Matrix element electrostatic interaction

Energies of terms by tensor operators The formulas for evaluating the electrostatic interaction matrix elements within a two electron configuration are given by... 

Each diagonal electrostatic free energy interaction matrix element is the difference between the free energy change of ionization for group i in the otherwise un-ionized protein, and in a model compound in water of p modei,/ ... 

In a subsequent communication, Elliott and coworkers found that uniaxially oriented membranes swollen with ethanol/water mixtures could relax back to an almost isotropic state. In contrast, morphological relaxation was not observed for membranes swollen in water alone. While this relaxation behavior was attributed to the plasticization effect of ethanol on the fluorocarbon matrix of Nafion, no evidence of interaction between ethanol and the fluorocarbon backbone is presented. In light of the previous thermal relaxation studies of Moore and co-workers, an alternative explanation for this solvent induced relaxation may be that ethanol is more effective than water in weakening the electrostatic interactions and mobilizing the side chain elements. Clearly, a more detailed analysis of this phenomenon involving a dynamic mechanical and/ or spectroscopic analysis is needed to gain a detailed molecular level understanding of this relaxation process. 

Among all the possible two-particle operators for physical quantities for lN configuration we have only considered in detail the electrostatic interaction operator for electrons here too we shall confine ourselves to the examination of this operator. The explicit form of the two-electron matrix elements of the electrostatic interaction operator for electrons (the... 

These operators can be averaged in the same manner as in Chapter 14 where we have introduced the average operator of electrostatic interaction of electrons in a shell. The main departure of the case at hand is that the Pauli exclusion principle, owing to the fact that electrons from different shells are not equivalent, imposes constraints neither on the pertinent two-particle matrix elements nor on the number of possible pairing states, which equals (4/i + 2)(4/2 + 2). The averaged submatrix element of direct interaction between the shells will then be... 

The coefficients of the exchange integrals of electrostatic interaction are equal to the matrix elements of the operator... 

Corresponding expressions for matrix elements of relativistic corrections to the electrostatic energy and to two-electron parts of magnetic interactions are rather cumbersome and are not presented here. They may be found in [14]. In Chapter 27 we describe the simplified method of taking them into account. 

However, formulas of the kind (20.26) are rather inconvenient for calculations. Therefore, one has usually to insert explicit expressions for two-electron matrix elements, to perform, where it turns out to be possible, the summations necessary and to find finally the representation of the energy matrix element of the interaction between two subshells in the form of direct and exchange parts. Thus, for the electrostatic interaction we find... 

The matrix element of the operator of the energy of magnetic interactions between two subshells may be found in a similar way to the case of electrostatic interaction. The final result is as follows for direct dH 2 and exchange eH 2 parts, respectively ... 

One of the easiest ways to improve the results is through the replacement of the matrix element of the energy operator of electrostatic interaction by some effective interaction, in which, together with the usual expression of the type (19.29), there are also terms containing odd k values. This means that we adopt some effective Hamiltonian, whose matrix elements of the... 

For pN shells the effective Hamiltonian Heff contains two parameters F2 and 4>i, as well as the constant of spin-orbit interaction. The term with k = 0 causes a general shift of all levels, which is usually taken from experimental data in semi-empirical calculations, and can therefore be neglected. The coefficient at 01 is proportional to L(L + 1). Therefore, to find the matrix elements of the effective Hamiltonian it is enough to add the term aL(L + 1) to the matrix elements of the energy of electrostatic and spin-orbit interactions. Here a stands for the extra semi-empirical parameter. 

Such general expressions for matrix elements of electrostatic interactions, covering the cases of three and four open shells, may be found in Chapter 25 of [14]. However, they are rather cumbersome and, therefore of little use for practical applications. Quite often sets of simpler formulas, adopted for particular cases of configurations, are employed. Below we shall present such expressions only for the simplest interconfigurational matrix elements occurring while improving the description of a shell of equivalent electrons (the appropriate formulas for the more complex cases may be found in [14]) ... 

A two-electron matrix element of the operator of the electrostatic interaction energy is equal to... 

This term occurs for matrix elements differing only by a principal quantum number of one electron. It describes terms corresponding to kinetic energy and electrostatic interaction of an electron with a nuclear field (integral of the type (19.23)) as well as to the interaction with closed shells (summation over nok). 

These are produced by autoionization transitions from highly excited atoms with an inner vacancy. In many cases it is the main process of spontaneous de-excitation of atoms with a vacancy. Let us recall that the wave function of the autoionizing state (33.1) is the superposition of wave functions of discrete and continuous spectra. Mixing of discrete state with continuum is conditioned by the matrix element of the Hamiltonian (actually, of electrostatic interaction between electrons) with respect to these functions. One electron fills in the vacancy, whereas the energy (in the form of a virtual photon) of its transition is transferred by the above mentioned interaction to the other electron, which leaves the atom as a free Auger electron. Its energy a equals the difference in the energies of the ion in initial and final states ... 

Since we neglected overlap between orbitals of subunits, overlap densities a(l)b(l) and a (2)b (2) are negligible. The matrix element //, for the singlet state can be approximated by the electrostatic interaction between transition densities aa on subunit A and bb on subunit B ... 

The first term in eq. (1) Ho represents the spherical part of a free ion Hamiltonian and can be omitted without lack of generality. F s are the Slater parameters and ff is the spin-orbit interaction constant /<- and A so are the angular parts of electrostatic and spin-orbit interactions, respectively. Two-body correction terms (including Trees correction) are described by the fourth, fifth and sixth terms, correspondingly, whereas three-particle interactions (for ions with three or more equivalent f electrons) are represented by the seventh term. Finally, magnetic interactions (spin-spin and spin-other orbit corrections) are described by the terms with operators m and p/. Matrix elements of all operators entering eq. (1) can be taken from the book by Nielsen and Koster (1963) or from the Argonne National Laboratory s web site (Hannah Crosswhite s datafiles) http //chemistry.anl.gov/downloads/index.html. In what follows, the Hamiltonian (1) without Hcf will be referred to as the free ion Hamiltonian. 

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