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Kohn-Sham density functional theory with the available exchange–correlation functionals is less accurate for strongly correlated systems, which require a multiconfigurational description as a zero-order function, than for weakly correlated systems, and available functionals of the spin densities do not accurately predict energies for many strongly correlated systems when one uses multiconfigurational wave functions with spin symmetry. Furthermore, adding a correlation functional to a multiconfigurational reference energy can lead to double counting of electron correlation. Multiconfiguration pair-density functional theory (MC-PDFT) overcomes both obstacles, the second by calculating the quantum mechanical part of the electronic energy entirely by a functional, and the first by using a functional of the total density and the on-top pair density rather than the spin densities. This allows one to calculate the energy of strongly correlated systems efficiently with a pair-density functional and a suitable multiconfigurational reference function. This article reviews MC-PDFT and related background information.

The density per particle σ carries the same information of the electron density ρ. However, σ and ρ exhibit different local behavior and consequently their respective topological functions should be different. Paradoxically, the differential geometry equations supporting their topological behavior are the same. This fact implies, in particular, the invariance of the molecular structure and that of the atomic basin. But other topologically-relevant functions—such as the gradient, the unit-less gradient, the Laplacian, and the one-electron potential, among others—may behave differently. The scale transformation ρ = N σ applied to these functions reveals that the only first-degree invariant function is the logarithmic derivative of the density. This function shows the local symmetry around the stationary points of the density. Present work exemplifies what seems to be a general principle: for determining a well-constructed density function it should be formulated equally for both the electron density and the density per particle.

The topology of the Coulomb potential density has been studied within the context of the theory of Atoms in Molecules and has been compared with the topologies of the electron density, the virial energy density and the Ehrenfest force density. The Coulomb potential density is found to be mainly structurally homeomorphic with the electron density. The Coulomb potential density reproduces the non-nuclear attractor which is observed experimentally in the molecular graph of the electron density of a Mg dimer, thus, for the first time ever providing an alternative and energetic foundation for the existence of this critical point. © 2017 Wiley Periodicals, Inc.

A link between density and pair density functional theories is presented. The method of Levy, Perdew and Sahni is used to derive a relationship between the effective potentials of the Euler equation of the density functional theory and the Pauli potential of the pair density functional theory and between the effective potential for the spherically and system-averaged pair density and the Pauli potential of the pair density functional theory. These expressions can be considered constraints that the Pauli potential and the effective potentials should fulfil. They might be very useful in constructing approximations.

Based on a computational procedure for determining the functional derivative with respect to the density of any antisymmetric N-particle wave function for a non-interacting system that leads to the density, we devise a test as to whether or not a wave function known to lead to a given density corresponds to a solution of a Schrödinger equation for some potential. We examine explicitly the case of non-interacting systems described by Slater determinants. Numerical examples for the cases of a one-dimensional square-well potential with infinite walls and the harmonic oscillator potential illustrate the formalism.

Density functional theory is currently the most widely applied method in electronic structure theory. The Kohn-Sham method, based on a fictitious system of non-interacting particles, is the work horse of the theory. The particular form of the Kohn-Sham wavefunction admits only idem-potent one electron density matrices whereas wavefunctions of correlated electrons in post-Hartree-Fock methods invariably have fractional occupation numbers. Here we show that by generalizing the orbital concept, and introducing a suitable dot-product as well as a probability density a non-interacting system can be chosen that can represent the one-electron density matrix of any system, even one with fractional occupation numbers. This fictitious system ensures that the exact electron density is accessible within density functional theory. It can also serve as the basis for reduced density matrix functional theory. Moreover, to aid the analysis of the results the orbitals may be assigned energies from a mean-field Hamiltonian. This produces energy levels that are akin to Hartree-Fock orbital energies such that conventional analyses based on Koopmans theorem are available. Finally, this system is convenient in formalisms that depend on creation and annihilation operators as they are trivially applied to single determinant wavefunctions., Corrections in proofs

