Primitive Microstates and Russell-Saunders Terms for Multielectron Atoms Several analytical procedures are available for deriving energy levels for atoms containing equivalent electrons (I6). In this note we present a numerical scheme for enumerating all primitive microstates (PM's) and Russell-Saunders (R-S) terms arising from an equivalent electron configuration.' We designate a PM by an integer containing 21 + 1 digits, where 1 is the angular momentum associated with individual electrons in a given orbital, i.e., 1 = 2 for d-electrons. Thus a PM arising from a (nd)Nconfiguration will be represented by a 5 digit integer. The positions of these digits in the integer represent the possible values of mi. The rightmost dipit corresponds to rnr = 1, and the leftmost digit to mi = -1. The intermediate digits correspond to ml values lying between 1 and -I. Now, for a given value of ml the number 2 will he used t o represent an electron with rn, = 'h, 3 for an electron with m, = -'h, and 5 for two paired electrons with hL = Thus the six possible PM's arising from (np)' configuration will be represented by 002,020,200,003,030, and 300. For the sake of arithmetical manipulations the PM M)2 will be represented by 2, the PM 020 by 20, ete. Similarly, the PM's arising from the (nd)' configurations are denoted by 2, 20,200,2000, 20000,3,30,300, 3000,30000. The fundamental rules (4, 5 ) for constructing PM's now become equivalent t o the following working steps in our numerical scheme. Step I. Write the numerical representation for 1-electron states directly following the above notation. Step 2. To form 2-electron PM's arrange the l-electron states along a raw and also along a column. Now form a diagonal matrix by adding each numher on the row to every number on a column. Discard the diagonal numbers which correspond to states not allowed by the Pauli ~ r i n e i p l eThe . ~ remaining off diagonal numbers (either above or below the diagonal) represent all possible distinct 2-electron PM's: Step 3. T o form PM's for configurations with more than two electrons the following procedure should be used. (1) Write all 1-electron states in a row (call it R). (2) Write all ( N 1)-electronstates in a column (call it C ) . (3) Form an array ofnumbers by edding the first element of R toall elements of C. (4) Eliminate all numbers containing a 4 , 6 , 7 , or 8. These correspond t o states not allowed by the Pauli p r i n ~ i p l e(5) .~ Repeat (3) and (4) above with the second, third, etc., elements of R. After completing (3) and (4) with a given element of R make sure to (a) eliminate any state that has been generated twice, and (bl count to see if all PM's have been generated. The number of possible PM's arising from a general configuration, (nON, is equal t o (41 + 2)!lN!(41 + 2 - N)!. Step 4. Repeat step 3 for N larger than 3. The R-S terms may now he derived in the usual manner from the collection of Mr. and Ms values obtained from the PM's (7). The scheme presented above is particularly well suited for implementation in a computer. Far use as a teaching aid, we have written an APL program which uses this seheme t o generate all PM's and R-S terms for an equivalent electron configuration. Copies of a detailed write-up, illustrative tables, computer program, and sample output are available from the author. Thanks are due t o Lee Bryant and Raymond Coco of our computer center for several helpful discussions regarding APL programming.
Literature Cited 111 Hyde, K. E., J. CHEM. EDUC.52.87 (19751. i2i Carman, M..J.CHEM. EDUC..SO. 189119731. Tuttlo, E. R..Arner. J. Phyr.. 35. 26119671.
I:Il 141 151 16) 17)
Pading, L.. and Wilson. E. B., "lnVodurtion toQuantum Mechanie8,"McCraw-Hill Book Co., lnc.. New York. 1935, pp. 230-235. b r i n g , H., Wa1ter.J.. and Kimball. G. E.. "Quantum Chemi~try."JohnWilry &Sans. Ine., New York, 19U. pp. 156162. Curl. R. F.,Jr., and Kilpafriek,J. E., Arne?. J. Phys.. 28.357 118601. Hanna. M. W.."Quantum Mechanics in Chemistry,"2nd ed., W. A. Benjamin. Inc.,NawYork. 1969. pp. 152.159.
'For terminology used in the generation of energy levels for multielectron atoms, see ref. (I), Table 1. It should be noted that a microstate. level. or a term must be an eieenfunction of the total orbital aneular momentum and the soin n n n ~ u l a rnnmcntum operaturs In gen~ral,all of the stale* generated 1 9 Hyde's pnredure, ur 1)st h e procedure giwn here. i l w nut cizenimmims iui rhrse rwc, .,pvrntc,rz. Wr., t h e m f o r e , rofw to these stater as the pnntiriw rntcrocfofr, 1 1 ' 8 1 ' ~T. h e PN'v can he used 10conslrucl propcv mirmctnws o f a syrrem. Our PM'h rorres~undl o the D.fnnrt,uns of ret: (5). 'The use of the number 2, t o represent an electron with a-spin (f), and 3, to represent an electron with 8-spin (i), is very convenient although somewhat arbitrary. In this notation, for electrons with the same value of n, 1, and rnl, the number of 4 (double sum: 2 + 2) represents the aa-arrangement of two electrons; the number 5 (double sum: 2 3) represents the a@-arrangement;the number 6 (double sum: 3 + 3) represents the 013-arrangement; the numher 7 (triple sum: 2 + 2 + 3) represents the aaB-arrangement of three electrons; and the number 8 (triple sum: 2 3 3) represents the rrj70-arrangement. Clearly, the PM's represented by integers containing the digits 4, 6, 7, or 8 are not allowed by the Pauli principle. It should be notedthat any other pair of numbers, such as 3 and 4, whose double and triple sums are unique would be equally satisfactory to represent electrons with a and 8 spins.
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SUNY at Geneseo Geneseo. 14454
Bhairav D. Joshi
Volume 53,Number 4, ADril 1976 / 245
