Kolos-Wolniewicz function

III. Two-electron molecules Kolos-Wolniewicz wave function... 

The simplest molecular system exhibiting effects of electron correlation is the hydrogen molecule. For this molecule the explicitly correlated wave function has been applied in the early days of quantum mechanics (James and Coolidge, 1933), It was later generalised by Kolos and Wolniewicz (Kcrfos and Wolniewicz, 1965) and successfully used to solve variety of problems in the ground and excited states of the hydrogen molecule. This wave function, called there the Kolos-Wolniewicz function (Kolos and Wolniewicz, 1965) is assumed in the form of an expansion ... 

Kolos-Wolniewicz function (p. 590) many body perturbation theory (p. 641)... 

The family of variational methods with explicitly correlated functions includes the Hyller-aas method, the Hylleraas Cl method, the James-Coolidge and the Kolos-Wolniewicz approaches, and the method with exponentially correlated Gaussians. The method of explicitly correlated functions is very successful for 2-, 3- and 4-electron systems. For larger systems, due to the excessive number of complicated integrals, variational calculations are not yet feasible. 

Hylleraas function (p. 506) harmonic helium atom (p. 507) James-Coolidge function (p. 508) Kolos-Wolniewicz function (p. 508) geminal (p. 513)... 

These functions have been regarded as the ultimate choice for two-electron diatomic molecules, " and the quality of the results has reached an unprecedented accuracy (nanohartree level, 1 n h = 0.00022 cm ). However, it has been shown that the accuracy of the expansion in a Kolos-Wolniewicz basis is also achieved by expansions in terms of the Gaussian functions (geminals)... 

For the systems studied so far (see Tables 7, 3, and 4), the Cencek-Rychlewski method has provided variational energies for molecules that are superior to all other variational calculations, including the Kolos-Wolniewicz-type calculations for H2. This success can be attributed to the rigorous and efficient optimization of the nonlinear parameters. Note that the Gaussian centers are not restricted to the positions of the nuclei, but are completely free to float. (Only s-type Gaussians are used currently, but the method can of course be extended to Cartesian functions by making available the corresponding computer code.)... 

Fig. 3. This figure shows the total energy of the H2 molecule as a function of intemuclear distance, calculated from the electronic energies shown in Figure 2. For small values of R, the calculation using 15 atomic orbitals per atom agrees well with the values of Kolos and Wolniewicz but another configuration would be needed for agreement at large R.

The wave function of Eqs. (14) and (15) was widely used to obtain BO potential energy curves and adiabatic corrections for the ground state (Kolos et al., 1986 Kotos and Rychlewski 1993, Wolniewicz 1993, 1995a) and electronically excited... 

Figure 2.1. Energies of H2 for various calculations using the H-atom Is orbital functions (b) covalent + ionic, scaled (c) covalent oidy, scaled (d) covalent + ionic, unsealed (e) covalent only, unsealed, (a) labels the curve for the accurate function due to Kolos and Wolniewicz[31]. This is included for comparison purposes.

Kolos and Wolniewicz (11) calculate from accurate wave functions of the ground state of H2 that the diagonal nuclear kinetic energy term is 5 cm at the equilibrium distance and - 38 cm" at a nuclear separation of 0.3 A. 

Table 3.2.1 summarizes the results of various approximate wavefunctions for the hydrogen molecule. This list is by no means complete, but it does show that, as the level of sophistication of the trial function increases, the calculated dissociation energy and bond distance approach closer to the experimental values. In 1968, W. Kolos and L. Wolniewicz used a 100-term function to obtain results essentially identical to the experimental data. So the variational treatment of the hydrogen molecule is now a closed topic. 

HJ point out that in the detailed work on H2 by Kolos and Wolniewicz,114 the first excited state 3 2 was found to have a very weak minimum at a large separation. This binding presumably arises from a van der Waals force which is not included in the density functional theory when a local approximation to exchange and correlation is employed. Nevertheless, as HJ point out, their study of the corresponding state of the dimers Li2-Cs2 revealed a weak, but definite maximum in each case. Rough estimates of binding energy and equilibrium separation are shown in Table 16. It is, of course, possible that these results are a consequence of the local spin-density approximation, so that further work will... 

Figure 16 Calculated potential energies for an isolated H2 molecule as a function of interatomic distance. The closed and open circles are results from spin-unpolarized and spin-polarized GGA calculations, respectively, the solid curve is a fit to a Morse potential, and the dashed curve shows the highly accurate results of Kolos and Wolniewicz from ref. 43...

Kolos and Wolniewicz applied the Ritz variational method (see Chapter 5) to the hydrogen molecule with the following trial function ... 

Relative total energy of H2 as a function of internuclear distance. The SCF, MP2, and full Cl calculations usea6-31G basis set, and the exact calculation refers to the work by Kolos and Wolniewicz [22]. 

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