Semi-empirical quantum chemistry method

Semi-empirical methods follow what are often called empirical methods where the two-electron part of the Hamiltonian is not explicitly included. For π-electron systems, this was the Hückel method proposed by Erich Hückel.{{cite journal | last=Hückel | first=Erich | title=Quantentheoretische Beiträge zum Benzolproblem I| journal=Zeitschrift für Physik | publisher=Springer Science and Business Media LLC | volume=70 | issue=3–4 | year=1931 | issn=1434-6001 | doi=10.1007/bf01339530 | pages=204–286 | bibcode=1931ZPhy...70..204H | s2cid=186218131 | language=de}}{{cite journal | last=Hückel | first=Erich | title=Quanstentheoretische Beiträge zum Benzolproblem II| journal=Zeitschrift für Physik | publisher=Springer Science and Business Media LLC | volume=72 | issue=5–6 | year=1931 | issn=1434-6001 | doi=10.1007/bf01341953 | pages=310–337 | bibcode=1931ZPhy...72..310H | language=de}}{{cite journal | last=Hückel | first=Erich | title=Quantentheoretische Beiträge zum Problem der aromatischen und ungesättigten Verbindungen. III | journal=Zeitschrift für Physik | publisher=Springer Science and Business Media LLC | volume=76 | issue=9–10 | year=1932 | issn=1434-6001 | doi=10.1007/bf01341936 | pages=628–648 | bibcode=1932ZPhy...76..628H | s2cid=121787219 | language=de}}{{cite journal | last=Hückel | first=Erich | title=Die freien Radikale der organischen Chemie IV| journal=Zeitschrift für Physik | publisher=Springer Science and Business Media LLC | volume=83 | issue=9–10 | year=1933 | issn=1434-6001 | doi=10.1007/bf01330865 | pages=632–668 | bibcode=1933ZPhy...83..632H | s2cid=121710615 | language=de}}Hückel Theory for Organic Chemists, C. A. Coulson, B. O'Leary and R. B. Mallion, Academic Press, 1978.Andrew Streitwieser, Molecular Orbital Theory for Organic Chemists, Wiley, New York, (1961) For all valence electron systems, the extended Hückel method was proposed by Roald Hoffmann.{{cite journal | last=Hoffmann | first=Roald | title=An Extended Hückel Theory. I. Hydrocarbons | journal=The Journal of Chemical Physics | publisher=AIP Publishing | volume=39 | issue=6 | date=1963-09-15 | issn=0021-9606 | doi=10.1063/1.1734456 | pages=1397–1412| bibcode=1963JChPh..39.1397H }}

Semi-empirical calculations are much faster than their ab initio counterparts, mostly due to the use of the zero differential overlap approximation. Their results, however, can be very wrong if the molecule being computed is not similar enough to the molecules in the database used to parametrize the method.

These methods exist for the calculation of electronically excited states of polyenes, both cyclic and linear. These methods, such as the Pariser–Parr–Pople method (PPP), can provide good estimates of the π-electronic excited states, when parameterized well.{{cite journal | last1=Pariser | first1=Rudolph | last2=Parr | first2=Robert G. | title=A Semi-Empirical Theory of the Electronic Spectra and Electronic Structure of Complex Unsaturated Molecules. I. | journal=The Journal of Chemical Physics | publisher=AIP Publishing | volume=21 | issue=3 | year=1953 | issn=0021-9606 | doi=10.1063/1.1698929 | pages=466–471| bibcode=1953JChPh..21..466P }}{{cite journal | last1=Pariser | first1=Rudolph | last2=Parr | first2=Robert G. | title=A Semi-Empirical Theory of the Electronic Spectra and Electronic Structure of Complex Unsaturated Molecules. II | journal=The Journal of Chemical Physics | publisher=AIP Publishing | volume=21 | issue=5 | year=1953 | issn=0021-9606 | doi=10.1063/1.1699030 | pages=767–776| bibcode=1953JChPh..21..767P }}{{cite journal | last=Pople | first=J. A. | title=Electron interaction in unsaturated hydrocarbons | journal=Transactions of the Faraday Society | publisher=Royal Society of Chemistry (RSC) | volume=49 | year=1953 | issn=0014-7672 | doi=10.1039/tf9534901375 | page=1375}} For many years, the PPP method outperformed ab initio excited state calculations.

