University of Chicago Laboratory of Molecular Structure and Spectra Technical Report: 1952-1953, Part 2 Page: 185
xxiv, 359 p. : diagrams, graphs, ill. ; 28 cm.View a full description of this report.
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TWO-CENTER INTEGRALS. III
index sum y+6+2e; such a group is then conveniently subdivided according to the dif-
ferent values of e. In order to calculate C-functions with the same upper index sum,
say n, from each other, we need also C-functions of the group n-1 if Eq. (3.15) is
used, and n-2 if Eq. (3.17) or (3.18) is used. Finally, it should be noted that
parallel arrows in Fig. 1 represent the same equation; this also applies to arrows
which are not drawn in Fig. 1, e.g.,an arrow from 310 to 111 would represent Eq.
(3.17). Another set of parallel arrows which is not shown at all in Fig. 1 is a set
of vertical arrows representing Eq. (3.16), e.g., from 310 to 201.
In the last analysis the methods described for raising the upper indices amount
of course to making the decomposition
(1+(~)7(1-(n)6 (2-1)(1-n2)E =i aiJ7((Eri))i( -)j ,
where the coefficients aij are constants. But the described systematic arrangement
of successive steps appears more convenient.
d. The Total Recurrence Procedure
From the foregoing it appears that two units of the lower index sum a+P are con-
sumed in order to raise the upper index sum y+6+2E by one unit. Now from Tables VII,
IX,and XI it is seen that the maximum value of the upper index sum is 4, and that this
occurs for the index pairs (a,B) = (0,1) and (a,P) = (1,1). Furthermore, it is seen
that the maximum value of "lower sum plus two times upper sum" is 10. For the cal-
culation of the needed C-functions we therefore used the following scheme (see Fig. 2).
First the functions C 000 (i.e.,upper sum = O) were calculated by means of Eq. (3.7)
for all index pairs a,B which are entered in Fig. 2. Then, in subsequent steps, the
C-functions with upper sum = 1, 2, 3, 4 were calculated by means of Eqs. (3.11-19) for
all the index pairs a,P situated above and to the left of the lines marked 1, 2, 3, 4,
respectively, in Fig. 2. In this manner explicit formulas were calculated for the
260 C-functions with a > 0 which are listed in Table XIIa. They were needed in order
to establish the 88 among them which appear in Tables VII, IX, and XI. The explicit
formulas for these 88 are given in Table XIII. Formulas for the 172 others can be
made available upon demand.
e. The Limiting Cases p = 0 and 7 = 0
For p = 0 and 7 = 0 the formulas of Table XIII cannot be employed directly for
numerical computations. Tables XIV and XV give the formulas for the functions Co6
for these two limiting cases. All these formulas were calculated in two ways. The185
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University of Chicago. Laboratory of Molecular Structure and Spectra. University of Chicago Laboratory of Molecular Structure and Spectra Technical Report: 1952-1953, Part 2, report, 1953~; Chicago, Illinois. (https://digital.library.unt.edu/ark:/67531/metadc228353/m1/77/: accessed July 16, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.