I\'m reading about Quantum Monte Carlo, and I see that some people are trying to
ID: 1381022 • Letter: I
Question
I'm reading about Quantum Monte Carlo, and I see that some people are trying to calculate hydrogen and helium energies as accurately as possible.
QMC with Green's function or Diffusion QMC seem to be the best ways to converge on the "exact" solution to Schrodinger's equation.
However, if one wants to be very exact, then the Born-Oppenheimer approximation must be removed. A lot of papers mention that the results are still not exact enough, and must be corrected for relativistic and radiative effects.
I'm pretty sure I know what relativistic effects are -- the non-relative Schrodinger's equation cannot account for GR as particles approach the speed of light (or even small but measurable effects at lower speeds). But what are radiative effects?
And I would think you would include these two things into your QMC calculation instead of applying a post-simulation correction factor if you wanted to be ultra-precise (e.g. use the Dirac equation instead for relativistic effects). So why don't most researchers do this? Does it raise the calculation time by orders of magnitude for an additional 4th decimal place of accuracy?
Finally, is there anything at a "deeper" level than relativistic and radiative effects? In other words, if I left a supercomputer running for years to compute helium's energies without the BO approximation, and with relativistic and radiative effects included in the MC calculations, would this converge on the exact experimental values?
(Actually, I just thought of one such left-out factor -- gravity ... and might you have to simulate the quarks within the protons individually? Anything else?)
Explanation / Answer
Relativistic effects are those that disappear in the non-relativistic approximation 1/c?0, usually small corrections to the non-relativistic approximate results that are proportional to 1/c2 or higher powers of the inverse speed of light.
Let me correct a typo: "cannot account for GR" should have read "cannot account for the special theory of relativity". When we talk about relativistic corrections, we always talk about the 1905 special theory of relativity, not about GR i.e. the 1915 general theory of relativity. Corrections that have something to do with general relativity are "gravitational" or "quantum gravitational" corrections and they're typically proportional to powers of Newton's constant G which makes them even more negligible.
For example, the Hydrogen atom may be described by the [=special] relativistic Dirac equation which reduces to the Pauli equation, i.e. the Schr