Semi-empirical quantum chemistry method

Semi-empirical quantum chemistry methods are based on the Hartree–Fock formalism, but make many approximations and obtain some parameters from empirical data. They are very important in computational chemistry for treating large molecules where the full Hartree–Fock method without the approximations is too expensive. The use of empirical parameters appears to allow some inclusion of electron correlation effects into the methods.

Electronic structure methods
Valence bond theory
Coulson–Fischer theory
Generalized valence bond
Modern valence bond theory
Molecular orbital theory
Hartree–Fock method
Semi-empirical quantum chemistry methods
Møller–Plesset perturbation theory
Configuration interaction
Coupled cluster
Multi-configurational self-consistent field
Quantum chemistry composite methods
Quantum Monte Carlo
Density functional theory
Time-dependent density functional theory
Thomas–Fermi model
Orbital-free density functional theory
Linearized augmented-plane-wave method
Projector augmented wave method
Electronic band structure
Nearly free electron model
Tight binding
Muffin-tin approximation
k·p perturbation theory
Empty lattice approximation
GW approximation
Korringa–Kohn–Rostoker method

Within the framework of Hartree–Fock calculations, some pieces of information (such as two-electron integrals) are sometimes approximated or completely omitted. In order to correct for this loss, semi-empirical methods are parametrized, that is their results are fitted by a set of parameters, normally in such a way as to produce results that best agree with experimental data, but sometimes to agree with ab initio results.

Type of simplifications used

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.[1][2][3][4][5][6] For all valence electron systems, the extended Hückel method was proposed by Roald Hoffmann.[7]

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.

Preferred application domains

Methods restricted to π-electrons

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.[8][9][10] 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.[11][12][13] 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.[14] These are MINDO, MNDO,[15] AM1,[16] PM3,[17] PM6,[18] PM7[19] 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.[20][21] The OMx (x=1,2,3) methods[22] 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[23] is an important tool for excited state molecular dynamics.
  • Tight-binding methods, e.g. a large family of methods known as DFTB,[24] 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.[25]
  • The NOTCH method[26] 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.

