Hylleraas

Figure B3.1.5. Probability (as a fimction of angle) for finding the seeond eleetron in He when both eleetrons are loeated at the maximum in the Is orbital s probability density. The bottom line is that obtained using a Hylleraas-type fiinetion, and the other related to a highly-eorrelated multieonfigurational wavefLinetion. After [22],...

Hylleraas E A 1963 Reminisoenoes from early quantum meohanios of two-eleotron atoms Rev. Mod. Phys. 35 421-31... 

This characteristic is commonly referred to as the bracketing theorem (E. A. Hylleraas and B. Undheim, Z. Phys. 759 (1930) J. K. E. MacDonald, Phys. Rev. 43, 830 (1933)). These are strong attributes of the variational methods, as is the long and rich history of developments of analytical and computational tools for efficiently implementing such methods (see the discussions of the CI and MCSCF methods in MTC and ACP). 

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). 

Alternative methods are based on the pioneering work of Hylleraas ([1928], [1964]). In these cases orbitals do not form the starting point, not even in zero order. Instead, the troublesome inter-electronic terms appear explicitly in the expression for the atomic wavefunction. However the Hylleraas methods become mathematically very cumbersome as the number of electrons in the atom increases, and they have not been very successfully applied in atoms beyond beryllium, which has only four electrons. Interestingly, one recent survey of ab initio calculations on the beryllium atom showed that the Hylleraas method in fact produced the closest agreement with the experimentally determined ground state atomic energy (Froese-Fischer [1977]). 

Hylleraas, E. A. [1964] The Schrodinger Two-Electron Atomic Problem , Advances in Quantum Chemistry, 1, p. 1. 

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. 

If a trial function 9 leads to a kinetic energy 1 and a potential energy Vx which do not fulfill the virial theorem (Eq. 11.15), the total energy (7 +Ei) is usually far from the correct result. Fortunately, there exists a very simple scaling procedure by means of which one can construct a new trial function which not only satisfies the virial theorem but also leads to a considerably better total energy. The scaling idea goes back to a classical paper by Hylleraas (1929), but the connection with the virial theorem was first pointed out by Fock.5 It is remarkable how many times this idea has been rediscovered and published in the modern literature. 

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... 

Hylleraas mentioned the possibility of using the (2/-f 2) order functions in a footnote in ref. 14, but he did not give any explicit formulas or numerical results. 

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... 

A great deal of attention has been paid to the question of the necessity of having logarithmic terms in the expansion (Gronwall 1937, Bartlett 1952, 1955, Fock 1954, Hylleraas 1955), but Kino-shita has pointed out that although such terms may be convenient from the. numerical point of view (Hylleraas and Midtdal 1956), they are not necessarily required by the form of the Schrodinger equation itself. 

We note that the power series expansion III. 119is a direct generalization of the Hylleraas form III. 114 to which it should go over in the limiting case Rab = 0. James and Coolidge obtained a value of the electronic energy, —1.17347 at.u., in excellent agreement with the experimental results available, and their work forms even today the best basis for our understanding of the electronic structure of the chemical bond. 

For two-electron systems the basic idea of using different orbitals for different electrons goes back to Hylleraas (1929) and to Eckart (1930) who both used it in treating He. The method was thoroughly discussed at the Shelter Island Conference 1951 in treatments of He and H2 (Kotani 1951, Taylor and Parr 1952, Mulliken 1952), but the circumstances are here exceptionally simple because of the possibility of separating space and spin according to Eq. III. 1. 

Hylleraas, E. A., Z. Physik 48, 469, "Uber den Grundzustand des Heliumatoms." Superpositions of configurations. 

Hylleraas, E. A., Z. Physik 54, 347, "Neue Berechnung der Energie des Heliums im Grundzustande, sowie des tiefsten Terms von Ortho-Helium/ Explicit inclusion of r12. Scaling procedure. Open shell idea (Is, Is") introduced. 

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, E. A., Svensk Kem. Tidskr. 67, 372, "Recent calculations of the energy values of two-electron atoms."... 

Shull, H., and Lowdin, P.-O., J. Chew. Phys. 23, 1362, "Role of the continuum in superposition of configurations." Criticism of Taylor and Parr (1952). Emphasis of the work by Hylleraas (1928). 

Hylleraas, E. A., and Midtdal, J., Phys. Rev. 103, 829, Ground state energy of two-electron atoms. ... 

Schwartz, H. M., Phys. Rev. 103, 110, "Ground state solution of the non-relativistic equation for He." More rapid convergence in the Ritz variational method by inclusion of half-integral powers in the Hylleraas function. 

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