A Numerical Algorithm Technique for Deriving Russell-Saunders [R-S) Terms E. M. R. Kiremire University of Zambia, P.O. Box 32379, Lusaka, Zambia Table 1.
The R-S terms' are very important in the interpretation of atomic and electronic spectra. A number of methods for deriving R-S terms have been advanced (1-11). Of direct influence to this article is the method put forward by McDaniel (12) in which R-S terms of a and 0 spins are multiplied to generate new ones in the process of deriving R-S terms of a given configuration. This article presents yet another simple, versatile, and systematic numerical method of deriving the R-S terms. The "m, Combination" Method In this method, a numerical algorithm has been developed whereby all possible mr combinations for equivalent electrons giving rise to positive or zero Mr. values are generated. ~o&idckfirstthe microstates of zek) or positive MI, value that arise when all of the electrons have the same m , value. i.e., when all of the electrons have the same spin. First, you begin by finding the ml combination that bas the highest Mr value. This is a numerical seauence in which the first electron is assigned an ml value of 1, the next avalu6 of 1 1.the next 1 - 2.. etc... UD . to the limit of a ~ossible21 + 1 e~ectrohs.For a d2 configuration with all electrons of the same spin, you obtain the first ml set of (2,l) having an Mr. value of 3 (that is, 2 1). T o go to the next Mr. value, step down the smaller ml value by unit and again sum the mi values to obtain the Mr. value. In this case the new ml set will be (2,O) and the corresponding Mr. value 2 0 = 2. This is repeated until either an MLvalue of zero is reached or the mr value being decreased reaches the limitingvalue of -1. When these operations are carried out on the d2 configuration we obtain theentiremlsets (2,1), (2,0), (2,-l),and (2,-2) inthis first "cvcle". T o obtain the mr sets of the second cvcle, we decrease by one unit each of the two original mr vaiues and repeat the vrevious ~rocedure.again until either an ML value of zero is reached or the smailer mr value has reached the limit of -1. This gives us the m, sets of (1,O) and (1,-1). The next cycle beginswith an ml set of (0,-l), which gives us an MLvalue of -1, and this is ignored since we are considering only Mr. 2 0 values. Thus, we have exhausted all the possible mi combinations containing different mi values for the d2 system. These allowed ml sets are shown in Table 1.
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Extractlon of R-S Terms of the Hlghesi Spin Multlpliclty We have generated the ml combinations when the two d electrons have the same spin ( a a or @). Hence we can, a t this stage, extract from Table 1the R-S terms of the highest spin multiplicity, in this case triplet terms. The mi sets (2,-2), (2,-I), (2,0), and(2,l) span the M~valuesO,1,2,and 3. From these ML values we derive a 3F R-S term. Similarly, the mi combinations (1,-1) and (1,0) will give us a s p term. Since we have restricted the two electrons to the same spin (M, = +I), the actual number of microstates arising from this will be less than the number of microstates implied by the terms. For completion, we have t o utilize some of the
' R-S
States.
terms are also known as spectroscopic terms or atomic
The m, Sets ol dZYlelding the Triplet R-S Terms 1st cycle m, sets
ML
12.1)
3
F
2nd cycle m, sets
microstates from the "pseudo" singlet terms (explained helow). Extractlon of Singlet R-S Terms We vroceed to the derivation of sinelet R-S terms when the two electrons are paired up (i.e., having a and 0 spins). Using a modification of the procedure developed by McDaniel (fi),the electrons with and 0 spins can be considered separately (spin factoring). The a electron will span the ml sets (2), (l), and (0) corresponding to Mr. values of 2,1, and 0. The MLvalues yield a D partial term. Similarly from the 0 clrrtron we shail get another I ) partial term. To obtain the singlet I
