Praying for Non-Christians

Do you pray for specific unbelievers to be saved?

After reading a short section of Augustine's Confession, I was struck with these words that I hope to incorporate into my prayers for unbelievers (if you like, read it in context here ch. 8, para. 12):

"It was pleasing in Your sight to reform my deformity, and by inward stings You didst disturb me so that I was impatient until You were made clear to my inward sight."

I want God to "disturb" the unbelievers I pray for, along with convicting them of "sin, righteousness and judgment" (John 16:8) and putting gospel-sharing believers around them so that He will be made clear to their inward sight, something I think only happens after salvation (cf. Romans 1:18-23, 8:5-8; 1 Corinthians 2:14; Ephesians 4:17-18).