Helium correlated wave functions

Mainly for considerations of space, it has seemed desirable to limit the framework of the present review to the standard methods for treating correlation effects, namely the method of superposition of configurations, the method with correlated wave functions containing rij and the method using different orbitals for different spins. Historically these methods were developed together as different branches of the same tree, and, as useful tools for actual applications, they can all be traced back to the pioneering work of Hylleraas carried out in 1928-30 in connection with his study of the ground state of the helium atom. 

On the helium problem, the connection between the method of correlated wave function and the method of superposition of configurations has also been investigated in detail.8... 

Power Series Expansions and Formal Solutions (a) Helium Atom. If the method of superposition of configurations is based on the use of expansions in orthogonal sets, the method of correlated wave functions has so far been founded on power series expansions. The classical example is, of course, Hyl-leraas expansion (Eq. III.4) for the ground state of the He atom, which is a power series in the three variables... 

Morrison, R. C., Q. Zhao, R. C. Morrison, and R. G. Parr. 1995. Solution of the Kohn-Sham equations using reference densities from accurate, correlated wave functions for the neutral atoms helium through argon. Phys. Rev. A51, 1980. 

Morrison, R. C., Zhao, Q., 1995, Solution to the Kohn-Sham Equations Using Reference Densities from Accurate, Correlated Wave Functions for the Neutral Atoms Helium Through Argon , Phys. Rev. A, 51, 1980. 

Improvement of Correlated Wave Functions for Helium by Means of Local-Scaling Transformations... 

Abstract. We have calculated the scalar and tensor dipole polarizabilities (/3) and hyperpolarizabilities (7) of excited ls2p Po, ls2p P2- states of helium. Our theory includes fine structure of triplet sublevels. Semiempirical and accurate electron-correlated wave functions have been used to determine the static values of j3 and 7. Numerical calculations are carried out using sums of oscillator strengths and, alternatively, with the Green function for the excited valence electron. Specifically, we present results for the integral over the continuum, for second- and fourth-order matrix elements. The corresponding estimations indicate that these corrections are of the order of 23% for the scalar part of polarizability and only of the order of 3% for the tensor part... 

Explicitly correlated wave function fheory [14] is anofher imporfanf approach in quantum chemistry. One introduces inter-electron distances together with the nuclear-electron distances and set up some presumably accurate wave function and applies the variation principle. The Hylleraas wave function reported in 1929 [15] was the first of this theory and gave accurate results for the helium atom. Many important studies have been published since then even when we limit ourselves to the helium atom [16-28]. They clarified the natures and important aspects of very accurate wave functions. However, the explicitly correlated wave function theory has not been very popularly used in the studies of chemical problems in comparison with the Hartree-Fock and electron correlation approach. One reason was that it was generally difficult to formulate very accurate wave functions of general molecules with intuitions alone and another reason was that this approach was rather computationally demanding. 

Key words Helium atom - Electron correlation -Explicitly correlated wave functions - Hylleraas expansion... 

Already in 1929 Hylleraas found that the orbital expansion of the wave function of helium converges extremely slowly. The problem could be overcome by including terms in the wave function that depend explicitly on interelectronic coordinates [6, 7]. The proposed explicitly correlated wave function was of the form... 

Table X gives an idea of the strength of the various expansion methods, and it shows that, by using the principal term only, one can hardly expect to reach even the above-mentioned chemical margin, even if the wave function W gO(D) is actually very close in the helium case. This means that one has to rely on expansions in complete sets, and the construction of the modern electronic computers has fortunately greatly facilitated the numerical solution of secular equations of high order and the calculation of the matrix elements involved. For atoms, the development will probably go very fast, but, for small molecules one has first to program the conventional Hartree-Fock scheme in a fully self-consistent way for the computers, before the next step can be taken. For large molecules and crystals, the entire situation is much more complicated, and it will hence probably take a rather long time before one can hope to get a detailed understanding of the correlation phenomena in these systems.

