Process for distilling a crude composition in a rectification plant including an indirect heat pump

A process for purifying a crude composition comprising rectifying the crude composition in a rectification plant. The rectification plant comprises a rectification column including: a first overhead condenser for condensing a head fraction, and a reboiler for evaporating a bottom fraction. A difference between a temperature of the head fraction and a temperature of the bottom fraction is less than or equal to 20° C. A heat pump is provided between the first overhead condenser and the reboiler. The heat pump is an indirect heat pump that is operated with water or methanol as a refrigerant, and the indirect heat pump includes an expansion valve and a compressor. The heat pump comprises a second condenser located upstream of the compressor.