A Handbook of Magnetochemical Formulae by Boca, Roman (Auth.)

By Boca, Roman (Auth.)

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21 5 490 1224 11 holds true. 5 Multi-electron Wave Function The wave function of a multi-electron system should be symmetry-invariant. The symmetry under consideration consists of: 1. the spatial symmetry of the atomic coordinates within the point group G; 2. the symmetry of the angular momentum within the groups R3 and SU2S11; 3. the permutation symmetry that accounts for permutations of individual particles (spins) within the symmetry group SN. The most general formulation of the Pauli principle is based upon the fact that the only allowed states refer to the one-dimensional permutational states.

For instance, C3vBD3BS3 (h 5 6); OBTdBS4 (h 5 24). The point group Td is more familiar than its isomorphous counterpart S4. e. m 52j SO3, R3, O1 3 ; SO(3) Sp2j 1 1 Rotation in three dimensions Symplectic, in (2j11) dimensions G2 SNb Symmetric πN Permutation the scalar product Used mainly in classification of the orbital angular momentum j is half-integral, infinitesimal operators are the ( j11)(2j11) tensor operators TqðkÞ of odd rank; used mainly in the jj-coupling scheme Special subgroup of R7; used mainly for classification of ATs formed of f-electrons Finite, formed of all permutations of a set (N!

