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A generalized Ewald method for lattice sums
Adler,
S.
Physica
1961
27
1193-1201
A GENERALIZED
EWALD METHOD FOR LATTICE SUMS by S. ADLER
General
Telephone
Sr Electronics
Laboratories
Inc.,
New York,
U.S.A. *)
Synopsis By a simple
and straightforward
procedure
an Ewald
C c~f(lrPl)[X A,Yz,(fV,4~*)1 m i, k is derived. table
The function
by a certain
f is arbitrary,
integral
1. I&rod&ion.
method
for the lattice
sum
@.+I’
subject only to the condition
that it be represen-
transform.
Lattice
sums of the form
@ = Z ckf(PPl)
[X &Yz&JP,
m
i,k
W)l eiK"fr
(1)
enter into many physical theories. Here, Yl, is a spherical harmonic and irjkj, C&k, t&k are the spherical polar coordinates of the kth basis atom in the jth unit cell, relative to an arbitrary Cartesian axis system. An Ewald method for sums of this type, with f(lrjkl) = jr$l-P, p > 0 has been derived by N ij boe r and De Wet t e 1). This paper will present an alternative derivation which leads to an Ewald method for the more general sum (1). The only restriction on f is that it can be expressed in the form
f(l = ,,j?z-“’ 4(t) dt) that is, the inverse
Laplace
transform
of
f v’*) 1112 must exist. A necessary (but not sufficient) condition 2, for this to be true is that for r > 0, f’(r) and all higher derivatives must exist, and that f(co)= 0. In general, there is no restriction on the weighting coefficients ck; but if 1 = 0, K is a reciprocal lattice vector and
s‘d(t) dt
t”‘”
*) Summer visitor,
1960; now at Physics
Department,
-
1193
Princeton
-
University,
Princeton,
NewJersey.
1194
S. ADLER
diverges,
then the condition c Q eiK.rrk = 0 1;
must be imposed.
(Here i is a free index).
2. Theory. The derivation to be presented treatment by Moliere 3) of the sums
is a generalization
of the
n = 1, 2, 3. Consider first the case in which the origin is not a lattice point; the case when the origin is a lattice point will be considered below. Abbreviate c &YZ?#P> ?n Write
dP) = Yz(V,
+I”).
the sum as (4
(3)
(4) E
(5) The splitting of the sum and the interchange of the order of integration and summation in @II are justified by the fact that for 6 > 0 the sums @I and @II are absolutely and uniformly convergent; in effect we are defining the conditionally
convergent
sum (1) by
@ = lim x CkYl(ejk, $3”) eiK’@ lrjklE/IP1rj”lzt 4(t) dt. 6 6-•o 3,lc By making the form
the change of variable
Expression
(5) can be written
(1’)
l,$12 t = d the sum (3) can be put into 00
G(t, r) = x ckYl(ejk, i,x-
as lim j+(t) 6+0 6
G(t, 0) dt where
$5”) eiK.(rj”-r)jrjk
_ r(l e-~‘~k-rlZ~
(7)
and 81” and &k are now the angular coordinates of rik - r. Clearly G(t, r) is, for all t, a cell periodic function of r, and so can be
A GENERALIZED
expanded
in a Fourier
EWALD
METHOD
FOR LATTICE
SUMS
1195
series G(t, r) = 2 C, e-ign’r. n
The sum is over all reciprocal The coefficient
space lattice
vectors
gn (see the Appendix).
Cn is given by
s
CTA= t
eign’r
G(t, r)dr.
unit cell V
is the unit cell volume.
Because 1
-7 s
The right-hand
G is cell periodic,
eign’r G(t, r) dr
side of (8) can be evaluated
(8)
(The cell index j is free). explicitly. Setting r,k - r = r’
(8) b ecomes Cn=T/
ei(K-gn)‘r’ yl(o’, 4’) /r’lZe-ir’l’t dr’.
