{"id":"4c7852a4-c1d0-427f-8469-5daf5ea19f2f","arxiv_id":"2505.07080","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A relativistic EOM-CCSD method using ADC(2)-derived state-specific frozen natural spinors with X2CAMF and Cholesky decomposition gives excitation energies close to the canonical method at reduced cost.","lead":"This paper presents a cheaper way to compute excited states of molecules containing heavy atoms using a relativistic coupled-cluster method. It generates state-specific frozen natural spinors from a lower-cost method and shows the truncated calculation matches the full calculation closely.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Transition dipole moment claim is weakly supported: up to 8% error at the chosen threshold, no molecular TDM test, and no property-specific correction.","rationale":"The paper's strongest claim bundles excitation energies, fine-structure splittings, and transition dipole moments under a single reduction-cost umbrella. The energy and splitting benchmarks are credible: agreement with canonical results is within 0.01 eV for most states, and even the 4p2 1S0 double-excitation-like states in Ga+, In+, and Tl+ are reproduced, which partially answers the reader's concern about ADC(2) failing for double excitations. The property side, however, is much thinner. Only one atom (Xe) is used for TDM validation, deviations reach 8%, and no correction is applied. Because the truncation threshold was chosen for energies, the property accuracy is an unsupported extrapolation. This does not invalidate the method's energy claims, but it does mean the central claim as stated is too broad. The reader's conditional verdict already captures this risk; my concern is a different facet of the same need for qualification, so I recommend keeping the verdict unchanged.","tokens_in":19525,"tokens_out":13937,"duration_ms":140056,"concrete_test":"Recompute the Xe 5p5(2P3/2)5d[1/2]1 transition dipole moment with X2CAMF-SS-FNS-EE-EOM-CCSD in the same triply augmented basis at thresholds 10^-3, 10^-4.5, 10^-5, and 10^-6. If the TDM changes by more than 0.01 a.u. between 10^-4.5 and 10^-6, the chosen threshold is not converged for properties, and the conclusion should be revised to state that the method targets excitation energies and fine-structure splittings, with TDMs requiring a stricter threshold or a property-specific correction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim includes \"excellent agreement ... for transition properties\" (Conclusion). Yet the only transition property benchmark is the Xe atom (Table V). At the universal truncation threshold 10^-4.5, SS-FNS-EE-EOM-CCSD transition dipole moments deviate from canonical 4c-DC-EE-EOM-CCSD by +0.020, +0.026, -0.010, and -0.011 a.u. for the four bright Rydberg states, corresponding to relative errors of 3.1%, 5.1%, -8.1%, and -1.6%. The largest error occurs for the 5p5(2P3/2)5d[1/2]1 state, where the truncated-basis TDM (0.114 a.u.) moves away from the experimental value (0.120 ± 0.003 a.u.) compared with the canonical result (0.124 a.u.). The paper explicitly states that no perturbative correction is applied to transition properties, and no molecular TDM benchmark is presented. The truncation threshold was selected on the basis of excitation-energy convergence (Section IV.A, Zn and AuH), so it is not known to be converged for one-electron property operators. If a similar or larger TDM error occurs in molecules, the property component of the headline claim is overstated.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript presents the theory, implementation, and benchmarks of a state-specific frozen natural spinor (SS-FNS) approach to reduce the cost of relativistic equation-of-motion coupled cluster singles and doubles (EOM-CCSD) for excited states. The SS-FNS virtual space is generated from the one-particle density matrix of the relativistic ADC(2) excited state (Eq. 31), and the method is implemented both with the four-component Dirac-Coulomb Hamiltonian and with the X2CAMF two-component Hamiltonian combined with Cholesky decomposition of the two-electron integrals. The authors show that SS-FNS converges faster in the virtual-space truncation threshold than the standard MP2-based FNS for Zn, Ga+, and AuH, that the 4c-DC and X2CAMF versions agree closely, and that excitation energies and fine-structure splittings for