What do you mean by coupled?

In your settings electron subsystems of two domes would be independent. Electrons do not tunnel through infinite potentials. 

Normally, plasmons in such systems are coupled via electromagnetic filed. They can be treated quantum mechanically or using a model for dielectric function. Electrodynamical problem is still very difficult.

Dear Prof.Igor Goliney,

Thanks for your response.In follow up to your question;what i mean by couples?

By coupling I mean the two separated spherical dome shells are connected with each other with a rod/wire in which the potential in the connecting rods is not the same or equal to zero as in the shell..?How possible is it to achieve and what can I do to achieve this?

SEE THE NEW GEOMETRY BASED ON YOUR ADVICE

I look forward to your reply.

Best regards,

Lateef; [email protected]

It is not possible to have two conducting shells at different DC potentials joined by a conductor.  They will quickly all arrive at the same potential.  It is possible with AC.  The shape will have a resonance at the frequency close to where the perimeter of the dog-bone shape is a wavelength, in which opposite ends have opposite potentials, alternating at the resonant frequency.

@ Malcolm White   Many thanks for sharing your thought with me.  

Can you Please give a geometry of two coupled spherical dome shells that is feasible using AC as against DC.or Using the same dc potential for both shells,how do I go about this

So, you want to calculate an antenna consisting of two shells connected with a rod.

Note that each  part have their resonant frequencies.  Oscillations when opposite ends of the rod have opposite charges, for example, are common. That's how usual rod antenna work.The same for spherical particles and shells.

The composite problem is quite complex. You have to familiarize yourself with available software for antenna calculations. 

You mentioned wave function, so I thought that you wanted to do it quantum mechanically. That is even more complex and I don't see a point. Just choose a model for the metal's dielectric function and do electodynamics. 

If you want to see how the quantum mechanical calculations are done, look up the nanoshell treatment by Norlander at al. Hope I did not misspell the name. I have a reference somewhere but not at the moment.

@Igor Goliney....many thanks for sharing your thoughts with me.

BUT i want to work with spherical nanoparticles to obtain dielectric function in the final analysis but not antenna.However, the idea is to solve schrodinger equation of time-independence.To achieve this,the wavefuction (s)(eigenfunction of the spherical dome shells must be well-formulated(which depends on the radial,polar and azimuthal components of the particle),then we need to substitute this wavefunction back to shrodinger eqn in order to obtain the corresponding eigenvalue(s).The task here is to couple two spherical dome shells together and write out the eigenfunctions and eigenvalues knowing full well there is going to be electron transitions after applying electromagnetic waves.How do I go about this?

If you want to achieve an artificial dielectric, then treating the particles as antennas is a valid approach.  They receive the incident radiation and re-transmit it with a phase change, which results in the different velocity observed in the artificial dielectric.  This was done 60 years ago when conducting discs were held on strings in a regular array to make an artificial dielectric at VHF/UHF.  It was a lot lighter than a solid dielectric lens.

See page 23 below for a more recent example.

https://ocw.mit.edu/resources/res-6-001-electromagnetic-fields-and-energy-spring-2008/chapter-6/06.pdf

Malcolm says reasonable things.   Antenna is just another term for your system. Electrodynamis is the most interesting part if you talk about photonics.

For plasmonics, you have to solve  electrodynamic part together with a model for plasmons. They are collective oscillations of electron density together with the electromagnetic field. any classical model would suffice, but a quantum mechanical would do as well. Yet the problem is much more complex.

if you want this you have to start with a many-body wavefunction which could be chosen as the antisymmerized product on single electron functions. You should take as many of them as there are electrons in the system. So, count electrons first. Fining even one single electron function for your geometry is a complex task. But if you are able to find one, you would probably be able to find as many as necessary.

So, find a software ,for exmaple, for DTF calculations. Set your geometry and solve the problem. DTF is good since it automatically builds the   antisymmerized product and then iterates with a new potential till the procedure converges.

But you'll be yet not there. Electrodynamic part is eminent for description of plasmons and DTF lacks it.