Electron densities calculated

Electron density represents the probability of finding an electron at a poin t in space. It is calcii lated from th e elements of th e den sity matrix. The total electron density is the sum of the densities for alpha and beta electrons. In a closed-shell RUE calculation, electron densities are the same for alpha and beta electrons. 

We have the makings of an iterative computer method. Start by assuming values for the matr ix elements and calculate electron densities (charge densities and bond orders). Modify the matr ix elements according to the results of the electron density calculations, rediagonalize using the new matrix elements to get new densities, and so on. When the results of one iteration are not different from those of the last by more than some specified small amount, the results are self-consistent. 

Having filled in all the elements of the F matr ix, we use an iterative diagonaliza-tion procedure to obtain the eigenvalues by the Jacobi method (Chapter 6) or its equivalent. Initially, the requisite electron densities are not known. They must be given arbitrary values at the start, usually taken from a Huckel calculation. Electron densities are improved as the iterations proceed. Note that the entire diagonalization is carried out many times in a typical problem, and that many iterative matrix multiplications are carried out in each diagonalization. Jensen (1999) refers to an iterative procedure that contains an iterative procedure within it as a macroiteration. The term is descriptive and we shall use it from time to time. 

Pugmire etal. have published calculated electron densities for pyrazine (68JA697), quinoxaline (69JA6381) and phenazine and the calculated total electron densities a + v) are shown in (10), (11) and (12). 

The precise numerical values of the calculated electron densities are unimportant, as the most important feature is the relative electron density thus, the electron density at the pyrazine carbon atom is similar to that at an a-position in pyridine and this is manifest in the comparable reactivities of these positions in the two rings. In the case of quinoxaline, electron densities at N-1 and C-2 are proportionately lower, with the highest electron density appearing at position 5(8), which is in line with the observation that electrophilic substitution occurs at this position. 

Pyrazine and its derivatives have been extensively studied by proton and NMR spectroscopy and conflicting reports on the reliability of additivity rules and/or correlation of chemical shifts with calculated electron densities have appeared. 

Table 2 Calculated Electron Densities of Pyrazole, its Anion and its Cation...

Figure 1.2 A schematic view of an atom. The dense, positively charged nucleus contains most of the atom s mass and is surrounded by negatively charged electrons. The three-dimensional view on the right shows calculated electron-density surfaces. Electron density increases steadily toward the nucleus and is 40 times greater at the blue solid surface than at the gray mesh surface.

Table 11. X-ray reflection data for -y-brass (CvuZna) and calculated electron densities.

What is then missing are merely the numerical values of the constants a and b (c is of no physical relevance). These values are best obtained from a linear regression analysis of calculated electron densities versus observed isomer shifts. Important early work along these lines was reported by Nieuwport who has already demonstrated the potential of this approach [16]. More recently, several workers have... 

Fig. 5.1 Linear regression analysis between calculated electron densities at the iron nucleus and measured isomer shifts for a collection of iron complexes. (From [19])...

A correlation of isomer shift, electronic configuration, and calculated -electron densities for a number of ruthenium complexes in analogy to the Walker-Wertheim-Jaccarino diagram for iron compounds has been reported by Clausen et al. [ 127]. Also useful is the correlation between isomer shift and electronegativity as communicated by Clausen et al. [128] for ruthenium trihalides where the isomer shift appears to increase with increasing Mulliken electronegativity. 

Fig. 7.92 Mercury compounds arranged according to increasing isomer shift values. Using the calculated electron densities, the isomer shifts expected for free Hg - and Hg -ions are shown. The experimental error is given by the bars at the right side (from [485])...

Borman, S. (1995) MEDLA Technique Calculates Electron Densities, Chem. Eng. News, 73, 29. 

Figure 6.7 shows the calculated electron density distributions for the H2 and N2 molecules in their equilibrium geometry together with the standard deformation densities. There is clearly a buildup of electron density in the bonding region in both molecules. In the N2 molecule there is also an increase in the electron density in the lone pair region and a de-... 

Further doubt about the validity of the original Grant-Cheney model was expressed by Seidman and Maciel (185), whose INDO calculations of proximity effects in hydrocarbons revealed that there is no simple correlation between carbon chemical shifts and calculated electron-density increases caused by steric C-H bond polarization they report the conformational relation of interacting bonds and groups to be at least equally important, if not more so (185). 

FMO theory according to calculated electron densities at different carbon atoms of the HOMO of the radical cation (Hiickel and MNDO programmes). 

Table VII. Calculated Electron Densities of Various Differently p-Substituted 2-phenyl-pyridazinones...

The authors are grateful to Drs. Becker, Hansen, Parg, and Zeeh, Central Laboratory of BASF Aktiengesellschaft, Ludwigshafen, Germany, for efficient cooperation and synthesis of the chemical compounds The authors particularly are indebted to Dr. Feichtmayr for calculating electron densities and helpful discussion, and to Dr. Haedicke for doing X-ray structural analysis. 

Compare the calculated electron density, nKs(r), with the electron density used in solving the Kohn-Sham equations, (r). If the two densities are the same, then this is the ground-state electron density, and it can be used to compute the total energy. If the two densities are different, then the trial electron density must be updated in some way. Once this is done, the process begins again from step 2. 

Be(OH)2 were the same as that in other compounds, they would all follow the same horizontal trend within error. Figure l-4b shows that ln( Mo) + 0.012977t versus t follows different trend for different compounds, indicating that the decay constant of Be depends on the kind of compound Be is in. The variation of the decay constant amounts to about 1.5%. Tossell (2002) raised doubt about the experimental results based on calculated electron density of Be in various compounds, but the reliability of the theoretical calculation has not been verified. 

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