with "Theoretical Chemists ......".  Was that TCP/IP?  Anyway I know it is
possible to link CPU/cores with different speeds and different memory-bus
speeds so my question is whether "Open MPI" can handle this situation?

sure.  nothing about MPI assumes that nodes are homogenous in speed,
just that they can somehow get packets from sender to receiver.

Specifically, suppose I set up:

the cluster you describe is basically a normal beowulf.

1. Can Open MPI handle different clock speeds across several node/cores?

of course.

2. Can Open MPI handle different memory bus clock speeds across several 
node/cores?

of course.  MPI itself doesn't know or care about what cpu/mem are in the
nodes, though individual applications may work best with homogenous nodes.
(consider if the app has a periodic global collective operation such as broadcast or reduce. the work done between these collectives should ideally
take the same amount of elapsed/wallclock time, or else some nodes will wind
up waiting for the slower nodes.)

3. Why not LINSPIRE instead of UBUNTU?

it doesn't matter.  an MPI application just wants basic OS functionality
like a network stack, process scheduler, memory manager. distros differ only in desktop and config features - they all use pretty much the same kernel, so from MPI's perspective are nearly equivalent.

possible, but is Open MPI the best choice or what else?

the MPI implementation won't have any effect on how well your application
tolerates heterogeneity of nodes.

regards, mark hahn.
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