Independent pair model

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. 

In the independent pair model, the average or expectation value of N is the survival probability of the radical pairs when each is distributed as a Gaussian. It can be calculated by convolution and will not be unity even at time t = 0 because some pairs are formed with r0 pair survival probability was called IIa(f). Assuming that the distances of separation of the reactant pairs is independent of all other distances (this is not strictly true) and that only one pair reacts at any one time, Clifford et al. developed the probability that a spur initially containing N0 reactants [and hence M0 = (N0/2) (N0 — 1) pairs] and IV at time t [and hence M — (AT/2) (IV —1) pairs] does not react further for a time r and this was shown to be... 

Great simplification is achieved by introducing the hypothesis of independent reaction times (IRT) that the pairwise reaction times evolve independendy of any other reactions. While the fundamental justification of IRT may not be immediately obvious, one notices its similarity with the molecular pair model of homogeneous diffusion-mediated reactions (Noyes, 1961 Green, 1984). The usefulness of the IRT model depends on the availability of a suitable reaction probability function W(r, a t). For a pair of neutral particles undergoing fully diffusion-con-trolled reactions, Wis given by (a/r) erfc[(r - a)/2(D t)1/2] where If is the mutual diffusion coefficient and erfc is the complement of the error function. 

The most widely used qualitative model for the explanation of the shapes of molecules is the Valence Shell Electron Pair Repulsion (VSEPR) model of Gillespie and Nyholm (25). The orbital correlation diagrams of Walsh (26) are also used for simple systems for which the qualitative form of the MOs may be deduced from symmetry considerations. Attempts have been made to prove that these two approaches are equivalent (27). But this is impossible since Walsh s Rules refer explicitly to (and only have meaning within) the MO model while the VSEPR method does not refer to (is not confined by) any explicitly-stated model of molecular electronic structure. Thus, any proof that the two approaches are equivalent can only prove, at best, that the two are equivalent at the MO level i.e. that Walsh s Rules are contained in the VSEPR model. Of course, the transformation to localised orbitals of an MO determinant provides a convenient picture of VSEPR rules but the VSEPR method itself depends not on the independent-particle model but on the possibility of separating the total electronic structure of a molecule into more or less autonomous electron pairs which interact as separate entities (28). The localised MO description is merely the simplest such separation the general case is our Eq. (6)... 

It can be shown that the virial type of activity coefficient equations and the ionic pairing model are equivalent, provided that the ionic pairing is weak. In these cases, it is in general difficult to distinguish between complex formation and activity coefficient variations unless independent experimental evidence for complex formation is available, e.g., from spectroscopic data, as is the case for the weak uranium(VI) chloride complexes. It should be noted that the ion interaction coefficients evaluated and tabulated by Cia-vatta [10] were obtained from experimental mean activity coefficient data without taking into account complex formation. However, it is known that many of the metal ions listed by Ciavatta form weak complexes with chloride and nitrate ions. This fact is reflected by ion interaction coefficients that are smaller than those for the noncomplexing perchlorate ion (see Table 6.3). This review takes chloride and nitrate complex formation into account when these ions are part of the ionic medium and uses the value of the ion interaction coefficient (m +,cio4) for (M +,ci ) (m +,noj)- Io... 

The conclusions of Ciraci et al. (1990a) can be understood with a simple independent-atom model, that is, by considering the tip as a single atom, and the total force is the sum of the forces on every atom on the sample surface. Assuming that the force between individual pairs of atoms can be represented as a Morse curve, the z component of the force to the nth atom at a point in space, r, is ... 

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 Chap. 2 and Chap. 3, Sect. 1.2, the appropriate boundary and initial conditions for reactions between statistically independent pairs of reactants were formulated to model a homogeneous reaction. In these cases, if there is no inter-reactant force, all that is required is one or other reactant to be in vast excess on the other. Since the excited donor or the electron donor has to be produced in situ by photostimulation or high-energy radiation, it is natural to choose [D ] < [A], though there are exceptions. Locating the donor at the origin in a sea of acceptor molecules distributed randomly leads to the initial condition, as before... 

