Is this book related to Freud's concept of a person's id (Hyde) and super-ego (Dr. Jekyll), or is there no relation at all?
There is a relationship between the Id and the Superego in this book. Dr. Jekyll, the moral citizen, a doctor, who lives within the confines of his society and follows the rules is suffering from repression, or a desire to ultimately be free of the boundaries that are imposed on him. Mr. Hyde is the outlet for Dr. Jekyll to express his primal desires.
Dr. Jekyll cannot pursue his darker desires and maintain his reputation. The author suggests that the very respectable Dr. Jekyll has entertained ideas as a young man of living a forbidden life full of vices, but is held in check by the moral restraints of his Superego.
"Jekyll transforms both his physical and his moral self into Edward Hyde, a diabolical man who wallows in his wickedness."
Mr. Hyde is all Id. He is a pleasure seeker, if it feels good, including killing, Mr. Hyde indulges without conscience in any activity.
Mr. Hyde can be viewed as Dr. Jekyll's subconscious yearning to be free from the restrictive Victorian society.
"Jekyll embraces the sense of freedom he experiences as Hyde, claiming, "when I looked upon that ugly idol in the glass, I was conscious of no repugnance, rather of a leap of welcome."
The only problem that Dr. Jekyll has with this division of his personality is that he recognizes that Mr. Hyde is far too dangerous.
"Yet as Hyde unleashes all of Jekyll's repressed desires, Jekyll cannot help but label him "pure evil"
I recommend a terrific essay by Nabokov (can be found in the collection Lectures on Literature.)
Consider these three quotes by Freud and a correlative argument by Nabokov:
Every normal person, in fact, is only normal on the average. His ego approximates to that of the psychotic in some part or other and to a greater or lesser extent.
Men are more moral than they think and far more immoral than they can imagine.
Like the physical, the psychical is not necessarily in reality what it appears to us to be.
Nabokov writes: It follows that Jekyll's transformation implies a concentration of evil that already inhabited him rather than a complete metamorphosis. Jekyll is not pure good, and Hyde (Jekyll's statement to tthe contrary) is not pure evil, for just as part of unacceptable Hyde dwells within acceptable parts of Jekyll, so over Hyde hovers a halo of Hekyll, horrified at his worser half's iniquity.
I am actually teaching a unit on Gothic literature at the moment, and we have finished studying The Picture of Dorian Gray. A fascinating quote that Dorian makes to Basil Hallward just before killing him is that "There is a heaven and a hell in all of us." This idea is central to Dr. Jekyll and his alter-ego as we consider such Freudian concepts as the id and the ego and similar Gothic themes such as doubles. What Dr. Jekyll of course managed to do was to isolate all of his natural evil and give it full expression, transforming himself into Mr. Hyde.