First-order pairs

This differential equation is analytically solvable in functions, but here it is solved numerically. Represent equation by the first order pair,... 

In Paper I of this series [5], the extremal pair functions for the systems He2, Ne, F, HF, H20, NH3 and CH4 were analyzed. We now follow a different line of thought that was also opened in Paper I, namely to use extremal pairs for the construction of correlated wave functions. We have already pointed out that there is a special set of extremal pair functions associated with MP2 (Moller-Plesset perturbation theory of second order). In fact we have shown, that there are two choices for which the Hylleraas functional of MP2 decomposes exactly into a sum of pair contributions. One choice is the conventional one of pairs of canonical spin orbitals, the other one the use of first-order pairs with extremal norm... 

With the first-order pair function iJjJ satisf3dng Eq. (70), Eq. (69) gives the energy to the second order as... 

Let us make the observation that Eqs. (23) and (24) represent an alternative partial wave expansion of the SAPF in which the individual terms are defined by the degrees I of the Legendre polynomials. To distinguish this expansion from the PW/m expansion (15), where the individual terms are defined by pairs of orbital momentum quantum numbers of one-electron wave functions employed in the Cl representation of the pair function, we shall refer to it as auxiliary PW expansion and denote it by the acronym PW/a. This expansion turned out to be well suited for representing the first-order pair functions at the interelectronic cusp. Unlike the PW/m expansion is it not directly related to the Cl approach. Let us stress the important fact that for pairs defined by other than s-electrons the PW/m and PW/a expansions of the second-order energies need not be the same. 

Exercise 5.3 Calculate the total first-order pair correlation energy for the dimer using Eq. (5.19) and show that it is twice the result obtained in Exercise 5.1. 

It is interesting to note, however, that the simplest form of pair theory, namely, first-order pairs (see Eq. (5.19)), is invariant to unitary transformations of degenerate orbitals. To see this, we approximate A = A" = 62 in Eqs. (5.38) and (5.45),... 

These results are equal to the total first-order pair correlation energy, obtained in Exercise 5.3, for the dimer using localized orbitals. The total first-order pair correlation energy is identical to the second-order many-body perturbation result for the correlation energy (see Chapter 6). The above results are a reflection of the fact that many-body perturbation theory is invariant to unitary transformations of degenerate orbitals. 

We have seen in Chapter 5 that cJP is the first-order pair energy. Thus at the level of first-order pairs, pair theory gives the same correlation energy as second-order perturbation theory. 

The long-range interactions between a pair of molecules are detemiined by electric multipole moments and polarizabilities of the individual molecules. MuJtipoJe moments are measures that describe the non-sphericity of the charge distribution of a molecule. The zeroth-order moment is the total charge of the molecule Q = Yfi- where q- is the charge of particle and the sum is over all electrons and nuclei in tlie molecule. The first-order moment is the dipole moment vector with Cartesian components given by... 

If the long-range mteraction between a pair of molecules is treated by quantum mechanical perturbation theory, then the electrostatic interactions considered in section Al.5.2.3 arise in first order, whereas induction and dispersion effects appear in second order. The multipole expansion of the induction energy in its fill generality [7, 28] is quite complex. Here we consider only explicit expressions for individual temis in the... 

A very successfiil first-order perturbation theory is due to Weeks, Chandler and Andersen pair potential u r) is divided into a reference part u r) and a perturbation w r)... 

Equation (B2.4.13) is a pair of first-order differential equations, so its fonnal solution is given by equation (B2.4.14)), in which exp() means the exponential of a matrix. 

A more convincing approach leads to an adaptive method based on the symmetric second order scheme (9). As a first step, we have to introduce a first order scheme substituting p of the previous section. In what follows, we use the following pair of schemes ... 

On subsciCuLlng (12.49) into uhe dynamical equations we may expand each term in powers of the perturbations and retain only terms of the zeroth and first orders. The terms of order zero can then be eliminated by subtracting the steady state equations, and what remains is a set of linear partial differential equations in the perturbations. Thus equations (12.46) and (12.47) yield the following pair of linearized perturbation equations... 

Another reagent which must be considered is the ion pair ACONO2H+ NO3", the species favoured by Fischer, Read and Vaughan. Its participation would make it possible to account for the dependence of rate of zeroth-order nitration upon the concentration of acetyl nitrate and acetic acid, and would lead to the prediction of similar dependencies in first-order nitration. It would not, however (pace Fischer, Read and Vaughan ), explain the anticatalytic effect of added nitrate. 

These ideas are readily applied to the mechanism described by reaction (5.F). To begin with, the rate at which ab links are formed is first order with respect to the concentration of entrapped pairs. In this sense the latter behaves as a reaction intermediate or transition state according to this mechanism. Therefore... 

Substituted 2-haloaziridines are also known to undergo a number of reactions without ring opening. For example, displacement of chlorine in (264) with various nucleophilic reagents has been found to occur with overall inversion of stereochemistry about the aziridine ring (65JA4538). The displacements followed first order kinetics and faster rates were noted for (264 R = Me) than for (264 R = H). The observed inversion was ascribed to either ion pairing and/or stereoselectivity. 

Equation (7) is a second-order differential equation. A more general formulation of Newton s equation of motion is given in terms of the system s Hamiltonian, FI [Eq. (1)]. Put in these terms, the classical equation of motion is written as a pair of coupled first-order differential equations ... 

These equations hold if an Ignition Curve test consists of measuring conversion (X) as the unique function of temperature (T). This is done by a series of short, steady-state experiments at various temperature levels. Since this is done in a tubular, isothermal reactor at very low concentration of pollutant, the first order kinetic applies. In this case, results should be listed as pairs of corresponding X and T values. (The first order approximation was not needed in the previous ethylene oxide example, because reaction rates were measured directly as the total function of temperature, whereas all other concentrations changed with the temperature.) The example is from Appendix A, in Berty (1997). In the Ignition Curve measurement a graph is made to plot the temperature needed for the conversion achieved. 

Kinetic studies of the addition of hydrogen chloride to styrene support the conclusion that an ion-pair mechanism operates because aromatic conjugation is involved. The reaction is first-order in hydrogen chloride, indicating that only one molecule of hydrogen chloride participates in the rate-determining step. ... 

To generalize, let x, represent the nth ab.scis.sa of the first distribution and Pj the corresponding ordinate of a normalized distribution. Then c is formed of the ordered pairs (equation 2.7-.30). As an example, for x,... 

The present chapter is organized as follows. We focus first on a simple model of a nonuniform associating fluid with spherically symmetric associative forces between species. This model serves us to demonstrate the application of so-called first-order (singlet) and second-order (pair) integral equations for the density profile. Some examples of the solution of these equations for associating fluids in contact with structureless and crystalline solid surfaces are presented. Then we discuss one version of the density functional theory for a model of associating hard spheres. All aforementioned issues are discussed in Sec. II. 

The formula for the first-order correction to the wave function (eq. (4.37)) similarly only contains contributions from doubly excited determinants. Since knowledge of the first-order wave function allows calculation of the energy up to third order (In - - 1 = 3, eq. (4.34)), it is immediately clear that the third-order energy also only contains contributions from doubly excited determinants. Qualitatively speaking, the MP2 contribution describes the correlation between pairs of electrons while MP3 describes the interaction between pairs. The formula for calculating this contribution is somewhat... 

A function is a set of ordered elements such that no two ordered pairs have the same first element, denoted as (x,y) where x is the independent variable and y is the dependent variable. A function is established when a condition exists that determines y for each x, the condition usually being defined by an equation such as y = f(x) [2]. 

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