The chapter is devoted to the study of density functional theory (DFT) which provides an understanding of a wide range of properties from energetics and geometries of molecules to reaction barriers and van der Waals interactions. Starting from the statements of the Hohenberg–Kohn theorem, it develops Kohn–Sham (KS) equations and the linear combination of atomic orbitals ansatz and describes the approach to solve these equations. Different types of exchange–correlation functionals such as LDA, LSDA, GGA, meta-GGA, and hybrid exchange functionals have been discussed. Since the choice of a functional determines the performance of the theory, guidelines for the selection of the right functional for calculations have been given. The limitations, shortcomings, and challenges before DFT, and new developments to address them have been discussed. In the later part of the chapter, the time-dependent density functional theory (TDDFT), which is probably the most promising theory for the treatment of problems of the excited states of molecules, has been given. This also includes topics such as Runge–Gross theorem, derivation of time-dependent KS equations, linear response theory, Casida’s matrix formulation of TDDFT, techniques to calculate electronic excitation, and an account of the various approximate exchange–correlation functionals used for TDDFT calculations.

The differential virial theorem (DVT) is an explicit relation between the electron density ρ(r), the external potential, kinetic energy density tensor, and (for interacting electrons) the pair function. The time-dependent generalization of this relation also involves the paramagnetic current density. We present a detailed unified derivation of all known variants of the DVT starting from a modified equation of motion for the current density. To emphasize the practical significance of the theorem for noninteracting electrons, we cast it in a form best suited for recovering the Kohn–Sham effective potential vs(r) from a given electron density. The resulting expression contains only ρ(r), vs(r), kinetic energy density, and a new orbital-dependent ingredient containing only occupied Kohn–Sham orbitals. Other possible applications of the theorem are also briefly discussed. © 2012 Wiley Periodicals, Inc.

The paper begins with the definition of the electron density given by Schrodinger in 1926 and traces its use and many applications directed at the understanding of the properties of matter to its present day role in density functional theory and the development of the quantum mechanics of an open system.

Conceptual insights from the density functional theory have been enormously powerful in the fields of physics, chemistry and biology. A natural outcome is the concept of “energy density” as has been developed recently: drop region, spindle structure, interaction energy density. Under external source of electromagnetic fields, charged particles can be accelerated by Lorentz force. Dissipative force can make the state of the charged particles stationary. In quantum mechanics, the energy eigenstate is another rule of the stationary state. Tension density of quantum field theory has been formulated in such a way that it can compensate the Lorentz force density at any point of space–time. This formulation can give mechanical description of local equilibrium leading to the quantum mechanical stationary state. The tension density is given by the divergence of stress tensor density. Electronic spin can be accelerated by torque density derived from the stress tensor density. The spin torque density can be compensated by a force density, called zeta force density, which is the intrinsic mechanism describing the stationary state of the spinning motion of electron.

We consider the extension of the standard single-determinant Kohn–Sham method to the case of a multi-configuration auxiliary wave function. By applying the rigorous Kohn–Sham method to this case, we construct the proper interacting and auxiliary energy functionals. Following the Hohenberg–Kohn theorem for both energy functionals, we derive the corresponding multi-configuration Kohn–Sham equations, based on a local effective potential. At the end of the analysis we show that, at the ground state, the auxiliary wavefunction must collapse into a single-determinant wave function, equal to the regular KS wavefunction. We also discuss the stability of the wavefunction in multi-configuration density functional theory methods where the auxiliary system is partially interacting, and the remaining (residual) correlation is evaluated as a functional of the density. As an example showing both the challenges and the possibilities, we implement such a procedure for the perfect pairing wavefunction, using a residual cor...

An explicit 1-particle density matrix functional is derived for the ground state of an (N – 1)-particle system when the ground state wavefunction of the N-particle system is known. The extended Koopmans' theorem is used to express the (N – 1)-particle wavefunction in terms of the N-particle wavefunction. The electron repulsion part of the energy expression is an integral of a product of the 1-particle density matrix of the (N – 1)-particle system and an explicit function that contains correlation and exchange information from the N-particle system.

A variational principle is presented for the ground state energy in terms of the one-particle density matrix. The functional expression of the energy in terms of the density matrix is obtained as the limit of zero temperature of the free energy. This is achieved by adding a nonlocal external potential to the N-body Hamiltonian and obtaining a renormalized equation for the density matrix instead of the Green's function. By eliminating the external source it was possible to express the energy as a functional of the density matrix.