(9)
au ArXV33 Here is the crystallographic structure factor. Let 0 be the angle between gn - K and r’. Let the polar angles of g, - K be &,, & and let the polar angles of r’ be W, I$‘, with respect to the axes to which 8 and 4 are referred. Then using e i(K-Bn).r’ = e --ilK-&!“III’)008 8 = jO(21 +
Pl(COS0) = *
1)(-V
Ir’l) PZ(COS0)
(1014)
,$_,y*zm(e’,47 yzden, $4
(1115)
iz(4 = it is immediate C
A
=
iz(lK -
vG
g,l
Jz+&)
that
”
4n(-v S(n) 2 VIK - &J’*
Jz++(lK 0
-
emrat dr.
84 11
(12)
1196
S. ADLER
The integral can be evaluated by term-by-term integration series expansion of the integrand 6) ; the result is
At this point it becomes
apparent
why the transform
of the Taylor
was defined by
f(r) = ~l,/e-~‘~ r$(t) dt instead
of f(r) =;;-+t
d(t) dt,
more like the usual Laplace transform. It is precisely the presence of the factor YZwhich causes the integral (12) to reduce to the simple expression (13) and to fall off as exp -‘Ii:
In order to see this, compare of reference 6. Using
(Is),
j#(t) 6
as
- g7J2 4t
Iii: -
equation
G(t, 0) dt
gfil +O”.
(2) with equation
becomes
(4) in section
13.3
--
IK-_gnl2
7c 452(-i)l
1/ 2
z S(n) lK V 2z+3’a rr
g,ll
Yz(&
$M J!(t)
e
,t+::,
d.8.
(14)
d
The limit of (14) as 6 --f 0 must be evaluated. When K is not a reciprocal lattice vector, (K - g,i > 0 for all 1z and the integral E s 8
-__Ix-&?,12
4(t) e t”+“,:
dt
converges at its lower limit. When K is a reciprocal lattice vector but I > 0 the facort /I( - g,[l causes the term of the sum for which lK - g,l = 0 to vanish. When K is a reciprocal lattice vector and 1 = 0 and 4(t) does not fall off sufficiently rapidly for
skt) p
dt
0
to converge
at the lower limit,
then the condition 2 k
Q
eiK.+Jk =
0
A GENERALIZED
EWALD
must be imposed. This guarantees
which diverges With
METHOD
FOR LATTICE
SUMS
1197
that there appears no term
as 6 --f 0.
the condition s Ck eiK.r~k = 0 k
imposed give
when necessary,
the limit can be taken
inside the summation
to
The ’ on the summation sign means that if K is a reciprocal lattice vector, say K = gN, the term of the sum with n = N should be omitted. The factor in the last term is to be taken as 0 if K is a reciprocal lattice vector g, and if either I=
(1)
0
and
s
’ b(t) dt ~ t”‘a
0
diverges,
so that
the condition 2 Ck eiK.rP = 0 x:
must be imposed to make S(N)
vanish,
or
(2) 1 > 0. In other words, in either case (1) or case (2) the lattice is electrically neutral, resulting in the vanishing of the expansion coefficient CN. The factor in the last term is to be taken as
(forYa=
l)ifK=@,l=Oand’
s‘4(t)
t”/” dt converges.
