## 1 Introduction

## 2 Density Functional Theory of Quantum and Classical Systems

**r**) which is defined, for an N-particle system, by integrating the N-particle distribution function $P({\mathbf{r}}_{1},{\mathbf{r}}_{2},\dots ,{\mathbf{r}}_{N})$ over N − 1 variables, as

**r**)d

**r**= N. This concept of single-particle density is valid for both quantum and classical systems and for the former, one has $P({\mathbf{r}}_{1},{\mathbf{r}}_{2},\dots ,{\mathbf{r}}_{N})$ = |ψ$({\mathbf{r}}_{1},{\mathbf{r}}_{2},\dots ,{\mathbf{r}}_{N})$|

^{2}where ψ is the manyelectron wavefunction. The possibility of describing a quantum or classical many-particle system completely in terms of the reduced one-particle density has been rigorously proved by Hohenberg and Kohn [1] and Mermin [2] (HKM) who demonstrated for the first time a one to one mapping between the density and the potential.

**r**) (arising due to the nuclei in the case of electrons and due to the walls or pores in the case of confined classical fluid particles), the ground state energy E

_{v}[ρ] (for a many-electron quantum system) or, the grand potential Ω

_{v}[ρ] (for a many-particle classical system) can be expressed as the unique functionals of density, given respectively by

_{s}[ρ] + E

_{coul}[ρ] + E

_{xc}[ρ]

_{s}[ρ], E

_{coul}[ρ] and E

_{xc}[ρ] represent the noninteracting kinetic energy, classical Coulomb energy and the exchange-correlation (XC) energy respectively. The energy functional E

_{v}[ρ] with the known exact expressions for T

_{s}[ρ] and E

_{coul}[ρ] as given by

**r**) is obtained as the sum

_{id}[ρ] + F

_{ex}[ρ]

_{id}[ρ] (corresponding to absence of internal interactions) is the analogue of the noninteracting kinetic energy T

_{s}[ρ] of Eqs. (5) and (6), and is given by the exact expression

_{0}(= 1/k

_{B}T) is the inverse temperature and Λ is the thermal de-Broglie wavelength. The quantity F

_{ex}[ρ] representing the excess free-energy for the classical system is analogous to the interaction energy functional (E

_{coul}[ρ] + E

_{xc}[ρ]) of the quantum system and is a universal functional of density for a specified interparticle interaction. The Euler-Lagrange equation (4) for this classical system is given by

**r**) = Λ

^{−3}exp(β

_{0}µ) exp{−β

_{0}v(

**r**) + c

^{(1)}(

**r**; [ρ])}

_{eff}(

**r**; [ρ]) = [v(

**r**) − ${\beta}_{0}^{-1}$c

^{(1)}(

**r**)], where the first order direct correlation function (DCF) c

^{(1)}(

**r**) is defined as the functional derivative

^{(1)}(

**r**; [ρ]) to the effective potential, arising from interparticle interactions and correlations is analogous to the exchange-correlation contribution to the Kohn-Sham potential for quantum systems. It may be noted that a spin-polarized quantum system and a two-component classical fluid mixture can be treated using an extended version of the above respective frameworks with two density components as the basic variables.

**r**) into another system of N noninteracting particles of the same density ρ(

**r**) but moving in an effective potential v

_{eff}(

**r**; [ρ]), which itself depends on the density, thus requiring a self-consistent iterative procedure for solution of the Kohn-Sham or the Boltzmann-like density equations.

_{eff}(

**r**; [ρ]) as a density functional for inhomogeneous systems, it is however essential to approximate the XC energy functional E

_{xc}[ρ] for quantum systems, and the excess free energy functional F

_{ex}[ρ] or its derivatives for classical systems. The knowledge of the functionals for specific systems with homogeneous density are often useful in approximating the functionals for the corresponding inhomogeneous systems. For many-electron quantum systems, a standard simple scheme is the local density approximation [10] (LDA) where the expression for the XC energy functional of the homogeneous system is directly evaluated using the inhomogeneous density, viz.

**r**) is smoothed out by coarse graining with a weight function w(

**r**,

**r**′; $\overline{\rho}$(

**r**)) to obtain an effective density $\overline{\rho}$(

**r**) as

## 3 Electron Density and Microscopic Modeling of Intra- and Inter-molecular Interaction Potential

_{α}(

**r**) and ρ

_{β}(

**r**), (which integrate to the corresponding number of electrons N

_{α}and N

_{β}), as the basic variables, the spin-polarised version of the energy functional of Eq. (2) can be written as

**r**) with consequent density changes δρ

_{α}(

**r**) and δρ

_{β}(

**r**), the resulting energy change can be expressed [17, 18] using the functional Taylor expansion (retaining terms upto second order) as

^{2}E/δρ

_{ν}(

**r**)δv(

**r**′)) = δ(

**r**−

**r**′), yields the up- and down-spin chemical potentials given (upto first order) by

_{α}) and (∂E/∂N

_{β}) respectively. These are the key equations providing a justification for the chemical potential equalisation within a molecular species. Here the hardness kernel [21] η

_{µν}(

**r**,

**r**′) represents the energy functional derivative

_{µν}(

**r**) as well as the cross hardness η

_{αβ}= (∂

^{2}E/∂N

_{α}∂N

_{β}) as

_{ν}(

**r**) = (∂ρ

_{ν}(

**r**)/∂N

_{ν}).

