Correlation functions trial wavefunction calculations

Though not discussed above, in all the studies mentioned the trial wavef unctions included pair correlation functions. J j. as prescribed by Reynolds et al. ( ). Moskowitz et al. (48.49) have shown that the product of a relatively simple multiconfiguration wavefunction with pair correlation functions can provide a rather accurate approximation to the exact wavefunction. In our calculations and in those of Hammond et al. (59) the many-electron local potential, has been obtained by allowing the REP to... 

This exercise involves a study of two electron systems ranging from H to Be2+. Five different trial wavefunctions are used to describe the behavior of the electrons beginning with a scaled hydrogen Is orbital. Various correlated wave-functions are studied next and the improvement in the agreement between theory and experiment is noted.i" > " " i i These calculations support lecture material on the orbital approximation and the early attempts to introduce electron correlation into the trial wavefunction. This laboratory is done either with QuickBASIC programs or Mathematica. 

The ideal trial wavefunction is simple and compact, has simple easily evaluated first and second derivatives, and is accurate everywhere. Because the local energy must be evaluated repeatedly, the computation effort required for the derivatives makes up a large part of the overall computation effort for many systems. The typical trial wavefunctions of analytic variational calculations are not often useful, since they are severely restricted in form by the requirement that they be amenable to analytic integrations. The QMC functions are essentially unrestricted in form, since no analytic integrations are required. First and second derivatives of trial wavefunctions are needed, but differentiation is in general much easier than integration, and most useful trial wavefunctions have reasonably simple analytical derivatives. In most analytic variational calculations to date, it has not been possible to include the interelectron distances r,y in the trial wavefunction, and these wavefunctions are not usually explicitly correlated, whereas for QMC calculations of all types, explicitly correlated functions containing are the norm. 

The typical trial wavefunction for QMC calculations on molecular systems consists of the product of a Slater determinant multiplied by a second function, which accounts to some extent for electron correlation with use of interelectron distances. The trial wavefunctions are most often taken from relatively simple analytic variational calculations, in most cases from calculations at the SCF level. Thus, for the 10-electron system methane," the trial function may be the product of the SCF function, which is a 10 x 10 determinant made up of two 5x5 determinants, and a Jastrow function for each pair of electrons. 

A correlated sampling method, known as reweighting [39,40] is much more efficient. One samples a set of configurations Rj (usually several thousand points at least) according to some distribution function, usually taken to be the square of the wavefunction for some initial trial function ] f T(-R q o) -Then, the variational energy (or the variance) for trial function nearby in parameter space can be calculated by using the same set of points ... 

The electron affinity, which is very small for the Fe atom (0.15 eV), has so far not been reliably calculated. However, even the essentially zero affinity obtained is a tremendous improvement from the uncorrelated value of -2.36 eV. One of the reasons for the small remaining errors is that only simple trial functions were used. In particular, the determinants were constructed from Hartree-Fock orbitals. It is known that the Hartree-Fock wavefunction is usually more accurate for the neutral atom than for negative ion, and we conjecture that the unequal quality of the nodes could have created a bias on the order of the electron affinity, especially when the valence correlation energy is more than 20 eV. One can expect more accurate calculations with improved trial functions, algorithms, and pseudopotentials. 

---