Ground-state electron density

The so-called Flohenberg-Kolm [ ] theorem states that the ground-state electron density p(r) describing an A-electron system uniquely detemiines tlie potential V(r) in the Flamiltonian... 

Results for these CEBEs are presented in Table 1. As can be seen, for the carvone variants I-V the various substitutions have absolutely no effect at the carbonyl C=0 core, and are barely significant at the chiral center that lies between the carbonyl and substituent groups in these molecules. Only upon fluorine substitution at the tail (molecule VI) does the C=0 CEBE shift by one-half of an electronvolt the second F atom substitution adjacent to the C=0 in the difluoro derivative, VII contributes a further 0.6-eV shift. This effect can be rationalized due to the electron-withdrawing power of an F atom. Paradoxically, it is these fluorine-substituted derivatives, VI, VII, that arguably produce b curves most similar to the original carvone conformer, I, yet they are the only ones to produce a perturbation of the ground-state electron density at the C li core. This contributes further evidence to suggest that, at least for the C li... 

This concludes the proof that there cannot be two different Vext that yield the same ground state electron density, or, in other words, that the ground state density uniquely specifies the external potential Vext. Using again the terminology of Section 1.2 we can simply add p0 as the property which contains the information about N, ZA, RA and summarize this as... 

Since the complete ground state energy is a functional of the ground state electron density so must be its individual components and we can write (where we revert to the subscript Ne to specify the kind of external potential present in our case, which is fully defined by the attraction due to the nuclei)... 

These results, as most related results of density functional theory, have direct connections to the fundamental statement of the Hohenberg-Kohntheorem the nondegenerate ground state electron density p(r) of a molecule of n electrons in a local spin-independent external potential V, expressed in a spin-averaged form as... 

As a consequence of the Hohenberg-Kohn theorem [14], a non-degenerate ground state electron density p(r) determines the Hamiltonian H of the system within an additive constant, implying that the electron density p(r) also determines all ground state and all excited state properties of the system. 

The non-degenerate ground state electron density p/(r) over any subset d of manifold S3, S3 zd d, where subset d has non-zero volume on S3, determines uniquely... 

Wesolowski, T. A. and J. Weber. 1996. Kohn-Sham equations with constrained electron density an iterative evaluation of the ground-state electron density of interacting molecules. Chem. Phys. Lett. 248,71. 

The important conclusion is that in principle, all ground-state molecular properties may be calculated from the ground-state electron density p(x, y, z). The challenge is to find the density and use it to calculate energies. A partial solution was found by Kohn and Sham [105]. 

On the other hand, when the solute is under the influence of the external perturbation, the effective energy functional is minimized by a new electron density p(r), which differs from the ground state electron density of the isolated solute p°(r) by an amount Sp(r). In other words, the effective energy functional E p may be written as [38] ... 

The implication of this theorem is that it gives a prescription for the variational determination of the ground-state electron density, since the latter minimizes the energy. 

Since the ground-state electron density minimizes the energy, subject to the normalization constraint, Jp(r)dr — N = 0, theEuler-Lagrange equation (see Equation 4.23) becomes... 

According to the Hohenberg-Kohn theorem of the density functional theory, the ground-state electron density determines all molecular properties. E. Bright Wilson [46] noticed that Kato s theorem [47,48] provides an explicit procedure for constructing the Hamiltonian of a Coulomb system from the electron density ... 

In Equation 9.21, T yields n and is orthogonal to the first i — 1 state of the Hamiltonian for which n0 is the ground-state density. Here, this Hamiltonian is the // in Equation 9.19. Note that instead of the ground-state electron density n0, we could use the external potential v or any ground-state Kohn-Sham orbital, etc. Thus we could use Ft[n, v]. The extension to degenerate states is studied in Section 9.4. 

The ground-state electronic density p(r) is uniquely related to the external potential Vext(r) as stated by the fundamental theorems of DFT [1,2,8]. At zero field, the external potential of an atom is due to its nuclei and vext(r) = —Ze2/r where Z is the nuclear charge. It is shifted by the quantity V... 

