Density Functional Theory‌ Sample Clauses

Density Functional Theory‌. As the wave function for an N-electron system is a function of 3N independent spatial variables, solving the electronic Schrödinger equation (2.

Density Functional Theory‌. Density Functional Theory (DFT) has become a standard tool for electronic calculations in quantum chemistry and material science. In 1964, Xxxxxxxxx and Xxxx [1] proposed to use the electron density ρ(→r) as single basic quantity for considering the N−electron system located in the external potential Vext [18]. This was the later known as the Xxxxxxxxx-Xxxx theorem, which I will introduce as follows: = − Σ Hˆ = Tˆ + Vˆee + Vext , where Tˆ k2 2m i 2 is the kinetic operator, Vˆee indicates the interaction potential yielded by the N -electrons, and Vext the external potential, including but not only the ionic potential. The Hamiltonian above is non-disputive, since the N - electrons and the external potential completely determine the properties of the system, and therefore the Hamiltonian. The total energy is expressed as: E[ρ(→r), Vext(→r)] = DΨ| Tˆ + Vˆee |ΨE + ∫ dr3Vext(→r)ρ(→r) (2.12) where ρ(→r) is the electron density corresponding to the squared modulus of the wave- fuctions. The HK theorem states that the external potential Vext(→r) is determined, within an additive constant, by electron density ρ(→r) [1]. Directly from the theorem, density ρ(→r) is therefore unique feature of the N -electron system. A simple proof is that, suppose there are two different external potentials Vext and Verxt that correspond to the same electron density ρ(→r). According to the variational principle, it is known that only the Eigen wave-function minimises the associated energy. Two ground-state energies E0 and E0r respectively. correspond to the external potentials Vext and Verxt, E0 = DΨ|Hˆ|ΨE < DΨr|Hˆ|ΨrE = E0r + ∫ dr3(Vext − Verxt)ρ(→r)  E0r = DΨr|Hˆr|ΨrE < DΨ|Hˆr|ΨE = E0 − ∫ dr3(Vext − Verxt)ρ(→r) Adding up two of the inequalities by each side, we obtain E0r + E0 < E0r + E0. This contradictory result suggests that, at ground state the N -electron Hamiltonians must be unique functional of electron density ρ. Based on this theorem, the ground state properties, such as wavefunctions and energies are all uniquely determined by the electron density ρ(→r). The total energy in Eq.2.12 at ground state is thus written as: E[ρ] = F [ρ] + ∫ dr3Vext(→r)ρ(→r) (2.13) D E with F [ρ] corresponding to F [ρ] = Ψ| Tˆ + Vˆee |Ψ and indicating the sum of the kinetic energy and electron-electron interaction energy for a given electron density distribution of ρ. This F [ρ] is a universal functional in the sense that it is only determined by the electron density and independent of any exter...

Density Functional Theory‌. Whilst the BO approximation simplifies the problem considerably, the resulting many- electron Schrödinger equation in Eq.