backflow

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).

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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   # η is antisymmetric in the pair

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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.

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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.

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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.

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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).