Ga+, In+, Tl+, and I3- agree with canonical EOM-CCSD to within about 0.01-0.04 eV. A single transition-dipole benchmark is reported for Xe, and a timing demonstration is given for [I3(H2O)6]-. The central claim is that the method reproduces canonical excitation energies, spin-orbit splittings, and transition properties at a fraction of the virtual-space size.","tokens_in":19831,"tokens_out":8184,"duration_ms":80694,"significance":"If the accuracy claims hold, the method is a practically useful compromise for heavy-element excited-state calculations, because it removes a large fraction of the virtual space while preserving EOM-CCSD accuracy for energies and splittings. The strengths are the clear formal derivation, the use of ADC(2)-based state-specific densities rather than MP2 densities to describe excited states, the inclusion of a perturbative correction (Eq. 32), and the demonstration of both 4c-DC and X2CAMF implementations. The benchmarks against canonical EOM-CCSD, FSCC, and experiment are appropriate and generally convincing. The main caveat is that the transition-property evidence is thin: only one atom is tested, with errors up to 8% at the chosen threshold, while the conclusion claims excellent agreement for transition properties.","major_comments":[{"comment":"The conclusion that the method provides 'excellent agreement with the canonical EOM-CCSD method for ... transition properties' is not supported by the evidence presented. The only transition-property benchmark is the Xe atom in Table V, and at the threshold eta_crit=10^-4.5 the SS-FNS transition dipole moments differ from the canonical 4c-DC-EE-EOM-CCSD values by +0.020, +0.026, -0.010, and -0.011 a.u. for the four bright states. The largest relative error, 8.1%, occurs for the 5p5(2P3/2)5d[1/2]1 state, where the truncated value 0.114 a.u. moves away from the experimental value 0.120 +/- 0.003 a.u. relative to the canonical value 0.124 a.u. No molecular transition dipole moment is reported, the truncation threshold was selected using excitation-energy convergence only, and the perturbative correction of Eq. (32) is applied only to energies. The authors should either add a molecular TDM benchmark together with a TDM convergence test in eta_crit, or explicitly restrict the central claim to excitation energies and fine-structure splittings.","section":"Section IV.B-IV.E"},{"comment":"The method's reliability rests on the assumption that the relativistic ADC(2) one-particle density matrix in Eq. (31) retains the essential character of the target excited state after truncation. The benchmarks cover singly excited valence and Rydberg states in atoms, AuH, I3-, and Xe, but they do not include states with strong double-excitation character, long-range charge transfer, or strongly mixed spin-orbit manifolds in a relativistic setting. A qualitative failure of ADC(2) for such a state would directly bias the truncated SS-FNS-EE-EOM-CCSD result, because the retained virtual spinors would be the wrong ones. This limitation should be stated explicitly, or at least one known difficult state should be benchmarked before the method is presented as a general low-cost relativistic excited-state approach.","section":"Section IV.B-IV.E"}],"minor_comments":[{"comment":"The text in Section IV.E says the Xe calculations use the triply augmented dyall.v3z basis set, while the Table V caption says triply augmented dyall.ae3z basis set; please reconcile this inconsistency.","section":"Section IV.E, Table V caption"},{"comment":"Step 2 of the X2CAMF-SS-FNS algorithm says 'generate three-centered two-electron integrals in the canonical natural spinor basis,' but natural spinors have not yet been constructed at that point; this should presumably read 'canonical spinor basis'.","section":"Section III, step 2"},{"comment":"In Table I, the standard FNS row leaves the 'Corrected' and 'Canonical' cells empty, which makes the comparison difficult to parse; explicit 'not applicable' entries would clarify that the perturbative correction is not defined for that variant.","section":"Table I"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and the formal development is sound, but the transition-property claim needs to be either strengthened with additional benchmarks or softened. The authors may also wish to include a brief statement about the absence of double-excitation and charge-transfer benchmarks in the relativistic framework. No concerns about novelty overlap or citation practices."