International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 38-39   | 1 | 2 |

Section 1.3.2.4.4.2. Gaussian functions and Hermite functions

G. Bricognea

aGlobal Phasing Ltd, Sheraton House, Suites 14–16, Castle Park, Cambridge CB3 0AX, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.2.4.4.2. Gaussian functions and Hermite functions

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Gaussian functions are particularly important elements of [{\scr S}]. In dimension 1, a well known contour integration (Schwartz, 1965[link], p. 184) yields[{\scr F}[\exp (- \pi x^{2})](\xi) = \bar{\scr F}[\exp (- \pi x^{2})](\xi) = \exp (- \pi \xi^{2}),]which shows that the `standard Gaussian' [\exp (- \pi x^{2})] is invariant under [{\scr F}] and [\bar{\scr F}]. By a tensor product construction, it follows that the same is true of the standard Gaussian[G({\bf x}) = \exp (- \pi \|{\bf x}\|^{2})]in dimension n:[{\scr F}[G]({\boldxi}) = \bar{\scr F}[G]({\boldxi}) = G({\boldxi}).]In other words, G is an eigenfunction of [{\scr F}] and [\bar{\scr F}] for eigenvalue 1 (Section 1.3.2.4.3.4[link]).

A complete system of eigenfunctions may be constructed as follows. In dimension 1, consider the family of functions[H_{m} = {D^{m}G^{2} \over G}\quad (m \geq 0),]where D denotes the differentiation operator. The first two members of the family[H_{0} = G,\qquad H_{1} = 2 DG,]are such that [{\scr F}[H_{0}] = H_{0}], as shown above, and[DG(x) = - 2\pi xG(x) = i(2\pi ix)G(x) = i{\scr F}[DG](x),]hence[{\scr F}[H_{1}] = (- i)H_{1}.]We may thus take as an induction hypothesis that[{\scr F}[H_{m}] = (-i)^{m}H_{m}.]The identity[D\left({D^{m}G^{2} \over G}\right) = {D^{m+1}G^{2} \over G} - {DG \over G} {D^{m}G^{2} \over G}]may be written[H_{m+1}(x) = (DH_{m})(x) - 2\pi xH_{m}(x),]and the two differentiation theorems give:[\eqalign{{\scr F}[DH_{m}](\xi) &= (2\pi i{\boldxi}) {\scr F}[H_{m}](\xi)\cr {\scr F}[-2\pi xH_{m}](\xi) &= - iD({\scr F}[H_{m}])(\xi).}]Combination of this with the induction hypothesis yields[\eqalign{{\scr F}[H_{m+1}](\xi) &= (-i)^{m+1}[(DH_{m})(\xi) - 2\pi \xi H_{m}(\xi)]\cr &= (-i)^{m+1} H_{m+1}(\xi),}]thus proving that [H_{m}] is an eigenfunction of [{\scr F}] for eigenvalue [(-i)^{m}] for all [m \geq 0]. The same proof holds for [\bar{\scr F}], with eigenvalue [i^{m}]. If these eigenfunctions are normalized as[{\scr H}_{m}(x) = {(-1)^{m}2^{1/4} \over \sqrt{m!}2^{m}\pi^{m/2}} H_{m}(x),]then it can be shown that the collection of Hermite functions [\{{\scr H}_{m}(x)\}_{m \geq 0}] constitutes an orthonormal basis of [L^{2}({\bb R})] such that [{\scr H}_{m}] is an eigenfunction of [{\scr F}] (respectively [\bar{\scr F}]) for eigenvalue [(-i)^{m}] (respectively [i^{m}]).

In dimension n, the same construction can be extended by tensor product to yield the multivariate Hermite functions[{\scr H}_{\bf m}({\bf x}) = {\scr H}_{m_{1}}(x_{1}) \times {\scr H}_{m_{2}}(x_{2}) \times \ldots \times {\scr H}_{m_{n}}(x_{n})](where [{\bf m} \geq {\bf 0}] is a multi-index). These constitute an orthonormal basis of [L^{2}({\bb R}^{n})], with [{\scr H}_{\bf m}] an eigenfunction of [{\scr F}] (respectively [\bar{\scr F}]) for eigenvalue [(-i)^{|{\bf m}|}] (respectively [i^{|{\bf m}|}]). Thus the subspaces [{\bf H}_{k}] of Section 1.3.2.4.3.4[link] are spanned by those [{\scr H}_{\bf m}] with [|{\bf m}| \equiv k\hbox{ mod } 4\ (k = 0, 1, 2, 3)].

General multivariate Gaussians are usually encountered in the nonstandard form[G_{\bf A}({\bf x}) = \exp (- {\textstyle{1 \over 2}} {\bf x}^{T} \cdot {\bf Ax}),]where A is a symmetric positive-definite matrix. Diagonalizing A as [{\bf E}\boldLambda{\bf E}^{T}] with [{\bf EE}^{T}] the identity matrix, and putting [{\bf A}^{1/2} = {\bf E}{\boldLambda}^{1/2}{\bf E}^{T}], we may write[G_{\bf A}({\bf x}) = G\left[\left({{\bf A} \over 2 \pi}\right)^{1/2} {\bf x}\right]]i.e.[G_{\bf A} = [(2\pi {\bf A}^{-1})^{1/2}]^{\#} G\hbox{\semi}]hence (by Section 1.3.2.4.2.3[link])[{\scr F}[G_{\bf A}] = |\!\det (2\pi {\bf A}^{-1})|^{1/2} \left[\left({{\bf A} \over 2 \pi}\right)^{1/2}\right]^{\#} G,]i.e.[{\scr F}[G_{\bf A}]({\boldxi}) = |\!\det (2\pi {\bf A}^{-1})|^{1/2} G[(2\pi {\bf A}^{-1})^{1/2}{\boldxi}],]i.e. finally[{\scr F}[G_{\bf A}] = |\det (2\pi {\bf A}^{-1})|^{1/2} G_{4\pi^{2}{\bf A}^{-1}}.]

This result is widely used in crystallography, e.g. to calculate form factors for anisotropic atoms (Section 1.3.4.2.2.6[link]) and to obtain transforms of derivatives of Gaussian atomic densities (Section 1.3.4.4.7.10[link]).

References

Schwartz, L. (1965). Mathematics for the Physical Sciences. Paris: Hermann, and Reading: Addison-Wesley.








































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