Overlap integral over densities

To get the potential energy of two overlapping charge clouds, consider the interaction of small elements of each cloud and then form a double integral over each of them. Calling the clouds 1 and 2, and the volume elements within them dvi and dv2, each of which carries a charge equal to the charge density of each cloud times the element s volume, the potential is ... 

The result of the integration will be less sensitive to the exact position of the boundary when the density in the boundary area is small, which means that integration over the deformation density is preferable. Because covalently bonded atoms overlap significantly, the method is suitable for obtaining ionic or molecular charges but not for separation of atoms in a molecule. It should not be used for the latter purpose. 

Goulomb interaction integrals over molecular orbitals can be written as a sum of similar interaction integrals with G-spinor overlap densities and... 

The repulsion energy, Erep, is modeled as proportional to intermolecular overlap. The total overlap integral between the charge densities of any two molecules, Sab, is calculated by numerical integration over the original uncondensed densities, and is... 

The term luminosity is used in connection with colliding beam experiments to determine the production rate of a collision. The instantaneous luminosity is proportional to the overlap integral to the densities of the two beams. The total number of individual collisions in an experiment is given by the cross section of the collision process times the integrated luminosity, which is defined as the instantaneous luminosity integrated over the total time of the experiment. More details can be found in the work of Conte and Mackay (1991b). 

Gavezzotti developed a method for calculating lattice energy by integrating over the molecular electron density in the crystal. The molecular electron density is typically taken from a molecular quantum mechanical calculation, although it is not restricted in this way as the density could come from solid state calculations too. The Coulomb interaction between the molecules is calculated by a numerical integration over the tabulated electron densities. The repulsion between molecules is calculated from the overlap between... 

First, we consider the classification of the integrals pq rs) into symmetry classes, (// /7), II JJ), IJ IJ), and IJ KL). For simplicity we consider only integrals over the Coulomb interaction. To make the classification, we need to know what the symmetry of a spinor overlap density is. We do not need to take the spin into account, as we did for the two-particle wave functions, because the spin is integrated out. Using the tools developed in section 10.3, and the result in (10.47), the symmetry of a charge density can be represented as follows ... 

Figure A10.8 The contributions to the three terms for the electron nuclear potential energy difference between H2 in the first excited state, with the electron in 2a ) and H integrated over planes perpendicular to the molecular axis, (a) The interaction of the density described by each s,) basis function with the corresponding nuclear centre (thin line) is compared with the same interaction for the isolated atoms scaled by a factor of 0.5 (dashed line) the difference gives a negative contribution to the bond formation energy (thick line), (b) The interaction of the density described wholly by the Si> basis function with the potential due to the H2 nucleus the corresponding integral for IS2) with H, is shown as a dotted line, (c) The integral value versus z from the interaction of the overlap density with H the interaction with Hi is shown as a dotted line.

The difference between this result and the molecular orbital result is the factor S (the overlap integral) in the second term of the valence-bond result. The probability density in the overlap region is smaller than with the LCAOMO function, since the overlap integral S is smaller than unity. However, the electron moves over the entire molecule in much the same way as with the LCAOMO wave function. 

In the connnonly used atomic sphere approximation (ASA) [79], the density and the potential of the crystal are approximated as spherically synnnetric within overlapping imifiBn-tin spheres. Additionally, all integrals, such as for the Coulomb potential, are perfonned only over the spheres. The limits on the accuracy of the method imposed by the ASA can be overcome with the fiill-potential version of the LMTO (FP-LMTO)... 

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