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?)


1 Answer 1


Relativistic effects are those that disappear in the non-relativistic approximation $1/c\to 0$, usually small corrections to the non-relativistic approximate results that are proportional to $1/c^2$ 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 Schrödinger equation with the spin, in the $c\to\infty$ limit. However, there are some relativistic corrections and they actually make the energy slightly depend on the angular momentum, too. This effect seen in the Dirac equation is both due to the corrections to the $p^2/2m$ formula for the kinetic energy as well as due to the spin-orbital coupling.

Radiative corrections are corrections due to the virtual particles that can only be seen in the language of quantum field theory. The hydrogen atom, for example, may emit a photon and reabsorb it: that is the Lamb shift. Or a particle may temporarily create a positron-electron pair. Those processes are expressed by Feynman diagrams and if they have loops in the middle, their loop diagrams and all effects due to Feynman diagrams with loops are know as radiative processes or corrections. Note that this goes beyond the simple Dirac equation.

Radiative corrections are typically smaller because of an extra factor such as the fine-structure constant $\alpha=1/137.036...$. Well, it's usually corrections like $1/2\pi\alpha$ so it's about 1,000 times smaller than the "main" term. So those things are small and even if you want to incorporate them, it doesn't matter at what stage you do so. You must only do it right.

Quantum field theory – which in principle contains all relativistic and radiative corrections – is enough to explain any observable lab experiment within any realistic error margin. If you care about things that can't be measured by current or realistic technology, you need the full theory of everything including the quantum gravity effects, i.e. string/M-theory.


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