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.