What you really need to do is ride the defense lawyer to arrange another evaluation. The type needed is called a forensic evaluation. This evaluates here state of mind at the time of the crime. I'm not surprised that the state hospital is saying that she was competent at the time.
Here's the basic difference for your question. Competency refers to the defendant's ability to understand the trial process and be capable of assisting their attorney to prepare and put on a defense at trial. Most mentally ill people can become competent with the right level and type of medication. Obviously some cannot, and depending upon the state, they will either be sent to an institution for treatment, or released.
Schizophrenia is a real problem from a legal standpoint. The biggest issue is usually a determination of when the illness set in (either before or after the crime), and also, if known, whether the person was on their medication. From my experience, many clients stop taking their medication which creates the aberrant behavior.
So, with so much at stake, I do recommend that you try and hire your doctor to perform another forensic evaluation. This is a very specific area of psychiatry and most practitioners are not certified to perform these exams. It would probably also be worth doing even if the state refuses to pay for it.
Finally, there is no substitute for a visit with a local highly recommended attorney in your area. If nothing else, you could determine whether or not the state and attorney are doing everything they are suppose to in order to properly address your daughter's case.
I wish you and her the best.
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