Small correlated systems

There is an important case which is intermediate between small bounded systems and macroscopic fully extended systems, namely the description of the surface region of a macroscopic metal. The correlation functions which describe density fluctuations in the surface region are extremely anisotropic and of long range, very unlike their counterparts in the bulk, and the thermodynamic limit must be taken with sufficient care. Consider the static structure factor for a large system of N particles contained within a volume Q,... 

The physical interpretation of the functional derivative vx(r) shows that it is comprised of a term Wx (r) representative of Pauli correlations, and a term wj (r) that constitutes part of the total correlation-kinetic contribution Wt (r). cThe exact asymptotic structure of these components in the vacuum has been determined and shown to also be image-potential-like. Although the structure of vx(r) about the surface and asymptotically in the vacuum and metal-bulk regions is comprised primarily of its Pauli component, the correlation-kinetic contribution is not insignificant for medium and low density metals. It is only for high density systems (rs < 2) that vx(r) is represented essentially by its Pauli component Wx (r). Thus, we see that the uniform electron gas result of -kF/ir for the functional derivative vx(r), which is the asymptotic metal-bulk value, is not a consequence of Pauli correlations alone as is thought to be the case. There is also a small correlation-kinetic contribution. The Pauli and correlation-kinetic contributions have now been quantified. 

First we will demonstrate highly accurate (errors less than 10-3cm-1) non-adiabatic calculations for H, D, and T accomplished with Method I. These calculations reflect the capability of the explicitly correlated gaussian basis to describe subtle non-adiabatic effects in small atomic systems. 

The validity of the CSP method for sub-picosecond processes in systems which exhibit moderate quantum effects was established by comparison with numerically exact quantum dynamical calculations for small modell systems [27,49], Good correlation with experiments for large realistic systems is also encouraging [50, 51], Currently, limitations of the CSP scheme given by its separable nature are being overcome by extending the method in the direction of configuration interaction [53],... 

Strictly speaking, an independence shown by the probability distributions derived from two states implies an absence of interaction between the systems or subsystems described by these two states however, in practice, the great success of models which combine the concept of independence with a non-negligible interaction shows that except for a relatively small correlation error, on average two systems may show an independent behaviour even if they interact rather strongly . The most obvious example of this is the Hartree-Fock method. 

The Coupled-clusters (CC) method[7] based on the cluster expansion of the wavefunction has been established as a highly reliable method for calculations of ground state properties of small molecules with the spectroscopic accuracy. When this method is used together with a flexible basis set it recovers the dominant part of the electron correlation. Typically, CC variant explicitly considering single and double excitations (CCSD) is used. In order to save computer time the contributions from triple excitations are often calculated at the perturbation theory level (notation CCSD(T) is used in this case). CCSD(T) method can be routinely used only for systems with about 10 atoms at present. Therefore, it cannot be used directly in zeolite modeling, however, results obtained at CCSD(T) level for small model systems can serve as an important benchmark when discussing the reliability of more approximate methods. 

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