Fitness density

Nevertheless, the formal A/ scaling has spawned approaches which reduce the dependence to A/. This may be achieved by fitting the electron density to a linear combination of functions, and using the fitted density in evaluating the J integrals in the Coulomb term. 

The density fitting functions may or may not be the same as those used in expanding the orbitals. The fitting constants a are chosen so that the Coulomb energy arising from the difference between the exact and fitted densities is minimized, subject to the constraint of charge conservation. The J integrals then become... 

The latter approach has the advantage that the exact J is approached strictly from above, however for technical reasons it is only applicable if Gaussian basis functions are employed (Dunlap, Connolly, and Sabin, 1979). Both schemes are of course subject to the constraint that the fitted density is normalized to the total number of electrons, i. e ... 

The coefficients defining the fitted density are obtained via equation (7-27). To avoid the difficulties of dealing with two-electron integrals in a Slater-type basis, is evaluated in this context by a numerical integration on a grid as... 

The efficiency of the methods outlined above has been tested by calculating the intermolecular Coulomb energies and forces for a series of water boxes (64,128,256, 512 and 1024) under periodic boundary conditions [15, 62], The electron density of each monomer is expanded on five sites (atomic positions and bond mid-points) using two standard ABSs, A2 and PI.These sets were used to fit QM density of a single water molecule obtained at the B3LYP/6-31G level. We have previously shown that the A1 fitted density has an 8% RMS force error with respect to the corresponding ab initio results. In the case of PI, this error is reduced to around 2% [15, 16], Table 6-1 shows the results for the 5 water boxes using both ABSs (Table 6-7). 

With this approximation, the evaluation of the Coulomb term scales as N2M, in contrast to the standard way, which scales as N4 (N and M are the number of primitive functions in the orbital and density basis sets, respectively). The expansion coefficients of the electronic density in Eq. (8) are chosen such as to minimize the error in the Coulomb term arising from the difference between the real density and the fitted density [25],... 

T. Korona, B. Jeziorski, Dispersion energy from denisty-fitted density susceptibilities of singles and doubles coupled cluster theory. J. Chem. Phys. 128, 144107 (2008)... 

Note that the added term, which involves the fit twice, is in that sense less accurate than Sambe and Felton s original term, which involves the fit only once. The point is, however, that if the fitted density is expressed as the exact density minus an error term, then the error in Eq. (5) is the Coulomb repulsion of the error with itself, which is always second order and positive [12]. This approach is readily extended to four-center Coulomb integrals [13],... 

Analysis of Peak Areas in the Fitted Density Profiles for 43.6 A n-Propylammonium Vermiculite Gels... 

Here P denotes the density matrix, h contains kinetic energy and electron-nuclear attraction operators [pqllr] and [rllt] are Coulomb repulsion integrals with 3 and 2 indices, respectively, [pqs] denote one-electron 3-index integrals and is the nuclear-nuclear repulsion term. The form of Equation 4 ensures (11b) that the Coulomb energies are accurate up to second order in the difference between the fitted density and the "exact" density obtained directly from the wavefunctioa... 

As shown in Tables 19.13 and 19.14, the fitting density varies with line size for the process area piping. The densities reflect actual experience on several projects and coincide with those used by others. 

The comprehensive piping units in Tables 19.21 and 19.22 were developed with the basic units in Table 19.20 and the fitting density used for the piping comprehensive units (Table 19.13) applying the linear-feet-per-fitting equivalence normally used in the trade ... 

Fitting Densities to Linear Cmnlnnations of Gaussian Functions... 

Within this fitting procedure, if Af is taken as the number of Gaussian functions used in the expansion of Eq. (4) and n is the number of basis functions used to expand the wavefunction, evaluation of the Similarity using a fitted density results in an Af depending proc as opposed to an n -depending process when using the exact density. 

The SCF wavefunctions from which the electron density is fitted was calculated by means of the GAUSSIAN-90 system of programs [18]. The program QMOLSIM [3] used in the computation of the MQSM allows optimization of the mutual orientation of the two systems studied in order to maximize their similarity by the common steepest-descent, Newton and quasi-Newton algorithms [19]. The DIIS procedure [20] has been also implemented for the steepest-descent optimizations in order to improve the performance of this method. The MQSM used in the optimization procedure are obtained from fitted densities. This speeds the process. The exact MQSM were obtained from the molecular orientation obtained in this optimization procedure. 

The goal of the discussion in the next sections is to show several examples of how fitted densities can be used in order to study large systems where their size prevents the calculation of exact MQSM. In the first section, a study of molecular properties ordering is discussed. Further, calculations of MQSM obtained from fitted densities are applied to the prediction of the activity for a series of metal-substituted enzyme models. Finally, the use of these MQSM as an interpretative tool in chemical reactivity is discussed. 

Second, fitted densities do not automatically conserve the number of electrons in the system. Though it is straightforward to enforce charge conservation via a Lagrange constraint, which is usually done, the fitted density... 

The approach works well but not perfectly. It certainly does not guarantee the absence of negative fitted density. In a crystal this usually does little harm but in a UTF or slab calculation, it can be a real problem. The issue is the work function (—Ef). In a UTF it can be energetically favorable to put a very small amount of negative density very far out in the most diffuse fitting function. The result is a dipole reversed fi om the physical situation which ruins the calculated work function. We monitor calculations and when this behavior occurs (infrequently), we eliminate it by careful study of computed output to see how to improve the Q-fitting basis. 

The accurate density of reactive states O + H2, J = 0 is shown in the top left panel of Fig. 5, and results of the quantized transition state theory fit are in Table 6, along with assignments discussed below. The quantal and fitted densities are indistinguishable to plotting accuracy (14), indicating that quantized transition states control the chemical reactivity. The density closely resembles that for the reaction of H with H2 up to about 1.3 eV. Analogous features are associated with the same sets of quantum numbers through the [06°] transition state at 1.218 ev. 

The three-center integrals [mn k]Erei in the definition of the DKH Hamiltonian, variationaUy obtained from the energy expression, are closely related to the three-center integrals [mnjjfelvrei defining the transformed Hartree potential of the fitted density. Indeed, analysis of the Hartree potential dependent terms in the Hamiltonians h oLeia > Eqs. (23), and > Eq. (24), establishes the equiva-... 

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