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The surface potential of point charge and point dipole lattices
Surface Science 110 (1981) L619-L624 North-Holland Publishing Company
L619
SURFACE SCIENCE LETTERS THE SURFACE POTENTIAL DIPOLE LATTICES
OF POINT CHARGE AND POINT
D.M. HEYES * Laboratory for Physical Chemistry, University of Amsterdam, IO18 WS Amsterdam, The Netherlands
Nieuwe Achtergracht
127,
Received 11 March 1981; accepted for publication 6 July 1981
New expressions for the potential in the surface region of point charge and point dipole lattices reveal that a modified direct summation scheme can converge with comparable rapidity to those of the more complicated Ewald form.
There is much interest at present in investigating the structural, thermodynamic and dynamical behaviour of solid ionic surfaces [ 1,2]. In order to develop models for such behaviour it is often essential to obtain rapidly converging expansions for the electrostatic potential in the surface region. Here a number of new and easy to use general expressions for calculating the potential energy of point charges and dipoles in the field of point charge [3] and point dipole [4] infinite laminae lattices are derived. Two methods are considered. The procedure of directly summing the interaction from each charge in a lattice of cubic unit cells is made more rapidly converging by the inclusion of a long range correction term which depends on the shape and size of the outermost shell of charges incorporated in the summation. These are compared with formulae which are similar to those of Ewald [5] and Ewald-Kornfeld [5] used for bulk point charge and point dipole lattices [6], respectively. First consider the potential energy of a test charge, 4, at a point, R, from a square planar lamina lattice of unit cell thickness and infinite extent in the (x, y) plane, V,(R), which is obtained from the following absolutely convergent series,
V,(R)=q
5 qj
P'B;'.
(1)
j=l
There are N charges of magnitude qj in the unit cell, which is overall neutral. The lattice vector, I, is the coordinate of the image unit cell relative to the chosen origin * Present address: Department of Chemistry, Royal Holloway College, University of London, Egham, Surrey TW20 OEX, UK.
0039-6028/81/0000-0000/$02.50
0 1981 North-Holland
D.M. Heyes / Surface potential of point charge and point dipole lattices
L620
cell. The charge separation, Bj equals IR- Rj+ 11, where Rj is the position of charge j in the unit cell. The prime on the 1 summation denotes the omission of any term: Bi = 0. For practical purposes this summation can be replaced by the following series,
(2)
V,(R) = lim Rc+=
where Cj = Bj ,
BjGR,,
(3)
cj=-,
Bi >R,,
(4)
A is the area of the unit cell in the (x, v) plane,
M& =
5
qj(O!-
(5)
~j)',
j=1
where x, Y, z, xi, Yj and Zj are the components of R and Rj.The z direction is perpendicular to the surface plane. The first term in eq. (2) represents a direct summation of the contributions from each lattice site summed in circular shells of image cells up to a cut-off radius, R,. The second term in eq. (2) approximates the contribution to the potential made from the charges beyond this truncation distance by replacing them with a uniform charge distribution. The effect on the potential is obtained by Taylor expanding their interactions about R - Ri= 0 and performing the angular and radial integrations in the surface plane. The series of eq. (1) can be represented by the following transform [3,.5], N T/l (R) = q ,g
t 7’
qi ((n/A)
G
Rj)l Fi
c’Js[h ’ C.R -
erfc(bj) By1 - 6 (Bj) 2n-“‘~-~)
,
(6)
where Ir is the reciprocal lattice vector in the surface plane, Fj = [exp@)
erfc(dj) + exp(Ej) erfc(ej)]h-‘,
(7)
bj = B/IQ t
(8)
Dj = hB,i)
(9)
Et = -Dj,
(10)
di = VhI2 + Bzj/Vj
(11)
ej = Vhl2 B,j = Z
-
Zj,
BzJ~,
(12) (13)
D.M. Heyes / Surface potential of point charge and point dipole lattices
where 1) is an arbitrary tion of x [7] and 6(x)=
1 =o
distance parameter,
erfc(x) is the complementary
L621
error func-
x=0,
(14)
xzo.
