We propose a general approach to reducing basis set incompleteness error in electron correlation energy calculations. The correction is computed alongside the correlation energy in a single calculation by modifying the electron interaction operator with an effective short-range electron-electron interaction. Our approach is based on a local mapping between the Coulomb operator projected onto a finite basis and a long-range interaction represented by the error function with a local range-separated parameter, originally introduced by Giner et al. [ J. Chem. Phys. 2018, 149, 194301]. Unlike the basis set incompleteness error correction proposed in that work, our method does not rely on short-range correlation density functionals. As a numerical demonstration, we apply the method with complete active space wave functions. Correlation energies are computed using two distinct approaches: the linearized adiabatic connection (AC0) method and n-electron valence state second-order perturbation theory (NEVPT2). We obtain encouraging results for the relative energies of test molecules, with accuracy in a triple-ζ basis set comparable to or exceeding that of uncorrected AC0 or NEVPT2 energies in a quintuple-ζ basis set.

Faculty of Chemistry University of Warsaw ul L Pasteura 1 02 093 Warsaw Poland

Institute of Physics Lodz University of Technology ul Wolczanska 217 221 93 005 Lodz Poland

J Heyrovský Institute of Physical Chemistry Academy of Sciences of the Czech Republic v v i Dolejškova 3 18223 Prague 8 Czech Republic

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Werner H.-J., Adler T. B., Manby F. R.. General orbital invariant MP2-F12 theory. J. Chem. Phys. 2007;126:164102. doi: 10.1063/1.2712434. PubMed DOI

Fliegl H., Klopper W., Hättig C.. Coupled-cluster theory with simplified linear-r12 corrections: The CCSD (R12) model. J. Chem. Phys. 2005;122:084107. doi: 10.1063/1.1850094. PubMed DOI

Werner H.-J., Knizia G., Manby F. R.. Explicitly correlated coupled cluster methods with pair-specific geminals. Mol. Phys. 2011;109:407–417. doi: 10.1080/00268976.2010.526641. DOI

Kong L., Bischoff F. A., Valeev E. F.. Explicitly correlated R12/F12 methods for electronic structure. Chem. Rev. 2012;112:75–107. doi: 10.1021/cr200204r. PubMed DOI

Shiozaki T., Werner H.-J.. Multireference explicitly correlated F12 theories. Mol. Phys. 2013;111:607–630. doi: 10.1080/00268976.2013.779393. DOI

Shiozaki T., Werner H.-J.. Communication: Second-order multireference perturbation theory with explicit correlation: CASPT2-F12. J. Chem. Phys. 2010;133:141103. doi: 10.1063/1.3489000. PubMed DOI

Guo Y., Sivalingam K., Valeev E. F., Neese F.. Explicitly correlated N-electron valence state perturbation theory (NEVPT2-F12) J. Chem. Phys. 2017;147:064110. doi: 10.1063/1.4996560. PubMed DOI

Boys S. F., Handy N. C.. The determination of energies and wavefunctions with full electronic correlation. Proc. R. Soc. Lond. A. 1969;310:43–61. doi: 10.1098/rspa.1969.0061. DOI

Ten-no S.. A feasible transcorrelated method for treating electronic cusps using a frozen Gaussian geminal. Chem. Phys. Lett. 2000;330:169–174. doi: 10.1016/S0009-2614(00)01066-6. DOI

Yanai T., Shiozaki T.. Canonical transcorrelated theory with projected Slater-type geminals. J. Chem. Phys. 2012;136:084107. doi: 10.1063/1.3688225. PubMed DOI

Cohen A. J., Luo H., Guther K., Dobrautz W., Tew D. P., Alavi A.. Similarity transformation of the electronic Schrödinger equation via Jastrow factorization. J. Chem. Phys. 2019;151:061101. doi: 10.1063/1.5116024. DOI

Giner E.. A new form of transcorrelated Hamiltonian inspired by range-separated DFT. J. Chem. Phys. 2021;154:084119. doi: 10.1063/5.0044683. PubMed DOI

Giner E., Pradines B., Ferté A., Assaraf R., Savin A., Toulouse J.. Curing basis-set convergence of wave-function theory using density-functional theory: A systematically improvable approach. J. Chem. Phys. 2018;149:194301. doi: 10.1063/1.5052714. PubMed DOI

Toulouse J., Gori-Giorgi P., Savin A.. A short-range correlation energy density functional with multi-determinantal reference. Theor. Chem. Acc. 2005;114:305–308. doi: 10.1007/s00214-005-0688-2. DOI

