Beyond the variational principle

Because the central idea in this section has been the variational principle, it is doubly important to emphasize its limits. In his book Basic Notions of Condensed Matter Physics, P.W. Anderson terms the naive application of the variational principle as Quantum Chemists Fallacy No. 1. The substance of the comment is that ideas which are completely sufficient in small systems can go astray in condensed matter. In particular, In the limit of a large system there may be vanishing overlap between the true ground state and a trial state that appears sensible. A clever choice of the nature of the state at the beginning can lead to the solution, whereas no amount of brute force will arrive at the answer starting from a wrong point. Cases in point include the development of a collective order parameter at a 

One of the important applications of the quantum Monte Carlo calculations has been for the interacting electron gas in a uniform positive 

The Monte Carlo calculations have also been reported hydrogen, which is the simplest of all condensed matter. The ability to carry out such computer simluations of real many-body problems in condensed matter is one of the major breakthroughs, which may lead to entirely new approaches to the theory of condensed matter. The results for the phase transition from the molecular solid to the metallic monatomic solid are of great interest at the present time and are discussed in Section V. 

For our purposes, there are three important aspects of the density functional method. First, it is in principle exact and provides an alternative to the direct treatment of the full many-body Hamiltonian discussed above. It is therefore relevant to establish rigorous expressions for other physical quantities, such as the stress, in the density functional formalism. Second, for any functional, variational solutions of the equations satisfy all the properties required to derive the requisite theorems for force, stress, and other derivatives. Third, there are local approximations to the exact 

Here we give a short review of the LDF equations and methods of solution. In the notation of Wendel and Martin and Martin and Kunc, the energy may be written 

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