{"id":"2f520874-122b-4aa5-99b3-99882efbed06","arxiv_id":"1908.04158","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Using the selected space from SCI as a fixed initiator space in i-FCIQMC allows a variational calculation in a much larger space, and the required walkers are only a small multiple of the selected space size.","lead":"One approach in quantum chemistry picks the most important electronic states (selected CI), while another uses random walkers with a sign trick (initiator FCIQMC). This paper combines the two, using the selected states as the fixed walker-initiator set, which gives much more accurate energies for the same memory on test molecules.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"PT2 estimator's cancellation of allowed-spawning terms in Eq. (27) rests on an unproven equilibrium relation, risking systematic bias.","rationale":"The reader identified the same weakest assumption: the 'on average' cancellation of allowed-spawning terms in the PT2 derivation is not proven with an error bound. This is the most load-bearing technical concern because the PT2 correction is a central component of the proposed method, and a systematic bias would undermine the near-exact energy claims even if the variational energies are accurate. The concern is mitigated by the paper's empirical benchmarks on formamide and butadiene, where Evar+PT2 matches exact references within statistical error, but those systems are weakly correlated. The unproven cancellation remains a genuine soft spot, though it does not invalidate the core plateau-scaling result. Since the reader already flagged this caveat and assigned MODERATE confidence, the verdict need not change; the paper would be strengthened by an explicit numerical check of the cancellation.","tokens_in":19698,"tokens_out":19305,"duration_ms":198402,"concrete_test":"On a small system (e.g., N2 or water in a small basis) where H_trunc can be diagonalized exactly, run i-FCIQMC(SCI) and compare the stochastic PT2 estimator of Eq. (30) with the exact expression of Eqs. (18)-(21) evaluated using the exact H_trunc ground state. If the two disagree by more than the statistical error, the allowed-spawning cancellation fails. Alternatively, accumulate the uncancelled terms in Eq. (25) for allowed determinants separately and check whether their average vanishes.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The final PT2 estimator in Eq. (30) is obtained from Eq. (27) by assuming that, for determinants where spawnings are allowed, the equilibrium relation C_i = -S_i/(Delta_tau (E - H_ii)) holds exactly, so the allowed-spawning terms cancel. This relation follows from the full FCIQMC propagation Eq. (2), but in i-FCIQMC(SCI) the sampled coefficient vector is the ground state of the truncated Hamiltonian H_trunc, not the full H. Even for a determinant i with no rejected incoming spawns (e.g., i in V or occupied), the right-hand side of Eq. (28) contains amplitudes C_j that are themselves biased by the initiator truncation on other determinants. Consequently, C_i = -S_i/(Delta_tau (E - H_ii)) holds only approximately, with an uncontrolled error of the order of the initiator bias. If this error does not vanish on average over the allowed set, the PT2 correction acquires a systematic bias that is not reflected in the reported statistical error bars. The paper provides no error bound or numerical verification of this cancellation, only empirical benchmarks for weakly correlated systems where the bias may be small.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes i-FCIQMC(SCI), a hybrid of selected CI and initiator FCIQMC in which the initiator space is fixed to be the SCI-selected space V rather than chosen by a population threshold. Initiators in V may spawn freely into the first-order interacting space (FOIS), so the simulation samples a variational energy in V ⊕ FOIS, while rejected spawns are used to construct a second-order perturbative correction. The manuscript derives the PT2 estimator, studies population plateaus as a function of basis set and orbital choice, and compares energies against standard i-FCIQMC and SHCI/SCI+PT2 for formamide, hexacene, butadiene, BPEA, and the water dimer. The central conclusion is that the walker population required to sample the FOIS is only a small multiple of |V|, and that i-FCIQMC(SCI)+PT2 provides near-exact energies for weakly correlated systems at memory cost comparable to SCI+PT2.","tokens_in":19908,"tokens_out":13428,"duration_ms":145468,"significance":"Should the claims hold, the method is a practically useful synthesis of SCI and FCIQMC: it offers a variational energy in V ⊕ FOIS without storing that space explicitly, a nearly-free PT2 correction built from rejected spawns, and a reduced need for extrapolation relative to SCI+PT2. The paper is strong on transparency: the PT2 estimator is derived from the Guo-Sharma effective-Hamiltonian expression, the interpolation between the two truncated Hamiltonians HA and HB is clearly formulated, and the benchmarks use independent DMRG or extrapolated SHCI references. No parameters are fitted to benchmark energies; the only tunable inputs are the SCI threshold and the walker population. The main weakness is the unproven average cancellation in the derivation of Eq. (30), which is the basis of all +PT2 results.","major_comments":[{"comment":"The cancellation of the allowed-spawning terms is asserted ‘on average’ via the equilibrium relation (28), but