The Hylleraas function

In 1929, two years after the birth of quantum chemistry, a paper by Hylleraas appeared, where, for the ground state of the helium atom, a trial variational function, containing the interelectronic distance explicitly, was applied. This was a brilliant idea, since it showed that already a small number of terms provide very good results. Even though no fundamental difficulties were encountered for larger atoms, the enormous numerical problems were prohibitive for atoms with larger numbers of electrons. In this case, the progress made from the nineteen twenties to the end of the twentieth century is exemplified by transition from two- to ten-electron systems. 

In this method, we exploit the Hylleraas idea in such a way that the electronic wave function is expressed as a linear combinations of Slater determinants, and in front of each determinant (1,2,3. A ) we insert, next to the variational coefficient Cj, correlational factors with some powers (v, u.) of the interelectronic 

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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. 

Equation (9) can be solved variationally by minimizing the Hylleraas functionals [22,23]... 

The unknown amplitudes (f-amplitudes) in the T operator are determined by minimization of the Hylleraas functional [40] ... 

There are many ways to improve this independent-particle model by incorporating electron correlation in the spatial part Hylleraas function [Hyl29] and the method of configuration interaction (Cl) will be used as illustrations. 

The Hylleraas function, with its improved properties as compared to a Hartree-Fock function, is called a correlated wavefunction, because it takes into account the mutual electron-electron interaction much better, and the motion of electrons beyond a mean-field average is termed correlated motion or the effect of electron correlations. (The definition of electron correlation is used here in the strict terminology. The mean-field average of electron-electron interactions is frequently also called electron correlation.) Comparing equ. (1.20) with equ. (1.16b) one has... 

This result states that the angular momenta ( in the expansion of the Hylleraas function impose the following symmetry properties on the individual angular momenta and (2 attached to the two electrons with spatial directions and r2, respectively ... 

In Paper I of this series [5], the extremal pair functions for the systems He2, Ne, F, HF, H20, NH3 and CH4 were analyzed. We now follow a different line of thought that was also opened in Paper I, namely to use extremal pairs for the construction of correlated wave functions. We have already pointed out that there is a special set of extremal pair functions associated with MP2 (Moller-Plesset perturbation theory of second order). In fact we have shown, that there are two choices for which the Hylleraas functional of MP2 decomposes exactly into a sum of pair contributions. One choice is the conventional one of pairs of canonical spin orbitals, the other one the use of first-order pairs with extremal norm... 

Let us insert (47) into the Hylleraas functional (19) (with H<, the two-electron Fock operator, g = H — H0, and 4>(0) the Hartree-Fock wave function)... 

These equations are expressed in the spin-orbital formalism and the products of orbitals are assumed to be antisymmetrized. The coefficients are the explicitly correlated analogues of the conventional amplitudes. The xy indices refer to the space of geminal replacements which is usually spanned by the occupied orbitals. The operator Q12 in Eq. (21) is the strong orthogonality projector and /12 is the correlation factor. In Eq. (18) the /12 correlation factor was chosen as linear ri2 term. It is not necessary to use it in such form. Recent advances in R12 theory have shown that Slater-type correlation factors, referred here as /12, are advantageous. Depending on the choice of the Ansatz of the wave function, the formula for the projector varies, but the detailed discussion of these issues is postponed until Subsection 4.2. The minimization of the Hylleraas functional... 

We obtain the same equation, if in the Hylleraas functional eq. (5.28), the variational function x s expanded as a linear combination (5.29), and then vary d,- in a similar way to that of the Ritz variational method described on p. 202. 

Since the local orbital basis does not diagonalize the zeroth-order HF Hamiltonian, an iterative procedure is required to determine the amplitudes T -. By minimizing the Hylleraas functional, one obtains the linear equations [130]... 

A hierarchy of functionals exists from which the Rayleigh-Schrodinger eneigies of even orders and wave functions of general orders may be obtained by a variational procedure. In the Hylleraas variation method, the energy of order 2n is calculated by a minimization of the Hylleraas functional of order 2n [3] ... 

At the minimum of the Hylleraas functional (to be constructed below), the variational parameters determine the wave function to order n. In some situations, the Hylleraas method represents a useful alternative to standard perturbation theory, as we shall see in the present subsection. 

The Hylleraas functional is easily related to the variational Lagrangian (14.1.53) of Rayleigh-Schrodinger theory. We recall that the RSPT Lagrangian C ) is symmetric with respect... 

In the process leading to this expression, we have abstained from modifying or eliminating any of the terms that contain C " The reduced Lagrangian (14.1.71) therefore remains variational with respect to C " Based on this observation, we introduce the Hylleraas functional as... 

In conclusion, we may calculate the nth-order Rayleigh-Schrodinger correction to the ground-state wave function by minimizing the Hylleraas functional The minimum of the functional... 

Inserting the approximate vector into the Hylleraas functional, we obtain... 

Thus, if the error in is of order 8, then the error in will be positive and proportional to 8 p provided the energy has been obtained from the Hylleraas functional. If instead we had used the standard Lagrangian, simplified to comply with the 2n - -1 and 2n - - 2 rules, an error of magnitude 8 in the wave function would give rise to an error (positive or negative) proportional to 8 in the energy. 

From this COTrection, calculate the second-ordCT energy correction from the standard expression and from the Hylleraas functional... 

Comparing with the exact second-order energy correction of —0.00723732 obtained from the exact first-order state in Exercise 14.3.3, we find that, whereas the usual expression gives an error of —0.00003088, the Hylleraas fimctional gives an error of only 0.00000013. Note that the Hylleraas functional gives a positive error. 

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