=== Methods restricted to all valence electrons. ===

These methods can be grouped into several groups:

:* Methods such as CNDO/2, INDO and NDDO that were introduced by John Pople.J. Pople and D. Beveridge, Approximate Molecular Orbital Theory, McGraw–Hill, 1970.Ira Levine, Quantum Chemistry, Prentice Hall, 4th edition, (1991), pg 579–580C. J. Cramer, Essentials of Computational Chemistry, Wiley, Chichester, (2002), pg 126–131 The implementations aimed to fit, not experiment, but ab initio minimum basis set results. These methods are now rarely used but the methodology is often the basis of later methods.

:* Methods that are in the MOPAC, AMPAC, SPARTAN and/or CP2K computer programs originally from the group of Michael Dewar.J. J. P. Stewart, Reviews in Computational Chemistry, Volume 1, Eds. K. B. Lipkowitz and D. B. Boyd, VCH, New York, 45, (1990) These are MINDO, MNDO,{{cite journal | title = Ground states of molecules. 38. The MNDO method. Approximations and parameters |author1=Michael J. S. Dewar |author2=Walter Thiel |name-list-style=amp | journal = Journal of the American Chemical Society | volume = 99 | pages = 4899–4907 | year = 1977 | doi = 10.1021/ja00457a004 | issue = 15 }} AM1,{{cite journal | title = Development and use of quantum molecular models. 75. Comparative tests of theoretical procedures for studying chemical reactions|author1=Michael J. S. Dewar |author2=Eve G. Zoebisch |author3=Eamonn F. Healy |author4=James J. P. Stewart | journal = Journal of the American Chemical Society | volume = 107 | pages = 3902–3909 | year = 1985 | doi = 10.1021/ja00299a024 | issue = 13 }} PM3,{{cite journal | title = Optimization of parameters for semiempirical methods I. Method | author = James J. P. Stewart | journal = The Journal of Computational Chemistry | volume = 10 | pages = 209–220 | year = 1989 | doi = 10.1002/jcc.540100208 | issue = 2 | s2cid = 36907984 }} PM6,{{cite journal | url=https://doi.org/10.1007/s00894-007-0233-4 | doi=10.1007/s00894-007-0233-4 | title=Optimization of parameters for semiempirical methods V: Modification of NDDO approximations and application to 70 elements | year=2007 | last1=Stewart | first1=James J. P. | journal=Journal of Molecular Modeling | volume=13 | issue=12 | pages=1173–1213 | pmid=17828561 | pmc=2039871 }} PM7{{cite journal | url=https://doi.org/10.1007/s00894-012-1667-x | doi=10.1007/s00894-012-1667-x | title=Optimization of parameters for semiempirical methods VI: More modifications to the NDDO approximations and re-optimization of parameters | year=2013 | last1=Stewart | first1=James J. P. | journal=Journal of Molecular Modeling | volume=19 | issue=1 | pages=1–32 | pmid=23187683 | pmc=3536963 }} and SAM1. Here the objective is to use parameters to fit experimental heats of formation, dipole moments, ionization potentials, and geometries. This is by far the largest group of semiempirical methods.

