| name | backflow |
| description | Use this skill when working on the backflow transformation in casino/backflow.py or casino/readers/backflow.py: the eta/mu/Phi/Theta displacement terms, the cutoff and truncation order C, AE-cutoff multiplier g(r), Kato cusp constraints, irrotational backflow, construct_c_matrix, or the backflow Jacobian used in the kinetic energy. This is the physics/math reference (López Ríos et al. 2006) mapped to the code. See also the qmc skill for how backflow enters the wave function and local energy.
|
Backflow transformation (casino/backflow.py)
Backflow (BF) replaces the electron coordinates fed into the Slater determinant with
collective quasiparticle coordinates, improving the nodal surface (and thus the fixed-node
DMC energy). The Jastrow factor is left untouched.
Ψ_BF(R) = exp(J(R)) · Φ(X), x_i = r_i + ξ_i(R)
ξ_i is the backflow displacement of electron i, depending on the whole configuration.
Primary reference: pdfs/backflow/backflow.pdf (López Ríos, Ma, Drummond, Towler,
Needs, Inhomogeneous backflow transformations in QMC, PRE 74, 066701 (2006)). The
equation numbers below refer to that paper. The constraint code is a direct port of
CASINO's pbackflow.f90 (pdfs/backflow/pbackflow.f90).
The three displacement terms
The total displacement is a sum of e-e, e-n and e-e-n contributions (Eqs. 4–6):
ξ_i = ξ_i^ee + ξ_i^en + ξ_i^een
| Term | Eq. | Function of | Code (impl) |
|---|
η e-e | 4 | η(r_ij)·r_ij | eta_term |
μ e-n | 5 | μ(r_iI)·r_iI | mu_term |
Φ,Θ e-e-n | 6 | Φ^{jI}·r_ij + Θ^{jI}·r_iI | phi_term |
Φ^{jI} r_ij + Θ^{jI} r_iI spans the plane of r_i, r_j, r_I, so no separate
perpendicular component is needed. Θ is often slightly more important than Φ.
CASINO also has a three-electron Ω (e-e-e) term and a Π (periodic) term. Ω is being
ported — its reader lives in casino/readers/backflow.py; the evaluation is not in
casino/backflow.py yet. See the omega-backflow skill. Note: the kappa_term WIP
branch left kappa_parameters references in casino/backflow.py commented out (no reader
support yet).
Polynomial + cutoff form (Eqs. 7–11)
Each radial function is a natural power expansion multiplied by a smooth cutoff:
f(r; L) = (1 - r/L)^C · H(L - r) # Eq. 7, truncation order C = self.trunc
η_ij = f(r_ij; Lη) · Σ_k c_k r_ij^k # Eq. 8
μ_iI = f(r_iI; Lμ) · Σ_k d_k r_iI^k # Eq. 9
Φ_i^jI = f(r_iI;LΦ)f(r_jI;LΦ) · Σ_klm φ_klm r_iI^k r_jI^l r_ij^m # Eq. 10
Θ_i^jI = f(r_iI;LΦ)f(r_jI;LΦ) · Σ_klm θ_klm r_iI^k r_jI^l r_ij^m # Eq. 11
C = self.trunc (only C = 2 and C = 3 are used; the C-th derivative of f is
discontinuous at r = L, so the Laplacian in the kinetic energy is discontinuous for C < 3).- Cutoffs:
eta_cutoff (Lη), mu_cutoff (Lμ), phi_cutoff (LΦ). Optimizable via the
*_cutoff_optimizable flags.
- Parameters are spin-pair dependent:
eta_set / mu_set / phi_set index into the
parameter array by counting how many of the pair are down-spin
(int(e1>=neu)+int(e2>=neu)), mod the number of spin channels.
- Powers
r^k are precomputed once per configuration: ee_powers → e_powers,
en_powers → n_powers, then poly = parameters @ powers[:k_max].
Code pattern (eta_term, all terms follow it):
poly = parameters[eta_set] @ e_powers[e1, e2, :k_max]
bf = (1 - r / L) ** C * poly * r_vec
res[ae_cutoff_condition, e1] += bf
res[ae_cutoff_condition, e2] -= bf
AE-cutoff multiplier g(r) and the size-2 axis
Near an all-electron (AE) nucleus the e-n Kato cusp cannot be satisfied unless the
total displacement vanishes at the nucleus. So every contribution that is not
already centered on that nucleus is multiplied by a smooth cutoff g(r_iI) (Eq. B1):
g(r) = (r/Lg)² · [6 - 8(r/Lg) + 3(r/Lg)²], r < Lg; g = 1 otherwise # Lg = ae_cutoff
g, g' are continuous at r = Lg, and g(0) = g'(0) = 0. Implemented in
ae_multiplier / ae_multiplier_gradient / ae_multiplier_laplacian.