(15)
Note that here the unit of n is distance whereas this symbol represents an inverse length in ref. [5]. The last term in eq. (6) is a self energy component which contributes if R coincides with a lattice point and is derived from the limit of the real space term as Bi+
0 PI. The h = 0 component of eq. (6) is obtained by taking the limit h * 0; the terms proportional to h-r cancel because the unit cell is a neutral entity. Thus,
[Bzj erf(B,j/n) + vn-“’
Fj(h = 0) = -2
exp(-(Bzj/q)2)].
(16)
The potential energy of a point charge CJin the field of a point dipole lattice, V,(R), is obtained by making the substitution qj ‘-pj . V in eq. (1) where pj is the point dipole at lattice site j [2]. Hence, N
V2 (R) = q ,z
T’
bj * Bj)By3 9
which can be approximated
(17)
as before by,
V2 (R)= lim Rc-*= + (qnlARc)
i2PzjCz
- zj) -
Pxj(X
- Xj) -
Pyj(V
-
Yj)l
)I9
where naj is the (Ycomponent of pj. A uniform distribution of dipoles beyond the truncation radius is now considered. The same operation on eq. (6) gives, V,(R) = q 5 j=l -yzj
((*//I)
COS(~ *
F
[(pj * h) sin(h
*(R - Rj))Fj
(R -Rj))Hj] + q' (Pi*Rj)G(Bj)'
(19)
where
Hi= exp(Dj) erfc(dj) - exp(Ej) erfc(ej) t27r-1’2n-1h-’
[eXp(Ej - eT)-
eXp(Dj - dj)],
(20)
and
G(Bj) = erfc(bj)By3 + 2n-1’2n-1 exp(-L$)B,Y2.
(21)
L622
D.M. Heyes / Surface potential of point charge and point dipole lattices
The potential energy of a test dipole p at Ii from the above dipole lattice, V#), is obtained from eq. (17) by replacing 4 by the operator p . V,
p’ [(p * pip;3
,,I=]$
- 3(p* Bj)(JAj - Bj)B;5] )
(24
and
where par is the (Ycomponent eq. (19) yields,
of c and (Y= x, y or z. The same operation
applied to
N
17, (R) =
c
(r/A)
j=1
F
[cos(h (R - R/))((P h)hj * h)Fj l
l
+ sin(h . (R - Rj))&-(pj a h) + P&
* h))Hil
PzPzjKj)
+ 5;’ I(!$) - W$)
Vs) ,
(24)
where Kj = h exp(Di) erfc(dj) + h exp(Ej) - 4n-”
erfc(ej)
n-r [exp(Dj - d; ) + exp(Ej
t 41r-“~n-~ h-r [di exp(L$ - d/) I(Bj)’ +
(p ‘~j)G(Bj)
- (cl’
Bj)@j
- ef
)]
+ ej exp(Ej Bj)[3
’
27~-“‘77- (2~~~ t 3By2) exp(-b~)B~2
- ef)]
,
(25)
dC(bj)ET’
],
(26)
and the dipolar self energy V, is obtained by taking the following limit [5], Vs = f’lio
- I(Bj)]
[-2/J2/Bj
ri”r = L$$ 7)-3
For h = 0 and ej from omitted. Eqs. (19) in a computer Alternative
a-rf2/3
.
