Resolution-of-the-identity technique

Table 1.8 P3 Ionization energies (eV) calculated with and without resolution of the identity techniques...

Hamel S, Casida ME, Salahub DR (2002b) Exchange-only optimized effective potential for molecules from resolution-of-the-identity techniques Comparison with the local density approximation, with and without asymptotic correction, J Chem Phys, 116 8276-8291... 

Now, these equations have been written with formal integration, but of course in the numerical implementation only the evaluation of the integrand at the grid points is required. Therefore, it is evident that the derivatives lyl, (Vpo)M and (Vpp)tyl can be evaluated on the grid independently of the indices pv, and so the four-index problem is decomposed into two independent two-index procedures, avoiding the potential computational bottleneck. By comparison, the resolution of the identity technique proposed by Komornicki and Fitzgerald [66] gives an approximate result in terms of a product of one two-index and two three-index quantities. 

A recent development in ab initio quantum theory has been the introduction of (partially) numerical schemes for dealing with the two-electron integrals in a way that reduces the scaling with the size of the system. One of these is the pseudospectral (PS) [12, 13] technique, which is closely related to another procedure known as resolution of the identity (RI) [14-16]. The use of these schemes in conjunction with the LSA has been discussed in detail elsewhere [17]. Here we present just the basic idea behind the PS approach. 

In recent years there has been a growing interest in numerical techniques which can speed up quantum chemical computations. Various methods are available to approximate the four-index electron repulsion integrals as products of three-index intermediates. These methods are called density fitting (DF) or resolution of the identity (Rl), and Cholesky decomposition (CD) techniques. A general comparison... 

From the above discussion it follows that the calculation of the Coulomb repulsion energy represents the most demanding computational task in Eq. 16.18. The introduction of the variational approximation of the Coulomb potential (Dunlap et al. 1979 Mintmire and Dunlap 1982 Mintmire et al. 1982) reduces the formal scaling of this term to x M, where M is the number of auxiliary functions which is usually three to five times N. This technique is nowadays used in most LCGTO-DFT programs. It is identical to the so-called resolution of the identity (RI) (Flores-Moreno and Ortiz 2009 Hamel et al. 2001 Vahtras et al. 1993) that cropped up in wave function methods, too. The variational approximation of the Coulomb potential, as implemented in deMon2k, is based on the minimization of the following self-interaction term ... 

The resulting operators are products of algebraic functions (square root expressions) of p and are usually not evaluated analytically in the framework of a basis set in the LCAO (linear combination of atomic orbitals) approximation. Rather, a matrix technique employing the approximate resolution of the identity is used to evaluate the operators. ... 

In this form, it is necessary to perform the inversion of a superoperator which is very cumbersome, so an inner p ojection technique (79-81) or fte resolution of the superoperator identity (49),... 

The primary analytical applications of RPLC in the development of biopharmaceuticals are the determination of protein purity and protein identity. Purity is established by analysis of the intact protein, and RPLC is useful in detecting the presence of protein variants, degradation products, and contaminants. Protein identity is most often established by cleavage of the protein with a site-specific protease followed by resolution of the cleavage products by RPLC. This technique, termed peptide mapping, should yield a unique pattern of product peptides for a protein that is homogeneous with respect to primary sequence. 

Recently, a novel method for determining the microstructure of crosslinked polybutadiene in latex using solution 13C-NMR technique was reported [133]. The surfactant and polymer concentrations in the latex were adjusted to give a good signal resolution of the latex sample, as indicated by half-width of the resonance peak at 32.7 ppm. Under these conditions, the S/N ratio was almost identical to that of sample in solution, as shown in Figure 11.31. The microstructure of sol and gel fractions in a radical initiated polybutadiene, determined by this technique, was similar to that of solution measurements. 

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