CC-Neg Benchmark: aVnZ-F12 Advances

• CC-Neg Benchmark evaluates aVnZ-F12, a series of diffuse-augmented correlation-consistent basis sets optimized for explicitly correlated (F12) methods.
• Method enhancements include adding diffuse functions to higher angular momentum shells, which significantly reduce errors in electron affinity and dissociation energy benchmarks.
• Practical tests on atomic anions and water clusters demonstrate rapid convergence and efficiency, surpassing conventional cc-pVnZ-F12 sets in accuracy and computational cost.

The aug-cc-pVnZ-F12 basis set family ("aVnZ-F12") constitutes a sequence of correlation-consistent basis sets specifically designed for explicitly correlated (F12) methods. These sets enhance the established cc-pVnZ-F12 orbital foundations by introducing additional diffuse functions on higher angular momentum shells, thereby addressing the radial flexibility demands inherent in calculations on atomic anions and noncovalent complexes. The construction, benchmarking, and application of the aVnZ-F12 sets foster systematic improvements in accuracy and efficiency, especially for difficult cases involving weakly-bound electrons and extended intermolecular interactions (Sylvetsky et al., 2017).

1. Basis Set Design and Construction

The aVnZ-F12 family, with n = D (double-zeta), T (triple-zeta), Q (quadruple-zeta), and 5 (quintuple-zeta), augments parent cc-pVnZ-F12 sets by adding diffuse functions to higher angular momenta (d, f, g, etc.). For non-hydrogen atoms, diffuse d exponents are added at the DZ level, d and f at TZ, and d, f, g at QZ and 5Z. For hydrogen, diffuse p is added for DZ, p and d for TZ, p, d, and f for QZ, and so on. The exponent values of these augmentations are optimized for MP2-F12 energies of the target atomic anions by minimizing the total energy:

$$\min_{\{\zeta_L\}} E_{\mathrm{MP2-F12}}[A^-; \{\zeta_L\}]$$

where $E_{MP2-F12}$ comprises both conventional second-order correlation and F12 geminal corrections. For atoms such as N, Ne, and Ar, which lack bound anions, exponents are interpolated or extrapolated from nearby elements. Extensive auxiliary sets (awCV5Z/MP2fit for density-fitting and aV5Z-based JKfit/OptRI for F12 integrals) support the optimization process.

Element	Nbas(VDZ-F12)	Nbas(aVDZ-F12)	$\zeta$(d)
H	13	15	0.0743
B	49	51	0.0730
C	49	51	0.0968
...	...	...	...

2. Explicitly Correlated Formalism (F12)

The F12 methodology introduces a geminal operator $F_{12}$—dependent on the interelectronic distance $r_{ij}$—into the wavefunction, improving basis set convergence for correlation energies. Under the "3C(Fix)" ansatz,

$$E_{\text{tot}} = E_{\text{HF}} + E_c(\text{MP2-F12}) + \Delta\text{CCSD(F12*)} + ...$$

with the MP2-F12 term written as:

$$E_c(\text{MP2-F12}) = \langle 0 | \hat{H}_N T_2 + \hat{H}_N F_{12} + F_{12} \hat{H}_N | 0 \rangle$$

where $F_{12} = \sum_{i<j} f(r_{ij})$, $f(r) = e^{-\beta r}$, and typical $\beta$ values are 0.9 (DZ-F12), 1.0 (TZ- and QZ-F12), and 1.2 (V5Z-F12). Resolution-of-the-identity (RI) techniques circumvent computationally expensive higher-order integrals. Three auxiliary sets serve crucial roles: JKfit (Coulomb/exchange), MP2fit (correlation), and OptRI (F12-specific and CABS corrections). OptRI+ variants with extra diffuse functions optimally improve the HF+CABS component.

3. Benchmarking on Atomic Anion Electron Affinities

Electron affinities (EAs) were evaluated using CCSD-F12b with augmented d-aug-cc-pwCV5Z as the reference. Relative to this reference, the unaugmented VnZ-F12 converges slowly for anions (MAD for VQZ-F12 ≈ 0.04 eV). The augmentation accelerates convergence significantly; aVDZ-F12 drops RMS error from 0.188 eV (VDZ-F12) to 0.053 eV. The aVTZ-F12 set yields RMS ≈ 0.016 eV, MAD ≈ 0.012 eV, and aVQZ-F12 approaches the CCSD limit (RMS ≈ 3 meV).