See also

References
  1. Hückel, Erich (1931). "Quantentheoretische Beiträge zum Benzolproblem I". Zeitschrift für Physik (in German). 70 (3–4). Springer Science and Business Media LLC: 204–286. Bibcode:1931ZPhy...70..204H. doi:10.1007/bf01339530. ISSN 1434-6001. S2CID 186218131.
  2. Hückel, Erich (1931). "Quanstentheoretische Beiträge zum Benzolproblem II". Zeitschrift für Physik (in German). 72 (5–6). Springer Science and Business Media LLC: 310–337. Bibcode:1931ZPhy...72..310H. doi:10.1007/bf01341953. ISSN 1434-6001.
  3. Hückel, Erich (1932). "Quantentheoretische Beiträge zum Problem der aromatischen und ungesättigten Verbindungen. III". Zeitschrift für Physik (in German). 76 (9–10). Springer Science and Business Media LLC: 628–648. Bibcode:1932ZPhy...76..628H. doi:10.1007/bf01341936. ISSN 1434-6001. S2CID 121787219.
  4. Hückel, Erich (1933). "Die freien Radikale der organischen Chemie IV". Zeitschrift für Physik (in German). 83 (9–10). Springer Science and Business Media LLC: 632–668. Bibcode:1933ZPhy...83..632H. doi:10.1007/bf01330865. ISSN 1434-6001. S2CID 121710615.
  5. Hückel Theory for Organic Chemists, C. A. Coulson, B. O'Leary and R. B. Mallion, Academic Press, 1978.
  6. Andrew Streitwieser, Molecular Orbital Theory for Organic Chemists, Wiley, New York, (1961)
  7. Hoffmann, Roald (1963-09-15). "An Extended Hückel Theory. I. Hydrocarbons". The Journal of Chemical Physics. 39 (6). AIP Publishing: 1397–1412. Bibcode:1963JChPh..39.1397H. doi:10.1063/1.1734456. ISSN 0021-9606.
  8. Pariser, Rudolph; Parr, Robert G. (1953). "A Semi‐Empirical Theory of the Electronic Spectra and Electronic Structure of Complex Unsaturated Molecules. I.". The Journal of Chemical Physics. 21 (3). AIP Publishing: 466–471. Bibcode:1953JChPh..21..466P. doi:10.1063/1.1698929. ISSN 0021-9606.
  9. Pariser, Rudolph; Parr, Robert G. (1953). "A Semi‐Empirical Theory of the Electronic Spectra and Electronic Structure of Complex Unsaturated Molecules. II". The Journal of Chemical Physics. 21 (5). AIP Publishing: 767–776. Bibcode:1953JChPh..21..767P. doi:10.1063/1.1699030. ISSN 0021-9606.
  10. Pople, J. A. (1953). "Electron interaction in unsaturated hydrocarbons". Transactions of the Faraday Society. 49. Royal Society of Chemistry (RSC): 1375. doi:10.1039/tf9534901375. ISSN 0014-7672.
  11. J. Pople and D. Beveridge, Approximate Molecular Orbital Theory, McGraw–Hill, 1970.
  12. Ira Levine, Quantum Chemistry, Prentice Hall, 4th edition, (1991), pg 579–580
  13. C. J. Cramer, Essentials of Computational Chemistry, Wiley, Chichester, (2002), pg 126–131
  14. J. J. P. Stewart, Reviews in Computational Chemistry, Volume 1, Eds. K. B. Lipkowitz and D. B. Boyd, VCH, New York, 45, (1990)
  15. Michael J. S. Dewar & Walter Thiel (1977). "Ground states of molecules. 38. The MNDO method. Approximations and parameters". Journal of the American Chemical Society. 99 (15): 4899–4907. doi:10.1021/ja00457a004.
  16. Michael J. S. Dewar; Eve G. Zoebisch; Eamonn F. Healy; James J. P. Stewart (1985). "Development and use of quantum molecular models. 75. Comparative tests of theoretical procedures for studying chemical reactions". Journal of the American Chemical Society. 107 (13): 3902–3909. doi:10.1021/ja00299a024.
  17. James J. P. Stewart (1989). "Optimization of parameters for semiempirical methods I. Method". The Journal of Computational Chemistry. 10 (2): 209–220. doi:10.1002/jcc.540100208. S2CID 36907984.
  18. Stewart, James J. P. (2007). "Optimization of parameters for semiempirical methods V: Modification of NDDO approximations and application to 70 elements". Journal of Molecular Modeling. 13 (12): 1173–1213. doi:10.1007/s00894-007-0233-4. PMC 2039871. PMID 17828561.
  19. Stewart, James J. P. (2013). "Optimization of parameters for semiempirical methods VI: More modifications to the NDDO approximations and re-optimization of parameters". Journal of Molecular Modeling. 19 (1): 1–32. doi:10.1007/s00894-012-1667-x. PMC 3536963. PMID 23187683.
  20. M. Zerner, Reviews in Computational Chemistry, Volume 2, Eds. K. B. Lipkowitz and D. B. Boyd, VCH, New York, 313, (1991)
  21. Nanda, D. N.; Jug, Karl (1980). "SINDO1. A semiempirical SCF MO method for molecular binding energy and geometry I. Approximations and parametrization". Theoretica Chimica Acta. 57 (2). Springer Science and Business Media LLC: 95–106. doi:10.1007/bf00574898. ISSN 0040-5744. S2CID 98468383.
  22. Dral, Pavlo O.; Wu, Xin; Spörkel, Lasse; Koslowski, Axel; Weber, Wolfgang; Steiger, Rainer; Scholten, Mirjam; Thiel, Walter (2016). "Semiempirical Quantum-Chemical Orthogonalization-Corrected Methods: Theory, Implementation, and Parameters". Journal of Chemical Theory and Computation. 12 (3): 1082–1096. doi:10.1021/acs.jctc.5b01046. PMC 4785507. PMID 26771204.
  23. Tuna, Deniz; Lu, You; Koslowski, Axel; Thiel, Walter (2016). "Semiempirical Quantum-Chemical Orthogonalization-Corrected Methods: Benchmarks of Electronically Excited States". Journal of Chemical Theory and Computation. 12 (9): 4400–4422. doi:10.1021/acs.jctc.6b00403. PMID 27380455.
  24. Seifert, Gotthard; Joswig, Jan‐Ole (2012). "Density‐functional tight binding—an approximate density‐functional theory method". WIREs Computational Molecular Science. 2 (3): 456–465. doi:10.1002/wcms.1094. S2CID 121521740.
  25. Bannwarth, Christoph; Ehlert, Sebastian; Grimme, Stefan (2019-03-12). "GFN2-xTB—An Accurate and Broadly Parametrized Self-Consistent Tight-Binding Quantum Chemical Method with Multipole Electrostatics and Density-Dependent Dispersion Contributions". Journal of Chemical Theory and Computation. 15 (3): 1652–1671. doi:10.1021/acs.jctc.8b01176. ISSN 1549-9618. PMID 30741547. S2CID 73419235.
  26. Wang, Zikuan; Neese, Frank (2023). "Development of NOTCH, an all-electron, beyond-NDDO semiempirical method: Application to diatomic molecules". The Journal of Chemical Physics. 158 (18): 184102. Bibcode:2023JChPh.158r4102W. doi:10.1063/5.0141686. PMID 37154284. S2CID 258565304.
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