Quantum electronic currents can be determined directly from off-diagonals of the one electron reduced density matrix. Behaviour of the density matrix in a natural orbital expansion is well understood, whereas very little is known about the current density expressed in any orbital basis. Given the relationship of the current to the density matrix, it seems plausible that improved approximations to the reduced density matrix should lead to better descriptions for the current density. To examine how improved approximations to the reduced density matrix are related to the current density, an upper limit to the error resulting from truncating a natural orbital expansion of the current is found and related to the density matrix and kinetic energy expressed in the same basis. As a consequence, it is confirmed that quantum transport formulations leading to an accurate one-electron reduced density matrix will deliver controllable and well-behaved approximations for quantum electronic currents.

The fundamental theorem of time-independent/time-dependent density functional theory due to Hohenberg–Kohn (HK)/Runge–Gross (RG) proves the bijectivity between the density ρ(r)/ρ(rt) and the Hamiltonian /(t) to within a constant C/function C(t), and wave function Ψ/Ψ (t). The theorems are each proved for scalar external potential energy operators. By a unitary or equivalently a gauge transformation that preserves the density, we generalize the realm of validity of each theorem to Hamiltonians, which additionally include the momentum operator and a curl-free vector potential energy operator defined in terms of a gauge function α (R)/α (Rt). The original HK/RG theorems then each constitute a special case of this generalization. Thereby, a fourfold hierarchy of such theorems is established. As a consequence of the generalization, the wave function Ψ/Ψ (t) is shown to be a functional of both the density ρ(r)/ρ(rt), which is a gauge-invariant property, and a gauge function α(R)/α(Rt). The functional dependence on the gauge function ensures that as required by quantum mechanics, the wave function written as a functional is gauge variant. The hierarchy and the dependence of the wave function functional on the gauge function thus enhance the significance of the phase factor in density functional theory in a manner similar to that of quantum mechanics. Various additional perspectives on the theorem are arrived at. These understandings also address past critiques of time-dependent theory. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008

The problems in the density functional theory arising when it is applied to the spin- and orbitally-degenerate states are discussed. It is rigorously proved that the electron density of an arbitrary N-electron system cannot, in principle, depend upon the total spin S and for all values of S has the same form as it has for a single-determinantal wave function. It is also proved that the diagonal element of the density matrix is invariant with respect to the symmetry of the state and in the frame of density matrix description there is no difference between degenerate and nondegenerate states. Thus, the problems in DFT connected with the total spin and degenerate states cannot be rigorously solved within the framework of the density matrix formalism.

This chapter discusses the continuity equation and continuity properties of the wavefunction and its spatial derivatives. In addition, it discusses other theorems which are useful for calculations, such as the variational theorem.

A long-standing challenge in the time-dependent density functional theory is to efficiently solve the exact time-dependent optimized effective potential (TDOEP) integral equation derived from orbital-dependent functionals, especially for the study of nonadiabatic dynamics in time-dependent external fields. In this Letter, we formulate a completely equivalent time-local TDOEP equation that admits a unique real-time solution in terms of time-dependent Kohn-Sham and effective memory orbitals. The time-local formulation is numerically implemented, with the incorporation of exponential memory loss to address the unaccounted for correlation component in the exact-exchange-only functional, to enable the study of the many-electron dynamics of a one-dimensional hydrogen chain. It is shown that the long time behavior of the electric dipole converges correctly and the zero-force theorem is fulfilled in the current implementation.

For a many-electron system, whether the particle density $\rho(\mathbf{r})$ and the total current density $\mathbf{j}(\mathbf{r})$ are sufficient to determine the one-body potential $V(\mathbf{r})$ and vector potential $\mathbf{A}(\mathbf{r})$, is still an open question. For the one-electron case, a Hohenberg-Kohn theorem exists formulated with the total current density. Here we show that the generalized Hohenberg-Kohn energy functional $\mathord{\cal E}_{V_0,\mathbf{A}_0}(\rho,\mathbf{j}) = \langle \psi(\rho,\mathbf{j}),H(V_0,\mathbf{A}_0)\psi(\rho,\mathbf{j})\rangle$ can be minimal for densities that are not the ground-state densities of the fixed potentials $V_0$ and $\mathbf{A}_0$. Furthermore, for an arbitrary number of electrons and under the assumption that a Hohenberg-Kohn theorem exists formulated with $\rho$ and $\mathbf{j}$, we show that a variational principle for Total Current Density Functional Theory as that of Hohenberg-Kohn for Density Functional Theory does not exist. The reason is that the assumed map from densities to the vector potential, written $(\rho,\mathbf{j})\mapsto \mathbf{A}(\rho,\mathbf{j};\mathbf{r})$, enters explicitly in $\mathord{\cal E}_{V_0,\mathbf{A}_0}(\rho,\mathbf{j})$., Comment: 7 pages

