Gaussian-1 method

In hybrid DET-Gaussian methods, a Gaussian basis set is used to obtain the best approximation to the three classical or one-election parts of the Schroedinger equation for molecules and DET is used to calculate the election correlation. The Gaussian parts of the calculation are carried out at the restiicted Hartiee-Fock level, for example 6-31G or 6-31 lG(3d,2p), and the DFT part of the calculation is by the B3LYP approximation. Numerous other hybrid methods are currently in use. 

The net result of all these operations is that in place of the system Ax = h, the system CAx = Ch is obtained, where CA is an upper triangular matrix. Such a matrix is easfiy inverted (the inverse will be exhibited below), and the triangular system is even more easily solved. With this explanation of the gaussian method, the basic theory of this and related methods will now be developed. 

PROC NLIN METHOD=GAUSS / Gaussian method is default. / PARAMETERS RMAX=0.5 KM=1 / Starting estimates of parameters. / MODEL R=RMAX CS/(KM+CS) / dependent=expression /... 

Note, as with Gaussian methods, the semiempirical aspect of CBS methods. 

The second advantage of the DV-Xa method is that the basis functions of the DV-Xa method are atomic orbitals. Thus the number of nodes is exact as shown in Fig. 4, where Si 2s GTO in GAUSSIAN method is compared with the numerical basis function used in the DV-Xa method [17]. The use of the atomic orbital wavefunction makes it possible to perform the direct calculation of the electric dipole matrix elements, e.g. , using the DV-Xa MO, yielding better result than when using a GTO basis MO. 

Much has been said (see chapters by Tukey, Andrews, and Nicholson, this volume) concerning the Gaussian method of least squares (characterized repeatedly during discussion as the method of ill repute ) and possible modifications or replacements of it. Nevertheless, it is by far the most frequently used method for crystal structure refinement and will continue to be for some time to come. 

PROC NLIN METHOD=GAUSS / Gaussian method is default. / PARAMETERS RMAX=0.5 KM=1 / Starting estimates of parameters. / MODEL R=RMAX CS/(KM+CS) / dependent=expression / DER.RMAX=CS/(KM+CS) / Partial derivatives of the model / DER.KM=-RMAX CS/ ((KM+CS) (KM+CS)) / with respect to each parameter. /... 

Solve the following set of equations using the Gaussian method. 

One of the most important reasons for the impact of Becke s papers was his decision to assess the DFT methods systematically on the same set of test molecules assembled by Pople and coworkers in their development of the Gaussian- methods [15-18]. The successful G1 [15, 16] and G2 [17] methods were the first two methods developed for the prediction of the thermochemistry of small molecules within chemical accuracy ( 2kcal/ mol). In order to assess the performance of these theoretical methods, Pople and coworkers also developed a set of test molecules for which accurate experimental thermochemical information is known. Becke adopted the initial G1 test set containing 55 neutral molecules as the standard test set for the development and assessment of new density functionals. This immediately provided a mechanism for an unbiased comparison of the performance of DFT methods with the state-of-the-art ab initio quantum chemical techniques for the same set of molecules. 

The number of stoichiometrically independent reactions is given by the rank of the matrix Rp, which can be determined with e.g. the aid of the Gaussian method of elimination. As a result, the stoichiometrical coefficients of linearly independent equations for the reaction system are necessary and sufficient for, for example, calculation of the conversion of the key variables and therefore also for all other components. Thus... 

Fig. 11.4 Fitting curves by bi-Gaussian method of Cao.625[Alo.75Sii.25JN2(No.850o.i5) 0.02Eu. The inset figures show EuN 2N 3 and EuN 2N 20 molds corresponding to PI (1.910 eV) and P2 (2.115 eV), respectively. Reprinted from Ref. [27], copyright 2013, with permission from Elsevier...

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