International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 |
International Tables for Crystallography (2010). Vol. B, ch. 3.5, pp. 460-471
Section 3.5.2. Lattice sums of point charges^{a}Laboratory of Structural Biology, National Institute of Environmental Health Sciences, 111 T. W. Alexander Drive, Research Triangle Park, NC 27709, USA |
We begin by discussing an idealized infinite crystal C, made up of point charges.
The periodicity of an idealized crystal C is specified by lattice basis vectors , and , which are linearly independent as vectors and thus form a basis for the usual vector space of points in three dimensions. The conjugate reciprocal vectors are defined by the relations (the Kronecker delta) for . Thus for example is given by = . General lattice vectors n are given by integral combinations of the lattice basis vectors:where are integers. General reciprocal-lattice vectors m are given by integral combinations of the reciprocal-lattice basis vectors:
An arbitrary point r in the crystal (r also denotes the vector from the origin of coordinates to the point) is given bywhere are the fractional coordinates of the point. From the above equation and the defining relations for the reciprocal basis we see that for . The unit cell U is the set of points r with associated fractional coordinates satisfying , . The idealized infinite crystal C is generated by the union of all periodic translations of the unit cell U, using the set of general lattice vectors n. A point charge q at position r in U has periodic `image' charges at positions for all lattice vector n. The periodic image of U, denoted , is given by the periodic images of all point charges q in U. Thus, the crystal is made up of the union of images , including , or
A point charge in U at position interacts with other unit-cell charges at positions as well as with all of their periodic images at positions for all lattice vectors n. It also interacts with its own periodic images at for all nonzero lattice vectors n, that is all with . Suppose the unit cell U is made up of N point charges at positions . Then the electrostatic energy of U, given by the sum of the Coulombic interactions of charges in U with each other and with the rest of C, is written aswhere here and below denotes the set of points and where the outer sum on the right is over the general lattice vectors n, the prime indicating that terms with i = j and n = 0 are omitted. The treatment of this infinite sum requires some care. The energy diverges unless the unit cell is neutral (i.e. unless ). When U is neutral, the sum, if carried out in a naive fashion, can still converge quite slowly (see Chapter 3.4 ). In addition, the convergence is conditional, that is the resulting energy can vary depending on the order in which the sum is carried out! To better understand these phenomena, it is helpful to re-express the above energy. Let E(U, U) denote the Coulomb interaction energy of U with itself and the Coulomb interaction energy of U with , that isandThen we can abbreviate equation (3.5.2.3) byNote that for large n (that is, n with large norm ) can be approximated by a Taylor expansion. Let denote the centroid of points in U, given by , and let denote the centroid of . Then, expanding in terms of and (which are small compared to ) we have
Let and denote the net charge and dipole moment of the unit cell U, and let = denote the quadrupole moment of the unit cell. Using the expansion in the above equation, we can Taylor expand the interaction energy asNote that the first-order term in the expansion has dropped out, since the dipole moments D of U and are the same. If then for large the leading term in the Taylor expansion of is . Since , then using a standard integral comparison is infinite, and so the resulting electrostatic energy of U diverges. If we see that the leading term in the expansion of is given by , that is a quadratic expression in the dipole moment D times a prefactor that decays like . Since , also diverges (logarithmically), and so the electrostatic energy of U cannot converge absolutely in this case either. Hence, if it converges at all, it does so conditionally, that is the energy of U depends on the order in which we sum the contributions from image cells . If Q = 0 and D = 0, the leading term in the Taylor expansion of depends on the quadrupole moment of the unit cell and decays like for large , and thus since does converge, the electrostatic energy of U converges absolutely. From these considerations we might hope to express the energy of a neutral unit cell U in terms of an absolutely convergent sum plus a term quadratic in the unit-cell dipole moment D which also depends on the method of summation. As we will see below this is precisely what the Ewald summation method gives us.
