Indian Journal of Chemistry Vol. 53A, Aug-Sept 2014, pp. 936-939 Hypervirial theorems and scaling relations in density functional theory Swapan K Ghosh Theoretical Chemistry Section, Bhabha Atomic Research Centre, Mumbai 400 085, India Email: skghosh@barc.gov.in Received 29 May 2014: accepted 29 May 2014 Within the framework of density functional theory, a hypervirial theorem for an operator ˆ A is shown to be given by ∫dr ρ(r) 1/2 [μ, ˆ A ] ρ(r) 1/2 = 0, where ρ(r) is the single particle electron density of a many-electron system. The chemical potential operator μ expressed as either a multiplicative or a differential operator, depending on the variant of DFT, is shown to yield equivalent results. The consequence of using approximate forms of the energy density functionals are discussed and for special cases of the operator ˆ A , the virial theorem in DFT is recovered. Generalised scaling relations for the energy density functionals are obtained and the implications towards tensor virial theorem are discussed. Keywords: Theoretical chemistry, Density functional calculations, Hypervirial theorem The hypervirial theorem (HVT) 1 can be regarded as a powerful theorem, as applied to the quantum mechanics of atoms, molecules and solids. In the conventional wavefunction theory, the HVT for an operator ˆ A , introduced originally by Hirschfelder 1 , is given by ˆ ˆ , 0 HA ψ ψ   =   … (1) where ψ is the eigenfunction for the Hamiltonian operator H. The theorem can be used to check the accuracy of an approximate wavefunction or can be imposed on the latter to improve the accuracy. If a function satisfies Eq. (1) for any arbitrary operator ˆ A , it must be a true eigenfunction 2 . The HVT also yields the Hellmann-Feynman (H-F) 3 and the virial theorem (VT) 4 as special cases for selective forms of the operator ˆ A . The wide applicability of HVT has, however, been discussed mainly in the context of wavefunction theory. In view of the upsurge of interest in density functional theory (DFT) 5-8 , it would be of importance to look at the HVT within the framework of DFT, which is established as a conceptually simple and computationally economic route to a quantum mechanical description. The special cases of H-F and VT within DFT have been dealt with by Ghosh and Parr 9 and subsequently HVT in one and three dimensions have also been discussed by Baltin 10, 11 . For an N-electron system, characterized by an external potential v(r), the energy density functional, E v [ρ], given by Eq. (2), v xc E [ ] = v( ) ( )d + T[ ] + U [ ] ρ ρ ρ ρ ∫ r r r … (2) is known to possess a minimum value for the true density, thus enabling one to calculate the density from the relevant Euler equation. [ ] = () [ ] [ ] [ ] () () () () v xc E U T J v δ ρ μ δρ δ ρ δ ρ δ ρ δρ δρ δρ = + + + r r r r r … (3) Here, μ is the Lagrange multiplier corresponding to the normalization constraint ∫dr ρ(r) = N and represents the chemical potential of the electron cloud, identified as the electronegativity parameter by Parr et al. 12 The density functionals T[ρ], J[ρ], and U xc [ρ] represent the kinetic energy, classical Coulomb energy and exchange-correlation energies respectively. While the classical Coulomb energy functional is given by Eq. (4) (atomic units are used throughout), 1 ()(') J[ ] = dd' , 2 - ' ρ ρ ρ ∫∫ r r rr r r … (4) the exact forms of the kinetic and exchange- correlation energy functionals are unknown. For the latter, various local density approximations and gradient corrections are found to be quite satisfactory.