can equate the coefficients of different orders of n2 on both sides of (16). In this way we get a series of relations between the kernels in fi. Thus from the first of (16) we get 2&O, P, n,)
s3, n,) (13)
in the usual
+
x fiiz(p) + .
sj, n,,)
s2, r -
fi,(p)fi,(q)
and,
.
+
sI, r -
+
x
s2, n,,)n2(s2)dsz s2, r -
x n2(s&zZ(s3)dsldsj
(15)
then as usual. the coefficients in (14) can be related to the terms in the expansion of b,, 6, etc. about pi = 0. As vi,(q) is localised about q = 0 then this expansion should be valid. We obtain then the “sum rules” involving the leading term a as
(11)
are given by,
s, r -
r, hl, no),
no)
.
+
s, r, [nz], no)
(14)
n,)A,(s)ds
x A,(s,)A,@Jds,ds, (2) the kernels
using (11). in the
a(r, [n,]. n,(r))
+ 3 J $$
From
Vol. 63, No. 11
SERIES
series in the part n, (r) of the density, form,
8, (0, r, [hl, no)
fi(r, En, + 41)
1 D,(r
GRADIENT
=
340, p, q, %I) =
WP,
4
a I 10
,
d&P, q> no) an
0
,
(20)
Vol. 63, No. 11
THE THEORY
OF DENSITY
FUNCTIONAL
and so on from higher orders. From the second of (16) we get, 31^OOP,%I) =
; 2
(21)
(p, n,), 0
etc., which are all obtainable from (20). We have then the first term in (14) as,
a@, [n,l, n,(r))
= n,(r) + j
x e’q”G(q, n,(r))&(q) x
eGll+s2J.r
*
Nq,,
q2.
$jj
+ J +$
nl(W2(qlF2(q2> +
. . (22)
The n, dependence of a can be dealt with in a similar way. We postulate a gradient series in n?(r) and obtain the coefficients in the series by comparison in the usual way with the perturbation series (22). As the part of the density i,(q) is assumed to be localised around q = K then this time we write the gradient series in the form, a(r,
+
. .,
j&
(q - k)2e’q’rA2(q).
N
@b(r),
n*(r))
=
+ &K, K, nl(r>)(n2(r))*
4(r)
+ WG 4(r))n2(r)
+ . . .
=
j o(r
These summed
- s, nl (r))(n) (4 + n2(s))ds. terms are the ones with coefficients
=
j &(n,)dr
+ 5 drdsB(r
x A(
+
-
s. no)
.,
3
no
0
h*(q,no) +
2
0
(27)
h(q, no)
20 Qq,q,no>=
B(q,no).
no
+
(25)
(26) in
(28)
If we write, &I,,
q2, a,,)
=
&(qi,
qZ, no) + A&q,, q2, no), (29)
where the part Ak^ has the property Ak(q,, 0)
=
A&Oq,)
=
0,
(30)
then k^, is given by the relations
This is unfortunately in the form of a perturbation series in n*(r). However, because of the relations (6) it is expected that the first two terms in (25) will be the dominant ones. In addition, as is shown later, the kernel G(q, no) is only weakly dependent on no, and so (2) would indicate again that k^, fete. must be smaller than oi. For a density G,(q) peaked at a number of q values the generalization of (25) is obvious. In fact it is easy to see that we obtain a summation of a well defined subset of the “gradient” terms as defined above by putting, ii(r)
E,,.[n]
(24)
Again we are interested in just the dominant term, and so shall not elaborate on the general form of (23). To obtain the coefficients tl, /I etc., the kernels k(q, n, (r)) etc. in (22) must be expanded, this time about q = K. Thus now neglecting terms in a which involve integrals like (24) we get, ii(r)
either the series in (Vn,) or (V - irc)n,(r) given by the expansion in powers of q, of oi(q, no). As this is the only kernel known with any accuracy this is the best we can do. Thus in (26) we have an effective density approximation in which is neglected small gradient terms and those coming from &(K, K, n, )(lt?(r))* etc. In this expression ii,(q) can be any part of the density localized round q = 0, provided C(q) is reasonably small. Then in (26) the perturbation terms neglected, &n,(r))’ etc, involve powers of nz and not the much larger gradients of this part. In practice the electron densities one meets in practice take this form. The higher q components &(y) which invalidate the usual gradient series (7) are usually much smaller than the ones localized round q = 0. A reliable form of the kernel &(q, n) can be obtained as follows. A standard perturbation series for Ex, is written,
f E
1069
(23)
where On,(r)
SERIES
and the function b(q, no) is known with some accuracy (3). From (1) and (2) we obtain,
hl, nlW> = ah W, MN + Bh (4, x G9)(Wr))
GRADIENT
h(q,,
42, no)
=
k &
(20) as, ^
(q,, no) + k&f 0
This is the only part assume the dominant becomes,
(q2, 12,). (31) 0
of I$ which is known, and we part. Thus the equation for h
I
inosi12
+ $&I 0
+ no!$*g
0
0
=
B(q,no).(32)
0
Using l?(q, no) obtained self-consistently in earlier work we solve this for h(q, no). If turns out that the first two terms in (32) are dominant, the dependence of &J on no being relatively small. With the numerical values of 8 obtained from the self-consistency procedure the solution for & can be well represented in the form, G(q, no)
=
1 + b(q)n’,,
(33)
1070
THE THEORY
OF DENSITY
FUNCTIONAL
where b(0) = 0 and b(q) and z are almost independent of n,. In fact, b(q) can be well approximated by b(q) - (1 - ee”.‘76q2) and z - 0.12. In conclusion we have shown how the density gradient series for the effective potential ii(r) can be re-formulated so as to sum a well-defined subset of dominant expansion terms. A self-consistent determination of the amplitude involved indicates that it is almost independent of the density.
