Results

Dataset inventory

The current data layer contains four generated profile/radii/QA datasets. The two primary datasets cover all curated states within their element ranges. The supplemented/augmented branches contain neutral and anion states only; cations are not repeated because compact positive ions are not the main target of basis-tail sensitivity analysis.

dataset ID	basis	branch role	state scope	profile rows	QA rows
pbe0_sfx2c_x2cqzvpall_h-rn_spherical_v2	x2c-QZVPall	primary H--Rn	H-Rn all curated states	430	430
pbe0_sfx2c_dyallv4z_h-lr_spherical_v2	dyall-v4z	primary H--Lr	H-Lr all curated states	501	501
pbe0_sfx2c_x2cqzvpalls_h-rn_spherical_v2	x2c-QZVPall-s	supplemented H--Rn	H-Rn neutral and anion states with x2c-QZVPall-s; cations excluded	192	192
pbe0_sfx2c_dyallav4z_h-ba_hf-ra_spherical_v2	dyall-av4z	augmented available intervals	H-Ba and Hf-Ra neutral atoms plus selected anions in the same intervals with dyall-av4z; cations excluded	166	166

Together, these datasets contain 1289 generated dataset-state rows. A dataset-state row means one state emitted in one basis branch; the same curated state can appear in more than one branch when it participates in a basis comparison.

State and charge coverage

The curated table contains 501 state records. The charge distribution shows the intended scope: neutral atoms, cations through the current charge policy, accepted/provisional monoanions, source-backed diagnostic monoanions, and formal anions.

charge	curated states
-3	6
-2	20
-1	86
0	103
+1	102
+2	95
+3	89

The state-role table separates source-backed references from explicitly formal rows. This distinction is essential for interpretation: formal anion rows are useful reference gauges, not claims of isolated stable anions.

state role	curated states
reference	229
reference_uncertain	160
bound_experimental	65
bound_provisional	4
diagnostic_theory	3
formal_monoanion	14
formal_multianion	26

Because the profile branches overlap, the generated-row counts are larger than the curated-state counts. The generated-row distribution below is the effective coverage that downstream consumers see when they read all profile/radii/QA datasets.

charge	generated dataset-state rows
-3	24
-2	80
-1	313
0	348
+1	187
+2	174
+3	163

state role	generated dataset-state rows
reference	594
reference_uncertain	278
bound_experimental	247
bound_provisional	11
diagnostic_theory	4
formal_monoanion	51
formal_multianion	104

The generated layer therefore exposes the formal anion rows explicitly rather than hiding them in a generic anion class. This is useful for sensitivity analysis because the formal rows are exactly where low-density tail behavior is expected to be most delicate.

Validation outcomes

The validation summary below reports one row per generated dataset. max |ΔN| is the maximum independent electron-count error in electrons. The angular quantity is the maximum relative angular standard-deviation diagnostic above the QA density floor. Linear-dependency warnings count rows where the backend reported basis linear-dependency handling or dropped primitive behavior.

dataset ID	basis	rows	failed rows	max |ΔN|	max angular σ/ρ	linear-dependency warnings
pbe0_sfx2c_x2cqzvpall_h-rn_spherical_v2	x2c-QZVPall	430	0	2.30e-12	1.56e-14	0
pbe0_sfx2c_dyallv4z_h-lr_spherical_v2	dyall-v4z	501	0	2.43e-12	3.75e-15	266
pbe0_sfx2c_x2cqzvpalls_h-rn_spherical_v2	x2c-QZVPall-s	192	0	2.33e-12	3.83e-15	38
pbe0_sfx2c_dyallav4z_h-ba_hf-ra_spherical_v2	dyall-av4z	166	0	2.30e-12	2.68e-15	68

All generated rows pass the current validation criteria. The electron-count and angular-sphericity errors are near numerical precision. Linear-dependency warnings are most common in the large Dyall branches and in the supporting comparison branches, but they do not coincide with validation failures in the committed data layer.

basis	QA rows	LD-warning rows	fraction
x2c-QZVPall	430	0	0%
dyall-v4z	501	266	53.1%
x2c-QZVPall-s	192	38	19.8%
dyall-av4z	166	68	41.0%
all datasets	1289	372

The validation result supports using the committed profile/radii/QA layer as the baseline for downstream export and interoperability work. It does not mean that every formal anion is physically stable; it means the generated density row is internally consistent under the declared reference convention.