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nThe paper is a genuine step forward for relativistic excited-state calculations, but its headline claim about transition properties outruns the evidence.\n\nWhat is new: the authors build the truncated virtual space for EOM-CCSD from state-specific ADC(2) densities, rather than from MP2, and implement it in both four-component and X2CAMF two-component forms with Cholesky decomposition. The state-specific idea itself is known in non-relativistic theory—Hättig, Kállay, and Valeev have done similar things—but the relativistic spinor implementation, with the D_ab expression in Eq. (31) and the appendix giving the density equations, is new and useful.\n\nWhat the paper does well: the benchmarks are credible. SS-FNS converges much faster than MP2-based FNS for Zn and AuH. For Ga+, In+, and Tl+, the excitation energies and fine-structure splittings match canonical EOM-CCSD to about 0.01 eV while retaining under 40% of the virtual space. The I3- benchmark gives RMSD 0.021 eV with the perturbative correction. The solvated triiodide complex calculation (21 atoms, 220 electrons) is exactly the kind of large-scale test this method is meant to enable.\n\nSoft spots: the transition property claim is the weakest section. The only property test is the Xe atom, and at the universal threshold the TDM errors are up to 8% for the 5d[1/2]_1 state, moving away from experiment relative to the canonical result. No perturbative correction is applied to properties, and no molecular TDM is computed. The threshold itself was chosen by watching excitation-energy convergence, not one-electron properties, so the \"excellent agreement ... for transition properties\" sentence in the conclusion is overstated. A referee should ask for more property benchmarks or a softened claim. Also, the truncation threshold is a free parameter tuned on the same kind of systems the method is later benchmarked on; not fatal, but it limits the predictive claim. No code or data is released, so the large-scale calculation cannot be independently checked.\n\nThe method assumes ADC(2) densities capture the excited-state character well. That is plausible for the tested valence and Rydberg states, but untested for double-excitation, charge-transfer, or strongly spin-orbit-mixed states in the relativistic setting.\n\nWho this is for: people working on heavy-element excited-state theory and reduced-cost coupled cluster. It deserves a serious referee, and the revision should address the property claim directly. The core method appears sound; the paper is publishable after the property benchmarks are strengthened or the claim is appropriately narrowed.\n\nRecommendation: engage with it in peer review, with the TDM issue as the main revision point.","headline":"Useful relativistic extension of state-specific FNS to EOM-CCSD, but the transition-dipole claim is under-supported.","tokens_in":20409,"tokens_out":3096,"would_cite":true,"duration_ms":29368,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"State-specific frozen natural spinors derived from ADC(2) excited-state densities let a relativistic EOM-CCSD calculation retain the accuracy of the full canonical calculation while using only about a third of the virtual spinors.","keywords":["relativistic coupled cluster","equation-of-motion CCSD","frozen natural spinors","state-specific virtual orbitals","ADC(2)","excitation energies","fine-structure splitting","X2CAMF Hamiltonian"],"falsifier":"Run SS-FNS-EE-EOM-CCSD at the recommended $10^{-4.5}$ occupation threshold on a state known to be dominated by double excitations or a strong charge-transfer state in a heavy-element molecule, and compare with the untruncated canonical calculation; a deviation much larger than 0.01 eV in the excitation energy would show the ADC(2)-derived spinors do not always span the needed virtual space.","tokens_in":19364,"feed_emoji":"⚛️","tokens_out":9151,"duration_ms":78292,"temperature":0.7,"pith_summary":"This paper aims to make relativistic excited-state calculations for heavy-element systems dramatically cheaper without losing the accuracy of the full relativistic equation-of-motion coupled-cluster method. It proposes building frozen natural spinors, reduced sets of virtual orbitals tailored to a given excited