0
1198
S. ADLER
Making the change of variable
IK - &I2
~
4t
= ua (15) becomes
S(n) Yz(%&, &J IK - gd+l IK-gnl
2d&
+ According
n3’a1 0
v 1S(N)
‘+(t) dt. s”
t”‘”
to (2)) (6) and (16)
(17)
S(f4 Yz(bh&a) IK
_
g,,l+l
/e
_
uzu2~$( IKip”)du
IK-g,l -24&
0
Note that (17) is valid only when the origin is not a lattice point. Often the sum C’
i,k
ckf(lrjkl) Yz(W, 43”)eiK’@
(18)
is wanted, where the origin is the M th basis atom of a unit cell and the ’ denotes that the sum is over all lattice points except the origin. (This is analogous to the “exciting” (erregendes) potential of Ewald). This can be obtained from (17) as follows: Set
rjk = Rjk - r,
(19)
where Rjk is the vector from the lattice point which we wish to be the new origin to the k th basis atom in the jth unit cell. The potential of the lattice point at the new origin is subtracted from (17) and the limit r + 0 is taken. Because 00
A GENERALIZED
EWALD
-
METHOD
FOR LATTICE
cMf(lrl) Yz(~,4) e--iK’r 1
I =
1
SUMS
1199
(20)
I>0
O
-CM;+(Z) dz
I=0
0
=
-CM
(taking
dz
&O;+(Z)
Ye “= l), the sum (18) is given by
IK-grill -__ 2\/E E
0
3. Discussion, Equation (17) with substitution (19) permits the evaluation of the sum (1) at a point in the lattice which is not a lattice point; equation (21) gives the value of the sum at a lattice point, when the term contributed by that point to the sum is omitted. The parameter F contols the rapidity of convergence of the two sums in (17). Increasing E causes the real space sum to converge more rapidly and the reciprocal space sum to converge less rapidly; decreasing E has the opposite effect. In computation F is adjusted so as to minimize the labor required and /or to maximize the accuracy. Thus, suppose the rela space sum in (17) is carried out within a sphere of radius R in real space and the reciprocal space sum within a sphere of radius G in reciprocal space. Then for the errors in the two sums due to neglect of the remaining terms to be roughly the same, the parameter c should be chosen so that eThe integrals
eI12
-
eVGla14’, that is,
E N + g.
1200
S.
__~-_
ADLER
___--_.
~
and
(23) involve three parameters and two parameters, respectively. However, F and 1 can be fixed in advance of the numerical calculation, so that only a table of numerical values of the integrals versus one parameter (IK - g,i and Ir~kl respectively for (22) and (23)) would have to be computed. For certain functions f(r) the integrals simplify considerably. When f(r) = yPP, p > 0, the inverse
Laplace
transform
of
f Via) is given by P+z
L-l[r
--
Ptl
l
t2
Z]=
=
P+l
( >
ri
W).
(24)
2
In this case, which is the one treated by integrals become complementary gamma tabulated. Equation (21) becomes *)
+ $f;
(-i)Z 2”‘“_0 ~‘S(n)
Nijboer and De Wette, functions, which have
Yz(O,, I&) IK-g,JP-3 0
__cM
2_
p
SZO +
4
g
i
f+)
the been
Tf*)
p-3 & 2
S(N)
___
P-3 2
1
where r(cr, x) = f-e& ~(cc, x) satisfies
the
recursion
T(l, x) = e-“,
relation
tap1 dt. 1) r(cc +
T(O, x) = --Ei(--x),
1, X) = crT(cr, x) + x” e-“;
T($, x) = d7c erfc 2/x.
*) The result (25) differs from that given by Nijboer and De Wette by a factor of (--1)t in the second term. Their result is in error, as a result of an incorrect generaIization of their equation (38). (F. W. de Wette, private communication).
A
GENERALIZED
EWALD
METHOD
FOR
LATTICE
The factor in the last term of (25) is to be chosen as explained condition for
to converge
1201
SUMS
above;
the
is p > 3.
4. Acknowledgement. L. Birman for many treated here arose.
The writer wishes to thank Dr. Joseph stimulating discussions, from which the problem
APPENDIX
Reciprocal lattice vectors.
lattice.
Then the reciprocal
Let al, us, us be the basis vectors for the real lattice vector g, introduced above is given by
gn = lrgr +
l2g2
+
J3g3
where 11, 12, 1s are integers and (a2 g1
=
2n
g1
=
2n
g2
=
2?z
Received
Q3)
V (a3
V = (al x a$. u3 is the volume
x
x
al)
V (a1
x
a2)
.
v
of the unit cell.
22-3-61
REFERENCES 1) Nijboer,
B. R. A. and De Wette,
2)
Widder,
3)
MoliBre,
D. V., “Advanced Calculus”, G., Z. Krist. 101 (1939) 393.
4)
Morse, P. M. and Feshbach, H., “Methods of Theoretical 1953, p. 1466, eq. (11.3.45). Ibid, p. 1274. Watson, G. N., “Theory of Bessel Functions”, Macmillan,
5) 6)
F. W., Physica
23 (1957)
Prentice-Hall,
309.
New York
1947,
Physics”,
New York
p. 375,
McGraw
379. Hill, New York
1948, p. 394, eq. 4.