Density functional theory (DFT) uses the electron density p(r) as the basic source of information of an atomic or molecular system instead of the many-electron wave function T [1-7]. The theory is based on the Hohenberg-Kohn theorems, which establish the one-to-one correspondence between the ground state electron density of the system and the external potential v(r) (for an isolated system, this is the potential due to the nuclei) [6]. The electron density uniquely determines the number of electrons N of the system [6]. These theorems also provide a variational principle, stating that the exact ground state electron density minimizes the exact energy functional F[p(r)]. 

An approximate quantum mechanical expressions- that allows one to calculate the electrostatic surface potential around atoms, radicals, ions, and molecules by assuming that the ground-state electron density uniquely specifies the Hamiltonian of the system and thereby all the properties of the ground state. This approach greatly facilitates computational schemes for exact calculation of the ground-state energy and electron density of orbitals. 

The determination of the ground state energy and the ground state electron density distribution of a many-electron system in a fixed external potential is a problem of major importance in chemistry and physics. For a given Hamiltonian and for specified boundary conditions, it is possible in principle to obtain directly numerical solutions of the Schrodinger equation. Even with current generations of computers, this is not feasible in practice for systems of large total number of electrons. Of course, a variety of alternative methods, such as self-consistent mean field theories, also exist. However, these are approximate. 

N. H. March, The local potential determining the square root of the ground-state electron-density of atoms and molecules from the schriklinger equation. Phys. Lett. A 113, 476 78 (1986). 

Density functional theory is originally based on the Hohenberg-Kohn theorem [105, 106]. In the case of a many-electron system, the Hohenberg-Kohn theorem establishes that the ground-state electronic density p(r), instead of the potential v(r), can be used as the fundamental variable to describe the physical properties of the system. In the case of a Hamiltonian given by... 

Compare the calculated electron density, nKs(r), with the electron density used in solving the Kohn-Sham equations, (r). If the two densities are the same, then this is the ground-state electron density, and it can be used to compute the total energy. If the two densities are different, then the trial electron density must be updated in some way. Once this is done, the process begins again from step 2. 

Density functional theory purists are apt to argue that the Hohenberg-Kohn theorem [1] ensures that the ground-state electron density p(r) determines all the properties of the ground state. In particular, the electron momenmm density n( ) is determined by the electron density. Although this is true in principle, there is no known direct route from p to IT. Thus, in practice, the electron density and momentum density offer complementary approaches to a qualitative understanding of electronic structure. 

Now, what if we abandon the orbital-by-orbital electron partitioning in favor of a description based on the stationary ground-state electron density p(r) Clearly, this will oblige us to redefine the coie-valence separation. In sharp contrast with what was done in orbital space, we need a partitioning in real space. Let us begin with isolated atoms. 

The spirit of the Hohenberg-Kohn theorem is that the inverse statement is also true The external potential v(r) is uniquely determined by the ground-state electron density distribution, n(r). In other words, for two different external potentials vi(r) and V2(r) (except a trivial overall constant), the electron density distributions ni(r) and 2(r) must not be equal. Consequently, all aspects of the electronic structure of the system are functionals of n(r), that is, completely determined by the function (r). 

The total energy E of the system is also a functional of the density distribution, E = [] (r)]. Therefore, if the form of this functional is known, the ground-state electron density distribution n t) can be determined by its Euler-Lagrange equation. However, except for the electron gas of almost constant density, the form of the functional [ (r)] cannot be determined a priori. 

For a fixed external potential v(f), the ground-state electron density po satisfies the variational equation ... 

More recently, a research group at the DuPont de Nemours Company has studied the benzo derivatives 254b, 255, 328, and 329.336,34°, 388 Scheme 20 shows the preferred substitution positions in these compounds experimental results are generally in accord with ground-state electron density calculations388 (Section V,B). Compounds 255340 and 254b336 have been particularly closely studied. 

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