Direct electron correlation methods

P PjV 2 (pp4>q)) [ qj / ) reading/writing to disk, it is desirable also to have direct algorithms for electron correlation method. Direct in this context means that the integrals are calculated as needed and then discarded. The need for integrals over MOs instead of AOs, however, makes the development of direct methods in electron correlation somewhat more complicated than at the HF level. Consider for example the MP2 energy expression." ... 

Linear-scaling electron-correlation methods were developed by 1) combining the Pulay-Saebo local correlation variant [136] with integral-direct techniques [131] and consequently exploiting the spatial locahty of the electron-correlation effect 2) Laplace-transform techniques suggested in [137] and apphed in [129]. 

A is a parameter that can be varied to give the correct amount of ionic character. Another way to view the valence bond picture is that the incorporation of ionic character corrects the overemphasis that the valence bond treatment places on electron correlation. The molecular orbital wavefimction underestimates electron correlation and requires methods such as configuration interaction to correct for it. Although the presence of ionic structures in species such as H2 appears coimterintuitive to many chemists, such species are widely used to explain certain other phenomena such as the ortho/para or meta directing properties of substituted benzene compounds imder electrophilic attack. Moverover, it has been shown that the ionic structures correspond to the deformation of the atomic orbitals when daey are involved in chemical bonds. 

The above is an example of how direct algorithms may be formulated for methods involving electron correlation. It illustrates that it is not as straightforward to apply direct methods at the correlated level as at the SCF level. However, the steady increase in CPU performance, and especially the evolution of multiprocessor machines, favours direct (and semi-direct where some intermediate results are stored on disk) algorithms. Recently direct methods have also been implemented at the coupled cluster level. 

The HF level as usual overestimates the polarity, in this case leading to an incorrect direction of the dipole moment. The MP perturbation series oscillates, and it is clear that the MP4 result is far from converged. The CCSD(T) method apparently recovers the most important part of the electron correlation, as compared to the full CCSDT result. However, even with the aug-cc-pV5Z basis sets, there is still a discrepancy of 0.01 D relative to the experimental value. 

The Hartree-Fock approach derives from the application of a series of well defined approaches to the time independent Schrodinger equation (equation 3), which derives from the postulates of quantum mechanics [27]. The result of these approaches is the iterative resolution of equation 2, presented in the previous subsection, which in this case is solved in an exact way, without the approximations of semiempirical methods. Although this involves a significant increase in computational cost, it has the advantage of not requiring any additional parametrization, and because of this the FIF method can be directly applied to transition metal systems. The lack of electron correlation associated to this method, and its importance in transition metal systems, limits however the validity of the numerical results. 

The ideal calculation would use an infinite basis set and encompass complete incorporation of electron correlation (full configuration interaction). Since this is not feasible in practice, a number of compound methods have been introduced which attempt to approach this limit through additivity and/or extrapolation procedures. Such methods (e.g. G3 [14], CBS-Q [15] and Wl [16]) make it possible to approximate results with a more complete incorporation of electron correlation and a larger basis set than might be accessible from direct calculations. Table 6.1 presents the principal features of a selection of these methods. 

Methods are introduced for generating many-electron Sturmian basis sets using the actual external potential experienced by an N-electron system, i.e. the attractive potential of the nuclei. When such basis sets are employed, very few basis functions are needed for an accurate representation of the system the kinetic energy term disappears from the secular equation solution of the secular equation provides automatically an optimal basis set and a solution to the many-electron problem is found directly, including electron correlation, and without the self-consistent field approximation. In the case of molecules, the momentum-space hyperspherical harmonic methods of Fock, Shibuya and Wulfman are shown to be very well suited to the construction of many-electron Sturmian basis functions. 

In this paper, the main features of the two-step method are presented and PNC calculations are discussed, both those without accounting for correlation effects (PbF and HgF) and those in which electron correlations are taken into account by a combined method of the second-order perturbation theory (PT2) and configuration interaction (Cl), or PT2/CI [100] (for BaF and YbF), by the relativistic coupled cluster (RCC) method [101, 102] (for TIF, PbO, and HI+), and by the spin-orbit direct-CI method [103, 104, 105] (for PbO). In the ab initio calculations discussed here, the best accuracy of any current method has been attained for the hyperfine constants and P,T-odd parameters regarding the molecules containing heavy atoms. 

These properties of the d-shell chromophore (group) prove the necessity of the localized description of d-electrons of transition metal atom in TMCs with explicit account for effects of electron correlations in it. Incidentally, during the time of QC development (more than three quarters of century) there was a period when two directions based on two different approximate descriptions of electronic structure of molecular systems coexisted. This reproduced division of chemistry itself to organic and inorganic and took into account specificity of the molecules related to these classical fields. The organic QC was then limited by the Hiickel method, the elementary version of the HFR MO LCAO method. The description of inorganic compounds — mainly TMCs,— within the QC of that time was based on the crystal field... 

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