S.
Physica
1961
27
1193-1201
A GENERALIZED
EWALD METHOD FOR LATTICE SUMS by S. ADLER
General
Telephone
Sr Electronics
Laboratories
Inc.,
New York,
U.S.A. *)
Synopsis By a simple
and straightforward
procedure
an Ewald
C c~f(lrPl)[X A,Yz,(fV,4~*)1 m i, k is derived. table
The function
by a certain
f is arbitrary,
integral
1. I&rod&ion.
method
for the lattice
sum
@.+I’
subject only to the condition
that it be represen-
transform.
Lattice
sums of the form
@ = Z ckf(PPl)
[X &Yz&JP,
m
i,k
W)l eiK"fr
(1)
enter into many physical theories. Here, Yl, is a spherical harmonic and irjkj, C&k, t&k are the spherical polar coordinates of the kth basis atom in the jth unit cell, relative to an arbitrary Cartesian axis system. An Ewald method for sums of this type, with f(lrjkl) = jr$l-P, p > 0 has been derived by N ij boe r and De Wet t e 1). This paper will present an alternative derivation which leads to an Ewald method for the more general sum (1). The only restriction on f is that it can be expressed in the form
f(l = ,,j?z-“’ 4(t) dt) that is, the inverse
Laplace
transform
of
f v’*) 1112 must exist. A necessary (but not sufficient) condition 2, for this to be true is that for r > 0, f’(r) and all higher derivatives must exist, and that f(co)= 0. In general, there is no restriction on the weighting coefficients ck; but if 1 = 0, K is a reciprocal lattice vector and
s‘d(t) dt
t”‘”
*) Summer visitor,
1960; now at Physics
Department,
-
1193
Princeton
-
University,
Princeton,
NewJersey.
1194
S. ADLER
diverges,
then the condition c Q eiK.rrk = 0 1;
must be imposed.
(Here i is a free index).
2. Theory. The derivation to be presented treatment by Moliere 3) of the sums
is a generalization
of the
n = 1, 2, 3. Consider first the case in which the origin is not a lattice point; the case when the origin is a lattice point will be considered below. Abbreviate c &YZ?#P> ?n Write
dP) = Yz(V,
+I”).
the sum as (4
(3)
(4) E
(5) The splitting of the sum and the interchange of the order of integration and summation in @II are justified by the fact that for 6 > 0 the sums @I and @II are absolutely and uniformly convergent; in effect we are defining the conditionally
convergent
sum (1) by
@ = lim x CkYl(ejk, $3”) eiK’@ lrjklE/IP1rj”lzt 4(t) dt. 6 6-•o 3,lc By making the form
the change of variable
Expression
(5) can be written
(1’)
l,$12 t = d the sum (3) can be put into 00
G(t, r) = x ckYl(ejk, i,x-
as lim j+(t) 6+0 6
G(t, 0) dt where
$5”) eiK.(rj”-r)jrjk
_ r(l e-~‘~k-rlZ~
(7)
and 81” and &k are now the angular coordinates of rik - r. Clearly G(t, r) is, for all t, a cell periodic function of r, and so can be
A GENERALIZED
expanded
in a Fourier
EWALD
METHOD
FOR LATTICE
SUMS
1195
series G(t, r) = 2 C, e-ign’r. n
The sum is over all reciprocal The coefficient
space lattice
vectors
gn (see the Appendix).
Cn is given by
s
CTA= t
eign’r
G(t, r)dr.
unit cell V
is the unit cell volume.
Because 1
-7 s
The right-hand
G is cell periodic,
eign’r G(t, r) dr
side of (8) can be evaluated
(8)
(The cell index j is free). explicitly. Setting r,k - r = r’
(8) b ecomes Cn=T/
ei(K-gn)‘r’ yl(o’, 4’) /r’lZe-ir’l’t dr’.
