The calculation of intermolecular energies in crystals

When close-packing ideas were formulated, there was some hope that these concepts might also help in the prediction and control of crystal sUmctures, but this expectation was not fulfilled. The complexity of crystal packing is too vast to be understood entirely by such simple and wide generalizations. Close packing is a necessary but not sufficient condition for the prediction of crystal structure, because all observed crystal stmctures are close packed, but not all close packed crystal sfiuctures are observed. These points will be taken up again in Chapter 14. 

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The calculation of intermolecular energies in crystals 8.7.1 Lattice energies Some basic concepts... 

In spite of all the evolution, there are still important problems for the calculation of intermolecular energies. Hartree-Fock (HF) methods use one-electron orbitals and therefore cannot account for those phenomena that depend on the simultaneous behavior of several electrons. Thus, HF energies may correctly represent the kinetic energies of electrons and the electrostatic effects between electrons and nuclei, but cannot take into account electron correlation. The results obtained at the limit of a complete (i.e. infinitely rich) basis set are called HF-limit energies and wavefunctions, the ideal best that can be obtained with one-electron orbitals. This intrinsic limitation forbids the treatment of dispersion energy, a crucial part of the intermolecular potential (see Chapter 4). Thus, for example, HF methods are intrinsically unsuitable for the calculation of the lattice energies of organic crystals. 

In a (rather crude) summary, it might be said that DFT does not rigorously solve the problem of correlation energy, but displaces the problem to a well defined location, the Sex(p), where a parametric ambush can be set for its solution. The method, whose computational demand is similar or sometimes more modest than those of MO methods, has been very successful in applications to isolated molecules, where the introduction of electron correlation corrections has been seen to improve, for example, calculated optimized molecular geometries, but has not yet proved completely satisfactory for the calculation of intermolecular interaction energies in systems where coulombic contributions are not overwhelmingly dominant. Molecular crystals are a typical example. 

Several examples of the MOLPAK + WMIN structure prediction procedures are given in the next sections. The problem of identifying the correct crystal structure from literally thousands of possible structures remains. The which is the best/correct solution was an important topic during the 1999 and 2001 CCDC sponsored blind tests [12]. Calculated lattice energies are adequate in many cases, but in others small lattice energy differences make it difficult to separate the wheat from the chaff. Other criteria, such as good or bad patterns of intermolecular contacts in comparison with known crystal structures could be helpful. 

The connection between energies and geometry must be empirical or indirect through theoretical hypotheses and calculations. This connection is further complicated by the other intermolecular interactions in crystals. The hydrogen-bond lengths observed in crystals, which provide the vast majority of the data available, are modulated by the crystal field effects and any one observation is therefore not necessarily representative of the potential energy minima of the isolated hydrogen-bonded adducts. 

The quantum mechanical approach cannot be used for the calculation of complete lattice energies of organic crystals, because of intrinsic limitations in the treatment of correlation energies. The classical approach is widely applicable, but is entirely parametric and does not adequately represent the implied physics. An intermediate approach, which allows a breakdown of the total intermolecular cohesion energy into recognizable coulombic, polarization, dispersion and repulsion contributions, and is based on numerical integrations over molecular electron densities, is called semi-dassical density sums (SCDS) or more briefly Pixel method. [12-14]... 

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