Independent Pairs Approximation

The IRT method was applied initially to the kinetics of isolated spurs. Such calculations were used to test the model and the validity of the independent pairs approximation upon which the technique is based. When applied to real radiation chemical systems, isolated spur calculations were found to predict physically unrealistic radii for the spurs, demonstrating that the concept of a distribution of isolated spurs is physically inappropriate [59]. Application of the IRT methodology to realistic electron radiation track structures has now been reported by several research groups [60-64], and the excellent agreement found between experimental data for scavenger and time-dependent yields and the predictions of IRT simulation shows that the important input parameter in determining the chemical kinetics is the initial configuration of the reactants, i.e., the use of a realistic radiation track structure. 

In order to study the influence of electron concentration on the observed dynamics, we performed experiments with different laser power densities. As an illustration, the transient absorption signals recorded at 715 nm in ethylene glycol upon photoionisation of the solvent at 263 nm with three different laser power densities are presented in Fig.3. As expected for a two-photon ionization process, the signal intensity increases roughly with the square of the power density. However, the recorded decay kinetics does not depend on the 263 nm laser power density since the normalised transient signals are identical (Cf. Fig.3 inset). That result indicates that the same phenomena occur whatever the power density and consequently that the solvation dynamics are independent of the electron concentration in our experimental conditions i.e. we are still within the independent pair approximation as opposed to our previous work on hydrated electron [8]. 

Bulk reaction is the reaction between two particles, say A and B which are uniformly distributed in the chemical system. In this situation it becomes necessary to define the concentrations ca and Cb of both species. In this section the bulk reaction rate is derived in terms of the pair survival probability and it is demonstrated how Smoluchowski s time dependent rate constant can be obtained by making use of the independent pairs approximation. 

It has been reported in the literature [48] that Smoluchowski s rate constant overestimates the rate of scavenging for a single target that can be hit multiple times (for example DNA). In their work, the authors found that Smoluchowski s rate constant overestimated the scavenging yield in comparison to Monte Carlo random flights simulation (which makes no assumptions on the rate of scavenging as they are explicitly treated). The authors have found that a modiflcation to Smoluchowski s rate constant is required in order to properly take the correlation of reaction times into account however, the independent pairs approximation is still made. 

This approximation was denoted initially by the acronym IQG [34] and later on by IP (Independent Pairs) [35]. It gave satisfactory results in the study of the Beryllium atom and of its isoelectronic series as well as in the BeH system. The drawback of this approximation is that when the eigen-vectors are diffuse, i.e. there is more than one dominant two electron configuration per eigen-vector, the determination of the corresponding nj is ambiguous. In order to avoid this problem the MPS approximation, which does not have this drawback, was proposed. 

In order to get significant results, the initial data must be formed by a set of clearly non-A -representable second-order matrices, which would generate upon contraction a closely ensemble A -representable 1-RDM. It therefore seemed reasonable to choose as initial data the approximate 2-RDMs built by application of the independent pair model within the framework of the spin-adapted reduced Hamiltonian (SRH) theory [37 5]. This choice is adequate because these matrices, which are positive semidefinite, Hermitian, and antisymmetric with respect to the permutation of two row/column indices, are not A -representable, since the 2-HRDMs derived from them are not positive semidefinite. Moreover, the 1-RDMs derived from these 2-RDMs, although positive semidefinite, are neither ensemble A -representable nor 5-representable. That is, the correction of the N- and 5-representability defects of these sets of matrices (approximated 2-RDM, 2-HRDM, and 1-RDM) is a suitable test for the two purification procedures. Attention has been focused only on correcting the N- and 5-representability of the a S-block of these matrices, since the I-MZ purification procedure deals with a different decomposition of this block. 

Finally, in order to illustrate the role of the 1-MZ purification procedure in improving the approximated 2-RDMs obtained by application of the independent pair model within the framework of the SRH theory, all the different spin-blocks of these matrices were purified. The energy of both the initial (non-purified) and updated (purified) RDMs was calculated. These energies and those corresponding to a full configuration interaction (full Cl) calculation are reported in Table 111. As can be appreciated from this table, the nonpurified energies of all the test systems lie below the full Cl ones while the purified ones lie above and very close to the full Cl ones. 

In order to analyze the performance of this purification procedure and to compare it with those reported in the previous section, the same atomic and molecular systems in their ground state were selected as test systems. Again, the basis sets used were formed by Hartree-Eock molecular orbitals built out of minimal Slater orbital basis sets and the initial data were chosen to be the approximate 2-RDMs built by application of the independent pair model within the framework of the SRH theory. 

Hartree-Fock with Proper Dissociation Internally Consistent Self Consistent Orbitals Independent Electron Pair Approximation Intermediate Neglect of Differential Overlap Intermediate Retention of Differential Overlap Iterative Natural Orbital Ionization Potential... 

The electronic structure methods are based primarily on two basic approximations (1) Born-Oppenheimer approximation that separates the nuclear motion from the electronic motion, and (2) Independent Particle approximation that allows one to describe the total electronic wavefunction in the form of one electron wavefunc-tions i.e. a Slater determinant [26], Together with electron spin, this is known as the Hartree-Fock (HF) approximation. The HF method can be of three types restricted Hartree-Fock (RHF), unrestricted Hartree-Fock (UHF) and restricted open Hartree-Fock (ROHF). In the RHF method, which is used for the singlet spin system, the same orbital spatial function is used for both electronic spins (a and (3). In the UHF method, electrons with a and (3 spins have different orbital spatial functions. However, this kind of wavefunction treatment yields an error known as spin contamination. In the case of ROHF method, for an open shell system paired electron spins have the same orbital spatial function. One of the shortcomings of the HF method is neglect of explicit electron correlation. Electron correlation is mainly caused by the instantaneous interaction between electrons which is not treated in an explicit way in the HF method. Therefore, several physical phenomena can not be explained using the HF method, for example, the dissociation of molecules. The deficiency of the HF method (RHF) at the dissociation limit of molecules can be partly overcome in the UHF method. However, for a satisfactory result, a method with electron correlation is necessary. 

There is, however, a complication in the treatment of the a bonds. Because the states are no longer purely gerade and ungerade, the four simultaneous equations cannot be reduced to two sets of two. In a diatomic molecule this would not be much of a complication, but it is very serious in solids. Fortunately, for many solids containing a bonds, hybrid basis states can be made from s and p states, and these can be treated approximately as independent pairs, which reduces the problem to that of finding two unknowns for each bond. In other cases, solutions can be approximated by use of perturbation theory. The approximations that are appropriate in solids will often be very dilTerent from those appropriate for diatomic molecules, Therefore, we will not discuss the special case of ir-bonded hetero-polar molecule. 

This defines an independent electron pair approximation in terms of extremal pairs, which can be regarded as a generalization of the independent electron pair approximation (IEPA) [4, 8] in terms of pairs (ij) constructed form (preferably) localized orbitals. As in the discussion in Paper I for MP2 [5], one can show that the extremal pairs defined in this section are related to approximate natural geminals corresponding to the coupled-cluster wave function. 

OVC stands for optimum-valence configuration", lEPA for "independent-electron-pair approximation" h) Equilibrium distances (in a. u.)... 

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