Constrained electron density

Wesolowski, T. A. and J. Weber. 1996. Kohn-Sham equations with constrained electron density an iterative evaluation of the ground-state electron density of interacting molecules. Chem. Phys. Lett. 248,71. 

The label KSCED standing for Kohn-Sham Equations with Constrained Electron Density is used here to indicate that, despite die similarity to Kohn-Sham equations, the effective potential and die one-electron functions differ from die corresponding quantities in these two frameworks. 

To underline these differences, we refer to each of the coupled equations (Eqs. 31 or 32) as Kohn-Sham Equations with Constrained Electron Density (KSCED). For the same reason, the effective potential in these equations is referred to by KSCED effective potentialc. 

Throughout this chapter, equations are written in atomic units. For the sake of simplicity, equations are given for spin-compensated electron densities hence the factor 2 in Eq. (3). The acronym KSCED stands for the Kohn-Sham equations with constrained electron density and is used to distinguish the two effective potentials expressed as density functionals the one in the considered one-electron equations, which involves an additional constraint (see Eq. (5) below), from that in the Kohn-Sham equations. 

This trend could be qualitatively related to the symmetry of frontier orbitals, whose populations differ between the two spin states. To go deeper into the analysis of this phenomenon, we then employed cDFT to impose spin density at the Cn02 core and to investigate the ELF topology of the constrained electronic densities. As shown in Figure 10.5, when the spin density at the CUO2 core increases, LP rotate toward the CuOO plane (+ symbols)—a process associated with the expansion of... 

Casida ME, Wesolowski TA (2003) Generalization of the Kohn-Sham equations with constrained electron density formalism and its time-dependent response theory formulation. Int J Quantum Chem 96 577-588... 

The connection to HF theory has been accomplished in a rather ingenious way by Kohn and Sham (KS) by referring to a fictitious reference system of noninteracting electrons. Such a system is evidently exactly described by a single Slater determinant but, in the KS method, is constrained to share the same electron density with the real interacting system. It is then straightforward to show that the orbitals of the fictitious system fulfil equations that very much resemble the HF equations ... 

Both the energy as well as the one- and two-electron properties of an atom or molecule can be computed from a knowledge of the 2-RDM. To perform a variational optimization of the ground-state energy, we must constrain the 2-RDM to derive from integrating an A -electron density matrix. These necessary yet sufficient constraints are known as A -representability conditions. 

In protein crystallography we assume that all electron density is real, and does not have an imaginary component. In reciprocal space this observation is known as Friedel s law, which states that a structure factor F(h) and its Friedel mate F(—h) have equal amplitudes, but opposite phases. The correspondence of these two assumptions follows straight from Fourier theory and, in consequence, explicitly constraining all electron density to be real is entirely equivalent to introducing Nadditional equalities of... 

As anticipated, the multipolar model is not the only technique available to refine electron density from a set of measured X-ray diffracted intensities. Alternative methods are possible, for example the direct refinement of reduced density matrix elements [73, 74] or even a wave function constrained to X-ray structure factor (XRCW) [75, 76]. Of course, in all these models an increasing amount of physical information is used from theoretical chemistry methods and of course one should carefully consider how experimental is the information obtained. 

The G2 and G3 methods go beyond extrapolation to include small and entirely general empirical corrections associated with the total numbers of paired and unpaired electrons. When sufficient experimental data are available to permit more constrained parameterizations, such empirical corrections can be associated with more specific properties, e.g., with individual bonds. Such bond-specific corrections are employed by the BAG method described in Section 7.7.3. Note that this approach is different from those above insofar as the fundamentally modified quantity is not Feiec, but rather A/7. That is, the goal of the method is to predict improved heats of formation, not to compute more accurate electronic energies, per se. Irikura (2002) has expanded upon this idea by proposing correction schemes that depend not only on types of bonds, but also on their lengths and their electron densities at their midpoints. Such detailed correction schemes can offer very high accuracy, but require extensive sets of high quality experimental data for their formulation. 

Finally, ordered water molecules were added to the model where unexplained electron-density was present in chemically feasible locations for water molecules. Temperature factors for these molecules (treated as oxygen atoms) were allowed to refine individually. If refinement moved these molecules into unrealistic positions or increased their temperature factors excessively, the molecules were deleted from the model. Occupancies were constrained to 1.0 throughout the refinement. This means that B values reflect both thermal motion and disorder (Section II.C). Because all B values fall into a reasonable range, the variation in B can be attributed to thermal motion. Table 8.2 shows the progress of the refinement. 

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