This work deals with a molecular dynamics simulation analysis of the intramolecular vibrational energy transfer in a system of two chromophores, azulene and anthracene, bridged by an aliphatic chain and is motivated by corresponding laser experiments. After selective excitation of the azulene chromophore, the subsequent intramolecular vibrational energy redistribution is monitored by analyzing the transient temperatures of the two chromophores and the chain between them. The main focus concerns the heat conduction process in the chain. Therefore, the chain length was varied from 0 to 19 CH2 units. In addition, methoxymethyl, 1,2-dimethoxyethyl, and a thiomethoxymethyl chains were studied. The investigation of the intramolecular vibrational energy process was decomposed into a temporal analysis and a spatial analysis. For short alkyl chains, the time constant of energy relaxation increases proportionally to the chain length. However, for longer chains, the time constant characterizing the energy decay of the azulene chromophore saturates and becomes independent of the chain length. This behavior is consistent with experimental findings. The spatial analysis shows more or less exponential decay of the temperature along the chain near the excited chromophore. In additional simulations, the two chromophores were thermostatted at different temperatures to establish a constant heat flux from the azulene to the anthracene side. The steady-state temperature profiles for longer alkyl chains show strong gradients near the two chromophores and constant but weak gradients in the central part of the chain. Both simulation methods indicate that strong Kapitza effects at the boundaries between each chromophore and the molecular chain dominate the intramolecular energy flux.