Hylleraas wave function

Hydrogen bond, in force field energies, 23 Hylleraas wave function, 140 Hyperpolarizability, 236... 

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. 

Figure 2 The wave function of helium atom in its electronic ground state. The upper part (a) represents the difference between the Hartree-Fock and Hylleraas wave functions in a plane that contains the nucleus and fixed electron, called in the literature the Coulomb hole [5]. In the middle part (b) the F12 geminal function with 7 = 1.0 is plotted. The bottom part (c) represents the difference between (a) and (b).

For the He atom and two-electron systems the Hylleraas wave function has been recently improved including logarithmic terms by Schwartz and Nakatsuji et al. 

Recently, impressive calculations using Hylleraas wave functions have been done for the H2 molecule by the Hylleraas method [44,63], the Iterative Complement Iteration (ICI) [36], and Explicitly Correlated Gaussian (ECG) [12] methods. Few molecules have yet been calculated using Hylleraas-type wave functions HeH+ and some other species [72] using the Hylleraas method, the helium dimer He2 interaction energy [46] and the ground state of the BH molecule [7], both using the ECG method. 

A method that avoids making the HF mistakes in the first place is called quantum Monte Carlo (QMC). There are several types of QMC variational, dilfusion, and Greens function Monte Carlo calculations. These methods work with an explicitly correlated wave function. This is a wave function that has a function of the electron-electron distance (a generalization of the original work by Hylleraas). 

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. 

The three basic methods introduced by Hylleraas in his work on the He series have in modern terminology obtained the following names (a) Superposition of configurations (b) Correlated wave functions (c) Different orbitals for different spins. The first two approaches are developed almost to the full extent, whereas the last method is at least sketched in the 1929 paper. 

The outcome was certainly good but, according to Hylleraas opinion, the series (Eq. III.2) converged too slowly. In 1929, Hylleraas tried instead to introduce the interelectronic distance r12 in the wave function itself, which is then called a correlated wave function. In treating the S ground state, he actually used the... 

In the preliminary discussions in the 1929 paper (Eq. 11), Hylleraas also discussed some lower approximations and pointed out the importance of a configuration where there exist one "inner electron and one "outer electron. In modern terminology, this corresponds to a splitting of the closed shell (Is)2 into an open shell (Is, Is), or to the use of "different orbitals for different electrons. Hylleraas reported the good result E = —2.8754 at.u. for such a configuration, but pointed also out that a "correlated wave function of the form... 

To test the accuracy and convenience of the method of superposition of configurations, the problem of the ground state of the helium atom has recently been reexamined by several authors. According to Hylleraas (1928), the total wave function may be expressed in the form... 

Hylleraas, E., Avhandl. Norske Videnskaps Akad. Oslo. I. Mat.-Naturv. Kl. No. 5, Two-electron angular wave functions."... 

Green, L. C., Mulder, M. M., Milner, P. C., Lewis, M. N., Woll, J. W., Jr., Kolchin, E. K., and Mace, D., Phys. Rev. 96, 319, (iii) Analysis of the three parameter wave function of Hylleraas for the He I ground state in terms of central field wave-functions/ Configurational interaction. 

Hylleraas was first to use a wave function with explicit r, -dependence in his 1929 breakthrough study of the helium atom [16,17], Applications of these... 

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. 

A suitable representation of the helium wave function was taken to be the Hylleraas form... 

Electron-positron annihilation in Ps2 was investigated by Tisenko (1981), who, using the relatively simple wave function of Hylleraas and Ore (1947), obtained the annihilation rates into two and three gamma-rays as 16 ns-1 and 0.043 ns-1 respectively. No such calculations have been performed using the more elaborate wave function of Ho. 

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]... 

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,... 

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... 

The presence of the l/ri2 Coulomb repulsion term in Eq. (1) makes the Schrodinger equation nonseparable, and so exact analytic solutions cannot be found. Early in the history of quantum mechanics, Hylleraas [23] suggested expanding the wave function in the form (generalized for states of arbitrary angular momentum L)... 

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