First-order interaction

The first order interelectron interaction in the Coulomb gauge is described by the two second-order in e Feynman graphs Fig.2. These graphs represent the one-photon exchange between the atomic electrons. They are irreducible and for the evaluation of the corresponding energy correction the simplified formulas (153)-(154) can be used. The iS-matrix element that corresponds to the Feynman graph Fig.2a looks like  

Antisymmetrization of the many-electron wave function corresponds to the permutation of the one-electron indices in the upper (or lower) part of the Feynman graph. The exchange graphs have the additional factor (— ) where (, is the number of permutations of pairs of indices. 

Being applied to the first-order Coulomb graph and its exchange counterpart, this yields  

The analogous evaluations for the Breit interaction described by the graph Fig.2b, result in  

Eq(160) includes magnetic interaction and retardation effects. In case of the ground state of the two-electron atom, when Ea = Eb, Eq(160) reduces to  

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Figure Al.5.3 First-order interaction energy for He-He. Based on data from Komasa and Thakkar [70],...

The first-order interaction of the two bonding levels should be the controlling interaction. The calculated PMO energies for concerted reactions are 0.61/ for the observed orientation and 0.53y for the other orientation. Calculated energies for biradical reactions are much smaller. 

M Values for the interdiad statistical weight matrix, Up, in which all first-order interactions are included, are given in the original paper. Table VI. 

First Experimental Matrix. In the present case we planned to determine the main effects of 5 variables (b bs, Table 2), and the following first-order interactions were considered most probable ... 

Discussion of the Results. Table 4 shows that first-order interactions between the three factors investigated are almost as important as their main effects. Hence, conclusions can only be drawn if the influence of these factors on the response is studied by considering them two by two. In fact, this is a typical... 

Frontier Molecular Orbital Theory can be used to describe qualitatively the trajectory of a nucleophile when it attacks a jt centre. Two sets of first-order interactions are considered. Firstly the stabilizing interaction of the HOMO of the nucleophile with the LUMO (jt and a orbitals) of the jt system and secondly the destabilizing interaction of the HOMO of the nucleophile with the HOMO (jt and a molecular orbitals) of the jt system, as shown in Figure 6. 

The model Mr+1 contains the r-th degree term in the mixture components only along with the product of this term with the first degree terms in the Zj s. For example, a planar or first-degree model in the mixture components, and a main effects only model in the process variables, is y=M1+i+e. A planar model in the Xj s, combined with a main effect plus first-order interaction effects model in the Zj s, would be y=Mi+i+Mi+2+ . The model containing up to quadratic blending terms by main effects in the Zfs is defined as y=Mi+i+M2+l+H. This continues, up to the complete 2q+n-2n term model that is defined as ... 

Figure 13 Schematic o orbital interaction diagrams for PC—CP (5a), CP—PC (5c), NC—CN (4a), and CN—NC (4c). MO and FMO energies in electron volts. Left panels FMOs central panels primary or first-order interaction (i.e., no °somo homo mixing) tight panels final situation including all interactions.

Figure 3.4 Interaction of two alternant radicals. Only the nonbonding orbitals give a first-order interaction (unbroken line).

In the PMO method, we need only to examine one interaction. In the FMO approach, two are sometimes necessary. The PMO method has firm theoretical foundations because the only first-order interaction is retained all the neglected interactions are second order in PAB. 

The first term (a + P) is the energy of the ncc orbital. The second term (0.707 x 0.620) is the correction due to first-order interaction with rcCN and the third term is the second-order correction due to 7t CN. In the same manner, the 1-azabutadiene LUMO energy is given by... 

Zero-order interaction in a heteroatomic pair is calculated in essentially the same way. The C- H interaction serves to illustrate the method. Starting from the first-order interaction d = 1.08 A, d = 0.90, be = 0.1377, D = 0.3983. Hence... 

Van Duijneveldt-Van de Rijdt JGCM, Van Duijneveldt FB (1972) Double-exchange contributions to the first-order interaction energy between closed-shell molecules. Chem Phys Lett 17 425—427... 

According to SAPT formulation of the first-order interaction energy, the Heitler-London term consists of electrostatic and exchange contributions (the former obtained from the perturbation theory formula) ... 

Henry s law (yB nearly constant and equal to yg5) holds for A-B alloys dilute in B. For this range of composition, another quantity can be introduced which takes into account interactions between dissolved O and dissolved B in an A matrix. This is the Wagner s first-order interaction parameter, e , defined by an equation in which high-order terms are neglected ... 

Chalasinski, G. and Gutowski, M., Dimer centred basis set in the calculations of the first-order interaction energy with Cl wavefunction. The He dimer, Mol. Phys. 54, 1173-1184 (1985). 

The first order interactions measure the extent to which the effect of one factor depends upon the value of the other factor. Thus there are three first order interactions involving S, namely S X W, S X B, and S X H. 

The Sum of Squares corresponding to the Residual is obtained by subtracting the four main effects, the six first order interactions, and the four second order interactions from, the Total Sum of Squares. 

Here we have a three factor analysis with a significant first order interaction, W X B. Accordingly we need to break the three factor experiment down, either into W X H experiments for Bi and for Ba or into B X H experiments for Wi and for Wj. 

It will be noted that this experiment has been complicated in the sense that there have been second order interactions significant. It would have been simpler if only the main effects, or first order interactions, had been significant. With second order interactions existing, however, the operation of the system is complicated, so it is inevitable that its analysis should be also. 

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