(9)
au ArXV33 Here is the crystallographic structure factor. Let 0 be the angle between gn - K and r’. Let the polar angles of g, - K be &,, & and let the polar angles of r’ be W, I$‘, with respect to the axes to which 8 and 4 are referred. Then using e i(K-Bn).r’ = e --ilK-&!“III’)008 8 = jO(21 +
Pl(COS0) = *
1)(-V
Ir’l) PZ(COS0)
(1014)
,$_,y*zm(e’,47 yzden, $4
(1115)
iz(4 = it is immediate C
A
=
iz(lK -
vG
g,l
Jz+&)
that
”
4n(-v S(n) 2 VIK - &J’*
Jz++(lK 0
-
emrat dr.
84 11
(12)
1196
S. ADLER
The integral can be evaluated by term-by-term integration series expansion of the integrand 6) ; the result is
At this point it becomes
apparent
why the transform
of the Taylor
was defined by
f(r) = ~l,/e-~‘~ r$(t) dt instead
of f(r) =;;-+t
d(t) dt,
more like the usual Laplace transform. It is precisely the presence of the factor YZwhich causes the integral (12) to reduce to the simple expression (13) and to fall off as exp -‘Ii:
In order to see this, compare of reference 6. Using
(Is),
j#(t) 6
as
- g7J2 4t
Iii: -
equation
G(t, 0) dt
gfil +O”.
(2) with equation
becomes
(4) in section
13.3
--
IK-_gnl2
7c 452(-i)l
1/ 2
z S(n) lK V 2z+3’a rr
g,ll
Yz(&
$M J!(t)
e
,t+::,
d.8.
(14)
d
The limit of (14) as 6 --f 0 must be evaluated. When K is not a reciprocal lattice vector, (K - g,i > 0 for all 1z and the integral E s 8
-__Ix-&?,12
4(t) e t”+“,:
dt
converges at its lower limit. When K is a reciprocal lattice vector but I > 0 the facort /I( - g,[l causes the term of the sum for which lK - g,l = 0 to vanish. When K is a reciprocal lattice vector and 1 = 0 and 4(t) does not fall off sufficiently rapidly for
skt) p
dt
0
to converge
at the lower limit,
then the condition 2 k
Q
eiK.+Jk =
0
A GENERALIZED
EWALD
must be imposed. This guarantees
which diverges With
METHOD
FOR LATTICE
SUMS
1197
that there appears no term
as 6 --f 0.
the condition s Ck eiK.r~k = 0 k
imposed give
when necessary,
the limit can be taken
inside the summation
to
The ’ on the summation sign means that if K is a reciprocal lattice vector, say K = gN, the term of the sum with n = N should be omitted. The factor in the last term is to be taken as 0 if K is a reciprocal lattice vector g, and if either I=
(1)
0
and
s
’ b(t) dt ~ t”‘a
0
diverges,
so that
the condition 2 Ck eiK.rP = 0 x:
must be imposed to make S(N)
vanish,
or
(2) 1 > 0. In other words, in either case (1) or case (2) the lattice is electrically neutral, resulting in the vanishing of the expansion coefficient CN. The factor in the last term is to be taken as
(forYa=
l)ifK=@,l=Oand’
s‘4(t)
t”/” dt converges.
