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Gaussian orbital

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Title: Gaussian orbital  
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Subject: Roothaan equations, Crystal (software), S. Francis Boys, Basis set (chemistry), Molecular physics
Collection: Molecular Physics, Quantum Chemistry
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Gaussian orbital

In computational chemistry and molecular physics, Gaussian orbitals (also known as Gaussian type orbitals, GTOs or Gaussians) are functions used as atomic orbitals in the LCAO method for the representation of electron orbitals in molecules and numerous properties that depend on these.[1]


  • Rationale 1
  • Mathematical form 2
  • Molecular integrals 3
  • The POLYATOM System 4
  • See also 5
  • References 6
  • External links 7


The use of Gaussian orbitals in electronic structure theory (instead of the more physical Slater-type orbitals) was first proposed by Boys[2] in 1950. The principal reason for the use of Gaussian basis functions in molecular quantum chemical calculations is the 'Gaussian Product Theorem', which guarantees that the product of two GTOs centered on two different atoms is a finite sum of Gaussians centered on a point along the axis connecting them. In this manner, four-center integrals can be reduced to finite sums of two-center integrals, and in a next step to finite sums of one-center integrals. The speedup by 4—5 orders of magnitude compared to Slater orbitals more than outweighs the extra cost entailed by the larger number of basis functions generally required in a Gaussian calculation.

For reasons of convenience, many quantum chemistry programs work in a basis of Cartesian Gaussians even when spherical Gaussians are requested, as integral evaluation is much easier in the cartesian basis, and the spherical functions can be simply expressed using the cartesian functions.[3]

Mathematical form

The Gaussian basis functions obey the usual radial-angular decomposition

\ \Phi(\mathbf{r}) = R_l(r) Y_{lm}(\theta,\phi),

where Y_{lm}(\theta,\phi) is a spherical harmonic, l and m are the angular momentum and its z component, and r,\theta,\phi are spherical coordinates.

While for Slater orbitals the radial part is

\ R_l(r) = A(l,\alpha) r^l e^{-\alpha r},

A(l,\alpha) being a normalization constant, for Gaussian primitives the radial part is

\ R_l(r) = B(l,\alpha) r^l e^{-\alpha r^2},

where B(l,\alpha) is the normalization constant corresponding to the Gaussian.

Because an individual primitive Gaussian function gives a rather poor description for the electronic wave function near the nucleus, Gaussian basis sets are almost always contracted:

\ R_l(r) = r^l \sum_{p=1,P} c_p A(l,\alpha_p) \exp(-\alpha_p r^2),

where c_p is the contraction coefficient for the primitive with exponent \alpha_p. The coefficients are given with respect to normalized primitives, because coefficients for unnormalized primitives would differ by many orders of magnitude. The exponents are reported in atomic units. There is a large library of published Gaussian basis sets optimized for a variety of criteria available at the EMSL basis set exchange.

Molecular integrals

Taketa et al. (1966) presented the necessary mathematical equations for obtaining matrix elements in the Gaussian basis.[4] Since then much work has been done to speed up the evaluation of these integrals which are the slowest part of many quantum chemical calculations. Živković and Maksić (1968) suggested using Hermite Gaussian functions,[5] as this simplifies the equations. McMurchie and Davidson (1978) introduced recursion relations,[6] which greatly reduces the amount of calculations. Pople and Hehre (1978) developed a local coordinate method.[7] Obara and Saika introduced efficient recursion relations in 1985,[8] which was followed by the development of other important recurrence relations. Gill and Pople (1990) introduced a 'PRISM' algorithm which allowed efficient use of 20 different calculation paths.[9]


The POLYATOM System[10] was the first package for ab initio calculations using Gaussian orbitals that was applied to a wide variety of molecules.[11] It was developed in Slater's Solid State and Molecular Theory Group (SSMTG) at MIT using the resources of the Cooperative Computing Laboratory. The mathematical infrastructure and operational software were developed by Imre Csizmadia,[12] Malcolm Harrison,[13] Jules Moskowitz[14] and Brian Sutcliffe.[15]

See also


  1. ^ Gill, Peter M.W. (1994). "Molecular integrals Over Gaussian Basis Functions" (PDF). Advances in Quantum Chemistry 25: 141–205.  
  2. ^ Boys, S. F. (1950). Proc. R. Soc. London Ser. A 200. p. 542.  
  3. ^ Schlegel, H.; Frisch, M. (1990). "Transformation between Cartesian and pure spherical harmonic Gaussians". International Journal of Quantum Chemistry 54 (2): 83–87.  
  4. ^ Taketa, Hiroshi; Huzinaga, Sigeru; O-ohata, Kiyosi (1966). "Gaussian-Expansion Methods for Molecular Integrals". Journal of the Physical Society of Japan 21 (11): 2313–2324.  
  5. ^ Živković, T.; Maksić, Z. B. (1968). "Explicit Formulas for Molecular Integrals over Hermite-Gaussian Functions". Journal of Chemical Physics 49 (7): 3083–3087.  
  6. ^ McMurchie, Larry E.; Davidson, Ernest R. (1978). "One- and two-electron integrals over Cartesian Gaussian functions". Journal of Computational Physics 26 (2): 218–31.  
  7. ^ Pople, J. A.; Hehre, W. J. (1978). "Computation of electron repulsion integrals involving contracted Gaussian basis functions.". J. Comp. Phys. 27 (2): 161–168.  
  8. ^ Obara, S.; Saika, A. (1986). "Efficient recursive computation of molecular integrals over Cartesian Gaussian functions". J. Chem. Phys. 84 (7): 3963–74.  
  9. ^ Gill, Peter M. W.; Pople, John A. (December 1991). "The Prism Algorithm for Two-Electron Integrals" (PDF). International Journal of Quantum Chemistry 40 (6): 753–772.  
  10. ^ I.G. Csizmadia, M.C. Harrison, J.W. Moskowitz and B.T. Sutcliffe, Nonempirical LCAO-MO-SCF-CI calculations on organic molecules with gaussian-type functions. Introductory review and mathematical formalism, Theoretica Chimica Acta, 6, 191, 1966.
  11. ^ A.C. Wahl, Chemistry by computer, Scientific American, pages 54-70, April, 1970.
  12. ^ , Professor Emeritus of Chemistry, University of Toronto, in Reviews in Computational Chemistry, vol.15, p.248Imre Csizmadia
  13. ^ , Professor of Computer Science, New York UniversityMalcolm C. Harrison
  14. ^ , Professor Emeritus of Chemistry, New York UniversityJules W. Moskowitz
  15. ^ , Professor of Chemistry, York UniversityBrian T. Sutcliffe

External links

  • A visualization of all common and uncommon atomic orbitals, from 1s to 7g (Note that the radial part of the expressions given corresponds to Slater orbitals rather than Gaussians. The angular parts, and hence their shapes as displayed in figures, are the same as those of spherical Gaussians.)
  • Explanation of Gaussian basis set
  • Basis set exchange
  • Petersson, T; Hellsing, B. (2010). "Detailed derivation of Gaussian orbital based matrix elements in electron structure calculations". Eur. J. Phys 31. p. 37.  
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