GRADIENT
SERIES
Vol. 63, No. 11
REFERENCES 1. 2. 3. 4.
J.C. Stoddart, Phys Lett. 95A, 484 (1983). J.C. Stoddart, Solid State Commun. 52, 1047 (1985). P. Vashishta & K.S. Singwi, Phys Rev. 86, 875 (1972). J.C. Stoddart, Solid State Cornmun. 56, 465 (1985).
Vol. 63, NO. 11, pp. 106771070,
003%1098/87 $3.00 + .OO 1987 Pergamon Journals Ltd.
1987. 0
THE THEORY
OF DENSITY
FUNCTIONAL
GRADIENT
SERIES
J.C. Stoddart Department
of Physics, (Received
University
of Sheffield, Sheffield S3 7RH UK
29 September
1986 by C. W. McCombie)
It is shown how a density functional correlation energy can be formulated terms.
IN A PREVIOUS PAPER the density gradient series for the exchange and correlation energy in density functional theory was investigated through the use of the effective density n(r). In this approach the exchange and correlation energy is written &c
bl = I We0 @(r))dr = j n(rkxc(r)dr,
(1)
expansion for the exchange and to include well defined dominant
B,(O,no) =
2,0
&00~2,)
;2,
2 =
etc. are easily proved, h, (4, no)
where eO(n) is the exchange and correlation energy per particle of a uniform electron gas of density n. The effective density ii was obtained in a perturbation series as,
=
=
n,
+ j W
+
1 o(r
sI, r -
-
-
s,
s2, n,Ms, s,, r -
x A(s,)A(s2)A(s~)
ds,ds,ds,
etc., then this means that fqOOn,)
=
&ooo, no)
eO(nO) + j B,(r
kW2Ws, ds2
s3, noI + . . .,
(2)
-
n(r)
=
-
sI, r -
x A(s,)A(s,)ds,ds,
no>=
i(r)
s2, no) + . . .,
=
no
x (3)
s, no)
+
i
t$~w(r)w(s). 0
(4) etc. As the sum rules
. .
+
(7)
+
sdq m
me @q,)&q2)
e'q'r3q,
n,@(q)
iq,+nz).r 4,,
+
q2, no>
. . . >
(8)
and expanding G(q, no) and &q, , q2, no) in powers of q, and comparing with (7) we get,
@(q,no> =
0
2.4, 0
1.
dq,dq2
2 -w(r, no),
B2(r, s, no>=
(6)
In accordance with the usual procedure (1) the gradient coefficients g(n) etc. can be obtained from the perturbation kernels in (2). We have,
s, n,)A(s)ds
and we obtain,
4@,
0 etc.,
n(r) + g(n(r))(Vr2n(r))
+s
+ IBzcr
=
=
The problem with using (1) in practice is that 5 is not in the right form to apply to a localized atomic density. One obvious way to remedy this is to write the associated gradient expansion for ii in the form,
with n(r) = n, + A(r). The formulation in this way corresponds to a normalized exchange and correlation hole. The kernels in this expansion are related to the ones in the usual expansion of e,,-(r), given by, =
no)
n,)A(s)ds
+ j l(r - sI, r -
e&r)
where
( drerq”B, (r,
and &(O, no) fi(r, [n]
(5) 0
1 + 02(no)q2 + . . . ,
(9)
and g(n) = -o,(n), with similar relations for other coefficients. This produces through (1) essentially a local approximation to the exchange and correlation energy, and is obviously just a re-arrangement of the usual gradient series terms, so that correction terms may be argued to be small (2).
