Matrix elements coulomb operator

The only way to take into account configuration interaction is to perform an exact calculation of the total system including the two interacting configurations. In the case of 4f /4f5d interaction, the only coupling operator is the crystal-field Hamiltonian since the two configurations are of opposite parity, Coulombic interaction matrix elements are zero. On the contrary, the matrix elements of the crystal field between 4f and 4f " 5d contains 5 parameters with odd k values. 

When the Coulomb and exchange operators are expressed in terms of the basis functions and the orbital expansion is substituted for xu then their contributions to the Fock matrix element take the following form ... 

This paper considers the hyperspherical harmonics of the four dimensional rotation group 0(4) in the same spirit ofprevious investigations [2,11]), where the possibility is considered of exploiting different parametrizations of the 5" hypersphere to build up alternative Sturmian [12] basis sets. Their symmetry and completeness properties make them in fact adapt to solve quantum mechanical problems where the hyperspherical symmetry of the kinetic energy operator is broken by the interaction potential, but the corresponding perturbation matrix elements can be worked out explicitly, as in the case of Coulomb interactions (see Section 3). A final discussion is given in Section 4. 

The Bethe logarithm is formally defined as a certain normalized infinite sum of matrix elements of the coordinate operator over the Schrodinger-Coulomb wave functions. It is a pure number which can in principle be calculated with arbitrary accuracy, and high accuracy results for the Bethe logarithm can be found in the literature (see, e.g. [13, 14] and references therein). For convenience we have collected some values for the Bethe logarithms [14] in Table 3.1. 

Calculation of the nonlogarithmic polarization operator contribution is quite straightforward. One simply has to calculate two terms given by ordinary perturbation theory, one is the matrix element of the radiatively corrected external magnetic field, and another is the matrix element of the radiatively corrected external Coulomb field between wave functions corrected by the external magnetic field (see Fig. 9.13). The first calculation of the respective matrix elements was performed in [34]. Later a number of inaccuracies in [34] were uncovered [22, 23, 40, 43, 44, 45] and the correct result for the nonlogarithmic contribution of order a Za) EF to HFS is given by... 

The occupation number vectors are basis vectors in an abstract linear vector space and specify thus only the occupation of the spin orbitals. The occupation number vectors contain no reference to the basis set. The reference to the basis set is built into the operators in the second quantization formalism. Observables are described by expectation values of operators and must be independent of the representation given to the operators and states. The matrix elements of a first quantization operator between two Slater determinants must therefore equal its counterpart of the second quantization formulation. For a given basis set the operators in the Fock space can thus be determined by requiring that the matrix elements between two occupation number vectors of the second quantization operator, must equal the matrix elements between the corresponding two Slater determinants of the corresponding first quantization operators. Operators that are considered in first quantization like the kinetic energy and the coulomb repulsion conserve the number of electrons. In the Fock space these operators must be represented as linear combinations of multipla of the ajaj... 

Making use of the properties of the eigenvalues of Casimir operators, mentioned in Chapter 5, we are in a position to find a number of interesting features of the matrix elements of the Coulomb interaction operator. Thus, it has turned out that for the pN shell there exists an extremely simple algebraic expression for this matrix element... 

We discussed in detail the properties of the matrix elements of the electrostatic energy operator for shell lN. The corresponding expressions for the remaining two-electron operators may be found in a similar way, therefore, here we shall present only final results. For the case of relativistic corrections H2, H and H s to the Coulomb energy (formulas (19.8), (19.11) and (19.12), respectively) we have... 

The formula for a matrix element of the energy operator of magnetic interactions (its irreducible form is presented by Eq. (19.66)) can be found in a similar way to that for the Coulomb interaction. It is... 

Matrix elements for the valence functions were taken with the effective core potential the coulomb and exchange terms were handled exactly, numerically, without any parameterization and a Phillips-Kleinman projection operator term was also used. Spin-orbit coupling effects amongst the valence orbitals were treated semi-empirically using the operator... 

