All-Temperature Partition Function from Variationally Optimized Canonical Density Matrix

Abstract

Abstract For the canonical density matrix C(r, r0,β) a variational ansatz C̄̄f = (1 - f̄) Ccl + f̄ Cgr is made where Ccl and Cgr are the classical and the ground state expressions which are exact in the high temperature (β → 0) and in the low-temperature limits (β → + ∞), respectively, and f̄ is a trial function subject to the restriction that f̄ → 0 for β → 0 and f̄ → 1 for β → ∞. With the approximation that f̄ be dependent only upon β, not upon spatial variables, the mean square error arising when Cf is inserted into the Bloch equation is made a minimum. The Euler equation for this variational problem is an ordinary second order differential equation for f̄=f(β) to be solved numerically. The method is tested for the exactly solvable case of the one dimensional harmonic oscillator.