Molecular self-density

The density of a condensed phase is mainly determined by the weight of the nuclei, which take only a small fraction of space, and thus depends only marginally on space occupation. A molecular self-density. Dm, can be defined as the ratio of molecular mass to molecular volume [13] ... 

As seen in Fig. 1.3, crystal density correlates very strictly with molecular self-density, so that a high density in a crystalline material depends primarily on the heavy elements contained in the molecule, and only to a minor extent on efficient space occupation. 

Fig. 1.3. The crystal density (g cc ) against molecular self-density, equation 1.4, for a large sample of crystalline compounds. The least-squares equation is y = 1.33 x — 0.17.

We add the operator —p, ER to the total molecular Hamiltonian. According to Eq. (3.1), the electronic Hamiltonian of the molecule in the field due to the solvent is then He — p ER. The electronic Schrodinger equation is then solved using this modified Hamiltonian. This leads to a self-consistent solution where the electronic wave function and the electronic energy are modified due to the solvent field. Thus, polarization of the molecular electronic density (as described approximately above) is automatically included in this approach. 

Finally, in careful comparative studies of the molecular electron densities generated by HF, correlated ab initio, and pure, self-interaction corrected, and hybrid DFT calculations, Cremer et al have made a very interesting observation [72, 73]. They found that the pure DFT generated densities differed from those obtained with accurate ab initio methods in a particular way, and that both hybrid, and self-interaction corrected DFT methods, yielded densities closer to the correct ones. Based on this observation, they suggested that mixing in of exact exchange in hybrid functionals serves as a proxy for the self-interaction correction. 

Physical parameters Molecular 0.1-1 nm Dipole-dipole interaction Second moment, fourth moment of lineshape Incoherent magnetization transfer characteristic times for cross-polarization and exchange Mesoscopic lnm-0.1 p,m Longitudinal relaxation time Ti Transverse relaxation time T2 Relaxation time Tip in the rotating frame Solid-echo decay time T2e Spin-diffusion constant Microscopic 0.1-lOp.m Molecular self-diffusion constant D Macroscopic 10 p,m and larger Spin density... 

The SCF LCAO ab initio representation of the molecular electronic density p(r) is a function of the 3D position variable r, and is defined in terms of a set of h atomic orbitals (p,(r), i= 1,2,..., n. The n x n dimensional density matrix P can be computed from the set of self-consistent coefficients of atomic orbitals in the occupied molecular orbitals. In terms of this density matrix P, the electronic density p(r) of the molecule can be written as... 

The Use of Ab Initio Quantum Molecular Self-Similarity Measures to Analyze Electron Charge Density Distributions. 

In Fig. 2 we see the result of using self-consistent-field (SCF) molecular electron densities. The final result (bottom) approaches quantitative agreement with experiment and indicates that the MT approximation and the lack of realistic electron densities were the two most serious deficiencies in the theoretical approach. The results are the best so far achieved within the MSW theoretical framework and underline the necessity of both self-consistent potentials and the non-MT (full potential) treatment. More recently, Joly et al. have investigated a solid state TiOe cluster using their finite difference techniques and have reached similar conclusions. 

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