Density Functional Theory through Legendre Transformation

Abstract

Theoretical foundation of density functional theory in terms of field theoretical Legendre trans­ formation is presented. The ground state energy is first written as a functional of local probe coupled to the density operator. The density functional is then defined by the functional Legendre transformation, which leads to a systematic formulation of density functional theory. Excitation spectrum is also determined within the same formalism in a unified way. The diagrammatic evaluation is most conveniently done by using auxiliary field method. Several generalizations of the formalism and extension to the case other than the density operator are also discussed. The density functional theory is now one.of the most commonly used methods in discussing various many particle systems. 1 l In this formalism the energy of the system is written as a functional of the density which is a function of single variable x instead of N coordinates X1 ~ XN of N particles. In spite of the usefulness, its theoretical formulation is rather involved and sometimes difficult to achieve a system­ atic approximation scheme. The purpose of this paper is to present a clear formulation of the density func­ tional theory in terms of full use of the Legendre transformation applied to the quantum system, especially to the system described by the second quantized field theory. 2 l It presents a theoretical basis of the density functional formalism which is both exact and systematic. These can be achieved by a straightforward application of our technique called on-shell expansion. The discussion is organized as follows: 1. Ground state 2. Excited states 3. Generalization to the case other than the density operator 4. Finite temperature case (equilibrium and non-equilibrium), etc. They are all formulated in terms of the field theoretical Legendre transformation in a unified way. We have in mind, in the following, atomic system with N electrons but the arguments can be applied to any many particle system with minor modifications. The essential feature of the formulation of the density functional theory in terms of Legendre transformation is the following. In the usual approach the ground state wave function lf! is used to connect the potential v and the density n;