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. 471-474
Section 3.5.3. Generalization to Gaussian- and Hermite-based continuous charge distributions^{a}Laboratory of Structural Biology, National Institute of Environmental Health Sciences, 111 T. W. Alexander Drive, Research Triangle Park, NC 27709, USA |
In this section we develop the generalization of Ewald sums to interacting molecular densities modelled by linear combinations of spherical or Hermite Gaussians. In Coppens (1997) the electrostatic potential and energy for crystal densities modelled by linear combinations of Slater-like density elements is discussed. As with quantum-chemical programs, there are trade-offs in the choice of density elements. The use of Slater-like densities provides accuracy with the use of fewer elements, but the Gaussian and Hermite Gaussian elements are far more convenient analytically. In particular the Ewald summation, as well as the fast-Fourier-transform-based methods discussed in Section 3.5.4, generalize straightforwardly to these elements. We note ahead of time that the case of lattice sums of point multipoles are obtained by specializing the results for Hermite Gaussians, just as the results for spherical Gaussians specialize to the above lattice summation results for point charges.
A somewhat different approach to Ewald summation over Hermite Gaussian elements, focused on application to periodic density-functional calculations, is given in Trickey et al. (2004). The continuous fast multipole method (Challacombe et al., 1997) provides yet a different but highly efficient approach to lattice summation of continuous charge densities.
We begin by discussing the electrostatic interaction energy between a normalized spherical Gaussian charge distribution , centred at a point and having exponent , and a second normalized spherical Gaussian charge distribution , centred at and having exponent , together with the latter's periodic images, centred at , . Their interaction energy can be writtenwhere , j = 1, 2. We now recall equation (3.5.2.16), applying its two right-hand components separately, to get , whereand
The direct pair energy involves the damped Coulomb interaction, which has been discussed by Gill & Adamson (1996). We can calculate using the following standard results from quantum chemistry, which are derived in Helgaker et al. (2000): Let , be two normalized spherical Gaussian charge distributions having exponents p and q and centred at points P and Q, i.e. and = . Then the electrostatic potential at a point R due to the charge distribution is given bywhere is the zeroth-order Boy's function, with for x > 0. The Coulomb interaction energy between and is given bywhere . The Boy's function can be related to the error function:Substituting equation (3.5.3.4) into (3.5.3.2), we see thata result which we had derived above by another method [see equation (3.5.2.7)]. From equations (3.5.3.3) and (3.5.3.5), substituting for p and for , we have for any point where . Iterating this result with replacing and replacing we get the damped interaction energy between and ,where and . Notice that in the limit , as expected, since as , and that in the limit , , , the damped interaction between two unit point charges at P and Q, as in the standard Ewald direct sum.
To calculate the reciprocal pair energy we first apply the first right-hand component of equation (3.5.2.16) together with the Fourier transforms of the normalized Gaussians and from equation (3.5.2.18) to obtainwhere we have used the fact that = , with the three-dimensional delta function, and similarly for the integral over . Note that in the limit , , reduces to the equivalent expression for unit point charges at P and Q, and in the limit , reduces to the Fourier-space expression for the direct Coulomb interaction between the Gaussians and .
Recalling the developments leading to equation (3.5.2.28), and using equations (3.5.3.6) and (3.5.3.7), we can write the interaction energy from equation (3.5.3.1) aswhere , and .
Note that although the energy of a finite-exponent Gaussian charge distribution interacting with itself is finite, when considering a generalization of lattice sums involving point charges it is necessary to remove the direct Coulomb interaction of the Gaussian with itself. From equations (3.5.3.3) and (3.5.3.6), if p = q, then as , . We can now generalize equation (3.5.2.29). Let be normalized spherical Gaussian charge distributions with Gaussian exponents , centred at (point charges are included by taking ), and let satisfy (neutral unit cell). Then the energy of the central unit cell within a large spherical crystal, due to interactions of the Gaussian charge distributions with each other and with their periodic images within the crystal, is given bywhere , , and as before is the unit-cell dipole.
