The way your question is phrased, the short answer is no.

First of all, there is the problem with the SS as an organization. /u/kieslowskifan succinctly summed up the nature of the SS in this post:

While the SS became the institutional face of the Third Reich's barbarism and inhumanity, it was an incredibly diverse organization composed of career civil servants amalgamated into the SS, young professionals who gravitated to the NSDAP, youthful volunteers to a self-styled military elite, and old Kampfers from the movement's early days.

The second problem is that of what is a psychopath? /u/Subs-man is more knowledgeable in that area but as far as I am aware, while the term "psychopath" is often used in popular media to portray a certain type of psycho-pathology, it does not actually exist as a diagnosis. Rather, both the DSM as well as the International Classification of Disease only recognize certain symptoms that are commonly associated with what is popularly referred to as psychopath or sociopath. These include among others persistent antisocial behavior, impaired empathy and remorse, and disinhibited, egotistical traits.

Regarding the first point: When talking about the SS, historians of the Third Reich generally distinguish between various factions within the SS which in turn present a – sometimes radically – different social profile. Basically, you see those running the Concentration Camps, those who are policemen and other civil servants, and those in the Waffen-SS, the fighting troops. While it is true that the organizational origins of all three can be found in the beginning of Nazi rule, specifically in the merging of the SS and the Bavarian police in 1934 and that the Waffen-SS developed out of the camp-SS, they begin to differ from each other in their social make-up the further Nazi rule progresses, especially after the police merges with the SS in 1936.

While the SS in the security apparatus was comprised generally speaking of people who had been socialized in right-wing academic circles of the Weimar Republic and who had developed a highly ideological world-view historian Michael Wildt describes as "uncompromising" and whose mindset was that of the "fighting bureaucrats", administering the security apparatus but also standing on the front-lines themselves in form of the Einsatzgruppen, other factions of the SS differ. The camp-SS as described by Karin Orth was often recruited locally from men from a lower socio-economic background than the security-apparatus SS. Reasons for signing up were often less ideological and more economically motivated than with say, someone joining the Gestapo. Similarly, the Waffen-SS is another different picture: On the one hand, people signed up there precisely because of their ideological convictions, on the other hand, they did start recruiting and drafting people, especially foreigners, into Waffen-SS at some point into the war.

The non-armed SS had in 1941 271.060 members while the Waffen-SS membership only took off during the war and reached 594.443 in June 1944. Just from a statistical stand-point it is highly unlikely that of these some 865.000 members, a considerable number were psychopaths. Even discounting the non-Germans and Volksdeutsche bringing it down to about 650.000 members still makes it incredibly unlikely. In May of 1939 (so including Austria and the Sudetenlands), the Thrid Reich had a population of 79 million people. Assuming a "normal" prevalence of psychopathic tendencies that most researchers today say is displayed by about 1% of a given population of sufficient size and discounting the female population as they would not have been officially part of the SS this means that virtually all German males with psychopathic tendencies would have joined the SS, even if one assumes that the male to female ration of these tendencies is 75:25. Which is statically speaking as well from practical experience highly unlikely.

But this is all a lot of assumptions that are very hard to impossible to back up with empirical evidence. However, when reviewing the empirical evidence we have, the answer remains still no. One huge factor showing that the SS could not have been made up by a significant number of psychopaths is its post war history. Again, citing from /u/kieslowskifan 's answer above, we can observe that most of the former members of the SS integrated perfectly into post-war democratic society. While psyhopathologies, especially those that depend in their definition on anti-social behavior always have a social component and depend heavily on their social surroundings, it is highly unlikely that several thousand and more psychopaths would have stopped being psychopaths on May 8, 1945.

Furthermore, the very idea of psychopaths and sociopaths joining an organization, when one of the strongest factors in their cluster of symptoms is anit-social behavior, is somewhat odd.

The sad reality of the SS and Nazi crimes in general is that they were committed by people who in the vast majority were completely normal. While we might speculate about certain outliers who might have been clinical psychopaths (Oskar Dirlewanger e.g.), we will also never know since we can't diagnose them. In general though, all indications show that the people shooting and gassing countless numbers of Jews were not ill in a clinical sense. Research such as that of Christopher Browning and Harald Welzer shows that rather than anti-social behavior, it was social behavior that played a huge part in why normal men went along with murdering men, women and children. As Browning shows in his study of police-battalion 101 in Poland, about 20% were ideological Nazis, about 20% refused to take part in mass shootings and the rest went along with them because of social pressure through camaraderie, subtly hints, and because they didn't want their colleagues facing this grim task by themselves.

Similarly, Welzer, who is a sociologist and social psychologist, demonstrates concerning the Einsatzgruppen that the perpetrators of the Holocaust developed a mentality and rationality that perceived these deeds as legitimate out of a sense of self-defense before the dangerous foe of an international Jewish conspiracy. Killing became work for them and like the surgeon whose job it is to cut people open, they developed a sense of professional detachment towards their work, resp. towards the act of killing in their work.

Lastly, as first described by Gerald Reitlinger in The SS: Alibi of a Nation, making the SS out to be a bunch of insane killers is a strategy frequently employed in the undertaking of absolving Germans and German society at large from the crimes of National Socialism. By making the perpetrators of the Holocaust out as psychopaths not only do we remove them from historical understanding – seeing as we by definition can't comprehend the minds of psychopaths –, we also exterritorilize them from history. By declaring them psychopaths, we neglect our duty to understand in order to prevent – Nazism and the Holocaust become accidents of history perpetrated by a gang of sociopaths rather than the result of a mixture of historical factors and developments.

Sources:

• Christopher Browning: Ordniary Men.

• Harald Welzer: Täter. Wie aus ganz normalen Menschen Massenmörder werden.

• Harald Welzer et. al. (ed.): Die Täter der Shoah.