:* Methods whose primary aim is to calculate excited states and hence predict electronic spectra. These include ZINDO and SINDO.M. Zerner, Reviews in Computational Chemistry, Volume 2, Eds. K. B. Lipkowitz and D. B. Boyd, VCH, New York, 313, (1991){{cite journal | last1=Nanda | first1=D. N. | last2=Jug | first2=Karl | title=SINDO1. A semiempirical SCF MO method for molecular binding energy and geometry I. Approximations and parametrization | journal=Theoretica Chimica Acta | publisher=Springer Science and Business Media LLC | volume=57 | issue=2 | year=1980 | issn=0040-5744 | doi=10.1007/bf00574898 | pages=95–106| s2cid=98468383 }} The OMx (x=1,2,3) methods{{cite journal | url=https://doi.org/10.1021/acs.jctc.5b01046 | doi=10.1021/acs.jctc.5b01046 | title=Semiempirical Quantum-Chemical Orthogonalization-Corrected Methods: Theory, Implementation, and Parameters | year=2016 | last1=Dral | first1=Pavlo O. | last2=Wu | first2=Xin | last3=Spörkel | first3=Lasse | last4=Koslowski | first4=Axel | last5=Weber | first5=Wolfgang | last6=Steiger | first6=Rainer | last7=Scholten | first7=Mirjam | last8=Thiel | first8=Walter | journal=Journal of Chemical Theory and Computation | volume=12 | issue=3 | pages=1082–1096 | pmid=26771204 | pmc=4785507 }} can also be viewed as belonging to this class, although they are also suitable for ground-state applications; in particular, the combination of OM2 and MRCI{{cite journal | doi=10.1021/acs.jctc.6b00403 | title=Semiempirical Quantum-Chemical Orthogonalization-Corrected Methods: Benchmarks of Electronically Excited States | year=2016 | last1=Tuna | first1=Deniz | last2=Lu | first2=You | last3=Koslowski | first3=Axel | last4=Thiel | first4=Walter | journal=Journal of Chemical Theory and Computation | volume=12 | issue=9 | pages=4400–4422 | pmid=27380455 | doi-access=free }} is an important tool for excited state molecular dynamics.

:* Tight-binding methods, e.g. a large family of methods known as DFTB,{{cite journal | url=https://doi.org/10.1002/wcms.1094 | doi=10.1002/wcms.1094 | title=Density-functional tight binding—an approximate density-functional theory method | year=2012 | last1=Seifert | first1=Gotthard | last2=Joswig | first2=Jan-Ole | journal=WIREs Computational Molecular Science | volume=2 | issue=3 | pages=456–465 | s2cid=121521740 | url-access=subscription }} are sometimes classified as semiempirical methods as well. More recent examples include the semiempirical quantum mechanical methods GFNn-xTB (n=0,1,2), which are particularly suited for the geometry, vibrational frequencies, and non-covalent interactions of large molecules.{{Cite journal |last1=Bannwarth |first1=Christoph |last2=Ehlert |first2=Sebastian |last3=Grimme |first3=Stefan |date=2019-03-12 |title=GFN2-xTB—An Accurate and Broadly Parametrized Self-Consistent Tight-Binding Quantum Chemical Method with Multipole Electrostatics and Density-Dependent Dispersion Contributions |journal=Journal of Chemical Theory and Computation |language=en |volume=15 |issue=3 |pages=1652–1671 |doi=10.1021/acs.jctc.8b01176 |pmid=30741547 |s2cid=73419235 |issn=1549-9618|doi-access=free }}

:* The NOTCH method{{cite journal | doi=10.1063/5.0141686 | title=Development of NOTCH, an all-electron, beyond-NDDO semiempirical method: Application to diatomic molecules | year=2023 | last1=Wang | first1=Zikuan | last2=Neese | first2=Frank | journal=The Journal of Chemical Physics | volume=158 | issue=18 | page=184102 | pmid=37154284 | bibcode=2023JChPh.158r4102W | s2cid=258565304 | doi-access=free }} includes many new, physically-motivated terms compared to the NDDO family of methods, is much less empirical than the other semi-empirical methods (almost all of its parameters are determined non-empirically), provides robust accuracy for bonds between uncommon element combinations, and is applicable to ground and excited states.

• List of quantum chemistry and solid-state physics software

{{Reflist}}