D. J. THOMAS 343 THOMAS, D.J. (1985). Computing for the Enraf-Nonius FAST system. In Crystallographic Computing 3: Data Collection, Structure De- termination, Proteins, and Databases, edited by G.M. SHELDRICK, C. KROGER & R. GODDARD, pp. 52-60. Oxford: Clarendon Press. THOMAS, D. J. (1989). Calibrating an area-detector diffractometer. Imaging geometry. Proc. R. Sot'. London Set" A, 425, 129-167. THOMAS, D.J. (1990a). Calibrating an area-detectordiffractometer. In- tegral response. Proc. R. Sot'. London Ser A, 428. 181-214. THOMAS, D.J. (1990b). Modern equations of diffractometry. Diffrac- tion geometry. In preparation. WONACOTT, A.J. (1977). Geometry of the rotation method. In The Ro- tation Method in Crystallography, edited by U. W. ARNDT & A. J. WONACOTT, pp. 75-103. Amsterdam: North-Holland. YANG, A.T. (1969). Displacement analysis of spatial five-link mech- anisms using (3 × 3) matrices with dual number elements. Trans. ASME, 91(B) (J. Eng. Ind.), 152-157. YANG, A.T. & FREUDENSTEIN,E (1964). Application of dual-number quaternion algebra to the analysis of spatial mechanisms. Trans. ASME, 86(E) (J. Appl. Mech.), 300-308. Acta Cryst. (1990). A46, 343-351 Determination of Electrostatic Potentials in Crystalline Compounds. The Application to Boric Acid BY PETER SOMMER-LARSEN,* ANDERS KADZIOLA AND MICHAEL GAJHEDE Department of Physical Chemistry, The HC Orsted Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark (Received 28 June 1989; accepted 15 November 1989) Abstract A formalism for deriving electrostatic potentials in crystals is presented, with emphasis on the choice of origin and the determination of the mean inner poten- tial. Conditions for applying the conventional origin chosen for isolated molecules are specified. The for- malism is applied to orthoboric acid, and maps of the electrostatic potentials are presented. Extinction appears to be a severe problem in mapping electro- static potentials, and its effects are investigated with a multipole expansion of the electron density. The effects of thermal motion are seen to be small at points far from the atomic core regions. Introduction It has been shown that electrostatic potentials in crystals can be derived from X-ray diffraction data (Stewart, 1979). This paper will present our applica- tion of this formalism. Even though electrostatic potentials are of major importance in the investigation of chemical dynamics, only a few groups work, or have worked, with the mapping of potentials from diffraction data (Moss & Coppens, 1980; Moss & Feil, 1981; Feil & Moss, 1983; Stewart, 1982; Swaminathan, Craven & McMullan, 1985). Bertaut (1952, 1977) has worked with the mapping of potentials in ionic crystals. The most frequently applied method for determin- ing electrostatic potentials from crystallographic data * Present address: Department of General Chemistry, Royal Danish School of Pharmacy, Universitetsparken 2, DK-2100 Copenhagen, Denmark. 0108-7673/90/050343-09503.00 (e.g. Swaminathan, Craven & McMullan, 1985) is based on a multipole expansion of the electron density. So far, this method has been used to derive the potential of a single molecule from the multipole functions and the expansion coefficients. In this paper we concentrate on the potential inside a crystal. The first section discusses the distinction between the electrostatic potential in a crystal, calcu- lated either by a Fourier sum in reciprocal space or by a superposition in direct space of the potentials from the units which build up the crystal. The charge density of such a unit - the building block for the crystal - could be the charge density inside a single unit cell or it could be the charge density from the atoms or molecules in the unit cell. The Fourier coefficient of the potential for the reciprocal-lattice vector of length zero is of special importance when discussing the two different ways of expressing the potential as this coefficient is the average value of the potential inside the crystal. It is known as the mean inner potential. Changing this Fourier coefficient corresponds to changing the origin for the potential inside the crystal. So in the first section we pay special attention to the question of how to determine the origin of the potential and how to calculate the mean inner potential. An important part of this section is found in Appendix A. In the second chapter we discuss some of the poten- tial maps and the chemical information which can be derived from X-ray diffraction data. Both maps of the potential in a crystal and the potential of a single molecule and their relations are discussed. The actual calculations deal only with the potential in a crystal. We use the algorithm of Stewart (1982) © 1990 International Union of Crystallography