Projection weights

Revised versions of the DFT codes developed at Yang s Lab are used. Multicenter integrations are evaluated using Becke s decomposition scheme [59]. Electrostatic potential is computed via Delley s method [60], A simple damping is used to achieve the convergence. All calculations are for fixed geometries only. The bond lengths used are C-H 1.06A C-C 1.36A OC 1.20A. 

For simplicity, the LDA exchange-correlation functional is used. The correlation is the VWN parametrization of Monte Carlo result of Ceperley and Alder for a free electron gas [48,61]. The calculation is not spin-polarized. The purpose here is to show the mechanism of the divide-and-conquer method. While nonlocal corrections to Exc[p] and spin-polarization are instrumental to get results of chemical accuracy, none of these is expected to affect the basic mechanism of the method. 

The projection weights need to be tailored to get a faithful projection of the KS density and energy density for each subsystem. This section investigates several different possibilities for P ( r). The criterion for selecting a particular functional form for P ( r) is the agreement of the computed energy with the KS energy with the same basis set. 

It is easier to work with g ( r ) than with P ( r) (see eq.(18)). There are many classes of (g (r) that can produce a prescribed degree of matchup with the corresponding basis sets. Among possible candidates the following eight types are 

Basis sets affect the performance of P ( r) in a subtle way. In fact, the requirement for (g (r) to give a good matchup is that g (r) decays much faster than the partial molecular density contributed by the basis functions centered at atoms in subsystem a. The bigger the basis sets are the less important the matchup will be. The extreme case will be that all subsystem basis sets are complete. In this case the matchup problem disappears. This is the KS limit solution. Or, all subsystems use a common basis set. The divide-and-conquer method reduces to the conventional KS method. The projection weights do not play any role in this situation. 

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Now define a set of positive-definite projection weights P (r) such that... 

This set of projection weights can be used to project out solutions. Projection weights are previously called partition functions. They are different from projection operators. Projection ojierators are idempotent while projection weights are not. For the current purpose g ( r) will be chosen to be a positive function that decreases faster than the sum of the atomic densities within subsystem a. Hence the projection weights as defined by eq.(18) will be able to project out local properties near subsystem a. Section 4 of this chapter discusses particular choices for g ( r). The projected charge density and energy density are... 

Several remarks are in order about the divide-and-conquer method. First, the matchup of basis set with projection weight in eqs.(19) and (20) is critical for accurately projecting the KS solution. For complete basis sets this is not a problem. 

Whenever the truncated basis sets are used, matching will be important, because the profiles of charge density and energy density are better represented in one region than others in this case. The subsystem orbitals with the lowest eigenvalues are attributed most to the most localized basis functions for the subsystem. The diffuse basis functions contribute much less to these orbitals. If a mismatch between basis set and projection weight occurs, the energy obtained from eqs.(10), (13), (20) and (22) will not be the optimal one. 

Finally, the Fermi energy controls directly the charge transfers between subsystems. It determines which subsystem orbitals to be utilized and also regulates their weights in both p ( r) and fi (r). The projection weights pick up subsystem densities and energy densities according to their profiles in real space. The Fermi function is the only approximation beyond the basis set truncation error in this construction. 

Note that two types of operations are involved in this construction the operations of 0 on the KS potential and the projections of solutions by projection weights. Take, for example, identical cubic boxes of size L3 in space as subsystems and... 

For L — 0 case eq.(34) will be equivalent to the eigenequation for a free noninteracting electron gas. The projection weights will become Dirac delta functions P ( r) = 5(r-ra). Inserting the exact plane wave solutions into eqs.(19) and (21) leads to [56]... 

The usefulness of this divide-and-conquer method is illustrated below by a series of calculations on the model system H-(-C=C-)-H. This system is convenient for testing various choices of projection weights, or basis sets, or Fermi parameter. The tests given here extend previous tests [43,57]. 

The outcome of the divide-and-conquer calculation depends on the definition of the projection weights, the Fermi parameter /3, and the choices of subsystem basis sets. These quantities were chosen as follows unless exceptions are specifically noted. 

It is appropriate to point out here that the density and energy density obtained via eqs.(21) and (22) are continuous across the subsystems since every term in the formulas is continuous. The projection weights have cusps in them, and so do the projected densities. The projected densities do not have the right number of cusps while the total density does. 

Effects of projection weights on total energy and kinetic energy (in hartree) for H-(-C=C-) H. The projection weights are generated from eqs.(18) and (42)-(49). 

Analysis of electron distributions among orbitals for H-(-C=C-)t,H after the self-consistency is achieved. The number of electrons in this molecule is 38. Projection weights are used in all cases. K — number of orbitals picked without Fermi fraction occupation numbers. 

The performance of basis sets also depends on the projection weights used. Using fewer buffer atoms requires faster decaying g ( r ) s. When many buffer atoms are used the sharpness of the g ( r ) s is not expected to be that important. After all, in the limiting case all basis sets include contributions from all atoms in the molecule. The conventional KS results will emerge no matter what form is used for g ( r ) s. 

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