(27) to h-r in Hj and Kj, after substituting for dj do not contribute to the potential and can be
the terms proportional
eqs. (11) and (12)
and (24) are lengthy but ar not significantly more difficult to program than their parent, eq. (6). expressions of the form of eq. (6) which consist of series in real and
2.3094 2.2611 2.2627 2.2610 2.2603 2.2601 2.2600
3.1929 2.4628 2.2947 2.2706 2.2600 2.2600 2.2600
4.0520 3.8022 3.7362 3.7296 3.7248 3.7248 3.7248
FT Eq.(6)
DS Eq.(2)
3.8037 3.7417 3.7240 3.7341 3.7226 3.7248 3.7248
(42L-l)
-VI(R)
5.3333 5.7487 5.7377 5.7528 5.7600 5.7620 5.7621
10.1445 10.4060 10.4833 10.4358 10.4876 10.4769 10.4770
DS Eq.(l8)
Vz(R)
4.0635 5.8056 5.8056 5.7830 5.7620 5.7621 5.7621
9.3036 10.2071 10.4384 10.4610 10.4770 10.4770 10.4770
FT Eq.(l9)
(qPL-2)
21.3520 18.9990 18.7650 18.7548 18.6042 18.6044 18.6035
28.1720 27.9018 27.8063 27.8877 27.8134 27.8326 27.8326
DS Eq.(23)
-V3W
9.7371 16.5598 18.4263 18.7335 18.6034 18.6035 18.6035
34.8375 30.2394 28.4081 28.1010 27.8328 27.8326 27.8326
FT Eq.(24)
(P2L-3)
0.02 0.04 0.05 0.09 0.2 1.8 28
0.02 0.04 0.05 0.09 0.2 1.8 28
Computing time for all columns (s)
The unit cell which gives rise to VI(R) has a charge, 4, at (O,O,O)L and a charge, -q, at (x,y,z)L which is also the test charge. For V,(R) the unit cell contains a dipole, p, of (XJ,Z) components, ~(1,1,1)3-“2 at (O,O,O)L. There is a test charge 4 at (x,y,z)L. A unit cell having a dipole of components ~(1,1,1)3-1~Z at (O,O,O)L gives rise to a potential energy Vs(R) of a test dipole, p’, of components ~.1)1,2,3)(14)-‘1’ at (x,y,z)L. The distance parameter r) has the value of 0.3L. The real and reciprocal space cells are included in the order of their proximity to the origin unit cell, i.e., they are added in expanding circular shells of image cells. (a) (x,y,z)L = (O.OO’, O.OO’, 0.25)L and (b) (x,y,z)L = (0.25,0.25, 0.25)L. DS and FT denote methods of direct summation and Fourier transformation, respectively. R, = (12Lw2 + 1)‘j2L in eqs. (2), (18) and (23).
0 1 2 4 10 100 1600
(b)
1 2 4 10 100 1600
0
(a)
IzLm2 and h2L2/4n2
Table 1 The potential energy of a test charge or dipole at (x,y,z)L from an infinite lamina lattice consisting of a unit cell with a square cross-section of sidelength L
L624
D.M. Heyes /Surface
potential of point charge and point dipole lattices
reciprocal space have been derived [9,10]. Therefore eq. (24) is not a unique solution of the real and reciprocal space series form for this dipole lattice. The potential from a semi-infinite lattice is obtained by adding the effects of such laminae of unit cell thickness that are positioned adjacent and parallel, so that they occupy a half-space. The convergence characteristics of the direct summation formulae of eqs. (2) (18) and (23) are compared with the partial Fourier space transforms of eqs. (6) (19) and (24) in table 1. The computing times required by the two different methods are, for modern computers, effectively the same because most of the time is consumed in generating the lattice vectors. The computing times on the ULCC CDC 7600 for the various methods are shown in table 1 and are the same to within the number of significant figures quoted (although the direct summation method is always marginaly faster). Perhaps the most significant conclusion to be drawn from this table is that the modified direct summation expressions converge almost as rapidly as the Fourier transformations. The long range corrections contribute approximately 5% to 1% of the total potential energy on increasing the number of shells of replica cells from 1 to 10 cells in radius. Thus the considerably simpler direct summation method, with the appropriate long range correction, could be used instead of a Fourier space transformation in many applications.
References [l] [2] [3] [4] [5] [6] [7]
M.A. van Hove and P.M. Echenique, Surface Sci. 82 (1979) L298. S. Iannotta and U. Valbusa, Surface Sci. 100 (1980) 28. D.M. Heyes, M. Barber and J.H.R. Clarke, JCS Faraday II, 73 (1977) 1485. F.W. de Wette and G.E. Schacher, Phys. Rev. Al37 (1965) 78. D.J. Adams and I.R. McDonald, Mol. Phys. 32 (1976) 931. E.F. Bertaut, J. Phys. 39 (1978) 1331. M. Abramowitz and LA. Stegun, Eds., Handbook of Mathematical Functions York, 1970) p. 295. [8] M.P Tosi, Solid State Phys. 16 (1964) 1. [9] D.M. Heyes, .I. Phys. Chem. Solids41 (1980) 291. [lo] D.M. Heyes and F. van Swol, J. Chem. Phys., submitted.
