Give a psychological explanation of murder (homicide)?
Not sure exactly what you're looking for here, definitions or psychological motivations or how murder is looked at by psychologists. I'll assume the last.
Psychologists who study murder seem to divide separate it by motivation and mental state. First, there is the crime of passion, or what some people call temporary insanity. Emotion overpowers reason and normal inhibitors such as conscience and social mores and the crime is committed. This type of murderer almost always feels remorse and is much more likely never to have committed such a crime before.
Second there is the clinically and legally insane killer, who through conditioning or injury have become divorced from conscience because they can no longer tell right from wrong, real from fantasy. This would include serial killers who have a compulsion to kill.
Lastly there is the motivational killer, who murders for a premeditated purpose, to cover up a crime, silence a witness, to commit a robbery, etc.
Central to all three of these main types is the psychological reality that the person has become divorced from reason and conscience, or that it is at least temporarily suspended. Some psychology tries to identify patterns and warning signs in a person's psychological state that early intervention may prevent them from becoming a murderer.