state, from the excited-state density of the cheaper ADC(2) method, one set per state, instead of from the ground-state MP2 density. With these state-specific spinors, the virtual space can be cut to roughly a third of the canonical basis while excitation energies stay within about 0.01 eV of the full relativistic EOM-CCSD results, and fine-structure splittings and transition dipole moments are similarly preserved. The claim matters because fully relativistic EOM-CCSD is one of the most accurate tools for heavy-element excited states but scales steeply, limiting it to atoms and small molecules; the truncated method is demonstrated on a solvated triiodide complex with over a thousand virtual spinors.","feed_headline":"One-third of the virtual space keeps EOM-CCSD accuracy","feed_subtitle":"State-specific ADC(2) spinors reproduce full relativistic excitation energies within about 0.01 eV.","key_machinery":"The load-bearing object is the state-specific frozen natural spinor (SS-FNS) basis: eigenfunctions of $D^{SS}_{ab}(k) = D^{MP2}_{ab} + D^{EE-ADC(2)}_{ab}(k)$, the virtual-virtual block of the correlated one-particle density for excited state $k$ built from MP2 plus an ADC(2) difference density. Diagonalizing this matrix, retaining spinors above an occupation threshold $\\eta_{crit}$, and semi-canonicalizing gives a compact one-particle basis in which the EOM-CCSD equations are solved root-wise. A perturbative correction, $\\omega_{corrected} = \\omega_{uncorrected} + (\\omega^{EE-ADC(2)}_{canonical} - \\omega^{EE-ADC(2)}_{SS-FNS})$, transfers the cheap method's truncation error into the expensive answer. The practical cost reduction comes from joining this basis with the X2CAMF two-component Hamiltonian, which avoids relativistic two-electron integrals, and Cholesky decomposition, which avoids forming integrals with three or four external indices in the canonical basis.","core_discovery":"The central claim is that the character of an excited state is encoded in the virtual-space one-particle density of a cheap but qualitatively reliable excited-state method, ADC(2), and that diagonalizing that density yields natural spinors in which the subsequent EOM-CCSD calculation converges with far fewer virtuals. Concretely, the state-specific density is the sum of the MP2 ground-state density and the EE-ADC(2) difference density for that state (Eq. 31). Diagonalizing this virtual-virtual block and truncating on occupation number produces a compact, state-adapted spinor basis; an optional perturbative correction, the difference between the ADC(2) excitation energy in the full and truncated bases, removes most of the remaining truncation bias. Across zinc, gallium, indium, and thallium cations, AuH, the triiodide ion, and xenon, the truncated method reproduces canonical relativistic EOM-CCSD excitation energies, spin-orbit fine-structure splittings, and transition dipoles to within about 0.01 eV (RMSD 0.021 eV over 18 states of I3−), while keeping only about 30–40% of the virtual spinors.","pith_inferences":["If the SS-FNS construction is as transferable as the benchmarks suggest, the same ADC(2)-density recipe could be paired with other high-level excited-state methods, such as EOM-CCSD(T) or algebraic diagrammatic construction at third order, to cut their cost in heavy-element applications.","The perturbative-correction idea is a template for a general two-level scheme: use a cheap method to measure the bias introduced by any basis truncation, then add that bias to an expensive method computed in the truncated space; this could reduce the cost of other truncated expansions, such as natural transition orbital bases for response properties.","The method's accuracy likely degrades for states where ADC(2) is a poor zero-order description, such as strong double excitations, charge transfer with large orbital relaxation, or highly multireference spin-orbit-mixed states, so a practical implementation would want a diagnostic flag in the ADC(2) step that warns when the SS-FNS subspace is too small."],"forward_implications":["Relativistic EOM-CCSD becomes practical for molecules with well over a thousand virtual spinors: the solvated triiodide complex benchmark keeps only 517 of 1394 virtuals and completes in about five days on a workstation.","Fine-structure splittings of heavy-element cations are preserved to within about 0.01–0.08 eV of experiment, so