In this section we give the standard Ewald-sum derivation found in textbooks such as Kittel (1986). To do this we need two preliminary results, namely the electrostatic potential due to an isolated Gaussian charge distribution having total charge of one, and then that due to a periodic array of such distributions.
First we derive the electrostatic potential due to an isolated Gaussian density given byOwing to the spherical symmetry of we can define the one-dimensional function by for any r with . We next invoke Gauss' law (Eyges, 1980) under spherical symmetry to evaluate the electric field due to at any distance . In the case of spherical symmetry Gauss' law states that the electric field at a point r, due to a spherically symmetric charge distribution centred at the origin, is the same as if all the charge of the distribution from the origin out to were collected into a point charge at the origin. More specifically, if S is the sphere of radius with surface , Gauss' law statesand thus, defining the scalar quantity , we getand thus, since (again by symmetry) , we can verify thatwhere C is an arbitrary constant of integration. We obtain a potential with the desired asymptotic behaviour, i.e. as , by setting C = 0. Note then that for large r as expected.
Next we derive the electrostatic potential due to a periodic array of Gaussian charge distributions, each of the form just studied. That is, for points r in the unit cell U, define the periodic density bywhere the density is defined in equation (3.5.2.6) and the sum is over all general lattice vectors n, defined in equation (3.5.2.1). Since is periodic, i.e. for any general lattice vector , we can expand it in a Fourier series over the unit cell:where the sum is over all reciprocal-lattice vectors m, defined in equation (3.5.2.2), and the Fourier coefficient is given bywhere V is the volume of the unit cell U. The periodicity of implies that of , so we can also expand the latter in a Fourier series:Next we recall Poisson's equation (Eyges, 1980) for in terms of :Using equations (3.5.2.10) and (3.5.2.8) we can rewrite equation (3.5.2.11) asthus using the uniqueness of Fourier-series coefficients . If this latter equation can be solved for . However, for m = 0 it requires that , whereas from equation (3.5.2.9) . To avoid this contradiction we modify by subtracting 1/V (uniform neutralizing plasma). That is, is now defined byNote that is still periodic. Also note that for is still given by equation (3.5.2.9), whereas now . Although we have avoided the contradiction, the value of is still not determined. Some authors define it by considering m as a real variable and taking the limit as (this depends on which direction the origin is approached from), but most treatments take it to be 0 (however, see Section 3.5.2.3). In the latter case the electrostatic potential due to is given by
Now as before let be point charges located at positions within the unit cell U. From the discussion in Section 3.5.2.1 we assume the unit cell is neutral, i.e. . Let r be a point in U with . We want to obtain the electrostatic potential due to the point charges in U together with their periodic images in the remainder of the crystal C. That is,We next supplement the potential due to the point charge at by the potential due to an equally charged Gaussian counterion, that is times a density of the form in equation (3.5.2.6), centred at . Using equation (3.5.2.7) the resulting direct sum potential is given bywhere . Note that since decays exponentially fast to zero as , the sum in the above equation converges rapidly and absolutely.
However, having added the potential due to counterions we now need to cancel it. That is, we must consider the reciprocal electrostatic potential due to times a density of the form , centred at , for all direct-lattice vectors n. Since the unit cell is neutral, the sum of these reciprocal potentials over and all n is the same as the sum over of times the potential due to centred at where is given by equation (3.5.2.12). Note that the extra term involving 1/V drops out by neutrality. [In the case of non-neutrality, such as discussion of the free energy of charging an ion (Hummer et al., 1996), the potential due to the uniform compensating plasma must be included (see below). Hummer has derived the potential due to a uniformly charged unit cell in the special case of a cubic unit cell (Hummer, 1996).] Thus, using equation (3.5.2.13) we arrive at the reciprocal sum potential:where the structure factor is given by
We have now expressed as the sum of plus (with the caveat that in the process some infinite series have been casually rearranged and that the contribution of the reciprocal vector m = 0 has been ignored). To get a valid electrostatic potential at the position of a charge in U we must remove the Coulomb potential due to and then take the limit as . Noting that and that , we haveThus we can write the electrostatic energy of U from equation (3.5.2.3) aswhere as above the prime indicates that terms with i = j and n = 0 are omitted.