Multiwfn WFN interoperability validation

A representative local H/O/H2O validation checks the WFN boundary before full Multiwfn artifact generation. The validation writes H and O atom WFNs and an H2O molecule WFN, reads the saved files with the package-side WFN evaluator, and compares Multiwfn plane output against direct PySCF density references. This is an interoperability test for a fixed small system, not a universal validation over the full state table.

check	points	reference	maximum absolute error	p95 absolute error	RMSE
H2O WFN read-back spot density	5	direct PySCF density	2.55e-7	NA	NA
O atom spin point, total density	1	package WFN evaluator	2.23e-11	NA	NA
O atom spin point, alpha density	1	package WFN evaluator	3.73e-11	NA	NA
O atom spin point, beta density	1	package WFN evaluator	1.50e-11	NA	NA
H2O deformation plane	14641	direct PySCF molecular-minus-atomic density	1.51e-4	3.08e-7	2.0e-6
package WFN deformation plane	14641	direct PySCF molecular-minus-atomic density	2.55e-7	5.03e-10	4.24e-9

The O-atom point diagnostic confirms that the current Multiwfn build interprets the generated atom WFN as spin resolved rather than alpha/beta averaged.

source	total density	alpha density	beta density	spin density
package WFN evaluator	0.502407	0.337266	0.165141	0.172125
Multiwfn point output	0.502407	0.337266	0.165141	0.172125

The standalone molecule and atom plane checks isolate the WFN-density interpretation from the deformation-density arithmetic. For the H2O total-density plane, the package WFN evaluator agrees with direct PySCF to an RMSE of 7.93e-9 e bohr^-3; the larger Multiwfn-minus-reference maximum occurs at the near-nuclear grid point and remains small relative to the maximum plane density. The H and O atom planes show the same pattern at smaller scale. The documentation notebook docs/notebooks/multiwfn_wfn_plane_validation.ipynb records the local workflow and the compact result tables.

Primary basis-family comparison

The primary comparison matches x2c-QZVPall and dyall-v4z over the H--Rn overlap by exact state_id and state-record digest. The comparison tests how two primary all-electron scalar-relativistic basis families change the same spherical reference state. It is not a diffuse-basis test.

comparison	matched states	integrity/validation failures	low	moderate	high	outliers	max relative L1	max sup |ΔN(r)| / e	max |ΔR_cut| / Å
x2c-QZVPall → dyall-v4z	430	0	418	11	1	1	0.163	0.811	1.040

Most matched states are in the low-difference tier. The single high-difference row is a formal multianion, C_qm3_mult2_formal. The table below groups the same comparison by charge. The relative L1 columns summarize redistribution in the radial distribution \(D(r)=4\pi r^2\rho(r)\), while sup |ΔN(r)| summarizes the largest cumulative electron-count separation at any radius.

comparison	charge	n	low	moderate	high	median rel. L1	p95 rel. L1	max rel. L1	max sup |ΔN(r)| / e	max |ΔR_cut| / Å
x2c-QZVPall → dyall-v4z	-3	6	1	4	1	0.0567	0.1561	0.1631	0.8106	0.6431
x2c-QZVPall → dyall-v4z	-2	20	15	5	0	0.0209	0.0938	0.0946	0.5924	0.7169
x2c-QZVPall → dyall-v4z	-1	80	78	2	0	0.0063	0.0258	0.1165	0.4680	1.0401
x2c-QZVPall → dyall-v4z	0	86	86	0	0	0.0011	0.0022	0.0046	0.0136	0.0309
x2c-QZVPall → dyall-v4z	+1	85	85	0	0	0.0009	0.0022	0.0040	0.0136	0.0363
x2c-QZVPall → dyall-v4z	+2	79	79	0	0	0.0008	0.0017	0.0021	0.0136	0.0218
x2c-QZVPall → dyall-v4z	+3	74	74	0	0	0.0009	0.0024	0.0040	0.0243	0.0234

The distribution table gives all-state quantiles for the main scalar diagnostics. Median primary-basis differences are small. The upper tail is driven by anionic and especially formal anionic references, where radial tails and weak binding are expected to be more basis-sensitive than compact neutral or cationic states.

