The theoretical background and practical procedures for phase determination by the symbolic addition method are discussed. Phase determining formulas are presented for centrosymmetric and noncentrosymmetric crystals. A probability formula is employed to evaluate the reliability of phase determination for centrosymmetric crystals and a formula for the variance is utilized for the same purpose in noncentrosymmetric ones. These probability measures play a key role in overcoming the main problems involved in carrying out the procedure, namely the nonuniqueness of the internal consistency criterion as applied to the phase determining formulas, and questions concerning the proper circumstances for assigning symbols. The method is generally applicable to centrosymmetric crystals and has been successful in several applications to noncentrosymmetric ones. Some auxiliary phase information is probably required to make the symbolic addition procedure a general one for noncentrosymmetric crystals.

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Joint probability distributions and relevant expected values and variances are obtained for selected (but typical) non-centrosymmetric space groups belonging to the four types 1P222, 2P22, 3P12, 3P22. These lead to formulas for phase determination the analysis and interpretation of which constitute the major goal of this paper. The analysis is strongly dependent on the theory of invariants and seminvariants, and the agreement between this theory and certain consequences of the probability theory is noteworthy.

ABSTRACT:We describe a kernel energy method (KEM) for applying quantum crystallography to large molecules, with an emphasis on the calculation of the molecular energy of peptides. The computational difficulty of representing the system increases only modestly with the number of atoms. The calculations are carried out on modern parallel supercomputers. By adopting the approximation that a full biological molecule can be represented by smaller "kernels" of atoms, the calculations are greatly simplified. Moreover, collections of kernels are, from a computational point of view, well suited for parallel computation. The result is a modest increase in computational time as the number of atoms increases, while retaining the ab initio character of the calculations. We describe a test of our method, and establish its accuracy using 15 different peptides of biological interest.

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Using quantum mechanics and exploiting known crystallographic coordinates of tRNA substrate located in the ribosome peptidyl transferase center around the 2-fold axis, we have investigated the mechanism for peptide-bond formation. The calculation is based on a choice of 50 atoms assumed to be important in the mechanism. We used density functional theory to optimize the geometry and energy of the transition state (TS) for peptide-bond formation. The TS is formed simultaneously with the rotatory motion enabling the translocation of the A-site tRNA 3′ end into the P site, and we estimated the magnitude of rotation angle between the A-site starting position and the place at which the TS occurs. The calculated TS activation energy, E a , is 35.5 kcal (1 kcal = 4.18 kJ)/mol, and the increase in hydrogen bonding between the rotating A-site tRNA and ribosome nucleotides as the TS forms appears to stabilize it to a value qualitatively estimated to be ≈18 kcal/mol. The optimized geometry corresponds to a structure in which the peptide bond is being formed as other bonds are being broken, in such a manner as to release the P-site tRNA so that it may exit as a free molecule and be replaced by the translocating A-site tRNA. At TS formation the 2′ OH group of the P-site tRNA A76 forms a hydrogen bond with the oxygen atom of the carboxyl group of the amino acid attached to the A-site tRNA, which may be indicative of its catalytic role, consistent with recent biochemical experiments.

We have investigated the heretofore unknown unimolecular decomposition pathway of the explosive molecule diaminodinitroethylene (DADNE). With the use of DFT methods, whose accuracy has been calibrated by means of ab initio calculations (MP2, MP4, G2) on a simpler but related molecule, nitroethylene, we have been able to characterize the entire decomposition reaction pathway. Importantly, we find that the reaction is initiated by a nitro-to-nitrite rearrangement with a calculated energy barrier of magnitude 59.1 kcal/mol obtained by use of B3LYP (59.7 kcal/mol B3P86) which is very close to the experimental activation energy of 58 kcal/mol. We have been able to characterize every step in the decomposition reaction path leading to fragments NO, HONO, CO, NH2, and HNC. These may interact to yield final stable products, CO, N2, H2O with an energy release that on average is adequate to initiate two additional DADNE molecular decompositions, and thus, sustain a chain reaction. The structural parameters we have calculated for DADNE are consistent with the known experimental crystallographic structure, and also with previous theoretical calculations. Additionally, we have obtained the structural parameters of the initial transition state, as well as each subsequent step along the decomposition pathway. Thus we consider the unimolecular decomposition of DADNE to be well characterized.