The derivation in Section 3.5.2.2 is incomplete. In addition to the caveats concerning the expression for in terms of plus , there is no mention of a quadratic term involving the unit-cell dipole moment M. In this section we provide a more complete derivation of the Ewald sum, which will in addition explain the conditional convergence discussed previously. We follow the developments in Smith (1981, 1994).
The derivation depends on the following identity:where is the reciprocal unit cell, i.e. the set of points with , . Actually we derive (following Essmann et al., 1995) a generalization of identity (3.5.2.16), valid for general inverse powers , p > 0. For this first note that for u > 0where is the Euler gamma function. Next note that for a > 0which is the Fourier transform of the Gaussian, as derived above within equation (3.5.2.9). In equation (3.5.2.17) substitute and u = p/2 to get, for α > 0,In we substitute , whereas in we use equation (3.5.2.18), change the order of integration, and substitute to obtainwhereandSpecializing to the case p = 1 we see that and that , which proves identity (3.5.2.16). Another important case is p = 6, for which = and = .
Note that from equation (3.5.2.17), substituting t = s^{2}, λ = r^{2} and u = p/2 we haveAlso, substituting into equation (3.5.2.20) we haveand so
Note that is a smooth function of x for , and that its derivatives are uniformly bounded for x bounded away from 0. To show this, substitute s = xt in equation (3.5.2.21) to write
In the case that p = 6, is smooth with uniformly bounded derivatives for all x, but this is unfortunately not true for all p. However, from the above expression we can show that for p > 3, is bounded and continuous as , whereas for as . Thus for p > 3 we see that for any x > 0, . However for , for any R > 1 if x > 0 sufficiently small ; thus and thus is unbounded as .
Despite this latter result, the integral of over is bounded for . This result is important below for the validity of rearranging the order of summation or taking limits inside infinite sums involving . To prove boundedness, use the above expression for and integrate by parts to write
We first discuss the lattice sum for the case p > 3. In this case the lattice sum converges absolutely, since the tail of this sum can be bounded from above by a constant times the integralfor some K > 0. Although the sum can thus be evaluated directly, it can be significantly accelerated using the same Ewald direct- and reciprocal-space decomposition. For , using equation (3.5.2.19) we havewhere . Since is bounded, the first of the above sums converges absolutely and we turn to the second. We recognize the integral as (1/V) times , the nth Fourier coefficient of , where V is the volume of the unit cell U (1/V is the volume of ). From the above results for it is clear that for , are smooth and have uniformly (in v, r and ) bounded derivatives. Thus the sum of the Fourier coefficients can be shown to converge absolutely and uniformly in m, which guarantees uniform (over ) convergence of the sum of these integrals to , which in turn equals .
The case of m = 0 is not as straightforward, since the uniform boundedness of derivatives of does not generally hold in this case. One way of handling it is as follows. Since everything else in the above equation converges as , the sum of the Fourier coefficients for converges to something. That is, iffor , , then for some s, as . Ifthen it is straightforward to show that also converges to s as [see, for example, Chapter 1 of Körner (1988) for the one-dimensional proof]. On the other hand, due to the boundedness and continuity of , as . [This involves the three-dimensional version of the Fejér kernel, which converges to the three-dimensional Dirac delta function. See, for example, Chapter 2 of Körner (1988) for the one-dimensional proof.] Thus as . Hence the above sum of integrals with m = 0 converges to .
Thus from the above arguments we have
Using the limiting behaviour given in (3.5.2.22) we can extend this result to the case r = 0. By removing the singularity when n = 0 we get
Now assume particles i and j in the unit cell U interact by the potential . Using equations (3.5.2.24) and (3.5.2.25) (the latter for particles interacting with their own images) the total energy for the unit cell U consisting of N particles interacting via the potential with each other and all of the image cells is given bywhere as before the prime on the first summation means that terms where i = j and n = 0 are omitted.