Critical convention — the leading size-2 axis of the term arrays (res[2, nelec, 3])
is NOT spin. It is the AE-cutoff branch:
- index
0 — "AE cutoff exactly not applied" (contribution centered on this nucleus,
or r > ae_cutoff);
- index
1 — "AE cutoff maybe applied" (multiply by g(r_iI)).
ae_cutoff_condition = int(r > self.ae_cutoff[label]). The value/gradient/laplacian
methods combine the two branches with the ae_multiplier. For PP atoms ae_cutoff = 0,
so everything lands in branch 1 with g = 1 (no-op). See memory
backflow-ae-multiplier-axis.
Cusp & irrotational constraints (construct_c_matrix)
The BF parameters are constrained so the transformation does not spoil the Kato cusp
conditions already enforced by the Slater/Jastrow parts (Appendix A). construct_c_matrix
builds the homogeneous constraint matrix C·p = cutoff_constraints; the null space
(rref / parameters_projector) gives the free parameters. This is a line-by-line port of
SUBROUTINE construct_C in pbackflow.f90.
Two boolean flags per Φ/Θ channel drive the constraint count:
phi_cusp — AE atom (True) vs PP atom (False).
- AE:
phi_constraints = 6·(en+ee)-2-type rows enforce that Φ,Θ and their gradients
vanish appropriately at the nucleus (Eqs. A1, A4, A5).
- PP: fewer rows, and the cutoff enters via
-C/phi_cutoff terms (cutoff_constraints).
phi_irrotational — impose ∇×ξ = 0 so ξ = ∇Y derives from a scalar backflow
potential (Appendix A2, Eqs. A6–A8). Adds ((eN+3)(ee+2)-4)(eN+1) rows (minus a block
when trunc == 0), roughly halving the free Φ/Θ parameters. Gives poor results for
atoms (e.g. carbon) — use with care.
Cusp-related building blocks in the matrix:
- e-e cusp (EKCC, Eqs. A2/A3): rows active for like-spin /
spin_dep in (0,2) channels,
keyed on m == 1 (the r_ij^1 term).
- e-n cusp (NKCC): rows keyed on
k == 0/1 (Φ) and l == 0/1 (Θ), i.e. the low e-n
power terms, with -trunc/phi_cutoff factors for PP.
When editing construct_c_matrix, keep the offset bookkeeping (ee_constrains,
en_constrains, offset) in exact correspondence with pbackflow.f90 — the row indices
are hand-computed and easy to break silently.
Value, gradient, Laplacian and the kinetic energy
value(e_vectors, n_vectors) → total displacement (nelec, 3) added to n_vectors
before the Slater determinant (wfn.py).gradient → the Jacobian ∂x/∂r of shape (nelec·3, nelec·3) (b_g).laplacian → returns (b_l, b_g, b_v): backflow Laplacian, Jacobian, displacement.
Because each quasiparticle depends on all electrons, the kinetic energy needs the
Slater Hessian and the chain rule (see the qmc skill):
b_l, b_g, b_v = self.backflow.laplacian(e_vectors, n_vectors)
s_h, s_g = self.slater.hessian(b_v + n_vectors)
s_l = np.sum(b_g * (s_h @ b_g)) + s_g @ b_l
This also breaks the EBEA fast determinant updates: with BF every electron move
recomputes the determinants via LU (CBCA-style), which is the main cost of BF.
Parameter optimization (varmin/emin) goes through get_parameters / set_parameters /
get_parameters_mask / set_parameters_projector; the projector keeps moves inside the
constraint null space. The _d1 methods (eta_term_d1, …) are parameter derivatives used
by the linear/emin method.
Reference PDFs (pdfs/backflow/)
Primary — matches the code directly:
backflow.pdf / lopezrios_backflow.pdf — López Ríos et al., PRE 74, 066701 (2006).
Eqs. 4–11 = the displacement terms; Appendix A = the constraints in construct_c_matrix;
Appendix B (Eq. B1) = ae_multiplier. Start here.pbackflow.f90 — CASINO Fortran source that construct_c_matrix and the term
evaluators were ported from. The authoritative index bookkeeping.
Background / origins:
pandharipande1973.pdf — Pandharipande & Itoh, PRA 8, 2564 (1973): effective mass of
³He, the quantum backflow idea (Feynman–Cohen lineage).pandharipande1986.pdf — Pandharipande, Pieper, Wiringa, PRB 34, 4571 (1986): VMC of
⁴He/³He drops with Feynman–Cohen backflow.schmidt1990.pdf — Schmidt & Moskowitz, JCP 93, 4172 (1990): correlated wave
functions for atoms He–Ne (e-e-n correlation forms).holzmann2003.pdf — Holzmann, Ceperley, Pierleoni, Esler, PRE 68, 046707 (2003):
analytic backflow + three-body for the electron gas / metallic hydrogen.
Extensions / theses:
single_vector.pdf — P. López Ríos PhD thesis, Backflow and pairing wave function for
QMC (Cambridge, 2006). Full derivations behind the paper.backflow_wfn.pdf — P. Seth PhD thesis, Improved wave functions for QMC (Cambridge,
2012).orbital_backflow.pdf — Holzmann & Moroni, Orbital-dependent backflow (arXiv:1910.07167).lopezrios_nov12.pdf — Seth/López Ríos/Needs slides on orbital-dependent backflow (2012).
