Electron density integrals

Electron Density Integrals and Atoms-in-Molecules Methods... 

Equation 2.133 is most commonly used to calculate the distributions of the electron (nuclear) density in the unit cell, which are also known as Fourier maps, from x-ray (neutron) diffraction data, respectively. The locations of peaks on the Fourier map calculated using x-ray diffraction data represent coordinates of atoms, while the electron density integrated over the range of the peak corresponds to the number of electrons in the atom. The major problem in using Eq. 2.133 is that only the absolute values of the... 

Contreras-Garcia J, Johnson ER, Yang W (2011) Analysis of hydrogen-bond interaction potentials from the electron density integration of noncovalent interaction regions. J Phys Chem A 115 12983-12990... 

Figure 7.9 The contributions of the electron kinetic and potential energy expectation values to the interatomic potential curve for H2. The internuclear term is also added to ensure the potential goes to zero at large R - The inset plot (bottom right) shows the electron density integrated over planes perpendicular to the bond axis with the basis set decay factor optimized at the potential minimum the dotted curve shows the density that would be obtained using the decay factor for an isolated H atom.

Faraday rotation also occurs in a free electron gas where the amount of rotation depends on the product of B and the electron density integrated along the path. This is a tool used in astrophysics for estimating the magnetic field on the sim, stars, and in the interstellar medium. It also provides a method of deterinming the electron density in the ionosphere. 

The advantage of using electron density is that the integrals for Coulomb repulsion need be done only over the electron density, which is a three-dimensional function, thus scaling as. Furthermore, at least some electron correlation can be included in the calculation. This results in faster calculations than FIF calculations (which scale as and computations that are a bit more accurate as well. The better DFT functionals give results with an accuracy similar to that of an MP2 calculation. 

A more complex set of functionals utilizes the electron density and its gradient. These are called gradient-corrected methods. There are also hybrid methods that combine functionals from other methods with pieces of a Hartree-Fock calculation, usually the exchange integrals. 

Molecular volumes are usually computed by a nonquantum mechanical method, which integrates the area inside a van der Waals or Connolly surface of some sort. Alternatively, molecular volume can be determined by choosing an isosurface of the electron density and determining the volume inside of that surface. Thus, one could find the isosurface that contains a certain percentage of the electron density. These properties are important due to their relationship to certain applications, such as determining whether a molecule will fit in the active site of an enzyme, predicting liquid densities, and determining the cavity size for solvation calculations. 

The subscripts i and j denote two nuclei one in the QM region and one in the MM region. The atomic charges for the MM atoms are obtained by any of the techniques commonly used in MM calculations. The atomic charges for the QM atoms can be obtained by a population analysis scheme. Alternatively, there might be a sum of interactions with the QM nuclear charges plus the interaction with the electron density, which is an integral over the electron density. 

The Extended Hiickel method neglects all electron-electron interactions. More accurate calculations are possible with HyperChem by using methods that neglect some, but not all, of the electron-electron interactions. These methods are called Neglect of Differential Overlap or NDO methods. In some parts of the calculation they neglect the effects of any overlap density between atomic orbitals. This reduces the number of electron-electron interaction integrals to calculate, which would otherwise be too time-consuming for all but the smallest molecules. 

The NDDO (Neglect of Diatomic Differential Overlap) approximation is the basis for the MNDO, AMI, and PM3 methods. In addition to the integralsused in the INDO methods, they have an additional class of electron repulsion integrals. This class includes the overlap density between two orbitals centered on the same atom interacting with the overlap density between two orbitals also centered on a single (but possibly different) atom. This is a significant step toward calculatin g th e effects of electron -electron in teraction s on different atoms. 

Hohenberg and Kohn demonstrated that is determined entirely by the (is a functional of) the electron density. In practice, E is usually approximated as an integral involving only the spin densities and possibly their gradients ... 

Integration of P(r) with respect to the coordinates of this electron (now written r) gives the number of electrons, 2 in this case. In the case of a many-electron wavefunction that depends on the spatial coordinates of electrons 1,2,..., m, we define the electron density as... 

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