In the case when factorizes, i.e. , the above lattice sum can be further simplified and accelerated using the structure factor given byIn this case,
Note that as expected, due to the absolute convergence of the lattice sums, there is no shape-dependent correction term to the Ewald sum. This is connected with the fact that is bounded and continuous as . Next we see what additional assumptions are necessary when this is no longer true, and what form the correction term takes. We cover the most interesting case, namely the Coulombic case p = 1.
In the case , the lattice sum does not converge absolutely, since the integral in equation (3.5.2.23) diverges. Thus extra conditions are needed and the order of summation must be specified. We restrict discussion to the case p = 1, that is to a crystal of point charges interacting via Coulomb's law. We consider a large but finite crystal, taking the limit as it grows larger while maintaining a specific shape. To that end, let P be a closed, bounded region in centred on the origin, such as a cube, sphere or ellipsoid. For a positive integer K, let denote the set of lattice vectors such that is in P, that is . Charges in the central unit cell U_{0} interact with each other and with image charges in U_{n} for . The electrostatic energy of the central cell in this finite crystal is given byand we want the limit of as .
Recalling that for p = 1, if we define bythen for , is bounded and smooth with uniformly (in ) bounded derivatives. For m = 0 we have = , where is smooth with bounded derivatives, and with . Then, using the same arguments that lead to equation (3.5.2.24), we can write for where here and below denotes a quantity (the remainder in the equation) that converges to zero as and
The Ewald potential , given for byhas a couple of noteworthy properties. First, note that since the left-hand side of equation (3.5.2.28) is independent of α, as is , is itself independent of α. Secondly, the average of over the unit cell U is zero. To establish this latter property, first note that for , . Thus the reciprocal sum in the above equation integrates to zero over U. Next, the integral of −π/(αV) over U is clearly −π/α. Finally, note thatcompleting the proof. The self or Wigner potential with given byis the potential of a charge q in a periodic array of images of itself together with a uniform neutralizing plasma throughout the crystal. For cubic unit cells with side L, (Nijboer & Ruijgrok, 1988). Owing to the above properties, the Ewald potential provides a consistent approach to the treatment of non-neutral unit cells, such as when calculating ionic charging free energies (Hummer et al., 1996; Darden et al., 1999).
If we define the structure factor byand the unit-cell dipole moment D bythen assuming the unit cell is neutral we can write the energy asThus, recalling equation (3.5.2.15), we can writewhereNote that if D = 0, and the Coulomb lattice sum converges absolutely, i.e. the result is the same as the Ewald sum regardless of the shape P, i.e. order of convergence. Expressions for have been derived in some cases by Smith (1981). In particular, if P is the unit sphere . Following Smith's approach we show this by first deriving an integral representation for and then evaluating it for the case P is the unit sphere.
First note that for any continuous function f, converges to as . Then, in the above integral for we can substitute w = Kv to getand thus
Specializing to the case P = S, the unit sphere, we see by using spherical coordinates for with the pole parallel to w that is a function only of the norm of w, that is . Then, again using spherical coordinates, this time for w with the pole parallel to D, and noting that , we havewhere we have switched the order of integration in the last step. Finally, since , the three-dimensional Dirac delta function, this last double integral equals one, completing the derivation of the Ewald correction term in the case of a large but finite crystal having a macroscopically spherical shape.
Above we have derived the form of the electrostatic energy in the case of a macroscopically spherical crystal in terms of the Ewald sum plus a correction term. In this section, following the approach of van Eijck & Kroon (1997), we develop a surface-integral-based expression for the shape-dependent correction term for more general macroscopic shapes.