Note that from the above derivation, the in is allowed to be different for each pair ij. One choice that leads to an efficient algorithm (particularly when combined with the fast-Fourier-transform-based methods discussed below) is to separate the Gaussian charge distributions into compact and diffuse sets, based on their exponents , and choose to be infinite for ij pairs where at least one of the two Gaussians is diffuse. These pair interactions are then evaluated entirely in reciprocal space. Next choose so that is constant for all compact pairs ij. More specifically, given , a Gaussian is classified as compact if and diffuse otherwise. We denote these cases as and , respectively. Then for , choose so that = . Otherwise, choose to be infinite. Then the Coulomb energy of the central unit cell can be re-expressed as
Note that in the limit that for all i, all of the Gaussians become compact and this formula reduces to equation (3.5.2.30) for a large spherical crystal of point charges. The volume-dependent term is somewhat surprising, and would not appear in a more straightforward Ewald derivation such as our first Ewald derivation. It vanishes by unit-cell neutrality for the case of point charges or multipoles, and would also vanish if we chose a single for all Gaussian pairs.
More complex crystal charge distributions can be realized by considering normalized Hermite Gaussian distributions in place of the normalized spherical Gaussian distributions discussed above. Let be a normalized spherical Gaussian charge distribution with exponent , centred at , i.e. . The associated normalized Hermite Gaussians , where t, u, v are non-negative integers, are defined byNote that this definition of Hermite Gaussians differs from that found in some textbooks, e.g. Helgaker et al. (2000), in that we have normalized the Gaussian it is derived from in order to allow exponents to tend to infinity to arrive at ideal point charges or point multipoles. We will refer to these Hermite Gaussians as normalized Hermite Gaussians to emphasize this distinction. The Hermite definition reduces to the spherical Gaussian when t, u and v are all zero, i.e. .
Above we discussed the interaction between spherical Gaussian charge distributions . Here we generalize these to higher angular momentum charge distributions given bywhere c is a vector of coefficients, a finite number of which are nonzero. For example, if is the only nonzero coefficient, we are reduced to the spherical charge distributions discussed above.
The fact that the partial derivatives within are with respect to the position of the centre of the distribution allows us to easily calculate the Coulomb interaction energy between two normalized Hermite Gaussian distributions. The partial derivatives can be pulled out of the double integral, so that the interaction energy is given by the corresponding partial derivatives of the interaction energy of the two spherical Gaussian charge distributions:
Next we consider the interaction energy between a generalized charge distribution centred at and another generalized charge distribution centred at , together with all the latter's periodic images, centred at , . Examining equation (3.5.3.8), we note that the terms to be differentiated all depend on . Thuswhere , i =1, 2 are given byandNote that by the arguments leading to equation (3.5.2.28), is quadratic in r, i.e. derivatives of higher than second order vanish.
We now can discuss the Coulomb energy of the central unit cell U within a large spherical crystal when the charge density in the unit cell is described by a basis of normalized Hermite Gaussians and a set of expansion points within U that are repeated periodically in the crystal (for example, some of the expansion points could be atomic nuclei). Let be positive Gaussian exponents (or infinite, in the case of nuclear charge or point charges or ideal multipoles) and let the charge density about the expansion point be given bywhere, as above, for each i, l the coefficients are nonzero for a finite number of tuv. As above we can separate the exponents into compact and diffuse, i.e. or , according to whether or , respectively. Let denote the net charge at the expansion point .
We are interested in the Coulomb energy of charge densities , , interacting with each other and their periodic images at expansion points , for large K. As above, the direct Coulomb interaction of with itself is not permitted. This can be modified for more sophisticated treatments e.g. in the context of the Coulomb integrals in a periodic density-functional-theory approach, but clearly a nuclear charge cannot interact with itself. The unit cell is assumed to be neutral, i.e. . Then, using equations (3.5.3.9) and (3.5.3.10) we can write the Coulomb energy of = within the spherical crystal aswhere the structure factors are given by is given bywhere , and the unit-cell dipole components are given by
In the limit that all Gaussians are compact with , reduces to the Coulomb energy of ideal multipoles positioned at the expansion points.
References
Challacombe, M., White, C. & Head-Gordon, M. (1997). Periodic boundary conditions and the fast multipole method. J. Chem. Phys. 107, 10131–10140.Coppens, P. (1997). X-ray Charge Densities and Chemical Bonding. New York: Oxford University Press.
Gill, P. M. W. & Adamson, R. D. (1996). A family of attenuated Coulomb operators. Chem. Phys. Lett. 261, 105–110.
Helgaker, T., Jorgensen, P. & Olsen, J. (2000). Molecular Electronic-Structure Theory. Chichester: John Wiley and Sons.
Trickey, S. B., Alford, J. A. & Boettger, J. C. (2004). Methods and implementation of robust, high-precision Gaussian basis DFT calculations for periodic systems: the GTOFF code. In Theoretical and Computational Chemistry, edited by J. Leszcynski, Vol. 15, ch. 6. Amsterdam: Elsevier.