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 (2.10) using traditional ab-initio quantum chemistry approaches, such as configuration interaction (CI), becomes computationally very demanding for large molecular complexes. Density functional theory (DFT) offers a valuable alternative that is computationally more efficient than ab-initio methods and yet is quite accurate. This theory is based on the Xxxxxxxxx-Xxxx theorem [2] showing that it is possible to use just the electronic density to calculate the ground-state energy E0. Hence the complexity of the problem decreases drastically, since the electronic density is a function of only three coordinates, and is independent of the number of electrons. Given an external potential vext(r), the total electronic energy can be expressed as a functional of the electron density ρ(r), E ≡ Eν [ρ(r)]≡ ∫ν ext (r)ρ(r)dr + F [ρ(r)], (2.13) where F[ρ(r)] is an universal functional and the integral runs over the entire space. For molecular systems the external potential vext(r) is usually the potential generated by the nuclei (see equation 2.9). The Xxxxxxxxx-Xxxx theorem also proves that a variational principle for the energy functional holds: For any density, the energy given by the corresponding energy functional is never smaller than the ground-state energy, and the ground-state energy is obtained only using the exact ground-state electron density. The total energy functional can be written as a sum of different terms: Eν [ρ] = T [ρ]+Ve-N [ρ]+Ve-e [ρ], (2.14) where T[ρ] is the electronic kinetic energy, Ve-N[ρ] is the energy due to the electron-nuclear attraction, and Ve-e[ρ] is the electron-electron repulsion energy, which can be decomposed into classical Coulomb and non-classical contributions: Ve-e [ρ] = J [ρ]+V nc [ρ]. (2.15) e-e While Ve-N[r] and J[ρ] can be calculated in terms of the electron density as and Ve-N [ρ]= ∫ν ext (r)ρ(r)dr (2.16) ∫ J [ρ]= ρ(r)ρ(r') drdr' , (2.17) r − r' the explicit form for the kinetic energy functional and the non-classical electron-electron repulsion energy are unknown. In order to simplify the calculations of the kinetic energy term, Xxxx and Sham proposed to compute Ts[ρ], which is the kinetic energy for a set of independent particles. This term can be easily written in terms of a set of orbitals, known as the Xxxx-Sham orbitals. The remaining unknown part of the kinetic energy Tc[ρ] is th...

Density Functional Theory‌. Whilst the BO approximation simplifies the problem considerably, the resulting many- electron Schrödinger equation in Eq.(2.5) is intractable for all but the simplest of atomic and molecular systems. The difficulty arises due to the dimension of the many-electron Xxxxxxx space. This is in direct analogy to the situation in the case of the total many-body Schrödinger equation. Like in the BO approximation, we first make the product-state ansatz O{ } Ne ψe ( r ) = ψα (σα, rα) (2.9) α=1 where Ne = ZaN is the number of electrons, ψα are one-electron wavefunctions and σα is the electron spin. In terms of the one-electron orbitals, we may write the wave- functions as ψα (σα, rα) = ψ˜α (rα) χα (σα) (2.10) where χ are the one-electron spinors and ψ˜ are the one-electron orbitals. We will ignore this distinction between the one-electron orbitals and the one electron wavefunctions. Furthermore, we shall implicitly include the spin within r. We have essentially re- stricted the solution to an unentangled subspace within the many-electron Xxxxxxx space. This approximation is referred to as the Hartree approximation [42, 43]. We recognise that Ee can be written as a functional of the one-electron orbitals Ee [{ψα}] = drα ψ1 (r1) · · · ψNe (rNe ) Heψ1 (r1) · · · ψNe (rNe ) (2.11) ∫ Σ Ne Σ ∗ ∗ α=1 subject to the orthonormality constraint ∫ dr ψα∗ (r) ψβ (r) = δαβ (2.12) where δ is the Kronecker delta. Taking the functional derivative of the above δEe/δψα∗ , and introducing the orthonormality constraint through the Lagrange multiplier Eα, yields the following equation for the one-electron orbitals Σ ℏ2 2 − 2m∇ + Veff (r)Σ ψα (r) = Eαψα (r) (2.13) which is equivalent to the Rayleigh-Ritz principle. Interpreting this equation is rather simple; it is the Schrödinger equation for a non-interacting electron moving in an effective potential Veff . The effective potential Veff depends on all of the occupied one- electron orbitals being calculated through the electron density n(r). This means that this partial differential equation is non-linear and must be solved self-consistently. In the Hartree approximation, the effective potential is given by Veff = Ven + VH (2.14) where Ven is the external potential due to the nuclei configuration and VH is the Hartree potential. The Hartree potential VH(r0) is the classical Coulomb potential arising at the point r0 from the mean charge density −en(r) of all the other electrons in the system. The electron density n(r) can be written as Σ n...

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 Xxxxxxxxx-Xxxx (HK) theorem. Let us start from writing the Hamiltonian as Hˆ = Tˆ + Vˆee + Vext , where Tˆ k2 2m i 2 is the kinetic operator, Vˆee indicates i 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 t...