If we imagine the nuclei to be forced together to = 0, the wave function Is A + Iss will approach, as a limit, a charge distribution around the united atom that has neither radial nor angular nodal planes. This limiting charge distribution has the same symmetry as the Is orbital on the united atom, Helium. On the other hand, the combination Isa Iss has a nodal plane perpendicular to the molecular axis at all intemuclear separations. Hence its limit in the united atom has the symmetry properties of a 2p orbital. A simple correlation diagram for this case is ... 

In a few cases, the wave-function F of a monatomic entity can be used for calculating a, e.g. 4.5 bohr3 for the hydrogen atom, or 0.205 A3 for the helium atom in agreement with the experimental value. Gaseous H does not have a Hartree-Fock function stable relative to spontaneous loss of an electron, and it is necessary to introduce correlation effects in order to calculate a which is said to be 31 A3. The value 1.8 A3 for H(-I) in Table 2 derives from NaCl-type LiH, NaH and KH. The anion B2Hg2 has a = 6.3 A3 to be compared with the isoelectronic C2H6 4.47 A3. Since CH4 has a =... 

Substantial differences exist between the results for the simple uncorrelated helium function HI and those for the more elaborate correlated helium functions H5 and H14, but the very close agreement between these latter two sets of results strongly suggests that they are both close to the exact values. Further evidence in support of this claim has recently been provided by Van Reeth and Humberston (1995a), who obtained very similar results using even more accurate helium wave functions, both with and without the use of the method of models. 

Several other calculations of the first few partial-wave phase shifts for positron-helium scattering have been carried out using a variety of approximation methods in all cases, however, rather simple uncorrelated helium wave functions have been used. Drachman (1966a, 1968) and McEachran et al. (1977) used the polarized-orbital method, whereas Ho and Fraser (1976) used a formulation based on the static approximation, with the addition of several short-range correlation terms, to determine the s-wave phase shifts only. The only other elaborate variational calculations of the s-wave phase shift were made by Houston and Drachman (1971), who employed the Harris method with a trial wave function similar to that used by Humberston (1973, 1974), see equation (3.77), and with the same helium model HI. Their results were slightly less positive than Humberston s HI values, and are therefore probably less... 

Comparisons of calculated and measured quenching rates provide a useful measure of the accuracy of the wave function used for the system. As an example, the value of Ze for helium calculated from the zero energy static-exchange wave function of Barker and Bransden (1968) is 0.0347, or 0.0445 when the van der Waals potential is added to the static-exchange equation however, the experimental value obtained by Coleman et al. (1975b) at room temperature is 0.125 0.002 (see section 7.3). This rather large discrepancy, a factor of three, shows that the static-exchange wave function provides a poor representation of the electron-positron correlations in this system. 

The ground state of helium is itself a rather special case as the wave function is relatively compact. It is thus not difficult to get a reasonable representation of this wave function with a rather modest, correlated basis set. Hylleraas[16]... 

The energy of the helium atom calculated above is the first-order energy, which differs from the true energy by an amount called the correlation energy this is a measure of the tendency of the electrons to avoid each other. The simplest improvement to the trial wave function is to allow Z in (6.29) to be a variable parameter, which we call (not to be confused with the spin-orbit coupling parameter in equation (6.20)) Z in the Hamiltonian (6.23) remains the same. The expression for the calculated energy,... 

In two previous papers [8,9] we have calculated the static polarizabilities and hyperpolarizabilities for ls3p Pj (J = 0, 2)-states of helium. The method was based on degenerate perturbation-theory expressions for these quantities. The necessary dipole matrix elements were found by using the high-precision wave function on framework of the configuration-interaction (Cl) method [10]. The perturbed wave functions are also expanded in a basis of accurate variational eigenstates [11]. These basis sets of the wave functions explicitly take account of electron correlation. To control the result we have also carried out similar calculations with Fues model potential method. 

The correlation energies for free (unconfined) H, He, Li+ and Be++ were well known at that time, and so Gimarc wanted to analyze, in particular, how the correlation energy changes as a function of the box radius for the confined helium atom isoelectronic series. Gimarc performed a number of variational calculations based on the following wave functions ... 

Table 2 Correlation energy (CE) estimated as the difference between the wave function expanded with 40 Hylleraas functions (H-WF) and the HF wave function obtained with optimized exponents, for the lowest singlet state of confined helium atom. All quantities are in hartrees...

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