Because the Coulomb operator is a two-particle operator, the transition matrix element Mn is non-zero only for cases in which at most two orbitals differ in the initial- and final-state wavefunctions. For normal Auger transitions it will turn out that these are just the electron orbitals used to characterize the Auger transition, including the Auger electron itself. To show this for the K-LfLf 0 transition one starts with the matrix element... 

From this matrix element it can be seen first, that the 2p orbitals on either side of the matrix element are the same second, that two 2s orbitals present on the right-hand side are absent on the left-hand side and third, that on the left-hand side two orbitals are present, ip(ss, ms) and one of the lsOmi- orbitals, which are absent on the right-hand side. (One of the spin-orbitals, lsO+ or IsO-, on the left-hand side must coincide with lsOM< on the right-hand side.) Therefore, exactly two different orbitals remain on each side of this matrix element. They are connected by the Coulomb operator and determine the value of the matrix element. As with photoionization, the two electrons relevant for the Auger transition are called active electrons while the remaining electrons which lead to an overlap integral are termed passive electrons. Hence, the calculation yields... 

This result shows that the original matrix element containing the orbitals of all electrons factorizes into a two-electron Coulomb matrix element for the active electrons and an overlap matrix element for the passive electrons. Within the frozen atomic structure approximation, the overlap factors yield unity because the same orbitals are used for the passive electrons in the initial and final states. Considering now the Coulomb matrix element, one uses the fact that the Coulomb operator does not act on the spin. Therefore, the ms value in the wavefunction of the Auger electron is fixed, and one treats the matrix element Mn as... 

Finally, it should be noted that treating one of the otherwise equivalent electrons in equ. (7.46) individually is frequently used for calculating matrix elements with a one-electron operator (e.g., the photon operator) acting on equivalent electrons. Similarly, if two-electron operators play a role, like in the Coulomb interaction between electrons, then it is convenient to separate two electrons from the equivalent electrons. This is done using the coefficients of fractional grandparentage (for more details see [Cow81]). 

All three forms of the dipole matrix element are equivalent because they can be transformed into each other. However, this equivalence is valid only for exact initial- and final-state wavefunctions. Since the Coulomb interaction between the electrons is responsible for many-body effects (except in the hydrogen atom), and the many-body problem can only be solved approximately, the three different forms of the matrix element will, in general, yield different results. The reason for this can be seen by comparing for the individual matrix elements how the transition operator weights the radial parts R r) and Rf(r) of the single-particle wavefunction differently ... 

In order to calculate the matrix elements with the Coulomb operator Vc, one again uses Slater determinantal wavefunctions, for the intermediate state xp(Mp, t) as well as for the complete final state which contains the doubly charged ion, f, and the two ejected electrons, x<, (Ka, Kb). Assuming that there is no correlation between the two escaping electrons and that their common boundary condition applies separately to each single-particle function, the directional emission property is included in the factors f( ka) and f( kb), and one gets for this Coulomb matrix element C... 

The Coulomb operator is a two-particle operator, i.e., it describes an interaction between at most two different orbitals on each side of its matrix element. Therefore, these matrix elements vanish unless the energies and spatial parts of the wave-function in the orbitals a> or b> coincide with t>. This gives... 

Couplings for Auger electron emission In this case one has to consider the decay of the intermediate photoionized state (J,) to the final ionic state J( by emission of the Auger electron (j2) taking care also of the Coulomb matrix elements (operator Op2) ... 

It leaves intact the fermion operators related to the /1-th group itself. By virtue of this the two-electron operators WBA result in a renormalization of one-electron terms in the Hamiltonians for each group. <4 = 1,..., M. The expectation values ((b+b ))B are the one-electron densities. The Schrodinger equation eq. (1.193) can be driven close to the standard HFR form. This can be done if one defines generalized Coulomb and exchange operators for group A by their matrix elements in the carrier space of group A ... 

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