What is the Shar'i ruling regarding a husband who left his first wife in his home with their children and has not approached her since his marriage to another woman, despite her desire to remain to raise the children? Is he thereby burdened with sin?

You have done well by keeping your first wife with your children. It is permissible for a wife to relinquish her right to stay overnight. Therefore, there is no sin upon you in keeping her under your marital bond and granting her her legitimate rights, without dividing the nights, as long as she is content.