Given a closed bounded region P that contains the unit sphere S in its interior and a K > 0, and recalling equations (3.5.2.4) and (3.5.2.27), we can write the electrostatic energy of the central unit cell in the finite crystal as
The asymptotic form of has been developed above. Recalling equation (3.5.2.5), for large K and neutral unit cells, we can approximate bywhere we have used Gauss' divergence theorem (Arfken & Weber, 2000) together with
Since, considering the outward unit normal on S,and since from the previous resultswe can now generalize the lattice summation to asymptotic shapes P:
The latter surface integral is referred to as the `surface term' and clearly must equal the quantity discussed above. It was derived previously by Deem et al. (1990) by different arguments. Van Eijck and Kroon calculate it explicitly for some specific crystal shapes. They consider, however, that in actual crystal growth, surface charge distributions should develop to cancel the surface-integral contribution (see also Smith, 1981). In contrast, Scheraga and co-workers argue for the importance of considering this term, which they refer to as the `Lorentz–Ewald correction', in crystal-structure prediction (Pillardy et al., 2000), although in this they are disputed by van Eijck & Kroon (2000) (see also the response by Wedemeyer et al., 2000).
The results in this section are all derived assuming the crystal is surrounded by vacuum, or at least by a medium with no dielectric screening capability. In the following section we study the effect of a surrounding continuum dielectric medium, assuming a spherical crystal (to make the calculations tractable).
In this section, following the approach in Smith (1981, 1994), we derive the polarization correction to the electrostatic energy of U due to immersing the large but finite crystal in a uniform dielectric with dielectric constant , assuming the macroscopic shape of the crystal is spherical. We first derive the dielectric reaction potential at a point due to the charges and their periodic images in , assuming the unit cell is neutral, i.e. , that P = S, the unit sphere, and that the sphere KS is immersed in the dielectric.
To begin with, given a point and a point , with spherical polar coordinates and , respectively, the electrostatic potential at r due to a unit charge at is given by the solution to Laplace's equation. Let β be the angle between r and , that is . Owing to the axial symmetry, the general solution to Laplace's equation in this case can be writtenwhere are the Legendre polynomials (Böttcher, 1973). The contributions can be thought of as due to charges further from the origin than r, while are due to charges closer to the origin. A particular solution to Laplace's equation is given by . Thus, we can write the potential due to the unit charge at in the presence of the dielectric continuum surrounding KS asWe now apply boundary conditions, noting that are an orthogonal basis of functions, so that boundary conditions can be applied term by term. Then we have
We expand (for ) in Legendre polynomials to allow comparison of with as from below:Using this we get:
Finally, since we need the solution for the potential due to each of the charges in , we invoke the spherical harmonic addition theorem (Arfken & Weber, 2000) applying it to the points r, :where (Arfken & Weber, 2000)and the associated Legendre polynomials , are defined by Rodrigues' formula,andIn particular, the first-order spherical harmonics are given byThe spherical harmonics satisfy an orthogonality relation:Then, for we express the polarization potential due to the dielectric response to a unit charge at aswhereand
We now examine the total polarization potentialwhere D is the unit-cell dipole and we have used unit-cell neutrality in the next to last equation.
The gradient with respect to the usual Cartesian directions, expressed in spherical polar coordinates (Arfken & Weber, 2000) can be re-expressed as
Using spherical polar coordinates to carry out the integrations over the unit sphere in equation (3.5.2.35), we see after the first partial integration over thatWe recognize the factors multiplying above as being proportional to , and , respectively, so that, using the above orthogonality of spherical harmonics, these contribute nothing to the integrals unless l = 1.
Next, examining I given byintegrating over and then applying integration by parts over to the first term, expressing using the Rodrigues' formulas (3.5.2.32) and (3.5.2.33), and finally substituting we see that I is proportional to the integral J given bywhich in turn is zero unless l = 1. Putting the above results together we see thatSimilarlyFinally,is proportional toand J is zero unless l = 1. Thus
Expressing in terms of the explicit forms for given in equations (3.5.2.34), we getand thusSimilarlyand finallySubstituting these into equation (3.5.2.35) we get
Recalling the shape-dependent term from equation (3.5.2.31) and adding the reaction energy , we see that the electrostatic energy of the unit cell U inside a large spherical crystal which in turn is immersed in a dielectric continuum with dielectric constant is given bywhere is given in equation (3.5.2.15). Note that we recover the usual Ewald sum in the limit as (`tin-foil' boundary conditions).