For systems with more than one electron pair, the simple picture illustrated above obviously breaks down. The approximate validity of the independent electron-pair model, however, still makes it possible to estimate different correlation effects also in many-electron systems from an inspection of the natural orbital occupation numbers. 

The first discovery of chemically induced dynamic electron polarization (CIDEP) was made by Fessenden and Schuler in 1963 (58). These authors observed the abnormal spectra of the H atoms produced during the irradiation of liquid methane. The low-field line in the esr spectrum was inverted compared to the corresponding high-field line. The related chemically induced dynamic nuclear polarization effect (CIDNP) was reported independently four years later by Bargon et al. (22) and by Ward and Lawler (134). Because of the wider application of nmr in chemistry, the CIDNP effect immediately attracted considerable theoretical and experimental attention, and an elegant theory based on a radical-pair model (RPM) was advanced to explain the effect. The remarkable development of the radical-pair theory has obviously brought cross-fertilization to the then-lesser-known CIDEP phenomenon. 

This planar structure is the one expected for three pairs of electrons around a central atom, which means that a double bond should be counted as one effective pair in using the VSEPR model. This makes sense because the two pairs of electrons involved in the double bond are not independent pairs. Both of the electron pairs must be in the space between the nuclei of the two atoms to form the double bond. In other words, the double bond acts as one center of electron density to repel the other pairs of electrons. The same holds true for triple bonds. This leads us to another general rule For the VSEPR model multiple bonds count as one effective electron pair. 

If the Coulomb interaction between electrons of different pairs is ignored, each localized bond and lone pair contribute independently to the total energy, which implies a perfect additivity of bond energies. In the independent-particle model, the localized bond function can be visualized as a two-center molecular orbital occupied by two electrons. Nevertheless, it is possible to reproduce deviations from additivity rules within this scheme, for instance, by taking into account overlap (for a review, see e.g. 2>). 

If the Hartree-Fock determinant dominates the wavefunction, some of the occupation numbers will be close to 2. The corresponding MOs are closely related to the canonical Hartree-Fock orbitals. The remaining natural orbitals have small occupation numbers. They can be analysed in terms of different types of correlation effects in the molecule . A relation between the first-order density matrix and correlation effects is not immediately justified, however. Correlation effects are determined from the properties of the second-order reduced density matrix. The most important terms in the second-order matrix can, however, be approximately defined from the occupation numbers of the natural orbitals. Electron correlation can be qualitatively understood using an independent electron-pair model . In such a model the correlation effects are treated for one pair of electrons at a time, and the problem is reduced to a set of two-electron systems. As has been shown by Lowdin and Shull the two-electron wavefunction is determined from the occupation numbers of the natural orbitals. Also the second-order density matrix can then be specified by means of the natural orbitals and their occupation numbers. Consider as an example the following simple two-configurational wavefunction for a two-electron system ... 

Exclusion Principle. The energy associated with the filled vacuum is an unobservable constant which should be subtracted from a given physical model. Calculations which go beyond an independent particle model but are carried out using only the positive energy branch of the Dirac spectrum are said to be carried out within the no virtual pair approximation. Such calculations essentially follow the procedures adopted in non-relativistic studies. The relativistic and non-relativistic correlation energy calculations differ only in the model used to defined the reference independent particle model. 

In summary, examination of CO2 leads us to the conclusion that in using the VSEPR model for molecules with double bonds, each double bond should be treated the same as a single bond. In other words, although a double bond involves four electrons, these electrons are restricted to the space between a given pair of atoms. Therefore, these four electrons do not function as two independent pairs but are "tied together" to form one effective repulsive unit. 

The association constants of table III can be compared with those from conductance measurements, and are found to be in perfojt agreement, e.g. K (MgS04/H20) = 160 dm mol . The agreement of the R -values of Table III for aqueous solutions with those of the ion-pair model, Eq. (20), ould be stressed as an important result. The calculated values, Rcaic. correspond to R = a + 2s (here s = don. dimension of OH) according to this model. The agreement of R with Bjerrum s distaiKe parameter q, which is often used as the upper limit of association and which depends only on the permittivity of the solvent [cf. Eq. (19b)], is less satisfactory. For aqueous solutions of 2,2-electrolytes at 25 °C q equals 1.43 nm, independent of the ionic radii. 

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