(Dover,
New
L619
SURFACE SCIENCE LETTERS THE SURFACE POTENTIAL DIPOLE LATTICES
OF POINT CHARGE AND POINT
D.M. HEYES * Laboratory for Physical Chemistry, University of Amsterdam, IO18 WS Amsterdam, The Netherlands
Nieuwe Achtergracht
127,
Received 11 March 1981; accepted for publication 6 July 1981
New expressions for the potential in the surface region of point charge and point dipole lattices reveal that a modified direct summation scheme can converge with comparable rapidity to those of the more complicated Ewald form.
There is much interest at present in investigating the structural, thermodynamic and dynamical behaviour of solid ionic surfaces [ 1,2]. In order to develop models for such behaviour it is often essential to obtain rapidly converging expansions for the electrostatic potential in the surface region. Here a number of new and easy to use general expressions for calculating the potential energy of point charges and dipoles in the field of point charge [3] and point dipole [4] infinite laminae lattices are derived. Two methods are considered. The procedure of directly summing the interaction from each charge in a lattice of cubic unit cells is made more rapidly converging by the inclusion of a long range correction term which depends on the shape and size of the outermost shell of charges incorporated in the summation. These are compared with formulae which are similar to those of Ewald [5] and Ewald-Kornfeld [5] used for bulk point charge and point dipole lattices [6], respectively. First consider the potential energy of a test charge, 4, at a point, R, from a square planar lamina lattice of unit cell thickness and infinite extent in the (x, y) plane, V,(R), which is obtained from the following absolutely convergent series,
V,(R)=q
5 qj
P'B;'.
(1)
j=l
There are N charges of magnitude qj in the unit cell, which is overall neutral. The lattice vector, I, is the coordinate of the image unit cell relative to the chosen origin * Present address: Department of Chemistry, Royal Holloway College, University of London, Egham, Surrey TW20 OEX, UK.
0039-6028/81/0000-0000/$02.50
0 1981 North-Holland
D.M. Heyes / Surface potential of point charge and point dipole lattices
L620
cell. The charge separation, Bj equals IR- Rj+ 11, where Rj is the position of charge j in the unit cell. The prime on the 1 summation denotes the omission of any term: Bi = 0. For practical purposes this summation can be replaced by the following series,
(2)
V,(R) = lim Rc+=
where Cj = Bj ,
BjGR,,
(3)
cj=-,
Bi >R,,
(4)
A is the area of the unit cell in the (x, v) plane,
M& =
5
qj(O!-
(5)
~j)',
j=1
where x, Y, z, xi, Yj and Zj are the components of R and Rj.The z direction is perpendicular to the surface plane. The first term in eq. (2) represents a direct summation of the contributions from each lattice site summed in circular shells of image cells up to a cut-off radius, R,. The second term in eq. (2) approximates the contribution to the potential made from the charges beyond this truncation distance by replacing them with a uniform charge distribution. The effect on the potential is obtained by Taylor expanding their interactions about R - Ri= 0 and performing the angular and radial integrations in the surface plane. The series of eq. (1) can be represented by the following transform [3,.5], N T/l (R) = q ,g
t 7’
qi ((n/A)
G
Rj)l Fi
c’Js[h ’ C.R -
erfc(bj) By1 - 6 (Bj) 2n-“‘~-~)
,
(6)
where Ir is the reciprocal lattice vector in the surface plane, Fj = [exp@)
erfc(dj) + exp(Ej) erfc(ej)]h-‘,
(7)
bj = B/IQ t
(8)
Dj = hB,i)
(9)
Et = -Dj,
(10)
di = VhI2 + Bzj/Vj
(11)
ej = Vhl2 B,j = Z
-
Zj,
BzJ~,
(12) (13)
D.M. Heyes / Surface potential of point charge and point dipole lattices
where 1) is an arbitrary tion of x [7] and 6(x)=
1 =o
distance parameter,
erfc(x) is the complementary
L621
error func-
x=0,
(14)
xzo.
