1600:, there are six possible GTOs that may be constructed; this is one more than the five canonical d orbital functions for a given angular quantum number. To address this, a linear combination of two d-type GTOs can be used to reproduce a canonical d function. Similarly, there exist 10 f-type GTOs, but only 7 canonical f orbital functions; this pattern continues for higher angular quantum numbers.
91:
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
1617:
and Hehre (1978) developed a local coordinate method. Obara and Saika introduced efficient recursion relations in 1985, 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
1608:
Taketa et al. (1966) presented the necessary mathematical equations for obtaining matrix elements in the
Gaussian basis. 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)
99:
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.
1421:
2191:
948:
776:
2115:
Csizmadia, I.G.; Harrison, M.C.; Moskowitz, J.W.; Sutcliffe, B.T. (1966). "Nonempirical LCAO-MO-SCF-CI calculations on organic molecules with gaussian-type functions. Introductory review and mathematical formalism".
1110:
193:
1638:
using the resources of the
Cooperative Computing Laboratory. The mathematical infrastructure and operational software were developed by Imre Csizmadia, Malcolm Harrison, Jules Moskowitz and Brian Sutcliffe.
569:
433:
60:
1008:. 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
804:
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:
242:
338:
680:
645:
607:
470:
1006:
1613:
Gaussian functions, as this simplifies the equations. McMurchie and
Davidson (1978) introduced recursion relations, which greatly reduces the amount of calculations.
1598:
1560:
1522:
1102:
979:
810:
1484:
1464:
1444:
1082:
1062:
1042:
799:
306:
286:
266:
688:
1416:{\displaystyle \Phi (x,y,z;\alpha ,i,j,k)=\left({\frac {2\alpha }{\pi }}\right)^{3/4}\left^{1/2}x^{i}y^{j}z^{k}e^{-\alpha (x^{2}+y^{2}+z^{2})}}
1104:
controlling the width of the orbital. The expression for a
Cartesian Gaussian-type orbital, with the appropriate normalization coefficient is
92:
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
114:
1648:
2232:
rather than
Gaussians. The angular parts, and hence their shapes as displayed in figures, are the same as those of spherical Gaussians.
1879:
1704:
478:
2308:
2004:
Pople, J. A.; Hehre, W. J. (1978). "Computation of electron repulsion integrals involving contracted
Gaussian basis functions".
349:
2163:
2039:
Obara, S.; Saika, A. (1986). "Efficient recursive computation of molecular integrals over
Cartesian Gaussian functions".
96:
outweighs the extra cost entailed by the larger number of basis functions generally required in a Gaussian calculation.
2303:
2298:
1930:Živković, T.; Maksić, Z. B. (1968). "Explicit Formulas for Molecular Integrals over Hermite-Gaussian Functions".
1895:
Taketa, Hiroshi; Huzinaga, Sigeru; O-ohata, Kiyosi (1966). "Gaussian-Expansion Methods for Molecular Integrals".
1729:"Electronic Wave Functions. I. A General Method of Calculation for the Stationary States of Any Molecular System"
1012:. There is a large library of published Gaussian basis sets optimized for a variety of criteria available at the
2167:, Professor Emeritus of Chemistry, University of Toronto, in Reviews in Computational Chemistry, vol.15, p.248
28:
1782:
Schlegel, H.; Frisch, M. (1990). "Transformation between Cartesian and pure spherical harmonic Gaussians".
202:
1630:
calculations using Gaussian orbitals that was applied to a wide variety of molecules. It was developed in
52:
2205:
88:
1024:
In Cartesian coordinates, Gaussian-type orbitals can be written in terms of exponential factors in the
2261:
2048:
2013:
1978:
1939:
1904:
1836:
1740:
1684:
311:
245:
80:
2250:"Detailed derivation of Gaussian orbital based matrix elements in electron structure calculations"
1669:
650:
615:
577:
440:
2277:
2133:
1852:
1826:
1799:
1764:
1756:
1631:
984:
2249:
1875:
1700:
64:
32:
1565:
1527:
1489:
1087:
2269:
2125:
2090:
2056:
2021:
1986:
1966:
1947:
1912:
1844:
1791:
1748:
1692:
1817:
Mathar, Richard J. (2002). "Mutual Conversion of Three Flavors of Gaussian Type Orbitals".
