Coulomb matrix

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... 

Challacombe, M., Schwegler, E., Almlof, J., 1996, Fast Assembly of the Coulomb Matrix A Quantum Chemical Tree Code , J. Chem. Phys., 104, 4685. 

Challacombe, M. E. Schwegler, and J. Almlof. 1996. Fast assembly of the Coulomb matrix A quantum chemical tree code. J. Chem. Phys. 104,4686. 

In this paper we present the first application of the ZORA (Zeroth Order Regular Approximation of the Dirac Fock equation) formalism in Ab Initio electronic structure calculations. The ZORA method, which has been tested previously in the context of Density Functional Theory, has been implemented in the GAMESS-UK package. As was shown earlier we can split off a scalar part from the two component ZORA Hamiltonian. In the present work only the one component part is considered. We introduce a separate internal basis to represent the extra matrix elements, needed for the ZORA corrections. This leads to different options for the computation of the Coulomb matrix in this internal basis. The performance of this Hamiltonian and the effect of the different Coulomb matrix alternatives is tested in calculations on the radon en xenon atoms and the AuH molecule. In the atomic cases we compare with numerical Dirac Fock and numerical ZORA methods and with non relativistic and full Dirac basis set calculations. It is shown that ZORA recovers the bulk of the relativistic effect and that ZORA and Dirac Fock perform equally well in medium size basis set calculations. For AuH we have calculated the equilibrium bond length with the non relativistic Hartree Fock and ZORA methods and compare with the Dirac Fock result and the experimental value. Again the ZORA and Dirac Fock errors are of the same order of magnitude. 

The internal basis can still be sizeable. Therefore the calculation of Coulomb matrix present in Vc, can be computationally very demanding. At the present time there are several possibilities implemented which will give different (in computational effort and accuracy) estimates of the Coulomb matrix. 

Projection of the density matrix onto the internal basis, which is, due to the way the internal basis is constructed, no extra approximation, and build the full Coulomb matrix. A further approximation can be made by calculating only the intra-atomic components of this matrix, which is not expected to produce a serious loss in accuracy, since V l Ic is only large near the nuclei. We will refer to these alternatives as the full and atomic Coulomb ZORA option. Of course these two options are equivalent in the atomic calculations presented in the next section. 

A significantly cheaper, but theoretical not well justified, alternative is to calculate the Coulomb matrix in the smaller external basis and project it subsequently to the internal basis (the projected ZORA option)... 

Experience has shown that it is not necessary to update the Coulomb matrix (in the inverse operator) every SCF cycle. Therefore we have chosen to compute the internal Coulomb matrix with a direct scf fock matrix builder, thereby avoiding the use of large two electron integral files. 

It is interesting to note that the Coulomb matrix and the matrix of the nuclear potential present in Vc are opposite in sign. This means that an underestimation, or complete neglect, of the Coulomb matrix will lead to a larger Vc and thus to an overestimation of the relativistic effect. If Vc is negligable compared to 2c the ZORA equation reduces to the non relativistic Schrodinger equation. 

In order to test the various approximations of the Coulomb matrix, all electron basis set and numerical scalar scaled ZORA calculations have been performed on the xenon and radon atom. The numerical results have been taken from a previous publication [7], where it should be noted that the scalar orbital energies presented here are calculated by averaging, over occupation numbers, of the two component (i.e. spin orbit split) results. Tables (1) and (2) give the orbital energies for the numerical (s.o. averaged) and basis set calculations for the various Coulomb matrix approximations. The results from table... 

Table 1 Xenon, comparison of orbital energies for numerical Dirac and ZORA and non relativistic calculations with basis set ZORA calculations in different Coulomb matrix approximations in the UGBS basis set...

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 ... 

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