Correlation wave function, exact

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

Many methods in chemistry for the correlation energy are based on a form of perturbation theory, but the positivity conditions are quite different. Traditional perturbation theory performs accurately for all kinds of two-particle reduced Hamiltonians, which are close enough to a mean-field (Hartree-Fock) reference. There are a myriad of chemical systems, however, where the correlated wave-function (or 2-RDM) is not sufficiently close to a statistical mean field. Different from perturbation theory, the positivity conditions function by increasing the number of extreme two-particle Hamiltonians in which are employed as constraints upon the 2-RDM in Eq. (50) and, hence, they exactly treat a certain convex set of reduced Hamiltonians to all orders of perturbation theory. For the... 

While any approximate wave function O that consists of Slater determinants violates this cusp condition, the product (1 + (1 /2)ri2)4> may satisfy it exactly. Inspired by this knowledge and by the fact that a reference wave function (O0) still dominates in the exact correlated wave function, Kutzelnigg has suggested... 

The assessment of the accuracy of fully correlated wavefunctions by means of variance methods requires computation which only varies as N, a lower power of N, the number of electrons, than had been expected. This seems to be dependent on an indirect approach first constructed for a trans correlated method. This means that various different variance tests could be used for the assessment of the accuracy of wavefunctions calculated by the transcorrelated method developed by Handy and Boys. These would require much less equipment in programmes and computer facilities, th the original calculations of such wavefunctions. Supplementary investigations on correlated wave-functions at this level might make possible a whole variety of informative experiments on very exact wavefunctions and energies. 

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... 

The exact exchange-correlation potential for light atoms has been computed from exact electron densities provided by CIs and quantum MC computations or other high-quality correlated wave functions. See, for instance. Ref. [41](b), or C. J. Umrigar and X. Gonze, Phys. Rev. A 50, 3827 (1994). 

In this system, we have N electrons, which also interact by Coulombic forces among themselves. All these interactions produce the ground-state electronic density distribution po (ideal i.e., that we obtain from the exact, 100% correlated wave function). Now let us consider... 

Finally we can construct in principle, a set of two-electron continuum states which have an incident boundary condition that two electrons are scattering from the nucleus. We can now make a crucial statement. The exact solution of the correlated two-electron wave function at a given total energy will in general be expressible as some linear combination of the channel states. As the best that we can hope for, when we diagonalize the two-electron problem to obtain the set that these pseudostates approximately represent the projection of the exact correlated wave function at energy %, we must assume that Xm is not pure. Its character is neither that of any single ionization channel, nor a double ionization channel. How then are we to proceed ... 

Cl calculations can be used to improve the quality of the wave-function and state energies. Self-consistent field (SCF) level calculations are based on the one-electron model, wherein each electron moves in the average field created by the other n-1 electrons in the molecule. Actually, electrons interact instantaneously and therefore have a natural tendency to avoid each other beyond the requirements of the Exclusion Principle. This correlation results in a lower average interelectronic repulsion and thus a lower state energy. The difference between electronic energies calculated at the SCF level versus the exact nonrelativistic energies is the correlation energy. 

RHF to UHF, or to a TCSCF, is almost pure static correlation. Increasing the number of configurations in an MCSCF will recover more and more of the dynamical correlation, until at the full Cl limit, the correlation treatment is exact. As mentioned above, MCSCF methods are mainly used for generating a qualitatively correct wave function, i.e. recovering the static part of the correlation. 

In other words, the exact wave function behaves asymptotically as a constant 4- l/2ri2 when ri2 is small. It would therefore seem natural that the interelectronic distance would be a necessary variable for describing electron correlation. For two-electron systems, extremely accurate wave functions may be generated by taking a trial wave function consisting of an orbital product times an expansion in electron coordinates such as... 

The problem of finding the best approximation of this type and the best one-electron set y2t. . ., y>N is handled in the Hartree-Fock scheme. Of course, a total wave function of the same type as Eq. 11.38 can never be an exact solution to the Schrodinger equation, and the error depends on the fact that the two-electron operator (Eq. 11.39) cannot be exactly replaced by a sum of one-particle operators. Physically we have neglected the effect of the "Coulomb hole" around each electron, but the results in Section II.C(2) show that the main error is connected with the neglect of the Coulomb correlation between electrons with opposite spins. 

The idea of constructing a good wave function of a many-particle system by means of an exact treatment of the two-particle correlation is also underlying the methods recently developed by Brueck-ner and his collaborators for studying nuclei and free-electron systems. The effective two-particle reaction operator and the self-consistency conditions introduced in this connection may be considered as generalizations of the Hartree-Fock scheme. 

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