What is the expiation for swearing falsely upon the Holy Qur'an to placate one's wife, and how can she be compensated for the harm caused?

Deliberately swearing a false oath is one of the gravest sins; it is called a "Ghamoos oath" because it plunges its perpetrator into sin or into the Fire. The majority of scholars hold that there is no expiation (kaffarah) for it, though if you do offer an expiation for an oath, it is better and more cautious. Praise Allah for your repentance and remorse, for repentance erases what came before it, and the one who repents from a sin is like one who has no sin. As for the harm you caused your wife, the remedy for wrongdoing is kindness; so be good to her.