Electron density defined

Calculate the electron density defined by the Kohn-Sham singleparticle wave functions from step 2, Ks(r) = 2 v (r)v ,(r). 

The density pa(r) may now be expressed as the sum of the average electron density (defined by the expectation value based on solute wavefuction i//a)5 and the nuclear density,... 

The density functional theory of Hohenberg, Kohn and Sham [173,205] has become the standard formalism for first-principles calculations of the electronic structure of extended systems. Kohn and Sham postulate a model state described by a singledeterminant wave function whose electronic density function is identical to the ground-state density of an interacting /V-clcctron system. DFT theory is based on Hohenberg-Kohn theorems, which show that the external potential function v(r) of an //-electron system is determined by its ground-state electron density. The theory can be extended to nonzero temperatures by considering a statistical electron density defined by Fermi-Dirac occupation numbers [241], The theory is also easily extended to the spin-indexed density characteristic of UHF theory and of the two-fluid model of spin-polarized metals [414],... 

The electron density defined by a MO can be decomposed into atomic and bond contributions. 

The electron distribution in a system is calculable using the coefficients. The Mulliken procedure (19), based on the fact that the integral of represents an electron density, defines the charge on atom r as... 

Extra information on electronic changes occurring when O2 binds to a complex may be obtained through the difference electron density, " defined as... 

Following the same reasoning as in Ref. [64], the energy functional given by Equation (1) may be constructed through an appropriate ansatz variational representation for the electron density defined as ... 

To summarize, the ground-state wavefunction, defined in the 4/V-dimensional configuration space, and the electron density, defined in the 3-dimensional physical space, constitute exactly equivalent definitions of the quantum-mechanical state of the iV-electron molecular system. They both carry the complete physical information about the system under consideration. The theory of electronic structure of matter can thus be rigorously based on the electronic density and this offers both computational and conceptual/interpretative advantages. 

The first box corresponds to p = 0.50, and only the core electrons for the carbon atoms are visible. For p = 0.32 the hydrogens also appear and bonds can be recognized for p = 0.20. Further reduction in the electron density level used for generating the surface obscures the bonding information, and for p = 0.001 there is little information about the underlying molecular structure left. A surface generated by a constant value of the electron density defines the size and shape of a molecule, but the exact size and shape clearly depend on the value chosen. It can be noted that an isocontour value of 0.001 has often been taken to represent a van der Waals type surface. 

Also shown in Figure 1.29(c) for the Is, 2s, and 3s orbitals are the corresponding boundary surface diagrams, defined as a surface containing 90 percent of the total electron density (defined as in the orbital. The boundary surface diagram serves as a useful representation of the shape of the orbital. All s orbitals are spherical but differ in size, which increases as the principal quantum number increases. 

We have seen that molecular electron density distributions are dominated by contributions of steeply falling electron densities around the nuclei, whieh we can attribute in the main to the core electrons. Superimposed on these are smaller contributions Ifom a few valence electrons, which may be bonding or nonbonding. These are loeal eharge concentrations. From the chemieal point of view the latter are the more interesting features, and if we derive the second derivative of the three-dimensional electron density, defined as the sum of partial seeond derivatives, Eq. 10.18, we can study them in more detail. 

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