You seem to want life to appear in a likely condition, without some external power source.
Whatever condition there was, it could as well be some extremely rare one, and I really can't imagine how some molecules could start to reproduce without their environment being constantly changed by some external energy flux.
Besides, I'm no expert, but I don't think there were huge amounts of pure water running around on the early Earth. There were so much more stuff to use those hydrogen and oxygen atoms, and Earth's water wasn't all here since the beginning.
Whatever condition there was, it could as well be some extremely rare one, and I really can't imagine how some molecules could start to reproduce without their environment being constantly changed by some external energy flux.
Besides, I'm no expert, but I don't think there were huge amounts of pure water running around on the early Earth. There were so much more stuff to use those hydrogen and oxygen atoms, and Earth's water wasn't all here since the beginning.