If your father is now collecting Social Securitydependents benefits (sometimes called spousal benefits) based on your mother's Social Security retirement payment, he is probably getting about half the amount she collects in Social Security retirement benefits. That arrangement will change when she dies. When you or your father notify Social Security of your mother's death (with a copy of the death certificate), they will switch your father to
Social Security survivors benefits, which is equal to the amount that your mother is now getting in retirement benefits. In other words, his Social Security benefits will double (or nearly). But there will then only be one Social Security payment, to your father.
Whether similar survivors benefits will be available to your father from your mother's private pension program depends entirely on the terms of the pension itself. Many pension programs have spousal survivors benefits similar to those paid by Social Security, but not all do.
Some pension programs have a lump-sum death benefit, paid when the retired worker dies. To find out whether there will be any type of survivors benefit paid to your father from your mother's pension program, you need to contact the pension program itself. You can probably find contact information for the pension program in your mother's papers, including the yearly tax information papers sent to her by the program.