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The investigation aims to identify potential approximative methodologies for expediting repeated calculations of Coulomb integral contributions to single-electron Hamiltonian elements during self-consistent field (SCF) iterations, thereby dramatically accelerating computationally intensive density functional theory (DFT) analyses of massive biomolecular structures. The research addressed several challenges: a) evaluating semi-empirical approaches for quantum chemical examination of enormous molecular systems; b) exploring how numerous distant molecular fragments could facilitate faster computation of Coulomb interaction contributions; c) examining contemporary approaches to fixed-geometry single-point molecular calculations; d) developing innovative methodologies for accelerated Coulomb integral contribution computation in DFT analyses of substantial bi-omolecular entities.
We present a novel suite of approximation techniques designed to substantially expedite calculations of Cou-lomb integral contributions to one-electron Hamiltonian elements in conventional DFT methodologies during SCF iterations-typically the rate-limiting phase of these essential yet computationally demanding calculations for exten-sive biomolecular systems, including thousands of docking complexes comprising thousands of atoms.
Our integrated approach features rapid and precise approximation of contribution modifications across innu-merable 4-center Coulomb integrals between successive SCF iterations through auxiliary density function-mediated transformation into linear combinations of 3-center integrals, subsequently converted to combinations of 2-center integrals. Contribution variations from non-multipole short-range components of these 2-center integrals are swiftly determined by modifying pre-computed spline contributions based on inter-atomic separations. The re-maining multipole-based long-range contributions undergo rapid computation for expansive molecular systems using a fast multipole method (FMM) framework, which strategically partitions extensive spatial domains into hi-erarchical regions (a technique originally pioneered for galactic dynamics simulations).
Each SCF iteration employs sophisticated screening to identify exclusively non-negligible integral combina-tions, particularly accounting for the progressively diminishing density matrix increments characteristic of con-verging SCF processes. The framework accommodates the unique characteristics of specific massive molecular systems or extensive collections thereof, such as thousands of docking arrangements between substantial protein structures and diverse small organic ligand molecules.
All bimolecular components-including approximations of two-center basis function overlaps via linear combi-nations of single-center auxiliary density functions-undergo efficient computation utilizing specialized database-stored inter-nuclear distance splines. For novel basis sets, the reference database can be promptly augmented through decomposition into universal exponential components with corresponding database enrichment.
We present a novel suite of approximation techniques designed to substantially expedite calculations of Cou-lomb integral contributions to one-electron Hamiltonian elements in conventional DFT methodologies during SCF iterations-typically the rate-limiting phase of these essential yet computationally demanding calculations for exten-sive biomolecular systems, including thousands of docking complexes comprising thousands of atoms.
Our integrated approach features rapid and precise approximation of contribution modifications across innu-merable 4-center Coulomb integrals between successive SCF iterations through auxiliary density function-mediated transformation into linear combinations of 3-center integrals, subsequently converted to combinations of 2-center integrals. Contribution variations from non-multipole short-range components of these 2-center integrals are swiftly determined by modifying pre-computed spline contributions based on inter-atomic separations. The re-maining multipole-based long-range contributions undergo rapid computation for expansive molecular systems using a fast multipole method (FMM) framework, which strategically partitions extensive spatial domains into hi-erarchical regions (a technique originally pioneered for galactic dynamics simulations).
Each SCF iteration employs sophisticated screening to identify exclusively non-negligible integral combina-tions, particularly accounting for the progressively diminishing density matrix increments characteristic of con-verging SCF processes. The framework accommodates the unique characteristics of specific massive molecular systems or extensive collections thereof, such as thousands of docking arrangements between substantial protein structures and diverse small organic ligand molecules.
All bimolecular components-including approximations of two-center basis function overlaps via linear combi-nations of single-center auxiliary density functions-undergo efficient computation utilizing specialized database-stored inter-nuclear distance splines. For novel basis sets, the reference database can be promptly augmented through decomposition into universal exponential components with corresponding database enrichment.
