Pair probability

N—H- 0=C hydrogen bonds show a tendency to occur in the directions of the oxygen sp2 lone pairs, probably because of an inherent preference for hydrogen bonding in a lone-pair direction and involving also steric factors75,76. 

This equation effectively imposes the condition that py — Pji = xaXb, with the important result that sro is totally excluded, because, when the pair probability is equal to x xb, the short-range-order parameter a becomes zero (see Section 7.1.2). 

Calcd. quantities A priori singlet and pair probabilities for infinite chains... 

The pair probability and initial condition have appeared on the right-hand side of the equation and P is not affected by z or by C and L, so expanding out z + Las a binomial... 

Northrup and Hynes [103] solved these equations for the case of a diffusion model and found the same results as Collins and Kimball [4] of eqn. (25). This case is reasonably easy to solve because the diffusion and reaction of the pair can be separated. When the motion of the pair involves a non-Markovian process, that is the reactants recall which direction they were moving a moment before (i.e. have a memory ) and the process is not diffusional, this elegant separation becomes very difficult or impossible to effect. Under these circumstances, eqn. (368) can only be solved approximately for the pair probability. The initial condition term, l(t), is non-zero if the initial distribution p(0) is other than peq. 

The last two lines are new, compared to the text output without the -p option and are a rough measure for the well-definedness of the MFE structure. The line before the last line shows a condensed representation of the pair probabilities, similar to the bracket notation, followed by the ensemble free energy in kcal mol-1. The structure string contains additional symbols coding for the pairing tendency of that position. 

Besides the text output the postscript file marry dp. ps is generated (if the input file contained no line with the sequence name the filename defaults to dot. ps). The dot plot shows the pair probabilities within the equilibrium ensemble as an n x n matrix and is an excellent way to visualize structural alternatives (Figure 12.1). A square at row i and column j indicates a base pair. The area of a square in the upper right half of the matrix is proportional to the probability of the base pair (i, j) within the equilibrium ensemble. The lower left half shows all pairs belonging to the mfe structure. The matrix of a sequence with well-defined MFE structure should show only a few very small additional squares in the upper right half compared to the lower left half. 

The resulting plot shows three curves two mountain plots derived from the MFE structure and the pairing probabilities and a positional entropy curve. Well-defined regions are identified by low entropy. By superimposing several mountain plots, structures can easily be compared. 

Scheme 5 depicts Winstein s complete solvolysis mechanism.29 Ion-pair return can be from the intimate ion pair (ion-pair return or internal return), from the external ion pair (external ion-pair return), or from the free ions (external ion return). The term external return refers to the sum of external ion-pair return and external ion return. The special salt effect operates by diversion of the external.ion pair, probably through the mechanism shown in Equation 519, so that it can no longer... 

The second equations of the kinetic system are those for 6jg(r) which determine the pair probability of covering sites numbered / and g with adspecies i and j at the distance r. With allowance for Eqs. (29) and (30), these equations have the following form ... 

This result for P2(ri,ra) can only be true in the limit that the motion of electron 1 and hence its probability of being at a certain point in space is totally independent of the motions of electron 2, i.e. there are no forces acting between the two electrons and their motions are totally uncorrelated. The best that can be done in a model which employs an expression of the form of equation (19) for the pair probability function is to determine the m (which in turn determine the one-particle probability function) in such a way that the electron it describes experiences the average field of the other electrons, a situation which is attained in the Hartree-Pock limit of the orbital model. 

The Hartree-Fock wavefunction is not a simple product function, but an antisymmetrized sum of such products. As pointed out above, the effect of the anti-symmetrization is to selectively remove, from the region of space about a given electron, charge density of electrons with the same spin. Thus, the pair probability function for electrons with the same spin is not of the form given in equation (19) but instead is... 

The ion pair (CCI3 11 Cl )solv being formed on ion recombination most likely corresponds to a solvent separated ion pair. It therefore does not show an extra band (no charge-transfer band). This solvent separated ion pair probably has a low dipole moment due to dielectric shielding from the solvation shell and therefore may not be detected by the microwave absorption technique. 

Finally, a referee has suggested that 8 could decay to product via a concerted pathway (i) [p. 76]. Although such a pathway cannot be ruled out, we feel that the large isotope effect suggests the radical pair, probably from B2. Both i and the radical mechanism would proceed with retention of configuration. 

Although Ceo is easily reduced, it is very difficult to oxidize [46, 53, 54, 72], The only definitive electrochemical oxidation of Cgo occurs at a potential of -F 1.76 V vs SCE in benzonitrile, and is irreversible [54]. The radical cation was reported to be produced by y-irradiation at 77 K in a glass, and to absorb near 980 nm [65, 66]. Attempts to generate the radical cation (Ceo) by electron transfer to singlet excited dicyanoanthracene, which has a reduction potential near + 2.0 V [73] were unsuccessful. This method has been used to generate, for example, trans-stilbene radical cation [73, 74]. The ion pair probably does not... 

The origin of the "repulsions" among electron pairs probably results more from the operation of the Pauli exclusion principle than from electrostatic effects, but we will not be concerned with that in this text. 

Pair probability of finding atoms of types / and J on lattice site m and n is given by the formula,... 

The so presented ELF is mainly based on an interpretation of the conditional pair probability density for electrons of the same spin. A conceptually different interpretation was put forward by Savin et al.23 who realized that the term Da could be generalized for any density p independent of the spin as... 

The problem of RNA formation is a chemical problem. This point is of cardinal importance. The hrst RNA molecules were the products, probably together with a number of similar compounds, of the early chemistry whereby life was launched they were not intended to serve as information carriers. It was their unique base composition that, by lending itself to pairing, probably allowed their specihc replication and amplihcation, which, in turn, caused them to emerge and to be molded by selection into the hrst bearers of genetic information and into the hrst catalysts of protein synthesis, as well as, perhaps, of other processes (de Duve, 2005). 

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