Davidson procedure

Finally, it should be noted that, with minor modification, the standard Davidson procedure is able to produce systematically the terms of the Moller-Plesset perturbation series to any order, provided that a full Cl space is used. This has been used to study the convergence properties of the perturbation series. 

The matrices A and 8 have been defined in Eqs (46) and (47). In the Davidson procedure, the matrix AR and the residual Y are obtained as linear combinations... 

Usually it is sufficient to calculate these elements only once per macroiteration with U = 1, such that the operators G need not be transformed each time U is updated. If the Davidson procedure has converged (i.e. Y is smaller than a certain threshold) the unitary matrix U is updated according to Eq. (44), and a new matrix B is calculated. The process is repeated until Eq. (53) is... 

Very similarly, higher-order processes can be shown to yield a size-consistent redistribution of the intensity of shake-up states among themselves, via multiple 2h-lp/2h-lp interactions. Any restriction on this balance will therefore yield a size-inconsistent description of correlation bands, which will tend to vanish with increasing system size (11). A nice example is provided here, with the necessary introduction of a lower limit on pole strengths in the block-Davidson diagonalization procedure. 

Thomas, R. R, Ure, A. M., Davidson, C. M., Littlejohn, D., Rauret, G., Rubio, R., and Lopez-Sanchez, J. F., Three-Stage Sequential Extraction Procedure for the Determination of Metal in River Sediments, Analytica Chimica Acta 286, 1994, 423 29. 

The procedure for the formation of diaminouracil bisulfite is slightly modified from that of Cain, Mallette, and Taylor,2 which in turn is derived from preparations of Bogert and Davidson,8 and Traube.4 The sulfate salt may be formed in lower yield than the hydrochloride described here by dissolving the bisulfite salt in aqueous base and precipitating with sulfuric acid.8-4 The hydrochloride is appreciably soluble in water, while the sulfate salt is only slightly soluble. 

Introduction of microbiological methods for the determination of amino acids made possible the estimation of the amount of both free and combined amino acids in urine. Dunn et al. (D4), Thompson and Kirby (Tl), Eckhard and Davidson (El), and Woodson et al. (W3) estimated the amount of amino acids liberated in the course of acid or, as in the case of tryptophan determination, alkaline hydrolysis. Microbiological and colorimetric methods used for the determination of certain amino acids present very little opportunity for evaluating the proper quantitative relations between free and combined amino acids, since under the applied condition both combined and free amino acids are equally involved in the reaction. In 1949 Albanese et al. (A3) applied such methods to the quantitative determination of free and combined amino acids in the nondiffusible fraction of urine, and subjected the procedures to broad criticism from just this point of view. 

Davidson CM, Duncan AL, Littlejohn D, Garden LM. A critical evaluation of the three-stage BCR sequential extraction procedure to assess the potential mobility and toxicity of heavy metals in industrially-contaminated land. Anal. Chim. Acta 1998 363 45-55. 

Mossop KF, Davidson CM. Comparison of original and modified BCR sequential extraction procedures for the fractionation of copper, iron, lead, manganese and zinc in soils and sediments. Anal. Chim. Acta 2003 478 111-118. 

There are even more possibilities for optimization procedures if one considers also a partitioning of the optimization problem. Work in this area is currently ongoing, and we present for the first time, in Section 5, results obtained using a Davidson-like strategy for the (linear) optimization of the VB structure coefficients [51, 52]. Further details of this new procedure will be presented in a future publication. 

Step 4 was suggested to me independently by Davidson and Erdahl. It is this step that simplifies the procedure and avoids a difficult variational approach. 

The parallelization of PEDICI mainly concerned itself with the direct Cl [10] part of the program, which genertdly is by far the most time consuming. In the iterative procedure first proposed by Davidson [11], or the many existing variants hereof, a central step consists of the efficient multiplication of the electronic Hamiltonian matrix onto a given trial vector ... 

Accurate measurement of pH is critically dependent on good analytical procedures, a fact that may not be appreciated by laboratory personnel [1,2]. The assumption is often made that if the electrode has been calibrated, there will be no variability in pH between laboratories. The pH measurement can erroneously be seen as merely dipping the electrode into the analyte and recording the value. In 1985, Davidson and Gardner [3] drew the following conclusion from their study Interlaboratory Comparisons of the Determination of pH in Poorly Buffered Fresh Waters ... 

It is not appropriate in this review to discuss the relative efficiency of minimization routines based on steepest descent procedures, but a few of the more important references are given. If the gradient can be evaluated analytically, and as McIver and Komomicki 25) have pointed out this is the case for many surfaces calculated by semi-empirical molecular orbital methods, then this is the best approach. The methods developed by Davidson 26), Fletcher and oweZZ (27), and AfurfagZz and Sargent 28) assume this to be the case. If the gradient has to be obtained numerically then the method developed by Powell 29) has found wide acceptance for its efficiency. 

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