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2. Zaleśny R., Papadopoulos M.G., Mezey P.G., Leszczynski J. (Eds.). (2011). Linear-Scaling Techniques in Computational Chemistry and Physics: Methods and Applications (Vol. 13). Springer Science & Business Media.
3. Ochsenfeld C., Kussmann J., Lambrecht D.S. Linear-scaling methods in quantum chemistry. Reviews in computational chemistry. 2007. No. 23. P. 1.
4. Kussmann J., Beer M., Ochsenfeld C. Linear-scaling self-consistent field methods for large molecules. Wiley Interdisciplinary Reviews: Computational Molecular Science. 2013. No. 3 (6). P. 614 – 636.
5. Vitale Valerio Computational methods for first-principles molecular dynamics with linear-scaling density functional theory. Diss. University of Southampton, 2017.
6. Niklasson, Anders MN. “Density matrix methods in linear scaling electronic structure theory.” Linear-Scaling Techniques in Computational Chemistry and Physics: Methods and Applications. Dordrecht: Spring-er Netherlands, 2011. P. 439 – 473.
7. Hu, Wei, Mohan Chen Advances in density functional theory and beyond for computational chemistry. Fron-tiers in Chemistry. 2021. No. 9. P. 705762.
8. Nakai H., Kobayashi M., Yoshikawa T., Seino J., Ikabata Y., Nishimura Y. Divide-and-conquer linear-scaling quantum chemical computations. The Journal of Physical Chemistry A. 2023. No. 127 (3). P. 589 – 618.
9. Nakata A., Baker J.S., Mujahed S.Y., Poulton J.T., Arapan S., Lin J., Bowler D.R. Large scale and line-ar scaling DFT with the CONQUEST code. The Journal of chemical physics. 2020. No. 152 (16).
10. H. A. Le T. Shiozaki Occupied-orbital fast multipole method for efficient exact exchange evaluation. Journal of Chemical Theory and Computation. 2018. No. 14:3. P. 1228 – 1234. https://doi.org/10.1021/acs.jctc.7b00880.
11. Li A., Muddana H.S., Gilson M.K. Quantum mechanical calculation of noncovalent interactions: A large-scale evaluation of PMx, DFT, and SAPT approaches. Journal of Chemical Theory and Computation. 2014. No. 10:4. P. 1563 – 1575. https://doi.org/10.1021/ct401111c
12. Hennemann M., Clark T. EMPIRE: a highly parallel semiempirical molecular orbital program: 1: self-consistent field calculations. Journal of Molecular Modeling. 2014. No. 20:7. P. 2331. https://doi.org/10.1007/s00894-014-2331-4
13. Oferkin I.V., Katkova E.V., Sulimov A.V. Evaluation of docking target functions by the comprehen-sive in-vestigation of protein-ligand energy minima. Advances in Bioinformatics. 2015. (2015). P. 126858. https://doi.org/10.1155/2015/126858
14. Anikin N.A., Anisimov V.M., Bugaenko V.L., Bobrikov V.V., Andreyev A.M. LocalSCF method for sem-iempirical quantum-chemical calculation of ultralarge biomolecules. Journal of Chemical Physics. 2004. Vol. 121. No. 3. P. 1266 – 1270. https://doi.org/10.1063/1.1764496
15. Anikin N.A., Andreev A.M., Kuzminsky M.B., Mendkovich A.S. High-speed method for mass semiempiri-cal calculations of docking complexes. Bulletin of the Academy of Sciences. Chemical Series. 2008. No. 9. P. 1759 – 1764.
16. Anikin N.A., Bugaenko V.L., Kuzminskii M.B., Mendkovich A.S. A Fast Method for Quantum Chemical Calculations of Large Molecules with DFT Hamiltonian Approximation. Bulletin of the Academy of Sciences. Chemical Series. 2014. No. 2. P. 346 – 349.
