Gaussian convention

For a.u., the unit of energy is the hartree, not the Rydberg, and the Gaussian convention is used an electromagnetic wave in vacuum has E and H components of equal magnitude. 

Fig. 9. Two-dimensional sketch of the 3N-dimensional configuration space of a protein. Shown are two Cartesian coordinates, xi and X2, as well as two conformational coordinates (ci and C2), which have been derived by principle component analysis of an ensemble ( cloud of dots) generated by a conventional MD simulation, which approximates the configurational space density p in this region of configurational space. The width of the two Gaussians describe the size of the fluctuations along the configurational coordinates and are given by the eigenvalues Ai.

By convention, n must be greater than nn for a system with an od number of electrons. Also, this counting should ignore the core electrons i the molecule (these are treated in step 6). Gaussian will indicate the numbt of electrons of each type. Look for the line containing NOB in the outpt from the single point energy calculation in step 3 ... 

Gaussian 94W menu options and display labels are set in boldface sans-serif type, following the usual Windows convention. 

Due is a certain non-uniqueness in the symmetry assignments in D2h symmetry, some authors, e.g. in ref [9], instead designate the compressed ground state as B2g and the elongated one as ig. In the present paper, we follow the convention set by the Gaussian program... 

As was shown, the conventional method for data reconciliation is that of weighted least squares, in which the adjustments to the data are weighted by the inverse of the measurement noise covariance matrix so that the model constraints are satisfied. The main assumption of the conventional approach is that the errors follow a normal Gaussian distribution. When this assumption is satisfied, conventional approaches provide unbiased estimates of the plant states. The presence of gross errors violates the assumptions in the conventional approach and makes the results invalid. 

Xr from the robust approach, as expected, still gives the correct answer however, the conventional approach fails to provide a good estimate of the process variables. Although the main part of the data distribution is Gaussian, the conventional approach fails in the task because of the presence of just one outlier. In a strict sense, the presence of this outlier results in the invalidation of the statistical basis of data reconciliation,... 

Very similar results were presented by Coffman et al. [2]. They introduced a yet different approach to the prediction of the efficiency of large molecule (proteins) separations on very short columns. Their approach is based on the fact that since short columns yield non-Gaussian effluent distributions, measuring the degree of binary separation using conventional chromatographic resolution is inadequate. Instead, they proposed the fractional purification P, of component z, defined as ... 

A Gaussian curve is characterised by peak position, peak height and peak width. Commonly, the half width is used, i.e. the width at half peak height. We accommodate this convention. In statistics, the Gaussian distribution is usually normalised to unit integral, however, this is not useful in the present context. 

The w-particle one-center correlated Gaussians, (j), can also be expressed in the more conventional form used in the electronic structure calculations as... 

The evaluation of matrix elements for exphcitly correlated Gaussians (46) and (49) can be done in a very elegant and relatively simple way using matrix differential calculus. A systematic description of this very powerful mathematical tool is given in the book by Magnus and Neudecker [105]. The use of matrix differential calculus allows one to obtain compact expressions for matrix elements in the matrix form, which is very suitable for numerical computations [116,118] and perhaps facilitates a new theoretical insight. The present section is written in the spirit of Refs. 116 and 118, following most of the notation conventions therein. Thus, the reader can look for information about some basic ideas presented in these references if needed. 

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