metric	n	p50	p90	p95	p99	max
relative L1	430	0.0011	0.0098	0.0185	0.0944	0.1631
sup |ΔN(r)| / e	430	0.0078	0.1978	0.2968	0.5974	0.8106
mean |radial shift| / Å	430	0.0001	0.0131	0.0215	0.0837	0.1511
max |ΔR_cut| / Å	430	0.0079	0.1655	0.3644	0.6766	1.0401
|Δ tail N(r>5 bohr)| / e	430	0.0006	0.0562	0.1691	0.4526	0.8069
|Δ tail N(r>10 bohr)| / e	430	5.9065e-06	0.0275	0.1421	0.2971	0.3697
|Δ tail N(r>15 bohr)| / e	430	9.8850e-09	0.0002	0.0063	0.0726	0.1061
|Δ tail N(r>20 bohr)| / e	430	8.0311e-12	3.5948e-07	0.0000	0.0033	0.0104

The outlier row is shown explicitly so that downstream users can decide whether it matters for their application. It should be treated as a scientific sensitivity flag for a formal reference state rather than as evidence of corrupted generated data.

state ID	element	charge	state role	tier	rel. L1	sup |ΔN(r)| / e	max |ΔR_cut| / Å	flags
C_qm3_mult2_formal	C	-3	formal_multianion	high	0.1631	0.7189	0.4980	relative_l1_outlier;cumulative_delta_watch;mean_radial_shift_watch

Supplemented/augmented basis sensitivity

The supplemented/augmented comparison uses the same matched-state contract but compares a primary branch with its supporting branch. The current neutral-plus-anion supporting branches are unified by basis identity rather than split into separate neutral and anion datasets. The two comparisons are not equivalent: dyall-av4z is an augmented Dyall branch, whereas x2c-QZVPall-s is an NMR-shielding-oriented supplemented x2c branch rather than a standard diffuse tail basis.

comparison	matched states	integrity/validation failures	low	moderate	high	outliers	max relative L1	max sup |ΔN(r)| / e	max |ΔR_cut| / Å
dyall-v4z → dyall-av4z	166	0	132	20	14	14	0.383	1.661	2.127
x2c-QZVPall → x2c-QZVPall-s	192	0	192	0	0	0	0.014	0.033	0.020

The x2c supplemented branch is uniformly low-sensitivity in the current data, which is consistent with treating it as a branch that most density-reference users can ignore. The Dyall augmented branch has 14 high-sensitivity rows, all in formal anion references. This pattern is visible when the data are grouped by charge.

comparison	charge	n	low	moderate	high	median rel. L1	p95 rel. L1	max rel. L1	max sup |ΔN(r)| / e	max |ΔR_cut| / Å
dyall-v4z → dyall-av4z	-3	6	0	0	6	0.1186	0.3615	0.3834	1.6607	1.8616
dyall-v4z → dyall-av4z	-2	20	3	10	7	0.0528	0.2630	0.2935	1.1532	1.6452
dyall-v4z → dyall-av4z	-1	67	56	10	1	0.0048	0.0805	0.2175	0.7016	2.1275
dyall-v4z → dyall-av4z	0	73	73	0	0	0.0000	0.0002	0.0004	0.0022	0.0071
x2c-QZVPall → x2c-QZVPall-s	-3	6	6	0	0	0.0018	0.0134	0.0138	0.0326	0.0135
x2c-QZVPall → x2c-QZVPall-s	-2	20	20	0	0	0.0004	0.0088	0.0095	0.0185	0.0197
x2c-QZVPall → x2c-QZVPall-s	-1	80	80	0	0	0.0005	0.0027	0.0042	0.0107	0.0134
x2c-QZVPall → x2c-QZVPall-s	0	86	86	0	0	0.0005	0.0015	0.0018	0.0107	0.0059

The state-role grouping makes the interpretation clearer: high sensitivity is concentrated in formal monoanion/multianion rows rather than neutral references or source-backed experimental monoanions.