17. Womack, J. C., Mardirossian, N., Head-Gordon, M., and Skylaris, Ch.-K. “Self-consistent Implementation of Meta-GGA Functionals for the ONETEP Linear-Scaling Electronic Structure Package”. Journal of Chemical Physics. 2016. No. 145:20. P. 204114. https://doi.org/10.1063/1.4967960
18. Higgins J.E., Probert M.I.J., Hasnip P.J., Refson K., Bush I.J. Hybrid OpenMP and MPI within the CASTEP code, 2015. http://www.archer.ac.uk/community/eCSE/eCSE01-017/eCSE01-017.php
19. Zhang J., Weisman A.L., Saitta P., Friesner R.A. Efficient simulation of large materials clusters using the jaguar quantum chemistry program: Parallelization and wavefunction initialization. International Journal of Quan-tum Chemistry. 2016. No. 116:5. P. 357 – 368. https://doi.org/10.1002/qua.25043
20. Horvath I., Jeszenoi N., Balint M., Paragi G., Hetenyi C. A fragmenting protocol with explicit hydration for calculation of binding enthalpies of target-ligand complexes at a quantum mechanical level. International Journal of Molecular Sciences. 2019. No. 20:18. P. 4384. https://doi.org/10.3390/ijms20184384
21. Brunk E., Rothlisberger U. Mixed Quantum Mechanical/Molecular Mechanical Molecular Dynamics Simu-lations of Biological Systems in Ground and Electronically Excited States. Chemical Reviews. 2015. No. 115 (12). P. 6217 – 6163. https://pubmed.ncbi.nlm.nih.gov/25880693/ doi:10.1021/cr500628b. PMID 2588069,
22. Nemukhin A.V., Polyakov I.V., Moskovsky A.I. Multi-scale supercomputing of large molecular aggregates: A case study of the light-harvesting photosynthetic center. Supercomputing Frontiers and Innova-tions. 2016. No. 2:4. P. 48 – 54. https://doi.org/10.14529/jsfi150403
23. Nakajima T., Katouda M., Kamiya M., Nakatsuka Yu. NTChem: A highperformance software package-age for quantum molecular simulation. I.J. Quant. Chem. 2015. No. 115:5. P. 349 – 359. https://www.researchgate.net/publication/270223372_NTChem_A_High-Performance_Software_Package_for_Quantum_Molecular_Simulation
24. Shao Y., Gan Z., Epifanovsky E., Gilbert A.T., Wormit M., Kussmann J., Rassolov V.A. Advances in mo-lecular quantum chemistry contained in the Q-Chem 4 program package. Molecular Physics. 2015. No. 113 (2). P. 184 – 215.
25. Van Voorhis Troy Software for the frontiers of quantum chemistry: An overview of developments in the Q-Chem 5 package. 2021.
26. de Freitas Rodrigues S.B., de Araújo, R.S.A., de Mendonça T.R.D., Mendonça-Júnior F.J.B., Zhan P., da Silva-Júnior E.F. Quantum chemistry in drug design: density function theory (DFT) and other quantum mechanics (QM)-related approaches. Applied Computer-Aided Drug Design: Models and Methods. 2023. No. 258.
27. Sarker N. Evaluation of computational chemistry software and density functional theory methods for elec-tronic structure computation of perovskites (Master's thesis, Itä-Suomen yliopisto). 2025.
28. Anikin N.A. A set of possible approximate methods for efficiently taking into account the contribution of Coulomb integrals for dramatic acceleration of DFT calculations of giant biomolecules: reduction to fast-computable short-range two-center splines plus long-range Coulomb FMM. Chemical Bulletin. 2024. T. 7. No. 3. P. 49 – 63. DOI: 10.58224/2619-0575-2024-7-3-49-63
29. Eichkorn K., Treutler O., Ohm H., Haser M., Ahlrichs R. Auxiliary basis sets to approximate Coulomb po-tentials // Chemical Physics Letters. 1995. No. 240. P. 283 – 290.
