Quantum pair correlations

Unlike the solid state, the liquid state cannot be characterized by a static description. In a liquid, bonds break and refomi continuously as a fiinction of time. The quantum states in the liquid are similar to those in amorphous solids in the sense that the system is also disordered. The liquid state can be quantified only by considering some ensemble averaging and using statistical measures. For example, consider an elemental liquid. Just as for amorphous solids, one can ask what is the distribution of atoms at a given distance from a reference atom on average, i.e. the radial distribution function or the pair correlation function can also be defined for a liquid. In scattering experiments on liquids, a structure factor is measured. The radial distribution fiinction, g r), is related to the stnicture factor, S q), by... 

The chemical shieldings were then recalculated in this same system using the QM/MM method [10], To this end, each molecule was considered individually. The water molecule of interest and its first solvation shell were treated quantum mechanically, whereas the surrounding water molecules were taken into account with an empirical force field representation (MM molecules). The first solvation shell was defined via a distance criterion on the oxygen-oxygen distance. As a threshold, the first minimum of the O-O pair correlation function was taken this occurs at 3.5 A [93]. All... 

First, the theory is valid for any kind of particles, not necessarily spherical particles. Only the spatial pair correlation function features in Gaig, even when the particles are not spherical. Second, no assumption on pairwise additivity of the total potential energy is invoked in the theory. Finally, we note that in this book, we discuss only classical systems the Kirkwood-Buff results, however, hold for quantum systems as well. 

The pair-correlation density is a property that arises due to the Pauli and Coulomb correlations between electrons. Thus it can also be interpreted as the density p(r ) at r plus the reduction in this density at r due to the electron correlations. The reduction in density about an electron which occurs as a result of the Pauli exclusion principle and Coulomb repulsion is the quantum-mechanical Fermi-Coulomb hole charge distribution p (r, r ). Thus we may write the pair-correlation density as... 

The field Coulomb correlations since its quantum-mechanical source charge distribution is the pair-correlation density g(r,r ). On the other hand, the field Z,Jr) arises from the kinetic-energy-density tensor tj (r). It is the difference of the fields derived from the tensor for the interacting and Kohn-Sham noninteracting systems, and is thereby representative of the correlation-kinetic-energy. 

At the bases of the second basic assumption made, e.g., that the fluids behave classically, there is the knowledge that the quantum effects in the thermodynamic properties are usually small, and can be calculated readily to the first approximation. For the structural properties (e.g., pair correlation function, structure factors), no detailed estimates are available for molecular liquids, while for atomic liquids the relevant theoretical expressions for the quantum corrections are available in the literature. 

Figure 26 Pair correlation function g r) for Li4. The upper panel shows the classical results for temperatures 50 K (solid line) 100 K (dashed line) and 200 K (dotted-dashed line), whereas the lower panel shows the quantum-mechanical (dashed line) and the classical (solid line) results for T — 50 K (Reproduced with permission from J. Chem. Phys., 108, 8848 1998 American Institute of Physics)...

A related study was reported by Mei et al. who studied proton transport along a chain of water molecules. Also they found clear signals of quantum effects of the protons. Instead of dwelling further on this study, we shall turn to another one where not protons but Li atoms were treated quantum-mechanically, i.e. a slightly heavier atom. Also here quantum effects were found to be important, in particular at lower temperatures (below, say, 50 K). This is examplified in Figure 26, which shows the pair correlation fimction g(r) for Li4. The differences between the classical and the quantum-mechanical results should be obvious. 

Condon, E. U., On pair correlation in the theory of atomic structure. Rev. Mod. Phys. 40 872 (1968) Nesbet, R. K., Electronic pair correlation in atoms and molecules, Int. J. Quantum Chem. S4 117 (1971). In the 1960 s there was a disagreement between Sin-anoglu and Nesbet about the origins of the lEPA. These two articles present complementary views of the history of the subject and make fascinating reading. 

The session on Theories of Liquid Structures treated the equilibrium properties of simple liquids from a quantum-statistical and statistical-thermodynamics point-of-view. The tenor in these presentations was the Pair-Correlation Function. The various equations that have been proposed to describe its behavior, and techniques for obtaining their solutions (in special cases) were reviewed. These concepts became more "concrete with a discussion of electrolytic solutions. 

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