The first area is the spacing of intermediate states. It is usually impossible to obtain an accurate estimate of a free energy difference with alchemical methods by sampling only at the end-states of interest. For overlap-based free energy estimators such as the Bennett Acceptance Ratio (BAR) or the Zwanzig equation, this is because sampling at one endstate almost never produces samples with non-negligible probability at the opposite end state. For thermodynamic integration (TI), errors are introduced because the potential of mean force (PMF) with respect to an alchemical variable (which smoothly interpolates the end state Hamiltonians) is poorly approximated. The problem is solved in both cases by sampling at intermediate states. The variable controlling the interpolation of intermediate states is usually called λ, and intermediate states may be referred to as λ windows.