The three centers and the chakras do not at all seem the same, but they are also obviously connected.
Three centers allowing the chakras to reorganize
It seems that the three centers allows the chakras to transform and reorganize, along with the human self in general. The belly center allows the three first chakras to reorganize within all as Spirit, within and allowing a deepening of a felt sense of all as Spirit. The heart center allows the heart chakra to reorganize to loving all as Spirit. The head center allows the 6th and 7th centers to reorganize. And my sense right now (which will probably change) is that all of the three centers together allows the throat center to reorganize.
The chakras setting the stage for soul center awakenings
It also seems that working with the appropriate chakras can set the stage for an awakening of the soul centers, inviting an awakening of the centers to take place. And, as mentioned above, that this awakening in turn allows the chakras to reorganize within this new context.
The relationship(s) between the hara and the belly center
For instance, Breema or other hara-oriented practices can reorganize the hara chakras (2nd, or 3rd, or as I experience it the three first) allowing a drop into endarkenment, which in turn continues the reorganization of the hara chakras within the new context of endarkenment, of a felt sense of all as Spirit, which in turn deepens and more fully embodies this felt sense of all as Spirit.
There must be many individuals out there who have explored this in far more depth, for instance in the different yogic traditions (maybe especially Taoist yoga) and it seems that Almaas too probably have some insights here (although I haven't gotten that far in reading his books yet.)