in i-FCIQMC(SCI) the coefficients are produced by a Hamiltonian truncation that depends on the instantaneous set of occupied determinants, so (28) is not an exact property of the full-H eigenvector and the error in the cancellation is uncontrolled. Since Eq. (30) is the basis of every ‘+PT2’ result in Section III, please add a direct numerical verification, for example comparing the full expression (26) with the rejected-only estimator (30) from the same simulation and showing that the difference is within statistical noise, or provide a rigorous bound on the neglected terms. The empirical agreement with DMRG does not isolate this step, because the PT2 contribution is small in the strongly correlated cases where it is least likely to mask a bias.","section":"Section II D, Eqs. (27)–(30)"},{"comment":"The conclusion that the walker population needed to sample the FOIS is ‘only a small factor of the size of V itself’ is not uniformly supported by the data: Table I reports plateau/NV = 34.7 for the water dimer with HF orbitals and no semi-stochastic adaptation. Please qualify the claim to the regime in which optimized orbitals and/or the semi-stochastic adaptation are used, or explicitly justify why a factor of roughly 35 is considered small in the context of the much larger FOIS.","section":"Section III B, Table I"}],"minor_comments":[{"comment":"The system list contains a typo: ‘actone’ should be ‘acetone’.","section":"Section III A"},{"comment":"The label ‘W alker pop.’ contains an extraneous space and should read ‘Walker pop.’.","section":"Figure 1 caption"},{"comment":"The sentence ‘these results are all exact within statistical errors’ would be more precise as ‘these results are all consistent with the benchmark within statistical errors.’","section":"Section III D"},{"comment":"The symbol S_i is used for two different quantities in Eq. (27): for rejected determinants it is the amplitude of rejected spawns, while for allowed determinants it is the accepted spawned amplitude. Please clarify this distinction in the text.","section":"Section II D, Eq. (27)"},{"comment":"The statement that the increase in plateau height with basis set cardinal number is ‘only a small factor’ is not immediately evident from the table; for HF orbitals with the semi-stochastic adaptation the ratio changes from 1.7 (cc-pVDZ) to 5.3 (cc-pVTZ). Reporting the ratios explicitly would help the reader assess the claim.","section":"Section III B, Table I"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the only substantive obstacle is the unverified cancellation in the PT2 derivation. The empirical benchmarks provide indirect support, but a direct numerical check of Eq. (30) against Eq. (26), or an equivalent consistency test, would close the gap identified in Section II D. If the authors provide such a check, I would view the paper as acceptable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Nick Blunt's paper is a genuinely useful contribution to the FCIQMC/SCI toolkit. The idea is straightforward: take the selected space V from an SCI calculation and use it as the fixed initiator space in i-FCIQMC, rather than letting the initiator set be determined dynamically by walker populations. That simple change lets you use much larger initiator spaces, and the paper shows convincing evidence that the walker population needed to sample V plus its first-order interacting space is often only a few times |V|, even when the FOIS is orders of magnitude larger. That's the kind of empirical result that makes you think about memory-limited active-space calculations.\n\nThe PT2 correction from rejected spawns is derived carefully and independently of previous work, though it is the same estimator. The algebra is clean, and the paper is refreshingly honest about the cost: SHCI is still an order of magnitude faster for a common selected space, and the statistical error on the PT2 correction is often large. The comparison to i-FCIQMC is fair and shows systematic improvement for the same walker population.\n\nNow the soft spots. The stress-test worry about the equilibrium cancellation in Eq. (27) is mostly misguided. For a determinant where no spawnings are rejected, the fixed-point equation C_i = -S_i/(dt (E-H_ii)) holds exactly for the simulated, possibly biased coefficients—it's an identity of the truncated dynamics, not a statement about the exact ground state. The real caveat is that the energy E in that relation is the shift or projected energy, while the estimator uses Evar. These are close when the truncation is mild, so the cancellation is approximate, but the paper doesn't quantify the error. The benchmarks show Evar+PT2 landing near exact energies, which suggests the bias is small in practice, but a formal statement would be nice. The plateau-scaling claim is empirical; the paper offers no theoretical argument that it holds universally, and it wouldn't surprise me to see exceptions in highly frustrated or metallic systems. That's a limitation, not a fatal one.