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The many-body expansion, where one computes the total energy of a supersystem as the sum of the dimer, trimer, tetramer, etc., subsystems, provides a convenient approach to compute the lattice energies of molecular crystals. We investigate approximate methods for computing the non-additive three-body contributions to the crystal lattice energy of the polar molecules acetic acid, imidazole, and formamide, comparing to coupled-cluster singles, doubles, and perturbative triples [CCSD(T)] level benchmarks. Second-order Møller–Plesset perturbation theory (MP2), if combined with a properly damped Axilrod–Teller–Muto dispersion potential, displays excellent agreement with CCSD(T) at a substantially reduced cost. Errors between dispersion-corrected MP2 and CCSD(T) are less than 1 kJ mol−1 for all three crystals. However, the three-body energy requires quite large distance cutoffs to converge, up to 20 Å or more. 

The focal-point approximation can be used to estimate a high-accuracy, slow quantum chemistry computation by combining several lower-accuracy, faster computations. We examine the performance of focal-point methods by combining second-order Møller–Plesset perturbation theory (MP2) with coupled-cluster singles, doubles, and perturbative triples [CCSD(T)] for the calculation of harmonic frequencies and that of fundamental frequencies using second-order vibrational perturbation theory (VPT2). In contrast to standard CCSD(T), the focal-point CCSD(T) method approaches the complete basis set (CBS) limit with only triple-ζ basis sets for the coupled-cluster portion of the computation. The predicted harmonic and fundamental frequencies were compared with the experimental values for a set of 20 molecules containing up to six atoms. The focal-point method combining CCSD(T)/aug-cc-pV(T + d)Z with CBS-extrapolated MP2 has mean absolute errors vs experiment of only 7.3 cm−1 for the fundamental frequencies, which are essentially the same as the mean absolute error for CCSD(T) extrapolated to the CBS limit using the aug-cc-pV(Q + d)Z and aug-cc-pV(5 + d)Z basis sets. However, for H2O, the focal-point procedure requires only 3% of the computation time as the extrapolated CCSD(T) result, and the cost savings will grow for larger molecules. 

Stochastic simulation can make the molecular processes of cellular control more vivid than the traditional differential equation approach by generating typical system histories, instead of just statistical measures such as the mean and variance of a population. Simple simulations are now easy for students to construct from scratch—that is, without recourse to black-box packages. In some cases, their results can also be compared directly with single-molecule experimental data. After introducing the stochastic simulation algorithm, this article gives two case studies involving gene expression and error correction, respectively. For gene expression, stochastic simulation results are compared with experimental data, an important research exercise for biophysics students. For error correction, several proofreading models are compared to find the minimal components necessary for sufficient accuracy in translation. Animations of the stochastic error correction models provide insight into the proofreading mechanisms. Code samples and resulting animations showing results are given in the online Supplemental Material .