**R**

_{i}} corresponding to a particular (say, equilibrium) configuration of the molecule and therefore the expressions of chemical potential given by Eqs. (22) and (23) are now recast to obtain the same for the i-th atomic or bond site as given by

_{β,i}obtained by interchanging α with β. This expression corresponds to the partitioning of the density changes δρ

_{ν}(

**r**) on molecule formation as a sum of the atomic and bond region components as δρ

_{ν}(

**r**) = ∑

_{i}δρ

_{ν,i}(

**r**) = ∑

_{i}δρ

_{ν,i}(

**r**

_{i}), with

**r**

_{i}denoting the atom or bond region around the i-th site location

**R**

_{i}. Also, it is assumed without any loss of generality that the density components δρ

_{ν,i}(

**r**) vanish outside this region

**r**

_{i}belonging to the i-th site. Introducing the assumption η

_{µν}(

**r**

_{i},

**r**′

_{j}) ≈ η

_{µν}(

**R**

_{i},

**R**′

_{j}) for the hardness kernel, the chemical potential for the i-th site as given by Eq. (28) can be further simplified as

_{β,i}given by

_{ν,j}denotes the atomic or bond site charges. One thus arrives at the lattice model of a molecular species with point charges (zeroeth moment of the density) located at the lattice sites. An equalisation of the effective chemical potentials of each spin as given by Eqs. (29) and (30) for all the M atomic and P bond sites leads to M + P − 1 linear equations in the charge variables for each spin which along with the charge conservation yield the individual site charges. The expression for the energy change as given by Eq. (21) as reexpressed in terms of these charges is given by

_{αα}(i, i), η

_{ββ}(i, i) and η

_{αβ}(i, i) of the i-th atom can be obtained from spin polarised DFT calculations as has been done earlier [26]. The bond site chemical potential as well as the hardness parameters can be approximated [28] by suitable averaging [29] of the corresponding values for the neighbouring atoms. For the off-diagonal (i ≠ j) elements of the hardness kernel η

_{µν}(i, j), one can essentially employ the atom-in-molecule hardness matrix concept of Nalewajski [30] generalised for the spin-dependence and model along similar lines [31] following basically an electrostatic analogy details of which have been discussed elsewhere [18]. For a nonbonded pair of sites (atom or bond), one considers a Coulomb potential η

_{αβ}(i, j) = 1/∊R

_{ij}, with ∊ as the dielectric constant of the electron cloud medium. while for bonded sites, a better modeling [31] involves the Mataga-Nishimoto-Ohno formula as η

_{αβ}(i, j) = 1/(R

_{ij}+ ${a}_{ij}^{\alpha \beta}$) with ${a}_{ij}^{\alpha \beta}$ = 2/(${\eta}_{\alpha ,i}^{0}$ + ${\eta}_{\beta ,i}^{0}$). Thus, the DFT framework with the simplifications through the associated concepts is shown to lead to a simple equation (Eq.(31)) for predicting the intra- as well as inter-molecular interaction energies.

## 4 Density Functional Theory of Soft Matter: A Mesoscopic Domain

_{µν}(

**r**

_{12}) with µ, ν = α, β denoting the two components. The fluid mixture is confined in a cavity (a slit between two planar walls or a spherical or cylindrical pore) which provides the external potential v

_{ν}(

**r**) and the resulting inhomogeneous density distribution is ρ

_{ν}(

**r**) for the ν-th component. This inhomogeneous fluid mixture is considered to be in equilibrium with the corresponding bulk phases of densities ${\rho}_{\alpha}^{0}$ and ${\rho}_{\beta}^{0}$ and chemical potentials µ

_{α}and µ

_{β}for the two components. The DFT for this system is quite analogous to the spin polarised DFT for the many-electron systems considered in the previous section.

_{α},ρ

_{β}] for this two-component system can be written as

_{ex}[ρ

_{α},ρ

_{β}] is the excess free energy arising from interparticle interaction and correlation effects. The grand potential, on minimisation, with respect to the component densities leads to the Euler-Lagrange equations, the final forms of which as obtained after equating the component chemical potentials with those of the bulk phase, are given by

**r**; [ρ

_{α},ρ

_{β}]) defined for the two component system as

**r**; [ρ

_{α},ρ

_{β}]) is unknown, one can employ a functional Taylor expansion for ${c}_{\nu}^{\left(1\right)}$(

**r**; [ρ

_{α},ρ

_{β}]) around the same for the corresponding homogeneous system in powers of the density deviation ∆ρ

_{ν}(

**r**) = [ρ

_{ν}(

**r**) − ${\rho}_{\nu}^{0}$] as

_{ν}(

**r**) represents the second order correction term given by

**r**

_{1},

**r**

_{2}) defined as

_{ν}(

**r**) denotes contributions from all the higher order terms. The corresponding density equation is given by

_{ν}(

**r**) can be evaluated easily while the higher order DCF’s are unknown and hence are approximated for practical implementation. Recently, we have proposed two approximate approaches to calculate ∆c

_{ν}(

**r**), one of which estimates a third order contribution [16] through an approximation for the third order DCF in the spirit of Kirkwood’s superposition approach, and also generalisation of the one-component result of Rickayzen and coworkers [37]. The conditions that the density functional as used here should yield the correct bulk partial pressures in the homogeneous limit are imposed to evaluate the parameters appearing in the approximation. The other alternative route that we have proposed [38] evaluates ∆c

_{ν}(

**r**) using the concept of bridge function used widely in the integral equation theory of homogeneous fluid mixtures.

## 5 Density Functional Theory of Dynamical Phenomena

**r**, t) and TD current density

**j**(

**r**, t). The TD DFT [43] for quantum systems involve the TD one-particle Kohn-Sham like equations of the form

_{eff}(

**r**, t) is the effective TD potential dependent on the densities determined by the orbitals. The equivalent quantum hydrodynamic formulation [44] consists of the continuity equation corresponding to the time evolution of the density given by

**r**,

**r**′; t).

## 6 Concluding Remarks

## Acknowledgments

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