Electronic density derivatives

The basis functions are normally the same as used in wave mechanics for expanding the HF orbitals, see Chapter 5 for details. Although there is no guarantee that the exponents and contraction coefficients determined by the variational procedure for wave functions are also optimum for DFT orbitals, the difference is presumably small since the electron densities derived by both methods are very similar. ... 

Breneman, C., Bennett, Bi, J., Song, M., and Embrechts, M. (2002) New electron density-derived descriptors and machine learning techniques for computational ADME and molecular design. MidAtlantic Computational Chemistry Meeting, Princeton University, Princeton, NJ. 

Special care must be taken with the term involving py, i.e. the electronic density derivatives such quantities are obtained, as in the gas phase, through a so-called coupled perturbed SCF procedure, in which the Schrodinger equation is iteratively solved for any... 

Many attempts have been made to quantify SIMS data by using theoretical models of the ionization process. One of the early ones was the local thermal equilibrium model of Andersen and Hinthome [36-38] mentioned in the Introduction. The hypothesis for this model states that the majority of sputtered ions, atoms, molecules, and electrons are in thermal equilibrium with each other and that these equilibrium concentrations can be calculated by using the proper Saha equations. Andersen and Hinthome developed a computer model, C ARISMA, to quantify SIMS data, using these assumptions and the Saha-Eggert ionization equation [39-41]. They reported results within 10% error for most elements with the use of oxygen bombardment on mineralogical samples. Some elements such as zirconium, niobium, and molybdenum, however, were underestimated by factors of 2 to 6. With two internal standards, CARISMA calculated a plasma temperature and electron density to be used in the ionization equation. For similar matrices, temperature and pressure could be entered and the ion intensities quantified without standards. Subsequent research has shown that the temperature and electron densities derived by this method were not realistic and the establishment of a true thermal equilibrium is unlikely under SIMS ion bombardment. With too many failures in other matrices, the method has fallen into disuse. 

FIGURE 10.17 Stereodiagram of a segment of helical double-stranded RNA, seven base pairs in length, is shown superimposed on its corresponding difference electron density derived from crystalline satelhte tobacco mosaic virus. 

In the following, we will denote the electron densities derived from Eqs. 31-32 by pESCED and pBSCED. Both of them are obviously noninteracting pure-state v-representable. Eqs. 31-32 can lead to exact ground-state electron density p0 only if it can be constructed as a sum of two noninteracting pure-state v-representable components po = Pa + Pb- Due to this observation, four possible cases for a given choice of NA and NB such that N = NA + NB are possible ... 

Let s denote the electron density derived from Eq. 31 for a given p a( by p m. The electron density p m is obviously noninteracting pure-state v-representable. Therefore, if po — p%%aX does not belong to the class of noninteracting pure-state v-representable electron densities, it cannot be obtained from Eq. 31. As a consequence... 

The fact that the exact ground-state electron density can be obtained as a sum of two components a trial electron density belonging to the class of functions, such that Px = Po — p1 0,1 is noninteracting pure-state v-representable, and the electron density derived from Eq. 31 provides a formal basis for the electron-density-partitioning based strategy in multi-level modelling. If subsystem A represents a small part of the... 

Pa and pB correspond to the monomers forming the complex. The pa,Pb overlap is small but not negligible in this model system. Moreover, forming a complex affects the electron density of each monomer. Especially, electron density of H2 becomes polarized due to the large dipole moment of NCH. Zeroth-order GEA, leads to a reasonably good approximation to — sp PB as indicated by the similar qualitative behavior of the electron density derived from the reference calculations (Kohn-Sham - not depending on approximations to T ad pA, Pb]) and Eqs. 31-32. The second-order contribution to T ad pA, Pb worsens qualitatively the results. 

The accuracy of the energies derived from Eq. 23 using electron densities derived from Eqs 31-32 depends on the accuracy of two approximate quantities STs p PB and T ad[pA, Pb - The discrepancy between the KSCED energy obtained with a given approximation for T ad[pA, pb] and the corresponding Kohn-Sham reference data can, however, be used to select the most appropriate approximation for applications concerning primarily interaction energies. The smaller are these deviations the better is the approximation. 

Other pharmacological activities have also been correlated with quantum-chemically derived descriptors. For instance, the quantitative structure-activity relationship developed for the antibacterial activity of a series of monocyclic (i-lactam antibiotics included the atomic charges, the bond orders, the dipole moment, and the first excitation energy of the compound [103]. The fungicidal activity of A3-l,2,4-thiadiazolines has been correlated with an index of frontier orbital electron density derived from semi-empirical PM3 molecular orbital calculations [104],... 

For example, consider the Nephelauxetie Effeet. Analysis of the interelectron repulsion parameters derived from analyzing the d-d speetrum invariably leads to lower values than in the free ion. The interpretation is that, in the eomplex, the d eleetrons are, on average, further apart which is consistent with expanded rf-funetions in the eomplex and/or with d electron delocalization onto the ligands. Analysis of the electron density distribution from X-ray diffraction in trans-[Ni(NH3)4(N02)2] yields a rf-orbital radius larger than that for free Ni. However, the unpaired electron density derived from polarized neutron diffraction (PND) data yields a if-orbital radius less than for free Ni " prompting Figgis to propose an anti-Nephelauxetic effect. DFT calculations support LFT in that the d-orbitals expand upon complex formation but also provide an explanation of the diffraction data. 

Figure 15 The HIVrt tivirapine, with its electronic van der Waals surface encoded with three electron density-derived properties and their respective property distributions.

Figure 16 WCDs are generated as illustrated for each electron density-derived property. The property distribution is deconstructed using the DWT (pyramid algorithm), allowing the isolation of the lowest frequency and coarsest approximation coefficients (ay and dy). These few coefficients are sufficient to reconstruct most of the original signal (via the inverse DWT) and contain the vital molecular property information needed for modeling. The WCDs replace the original TAE histogram descriptors and are orthogonal, consistent, and representative.

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