Correlation energy dynamic

Beyond the Boson peak, the reduced DOS reveals for all studied glasses a temperature-independent precisely exponential behavior, g(E)/E exp( / o) with the decay energies Eo correlating with the energies E of the Boson peak. This finding additionally supports the view that the low-energy dynamics of the glasses are indeed delocalized collective motions because local and quasilocal vibrations would be described in terms of a power law or a log-normal behavior [102]. 

Unfortunately, the dynamic correlation energy is not constant for a given molecule but may vary considerably between different electronic states. Thus, any procedure geared towards quantitative accuracy in predicting excited-state energies must in some way account for these variations. The most economical way to achieve this is to introduce a number of parameters into the model. By scaling those to a set of experimental data... 

In ab initio methods (which, by definiton, should not contain empirical parameters), the dynamic correlation energy must be recovered by a true extension of the (single configuration or small Cl) model. This can be done by using a very large basis of configurations, but there are more economical methods based on many-body perturbation theory which allow one to circumvent the expensive (and often impracticable) large variational Cl calculation. Due to their importance in calculations of polyene radical ion excited states, these will be briefly described in Section 4. 

Dynamic Correlation Energy Estimate and Analysis Based on Localized Orbitals... 

We now turn to the problem of simplifying the recovery of the dynamic correlation energy. We consider the simplest situation, viz., where the zeroth-order wavefunction can be chosen as the SCF approximation. A challenging disparity exists between the energetic smallness of these refinements and the complexity and magnitude of the computational efforts required for their variational determination. In order to reduce this disproportion, various semiempirical approaches have been proposed (56-61), notably in particular the introduction of semiempirical elements into MP2 theory which has led to the successful Gn methods (62-64). 

We have explored whether, on the basis of sound theoretical and physical reasoning, a semiempirical formula can be derived that would directly provide an accurate estimate of the dynamic correlation energy as a whole. Two known rigorous results are suggestive in this context (i) The dynamic correlation energy can be expressed as the expectation value of a perturbing correlation operator (65-67) and (ii) the correlation energy is known to be expressible (10) as sum of contributions of occupied orbital pairs, viz. 

The set of molecules for which the relationship of Eq. (3.3) has been tested with the reported accuracy, has certain sinq>lifying features in common They all have standard bonding coordinations around each atom and the shortfall of the SCF energy is entirely due to dynamic correlations. Modifications are to be expected for systems where these premises are not satisfied. Even with these limitations, however, the molecules in Table 2 represent a variety of atom and bond combinations. It is therefore remarkable that, for all of them, the correlation energy can be recovered by a sinq>le system-independent formula that allows for a physically meaningful interpretation. 

A close estimate of the dynamic correlation energy was obtained by a simple formula in terms of pair populations and correlation contributions within and between localized molecular orbitals. The orbital and orbital-pair correlation strengths rapidly decrease with the distance between the orbitals in a pair. For instance, the total valence correlation energy of diamond per carbon atom, estimated as 164 mh, is the result of 84 mh from intra-orbital contributions, 74.5 mh from inter-orbital closest neighbors contributions, and 6.1 mh from interorbital vicinal contributions. The rapid decay of the orbital correlation contributions with the distance between the localized orbitals explains the near-... 

The major advantage of a 1-RDM formulation is that the kinetic energy is explicitly defined and does not require the construction of a functional. The unknown functional in a D-based theory only needs to incorporate electron correlation. It does not rely on the concept of a fictitious noninteracting system. Consequently, the scheme is not expected to suffer from the above mentioned limitations of KS methods. In fact, the correlation energy in 1-RDM theory scales homogeneously in contrast to the scaling properties of the correlation term in DPT [14]. Moreover, the 1-RDM completely determines the natural orbitals (NOs) and their occupation numbers (ONs). Accordingly, the functional incorporates fractional ONs in a natural way, which should provide a correct description of both dynamical and nondynamical correlation. 

This chapter begins with a discussion of how to include non-dynamical and dynamical electron correlation into the wave function using a variety of methods. Because the mathematics associated with correlation techniques can be extraordinarily opaque, the discussion is deliberately restricted for the most part to a qualitative level an exception is Section 7.4.1, where many details of perturbation theory are laid out - those wishing to dispense with those details can skip this subsection without missing too much. Practical issues associated with the employ of particular techniques are discussed subsequently. At the end of the chapter, some of the most modem recipes for accurately and efficiently estimating the exact correlation energy are described, and a particular case study is provided. 

---