Density fitting approximations

H. J. Werner, F. R. Manby, and P. J. Knowles, Fast linear scaling second order Mpller Plesset perturbation theory (MP2) using local and density fitting approximations. J. Chem. Phys. 118, 8149 8160 (2003). 

Schultz, M., Werner, H-J., Lindh, R., Manby, F. (2004). Analytical energy gradients for local second-order Moller-Plesset perturbation theory using density fitting approximations. /. Chem. Phys. 121,737-750. 

Intermolecular energies and errors are evaluated at estimated CCSD(T)/CBS optimized geometries. All energies and errors in kcal mol . Density fitting approximations applied to all methods except for CCSD(T). Data from Ref. 88. 

Scaling Second-Order Moller-Plesset Perturbation Theory (MP2) Using Local and Density Fitting Approximations. 

Local (truncated) correlation methods, for example, LMP2, have the advantage of superior scaling behavior of the computation time with system size. In addition, they remove to large extend the basis set superposition error (explained below in O section Counterpoise Correction ). Such local methods have been applied to weak intermolecular interactions in combination with density fitting approximations (CoU et al. 2008 Hill and Platts 2008). 

Goll, E., Leininger, T., Manby, F., Mitrushchenkov, A., Werner, H.-J., 8c Stoll, H. (2008). Local and density fitting approximations within the short-range/long-range hybrid scheme Application to large non-bonded complexes. Physical Chemistry Chemical Physics, 10, 3353. 

Polly R, Werner HJ, Manby FR, Knowles PJ (2006) Fast Hattree-Fock theory using local density fitting approximations. Mol Phys 102 2311-2321. doi 10.1080/0026897042000274801... 

In summary, all intermediates can be evaluated from two-electron integrals at most, which renders the F12 methods feasible for molecular systems of any size that can be treated with conventional methods. We note that further reduction of the computational elfort is possible by the use of density-fitting approximations. ... 

The regression constants A, B, and D are determined from the nonlinear regression of available data, while C is usually taken as the critical temperature. The hquid density decreases approximately linearly from the triple point to the normal boiling point and then nonhnearly to the critical density (the reciprocal of the critical volume). A few compounds such as water cannot be fit with this equation over the entire range of temperature. Liquid density data to be regressed should be at atmospheric pressure up to the normal boihng point, above which saturated liquid data should be used. Constants for 1500 compounds are given in the DIPPR compilation. 

Transferred electron density fragments obtained by AFDF method can provide excellent approximations. One such approach, formulated in terms of transferability of fragment density matrices within the AFDF framework is a tool that has been suggested as an approach to macromolecular quantum chemistry [114, 115, 130, 142-146] and to a new density fitting algorithm in the crystallographic structure refinement process [161]. 

Figure 15 gives the superposition of RR (full line) and RY (dotted plot) spectral densities at 300 K. For the RR spectral density, the anharmonic coupling parameter and the direct damping parameter were taken as unity (a0 = 1, y0 = ffioo), in order to get a broadened lineshape involving reasonable half-width (a = 1 was used systematically, for instance, in Ref. 72). For the RY spectral density, the corresponding parameters were chosen aD = 1.29, y00 = 0.85angular frequency shift (the RY model fails to obtain the low-frequency shift predicted by the RR model) and a suitable adjustment in the intensities that are irrelevant in the RR and RY models. 

Werner, H.-J., Manby, F.R. Explicitly correlated second-order perturbation theory using density fitting and local approximations. J. Chem. Phys. 2006, 124, 054114. 

J. P. Perdew, A. Ruzsinszky, J. M. Tao, V. N. Staroverov, G. E. Scuseria, and G. I. Csonka, Prescription for the Design and Selection of Density Functional Approximations More Constraint Satisfaction with Fewer Fits, J. Chem. Phys. 123 (2005), 062201. 

Perdew, J. R Ruzsinszky, A. Tao, J. Staroverov, V. N. Scuseria, G. E. Csonka, G. I. Prescription for the design and selection of density functional approximations more constraint satisfaction with fewer fits, 7. Chem. Phys. 2005,123, 062201. 

Notice that the density of a complete p, d, f, atomic subshell, or an incomplete sub shell in the central field approximation is rotationally invariant [58]. Thus only s-type charge density fitting functions are needed in any atomic central-field calculation. However if the central-field approximation is not invoked then very-high angular momenta are required to fit the density. From a practical point of view it might be better to set off center s-type fitting functions. 

Constans, P. and Carbo, R. (1995). Atomic Shell Approximation Electron Density Fitting Algorithm Restricting Coefficients to Positive Values. JChem.Inf.Comput.Sci., 35,1046-1053. 

Finally, contrary to expectations, the apparent micelle radius decreases (by approximately 10 to 15%) as the polymerization progresses (see Figure 10). This trend was observed for all of the runs shown in Table 1. Not until almost total conversion, when coagulation and phase separation were observed in the view cell, was any increase in the apparent micelle size observed during polymerization. However, the growing polymer molecules could be in a highly collapsed state such that they fit inside a micelle core. (A one-million molecular weight molecule in its bulk state, density of approximately Ig/cc, would require a sphere of approximately 7 to 8 nm in radius). 

Constans P, Carbo R. Atomic shell approximation electron density fitting algorithm restricting coefficients to positive values. J Chem Inf Comput Sci 1995 35 1046-1053. 

Amat L, Carbo-Dorca R. Quantum similarity measures under atomic shell approximation first-order density fitting using elementary Jacobi rotations. J Comput Chem 1997 18 2023-2039. 

A pragmatic approach to developing density functional approximations is to expand the post-LDA correction in a set of suitably chosen parametrized functions and optimize the parameters by training the functional to reproduce certain calibration data as accurately as possible. Of course, fitted parameters appear in many of the previously discussed functionals, but there they were used only to clean up the constmction. In this survey, we call empirical or optimized only those functionals whose design is avowedly empirical. 

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