957:
943:{\displaystyle \ R_{l}(r)=r^{l}\sum _{p=1}^{P}c_{p}B(l,\alpha _{p})\exp(-\alpha _{p}r^{2})}
2229:
93:
84:
79:
The use of Gaussian orbitals in electronic structure theory (instead of the more physical
2237:
2265:
2052:
2017:
1982:
1943:
1908:
1840:
1744:
1688:
1562:, the GTO possesses axial symmetry along one axis and is considered a p-type GTO. When
1469:
1449:
1429:
1067:
1047:
1027:
784:
291:
271:
251:
56:
2075:
1696:
2292:
2281:
2273:
2137:
2025:
1990:
1856:
1768:
1803:
1009:
771:{\displaystyle \int _{0}^{\infty }\mathrm {d} r\,r^{2}\left|R_{l}(r)\right|^{2}=1}
17:
1614:
1524:, then the orbital has spherical symmetry and is considered an s-type GTO. If
1728:
2223:
2177:
1752:
472:
being a normalization constant, for Gaussian primitives the radial part is
2094:
1795:
2224:
A visualization of all common and uncommon atomic orbitals, from 1s to 7g
1916:
108:
The Gaussian basis functions obey the usual radial-angular decomposition
68:
1831:
2129:
1610:
1951:
1874:(2nd ed.). Chichester, West Sussex, England: Wiley. p. 167.
1848:
2060:
1760:
1967:"One- and two-electron integrals over Cartesian Gaussian functions"
188:{\displaystyle \ \Phi (\mathbf {r} )=R_{l}(r)Y_{lm}(\theta ,\phi )}
2228:
Note that the radial part of the expressions given corresponds to
2242:
1872:
Essentials of computational chemistry : theories and models
1013:
981:
is the contraction coefficient for the primitive with exponent
1635:
564:{\displaystyle \ R_{l}(r)=B(l,\alpha )r^{l}e^{-\alpha r^{2}},}
609:
is the normalization constant corresponding to the Gaussian.
428:{\displaystyle \ R_{l}(r)=A(l,\alpha )r^{l}e^{-\alpha r},}
2195:, Professor Emeritus of Chemistry, New York University
87:
in 1950. The principal reason for the use of Gaussian
1568:
1530:
1492:
1472:
1452:
1432:
1113:
1090:
1070:
1050:
1030:
987:
960:
813:
787:
691:
653:
618:
580:
481:
443:
352:
314:
294:
274:
254:
205:
117:
2181:, Professor of Computer Science, New York University
2074:Gill, Peter M. W.; Pople, John A. (December 1991).
1670:"Molecular integrals Over Gaussian Basis Functions"
1634:Solid State and Molecular Theory Group (SSMTG) at
1592:
1554:
1516:
1478:
1458:
1438:
1415:
1096:
1076:
1056:
1036:
1000:
973:
942:
793:
781:which in general does not impose orthogonality in
770:
674:
639:
601:
563:
464:
427:
332:
300:
280:
260:
236:
187:
1965:McMurchie, Larry E.; Davidson, Ernest R. (1978).
2154:, Scientific American, pages 54-70, April, 1970.
2076:"The Prism Algorithm for Two-Electron Integrals"
1626:The POLYATOM System was the first package for
71:and numerous properties that depend on these.
612:The normalization condition which determines
343:While for Slater orbitals the radial part is
8:
1084:directions as well as an exponential factor
2083:International Journal of Quantum Chemistry
1819:International Journal of Quantum Chemistry
1784:International Journal of Quantum Chemistry
2209:, Professor of Chemistry, York University
1830:
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1431:
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1029:
992:
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821:
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617:
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313:
293:
273:
253:
210:
204:
161:
142:
127:
116:
1897:Journal of the Physical Society of Japan
1660:
237:{\displaystyle Y_{lm}(\theta ,\phi )}
7:
2248:Petersson, T; Hellsing, B. (2010).
1649:Quantum chemistry computer programs
1114:
708:
702:
121:
25:
2238:Explanation of Gaussian basis set
288:are the angular momentum and its
1971:Journal of Computational Physics
128:
1870:Cramer, Christopher J. (2004).
