Boys method

One can notify, that using the method of Boys, the matrix of JV2 elements (representing the first order moment operator) should be transformed [8], This advantageous property is one of the reasons why the Boys method is often used for preparing LMOs. 

The localization procedure developed by Boys <60MI 209-01 > was also used and As,l (Boys) was calculated as above. Values calculated by both methods correlate well with experimental aromaticity criteria although the Boys method provides better correlation. The order of aromaticity as shown by these calculations is thiophene > pyrrole > furan which is not always the case by some of the other methods. 

Boys Method The Minimum Distance Between Electrons Occupying an MO... 

Minimization of the inteielectionic distance (Boys method) is in fact similar in concept to the maximization of the Coulcxnbic inteiacticm of two electrons in the same orbital (Ruedenberg method). 

Calculation of the Edmiston-Ruedenberg energy-localized MOs is very time consuming. Boys (and Foster) proposed a method to find localized MOs that is computationally much faster than the Edmiston-Ruedenberg method and that gives similar results in most cases see D. A. Kleier, J. Chem. Phys., 61, 3905 (1974). The Boys method defines the LMOs as those that maximize the sum of the squares of the distances between the centroids of charge of all pairs of occupied LMOs. The centroid of charge of orbital is defined as the point at (xc,yc>Zc), where Xc = ff>i x tf>i), yc = (I y I ). Zc = 4>i z (t>i) Tjj is, the distance between the centroids of LMOs i and j, the Boys LMOs maximize 2j>, 2, rfj. 

The most appropriate for the extension to crystal appears to be the Boys method. The Boys method [38] minimizes the sum of spreads of the locahzed orbitals (j>i r)... 

Thus, the Boys method can also be considered as maximization of the sum of distances between the orbital centroids I. The modifications of the Boys method are necessary to extend it for the localized crystalline orbitals generation. These modifications are considered in Sect. 3.3... 

Finally, Otto > solved the problem as follows. First, the canonical HF orbitals of the molecules were transformed into localized orbitals using Boys method. > For instance, in the case of molecule B the potential for every localized orbital was computed at points situated on spheres with different radii relative to the center of charge of the considered orbital. The range of the radii was varied from about 1.5 to 15 au, so that the short-, intermediate-, as well as long-range regions were included. Thus one can write... 

Boys S F 1950 Eleetronie wave funetions II. A ealeulation for the ground state of the beryllium atom Proc. R. See. A 201 125-37 Shavitt I 1977 The method of eonfiguration interaetion Modern Theoretical Chemistry vo 3, ed H F III Sehaefer (New York Plenum) pp 189-275... 

A General Method of Calculation for the Stationary States of any Molecular System S. F. Boys... 

There is little experience with the von Niessen method, but for most molecules the remaining three schemes tend to give very similar LMOs. The main exception is systems containing both a- and vr-bonds, such as ethylene. The Pipek-Mezey procedure preserves the cr/yr-separation, while the Edmiston-Ruedenberg and Boys schemes produce bent banana bonds. Similarly, for planar molecules which contain lone pairs (like water), the Pipek-Mezey method produces one in-plane cr-type lone pair and one out-of-plane yr-type lone pair, while the Edmiston-Ruedenberg and Boys schemes produce two equivalent rabbit ear lone pairs. 

Because of the success of the r12 method in the applications, one had almost universally in the literature adopted the idea of the necessity of introducing the interelectronic distances r j explicitly in the total wave function (see, e.g., Coulson 1938). It was there-fore essential for the development that Slater,39 Boys, and some other authors at about 1950 started emphasizing the fact that a wave function of any desired accuracy could be obtained by superposition of configurations, i.e., by summing a series of Slater determinants (Eq. 11.38) built up from a complete basic one-electron set. Numerical applications on atoms and molecules were started by means of the new modern electronic computers, and the results have been very encouraging. It is true that a wave function delivered by the machine may be the sum of a very large number of determinants, but the result may afterwards be mathematically simplified and physically interpreted by means of natural orbitals.22,17... 

