Kato theorem

IV) A self-contradiction (ad absurdum) of Eq. (22) might also mean that the to-be-refuted assumptions (i) or/and (ii) of the Hohenberg-Kohn theorem are selfcontradictory with Eq. (19) and this is precisely the case of many-electron Coulomb systems with Coulomb-type class of external potentials. In other words, the original reductio ad absurdum proof of the Hohenberg-Kohn theorem based on the assumption (19) is incompatible with the ad absurdum assumption (ii) since the Kato theorem is valid for such systems [18]. 

Deb [31], Smith [32], and E. Bright Wilson (quoted by Lowdin [33] for the recent applications of the Kato theorem to the Hohenberg-Kohn theorem see also [34—36]). Therefore, if a given pair of iV-electron systems with the Hamiltonians Hi and H2 of the type (1) are characterized by the same groxmd-state one-electron densities (= to-be-refuted assumption (ii)), their external potentials Vi(r) and V2(r) of the form (24) are identical. The latter contradicts (19) and hence, the assumption (ii) cannot be used in the proof via reductio ad absurdum of the Hohenberg-Kohn theorem together with the assumption (19). In other words, they are Kato-type incompatible with each other. 

Two theories for a single excited state [37—401 are the focus of this chapter. A nonvariational theory [37,38] based on Kato s theorem is reviewed in Section 9.2. Sections 9.3 and 9.4 summarize the variational density functional theory of a single excited state [39,40], Section 9.5 presents some application to atoms and molecules. Section 9.6 is devoted to discussion. 

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 ... 

Consequently, from the density the Hamiltonian can be readily obtained, and then every property of the system can be determined by solving the Schrodinger equation to obtain the wave function. One has to emphasize, however, that this argument holds only for Coulomb systems. By contrast, the density functional theory formulated by Hohenberg and Kohn is valid for any external potential. Kato s theorem is valid not only for the ground state but also for the excited states. Consequently, if the density n, of the f-th excited state is known, the Hamiltonian H is also known in principle and its eigenvalue problem ... 

Two approaches to the excited-state problem have been the focus of this chapter. The nonvariational one, based on Kato s theorem, is pleasing in that it does not require a bifunctional, but it presumes that the excited-state density is known. On the other hand, the bifunctional approach is appealing in that it actually generates the desired excited-state density, which results in the generation of more known constraints on the universal functional for approximation purposes. 

The method discussed so far originates from an analysis of n-electrons quantum states based on general mathematical theorems such as Kato s [18]. The particle model is not an element of the theory. In this sense, no direct relation to orbital methods including diabatic ones can be found at the present level of development. The diabaticity is related to a global conservation of the nodal planes. And the calculations reported here have been monitored to keep up to this level. 

To show this we first consider [Pg.40]

A better known relation, also involving the density, is the theorem of Kato.11 For a spherically symmetric atom or ion, it relates the electron density at the nucleus to the derivative of p also taken at the nucleus through... 

This important property has been proven by Ayers in two steps. The fact that nuclear positions, Ra, and the atomic numbers of the nuclei, Za, can be determined from the cusp conditions... 

Another alternative argument is valid only for Coulomb potentials. It is based on Kato s theorem, which states [29, 30] that for such potentials the electron density has a cusp at the position of the nuclei, where it satisfies... 

Here R/ denotes the positions of the nuclei, Zk their atomic number, and a0 = h2/me2 is the Bohr radius. For a Coulomb system one can thus, in principle, read off all information necessary for completely specifying the Hamiltonian directly from examining the density distribution the integral over n(r) yields N, the total particle number the position of the cusps of n(r) are the positions of the nuclei, R. and the derivative of n(r) at these positions yields Zf, by means of Eq. (19). This is all one needs to specify the complete Hamiltonian of Eq. (2) (and thus implicitly all its eigenstates). In practice one almost never knows the density distribution sufficiently well to implement the search for the cusps and calculate the local derivatives. Still, Kato s theorem provides a vivid illustration of how the density can indeed contain sufficient information to completely specify a nontrivial Hamiltonian.11... 

Note that, unlike the full Hohenberg-Kohn theorem, Kato s theorem does apply only to superpositions of Coulomb potentials, and can therefore not be applied directly to the effective Kohn-Sham potential. 

For an essentially self-adjoint operator, a theorem by Kato [14] ensures the existence of a complete set of basis functions, jfc(q), for any partitioning of the molecular Coulombic Hamiltonians. (See Refs. [12,14,15] for detailed discussions.) The present GED ansatz is defined by the existence of a unique set [ 0 t(q) that diagonalizes the Hamiltonian He(q, for all [10,12] ... 

Kato 1 ) has derived very interesting theorems that hold for the true solutions of the Schrodinger equation, e.g. for a two-electron system (y means the angular average of ip over the angle 0 between fi and... 

Bingel i ) has investigated the consequence of Kato s theorems for the pair density. It seems that generally the pair density has a cusp like that shown in Fig. 3. The Coulomb hole is not as deep as the Fermi hole and it has a cusp. 

Kato, T. 1950. On the adiabatic theorem of quantum mechanics. Journal of the Physical Society of Japan 5 435. 

A recently proposed theory for a single excited state based on Kato s theorem is reviewed. This theory is valid for Coulomb systems. The concept of adiabatic connection leads to Kohn-Sham equations. Differenticil virial theorem is derived. Excitation energies and inner-shell transition energies are presented. 

Prom the basic theorems of the density functional theory we know that the ground state electron density contains in principle all the information about the system. For Coulomb system this statement can be easily understood following Bright Wilson s argument Kato s theorem states that... 

In the following we restrict our study to Coulomb systems. Kato s theorem is valid not only for the ground state but also for the excited states. Consequently, if the density ni of the i-th excited state is known, the Hamiltonian H is also in principle known and its eigenvalue problem... 

Abstract The theory for a single excited state based on Kato s theorem is revisited. Density scaling proposed by Chan and Handy is used to construct a Kohn-Sham scheme with a scaled density. It is shown that there exists a value of the scaling factor for which the correlation energy disappears. Generalized OPM and KLI methods incorporating correlation are proposed. A KLI method as simple as the original KLI method is presented for excited states. 

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