5 e d f d e f (b) It is invariant by switching the upper and lower members of any two columns & ' & ' a b c d e c 5 5? ðA 1 B 2 c 2 nÞ! Š with ! ð2a 1 b 1 cÞ! 1=2 ða 1 b 1 c 1 1Þ! 4 Special Forms of the 6j-Symbols; Result 5 Num/Den & & & & & & & a b c d 0 e c c b b 1 a ' 5 (21)a 1 c 1 e [(2a 11)(2c 11)]21/2 δa,bδc,d ' c ðc 21Þ b b 1 a Num 5 ð21Þ2ða 1 b 1 cÞ ½aða 1 1Þ 2 bðb 1 1Þ 2 cðc 1 1ފ Den 5 2½bðb 1 1Þð2b 1 1Þcðc 1 1Þð2c 1 1ފ1=2 ' c ðb 21Þ ðc 21Þ b 1 a c ðb 22Þ ðc 22Þ b 2 a c ðb 21Þ ðc 22Þ b 2 a c ðc 22Þ b b 2 a ' ' ' ' Num 5 ð21Þ2ða 1 b 1 cÞ ½ð11 a 1 b 2 cÞða 2 b 1 cÞð2 a 1 b 1 cÞð1 1 a 1 b 1 cފ1=2 Den 5 2½bðb 1 1Þð2b 1 1Þcð2c 21Þð2c 1 1ފ1=2 Num 5 ð21Þ2ða 1 b 1 cÞ ½ð21 2 a 1 b 1 cÞð2a 1 b 1 cÞða 1 b 1 cÞð1 1 a 1 b 1 cފ1=2 Den 5 2½bð2b 21Þð2b 1 1Þcð2c 21Þð2c 1 1ފ1=2 Num 5 ð21Þ2ða 1 b 1 cÞ 3 ½ð23 2 a 1 b 1 cÞð22 2 a 1 b 1 cÞð21 2 a 1 b 1 cÞð2a 1 b 1 cފ1=2 3 ½ð22 1 a 1 b 1 cÞð21 1 a 1 b 1 cÞða 1 b 1 cÞð1 1 a 1 b 1 cފ1=2 Den 5 4½ðb 21Þbð2b 23Þð2b 21Þð2b 1 1ފ1=2 3 ½ðc 21Þcð2c 23Þð2c 21Þð2c 1 1ފ1=2 Num 5 ð21Þ2ða 1 b 1 cÞ 3 ½ð1 1 a 1 b 2 cÞða 2 b 1 cÞð22 2 a 1 b 1 cÞð21 2 a 1 b 1 cފ1=2 3 ½ð2 a 1 b 1 cÞð21 1 a 1 b 1 cÞða 1 b 1 cÞð11 a 1 b 1 cފ1=2 Den 5 2½2ðb 21Þbðb 1 1Þð2b 21Þð2b 1 1ފ1=2 3 ½ðc 21Þcð2c 23Þð2c 21Þð2c 1 1ފ1=2 Num 5 ð21Þ2ða 1 b 1 cÞ 3 ½3ð11 a 1 b 2 cÞð2 1 a 1 b 2 cÞð21 1 a 2 b 1 cÞða 2 b 1 cފ1=2 3 ½ð212 a 1 b 1 cÞð2a 1 b 1 cÞða 1 b 1 cÞð11 a 1 b 1 cފ1=2 Den 5 2½2bðb 1 1Þð2b 21Þð2b 1 1Þð2b 1 3ފ1=2 3 ½ðc 21Þcð2c 23Þð2c 21Þð2c 1 1ފ1=2 & & & ðc 21Þ b c ðb 21Þ c ðc 21Þ b b c c b b 2 a ' 2 a ' 2 a ' Num 5 ð21Þ2ða 1 b 1 cÞ 3 ½ð21 2 a 1 b 1 cÞð2a 1 b 1 cÞða 1 b 1 cÞð11 a 1 b 1 cފ1=2 3 ½ð1 1 aða 1 1Þ 2 b2 1 bc 2 c2 Š1=2 Den 5 ½2ðb 21Þbðb 1 1Þð2b 21Þð2b 1 1ފ1=2 3 ½2ðc 21Þcðc 1 1Þð2c 21Þð2c 1 1ފ1=2  Ã1=2 Num 5 ð21Þ2ða 1 b 1 cÞ 3=2 3 ½ð1 1 a 1 b 2 cÞða 2 b 1 cÞð2 a 1 b 1 cÞð1 1 a 1 b 1 cފ1=2 3 ½1 1 aða 1 1Þ 2 bðb 1 1Þ 2 c2 Š1=2 Den 5 ½bðb 1 1Þð2b 21Þð2b 1 1Þð2b 1 3ފ1=2 3 ½2ðc 21Þcðc 1 1Þð2c 21Þð2c 1 1ފ1=2 Num 5 ð21Þ2ðc 1 b 1 aÞ 2 3 f3½cðc 1 1Þ 1 bðb 1 1Þ 2 aða 1 1ފ 3 ½cðc 1 1Þ 1 bðb 1 1Þ 2 aða 1 1Þ 21Š 24cðc 1 1Þbðb 1 1Þg Den 5 ½ð2c 21Þð2cÞð2c 1 1Þð2c 1 2Þð2c 1 3ފ1=2 3 ½ð2b 21Þð2bÞð2b 1 1Þð2b 1 2Þð2b 1 3ފ1=2 60 A Handbook of Magnetochemical Formulae which can be expressed via 9j-symbols as h j1 j2 j12 j3 j4 j34 JðMÞj j1 j3 j13 j2 j4 j24 JðMÞi 8 > < j1 1=2 5 ½ð2j12 1 1Þð2j34 1 1Þð2j13 1 1Þð2j24 1 1ފ j3 > : j13 j2 j4 j24 9 j12 > = j34 > ; J ð2:17Þ A 9j-symbol can be expressed in terms of 3j-symbols 9 8 >    = X j < j11 j12 j13 > j21 j22 j23 j31 j32 j33 j12 j13 11 j21 j22 j23 5 > > m21 m22 m23 m31 m32 m33 ; all m m11 m12 m13 : j31 j32 j33     j11 j21 j31 j12 j22 j32 j13 j23 j33 3 m11 m21 m31 m12 m22 m32 m13 m23 m33 ð2:18Þ or 6j-symbols 9 8 & '& '& ' < j11 j12 j13 = X j12 j22 j32 j13 j23 j33 j j j j21 j22 j23 5 ð21Þ2j ð2j11Þ 11 21 31 j32 j33 j j21 j j23 j j11 j12 ; : j j31 j32 j33 ð2:19Þ In the last formula the index j runs over all the meaningful values for which the triangular conditions of 6j-symbols are satisfied.

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