(15)
Note that here the unit of n is distance whereas this symbol represents an inverse length in ref. [5]. The last term in eq. (6) is a self energy component which contributes if R coincides with a lattice point and is derived from the limit of the real space term as Bi+
0 PI. The h = 0 component of eq. (6) is obtained by taking the limit h * 0; the terms proportional to h-r cancel because the unit cell is a neutral entity. Thus,
[Bzj erf(B,j/n) + vn-“’
Fj(h = 0) = -2
exp(-(Bzj/q)2)].
(16)
The potential energy of a point charge CJin the field of a point dipole lattice, V,(R), is obtained by making the substitution qj ‘-pj . V in eq. (1) where pj is the point dipole at lattice site j [2]. Hence, N
V2 (R) = q ,z
T’
bj * Bj)By3 9
which can be approximated
(17)
as before by,
V2 (R)= lim Rc-*= + (qnlARc)
i2PzjCz
- zj) -
Pxj(X
- Xj) -
Pyj(V
-
Yj)l
)I9
where naj is the (Ycomponent of pj. A uniform distribution of dipoles beyond the truncation radius is now considered. The same operation on eq. (6) gives, V,(R) = q 5 j=l -yzj
((*//I)
COS(~ *
F
[(pj * h) sin(h
*(R - Rj))Fj
(R -Rj))Hj] + q' (Pi*Rj)G(Bj)'
(19)
where
Hi= exp(Dj) erfc(dj) - exp(Ej) erfc(ej) t27r-1’2n-1h-’
[eXp(Ej - eT)-
eXp(Dj - dj)],
(20)
and
G(Bj) = erfc(bj)By3 + 2n-1’2n-1 exp(-L$)B,Y2.
(21)
L622
D.M. Heyes / Surface potential of point charge and point dipole lattices
The potential energy of a test dipole p at Ii from the above dipole lattice, V#), is obtained from eq. (17) by replacing 4 by the operator p . V,
p’ [(p * pip;3
,,I=]$
- 3(p* Bj)(JAj - Bj)B;5] )
(24
and
where par is the (Ycomponent eq. (19) yields,
of c and (Y= x, y or z. The same operation
applied to
N
17, (R) =
c
(r/A)
j=1
F
[cos(h (R - R/))((P h)hj * h)Fj l
l
+ sin(h . (R - Rj))&-(pj a h) + P&
* h))Hil
PzPzjKj)
+ 5;’ I(!$) - W$)
Vs) ,
(24)
where Kj = h exp(Di) erfc(dj) + h exp(Ej) - 4n-”
erfc(ej)
n-r [exp(Dj - d; ) + exp(Ej
t 41r-“~n-~ h-r [di exp(L$ - d/) I(Bj)’ +
(p ‘~j)G(Bj)
- (cl’
Bj)@j
- ef
)]
+ ej exp(Ej Bj)[3
’
27~-“‘77- (2~~~ t 3By2) exp(-b~)B~2
- ef)]
,
(25)
dC(bj)ET’
],
(26)
and the dipolar self energy V, is obtained by taking the following limit [5], Vs = f’lio
- I(Bj)]
[-2/J2/Bj
ri”r = L$$ 7)-3
For h = 0 and ej from omitted. Eqs. (19) in a computer Alternative
a-rf2/3
.