The integrated Hellmann–Feynman theorem provides an expression for the energy as a functional of the density. However, this result is completely defined only for the densities corresponding to the eigen functions of the Schrodinger equation. Since this density functional is obtained by integrating the Hellmann–Feynman theorem, it is not possible to obtain bounds to the true energy by using the variation method together an approximate density. This point is discussed through some elementary examples. The relation between the integrated Hellmann–Feynman theorem and the Hohenberg and Kohn theorem, is also stressed in this Letter.

Density functional methods were developed, in which the Coulomb electron-electron interaction is split into a long- and a short-range part. In such methods, one term is calculated using traditional density functional approximations, like the local density approximation. The present paper tries to shed some light upon the best way to do it by comparing the accuracy of the local density approximation with accurate results for the He atom., Comment: 4 pages, 2 figures, to appear in J. Mol. Struct. (Theochem)

Can all the properties of a system composed of particles with masses, mi, and charges, qi, be determined from the charge density and the mass density? Subject to a small statistical caveat, this is true, and offers the prospect of an elegant extension of density-functional theory beyond the Born-Oppenheimer approximation.

Using the constrained search and Legendre-transform formalisms, one can derive “generalized” density-functional theories, in which the fundamental variable is either the electron pair density or the second-order reduced density matrix. In both approaches, theN-representability problem is solved by the functional, and the variational principle is with respect to all pair densities (density matrices) that are nonnegative and appropriately normalized. The Legendre-transform formulation provides a lower bound on the constrained-search functional. Noting that experience in density-functional and density-matrix theories suggests that it is easier to approximate functionals than it is to approximate the set ofN-representable densities sheds some light on the significance of this work.

A Lagrangian theorem-based density functional approximation [S. Zhou, New J. Phys. 4 (2002) 36] for hard sphere fluid is employed to describe non-uniform polymer melt in the framework of density functional theory. A required bulk second order direct correlation function (DCF) within the whole density range is obtained by solving the polymer-RISM integral equation, the associated adjustable parameter is specified by a hard wall sum rule, and is found to be a negative value when the bulk density is low and the number of chain segment is large. However, the mathematically meaningless value can be physically meaningful by the observation that the present recipe can produce out density profile in very good agreement with simulation data not only at the contact region, but also at the region far away from the surface, and that the predicted global quantities such as surface excess and surface tension are also in good agreement with the simulation data. It is considered that the LTDFA has a property of self correction, which enables the LTDFA-based DFA for non-uniform polymer melt performs quite well even with a not very accurate second order DCF as input. Potential applications of the self correction peculiarity are discussed.

The local-scaling density functional theory [J. Phys. Soc. Jpn. 72 (2003) 1926; J. Phys. Soc. Jpn. 73 (2004) 882] is extended to that for systems in magnetic fields of arbitrary strength. Formally ...

Exact time-dependent density functionals depend on both the entire history of the density and the initial wave function. Recently we have shown that the two effects are entangled by an exact condition that should be useful as a test in building accurate memory-dependent functionals. Here we consider this condition for small perturbations and derive an exact relation connecting the exchange-correlation kernel with initial-state derivatives. We discuss other exact conditions on the initial-state derivatives which shed some light on the elusive memory-dependence in time-dependent density functional theory. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

We present a density functional theory for inhomogeneous fluids at constant external pressure. The theory is formulated for a volume-dependent density, n(r,V), defined as the conjugate variable of a generalized external potential, nu(r,V), that conveys the information on the pressure. An exact expression for the isothermal-isobaric free-energy density functional is obtained in terms of the corresponding canonical ensemble functional. As an application we consider a hard-sphere system in a spherical pore with fluctuating radius. In general we obtain very good agreement with simulation. However, in some situations a peak develops in the center of the cavity and the agreement between theory and simulation becomes worse. This happens for systems where the number of particles is close to the magic numbers N=13, 55, and 147.

