Pair approximation

Each pair contains one purine and one pyrimidine base This makes the A T and G C pairs approximately the same size and ensures a consistent distance between the two DNA strands... 

Alpha helices in proteins are found when a stretch of consecutive residues all have the 0, y angle pair approximately -60° and -50°, corresponding to the allowed region in the bottom left quadrant of the... 

Coupled—Pair Functional (ACPF) and Coupled Electron Pair Approximation (CEPA). The simplest form of CEPA, CEPA-0, is also known as Linear Coupled Cluster Doubles (LCCD). 

We work with the pair approximation of a binary alloy system. The two... 

The continuous CVM can also be applied to an fee alloy. We use the pair approximation of continuous CVM within the NN pair approximation in the fee binary alloys. For the binary system, we use the chemical potential to control the composition as was done in the 2-D case. For convenience, we choose... 

The phase diagrams of the 2D binary alloys are shown in Fig.5. In Fig.6, we show the point distribution functions f and fg of the binary alloys. The dashed curve in Fig.5 shows the phase separation determined by the conventional CVM with the pair approximation The parameter is taken such that 4e = 2e g - ( aa bb)- The solid curve is calculated using the present continuous CVM, with the... 

The DC or DCB Hamiltonians may lead to the admixture of negative-energy eigenstates of the Dirac Hamiltonian in an erroneous way [3,4]. The no-virtual-pair approximation [5,6] is invoked to correct this problem the negative-energy states are eliminated by the projection operator A+, leading to the projected Hamiltonians... 

By adopting the no-pair approximation, a natural and straightforward extension of the nonrelativistic open-shell CC theory emerges. The multireference valence-universal Fock-space coupled-cluster approach is employed [25], which defines and calculates an effective Hamiltonian in a low-dimensional model (or P) space, with eigenvalues approximating some desirable eigenvalues of the physical Hamiltonian. The effective Hamiltonian has the form [26]... 

A fully relativistic extension of the scheme put forward in [12] has been introduced in [19], including the transverse electron-electron interaction (Breit +. .. ) and vacuum corrections. Restricting the discussion to the no-pair approximation [28] for simplicity, we here compare this perturbative approach to orbital-dependent Exc to the relativistic variant of the adiabatic connection formalism [29], demonstrating that the latter allows for a direct extraction of an RPA-like orbital-dependent functional for Exc- In addition, we provide some first numerical results for atomic Ec. 

In the electronic sector the presence of the potential leads to an inhomogeneous reference system. Within the no-pair approximation. 

As all quantities discussed in this publication are understood within the no-pair approximation, we will omit the index np in the following for brevity). In Eqs. (2.21, 2.22) bk and b are the annihilation and creation operators for positive energy KS states, which allow to write the electronic ground state as... 

In between these extremes lie a large number of CVM treatments which use combinations of different cluster sizes. The early treatment of Bethe (1935) used a pair approximation (i.e., a two-atom cluster), but this cluster size is insufficient to deal with fhistration effects or when next-nearest neighbours play a significant role (Inden and Pitsch 1991). A four-atom (tetrahedral) cluster is theoretically the minimum requirement for an f.c.c. lattice, but clusters of 13-14 atoms have been used by de Fontaine (1979, 1994) (Fig. 7.2b). However, since a comprehensive treatment for an [n]-member cluster should include the effect of all the component smaller (n — 1, n — 2...) units, there is a marked increase in computing time with cluster size. Several approximations have been made in order to circumvent this problem. 

One result of the simplification inherent in the CSA treatment is that the same expression is obtained for the entropy of both f.c.c. and b.c.c. lattices which clearly distinguishes it from the differences noted in Fig. 7.3(b). However, Fig. 7.10 shows that the overall variation of Gibbs energy derived from the CSA method agrees well with CVM, falling between the pair approximation, which overestimates the number of AB bonds, and the point approximation, where these are underestimated. As might be expected, if larger clusters are admitted to the CSA approximation the results become closer to the CVM result. However, this is counterproductive if the object is to increase the speed of calculation for multi-component systems. 

D. M. Silver, E. L. Mehler, and K. Ruedenberg, Electron correlation and separated pair approximation in diatomic molecules. 1. Theory. J. Chem. Phys. 52(3), 1174-1180 (1970). 

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. 

The Quasi-Chemical Approximation. The mean-field approximation ignores all correlation in the occupation of neighboring sites. This is incorrect when there is a strong interaction between adsorbates at such sites. The simplest way to include some correlation is to work with probabilities of occupations of two sites (XY) instead of one site (X). Approximations that do this are generally called pair approximations (not to be confused with pair interactions). There are more possibilities to reduce multi-site probabilities as in eqn. (8) to 2-site probabilities than to 1-site probabilities. This leads to different types of pair approximations. The best-known approximation that is used for Ising models is the Kirkwood approximation, which uses for example ... 

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