Ab-initio Quantum Chemistry Through Computer Algebra
Asian Journal of Research and Reviews in Physics, Volume 6, Issue 4,
Page 18-31
DOI:
10.9734/ajr2p/2022/v6i4126
Abstract
In this article, we demonstrate the first-principles computation of quantum chemistry through symbolic computation, using computational algebraic geometry. We generate symbolic formulas of one- and twoelectron integrals. The approximations of those integrals by multivariate polynomials yield the set of equations required by quantum chemistry. We solve these equations in hybrid ways where numeric and symbolic computations are intertwined. Thereby polynomials are converted into the Gronber basis; and it is decomposed to the primary ideals (each of which represents a quantum state). The primary ideals are equipped with triangular forms, which allows us to evaluate the roots robustly.
- Quantum mechanics
- quantum chemistry
- computational algebraic geometry
- gröbner basis
- symbolic computation
How to Cite
References
Kikuchi, Akihito. An approach to rst principle electronic structure computation by symbolicnumeri computation. QScience Connect 2013;2013(14 DOI: http://dx.doi.org/10.5339/connect.2013.14
Boys, S Francis. Electronic wave functions-I. general method of calculation for the stationar states of an molecular system. Proceedings of th Royal Society of London. Series A. Mathematical and Physical Sciences. The Royal Society London. 1950;200(1063):542-554.
Barca, Giuseppe MJ, Gill, Peter MW. Twoelectron integrals over gaussian geminals. Journal of Chemical Theory and Computation. ACS Publications. 2016;12(10):4915-4924.
Kikuchi, Akihito, Kikuchi, Ichio. Computational Algebraic Geometry and Quantum Mechanics: An Initiative toward Post Contemporary Quantum Chemistry. Journal of Multidisciplinary Research and Reviews. Innovationinfo. 2019;1:47-79. Available: https://www.innovationinfo.org/journalof-
multidisciplinary-research-and-reviews DOI: 10.3619/JMRR.1000118
Kikuchi, Ichio, Kikuchi, Akihito. Galois and Class Field Theory for Quantum Chemists. OSF Preprints; 2020. Available: https://osf.io/preprints/n46rf/
Kikuchi, Akihito. Quantum Mechanics built on Algebraic Geometry: Emerging Physics through Symbolic Computation. Eliva Press; 2021.
Eisenbud, David. Commutative Algebra: with a view toward algebraic geometry. Springer Science & Business Media. 2013;150.
Using algebraic geometry. Cox, David A, Little, John, O'shea, Donal. Springer Science & Business Media. 2006;185.
Cox, David, Little, John, OShea, Donal. Ideals, varieties, and algorithms: An introduction to computational algebraic geometry and commutative algebra. Springer Science & Business Media; 2013.
Ene, Viviana, Herzog, Jurgen. Grobner Bases in Commutative Algebra. American Mathematical Soc. 2011;130.
Gianni, Patrizia, Trager, Barry, Zacharias, Gail. Grobner bases and primary decomposition of polynomial ideals. Journal of Symbolic Computation. Elsevier. 1988;6(2-3):149-167.[12] Moller, H Michael. On decomposing systems of polynomial equations with nitely many solutions. Applicable Algebra in Engineering, Communication and Computing. Springer. 1993;4(4):217-230.
Lazard, Daniel. Solving zero-dimensional algebraic systems. Journal of Symbolic Computation. Elsevier. 1992;13(2):117-131.
Sottile, Frank. From enumerative geometry to solving systems of polynomial equations. Computations in algebraic geometry with Macaulay 2. Springer. 2002;101-129.
Decker, Wolfram, P ster, Gerhard. A rst course in computational algebraic geometry. Cambridge University Press; 2013.
Beach JY. Quantum-mechanical treatment of helium hydride molecule-ion HeH+. The Journal of Chemical Physics. American Institute of Physics. 1936;4(6):353-357.
Banyard KE, Baker CC. Molecular formation and electron correlation in HeH+. International Journal of Quantum Chemistry. Wiley Online Library. 1970;4(4):431-450.
Bovino, Stefano, Tacconi, Mario, Gianturco, Francesco A, Galli, Daniele. Ion chemistry in the early universe Revisiting the role of HeH+ with new quantum calculations. Astronomy & Astrophysics. EDP Sciences. 2011;529:A140.
Engel, Elodie A, Doss, Natasha, Harris, Gregory J, Tennyson, Jonathan. Calculated spectra for HeH+ and its e ect on the opacity of cool metal-poor stars. Monthly Notices of the Royal Astronomical Society. Blackwell Science Ltd
Oxford, UK. 2005;357(2):471-477.
Wang, Ya, Dolde, Florian, Biamonte, Jacob, Babbush, Ryan, Bergholm, Ville, Yang, Sen, Jakobi, Ingmar, Neumann, Philipp, Aspuru- Guzik, Alan, Whit eld, James D. Quantum simulation of helium hydride cation in a solidstate spin register. ACS nano. ACS Publications. 2015;9(8):7769-7774.
Oppermann F, Wustelt P, Florin T, Mhatre S, Grafe S, Paulus GG, Lein M. Dissociation and ionization of HeH+ in sub-cycle-controlled intense two-color elds. Journal of Physics B: Atomic, Molecular and Optical Physics. IOP Publishing. 2020;53(17):174001.
Dallas, Jax, Flint, Athena, Fortenberry, Ryan C. Solvation of HeH+ in neon atoms: Proton-bound complexes of mixed He and Ne. Chemical Physics. Elsevier. 2020;539:110927.
Szabo, Attila, Ostlund, Neil S. Modern quantum chemistry: Introduction to advanced electronic structure theory. Courier Corporation; 2012.
Faugere, Jean-Charles, Gianni, Patrizia, Lazard, Daniel, Mora, Teo. Ecient computation of zero-dimensional Grobner bases by change of ordering. Journal of Symbolic Computation. Elsevier; 1993;16(4):329-344.
Singular 4-3-0 - A computer algebra system for polynomial computations. Decker, Wolfram, Greuel, Gert-Martin, P ster, Gerhard, Schonemann, Hans. 2022. Available: http://www.singular.uni-kl.de
Kikuchi, Ichio, Kikuchi, Akihito. 2023. Available: http://github.com/kikuchiichio/AlgebraicQuantumChemistry
-
Abstract View: 28 times
PDF Download: 20 times