333:{\displaystyle r,\theta ,\phi }
1408:
1369:
1301:
1292:
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1277:
1271:
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1221:
1211:
1159:
1117:
937:
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902:
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748:
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622:
596:
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366:
231:
219:
182:
170:
154:
148:
132:
124:
1:
1697:10.1016/S0065-3276(08)60019-2
1677:Advances in Quantum Chemistry
2026:10.1016/0021-9991(78)90001-3
1991:10.1016/0021-9991(78)90092-X
675:{\displaystyle B(l,\alpha )}
640:{\displaystyle A(l,\alpha )}
602:{\displaystyle B(l,\alpha )}
465:{\displaystyle A(l,\alpha )}
1932:Journal of Chemical Physics
1001:{\displaystyle \alpha _{p}}
340:are spherical coordinates.
2325:
2274:10.1088/0143-0807/31/1/004
63:for the representation of
1668:Gill, Peter M.W. (1994).
1426:In the above expression,
1014:Basis Set Exchange portal
83:) was first proposed by
2309:Computational chemistry
2118:Theoretica Chimica Acta
1593:{\displaystyle i+j+k=2}
1555:{\displaystyle i+j+k=1}
1517:{\displaystyle i+j+k=0}
1097:{\displaystyle \alpha }
29:computational chemistry
1753:10.1098/rspa.1950.0036
1594:
1556:
1518:
1480:
1460:
1440:
1417:
1098:
1078:
1058:
1038:
1002:
975:
944:
869:
795:
772:
676:
641:
603:
565:
466:
429:
334:
302:
282:
262:
238:
189:
41:Gaussian type orbitals
2152:Chemistry by computer
2095:10.1002/qua.560400605
1796:10.1002/qua.560540202
1733:Proc. R. Soc. Lond. A
1595:
1557:
1519:
1486:must be integers. If
1481:
1461:
1441:
1418:
1099:
1079:
1059:
1039:
1020:Cartesian coordinates
1003:
976:
974:{\displaystyle c_{p}}
945:
849:
796:
773:
677:
642:
604:
566:
467:
430:
335:
303:
283:
263:
239:
190:
1917:10.1143/JPSJ.21.2313
1727:Boys, S. F. (1950).
1566:
1528:
1490:
1470:
1450:
1430:
1111:
1088:
1068:
1048:
1028:
985:
958:
811:
785:
689:
651:
616:
578:
479:
441:
350:
312:
292:
272:
252:
203:
115:
81:Slater-type orbitals
2266:2010EJPh...31...37P
2179:Malcolm C. Harrison
2053:1986JChPh..84.3963O
2018:1978JCoPh..27..161P
1983:1978JCoPh..26..218M
1944:1968JChPh..49.3083Z
1909:1966JPSJ...21.2313T
1841:2002IJQC...90..227M
1745:1950RSPSA.200..542B
1689:1994AdQC...25..141G
1622:The POLYATOM System
1618:calculation paths.
1604:Molecular integrals
706:
2243:Basis set exchange
2207:Brian T. Sutcliffe
2193:Jules W. Moskowitz
2130:10.1007/BF02394698
1590:
1552:
1514:
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791:
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672:
637:
599:
561:
462:
425:
330:
298:
278:
258:
246:spherical harmonic
234:
185:
2304:Quantum chemistry
2299:Molecular physics
1952:10.1063/1.1670551
1903:(11): 2313–2324.
1849:10.1002/qua.10085
1739:(1063): 542–554.
1479:{\displaystyle k}
1459:{\displaystyle j}
1439:{\displaystyle i}
1308:
1183:
1077:{\displaystyle z}
1057:{\displaystyle y}
1037:{\displaystyle x}
816:
794:{\displaystyle l}
484:
355:
301:{\displaystyle z}
281:{\displaystyle m}
261:{\displaystyle l}
120:
104:Mathematical form
65:electron orbitals
37:Gaussian orbitals
33:molecular physics
18:Gaussian orbitals
16:(Redirected from
2316:
2285:
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2061:10.1063/1.450106
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2001:
1995:
1994:
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1955:
1938:(7): 3083–3087.
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1674:
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1609:suggested using
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2230:Slater orbitals
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2176:
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2158:
2149:
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2006:J. Comput. Phys
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94:Slater orbitals
89:basis functions
77:
57:atomic orbitals
39:(also known as
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22:
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11:
5:
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2218:External links
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2165:Imre Csizmadia
2156:
2143:
2107:
2089:(6): 753–772.
2066:
2047:(7): 3963–74.
2031:
2012:(2): 161–168.
1996:
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1825:(1): 227–243.
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