Correlation function first-zero method

The First-Zero Method of Correlation Function Analysis. For the purpose of a practical graphical evaluation of the linear crystallinity, Eq. (8.67) can be applied to a renormalized correlation function y (x/Lapp). The method which has been proposed by Goderis et al. [162] is based on the implicit assumption that the first zero, Jto, of the real correlation function is shifted by the same factor as is the position of its first maximum, Lapp. 

The general inferiority of geometrical construction methods [162,163] as compared to more involved methods which consider polydispersity has first been demonstrated by Santa Cruz et al. [130], and later in many model calculations by Crist [ 165-167]. Nevertheless, in particular the first-zero method is frequently used. Thus, it appears important to assess its advantages as well as its limits. Validation can be carried out by graphical evaluation of model correlation functions [130,165],... 

Figure 8.22. Testing the first-zero method for the determination of the linear crystallinity, V[, from the linear correlation function, yi (x/Lapp) with Lapp being the position of the first maximum in yi (x) (not shown here - but cf. Fig. 8.21). Model tested Paracrystalline stacking statistics with Gaussian thickness distributions. The interval of forbidden zeroes is shown. An additional horizontal non-linear axis permits to determine the linear crystallinity directly. A corresponding vertical axis shows the variation of the classical valley-depth method ... 

The electron-nucleus (e-n) correlation function does not describe electron correlation per se because it is redundant with the orbital expansion of the antisymmetric function. If the correlation function expansion is truncated at Fi and the antisymmetric wave function is optimized with respect to all possible variations of the orbitals, then Fi would be zero everywhere. There remain two strong reasons for including Fi in the correlation function expansion. First, the molecular orbitals are typically expanded in Gaussian basis sets that do not satisly the e-n cusp conditions. The e-n correlation function can satisly the cusp conditions, but F/ influences the electron density in regions beyond the immediate vicinity of the nucleus, so simple methods for determining Fi solely from the cusp conditions may have a detrimental effect on the overall wave function. Careful optimization of a flexible form of is required if the e-n cusp is to be satisfied by the one-body correlation function [115]. 

In order to elucidate the correlation method it may be recalled that the viscosity 77 approaches asymptotically to the constant value r c with decreasing shear rate q. Similarly, the characteristic time t approaches a constant value xQ and the shear modulus G has a limiting value G0 at low shear rates. Bueche already proposed that the relationship between 77 and q be expressed in a dimensionless form by plotting 77/r]0 as a function of qx. According to Vinogradov, also the ratio t/tq is a function of qxQ. If the zero shear rate viscosity and first normal stress are determined, then a time constant x0 may be calculated with the aid of Eqs. (15.60). This time constant is sometimes used as relaxation time, in order to be able to produce general correlations between viscosity, shear modulus and recoverable shear strain as functions of shear rate. 

Of course, in the actual computation, one first has to go to all orders to establish the exact Xco, of Eq. (4) from which Eq. (7) follows. Nevertheless, in practice, for atomic and molecular ground states where the shell model holds well and the zero-order reference is a closed-shell, single deferminan-tal HE wavefunction, it remains true that the dominant contribution to the Ecorr comes from the double excitations (electron pair correlations), although for a given state, the exact magnitude of each term of Eq. (4) depends on the computational method and on the function spaces that are used. 

Modem quantum-chemical methods can, in principle, provide all properties of molecular systems. The achievable accuracy for a desired property of a given molecule is limited only by the available computational resources. In practice, this leads to restrictions on the size of the system From a handful of atoms for highly correlated methods to a few hundred atoms for direct Hartree-Fock (HF), density-functional (DFT) or semiempirical methods. For these systems, one can usually afford the few evaluations of the energy and its first one or two derivatives needed for optimisation of the molecular geometry. However, neither the affordable system size nor, in particular, the affordable number of configurations is sufficient to evaluate statistical-mechanical properties of such systems with any level of confidence. This makes quantum chemistry a useful tool for every molecular property that is sufficiently determined (i) at vacuum boundary conditions and (ii) at zero Kelvin. However, all effects from finite temperature, interactions with a condensed-phase environment, time-dependence and entropy are not accounted for. 

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