\n\nWho is this for? Anyone working on large active-space methods, especially if they want variational energies and RDMs in a space larger than they can store exactly. It deserves a serious referee. I'd recommend accept after a moderate revision that clarifies the energy used in the cancellation and adds a sanity check on the full vs simplified PT2 expression for a small system.","headline":"Solid, honest algorithmic paper that deserves serious refereeing; the PT2 estimator's derivation has a minor hidden approximation, but the benchmarks support its use.","tokens_in":20437,"tokens_out":6133,"would_cite":true,"duration_ms":63761,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper shows that fixing the initiator space in i-FCIQMC to a selected space from SCI lets the simulation sample the first-order interacting space with a walker population only a small multiple of |V|, and that rejected spawnings…","keywords":["i-FCIQMC(SCI)","selected configuration interaction","heat-bath CI","PT2 correction","population plateau","fermion sign problem","first-order interacting space","orbital optimization"],"falsifier":"Take a small active space where the H_B matrix in V+FOIS can be diagonalized exactly, run i-FCIQMC(SCI), and record for each non-rejected determinant the difference C_i + S_i/(Δτ(E-H_ii)) over equilibrated iterations; if this difference does not average to zero within statistical error while the variational energy is converged, the PT2 estimator is biased.","tokens_in":19487,"feed_emoji":"⚛️","tokens_out":8336,"duration_ms":79523,"temperature":0.7,"pith_summary":"i-FCIQMC(SCI) combines selected configuration interaction with initiator full configuration interaction quantum Monte Carlo by taking the selected space V from a prior SCI calculation as the fixed initiator space. Because initiators may spawn to any connected determinant, the simulation effectively performs a variational calculation in V plus its first-order interacting space (FOIS), even though the FOIS may be orders of magnitude larger than V. The paper shows that the population plateau, the sign-problem measure that sets the minimum walker memory, is typically only a small multiple of |V|. Rejected spawnings further supply a PT2 correction in the second-order interacting space, which brings weakly correlated systems to near-exact energies. A sympathetic reader would care because this yields SCI+PT2-like accuracy with rigorously variational energies and available reduced density matrices, at a memory cost set by V rather than by the full space sampled.","feed_headline":"Near-exact energies from a walker count close to the SCI list size","feed_subtitle":"The first-order space is sampled variationally at a cost set by the selected space, not the full space.","key_machinery":"The load-bearing object is the fixed initiator space V taken from a prior SCI calculation (here heat-bath CI), together with the initiator spawning rule that walkers in V may spawn to any connected determinant while non-initiators may spawn only to occupied determinants or to V. This construction makes the sampled space V plus its first-order interacting space without ever storing the FOIS vector. The other central object is the population plateau: its height measures how severe the fermion sign problem is, and the paper shows this plateau is a small multiple of |V| under semi-stochastic propagation and optimized orbitals. Finally, the PT2 estimator is assembled from spawnings rejected by the initiator rule, using an equilibrium relation C_i = -S_i/(Δτ(E-H_ii)) to cancel allowed-spawning terms, so the correction is obtained almost for free from data already present in the simulation.","core_discovery":"The central claim is that the initiator approximation in i-FCIQMC can be improved by fixing the initiator space to equal the selected space V of a prior SCI calculation instead of letting population thresholds pick initiators dynamically. In this hybrid method, the Hamiltonian acting on the sampled wave function interpolates between H_A (diagonal in the first-order interacting space) and H_B (the full V+FOIS Hamiltonian), so that as the walker population grows, the wave function converges toward the ground state of V+FOIS. The central numerical discovery is that the walker population needed to sample the FOIS accurately is, in every system studied, only a small factor of |V| itself, even when the FOIS is larger by several orders of magnitude; from those walkers, rejected spawning events give a near-free PT2 correction that removes most of the remaining error in weakly correlated systems. This makes i-FCIQMC(SCI)+PT2 a complementary alternative to SCI+PT2: more accurate for a common selected/initiator space, albeit slower, and able to supply variational energy estimates and reduced density matrices in the larger space.","pith_inferences":["Inference: the plateau-to-|V| scaling suggests a design rule: choose the initiator space as the top of a deterministic importance ranking, and the stochastic sampler will populate its connections at a cost set by the ranked space rather than by the connection space; this could generalize to other projector QMC formulations.","Inference: if the replica-trick statistical error on the PT2 correction is reduced by a single-replica estimator, the method's practical advantage over SCI+PT2 would grow, since the current bottleneck is noise on the correction, not the variational energy.","Inference: since the paper tests only heat-bath CI as the SCI generator, comparing CIPSI, ASCI, or other selection criteria would show whether the favorable plateau scaling is a property of the fixed-initiator construction itself or of the particular importance metric.","Inference: the interpolation between H_A and H_B implies a sharp testable prediction: variational energies should improve monotonically with walker population toward the H_B ground state, and deviations from that trend would signal that the initiator