Ferté A., Giner E., Toulouse J.. Range-separated multideterminant density-functional theory with a short-range correlation functional of the on-top pair density. J. Chem. Phys. 2019;150:084103. doi: 10.1063/1.5082638. PubMed DOI

Loos P.-F., Pradines B., Scemama A., Toulouse J., Giner E.. A Density-Based Basis-Set Correction for Wave Function Theory. J. Phys. Chem. Lett. 2019;10:2931–2937. doi: 10.1021/acs.jpclett.9b01176. PubMed DOI

Heßelmann A., Giner E., Reinhardt P., Knowles P. J., Werner H.-J., Toulouse J.. A density-fitting implementation of the density-based basis-set correction method. J. Comput. Chem. 2024;45:1247–1253. doi: 10.1002/jcc.27325. PubMed DOI

Giner E., Scemama A., Toulouse J., Loos P.-F.. Chemically accurate excitation energies with small basis sets. J. Chem. Phys. 2019;151:144118. doi: 10.1063/1.5122976. PubMed DOI

Giner E., Scemama A., Loos P.-F., Toulouse J.. A basis-set error correction based on density-functional theory for strongly correlated molecular systems. J. Chem. Phys. 2020;152:174104. doi: 10.1063/5.0002892. PubMed DOI

Mester D., Kállay M.. Basis set limit of CCSD (T) energies: explicit correlation versus density-based basis-set correction. J. Chem. Theory Comput. 2023;19:8210–8222. doi: 10.1021/acs.jctc.3c00979. PubMed DOI PMC

Mester D., Nagy P. R., Kállay M.. Basis-set limit CCSD (T) energies for large molecules with local natural orbitals and reduced-scaling basis-set corrections. J. Chem. Theory Comput. 2024;20:7453–7468. doi: 10.1021/acs.jctc.4c00777. PubMed DOI PMC

Savin, A. On degeneracy, near–degenaracy and density functional theory. In Recent Developments and Applications of Modern Density Functional Theory. Seminario, J. M. , Ed.; Elsevier: Amsterdam, The NEtherlands, 1996; pp 327–357.

Stoll, H. , Savin, A. . Density Functionals for Correlation Energies of Atoms and Molecules. In Density Functional Methods in Physics. Dreizler, R. , da Providência, J. , Eds.; Plenum, New York, NY, 1985; pp 177–207.

Toulouse J., Colonna F., Savin A.. Long-range-short-range separation of the electron-electron interaction in density-functional theory. Phys. Rev. A. 2004;70:062505. doi: 10.1103/PhysRevA.70.062505. DOI

Pollet R., Savin A., Leininger T., Stoll H.. Combining multideterminantal wave functions with density functionals to handle near-degeneracy in atoms and molecules. J. Chem. Phys. 2002;116:1250. doi: 10.1063/1.1430739. DOI

Pernal K., Hapka M.. Range-separated multiconfigurational density functional theory methods. WIREs Comput. Mol. Sci. 2022;12:e1566. doi: 10.1002/wcms.1566. DOI

Gori-Giorgi P., Savin A.. Properties of short-range and long-range correlation energy density functionals from electron-electron coalescence. Phys. Rev. A. 2006;73:032506. doi: 10.1103/PhysRevA.73.032506. DOI

Pernal K.. Electron Correlation from the Adiabatic Connection for Multireference Wave Functions. Phys. Rev. Lett. 2018;120:013001. doi: 10.1103/PhysRevLett.120.013001. PubMed DOI

Pastorczak E., Pernal K.. Correlation Energy from the Adiabatic Connection Formalism for Complete Active Space Wave Functions. J. Chem. Theory Comput. 2018;14:3493–3503. doi: 10.1021/acs.jctc.8b00213. PubMed DOI

Pastorczak E., Pernal K.. Electronic Excited States from the Adiabatic-Connection Formalism with Complete Active Space Wave Functions. J. Phys, Chem. Lett. 2018;9:5534–5538. doi: 10.1021/acs.jpclett.8b02391. PubMed DOI

Pastorczak E., Hapka M., Veis L., Pernal K.. Capturing the Dynamic Correlation for Arbitrary Spin-Symmetry CASSCF Reference with Adiabatic Connection Approaches: Insights into the Electronic Structure of the Tetramethyleneethane Diradical. J. Phys. Chem. Lett. 2019;10:4668–4674. doi: 10.1021/acs.jpclett.9b01582. PubMed DOI