Functional expansion of non-uniform first-order direct correlation function (DCF) around bulk density is rewritten with the concept of weighted density, it is found that the resulting first-order expansion term is actually the first-order expansion term of the Taylor series expansion of the uniform first-order DCF with weighted density as its density argument, the higher-order terms are incorporated into the first-order expansion approximation by Lagrangian theorem. Based on the functional integration view, the present Letter devises a numerical procedure to predict excess Helmholtz free energy from the approximation for the non-uniform first-order DCF. The physical foundation of the present discussion also applies to electronic density functional theory.

An ab initio density matrix algorithm for electronic structure computations of many-electron systems is proposed. The reduced single-electron density matrices are derived by mapping the density functional theory nonlinear optical response functions onto an effective multilevel system. These density matrices are then used as a zeroth order iteration into self-consistent equations whose solution should yield the exact energies and the complete set of (transition and diagonal) single-electron density matrices. Higher order (n electron) density matrices are not computed explicitly. The linear and nonlinear optical response functions may be obtained at a low computational cost. Application is made to constructing an exciton Hamiltonian for molecular aggregates using density matrices of isolated molecules, avoiding electronic structure calculations of the entire aggregate.

Multiresolution (or wavelet) analysis offers a strictly local basis set for a systematic introduction of new details into Hilbert space operators. Using this tool we have previously developed an expansion method for density matrices. The set of density operators providing a given electron density plays an essential role in density functional theory, in the minimization of energy expectation values with the constraint that the electron density is fixed. In this contribution, using multiresolution analysis, we present an excellent quality density matrix expansion yielding a prescribed electron density, and compare it to other known methods. Due to the strictly local nature of the applied basis functions, our construction has the specific advantage that the resulting density matrix is correlated and N-representable in the infinite resolution limit. As a further consequence of this scheme we can conclude that the deviation of the exact kinetic energy functional from the Weizsacker term is not a necessary consequence of the particle statistics.

A recently proposed density functional theory [New J. Phys. 4 (2002) 36], which is based on functional counterpart of Lagrangian theorem of differential calculus, is extended to calculate density distribution of an adhesive hard sphere fluid confined between two hard walls. The needed input parameters, i.e., bulk second order direct correlation function and equation of state, are employed from the analytical Percus–Yevick approximation and the compressibility equation. The present predictions are in very good agreement with available computer simulation data. Then the present formalism is applied to investigate the affecting factor for the adhesive hard sphere fluid adsorption behaviors. The continuity and differentiability of the density functional C (1) ( r ;[ρ]) , which underlies the application of the functional counterpart of Lagrangian theorem of differential calculus, is proven.

In this chapter, time-dependent density functional theory is introduced and an outline of the Runge–Gross theorem is presented. An equation for linear response within time-dependent density functional theory is derived. A key ingredient of this equation is the exchange-correlation kernel for which several modern-day approximation exist. These approximations are investigatead for their ability to capture the excitonic physics in absorption and electron loss spectra. To this end, results for medium (Si and diamond) to large (LiF and Ar) band-gap insulators are presented, which exhibit excitonic physics to varying degrees.

The differential virial theorem of March and Young for N fermions moving in a common one-dimensional potential energy V(x) is here combined with the Euler equation of density functional theory expressing the constancy of the chemical potential throughout the entire inhomogeneous particle density. The functional derivative of the single-particle kinetic energy is thereby expressed directly in terms of the kinetic energy density; a line integral being involved in establishing the connection. This result is then used to establish a formula measuring departures from simple homogeneity of the kinetic energy functional: a matter of current interest in density functional theory. Finally, the general theory of the functional derivative of the single-particle kinetic energy with respect to the particle density is exemplified for the case of harmonic confinement of fermions in one dimension. (C) 2001 American Institute of Physics.

I review recent advances in understanding the physical character of the exchange-correlation (xc) potential in time-dependent density functional theory. I show that in an electron gas of slowly varying density the dynamical part of the xc potential is an extremely nonlocal functional of the density. This difficulty can be circumvented by introducing an xc vector potential that is a function of the local current density. The form of this vector potential is entirely specified by symmetry. Its physical effects are analogous to those of the viscous force in the Navier-Stokes equation of classical hydrodynamics. Recent applications of the xc vector potential to the calculation of infrared absorption linewidths in quantum wells are described.

