Singlet correlation function

The singlet direct correlation function C r) is defined through the relationship C Ur) = + --/t]... 

Figure C 1.5.10. Nonnalized fluorescence intensity correlation function for a single terrylene molecule in p-terjDhenyl at 2 K. The solid line is tire tlieoretical curve. Regions of deviation from tire long-time value of unity due to photon antibunching (the finite lifetime of tire excited singlet state), Rabi oscillations (absorjDtion-stimulated emission cycles driven by tire laser field) and photon bunching (dark periods caused by intersystem crossing to tire triplet state) are indicated. Reproduced witli pennission from Plakhotnik et al [66], adapted from [118].

Let us begin our discussion from the model of Cummings and Stell for heterogeneous dimerization a + P ap described in some detail above. In the case of singlet-level equations, HNCl or PYl, the direct correlation function of the bulk fluid c (r) represents the only input necessary to obtain the density profiles from the HNCl and PYl equations see Eqs. (6) and (7) in Sec. II A. It is worth noting that the transformation of a square-well, short-range attraction, see Eq. (36), into a 6-type associative interaction, see Eq. (39), is unnecessary unless one seeks an analytic solution. The 6-type term must be treated analytically while solving the HNCl... 

Eq. (101) is the multidensity Ornstein-Zernike equation for the bulk, one-component dimerizing fluid. Eqs. (102) and (103) are the associative analog of the singlet equation (31). The last equation of the set, Eq. (104), describes the correlations between two giant particles and may be important for theories of colloid dispersions. The partial correlation functions yield three... 

We apply the singlet theory for the density profile by using Eqs. (101) and (103) to describe the behavior of associating fluids close to a crystalline surface [120-122], First, we solve the multidensity OZ equation with the Percus-Yevick closure for the bulk partial correlation functions, and next calculate the total correlation function via Eq. (68) and the direct correlation function from Eq. (69). The bulk total direct correlation function is used next as an input to the singlet Percus-Yevick or singlet hypernetted chain equation, (6) or (7), to obtain the density profiles. The same approach can be used to study adsorption on crystalline surfaces as well as in pores with walls of crystalline symmetry. 

The interfacial pair correlation functions are difficult to compute using statistical mechanical theories, and what is usually done is to assume that they are equal to the bulk correlation function times the singlet densities (the Kirkwood superposition approximation). This can be then used to determine the singlet densities (the density and the orientational profile). Molecular dynamics computer simulations can in... 

The calculation of the correlation functions is essentially the same as in Section 7.2, except for the additional complexity due to the existence of more states. Thus, the singlet distribution, i.e., the probability of finding any specific single site occupied, is... 

Having obtained two simultaneous equations for the singlet and doublet correlation functions, X and, these have to be solved. Furthermore, Kapral has pointed out that these correlations do not contain any spatial dependence at equilibrium because the direct and indirect correlations of position in an equilibrium fluid (static structures) have not been included into the psuedo-Liouville collision operators, T, [285]. Ignoring this point, Kapral then transformed the equation for the singlet density, by means of a Laplace transformation, which removes the time derivative from the equation. Using z as the Laplace transform parameter to avoid confusion with S as the solvent index, gives... 

Kapral next considered the various components of these equations and noted one class of collision is relatively unimportant. These are collision events when a reactant A collides with a solvent molecule S (particle 2) and then collides with another solvent molecule S (particle 3). A correlation in motion therefore exist between these two solvent molecules. While this is true, collision between solvent molecules even within a cage are more frequent than such events, and so this effect is ignored. Two equations can now be written for the doublet correlation functions XiS (12, z) and x B(12, z). Using these equations and eqn. (298) leads to an equation for the singlet density which bears a close resemblance to that of eqn. (298) itself... 

A similar equation may be written to describe the evolution of the singlet density of the reactant C. Cluster functions are introduced and, after using a super-position approximation again, the analysis follows that of Kapral [285] very closely with the complication that, instead of only singlet XA and doublet xAB correlation functions, it is necessary to consider the equation for C in tandem. This requires the use of the matrix notation for compactness. 

The paper is organized as follows. In See.2 we consider the frustrated spin chain at F-AF transition point and describe the exact singlet ground-state wave function as well as details of the spin correlation function calculations. We discuss the phase diagram of this model and its magnetic properties in the AF phase. In Sec.3 the special spin ladder will be considered. A two-dimensional frustrated spin model with the exact ground state is considered in Sec.4. Sec. 5 is devoted to the construction of the electronic models with the SB type of wave function. The results of this paper are summarized in Sec.6. 

Now we calculate the norm and correlation function of the wave function To (10). The norm of the singlet wave function To is... 

For the general case of model (2) the calculation of the singlet ground-state correlation functions can be performed in a similar way. The final result in the thermodynamic limit is [14, 11] ... 

In the particular case of zero singlet weight, y — 0, when spins on each rung form a local triplet, correlation functions (42) coincide with those obtained in [9, 29],... 

Having defined the different Interactions occurlng In [3.6.1], we now need to specify the probability of finding an Ion a at some position r. The one-particle (singlet) density p fr jls defined In sec. I.3.9d as the number of particles per volume at position r. Now we apply the definition to Ions. The radial distribution function g (r)and the ion-wall total correlation function h (r) follow from (1.3.9.22 and 23] as... 

The basic features of this protonation monitoring approach are illustrated in Fig. 8.2a-c. In Fig. 8.2a, a series of FCS curves are shown, recorded from FITC in aqueous solution at different pH, with no added buffer. The curves were fitted to (8.3) using (8.9) (also adding a second factor to the correlation function describing singlet-triplet transitions, which is described further in Sect. 8.3 of this chapter). 

Given that the singlet density is just a constant, the most important quantities characterizing the local structure within the adsorbed fluid are the two-particle correlation functions. We start by considering the pair correlation function ga q,q ) between two fluid particles or, equivalently, the corresponding total correlation function / ff q, = gg q, q ) — 1. The statis-... 

For coiiveiitioual fluids the outer (disorder) average of the first term on the right side is absent and each thermal average equals the singlet density. Thus, hb = 0 for systems without quenched disorder. In the presence of disorder, on the other hand, the blocked correlation function is usually nonzero, because the singlet density for a particular realization, (9 ) can be... 

We start with detailed definitions of the singlet and the pair distribution functions. We then introduce the pair correlation function, a function which is the cornerstone in any molecular theory of liquids. Some of the salient features of these functions are illustrated both for one- and for multicomponent systems. Also, we introduce the concepts of the generalized molecular distribution functions. These were found useful in the application of the mixture model approach to liquid water and aqueous solutions. 

For the single chain (w = w2 = 0), there are seven two-particle correlation functions, four of which are divergent. The divergent ones are the CDW (charge-density wave), SDW (spin-density wave), SS (singlet superconductor), and TS (triplet superconductor) response functions and are shown in Fig. (3). The... 

Watanabe and Klein have reported MD simulations of the hexagonal mesophase of sodium octanoate in water with hexagonal symmetry. The singlet (i.e., one atom) probability distribution functions of the carbon atoms on the hydrocarbon chains show close similarity to those in the micelle. The dynamics of water molecules close to the head groups shows lower mean square displacements, and their orientational correlation function decays more slowly than those of waters farther from the head groups, as was seen in a recent bilayer simulation.6 ... 

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