Spatial pair correlation function determination

Fluid microstructure may be characterized in terms of molecular distribution functions. The local number of molecules of type a at a distance between r and r-l-dr from a molecule of type P is Pa T 9afi(r)dr where Pa/j(r) is the spatial pair correlation function. In principle, flr (r) may be determined experimentally by scattering experiments however, results to date are limited to either pure fluids of small molecules or binary mixtures of monatomic species, and no mixture studies have been conducted near a CP. The molecular distribution functions may also be obtained, for molecules interacting by idealized potentials, from molecular simulations and from integral equation theories. 

The structure of a fluid is characterized by the spatial and orientational correlations between atoms and molecules determined through x-ray and neutron diffraction experiments. Examples are the atomic pair correlation functions (g, gj j ) in liquid water. An important feature of these correlation functions is that the thermodynamic properties of a... 

By analyzing the dependence of the echo amplitude on the delay t, it is possible to determine the spin-spin pair correlation function, which corresponds to the distribution of electron spins in the vicinity of the observed spin (35,36). Such spatial distributions of spin probes or spin labels can be interpreted by a model-free approach if they are reasonably narrow, or by simple models for the geometry of the system under investigation, if they are broad (37-39). The experiment requires that observer and pumped spins be excited separately, a condition that is easily achieved for nitroxide spin probes and labels by setting the difference of the two excitation frequencies to 65-70 MHz (Fig. 6c). 

Liquid structure is revealed by X-ray and neutron diffraction patterns. The measured diffraction is proportional to weighted sums of spatial Fourier transforms of the site-site pair correlation functions. The experimentally determined structure factors, are related to the pair functions by a... 

The neutron scattering cross section of an element can be broken into its coherent and incoherent components (there is also an absorption cross section which is umelated to scattering which we will not address here). Hie cross section reflects the number of neutrons scattered per second from the element divided by the intensity of the incident neutron beam. For coherent scattering events, there is a spatial correlation between the scatterings from different nuclei of the same type (with the same scattering length density). Hiese spatial correlations allow us to determine the Van Hove or the pair-pair correlation function, that is, the spatial correlations between the different atoms. For incoherent scattering events, this spatial correlation... 

The relationship between the Fermi correlation and the spatial localization of electrons is developed by defining, as McWeeny (1960) does, a correlation function/(rj, rj) in an expression which determines the extent to which the pair density deviates from a simple product of number densities,... 

UHF Methods. A major drawback of closed-shell SCF orbitals is that whilst electrons of the same spin are kept apart by the Pauli principle, those of opposite spin are not accounted for properly. The repulsion between paired electrons in spin orbitals with the same spatial function is underestimated and this leads to the correlation error which multi-determinant methods seek to rectify. Some improvement could be obtained by using a wavefunction where electrons of different spins are placed in orbitals with different spatial parts. This is the basis of the UHF method,40 where two sets of singly occupied orbitals are constructed instead of the doubly occupied set. The drawback is of course that the UHF wavefunction is not a spin eigenfunction, and so does not represent a true spectroscopic state. There are two ways around the problem one can apply spin projection operators either before minimization or after. Both have their disadvantages, and the most common procedure is to apply a single spin annihilator after minimization,41 arguing that the most serious spin contaminant is the one of next higher multiplicity to the one of interest. 

The electronic structure methods are based primarily on two basic approximations (1) Born-Oppenheimer approximation that separates the nuclear motion from the electronic motion, and (2) Independent Particle approximation that allows one to describe the total electronic wavefunction in the form of one electron wavefunc-tions i.e. a Slater determinant [26], Together with electron spin, this is known as the Hartree-Fock (HF) approximation. The HF method can be of three types restricted Hartree-Fock (RHF), unrestricted Hartree-Fock (UHF) and restricted open Hartree-Fock (ROHF). In the RHF method, which is used for the singlet spin system, the same orbital spatial function is used for both electronic spins (a and (3). In the UHF method, electrons with a and (3 spins have different orbital spatial functions. However, this kind of wavefunction treatment yields an error known as spin contamination. In the case of ROHF method, for an open shell system paired electron spins have the same orbital spatial function. One of the shortcomings of the HF method is neglect of explicit electron correlation. Electron correlation is mainly caused by the instantaneous interaction between electrons which is not treated in an explicit way in the HF method. Therefore, several physical phenomena can not be explained using the HF method, for example, the dissociation of molecules. The deficiency of the HF method (RHF) at the dissociation limit of molecules can be partly overcome in the UHF method. However, for a satisfactory result, a method with electron correlation is necessary. 

The Hartree Fock determinant describes a situation where the electrons move independently of one another and where the probability of finding one electron at some point in space is independent of the positions of the other electrons. To introduce correlation among the electrons, we must allow the electrons to interact among one another beyond the mean field approximation. In the orbital picture, such interactions manifest themselves through virtual excitations from one set of orbitals to another. The most important class of interactions are the pairwise interactions of two electrons, resulting in the simultaneous excitations of two electrons from one pair of spin orbitals to another pair (consistent with the Pauli principle that no more than two electrons may occupy the same spatial orbital). Such virtual excitations are called double excitations. With each possible double excitation in the molecule, we associate a unique amplitude, which represents the probability of this virtual excitation happening. The final, correlated wave function is obtained by allowing all such virtual excitations to happen, in all possible combinations. 

In the normal (probability theory) use of the term, two probability distributions are not correlated if their joint (combined) probability distribution is just the simple product of the individual probability distributions. In the case of the Hartree-Fock model of electron distributions the probability distribution for pairs of electrons is a product corrected by an exchange term. The two-particle density function cannot be obtained from the one-particle density function the one-particle density matrix is needed which depends on two sets of spatial variables. In a word, the two-particle density matrix is a (2 x 2) determinant of one-particle density matrices for each electron ... 

It might be anticipated that, just as the correlation between the spatial motions of electrons was recognized in the pair function, there ou t to be a 2-electron function that would recognize the correlation between spins of electrons located in different volume elements. Such functions have indeed been defined and used in discussion of spin-coupling effects that arise when relativistic terms are included in the Hamiltonian (McWeeny, 1965). Here we need only note that two functions are required, a spin-orbit coupling function Q, which fully determines spin-orbit interactions, and a spin-spin coupling function which... 

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