Correlated wave functions

Krishnan R, Binkley J S, Seeger R and Popie J A 1980 Self-consistent molecular orbital methods XX. A basis set for correlated wave functions J. Chem. Phys. 72 650-4... 

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). 

A second method is to use a perturbation theory expansion. This is formulated as a sum-over-states algorithm (SOS). This can be done for correlated wave functions and has only a modest CPU time requirement. The random-phase approximation is a time-dependent extension of this method. 

Explicitly correlated wave functions have been shown to give very accurate results. Unfortunately, these calculations are only tractable for very small molecules. 

It is usually observed that the CP correction for methods including electron correlation is larger and more sensitive to the size of the basis set, than that at the HE level. This is in line with the fact that the HE wave function converges much faster with respect to the size of the basis set tlian correlated wave functions. 

We may generalize this by introducing an occupation number (number of electrons), n, for each MO. For a single determinant wave function this will either be 0, 1 or 2, while it may be a fractional number for a correlated wave function (Section 9.5). 

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... 

Taking up the idea of "correlated wave functions containing r12, James and Coolidge (1933) made a careful study of the H2 problem and, after a great deal of numerical work, they obtained finally an energy value in complete agreement with experience. This was another successful test of the validity of the Schrodinger... 

For systems containing three or more electrons very little is so far known about the foundation for the method of correlated wave functions, and research on this problem would be highly desirable. It seems as if one could expect good energy results by means of a wave function being a product of a properly scaled Hartree-Fock function and a correlation factor" containing the interelectronic distances ru (Krisement 1957), but too little is known about the limits of accuracy of such an approach. 

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... 

In the three following sections we will try to sketch the mathematical foundation for the three approaches which are most closely connected with the Hartree-Fock scheme, namely the methods of superposition of configurations (a), correlated wave functions (b), and different orbitals for different spins (c). We will also discuss their main physical implications. 

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... 

He- like ions Z (w)2(J - ) Correlated wave function of type III. 125 Exact... 

Hartree, D. R., The Calculation of Atomic Structure, Wiley and Sons, New York, Chap. 10. Better approximation/ Brief survey of the two ways of improvement—Cl and correlated wave function. 

Hydrogen molecule, carbon oxide intramolecular energy, 110 clathrates, 12, 20 correlated wave function, 300... 

It is apparent that the Hartree-Fock level is characterized by an enormous average deviation from experiment, but standard post-HF methods for including correlation effects such as MP2 and QCISD also err to an extent that renders their results completely useless for this kind of thermochemistry. We should not, however, be overly disturbed by these errors since the use of small basis sets such as 6-31G(d) is a definite no-no for correlated wave function based quantum chemical methods if problems like atomization energies are to be addressed. It suffices to point out the general trend that these methods systematically underestimate the atomization energies due to an incomplete recovery of correlation effects, a... 

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

Figure 3. Comparison of the measured momentum distributions of the outermost valence orbital for wafer [6-8] with spherically averaged orbital densities from Hartree-Fock limit and correlated wave functions [6].

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. 

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