Basis	MAD (eV)	RMS (eV)
VDZ-F12	0.127	0.188
aVDZ-F12	0.040	0.053
aVTZ-F12	0.012	0.016
aVQZ-F12	0.002	0.003

Conventional aug-cc-pVnZ sets require at least AVQZ to match the aVTZ-F12 accuracy for MP2 or CCSD on anions.

4. Noncovalent Interaction Benchmarks

S66 Dataset

The MP2-F12 dissociation energies benchmark employed CP-corrected V{T,Q}Z-F12 as reference. Raw (uncorrected) VDZ-F12 underbinds (RMS ≃ 0.10 kcal/mol), while aVDZ-F12 improves this marginally. aVTZ-F12/CP achieves RMS ≈ 0.019 kcal/mol, outperforming VTZ-F12/CP (0.054 kcal/mol) and rivalling VQZ-F12. aVQZ-F12 (raw) realizes RMS ≈ 0.007 kcal/mol, matching V5Z-F12 accuracy at lower computational cost.

BEGDB Water Clusters

For 2- and 3-body decompositions involving water clusters, CCSD(F12*)/aVQZ-F12 matches or exceeds the accuracy of CCSD/haV5Z. For 2-body terms, RMS ≤ 0.004 kcal/mol; for 3-body, RMS ≈ 0.001 kcal/mol. Extrapolation to aV{T,Q}Z-F12 further reduces 2-body RMS to 0.002 kcal/mol.

WATER23 Subset

The WATER23 dataset involves neutral, protonated, and deprotonated clusters. Raw aVDZ-F12 RMS falls to ≈0.13 kcal/mol for deprotonated clusters compared to ≈0.40 kcal/mol (VDZ-F12). aVTZ-F12 and aVQZ-F12 further stabilize convergence. CCSD(F12*)/aV{T,Q}Z-F12 achieves the basis-set limit (RMS ≈0.02 kcal/mol) without counterpoise correction, critical for n-body expansion in large clusters.

5. CABS Transferability and Practical Recommendations

OptRI complementary auxiliary basis sets display notable transferability among matching orbital families, as confirmed by analyses on the S66 set. AVnZ/OptRI yields reliable results with both ano-pVnZ+ and sano-pVnZ densities. For small F12 sets (VDZ-F12, aVDZ-F12, aVTZ-F12), OptRI+—supplemented with extra diffuse layers—optimizes the HF+CABS correction, maintaining negligible impact on the MP2-F12 energy.

For electron affinities, aVTZ-F12 provides errors $\lesssim0.02$ eV, while aVQZ-F12 reaches the CCSD limit (errors $\lesssim0.003$ eV). In neutral and anionic noncovalent interactions, aVTZ-F12/CP and aVQZ-F12 offer RMS deviations of 0.019 kcal/mol and 0.007 kcal/mol, respectively. For water clusters, CCSD(F12*)/aVQZ-F12 reproduces 2- and 3-body terms with RMS $\leq0.004$ kcal/mol; for deprotonated WATER23, MP2-F12/aV{T,Q}Z-F12 plus CCSD/aV{D,T}Z-F12 HLC—with no CP—achieves RMS ≈0.021 kcal/mol.

6. Limitations, Convergence Trends, and Counterpoise Considerations

A key caution is that for total atomization energies of neutral molecules (W4-17), aVnZ-F12 yields negligible advantages over VnZ-F12, as additional diffuse functions do not enhance SCF or CCSD convergence. Counterpoise correction, while occasionally masking slow convergence (e.g., full-CP V{D,T}Z-F12 for WATER23, RMS ≃ 0.013 kcal/mol), proves impractical for large clusters in n-body expansions. The aVnZ-F12 sets exhibit monotonic convergence, obviating reliance on counterpoise artifacts.

7. Context and Impact

The aVnZ-F12 basis set family furnishes a systematically improvable and diffuse-augmented sequence of F12-optimized sets, addressing the critical needs of anion energetics and noncovalent interactions in extended systems. These sets enable accurate and efficient benchmarking for electron affinities, dissociation energies, and cluster energetics, substantiating their broad utility in computational chemistry contexts where extended radial flexibility and rapid convergence are required (Sylvetsky et al., 2017). A plausible implication is strengthened computational modeling in condensed-phase and supramolecular environments, where charge localization and noncovalent binding dominate.