Leading high-temperature density kernel ($\rho (x,x^{\prime })\simeq Z^{-1}\sqrt{m/(2\pi \beta {\hslash }^2)}{e}^{-m(x-x^{\prime }{)}^2/(2\beta {\hslash }^2)-\beta V(x)}$)

Neglecting commutators between kinetic and potential energy permits a leading product approximation to the imaginary-time evolution operator. The kinetic factor is a Gaussian Fourier integral; the potential factor multiplies it in position space. The unsymmetrized product places the potential at one endpoint and need not be exactly Hermitian at the discarded order. Splitting the potential equally between both endpoints gives a Hermitian approximation with exponent $-\beta [V(x)+V(x^{\prime })]/2$ and the same leading diagonal normalization.