They do not love their children, but will demand love in return.
And even when the children do adore their psychopathic-dad it will still not be enough, it never is. It will not be enough to appease him, not enough to trust the children and not enough to be there for them. On the other hand, any lack of complete adoration from the children will give him the ‘right’ to retaliate against you.
They will use their children at will to prove themselves to a new partner.
But sadly this also means that after totally discarding his own children and treating them as strangers, the psychopath will then be able to ‘prove’ to his new partner what an evil person you are and put the OW in a triangle with you that you never wanted. You get her anger (as do the children at times) and he gets her devotion and overflowing pity. Very rarely (not, in my experience) does he actually show himself to be proud of his children even though he sees himself as the best ever dad.