30. Berman H.M., Westbrook J., Feng Z., Gilliland G., Bhat T.N., Weissig H., Shindyalov I.N., Bourne P.E. The Protein Data Bank. Nucleic Acids Research. 2000. No. 28:1. P. 235 – 242.
31. Balandin M.Yu., Shurina E.P. Methods for solving large-scale linear equations. Novosibirsk: NSTU Pub-lishing House, 2000. 70 p.
32. Krotova E. L., Tsylova E. G., Osipov N. R., Filippov M. A. Review and study of the applicability area of numerical methods for solving linear equations. Science and education: modern trends. 2015. (X). P. 39 – 56.
33. Verzhbitsky V.M. Numerical Methods. Mathematical Analysis and Ordinary Differential Equations. Mos-cow: ONIX 21st Century. 2005, Ch. 4.
34. Laikov D. N. Development of an economical approach to calculating molecules by the density functional method and its application to solving complex chemical problems: Cand. Phys.-Math. Sciences in Specialty 02.00.17 – Quantum Chemistry. Moscow, 2000.
35. Anikin N.A., Andreev A.M., Kuzminsky M.B., Mendkovich A.S. A New Approach to Accelerating Mass Quantum Chemical Calculations of Docking Complexes. Bulletin of the Academy of Sciences. Chemical Series. 2018. No. 6. P. 1100.
36. Anikin N.A., Bugaenko V.L., Frash M.V., Gorb L., Leszczynski J. Localized Basis Orbitals: Minimization of 2-Electron Integrals Array and Orthonormality of Basis Set. Journal of Computational Chemistry. 2003. Vol. 24. No. 9. P. 1132 – 1141.
2. Zaleśny R., Papadopoulos M.G., Mezey P.G., Leszczynski J. (Eds.). (2011). Linear-Scaling Techniques in Computational Chemistry and Physics: Methods and Applications (Vol. 13). Springer Science & Business Media.
3. Ochsenfeld C., Kussmann J., Lambrecht D.S. Linear-scaling methods in quantum chemistry. Reviews in computational chemistry. 2007. No. 23. P. 1.
4. Kussmann J., Beer M., Ochsenfeld C. Linear-scaling self-consistent field methods for large molecules. Wiley Interdisciplinary Reviews: Computational Molecular Science. 2013. No. 3 (6). P. 614 – 636.
5. Vitale Valerio Computational methods for first-principles molecular dynamics with linear-scaling density functional theory. Diss. University of Southampton, 2017.
6. Niklasson, Anders MN. “Density matrix methods in linear scaling electronic structure theory.” Linear-Scaling Techniques in Computational Chemistry and Physics: Methods and Applications. Dordrecht: Spring-er Netherlands, 2011. P. 439 – 473.
7. Hu, Wei, Mohan Chen Advances in density functional theory and beyond for computational chemistry. Fron-tiers in Chemistry. 2021. No. 9. P. 705762.
8. Nakai H., Kobayashi M., Yoshikawa T., Seino J., Ikabata Y., Nishimura Y. Divide-and-conquer linear-scaling quantum chemical computations. The Journal of Physical Chemistry A. 2023. No. 127 (3). P. 589 – 618.
9. Nakata A., Baker J.S., Mujahed S.Y., Poulton J.T., Arapan S., Lin J., Bowler D.R. Large scale and line-ar scaling DFT with the CONQUEST code. The Journal of chemical physics. 2020. No. 152 (16).
10. H. A. Le T. Shiozaki Occupied-orbital fast multipole method for efficient exact exchange evaluation. Journal of Chemical Theory and Computation. 2018. No. 14:3. P. 1228 – 1234. https://doi.org/10.1021/acs.jctc.7b00880.
