[mxplxxx (OP)]

What is the gravitational energy of the elementary particles. May be a dumb question I know but I can't recall ever seeing the answer to this question. Does 1 graviton attract a single particle or multiple particles. If multi, what is the energy of each attraction. Does each particle in a molecule attract each other molecule and if so with what energy and does this attraction take place on a graviton basis.

[Janus]

Gravitational forces at that scale are insignificant.   Even two Uranium atoms placed so that their atomic radii are "touching" would only feel an acceleration of 1.7e-42g towards each other.  If you could ignore the repulsion of their electrons, it would take two hrs for the nuclei to fall together.  The escape velocity would be  only 2.6e-13 m/s.   Which is magnitudes smaller than the natural relative velocities of the atoms under any reasonable conditions.  The other forces acting between particles so overwhelm gravity that it need not be considered.

As far as gravitons go.  They, like photons have an infinite range in their  individual energies.   A single graviton could only react with a single particle, but the energy of this reaction depends on the energy of the graviton,
It should also be noted that gravitational attraction would be mediated by virtual gravitons, in the same way that the electromagnetic force is mediated by virtual photons.  The graviton itself is a quantum of gravitational radiation (gravitational waves) which is not the same thing as gravitational attraction.  In the same way, photons are quanta of electromagnetic radiation (Light waves for example), which is not the same as electromagnetic force.

[evan_au]

The energy of a graviton is not known; but it must be incredibly small compared to the particles with which we are familiar; some suggest < 10^-22eV.
There is no experimental technique at present which would be capable of detecting individual gravitons.
So for now, gravitons are listed as "hypothetical", ie credible, even likely, but not really proven.

See: https://en.wikipedia.org/wiki/Graviton#Energy_and_wavelength

[jeffreyH]

The wavelength of the graviton will be far longer than that of the photon. Therefore the individual energies of gravitons are less than those of photons. This long wavelength if you follow Susskind may mean that gravitons do not get trapped by black holes. If the  wavelength is longer than the diameter of the event horizon.

[Janus]

The energy of a graviton is not known; but it must be incredibly small compared to the particles with which we are familiar; some suggest < 10^-22eV.
There is no experimental technique at present which would be capable of detecting individual gravitons.
So for now, gravitons are listed as "hypothetical", ie credible, even likely, but not really proven.

See: https://en.wikipedia.org/wiki/Graviton#Energy_and_wavelength

While <1.2e-22 ev/c^2 has been set as the upper limit for the mass of the graviton, this does not preclude it being a zero mass particle like the photon. Likewise, the photon has had its upper mass set at <7e-17ev/c^2
Sometimes you will see these mass measures given as just ev,  but the /c^2 is implied and it is still an measure of mass not energy.

[mxplxxx (OP)]

The other forces acting between particles so overwhelm gravity that it need not be considered.

That is interesting, it implies a universe that has a continuum of gravitational states starting with tiny particles that mostly do their own thing, unaffected by gravity, and ending in massive structures (galactic clusters) where gravity is an incredibly powerful force. A cloud of intergalactic gas likely has huge amounts of particles doing their own thing and an increasing part being played by gravity as stars start to form. So it would seem gravity plays a big part in how abstractions are formed in the universe (one of which is you and I:)).  It may be that gravity plays a big part in allowing the universe to behave like a computer.