I guess that this may felt to be best translated by "simple negligence" and Tiny99 has found a link to show the phrase in use in the State of Hawaii.
However, just for the record, I am uncomfortable if the implication is that English and US Law know two forms of negligence, "simple negligence" and "gross negligence", which are simply degrees of each other.
The form of negligence known as "gross negligence" does exist. Gross negligence implies that the negligence is so severe, so "gross" as to be culpable even in the ordinary criminal law as something that the state will simply not allow a person or entity to do. (English law and derivatives are built up from the common law and not from a legal code - though of course states may choose to codify their law.)
The distinction between gross negligence and ordinary/simple negligence is, then, often one about the overlapping of negligence with the criminal law. The link re the State of Hawaii is indicating (without defining simple negligence) that it can furthermore be provided statutorily that negligence short of gross negligence can be made culpable under the criminal law. But there is no definition of simple negligence in the link given, which does not surprise me.
Turning to simple negligence, the distinction between simple negligence and something else is often a distinction between the law of negligence and another form of law, such as Contract Law or Land Law or some other form of tort/wrong. I have just found an English exam paper online which says "Limit your answer to simple negligence." It does not mean "Don't answer about gross negligence."! It means in the context "Don't answer on the basis of the Law of Landlord and Tenant as this question is on the Tort(civil wrongs) paper and not on the Land Law paper."!
This makes sense when one thinks that one can be really severely negligent, but can you really be negligent in a simple and ordinary way? It is NOT OK to be negligent at all. "Simple" relates to the way some negligence is regarded, rather than to its intrinsic severity.
Anyone interested in this topic may be interested in this link about an action following the events of 9/11:http://news.bbc.co.uk/2/hi/americas/1919431.stm
This is the BBC commenting upon an action in the USA. As I see it, simple negligence is being contrasted with much worse behaviour ("reckless misconduct") and is not a definition in itself.
Now to see if Leo will let me through.