Hylleraas-type basis functions

Calculations of total energies and widths of doubly excited states using the Breit-Pauli Hamiltonian and comparison with results from the use of Hylleraas-type basis functions [55]... 

A natural extension of a Hylleraas-type basis might include correlation in the exponential function. Hylleraas[24] introduced this type of exponentially correlated (EC) wave function, using a single term of the form,... 

The Hy-CI function used for molecular systems is based on the MO theory, in which molecular orbitals are many-center linear combinations of one-center Cartesian Gaussians. These combinations are the solutions of Hartree-Fock equations. An alternative way is to employ directly in Cl and Hylleraas-CI expansions simple one-center basis functions instead of producing first the molecular orbitals. This is a subject of the valence bond theory (VB). This type of approach, called Hy-CIVB, has been proposed by Cencek et al. (Cencek et.al. 1991). In the full-CI or full-Hy-CI limit (all possible CSF-s generated from the given one-center basis set), MO and VB wave functions become identical each term in a MO-expansion is simply a linear combination of all terms from a VB-expansion. Due to the non-orthogonality of one-center functions the mathematical formalism of the VB theory for many-electron systems is rather cumbersome. However, for two-electron systems this drawback is not important and, moreover, the VB function seems in this case more natural. 

We see that, for atoms or molecules more complex than beryllium, the use of correlated basis sets of Hylleraas type rapidly becomes infeasible and the Cl method and its variants become more attractive. For a correlated basis, the number of terms required to reach a particular metric order can become hopelessly large. In a 10-electron problem like neon, 1596 terms are needed to construct a wave function complete through u = 2. If instead the wave function is limited to a single rfj factor per term, this number is reduced to 561 terms, still a sizeable number for such a low metric order. It must also be kept in mind that antisymmetrizing the wave function means that each term... 

As a basis function for the FSS procedure, we used the Hylleraas-type functions [105] as presented by Yan and Drake [114] ... 

If the r-space wave function is a linear combination of Slater determinants constructed from a set of spin-orbitals 0/, then its p-space counterpart is the same linear combination of Slater determinants constructed from the spin-momentals j obtained as Fourier transforms, Eq. (9), of the spin-orbitals. The overwhelming majority of contemporary r-space wave functions can be expressed as a linear combination of Slater determinants, and in these cases only three-dimensional Eourier transforms, Eq. (9), of the spin-orbitals are necessary to obtain the corresponding A -electron wave function in p space. Use of the full Eq. (5) becomes necessary only for wave functions, such as Hylleraas- or Jastrow-type wave functions, that are not built from a one-electron basis set. Examples of transformations to momentum space of such wave functions for He and H, can be found elsewhere [20-22]. 

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... 

The coefficients Cj are determined variationally in an eigenvalue problem. The Norwegian physicist Egil Hylleraas was the first to calculate an accurate wave function for the He atom. He used six basis functions of the following type ... 

For two-electron atoms, many approaches have been applied a review made by Aquino reported the techniques used up to 2009 [20], To date, the expansion of the wave function in terms of Hylleraas-type functions is the technique that gives the lowest energies for several confinement radii [21-23], which can be used as reference when other techniques are proposed for the study of these systems. However, such a technique has not been used for atoms with several electrons, for example, beryllium. In this sense, in this chapter we test the many-body perturbation theory to second order, as a technique to estimate the CE for confined many-electron atoms. In the next section, we discuss the theory behind of the HF method, and the basis set proposed for its implementation for confined atoms. In the same section, the many-body perturbation theory to second order proposed by Moller and Plesset (MP2) [24] also is discussed, and we give some details about the implementation of our code implemented in GPUs [25]. Finally, we contrast our results for helium-like atoms with more sophisticated techniques in order to know the percent of correlation energy recovered by the MP2 method. 

The bound-state type of approach that was followed by Hylleraas—albeit fraught with a couple of erroneous conclusions—eventually evolved in the 1960s into methods aiming at the calculation and identification of resonance states from the behavior of roots of diagonalized energy matrices constructed from discrete basis sets, as a function of a scaling parameter, or size of discrete basis sets, or size of the artificial box inside which the continuous spectrum is artificially discretized. 

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