comparison	state role	n	low	moderate	high	median rel. L1	max rel. L1
dyall-v4z → dyall-av4z	reference	73	73	0	0	0.0000	0.0004
dyall-v4z → dyall-av4z	bound_experimental	56	52	4	0	0.0036	0.1030
dyall-v4z → dyall-av4z	bound_provisional	1	0	1	0	0.0138	0.0138
dyall-v4z → dyall-av4z	diagnostic_theory	1	1	0	0	0.0005	0.0005
dyall-v4z → dyall-av4z	formal_monoanion	9	3	5	1	0.0290	0.2175
dyall-v4z → dyall-av4z	formal_multianion	26	3	10	13	0.0549	0.3834
x2c-QZVPall → x2c-QZVPall-s	reference	86	86	0	0	0.0005	0.0018
x2c-QZVPall → x2c-QZVPall-s	bound_experimental	63	63	0	0	0.0005	0.0042
x2c-QZVPall → x2c-QZVPall-s	bound_provisional	3	3	0	0	0.0005	0.0005
x2c-QZVPall → x2c-QZVPall-s	formal_monoanion	14	14	0	0	0.0007	0.0039
x2c-QZVPall → x2c-QZVPall-s	formal_multianion	26	26	0	0	0.0004	0.0138

The high-sensitivity Dyall augmented rows are listed below. They are mainly light and p-block formal anions, with the strongest relative L1 response for C_qm3_mult2_formal. The large cutoff-radius shifts show that the augmented basis changes low-density tails and cumulative radial redistribution, not the integer electron count.

state ID	element	charge	state role	tier	rel. L1	sup |ΔN(r)| / e	max |ΔR_cut| / Å	flags
Be_qm1_mult2_formal	Be	-1	formal_monoanion	high	0.2175	0.5384	1.1900	relative_l1_outlier;cumulative_delta_watch;mean_radial_shift_outlier
B_qm2_mult4_formal	B	-2	formal_multianion	high	0.2935	1.0184	1.6333	relative_l1_outlier;cumulative_delta_outlier;mean_radial_shift_outlier
C_qm3_mult2_formal	C	-3	formal_multianion	high	0.3834	1.6607	1.8616	relative_l1_outlier;cumulative_delta_outlier;mean_radial_shift_outlier
C_qm2_mult3_formal	C	-2	formal_multianion	high	0.2251	0.8674	1.5880	relative_l1_outlier;cumulative_delta_watch;mean_radial_shift_watch
N_qm3_mult1_formal	N	-3	formal_multianion	high	0.2957	1.4680	1.6062	relative_l1_outlier;cumulative_delta_outlier;mean_radial_shift_outlier
N_qm2_mult2_formal	N	-2	formal_multianion	high	0.2614	1.1532	1.6452	relative_l1_outlier;cumulative_delta_outlier;mean_radial_shift_outlier
Al_qm2_mult4_formal	Al	-2	formal_multianion	high	0.1386	1.0191	1.2948	relative_l1_watch;cumulative_delta_outlier;mean_radial_shift_watch
P_qm3_mult1_formal	P	-3	formal_multianion	high	0.1587	1.4073	1.6637	relative_l1_outlier;cumulative_delta_outlier;mean_radial_shift_watch
Ga_qm2_mult4_formal	Ga	-2	formal_multianion	high	0.0667	1.0836	1.1706	relative_l1_watch;cumulative_delta_outlier;mean_radial_shift_watch
As_qm3_mult1_formal	As	-3	formal_multianion	high	0.0786	1.3931	1.6705	relative_l1_watch;cumulative_delta_outlier;mean_radial_shift_watch
In_qm2_mult4_formal	In	-2	formal_multianion	high	0.0418	1.0468	0.9184	cumulative_delta_outlier
Sb_qm3_mult1_formal	Sb	-3	formal_multianion	high	0.0510	1.3528	1.6587	relative_l1_watch;cumulative_delta_outlier
Tl_qm2_mult4_formal	Tl	-2	formal_multianion	high	0.0276	1.1325	1.1864	cumulative_delta_outlier
Bi_qm3_mult1_formal	Bi	-3	formal_multianion	high	0.0328	1.3868	1.7081	cumulative_delta_outlier

Practical result

The current data layer supports a clear default policy. Use the primary branches for reproducible default analyses: x2c-QZVPall for H--Rn when that element range is sufficient, and dyall-v4z for the broader H--Lr branch. Most users can ignore x2c-QZVPall-s; it is retained as an auditable supplemented comparison branch, not as the recommended tail-convergence reference. When low-density tails are central to the scientific question, prefer the Dyall primary/augmented comparison where its element coverage exists, and report formal-anion sensitivity explicitly.

The supplemented/augmented branches should not be silently substituted into the primary dataset. They are separate reference gauges. Their value is precisely that they make basis-tail sensitivity observable and auditable.

Next: Discussion.