11. Li A., Muddana H.S., Gilson M.K. Quantum mechanical calculation of noncovalent interactions: A large-scale evaluation of PMx, DFT, and SAPT approaches. Journal of Chemical Theory and Computation. 2014. No. 10:4. P. 1563 – 1575. https://doi.org/10.1021/ct401111c
12. Hennemann M., Clark T. EMPIRE: a highly parallel semiempirical molecular orbital program: 1: self-consistent field calculations. Journal of Molecular Modeling. 2014. No. 20:7. P. 2331. https://doi.org/10.1007/s00894-014-2331-4
13. Oferkin I.V., Katkova E.V., Sulimov A.V. Evaluation of docking target functions by the comprehen-sive in-vestigation of protein-ligand energy minima. Advances in Bioinformatics. 2015. (2015). P. 126858. https://doi.org/10.1155/2015/126858
14. Anikin N.A., Anisimov V.M., Bugaenko V.L., Bobrikov V.V., Andreyev A.M. LocalSCF method for sem-iempirical quantum-chemical calculation of ultralarge biomolecules. Journal of Chemical Physics. 2004. Vol. 121. No. 3. P. 1266 – 1270. https://doi.org/10.1063/1.1764496
15. Anikin N.A., Andreev A.M., Kuzminsky M.B., Mendkovich A.S. High-speed method for mass semiempiri-cal calculations of docking complexes. Bulletin of the Academy of Sciences. Chemical Series. 2008. No. 9. P. 1759 – 1764.
16. Anikin N.A., Bugaenko V.L., Kuzminskii M.B., Mendkovich A.S. A Fast Method for Quantum Chemical Calculations of Large Molecules with DFT Hamiltonian Approximation. Bulletin of the Academy of Sciences. Chemical Series. 2014. No. 2. P. 346 – 349.
17. Womack, J. C., Mardirossian, N., Head-Gordon, M., and Skylaris, Ch.-K. “Self-consistent Implementation of Meta-GGA Functionals for the ONETEP Linear-Scaling Electronic Structure Package”. Journal of Chemical Physics. 2016. No. 145:20. P. 204114. https://doi.org/10.1063/1.4967960
18. Higgins J.E., Probert M.I.J., Hasnip P.J., Refson K., Bush I.J. Hybrid OpenMP and MPI within the CASTEP code, 2015. http://www.archer.ac.uk/community/eCSE/eCSE01-017/eCSE01-017.php
19. Zhang J., Weisman A.L., Saitta P., Friesner R.A. Efficient simulation of large materials clusters using the jaguar quantum chemistry program: Parallelization and wavefunction initialization. International Journal of Quan-tum Chemistry. 2016. No. 116:5. P. 357 – 368. https://doi.org/10.1002/qua.25043
20. Horvath I., Jeszenoi N., Balint M., Paragi G., Hetenyi C. A fragmenting protocol with explicit hydration for calculation of binding enthalpies of target-ligand complexes at a quantum mechanical level. International Journal of Molecular Sciences. 2019. No. 20:18. P. 4384. https://doi.org/10.3390/ijms20184384
21. Brunk E., Rothlisberger U. Mixed Quantum Mechanical/Molecular Mechanical Molecular Dynamics Simu-lations of Biological Systems in Ground and Electronically Excited States. Chemical Reviews. 2015. No. 115 (12). P. 6217 – 6163. https://pubmed.ncbi.nlm.nih.gov/25880693/ doi:10.1021/cr500628b. PMID 2588069,
22. Nemukhin A.V., Polyakov I.V., Moskovsky A.I. Multi-scale supercomputing of large molecular aggregates: A case study of the light-harvesting photosynthetic center. Supercomputing Frontiers and Innova-tions. 2016. No. 2:4. P. 48 – 54. https://doi.org/10.14529/jsfi150403
23. Nakajima T., Katouda M., Kamiya M., Nakatsuka Yu. NTChem: A highperformance software package-age for quantum molecular simulation. I.J. Quant. Chem. 2015. No. 115:5. P. 349 – 359. https://www.researchgate.net/publication/270223372_NTChem_A_High-Performance_Software_Package_for_Quantum_Molecular_Simulation
24. Shao Y., Gan Z., Epifanovsky E., Gilbert A.T., Wormit M., Kussmann J., Rassolov V.A. Advances in mo-lecular quantum chemistry contained in the Q-Chem 4 program package. Molecular Physics. 2015. No. 113 (2). P. 184 – 215.
