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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.

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.

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Quantum crystallography is a developing technique for extracting quantum mechanically valid properties from X-ray diffraction experiments. Quantum mechanics and crystallography are joined through the fact that the electron distributions around atoms are the source of X-ray diffraction and electron density distributions are observables that lend themselves readily to quantum mechanical description. Direct contact with the X-ray diffraction data is made by equating the structure factor magnitudes, which are readily obtained from the measured X-ray diffraction intensities, with the magnitudes of certain Fourier transforms of the quantum mechanical description of the electron distribution. The quantum mechanical description of the electron density involves molecular orbitals and an associated matrix. By requiring that the associated matrix be a projector with a normalized trace, while optimizing the fit to the experimental structure factor magnitudes, strong constraints are imposed on the relationship between the X-ray data and the quantum mechanical description of the electron density distribution. The final result should be a wave function that is in good agreement with the X-ray diffraction information and from which a variety of properties, e.g., electron densities and electrostatic potentials, could be extracted. The method for making the fit to the X-ray data involves the use of least-squares calculations in which the defining equations are the structure factor equations and equations that arise from the conditions defining a projector matrix with a normalized trace. The variables are the elements of the projector. It is also possible to refine such parameters as atomic coordinates. The calculations are facilitated by use of good initial projector matrices. For this purpose, a method has been developed for generating projector matrices for large molecules from the sum of kernel matrices. Kernel matrices are obtained from the use of fragments of the known atomic coordinates of the substance of interest and the application of molecular orbital methods in quantum mechanics. The use of fragments is justified by the fact that overlap integrals rapidly approach zero as the distances between atoms increase.