The force on a charge in the central cell is obtained by taking the gradient of the energy with respect to for the finite spherical crystal, and then taking the limit as , or by taking the gradient of the limiting energy above (i.e. the limit of the gradient is the gradient of the limit):
Thus we can find tractable analytic expressions for the energy and force on particles within the central unit cell as long as the asymptotic shape of the crystal immersed in the dielectric medium is spherical. In Section 3.5.2.6 below we show that we can also arrive at a tractable result for the thermodynamic pressure under these circumstances, although surprisingly it does not seem to agree with the mechanical pressure as given by Smith (1994), which may reflect some not yet fully understood subtleties in the implementation of periodic boundary conditions.
We first discuss the scalar pressure, following the developments in Smith (1987, 1993). According to the thermodynamic definition, the scalar pressure P is defined bywhere A is the Helmholtz free energy, V is the volume and T is the temperature. In turn, A is given by where ( is Boltzmann's constant) and is the canonical ensemble partition function,where is the system Hamiltonian, h is Planck's constant and denotes the set of momenta with conjugate to . Introducing the scaling relations , and , we getwhere denotes the expectation or ensemble average. For the above system of N point charges having mass at positions in the central unit cell of a large spherical crystal (of radius K) immersed in a dielectric continuum, the Hamiltonian can be writtenand soThe direct- and reciprocal-lattice vectors are scaled as and . The derivative with respect to volume of the Ewald direct sum is straightforward using the chain rule, since if then and = . Since m and scale oppositely with V, . Continuing with the other terms in the Ewald reciprocal sum, and taking the limit as , we get
The scalar pressure has been generalized to the pressure tensor (Brown & Neyertz, 1995) using a more general scaling technique based on the expression for r in terms of unit-cell basis vectors and fractional coordinates, treating the standard Ewald sum without the surface correction or dielectric response term. To accommodate the latter, the energy of the central unit cell within a large ellipsoidal crystal immersed in a dielectric continuum must be studied. In Redlack & Grindley (1972, 1975) expressions are provided for the surface correction in ellipsoidal crystals in a vacuum. If this result were combined with the polarization response term (we are not aware of any papers giving this term for an ellipsoidal crystal immersed in a dielectric continuum) it is reasonable to assume that the standard Ewald sum would apply in this case as well, in the limits and . The trace of the pressure tensor defined by Nosé & Klein (1983) agrees with the above scalar pressure in the limit .
Interestingly, Smith (1994) derives a scalar pressure that differs from the above. In particular he finds a surface term that remains even in the limit of `tin-foil' boundary conditions, i.e. when . He gives a mechanical definition of the pressure tensor in terms of momentum flux across surfaces within and surrounding the crystal. Periodic boundary conditions are enforced using velocity constraints via the techniques of DeLeeuw et al. (1990). He derives forces on particles within the central unit cell (and on their images in other cells interior to the crystal) that agree in the large K limit with our result above in equation (3.5.2.36). However, combining the forces with particle positions in the scalar virial expression, after summing over particles and their images, and proceeding to the limit as leads to a vacuum surface correction involving the reciprocal sum term at m = 0 that, unlike our thermodynamic pressure expression above, is not cancelled by the dielectric response potential in the limit . To reiterate, his mechanical virial tensor, restricted to the standard Ewald sum part of the final expression, agrees completely with the results of Nosé & Klein (1983). Furthermore, his scalar virial, given by the trace of his virial tensor, has an -dependent part due to the dielectric response that agrees completely with our results above. However, the trace of the vacuum surface correction tensor within his mechanical pressure tensor does not agree with the equivalent term in our thermodynamic pressure derived above, and thus is not cancelled in the infinite dielectric limit.
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