25. Van Voorhis Troy Software for the frontiers of quantum chemistry: An overview of developments in the Q-Chem 5 package. 2021.
26. de Freitas Rodrigues S.B., de Araújo, R.S.A., de Mendonça T.R.D., Mendonça-Júnior F.J.B., Zhan P., da Silva-Júnior E.F. Quantum chemistry in drug design: density function theory (DFT) and other quantum mechanics (QM)-related approaches. Applied Computer-Aided Drug Design: Models and Methods. 2023. No. 258.
27. Sarker N. Evaluation of computational chemistry software and density functional theory methods for elec-tronic structure computation of perovskites (Master's thesis, Itä-Suomen yliopisto). 2025.
28. Anikin N.A. A set of possible approximate methods for efficiently taking into account the contribution of Coulomb integrals for dramatic acceleration of DFT calculations of giant biomolecules: reduction to fast-computable short-range two-center splines plus long-range Coulomb FMM. Chemical Bulletin. 2024. T. 7. No. 3. P. 49 – 63. DOI: 10.58224/2619-0575-2024-7-3-49-63
29. Eichkorn K., Treutler O., Ohm H., Haser M., Ahlrichs R. Auxiliary basis sets to approximate Coulomb po-tentials // Chemical Physics Letters. 1995. No. 240. P. 283 – 290.
30. Berman H.M., Westbrook J., Feng Z., Gilliland G., Bhat T.N., Weissig H., Shindyalov I.N., Bourne P.E. The Protein Data Bank. Nucleic Acids Research. 2000. No. 28:1. P. 235 – 242.
31. Balandin M.Yu., Shurina E.P. Methods for solving large-scale linear equations. Novosibirsk: NSTU Pub-lishing House, 2000. 70 p.
32. Krotova E. L., Tsylova E. G., Osipov N. R., Filippov M. A. Review and study of the applicability area of numerical methods for solving linear equations. Science and education: modern trends. 2015. (X). P. 39 – 56.
33. Verzhbitsky V.M. Numerical Methods. Mathematical Analysis and Ordinary Differential Equations. Mos-cow: ONIX 21st Century. 2005, Ch. 4.
34. Laikov D. N. Development of an economical approach to calculating molecules by the density functional method and its application to solving complex chemical problems: Cand. Phys.-Math. Sciences in Specialty 02.00.17 – Quantum Chemistry. Moscow, 2000.
35. Anikin N.A., Andreev A.M., Kuzminsky M.B., Mendkovich A.S. A New Approach to Accelerating Mass Quantum Chemical Calculations of Docking Complexes. Bulletin of the Academy of Sciences. Chemical Series. 2018. No. 6. P. 1100.
36. Anikin N.A., Bugaenko V.L., Frash M.V., Gorb L., Leszczynski J. Localized Basis Orbitals: Minimization of 2-Electron Integrals Array and Orthonormality of Basis Set. Journal of Computational Chemistry. 2003. Vol. 24. No. 9. P. 1132 – 1141.
Anikin N.A. A set of possible approximative methods for efficiently recalculating the contribution of coulomb integrals to the elements of the single-electron hamiltonian at SCF iterations to dramatically speed up extremely resource-intensive DFT calculations of giant biomolecules. Chemical Bulletin. 2025. 8 (3). 3. https://doi.org/10.58224/2619-0575-2025-8-3-3