1067
THE THEORY
1068
OF DENSITY
FUNCTIONAL
Density gradient series for the energy have been up to now the only form of local density approximation which has been available. Although useful, there are difficulties in obtaining the gradient correction terms, and attention has been focussed on obtaining non-local density functionals to represent the exchange and correlation energy. However, a local density form for the energy has considerable computational advantages, and we investigate here how the usual gradient series may be reformulated to improve considerably its usefulness. We start by recalling that a density gradient series is valid when the transform ri(q)is localised round q = 0, and as this is not usually the case large errors can enter. In fact, the coefficients in such a gradient series are obtained, under the above assumption, by expanding the perturbation kernels I?, (q,, no) etc. in (3) in power series in the q, about q, = 0. We will assume that we have an electron density n(q) which is locahsed in q around q = 0 and about another value q = K. Thus we have, n(r)
=
n, (r) + n?(r),
(10)
where n, ,n2 are derived from the two parts above. We look now for an expansion for exe(r) which uses these properties. Because of its association with a normalized exchange and correlation hole it is convenient to formulate the expansion in terms of the effective density h(r). We first write n,(r) = n, + A,(r) and expand the effective density as, ri(r. [n, + n2]) = +
-
s, r, [d
+ \&Cc
-
sI, r -
fi(r, In, + n?l
=
+ h(r, [~I~],n,(r)) V*m,(r))
+
D, (r -
s2, r, hl,
We are concerned just with the leading term in such an expansion and so will not elaborate upon the form of further terms. With the transforms b, and D”?defined by.
x
(pi3 p2,
+ 2 1 k(r + 3J”l( r -
=
s, r -
D2(r =
w(r -
G@, r, M no> =
hl, 4)
Mr,
’
c?n,
(16) etc., and from (14) u(r, [d, no> = fii(r,[n,,+ nd. From (12) and (13) we have, 1 + 2 J&
=
s2, r,
b,l, no>
k(r -
s, , r -
s2, n,)
3 J”Jr -
x
n2 @3 h-h.
We can now construct
(17)
eiP”40,
p, no)&(p)
s, r -
er(p+q)‘rkO, p, q, no)
s, no>
d2(0, 0, r, [n,], n,)
(12)
. . .>
(18)
=
3 s&
d’P”i(OOp, no)
way a gradient
.)
(19)
using the limiting values given by (6). From (2) we get a(r, [a,], rt,,) and as n, is arbitrary we can equate the coefficients of different orders of n2 on both sides of (16). In this way we get a series of relations between the kernels in fi. Thus from the first of (16) we get 2&O, P, n,)
s3, n,) (13)
in the usual
+
x fiiz(p) + .
sj, n,,)
s2, r -
fi,(p)fi,(q)
and,
.
+
sI, r -
+
x
s2, n,,)n2(s2)dsz s2, r -
x n2(s&zZ(s3)dsldsj
(15)
then as usual. the coefficients in (14) can be related to the terms in the expansion of b,, 6, etc. about pi = 0. As vi,(q) is localised about q = 0 then this expansion should be valid. We obtain then the “sum rules” involving the leading term a as
(11)
are given by,
s, r -
r, hl, no),
no)
.
+
s, r, [nz], no)
(14)
n,)A,(s)ds
x A,(s,)A,@Jds,ds, (2) the kernels
using (11). in the
a(r, [n,]. n,(r))
+ 3 J $$
From
Vol. 63, No. 11
SERIES
series in the part n, (r) of the density, form,
8, (0, r, [hl, no)
fi(r, En, + 41)
1 D,(r
GRADIENT
=
340, p, q, %I) =
WP,
4
a I 10
,
d&P, q> no) an
0
,
(20)
Vol. 63, No. 11
THE THEORY
OF DENSITY
FUNCTIONAL
and so on from higher orders. From the second of (16) we get, 31^OOP,%I) =
; 2
(21)
(p, n,), 0
etc., which are all obtainable from (20). We have then the first term in (14) as,
a@, [n,l, n,(r))
= n,(r) + j
x e’q”G(q, n,(r))&(q) x
eGll+s2J.r
*
Nq,,
q2.
