ABSTRACT
Ever since the advent of quantum mechanics attempts have been made to express it in terms of 3D classical-like quantities. According to density-functional theory (DFT) the single-particle density ρ(r) contains all information of an N -electron system in its ground state [1]. It is defined as
ρ(r) = N ∫
· · · ∫
Ψ∗(r, r2, . . . , rN)Ψ(r, r2, . . . , rN)dr2 · · · drN (14.1)
where Ψ(r1, r2, . . . , rN) is the N -electron wavefunction of the system with external potential υ(r) and total electronic energy E. The map between υ(r) and ρ(r) is invertible up to a trivial additive constant and all ground state properties are unique functionals of ρ(r) which is obtainable through the solution of the following EulerLagrange equation,
δE[ρ] δρ
= µ. (14.2) In Equation 14.2, E[ρ] is the energy functional and µ is the chemical potential which is the Lagrange multiplier associated with the normalization condition,∫
ρ(r)dr = N . (14.3)
T&F Cat # K11224, Chapter 14, Page 224, 15-9-2010
A time dependent (TD) variant of DFT is also available [2] which allows one to invert the mapping between the TD external potential and the TD density up to an additive TD function. All TD properties are unique functionals of density ρ(r, t) and current density j(r, t). The TDDFT may be used through the solution of a set of TD KohnSham type equations [2,3]. Alternatively, one may resort to a many-particle version of Madelung’s quantum fluid dynamics (QFD) [4] to obtain the basic QFD variables ρ(r, t) and j(r, t) and in turn all other TD properties. The backbone of QFD comprises two equations, viz., an equation of continuity,
∂ρ(r, t) ∂t
+ ∇.j(r, t) = 0 (14.4a) and an Euler type equation of motion,
∂j(r, t) ∂t
= Pυ(r,t)[ρ(r, t), j(r, t)]. (14.4b) Unfortunately TDDFT does not provide the exact form of the functional
Pv(r,t)[ρ(r, t), j(r, t)]. A quantum fluid density functional theory (QFDFT) [5] has been developed to tackle this problem by introducing an approximate version of this functional, writing the many-particle QFD equations as
∂ρ
∂t + ∇. (ρ∇ξ) = 0 (14.5a)
and ∂ξ
∂t + 1
2 (∇ξ)2 + δG[ρ]
δρ + ∫
ρ(r′, t) |r − r′|dr
′ + υext(r, t) = 0. (14.5b) In the above equations ξ is the velocity potential and the universal Hohenberg-Kohn functional G[ρ] is composed of kinetic and exchange-correlation energy functionals, while υext(r, t) contains any TD external potential apart from υ(r). These two equations and a 3D complex-valued hydrodynamical function Φ(r, t) provide the QFDFT whose basic equation is the following generalized nonlinear Schrödinger equation (GNLSE): [
−1 2 ∇2 + υeff (r, t)
] Φ(r, t) = i ∂Φ(r, t)
∂t , i = √−1 (14.6a)
where
Φ(r, t) = ρ(r , t) 12 exp[iξ(r, t)] (14.6b) ρ(r, t) = Φ∗(r, t)Φ(r, t) (14.6c) j (r, t) = [Φre∇Φim − Φim∇Φre] = ρ∇ξ (14.6d)
and the effective potential υeff (r, t) is given by
υeff (r, t) = δTNW[ρ] δρ
+ δExc[ρ] δρ
+ ∫
δρ(r, t) |r − r′|dr
′ + υext(r, t), (14.6e) TNW[ρ] and Exc[ρ] being respectively the non-Weizsäcker contribution to the kinetic energy and the exchange-correlation energy functionals.
