Wavefunction explicitly correlated

Kar and Ho [196] have estimated the oscillator strengths for different transitions, dipole and quadrupole polarizabilities of He for a wide range of the Debye screening parameters using explicitly correlated wavefunctions. Results presented by Kar and Ho [196] are very accurate and may be of substantial use for comparison with those from laser plasma experiments. The behavior of several singly and doubly excited states of He under screened potential was also accurately estimated by Kar and Ho [197] using correlated basis functions. Variation of the transition wavelength as a... 

First, let us consider the hydride anion, H. This is a highly correlated system since electron correlation is the sole source of the binding of the second electron. A CHF treatment therefore gives a spurious result for a of 91.40 au[27]. Pauling s value, see Table 1, was 68.26 au. A more reliable value is obtained from Eq. (11) and using explicitly correlated wavefunctions (ECW)[28]. Since this method will be frequently used for two-electron systems, we will give the relevant details. 

The procedure described here for configuration interaction wavefunctions can be extended to explicitly-correlated wavefunctions [73,74]. There is no conceptual difficulty in this extension other than the fact that one must perform all integrations numerically. 

We can distinguish between two broad classes of explicitly correlated wavefunctions polynomials in r(J and other inter-body distances, and exponential or Jastrow forms [15,16]... 

In most standard ab initio approaches, the parameters to minimize are the linear coefficients of the expansion of the wavefunction in some basis set. To make the problem tractable, one is usually forced to choose a basis set for which the integrals of Eq. (4.1) are analytically computable. However, as we have seen, it is practical to use very accurate explicitly correlated wavefunctions with VMC. 

The trial wavefunctions prolate spheroidal coordinates, properly symmetry adapted, and of the same explicitly correlated form as introduced by Kolos et al. [16]... 

Bases of fully exponentially correlated wavefunctions [1, 2] provide more rapid convergence as a function of expansion length than any other type of basis thus far employed for quantum mechanical computations on Coulomb systems consisting of four particles or less. This feature makes it attractive to use such bases to construct ultra-compact expansions which exhibit reasonable accuracy while maintaining a practical capability to visualize the salient features of the wavefunction. For this purpose, exponentially correlated functions have advantages over related expansions of Hylleraas type [3], in which the individual-term explicit correlation is limited to pre-exponential powers of various interparticle distances (genetically denoted r,j). The general features of the exponentially correlated expansions are well illustrated for three-body systems by our work on He and its isoelectronic ions, for... 

On May 27, 2003, at the age of 55, Jacek Rychlewski passed away suddenly, while working late at his office on a book presenting the theory and applications of explicitly correlated wavefunctions. This book was completed by his collaborators and will appear shortly in the PTCP bookseries. 

At first, aU these methods were developed for closed-shell systems only. Later research in this area was directed towards local methods for open-shell systems and excited states, local triples corrections beyond (T) (triples included in coupled cluster iterations), [138], local energy gradients for geometry optimizations of large molecules [139], combination of the local correlation method with explicitly correlated wavefunctions. It is evident from the discussion that these local 0 N) methods open the applications of coupled-cluster theory to entirely new classes of molecules, which were far ont-of-scope for such an accurate treatment before. Possible applications lie, for example, in the determination of the thermochemistry of reactions involving... 

The ideal trial wavefunction is simple and compact, has simple easily evaluated first and second derivatives, and is accurate everywhere. Because the local energy must be evaluated repeatedly, the computation effort required for the derivatives makes up a large part of the overall computation effort for many systems. The typical trial wavefunctions of analytic variational calculations are not often useful, since they are severely restricted in form by the requirement that they be amenable to analytic integrations. The QMC functions are essentially unrestricted in form, since no analytic integrations are required. First and second derivatives of trial wavefunctions are needed, but differentiation is in general much easier than integration, and most useful trial wavefunctions have reasonably simple analytical derivatives. In most analytic variational calculations to date, it has not been possible to include the interelectron distances r,y in the trial wavefunction, and these wavefunctions are not usually explicitly correlated, whereas for QMC calculations of all types, explicitly correlated functions containing are the norm. 

The first is benchmark studies of two-electron systems using explicitly correlated wavefunctions (see, e.g. Bishop (1994)). 

There are two possible solutions to this problem. We may either modify our ansatz for the wavefunction, including terms that depend explicitly on the interelectronic coordinates [26-30], or we may take advantage of the smooth convergence of the correlation-consistent basis sets to extrapolate to the basis-set limit [6, 31-39], In our work, we have considered both approaches as we shall see, they are fully consistent with each other and with the available experimental data. With these techniques, the accurate calculation of AEs is achieved at a much lower cost than with the brute-force approach described in the present section. 

Johnson and Rice used an LCAO continuum orbital constructed of atomic phase-shifted coulomb functions. Such an orbital displays all of the aforementioned properties, and has only one obvious deficiency— because of large interatomic overlap, the wavefunction does not vanish at each of the nuclei of the molecule. Use of the LCAO representation of the wavefunction is equivalent to picturing the molecule as composed of individual atoms which act as independent scattering centers. However, all the overall molecular symmetry properties are accounted for, and interference effects are explicitly treated. Correlation effects appear through an assigned effective nuclear charge and corresponding quantum defects of the atomic functions. 

The wavefunction is expanded in terms of Hylleraas basis sets incorporating explicitly the electron correlation effects... 

However, because of the correlated motion of the electrons, many-electron processes will also occur. (Looking at the many-particle effects in this way, the photon operator is a single-particle operator and electron-electron interactions have to be incorporated explicitly into the wavefunction. It is, however, also possible to describe the combined action of the electrons as an induced field which adds to the external field of the photoprocess, i.e., the transition operator becomes modified. Generally, the influence of the electron-electron interaction can be represented by modifying the wavefunction or the operator or by modifying both the wavefunction and the operator [DLe55, CWe87].) Of all the possible processes, only the important two-electron processes restricted to electron emission will be considered here. In many cases they can be divided into two different classes (see Fig. 1.3) t... 

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