Noninteracting atoms

The Finite Range Problem. The minimum image convention requires (1) the use of interaction energy functions that are of finite range, i.e. that are non-zero only for distances below a certain limit, R. As a consequence, only a fraction of all minimum image pairs actually interact with non-zero energy this fraction must be less than ir/2d, i.e., in two dimensions maximally 0.79, in three dimensions maximally 0.52 (and actually often less than 0.3). It is desirable to efficiently eliminate from consideration all noninteracting atom pairs. 

An alternative approach to calculating the free energy of solvation is to carry out simulations corresponding to the two vertical arrows in the thermodynamic cycle in Fig. 2.6. The transformation to nothing should not be taken literally -this means that the perturbed Hamiltonian contains not only terms responsible for solute-solvent interactions - viz. for the right vertical arrow - but also all the terms that involve intramolecular interactions in the solute. If they vanish, the solvent is reduced to a collection of noninteracting atoms. In this sense, it disappears or is annihilated from both the solution and the gas phase. For this reason, the corresponding computational scheme is called double annihilation. Calculations of... 

To describe the shifts and intensities of the m-photon assisted collisional resonances with the microwave field Pillet et al. developed a picture based on dressed molecular states,3 and we follow that development here. As in the previous chapter, we break the Hamiltonian into an unperturbed Hamiltonian H(h and a perturbation V. The difference from our previous treatment of resonant collisions is that now H0 describes the isolated, noninteracting, atoms in both static and microwave fields. Each of the two atoms is described by a dressed atomic state, and we construct the dressed molecular state as a direct product of the two atomic states. The dipole-dipole interaction Vis still given by Eq. (14.12), and using it we can calculate the transition probabilities and cross sections for the radiatively assisted collisions. 

Formation of the molecule from the free (noninteracting) atom cores and the free (noninteracting) electrons. 

The theory of Bose-Einstein condensation of A noninteracting atoms confined in a finite, three-dimensional cavity of size L [80, 126] starts from the... 

The atoms are held at the distance that separates them, which can either be calculated or obtained experimentally, and the molecular orbitals of HF are calculated. The calculations are nontrivial and beyond the scope of this book the result, however, is shown schematically in Fig. 2.Sb. The total number of electrons that have to be accommodated in the molecular orbitals is eight (seven from F and one from H). Placing two in each orbital fills the first four orbitals and results in an energy for the molecule that is lower (more negative) than that of the sum of the two noninteracting atoms, which in turn renders the HF molecule more stable relative to the isolated atoms. 

Parr and Pearson considered the simplest case of two noninteracting atoms exchanging electrons until their electronegativities are equal [4]. Then the charge transfer AN is ... 

The first step in the Heitler-London treatment of the H2 ground state is to approximate the molecule as two ground-state hydrogen atoms. The wave function for two such noninteracting atoms is... 

The cohesive energy is the energy necessary to dissociate a solid into separated entities, generally the atoms. Following this definition, the cohesive energy (AE) of a crystalline compound with formula unit AaBb, for example, is associated with the reaction of formation of AaB, in the solid state from the noninteracting atoms A and B in the gas phase... 

The cohesive (binding) energy Ecoh is dehned as the energy required to dissociate a sohd into noninteracting atomic species A with energies Ea... 

Figure 7.3 An example experimental electron density difference map for (Z)-N-methyl-C-phenylnitrone. The solid (dashed) contours are for an increased (decreased) electron density compared with a simple sum of noninteracting atomic densities. The inset in the top left shows a chemical structure of the same molecule in roughly the orientation in the contour plot, (Source Hibbs DE, Hanrahan JR, Hursthouse MB, Knight DW, Overgaard J, Turner P, Piltz RO, Waller MP (2003) Organic Blomolecular Chemistry 1 1034-1040.)...

The expressions considered above are valid for the phase transition to the Bose-Einstein condensate in an ideal gas of noninteracting atomic species. In any real atomic gas, atomic interactions alter the properties of the transition to the BEC. 

The DFTB method is based on the DFT of Hohenberg and Kohn [74] in the formulation of Kohn and Sham [75] and has been developed by Seifert et al. [76-78]. For more details we refer the reader to the original work. Within this approach, the total energy relative to the noninteracting atoms is given as... 

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