The one-particle reduced density matrix of correlated electronic systems is computed by expressing the many-electron wavefunction as a superpositon of generalized coherent states, each representing a single Slater determinant. This representation could be used for developing a density matrix functional theory. The sensitivity of the density matrix to the parameters is demonstrated.

We propose a novel density-functional approach to antiferromagnetism and spin-density waves. Nonlocal antiferromagnetic correlations and possible noncollinear spin configurations are incorporated via a staggered density, which enters the formalism on the same footing as the particle density. We establish the pertinent generalization of the Hohenberg-Kohn theorem, derive the associated Kohn-Sham equations, outline the construction of approximations for the corresponding exchange and correlation functional, discuss the relation of the present method to other density-functional approaches, and report a first application to a simple model system.

The intrinsic Helmholtz free energy, commonly used as a basis for density functional theories, is here given explicitly as a cluster diagram expansion with density field points. Also given are explicit variational procedures for determining the chemical potential for a given density, the pair potential for a given pair correlation function, and the pair correlation function for a given pair potential. The physical meaning of the density functional is established within the context of a new derivation of statistical mechanics based on entropy that supplies a variational principle for equilibrium by generalizing the thermodynamic potential to nonequlibrium states. This shows that the conventional density functional determines not only the equilibrium density, but also the probability of fluctuations about that density.

The density equation proposed previously for the direct determination of the density matrix, i.e. for the wave mechanics without wave, is extended to a time-dependent case. The time-dependent density equation has been shown to be equivalent to the time-dependent Schrodinger equation so long as the density matrix, included as a self-contained variable, is N-representable. Formally, it is obtainable from the previous time-independent equation by replacing the energy E with . The perturbation theory formulas for the density equation have also been given for both the time-dependent and time-independent cases.

In order to improve approximations to density functionals such as the exchange–correlation functional, Exc[ρ], and the independent Fermion kinetic energy functional, Ts[ρ], it is essential to have information on the behavior of the exact functionals. Any approximation should then be made to satisfy these conditions. With this in mind, constraints on functionals and functional derivatives which reflect stability against translation, rotation and uniform scaling of the charge density for electrons moving in a fixed external potential, are derived from the minimal property of the energy functional F[ρ]=Ts[ρ]+U[ρ]+Exc[ρ] when evaluated at a v-representable density ρ(r).

The time dependence of the exchange-correlation energy in density functional theory is given in terms of the exchange-correlation potential. The virial theorem for the exchange-correlation potential is shown to hold for time-dependent electronic systems and is illustrated by an exactly solved model: Hooke’s atom with a time-dependent force constant. A relation between the coupling constant and functionals evaluated on scaled densities is derived. [S0031-9007(98)08169-1]

Protective measurement, which was proposed as a method of observing the wavefunction of a single system, is extended to the observation of the density matrix of a single system. This provides a new meaning to the density matrix as having the same ontological status as the wave function describing a pure state. This also enables quantum entropy to be associated with a single system.

We present an improvement over the Wang-Teter, Perrot, and Smargiassi-Madden kinetic-energy functionals without going beyond linear-response theory and without introducing a density-dependent kernel. The improved functionals were tested on bulk aluminum, and excellent results were obtained. Accurate density-functional calculations using the new functionals on systems larger than one can study by traditional Kohn-Sham methods are demonstrated.

A potential flow formulation of the hydrodynamic equations with the quantum Bohm potential for the particle density and the current density is given. The equations are selfconsistently coupled to Poisson's equation for the electric potential. The stationary model consists of nonlinear elliptic equations of degenerate type with a quadratic growth of the gradient. Physically motivated Dirichlet boundary conditions are prescribed. The existence of solutions is proved under the assumption that the electric energy is small compared to the thermal energy. The proof is based on Leray-Schauder's fixed point theorem and a truncation method. The main difficulty is to find a uniform lower bound for the density. For sufficiently large electric energy, there exists a generalized solution (of a simplified system), where the density vanishes at some point. Finally, uniqueness of the solution is shown for a sufficiently large scaled Planck constant.

We consider the density functional theory from the point of view of one-dimensional two-particle systems, where the particles are either spin-1/2 fermions or spin-0 bosons. The basic equations are derived and applied to the system of two coupled harmonic oscillators. An exact density functional solution is constructed from the exactly known single-particle density for this system. We demonstrate how this solution may be applied to other two-particle systems to calculate the exact single-particle density and energy.