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

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#### 1.3.2.4.4.2. Gaussian functions and Hermite functions

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Gaussian functions are particularly important elements of . In dimension 1, a well known contour integration (Schwartz, 1965 , p. 184) yields which shows that the `standard Gaussian' is invariant under and . By a tensor product construction, it follows that the same is true of the standard Gaussian in dimension n: In other words, G is an eigenfunction of and for eigenvalue 1 (Section 1.3.2.4.3.4 ).

A complete system of eigenfunctions may be constructed as follows. In dimension 1, consider the family of functions where D denotes the differentiation operator. The first two members of the family are such that , as shown above, and hence We may thus take as an induction hypothesis that The identity may be written and the two differentiation theorems give: Combination of this with the induction hypothesis yields thus proving that is an eigenfunction of for eigenvalue for all . The same proof holds for , with eigenvalue . If these eigenfunctions are normalized as then it can be shown that the collection of Hermite functions constitutes an orthonormal basis of such that is an eigenfunction of (respectively ) for eigenvalue (respectively ).

In dimension n, the same construction can be extended by tensor product to yield the multivariate Hermite functions (where is a multi-index). These constitute an orthonormal basis of , with an eigenfunction of (respectively ) for eigenvalue (respectively ). Thus the subspaces of Section 1.3.2.4.3.4 are spanned by those with .

General multivariate Gaussians are usually encountered in the nonstandard form where A is a symmetric positive-definite matrix. Diagonalizing A as with the identity matrix, and putting , we may write i.e. hence (by Section 1.3.2.4.2.3 ) i.e. i.e. finally This result is widely used in crystallography, e.g. to calculate form factors for anisotropic atoms (Section 1.3.4.2.2.6 ) and to obtain transforms of derivatives of Gaussian atomic densities (Section 1.3.4.4.7.10 ).

### References

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