$jj
+ J +$
nl(W2(qlF2(q2> +
. . (22)
The n, dependence of a can be dealt with in a similar way. We postulate a gradient series in n?(r) and obtain the coefficients in the series by comparison in the usual way with the perturbation series (22). As the part of the density i,(q) is assumed to be localised around q = K then this time we write the gradient series in the form, a(r,
+
. .,
j&
(q - k)2e’q’rA2(q).
N
@b(r),
n*(r))
=
+ &K, K, nl(r>)(n2(r))*
4(r)
+ WG 4(r))n2(r)
+ . . .
=
j o(r
These summed
- s, nl (r))(n) (4 + n2(s))ds. terms are the ones with coefficients
=
j &(n,)dr
+ 5 drdsB(r
x A(
+
-
s. no)
.,
3
no
0
h*(q,no) +
2
0
(27)
h(q, no)
20 Qq,q,no>=
B(q,no).
no
+
(25)
(26) in
(28)
If we write, &I,,
q2, a,,)
=
&(qi,
qZ, no) + A&q,, q2, no), (29)
where the part Ak^ has the property Ak(q,, 0)
=
A&Oq,)
=
0,
(30)
then k^, is given by the relations
This is unfortunately in the form of a perturbation series in n*(r). However, because of the relations (6) it is expected that the first two terms in (25) will be the dominant ones. In addition, as is shown later, the kernel G(q, no) is only weakly dependent on no, and so (2) would indicate again that k^, fete. must be smaller than oi. For a density G,(q) peaked at a number of q values the generalization of (25) is obvious. In fact it is easy to see that we obtain a summation of a well defined subset of the “gradient” terms as defined above by putting, ii(r)
E,,.[n]
(24)
Again we are interested in just the dominant term, and so shall not elaborate on the general form of (23). To obtain the coefficients tl, /I etc., the kernels k(q, n, (r)) etc. in (22) must be expanded, this time about q = K. Thus now neglecting terms in a which involve integrals like (24) we get, ii(r)
either the series in (Vn,) or (V - irc)n,(r) given by the expansion in powers of q, of oi(q, no). As this is the only kernel known with any accuracy this is the best we can do. Thus in (26) we have an effective density approximation in which is neglected small gradient terms and those coming from &(K, K, n, )(lt?(r))* etc. In this expression ii,(q) can be any part of the density localized round q = 0, provided C(q) is reasonably small. Then in (26) the perturbation terms neglected, &n,(r))’ etc, involve powers of nz and not the much larger gradients of this part. In practice the electron densities one meets in practice take this form. The higher q components &(y) which invalidate the usual gradient series (7) are usually much smaller than the ones localized round q = 0. A reliable form of the kernel &(q, n) can be obtained as follows. A standard perturbation series for Ex, is written,
f E
1069
(23)
where On,(r)
SERIES
and the function b(q, no) is known with some accuracy (3). From (1) and (2) we obtain,
hl, nlW> = ah W, MN + Bh (4, x G9)(Wr))
GRADIENT
h(q,,
42, no)
=
k &
(20) as, ^
(q,, no) + k&f 0
This is the only part assume the dominant becomes,
(q2, 12,). (31) 0
of I$ which is known, and we part. Thus the equation for h
I
inosi12
+ $&I 0
+ no!$*g
0
0
=
B(q,no).(32)
0
Using l?(q, no) obtained self-consistently in earlier work we solve this for h(q, no). If turns out that the first two terms in (32) are dominant, the dependence of &J on no being relatively small. With the numerical values of 8 obtained from the self-consistency procedure the solution for & can be well represented in the form, G(q, no)
=
1 + b(q)n’,,
(33)
1070
THE THEORY
OF DENSITY
FUNCTIONAL
where b(0) = 0 and b(q) and z are almost independent of n,. In fact, b(q) can be well approximated by b(q) - (1 - ee”.‘76q2) and z - 0.12. In conclusion we have shown how the density gradient series for the effective potential ii(r) can be re-formulated so as to sum a well-defined subset of dominant expansion terms. A self-consistent determination of the amplitude involved indicates that it is almost independent of the density.
GRADIENT
SERIES
Vol. 63, No. 11
REFERENCES 1. 2. 3. 4.
J.C. Stoddart, Phys Lett. 95A, 484 (1983). J.C. Stoddart, Solid State Commun. 52, 1047 (1985). P. Vashishta & K.S. Singwi, Phys Rev. 86, 875 (1972). J.C. Stoddart, Solid State Cornmun. 56, 465 (1985).