Coulomb 7-matrix element

In the work of King, Dupuis, and Rys [15,16], the mabix elements of the Coulomb interaction term in Gaussian basis set were evaluated by solving the differential equations satisfied by these matrix elements. Thus, the Coulomb matrix elements are expressed in the form of the Rys polynomials. The potential problem of this method is that to obtain the mabix elements of the higher derivatives of Coulomb interactions, we need to solve more complicated differential equations numerically. Great effort has to be taken to ensure that the differential equation solver can solve such differential equations stably, and to... 

To compute the Coulomb matrix element, Eq. (8), we first note that the multiple integration with respect to the coordinates of all electrons of chromophore m and n can be reduced to a two-fold coordinate integration. This becomes possible because of the antisymmetric character of the chromophore electronic wave functions. Therefore, we introduce single electron densities of chromophore m ... 

To carry out the computation of Jmn for a particular pair of chromophores the Coulomb matrix element can be translated with high accuracy into the following form (see Fig. 5 and [30]) ... 

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... 

The integration over the angular part yields unity.) Before this integral is discussed further, it will be put into the more general frame of Coulomb matrix elements with different electron pairs. Using the symbols a, b, c and d to denote the orbitals, one then has to consider... 

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... 

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) ... 

If the one-electron and Coulomb matrix elements, which are the same for all basis functions of the configuration, are denoted by Eo, matrix elements of the Hamiltonian in the Ms = subconfiguration are... 

The Coulomb matrix elements for the HOs localized on different atoms A and B have the form ... 

For most charge transfer reactions the electronic coupling element is simply given by the coulomb matrix element defined by Eq. 61a. 

The connection of the effective intrachain g-ology couplings with the intramolecular or one-site repulsive interaction and the screened long-range Coulomb matrix element have been discussed in detail by Barisic et al. [110-112]. For gi, g2 and g, one has the following result. 

The point multipole expansion of the Coulombic matrix element governing the first-order electric dipole moment in Eq. (3) gives the selection rules of the ligand polarization model through Eq. (4),... 

From symmetry properties of the wave function and from the Coulomb matrix elements it is possible to distinguish between two classes of basis states, namely gerade (denoted by +) and ungerade (denoted by -) states. The corresponding wave functions and 0 may be obtained by replacing the spherical har monics T in equation (14) by... 

Coulomb matrix element and that of the soft potential enables us to... 

The creation/annihilation operators aj /a, denote the one-particle operators which diagonalize the Hamiltonian Hen. The summation indices i, j, k, l denote the usual set of one-electron quantum numbers and run over positive-energy states only. The quantities Vjju are two-electron Coulomb matrix elements and the quantities biju denote two-electron Breit matrix elements, respectively. We specify their static limit (neglecting any frequency dependence) ... 

To Equs. (1,2) we have to add the Fourier transform of the bare Coulomb matrix element... 

The relativistic segment, RGTOFF, is also straightforward, other than one conceptual cortqrlication. The incomplete DKH transformation discussed above involves evaluation of Coulomb matrix elements of the nuclear array without any... 

In the relativistic PS approach, the Coulomb matrix element (Eq. (82)) is given as... 

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