Hence, it is vital to know how electrons moving in nonlocal potentials couple to EM fields. To date, the attempts to do so have been heuristic or limited to weak or long wavelength fields.
Accomplishments --- We have been able to derive, through a Feynman path integral formalism, the exact, gauge-invariant coupling of nonlocal systems to arbitrary EM fields. Our results allow, for the first time, for systematic and accurate computation of response functions for nonlocal systems. Aside from first principles methods, many semi-empirical or phenomenological models use nonlocal Hamiltonians with tight-binding like hopping terms. Our results also justify the assumed forms of coupling used previously.
In order to ascertain the importance of treating the nonlocality correctly, we present two examples where the use of the correct nonlocal coupling is important for reliable calculation of response functions. The first example involves the strength of absorption of light by diamond as a function of the frequency of the light. The response function related to this physical quantity is the imaginary part of the RPA dielectric function. Below we show a plot of the absorption spectrum, where we computed the response in two ways: (1) using the textbook method (which is only appropriate to local potentials), where the momentum matrix elements are used (the red curve), and (2) using matrix elements of the velocity operator instead, which is what our formalism prescribes (blue curve). The importance of using the correct operator is on the level of 15% to 20% in this material, non negligible when seeking quantitative comparison to experimental results.
Our second example involves the response of matter to magnetic fields,
specifically the calculation of magnetic susceptibilities. We compute
the susceptibility of two representative atoms, carbon and neon in
various ways so as to clarify the nature and importance of the
nonlocality. For this case, we have chosen the gauge A=(0,x+a,0)
where a=4 a.u. as a representative value. The table below presents
the results for the atomic susceptibilities calculated in different
ways. Column 1 represents results from the traditional local-only
approach. Column 2 includes the correct nonlocal coupling to first
order in the magnetic field. Column 3 includes the full nonlocal
coupling, and is the method prescribed by our formalism. The last
column presents results all-electron calculations, which are
computationally expensive. As is easy to see, the nonlocal effects
are quite significant, and full inclusion of the correct
gauge-invariant nonlocal couplings provides for excellent reproduction
of the desired all-electron results.
| Atom | 1 | 2 | 3 | All-elec. |
|---|---|---|---|---|
| Carbon | -40.1 | 16.9 | 12.80 | 12.85 |
| Neon | -62.4 | 6.15 | 7.76 | 7.75 |
Significance --- Our work provides the explicit, closed-form expression for the coupling of nonlocal systems to external electromagnetic probes, and is thus directly relevant to a wide range of ab initio calculations of the response of materials and will be of direct use to a large number of workers in the field. Furthermore, our examples show that treating the nonlocality is indeed important for being able to have accurate and consistent values of the response when trying to compare to experimental results so as to understand what the experimental and theoretical results imply about the structure, behavior, and response of materials.
A preprint of this work is available at http://arXiv.org/abs/cond-mat/0101383.
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