the method can predict spin-orbit-resolved spectra without a full canonical relativistic calculation.","The perturbative ADC(2) correction is a cheap way to remove most of the remaining truncation bias, improving the I3− RMSD from 0.029 to 0.021 eV.","Because each excited state is solved in its own basis, ground-to-excited transition properties remain well defined and match canonical values, with Xe transition dipole moments within about 0.02 a.u. while dropping roughly 70% of the virtual spinors."],"supporting_citations":[{"why":"Provides the canonical 4c-DC-EE-EOM-CCSD excitation energies for I3− against which the truncated results are benchmarked, along with the virtual-energy cutoff scheme reused here.","marker":"[22]"},{"why":"Documented the failure of MP2-based natural spinors for excited states, the motivating result this work fixes, and supplies the Ga+ comparison case.","marker":"[49]"},{"why":"Supplies the Cholesky-decomposed X2CAMF frozen-natural-spinor coupled-cluster implementation that the present X2CAMF algorithm builds on.","marker":"[48]"},{"why":"Provides the programmable relativistic ADC(2) sigma vectors used to generate the state-specific densities.","marker":"[64]"},{"why":"Gives the canonical 4c-DC-EE-EOM-CCSD transition dipole moments for xenon used as reference values.","marker":"[23]"},{"why":"The non-relativistic predecessor showing ADC(2)-natural-orbital EOM-CCSD gives uniform accuracy, the philosophy transferred here to the relativistic case.","marker":"[61]"}],"fun_headline_variants":["State-specific spinors slash relativistic EOM-CCSD cost by two-thirds","Cheap ADC(2) spinors make relativistic EOM-CCSD practical","Relativistic EOM-CCSD with 60% fewer virtuals, same excitation energies","State-specific FNS: near-exact excited states at fraction of cost","Frozen natural spinors from ADC(2) preserve EOM-CCSD accuracy"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The method assumes the cheap ADC(2) calculation describes each excited state well enough that the spinors it keeps are exactly the ones the expensive coupled-cluster calculation needs; if ADC(2) misidentifies a state's character, the truncated space is biased no matter how accurate EOM-CCSD is.","fun_headline_variants_meta":{"raw":{"variants":["State-specific spinors slash relativistic EOM-CCSD cost by two-thirds","Cheap ADC(2) spinors make relativistic EOM-CCSD practical","Relativistic EOM-CCSD with 60% fewer virtuals, same excitation energies","State-specific FNS: near-exact excited states at fraction of cost","Frozen natural spinors from ADC(2) preserve EOM-CCSD accuracy"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000328,"raw_usage":{"total_tokens":1863,"prompt_tokens":1007,"completion_tokens":856,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":623,"completion_tokens_details":{"reasoning_tokens":749}},"tokens_in":623,"tokens_out":856,"duration_ms":7515,"temperature":1.0,"reasoning_tokens":749,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:24:52.411576+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run SS-FNS-EE-EOM-CCSD at the recommended $10^{-4.5}$ occupation threshold on a state known to be dominated by double excitations or a strong charge-transfer state in a heavy-element molecule, and compare with the untruncated canonical calculation; a deviation much larger than 0.01 eV in the excitation energy would show the ADC(2)-derived spinors do not always span the needed virtual space.","supporting_citations":[{"cited_title":"South , author A","cited_arxiv_id":null,"evidence_quote":"Provides the canonical 4c-DC-EE-EOM-CCSD excitation energies for I3− against which the truncated results are benchmarked, along with the virtual-energy cutoff scheme reused here."},{"cited_title":"Chamoli , author K","cited_arxiv_id":null,"evidence_quote":"Documented the failure of MP2-based natural spinors for excited states, the motivating result this work fixes, and supplies the Ga+ comparison case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Cholesky-decomposed X2CAMF frozen-natural-spinor coupled-cluster implementation that the present X2CAMF algorithm builds on."},{"cited_title":"Chamoli , author M","cited_arxiv_id":null,"evidence_quote":"The non-relativistic predecessor showing ADC(2)-natural-orbital EOM-CCSD gives uniform accuracy, the philosophy transferred here to the relativistic case."}],"review_version":1}