(27) to h-r in Hj and Kj, after substituting for dj do not contribute to the potential and can be
the terms proportional
eqs. (11) and (12)
and (24) are lengthy but ar not significantly more difficult to program than their parent, eq. (6). expressions of the form of eq. (6) which consist of series in real and
2.3094 2.2611 2.2627 2.2610 2.2603 2.2601 2.2600
3.1929 2.4628 2.2947 2.2706 2.2600 2.2600 2.2600
4.0520 3.8022 3.7362 3.7296 3.7248 3.7248 3.7248
FT Eq.(6)
DS Eq.(2)
3.8037 3.7417 3.7240 3.7341 3.7226 3.7248 3.7248
(42L-l)
-VI(R)
5.3333 5.7487 5.7377 5.7528 5.7600 5.7620 5.7621
10.1445 10.4060 10.4833 10.4358 10.4876 10.4769 10.4770
DS Eq.(l8)
Vz(R)
4.0635 5.8056 5.8056 5.7830 5.7620 5.7621 5.7621
9.3036 10.2071 10.4384 10.4610 10.4770 10.4770 10.4770
FT Eq.(l9)
(qPL-2)
21.3520 18.9990 18.7650 18.7548 18.6042 18.6044 18.6035
28.1720 27.9018 27.8063 27.8877 27.8134 27.8326 27.8326
DS Eq.(23)
-V3W
9.7371 16.5598 18.4263 18.7335 18.6034 18.6035 18.6035
34.8375 30.2394 28.4081 28.1010 27.8328 27.8326 27.8326
FT Eq.(24)
(P2L-3)
0.02 0.04 0.05 0.09 0.2 1.8 28
0.02 0.04 0.05 0.09 0.2 1.8 28
Computing time for all columns (s)
The unit cell which gives rise to VI(R) has a charge, 4, at (O,O,O)L and a charge, -q, at (x,y,z)L which is also the test charge. For V,(R) the unit cell contains a dipole, p, of (XJ,Z) components, ~(1,1,1)3-“2 at (O,O,O)L. There is a test charge 4 at (x,y,z)L. A unit cell having a dipole of components ~(1,1,1)3-1~Z at (O,O,O)L gives rise to a potential energy Vs(R) of a test dipole, p’, of components ~.1)1,2,3)(14)-‘1’ at (x,y,z)L. The distance parameter r) has the value of 0.3L. The real and reciprocal space cells are included in the order of their proximity to the origin unit cell, i.e., they are added in expanding circular shells of image cells. (a) (x,y,z)L = (O.OO’, O.OO’, 0.25)L and (b) (x,y,z)L = (0.25,0.25, 0.25)L. DS and FT denote methods of direct summation and Fourier transformation, respectively. R, = (12Lw2 + 1)‘j2L in eqs. (2), (18) and (23).
0 1 2 4 10 100 1600
(b)
1 2 4 10 100 1600
0
(a)
IzLm2 and h2L2/4n2
Table 1 The potential energy of a test charge or dipole at (x,y,z)L from an infinite lamina lattice consisting of a unit cell with a square cross-section of sidelength L
L624
D.M. Heyes /Surface
potential of point charge and point dipole lattices
reciprocal space have been derived [9,10]. Therefore eq. (24) is not a unique solution of the real and reciprocal space series form for this dipole lattice. The potential from a semi-infinite lattice is obtained by adding the effects of such laminae of unit cell thickness that are positioned adjacent and parallel, so that they occupy a half-space. The convergence characteristics of the direct summation formulae of eqs. (2) (18) and (23) are compared with the partial Fourier space transforms of eqs. (6) (19) and (24) in table 1. The computing times required by the two different methods are, for modern computers, effectively the same because most of the time is consumed in generating the lattice vectors. The computing times on the ULCC CDC 7600 for the various methods are shown in table 1 and are the same to within the number of significant figures quoted (although the direct summation method is always marginaly faster). Perhaps the most significant conclusion to be drawn from this table is that the modified direct summation expressions converge almost as rapidly as the Fourier transformations. The long range corrections contribute approximately 5% to 1% of the total potential energy on increasing the number of shells of replica cells from 1 to 10 cells in radius. Thus the considerably simpler direct summation method, with the appropriate long range correction, could be used instead of a Fourier space transformation in many applications.
References [l] [2] [3] [4] [5] [6] [7]
M.A. van Hove and P.M. Echenique, Surface Sci. 82 (1979) L298. S. Iannotta and U. Valbusa, Surface Sci. 100 (1980) 28. D.M. Heyes, M. Barber and J.H.R. Clarke, JCS Faraday II, 73 (1977) 1485. F.W. de Wette and G.E. Schacher, Phys. Rev. Al37 (1965) 78. D.J. Adams and I.R. McDonald, Mol. Phys. 32 (1976) 931. E.F. Bertaut, J. Phys. 39 (1978) 1331. M. Abramowitz and LA. Stegun, Eds., Handbook of Mathematical Functions York, 1970) p. 295. [8] M.P Tosi, Solid State Phys. 16 (1964) 1. [9] D.M. Heyes, .I. Phys. Chem. Solids41 (1980) 291. [lo] D.M. Heyes and F. van Swol, J. Chem. Phys., submitted.
(Dover,
New