0
1198
S. ADLER
Making the change of variable
IK - &I2
~
4t
= ua (15) becomes
S(n) Yz(%&, &J IK - gd+l IK-gnl
2d&
+ According
n3’a1 0
v 1S(N)
‘+(t) dt. s”
t”‘”
to (2)) (6) and (16)
(17)
S(f4 Yz(bh&a) IK
_
g,,l+l
/e
_
uzu2~$( IKip”)du
IK-g,l -24&
0
Note that (17) is valid only when the origin is not a lattice point. Often the sum C’
i,k
ckf(lrjkl) Yz(W, 43”)eiK’@
(18)
is wanted, where the origin is the M th basis atom of a unit cell and the ’ denotes that the sum is over all lattice points except the origin. (This is analogous to the “exciting” (erregendes) potential of Ewald). This can be obtained from (17) as follows: Set
rjk = Rjk - r,
(19)
where Rjk is the vector from the lattice point which we wish to be the new origin to the k th basis atom in the jth unit cell. The potential of the lattice point at the new origin is subtracted from (17) and the limit r + 0 is taken. Because 00
A GENERALIZED
EWALD
-
METHOD
FOR LATTICE
cMf(lrl) Yz(~,4) e--iK’r 1
I =
1
SUMS
1199
(20)
I>0
O
-CM;+(Z) dz
I=0
0
=
-CM
(taking
dz
&O;+(Z)
Ye “= l), the sum (18) is given by
IK-grill -__ 2\/E E
0
3. Discussion, Equation (17) with substitution (19) permits the evaluation of the sum (1) at a point in the lattice which is not a lattice point; equation (21) gives the value of the sum at a lattice point, when the term contributed by that point to the sum is omitted. The parameter F contols the rapidity of convergence of the two sums in (17). Increasing E causes the real space sum to converge more rapidly and the reciprocal space sum to converge less rapidly; decreasing E has the opposite effect. In computation F is adjusted so as to minimize the labor required and /or to maximize the accuracy. Thus, suppose the rela space sum in (17) is carried out within a sphere of radius R in real space and the reciprocal space sum within a sphere of radius G in reciprocal space. Then for the errors in the two sums due to neglect of the remaining terms to be roughly the same, the parameter c should be chosen so that eThe integrals
eI12
-
eVGla14’, that is,
E N + g.
1200
S.
__~-_
ADLER
___--_.
~
and
(23) involve three parameters and two parameters, respectively. However, F and 1 can be fixed in advance of the numerical calculation, so that only a table of numerical values of the integrals versus one parameter (IK - g,i and Ir~kl respectively for (22) and (23)) would have to be computed. For certain functions f(r) the integrals simplify considerably. When f(r) = yPP, p > 0, the inverse
Laplace
transform
of
f Via) is given by P+z
L-l[r
--
Ptl
l
t2
Z]=
=
P+l
( >
ri
W).
(24)
2
In this case, which is the one treated by integrals become complementary gamma tabulated. Equation (21) becomes *)
+ $f;
(-i)Z 2”‘“_0 ~‘S(n)
Nijboer and De Wette, functions, which have
Yz(O,, I&) IK-g,JP-3 0
__cM
2_
p
SZO +
4
g
i
f+)
the been
Tf*)
p-3 & 2
S(N)
___
P-3 2
1
where r(cr, x) = f-e& ~(cc, x) satisfies
the
recursion
T(l, x) = e-“,
relation
tap1 dt. 1) r(cc +
T(O, x) = --Ei(--x),
1, X) = crT(cr, x) + x” e-“;
T($, x) = d7c erfc 2/x.
*) The result (25) differs from that given by Nijboer and De Wette by a factor of (--1)t in the second term. Their result is in error, as a result of an incorrect generaIization of their equation (38). (F. W. de Wette, private communication).
A
GENERALIZED
EWALD
METHOD
FOR
LATTICE
The factor in the last term of (25) is to be chosen as explained condition for
to converge
1201
SUMS
above;
the
is p > 3.
4. Acknowledgement. L. Birman for many treated here arose.
The writer wishes to thank Dr. Joseph stimulating discussions, from which the problem
APPENDIX
Reciprocal lattice vectors.
lattice.
Then the reciprocal
Let al, us, us be the basis vectors for the real lattice vector g, introduced above is given by
gn = lrgr +
l2g2
+
J3g3
where 11, 12, 1s are integers and (a2 g1
=
2n
g1
=
2n
g2
=
2?z
Received
Q3)
V (a3
V = (al x a$. u3 is the volume
x
x
al)
V (a1
x
a2)
.
v
of the unit cell.
22-3-61
REFERENCES 1) Nijboer,
B. R. A. and De Wette,
2)
Widder,
3)
MoliBre,
D. V., “Advanced Calculus”, G., Z. Krist. 101 (1939) 393.
4)
Morse, P. M. and Feshbach, H., “Methods of Theoretical 1953, p. 1466, eq. (11.3.45). Ibid, p. 1274. Watson, G. N., “Theory of Bessel Functions”, Macmillan,
5) 6)
F. W., Physica
23 (1957)
Prentice-Hall,
309.
New York
1947,
Physics”,
New York
p. 375,
McGraw
379. Hill, New York
1948, p. 394, eq. 4.