Journey to the spring and drink the water by interacting with it. The player will then enter a hallucinatory state, and a ghostly, dark figure (known as the Mother of the Fog) covered in black smoke will appear. Following the figure will eventually lead the player to a shrine, which is guarded by feral ghouls.
After eliminating the feral ghouls, enter the shrine. The marker will point to an icon located behind a maglocked security door that can only be opened with a password-protected terminal. To get the terminal's password, read the note on the shelf, which hints at the password's location on the side of a locker near a periodic table. Retrieve the icon and bring it back to Richter, who will allow you to join the Children of Atom and enter the Nucleus.
The Sole Survivor may stray from the vision guide and travel on foot during the journey to the Children of Atom shrine. The hallucination effects will still be in place. Fast traveling will cause the effects to go away. After this, the player character may still discover the Children of Atom Shrine and continue the quest with no other changes to the quest lines.
This quest is much like the Walking with Spirits quest in Point Lookout, in that the player character must complete a rite of passage involving hallucinogens to infiltrate a cult.
It may be a good idea to dismiss companions before drinking from the spring, especially on Survival difficulty. Companions may die while hallucinated and be sent back to their last home location. If available, one solution may be to dismiss them to Echo Lake Lumber.
The following is based on unverified behind the scenes information and has not been confirmed by canon sources.
Every element in the MoThEr password for the shrine terminal (Molybdenum, Thallium and Erbium) has some connection with nuclear technology. Molybdenum is part of an alloy used to create missile parts, Thallium-201 is used in Nuclear cardiography, and Erbium is used as a control rod (item that absorbs neutrons) within nuclear reactors.
Molybdenum is listed as element number 99 in the shrine password, but it is actually element number 42 (element 99 is Einsteinium).