Beran P., Matoušek M., Hapka M., Pernal K., Veis L.. Density matrix renormalization group with dynamical correlation via adiabatic connection. J. Chem. Theory Comput. 2021;17:7575–7585. doi: 10.1021/acs.jctc.1c00896. PubMed DOI

Guo Y., Pernal K.. Spinless formulation of linearized adiabatic connection approximation and its comparison with second order N-electron valence state perturbation theory. Faraday Discuss. 2024;254:332–358. doi: 10.1039/D4FD00054D. PubMed DOI

Angeli C., Cimiraglia R., Evangelisti S., Leininger T., Malrieu J.-P.. Introduction of n-electron valence states for multireference perturbation theory. J. Chem. Phys. 2001;114:10252–10264. doi: 10.1063/1.1361246. DOI

Angeli C., Cimiraglia R., Malrieu J.-P.. N-electron valence state perturbation theory: a fast implementation of the strongly contracted variant. Chem. Phys. Lett. 2001;350:297–305. doi: 10.1016/S0009-2614(01)01303-3. DOI

Angeli C., Cimiraglia R., Malrieu J.-P.. n-electron valence state perturbation theory: A spinless formulation and an efficient implementation of the strongly contracted and of the partially contracted variants. J. Chem. Phys. 2002;117:9138. doi: 10.1063/1.1515317. DOI

Guo Y., Sivalingam K., Neese F.. Approximations of density matrices in N-electron valence state second-order perturbation theory (NEVPT2). I. Revisiting the NEVPT2 construction. J. Chem. Phys. 2021;154:214111. doi: 10.1063/5.0051211. PubMed DOI

Guo Y., Sivalingam K., Kollmar C., Neese F.. Approximations of density matrices in N-electron valence state second-order perturbation theory (NEVPT2). II. The full rank NEVPT2 (FR-NEVPT2) formulation. J. Chem. Phys. 2021;154:214113. doi: 10.1063/5.0051218. PubMed DOI

Dyall K. G.. The choice of a zeroth-order Hamiltonian for second-order perturbation theory with a complete active space self-consistent-field reference function. J. Chem. Phys. 1995;102:4909–4918. doi: 10.1063/1.469539. DOI

Werner H.-J., Knowles P. J., Knizia G., Manby F. R., Schütz M.. Molpro: a general-purpose quantum chemistry program package. WIREs Comput. Mol. Sci. 2012;2:242. doi: 10.1002/wcms.82. DOI

Pernal, K. ; Hapka, M. ; Przybytek, M. ; Modrzejewski, M. ; Sokół, A. ; Tucholska, A. . GammCor code. GitHub, 2020. https://github.com/pernalk/GAMMCOR (accessed 2025-01-23).

Hapka M., Pastorczak E., Pernal K.. Self-Adapting Short-Range Correlation Functional for Complete Active Space-Based Approximations. J. Phys. Chem. A. 2024;128:7013–7022. doi: 10.1021/acs.jpca.4c03299. PubMed DOI PMC

Modrzejewski, M. gammcor-integrals library. GitHub, 2022, https://github.com/modrzejewski/gammcor-integrals (accessed 2025-03-28).

Sun Q., Zhang X., Banerjee S., Bao P., Barbry M., Blunt N. S., Bogdanov N. A., Booth G. H., Chen J., Cui Z.-H.. et al. Recent developments in the PySCF program package. J. Chem. Phys. 2020;153:024109. doi: 10.1063/5.0006074. PubMed DOI

Dunning T. H. Jr. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J. Chem. Phys. 1989;90:1007–1023. doi: 10.1063/1.456153. DOI

Davidson E. R., Hagstrom S. A., Chakravorty S. J., Umar V. M., Fischer C. F.. Ground-state correlation energies for two- to ten-electron atomic ions. Phys. Rev. A. 1991;44:7071–7083. doi: 10.1103/PhysRevA.44.7071. PubMed DOI

Halkier A., Helgaker T., Jørgensen P., Klopper W., Koch H., Olsen J., Wilson A. K.. Basis-set convergence in correlated calculations on Ne, N2, and H2O. Chem. Phys. Lett. 1998;286:243–252. doi: 10.1016/S0009-2614(98)00111-0. DOI

Peterson K. A., Adler T. B., Werner H.-J.. Systematically convergent basis sets for explicitly correlated wavefunctions: The atoms H, He, B-Ne, and Al-Ar. J. Chem. Phys. 2008;128:084102. doi: 10.1063/1.2831537. PubMed DOI