Correlation function density expansion

Finally, we relate the gradient of the local density ViPq(fi) to the pair correlation functions. For this purpose we take the gradient of the expansion (5)... 

The equlibrium between the bulk fluid and fluid adsorbed in disordered porous media must be discussed at fixed chemical potential. Evaluation of the chemical potential for adsorbed fluid is a key issue for the adsorption isotherms, in studying the phase diagram of adsorbed fluid, and for performing comparisons of the structure of a fluid in media of different microporosity. At present, one of the popular tools to obtain the chemical potentials is an approach proposed by Ford and Glandt [23]. From the detailed analysis of the cluster expansions, these authors have concluded that the derivative of the excess chemical potential with respect to the fluid density equals the connected part of the fluid-fluid direct correlation function (dcf). Then, it follows that the chemical potential of a fluid adsorbed in a disordered matrix, p ), is... 

Findings with PDU. Work with the PDU largely paralleled the bench-scale reactor tests there was one important addition—extensive three-phase fluidization studies. As was mentioned, the PDU is equipped with a traversing gamma-ray density detector that is capable of measuring bed density to within dbO.Ol specific gravity units. Thus, we could measure and correlate fluidized bed expansion as a function of liquid and gas velocities and physical properties, and could also determine the... 

The advantage over the HF scheme is that whereas in conventional ah initio theory we must resort to costly perturbation theory or configuration interaction expansions, in DFT electron correlation is already included explicitly in the exchange-correlation functional. The key problem is instead to find an appropriate expression for xc. As stated above, when we have the correct functional we should be able to extract the exact energy, the exact ground state density, and all properties for our system. 

Consider a general system described by the Hamiltonian of Eq. (5), where = Huif) describes the interaction between the spin system (7) and its environment (the lattice, L). The interaction is characterized by a strength parameter co/i- When deriving the WBR (or the Redfield relaxation theory), the time-dependence of the density operator is expressed as a kind of power expansion in Huif) or (17-20). The first (linear) term in the expansion vanishes if the ensemble average of HiL(t) is zero. If oo/z,Tc <5c 1, where the correlation time, t, describes the decay rate of the time correlation functions of Huif), the expansion is convergent and it is sufficient to retain the first non-zero term corresponding to oo/l. This leads to the Redfield equation of motion as stated in Eq. (18) or (19). In the other limit, 1> the expan-... 

In order to calculate the band structure and the density of states (DOS) of periodic unit cells of a-rhombohedral boron (Fig. la) and of boron nanotubes (Fig. 3a), we applied the VASP package [27], an ab initio density functional code, using plane-waves basis sets and ultrasoft pseudopotentials. The electron-electron interaction was treated within the local density approximation (LDA) with the Geperley-Alder exchange-correlation functional [28]. The kinetic-energy cutoff used for the plane-wave expansion of... 

It was recently shown that a formal density expansion of space-time correlation functions of quantum mechanical many-body systems is possible in very general terms [297]. The formalism may be applied to collision-induced absorption to obtain the virial expansions of the dipole... 

The virial expansion of the time correlation functions is possible for times smaller than the mean time x between collisions. Accordingly, the spectral profiles may be expanded in powers of density, for angular frequencies much greater than the reciprocal mean time between collisions, co 1/r. Since at low density the mean time between collisions is inversely proportional to density, lower densities permit a meaningful virial expansion for a greater portion of the spectral profiles. 

Density expansion. The method of cluster expansions has been used to obtain the time-dependent correlation functions for a mixture of atomic gases. The particle dynamics was treated quantum mechanically. Expressions up to third order in density were given explicitly [331]. We have discussed similar work in the previous Section and simply state that one may talk about binary, ternary, etc., dipole autocorrelation functions. 

A. Raczynski. Density expansion of correlation functions. Acta Phys. Polonia, 62 A 303, 1982. 

Prof. Fleming, the expressions you are using for the nonlinear response function may be derived using the second-order cumulant expansion and do not require the use of the instantaneous normal-mode model. The relevant information (the spectral density) is related to the two-time correlation function of the electronic gap (for resonant spectroscopy) and of the electronic polarizability (for off-resonant spectroscopy). You may choose to interpret the Fourier components of the spectral density as instantaneous oscillators, but this is not necessary. The instantaneous normal mode provides a physical picture whose validity needs to be verified. Does it give new predictions beyond the second-order cumulant approach The main difficulty with this model is that the modes only exist for a time scale comparable to their frequencies. In glasses, they live much longer and the picture may be more justified than in liquids. 

One more relation is required to achieve closure, i.e., to determine the two types of correlation functions. The most commonly used relations are the Percus-Yevick (PY) and the hypernetted chain (HNC) approximations [47-49]. From graph or diagram expansion of the total correlation function in powers of the density n(r) and resummation, an exact relation between the total and direct correlation functions is obtained, namely... 

---