We develop the use of the population diagram method to analyze molecular emission in order to derive physical properties of interstellar clouds. We focus particular attention on how the optical depth affects the derived total column density and the temperature. We derive an optical depth correction factor that can be evaluated based on observations and that incorporates the effect of saturation on derived upper level populations. We present analytic results for linear molecules in local thermodynamic equilibrium (LTE). We investigate numerically how subthermal excitation influences the population diagram technique, studying how the determination of kinetic temperature is affected when the local density is insufficient to achieve LTE. We present results for HC3N and CH3OH, representative of linear and nonlinear molecules, respectively. In some cases, alternative interpretations to the standard optically thin and thermalized picture yield significantly different results for column density and kinetic temperature, and we discuss this behavior. The population diagram method can be a very powerful tool for determining physical conditions in dense clouds if proper recognition is given to effects of saturation and subthermal excitation. We argue that the population diagram technique is, in fact, superior to fitting intensities of different transitions directly, and we indicate how it can be effectively employed.
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