Stat inefficiency

[Pablo_Lopez_Rios]

The sweep does look like a quadric family: over the frames the surface goes from a closed ellipsoid, opens up, passes through something that looks like a hyperboloid, and closes back into an ellipsoid. That is what a family of second-order surfaces does, so at least in this region the node looks like a quadric in the scanned electron's coordinates rather than a curved plane.

Indeed, and there is probably a simple reason for the shapes - the trial wave function used in the plot is at heart a short-ish multideterminant expansion after all. What I tried to convey with the animation though is that a smooth, continuous backflow coordinate transformation x_i(R)=r_i+xi_i(R) cannot possibly take the Hartree-Fock node (a sphere) and make it go through this type of deformation because this would involve mapping points at infinity onto the surface of the sphere at some point in the animation, implying that xi_i(R) should diverge [i.e., so that Psi_HF is evaluated on the sphere (|xi_i|=r_sphere) when r_i is on the opened-up nodal surface at a point infinitely far from the origin].

[Mike Towler]

Vladimir's mechanism is a prediction, and it seems I've been accidentally
testing it. Neil and Pablo have a draft of my new DMC paper since last week;
the sequel derives, on solvable models, an exact propagator for a walker near a
node and an exact, bounded branching factor to go with it, and the sequel to
that puts them into CASINO. They are in my private version, verified against
exact transfer operators at the 1e-5 level, and a registered beryllium ladder
on the Hartree-Fock node has been running on my crappy computer since the
weekend in two arms that differ only in the near-node branching: one with the
usual limdmc-bounded exponent, one with the exact bounded ratio. If the
population-correlation warning comes from near-node excursions, the second
arm's spike fraction and population-weight autocorrelation fall; if it comes
from elsewhere, they don't. We'll see in a day or two.

Meanwhile, one thing about the code. On a determinant-only beryllium trial
under limdmc 4 the population lives about 3 per cent above target, all run,
both arms, and the growth energy is then exactly the mixed energy minus
ln(W/W_target): a limiter-driven population bias that scales as dt to about
0.63, and that a dt times N_w = const sequence cancels rather than removes. And
for calibration: on helium 2 3S, where the node is exact, the default CASINO
step is exact to 1e-5 across dt 0.001 to 0.02, which the DMC paper predicts.
Where the node is right, the code is already right; the question, as Dario's
figure says, is what happens on the sheets and at the crossing, and that is
what the instruments are for.

The follow-up puts limdmc 5 and 6 beside 4 at one rung, since on a bare
determinant the limiter, not the node, owns the linear term; if one of them is
what you would call production now, say so and it goes in the registration.

[Vladimir_Konjkov]

Mike — limdmc 4 here: the CASINO default, unset in every one of my 1673 local DMC directories, and the only scheme PyCasino implements. Put 4 in the registration.

What it is for. I want to optimize the backflow of a single-determinant Be-atom not by emin but on the nodal surface integral of Mitas and Annaberdiyev,
arXiv:2109.01734 — their Eq. (18), the fixed-node energy written as an integral over the node itself rather than as ⟨H⟩. Then compare the fixed-node energies of the two backflows, same determinant, same topology, only the functional they were optimized against differing. My reference point is the emin backflow on the HF node, which sits 2.379 ± 0.054 mHa above the exact non-relativistic energy, -14.667356.

So the comparison has to resolve a few tenths of a mHa between two wave functions of one system, and I would rather get the DMC protocol right the first time. Is dt·N_w = const at limdmc 4 still what you would recommend for that, or does the near-node propagator change what the best sequence is?

[Vladimir_Konjkov]

Bosonic chemistry, anyone?

A small confession from the nodal-surface corner. Eq. (20) of Mitas & Annaberdiyev is beautiful: the fixed-node energy is the bosonic ground-state energy plus one surface integral over the node, weighted by the exact bosonic ground state Φ_B. No curvature, no projection, just an integral.

Then you ask where Φ_B comes from. It is nodeless, so DMC is sign-problem-free for it — but DMC gives you an energy and a mixed distribution, and a Metropolis weight wants the function pointwise. So you write it down the only way anyone ever writes a wave function down: a product of one orbital times a Jastrow, and you optimize it. At which point you notice you have just built a machine for computing the ground-state energy of beryllium as if electrons were bosons.

Which, for any two-electron ground state, is the same number you started with. Congratulations: He and H₂ come out exactly right in the parallel universe too.

I am not complaining — the identity is exact, and the affordable version of §4.1 does earn its keep. But it does mean that anyone optimizing a node this way has to produce a bosonic ground state along the way, at least approximately, and that by-product strikes me as worth a look in its own right. Not for practical reasons: there is no shell structure, no valence and no periodicity to speak of, so the resulting table has one row and the discussion section writes itself. Out of curiosity, then. Sign-problem-free benchmarks for a whole row of the periodic table are cheap to produce and nobody seems to have bothered. Has anyone here?

By the way, in the Jastrow factor for bosons, the electron-electron cusp condition is the same as for antiparallel spins (opposite spins), not parallel spins.

If electrons were bosons, their total wavefunction would have to be completely symmetric under particle exchange. Without a spin degree of freedom to provide antisymmetry, the spatial part of the wavefunction must be strictly symmetric.

In the fermionic world, a symmetric spatial wavefunction corresponds exactly to the singlet state (antiparallel spins). Therefore, the behavior of the bosonic wavefunction at the electron-electron coalescence point (r→0) is mathematically identical to that of antiparallel fermions.

[Mike Towler]

Hi Vladimir, two answers and one result.

Protocol: for a few tenths of a mHa between two wavefunctions of one system, not dt·N_w = const. The constant that cancels the time-step term against the population term belongs to each wavefunction, so what cancels for one backflow does not cancel for the other. Fix the population, run the same dt ladder for both (four rungs), fit each as E0 + c1√dt + c2·dt and look at c1: on a Jastrow-free beryllium determinant I have just found a √dt term of 0.075 Ha au^-1/2 at eighteen standard errors, which no linear trick survives. With a Jastrow it is probably gone, but check. Then plot the difference against dt; if it is flat, its extrapolation is safe whatever the individual curvatures do.

The result I promised: the bounded near-node branching changed nothing in the population statistics on the Hartree-Fock node of beryllium. Your spike fraction was 5.2 to 5.9e-4 with and without it at five time steps from 0.00025 to 0.005, the measured inefficiency 0.95 to 1.16, no difference in either direction (Pablo's caveat on the measured number stands). On a bare determinant the spikes are not near-node excursions; the coalescence tail of a cuspless trial is the candidate, and your Jastrow-and-backflow runs may be a different regime. There is a reason, and it bears on your project: for a determinant of radial orbitals the node is the exchange surface r_i = r_j, and the Laplacian of an antisymmetric function of two radii vanishes there with the function, so the local energy has no 1/s pole on the node, however wrong its topology. The local residual is zero where your Mitas-Annaberdiyev integral is ten mHa; they measure different things. Backflow moves the node off the exchange surface and gives it a pole, which is where your trials and my propagator meet.

Bosonic chemistry: not that I know of, and it is cheap in CASINO (a nodeless product-times-Jastrow trial, DMC exact up to the time step). Your two-electron remark is the whole trap in one line. If you run the row, I would read it.