Density correlation functions self part

Finally, we comment on the difference between the self part and the full density autocorrelation function. The full density autocorreration function and the dynamical structure factor ire experimentally measured, while in the present MD simulation only the self pairt was studied. However, the difference between both correlation functions (dynamical structure factors) is considered to be rather small except that additional modes associated with sound modes appear in the full density autocorrelation. We have previously computed the full density autocorrelation via MD simulations for the same model as the present one, and found that the general behavior of the a relaxation was little changed. General trends of the relaxation are nearly the same for both full correlation and self part. In addition, from a point of numerical calculations, the self pMt is more easily obtained than the full autocorrelation the statistics of the data obtained from MD simulatons is much higher for the self part than for the full autocorrelation. 

The density-density correlation function G(r, t), generally known as the van Hove correlation function,describes the complete equilibrium and dynamic behaviour of the dispersion. Unlike h(r), G(r, t) also contains the correlation in the position of a single particle at time 0 and time t. This is called the van Hove self-correlation function Gs(r, t), to distinguish it from the distinct part Gd(r, t), which refers to correlations with other particles in the system. The behaviour of G r, t) and Gd(r, t) in a dense fluid system is shown in Figure 2. At very short times, Ga(r, t) approaches h(r), and Gj(r, t) approaches a delta function. The functions may be normalized by dividing by p. 

For quantitative comparison of theory and experiment several simple models have been developed which give analytical expressions for the pure dephasing time T2. These models are usually tested against the experimentally observed temperature, density, or concentration dependence of T2. Since these models have been intensively reviewed, we merely recall the results. Hydro-dynamic models have been proposed by Lynden-Bell - and Oxtoby. In the former calculation the vibrational correlation function is replaced by its self-part [see Eq. (96)], yielding... 

Figure 6. Test of potentiai separation of Schweizer and Chandier at moderateiy low density (pa = 0.3). Time scales for self part of vibrational correlation function for short- and long-range forces are well separated, but cross term 2 is not negligible. As density increases, cross term progressively cancels out, but not any faster than the force contribution , which also disappears eventually. (From Chesnoy and Weis. ° )...

Because the Fock matrix depends on the one-particle density matrix P constructed conventionally using the MO coefficient matrix C as the solution of the pseudo-eigenvalue problem (Eq. [7]), the SCF equation needs to be solved iteratively. The same holds for Kohn-Sham density functional theory (KS-DFT) where the exchange part in the Fock matrix (Eq. [9]) is at least partly replaced by a so-called exchange-correlation functional term. For both HF and DFT, Eq. [7] needs to be solved self-consistently, and accordingly, these methods are denoted as SCE methods. 

Hyperfine couplings, in particular the isotropic part which measures the spin density at the nuclei, puts special demands on spin-restricted wave-functions. For example, complete active space (CAS) approaches are designed for a correlated treatment of the valence orbitals, while the core orbitals are doubly occupied. This leaves little flexibility in the wave function for calculating properties of this kind that depend on the spin polarization near the nucleus. This is equally true for self-consistent field methods, like restricted open-shell Hartree-Fock (ROHF) or Kohn-Sham (ROKS) methods. On the other hand, unrestricted methods introduce spin contamination in the reference (ground) state resulting in overestimation of the spin-polarization. 

The evaluation to the desired numerical accuracy of the density functional total energy has been a major obstacle to such calculations for many years. Part of the difficulty can be related to truncation errors in the orbital representation, or equivalently to basis set limitations, in variational calculations. Another part of the difficulty can be related to inaccuracies in the solution of Poisson s equation. The problem of maximizing the computational accuracy of the Coulomb self-interaction term in the context of least-squares-fitted auxiliary densities has been addressed in [39]. A third part of the difficulty may arise from the numerical integration, which is unavoidable in calculating the exchange and correlation contributions to the total energy in the density functional framework. 

An interesting aspect of the density functional calculations of Penzar and Ekardt is that these include self-interaction corrections. It is well known that the local density approximation (LDA) to exchange and correlation effects is not sufficiently accurate to give reliable electron affinities of free atoms or clusters [47,48]. This defidency is due to the fact that, in a neutral atom for instance, the LDA exchange-correlation potential Vif (f) decays exponentially at large r, while the exact behavior should be — 1/r. As a consequence, some atomic and cluster anions become unstable in LDA. The origin of this error is the incomplete cancellation of the self-interaction part of the classical coulomb energy term... 

Here the first part is the electrostatic self-Coulomb interaction and the second part is the analog correction for the exchange-correlation part of the effective potential. Please note that both functionals depend on the total density n, and not on the spin densities as in the case of Perdew and Zunger who corrected the LSDA (Local Spin Density Approximation). Results for Cu are reproduced in Figure 1.15 [31]. Though with this simple functional quantitative agreement with experimental data is not to be expected, the experimentally observed shell effects in the electron affinity are qualitatively well reproduced. 

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