rules are not the only source of bias."],"forward_implications":["For any system where an SCI calculation can produce a selected space V, i-FCIQMC(SCI) can recover most of the correlation energy in V plus its first-order interacting space with memory proportional to a small multiple of |V|, not of the much larger FOIS.","The same selected space gives variational energies comparable to, and sometimes lower than, SCI+PT2 energies, with a rigorous variational bound and with exact reduced density matrices available for the V+FOIS wave function.","The rejected-spawn PT2 estimator brings weakly correlated systems to near-exact total energies at selected-space sizes where SCI+PT2 still has residual error.","The method extends to large active spaces and to basis sets up to quadruple-zeta, with plateau height growing only modestly with basis cardinal number when optimized orbitals and the semi-stochastic adaptation are used.","Using a fixed initiator space systematically lowers variational energies relative to standard i-FCIQMC for the same walker population, showing that less severe Hamiltonian truncation is possible while keeping the sign problem manageable."],"supporting_citations":[{"why":"Defines the FCIQMC projector and spawning/annihilation algorithm that the hybrid method modifies.","marker":"[1]"},{"why":"Introduces the initiator rules and population threshold that i-FCIQMC(SCI) replaces with a fixed selected space.","marker":"[5]"},{"why":"Adds the semi-stochastic adaptation whose deterministic space is set to V in the hybrid calculations.","marker":"[3]"},{"why":"Provides the heat-bath SCI selection used to generate the initiator space V.","marker":"[13]"},{"why":"Supplies the SHCI+PT2 reference method and energies against which the hybrid approach is compared.","marker":"[17]"},{"why":"Provides the second-order perturbative correction derivation whose A-term becomes the rejected-spawning PT2 estimator.","marker":"[47]"},{"why":"Introduces preconditioned propagation, used in the hybrid simulations to reach convergence quickly and stabilize PT2 statistics.","marker":"[40]"},{"why":"Establishes the SCI-CASSCF orbital optimization used to generate the optimized orbitals in all hybrid calculations.","marker":"[49]"}],"fun_headline_variants":["Hybrid SCI+FCIQMC samples first-order space variationally","Fixed initiators from SCI: bigger spaces at same walker cost","Variational FCIQMC in FOIS with walkers set by SCI size","SCI initiators cut walker need: near-exact energies, lower cost","Complement to SCI+PT2: variational first-order space sampling"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"In Section IID, the PT2 derivation assumes that at equilibrium the amplitude C_i and total spawning S_i on every non-rejected determinant satisfy C_i = -S_i/(Δτ(E-H_ii)), so the allowed-spawning terms in Eq. (27) cancel on average; the paper gives no error bound for this cancellation, so a systematic failure would bias the PT2 energies even when variational energies are accurate.","fun_headline_variants_meta":{"raw":{"variants":["Hybrid SCI+FCIQMC samples first-order space variationally","Fixed initiators from SCI: bigger spaces at same walker cost","Variational FCIQMC in FOIS with walkers set by SCI size","SCI initiators cut walker need: near-exact energies, lower cost","Complement to SCI+PT2: variational first-order space sampling"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000243,"raw_usage":{"total_tokens":1564,"prompt_tokens":1015,"completion_tokens":549,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":631,"completion_tokens_details":{"reasoning_tokens":454}},"tokens_in":631,"tokens_out":549,"duration_ms":5662,"temperature":1.0,"reasoning_tokens":454,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:50:01.416963+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a small active space where the H_B matrix in V+FOIS can be diagonalized exactly, run i-FCIQMC(SCI), and record for each non-rejected determinant the difference C_i + S_i/(Δτ(E-H_ii)) over equilibrated iterations; if this difference does not average to zero within statistical error while the variational energy is converged, the PT2 estimator is biased.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the FCIQMC projector and spawning/annihilation algorithm that the hybrid method modifies."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the initiator rules and population threshold that i-FCIQMC(SCI) replaces with a fixed selected space."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Adds the semi-stochastic adaptation whose deterministic space is set to V in the hybrid calculations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the heat-bath SCI selection used to generate the initiator space V."},{"cited_title":"Sharma , author A","cited_arxiv_id":null,"evidence_quote":"Supplies the SHCI+PT2 reference method and energies against which the hybrid approach is compared."},{"cited_title":"Guo , author Z","cited_arxiv_id":null,"evidence_quote":"Provides the second-order perturbative correction derivation whose A-term becomes the rejected-spawning PT2 estimator."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces preconditioned propagation, used in the hybrid simulations to reach convergence quickly and stabilize PT2 statistics."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the SCI-CASSCF orbital optimization used to generate the optimized orbitals in all hybrid calculations."}],"review_version":1}