For atoms with more than two electrons, it is very difficult to obtain such a small absolute error in the energy as in the helium case, but, within an isoelectronic sequence, the relative error will, of course, go down rapidly with increasing atomic number Z. The method of superposition of configurations has been used successfully in a number of applications, particularly by Boys (1950-) and Jucys (1947-), and, for a more detailed survey of the work on atoms, we will refer to the special table on atomic calculations in the bibliography. This is a field of rapid development, where one can expect important new results within the next few years. 

Boys, S. F., Proc. Roy. Soc. London) A200, 542, Electronic wave function. I. A general method of calculation for the stationary state of any molecular systems." a. 

Bernal, M. J. M., and Boys, S. F., Trans. Roy. Soc. [London) A245, 116, (i) Electronic wave functions. VII. Methods of evalua-ating the fundamental coefficients for the expansion of vector-coupled Schrodinger integrals and some values of these. ... 

For many years, the standard method for poUshing glass in the laboratory has been by rubbing the glass over a sheet of brown paper covered with a mixture of flom emery and a solution of camphor in turpentine. C. V. Boys (1927) has stated that he can find no reason other than tradition for the use of the solution of camphor in turpentine rather than pure turpentine for lubricating grinding media. 

The problem of drilling holes in glass has occupied the attention of scientists since the days of Faraday and many methods have been described. Articles reviewing the subject have been published by C. V. Boys (1927) and more recently by P. Grodzinski (1953). 

Epoi also relies on a local picture as it uses polarizabilities distributed at the Boys LMOs centroids [44] on bonds and lone pairs using a method due to Garmer et al. [35], In this framework, polarizabilities are distributed within a molecular fragment an therefore, the induced dipoles do not need to interact together (like in the Appleq-uist model) within a molecule as their value is only influenced by the electric fields from the others interacting molecules. 

Numerical LMOs of this work are determined by the natural localized-molecular-orbital (NLMO) method A. E. Reed and F. Weinhold, J. Chem. Phys. 83 (1985), 1736. The LMOs determined by other methods (e.g., C. Edmiston and K. Ruedenberg, Rev. Mod. Phys. 34 [1963], 457 and J. M. Foster and S. F. Boys, Rev. Mod. Phys. 32 [1960], 300) are rather similar, and could be taken as equivalent for present purposes. 

In recent years, increasing use has been made of in situ methods in EM—as is true of other techniques of catalyst characterization such as IR, Raman, and NMR spectroscopy, or X-ray diffraction. Although the low mean-free path of electrons prevents EM from being used when model catalysts are exposed to pressures comparable to those prevailing in industrial processes, Gai and Boyes (4) reported early investigations of in situ EM with atomic resolution under controlled reaction conditions to probe the dynamics of catalytic reactions. Direct in situ investigation permits extrapolation to conditions under which practical catalysts operate, as described in Section VIII. 

Unfortunately, extending Hylleraas s approach to systems containing three or more electrons leads to very cumbersome mathematics. More practical approaches, known as explicitly correlated methods, are classified into two categories. The first group of approaches uses Boys Gaussian-type geminal (GTG) functions with the explicit dependence on the interelectronic coordinate built into the exponent [95]... 

A single-determinant wave-function of closed shell molecular systems is invariant against any unitary transformation of the molecular orbitals apart from a phase factor. The transformation can be chosen in order to obtain LMOs. Starting from CMOs a number of localization procedures have been proposed to get LMOs the most commonly used methods are as given by the authors of (Edmiston et ah, 1963) and (Boys, 1966), while the procedures provided by (Pipek etal, 1989) and (Saebo etal., 1993) are also of interest. It could be stated that all the methods yield comparable results. Each LMO densities are found to be relatively concentrated in some spatial region. They are, furthermore, expected to be determined mainly by that part of the molecule which occupies that given region and its nearby environment rather than by the whole system. 

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