In all these cases the observed properties depend on the positions of the nuclei via the intermolecular potential. Other examples of situations where the intermolecular potential plays a role, are transport properties of fluids, molecular collision cross-sections, spectra of molecular complexes (dimers, trimers, …), and structures of molecular solids. Clearly, by a simple extension the same holds for more elaborate equations of state, i.e., for multiparameter P-V-T relations. After measurement of the P-V-T relation, the two parameters in the van der Waals equation can be determined and these parameters tell something about the interaction in a pair of molecules constituting the fluid. The standard example is the pressure-volume-temperature ( P-V-T) dependence of a van der Waals fluid. Experimental information is obtained from measurable properties that depend on the intermolecular potential. An intermolecular potential-a scalar function of the positions of the nuclei constituting a molecular complex- cannot be measured directly. Therefore, it is common to concentrate on the intermolecular potential rather than on its derivatives-the intermolecular force. In general, a force is minus the gradient (set of derivatives with respect to coordinates) of a potential. The position and depth of the Van der Waals minimum depends on distance and mutual orientation of the molecules. The sum of the attractive long-range and repulsive short-range forces gives rise to a minimum, referred to as Van der Waals minimum. They are partly caused by interaction between permanent molecular dipoles and higher multipoles, by induction effects, and by a quantum mechanical effect for which Fritz London coined the name dispersion effect. The long-range forces, on the other hand, fall off with inverse powers of the distance, R − n, typically 3 ≤ n ≤ 10, and are mostly attractive. In chemistry they are well known, because they give rise to steric hindrance, also known as Born or Pauli repulsion. Short-range forces fall off exponentially as a function of intermolecular distance R and are repulsive for interacting stable (closed-shell) moieties. As will be explained in this article, the former are mainly due to the intermolecular exchange of electrons, a phenomenon closely related to the quantum mechanical Pauli principle. In general one distinguishes short- and long-range intermolecular forces. 8 Anisotropy and non-additivity of intermolecular forces.7.1 Quantum mechanical theory of dispersion forces.2 Historical background and the need for quantum mechanics.We suggest (DFT-)D4 as a physically improved and more sophisticated dispersion model in place of DFT-D3 for DFT calculations as well as for other low-cost approaches like semi-empirical models. Especially for metal containing systems, the introduced charge dependence improves thermochemical properties. For various energy benchmark sets DFT-D4 slightly outperforms DFT-D3. Becke-Johnson-type damping parameters for DFT-D4 are determined for more than 60 common functionals. A common many-body dispersion expansion was extensively tested and an energy correction based on D4 polarizabilities is found to be advantageous for some larger systems. In addition to the two-body part, three-body effects are described by an Axilrod-Teller-Muto term. For a benchmark set of 1225 dispersion coefficients, the D4 model achieves an unprecedented accuracy with a mean relative deviation of 3.8% compared to 4.7% for D3. Similar to the D3 model, the dynamic polarizabilities are pre-computed by time-dependent DFT and elements up to radon are covered. A numerical Casimir-Polder integration of the atom-in-molecule dynamic polarizabilities yields charge- and geometry-dependent dipole-dipole dispersion coefficients. Classical charges are obtained from an atomic electronegativity equilibration procedure for which efficient analytical derivatives are developed. For this purpose, a new charge-dependent parameter-economic scaling function is designed. In this successor to the DFT-D3 model, the atomic coordination-dependent dipole polarizabilities are scaled based on atomic partial charges which can be taken from various sources. The D4 model is presented for the accurate computation of London dispersion interactions in density functional theory approximations (DFT-D4) and generally for atomistic modeling methods.
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