Generalized density-functional theory: Conquering the N -representability problem with exact functionals for the electron pair density and the second-order reduced density matrix

Using the constrained search and Legendre-transform formalisms, one can derive “generalized” density-functional theories, in which the fundamental variable is either the electron pair density or the second-order reduced density matrix. In both approaches, theN-representability problem is solved by the functional, and the variational principle is with respect to all pair densities (density matrices) that are nonnegative and appropriately normalized. The Legendre-transform formulation provides a lower bound on the constrained-search functional. Noting that experience in density-functional and density-matrix theories suggests that it is easier to approximate functionals than it is to approximate the set ofN-representable densities sheds some light on the significance of this work.     

Quantifying the intensity of vehicle impacts using vehicle tracking devices during live training exercises

The operation of off-road vehicles during military training exercises can affect the environmental conditions of training lands by removing or disturbing vegetation. To quantify the impact of vehicle based military training, global positioning system (GPS)-based vehicle tracking systems were used to characterize the movement of vehicles during live training exercises. Methods were developed to spatially estimate the tracking intensity (number of vehicle passes per area) resulting from the training exercises. This method was then combined with previous developed methods that identified off-road trail formation and vehicle dynamic properties to quantify the overall training mission impacts of specific training events on installation resources. This approach to characterizing training impacts results in mission impact profiles that more accurately quantify live training mission impacts.Search radius and output grid size are important parameters of the proposed traffic intensity approximation method. Traffic intensities estimated using a variety of search radii and grid sizes were compared. Results indicated that a 10 m search radius and a 10-by-10 m output grid size worked the best for the study dataset. Approximately, 89% accuracy was found for traffic intensity (number of passes) estimation when using a 10 m search radius and a 10-by-10 m output grid size.     

Assessing reproducibility of diffusion-weighted magnetic resonance imaging studies in a murine model of HER2 + breast cancer

Background and purpose

The use of diffusion-weighted magnetic resonance imaging (DW-MRI) as a surrogate biomarker of response in preclinical studies is increasing. However, before a biomarker can be reliably employed to assess treatment response, the reproducibility of the technique must be established. There is a paucity of literature that quantifies the reproducibility of DW-MRI in preclinical studies; thus, the purpose of this study was to investigate DW-MRI reproducibility in a murine model of HER2 + breast cancer.

Materials and methods

Test–Retest DW-MRI scans separated by approximately six hours were acquired from eleven athymic female mice with HER2 + xenografts using a pulsed gradient spin echo diffusion-weighted sequence with three b values [150, 500, and 800 s/mm²]. Reproducibility was assessed for the mean apparent diffusion coefficient (ADC) from tumor and muscle tissue regions.

Results

The threshold to reflect a change in tumor physiology in a cohort of mice is defined by the 95% confidence interval (CI), which was ± 0.0972 × 10^- 3 mm²/s (± 11.8%) for mean tumor ADC. The repeatability coefficient defines this threshold for an individual mouse, which was ± 0.273 × 10^- 3 mm²/s. The 95% CI and repeatability coefficient for mean ADC of muscle tissue were ± 0.0949 × 10^- 3 mm²/s (± 8.30%) and ± 0.266 × 10^- 3 mm²/s, respectively.

Conclusions

Mean ADC of tumors is reproducible and appropriate for detecting treatment-induced changes on both an individual and mouse cohort basis.     

Reactivity indicators for degenerate states in the density-functional theoretic chemical reactivity theory

Density-functional-theory-based chemical reactivity indicators are formulated for degenerate and near-degenerate ground states. For degenerate states, the functional derivatives of the energy with respect to the external potential do not exist, and must be replaced by the weaker concept of functional variation. The resultant reactivity indicators depend on the specific perturbation. Because it is sometimes impractical to compute reactivity indicators for a specific perturbation, we consider two special cases: point-charge perturbations and Dirac delta function perturbations. The Dirac delta function perturbations provide upper bounds on the chemical reactivity. Reactivity indicators using the common used "average of degenerate states approximation" for degenerate states provide a lower bound on the chemical reactivity. Unfortunately, this lower bound is often extremely weak. Approximate formulas for the reactivity indicators within the frontier-molecular-orbital approximation and special cases (two or three degenerate spatial orbitals) are presented in the supplementary material. One remarkable feature that arises in the frontier molecular orbital approximation, and presumably also in the exact theory, is that removing electrons sometimes causes the electron density to increase at the location of a negative (attractive) Dirac delta function perturbation. That is, the energetic response to a reduction in the external potential can increase even when the number of electrons decreases.     

Do the Local Softness and Hardness Indicate the Softest and Hardest Regions of a Molecule?

In this work, we will show that the largest values of the local softness and hardness do not necessarily correspond to the softest and hardest regions of the molecule, respectively. Based on our results, we will argue that it is more useful to interpret the local softness and the local hardness as functions that measure the "local abundance" or "concentration" of the corresponding global properties. This new point of view helps reveal how and when these local reactivity indices are most useful.     

Beyond electronegativity and local hardness: Higher-order equalization criteria for determination of a ground-state electron density

Higher-order global softnesses, local softnesses, and softness kernels are defined along with their hardness inverses. The local hardness equalization principle recently derived by the authors is extended to arbitrary order. The resulting hierarchy of equalization principles indicates that the electronegativity/chemical potential, local hardness, and local hyperhardnesses all are constant when evaluated for the ground-state electron density. The new equalization principles can be used to test whether a trial electron density is an accurate approximation to the true ground-state density and to discover molecules with desired reactive properties, as encapsulated by their chemical reactivity indicators.