Graphical Representation of Alpha and Beta Strings

where

There are several different methods for assigning the arc weights by
which we calculate the index of a string according to equation
(6.6).
Under the *lexical ordering* scheme, the tail
of an alpha string graph
is assigned a weight *x*=1.
Other vertex weights are computed according to the recursive formula

(tex2html_deferredtex2html_deferred6.tex2html_deferred7x(e, o) = x(e+1, o+1) + x(e, o+1) ) |

Using lexical ordering, typically all arc weights

(tex2html_deferredtex2html_deferred6.tex2html_deferred8Y_1(e,o) = x(e+1, o+1) + x(e+1, o) + + x(e+1, e+1) ) |

Figure 1a features vertex and arc weights computed in this manner. A result of the lexical ordering scheme is that paths with a fixed upper part and an arbitrary lower part have contiguous indices. The particular choice of

tex2html_{d}eferredtex2html_{d}eferred6.tex2html_{d}eferred9Y_{1}(e,o) |
= | 0 | (tex2html_deferredtex2html_deferred6.tex2html_deferred9Y_1(e,o) & = & 0 ) |

tex2html_{d}eferredtex2html_{d}eferred6.tex2html_{d}eferred10
Y_{0}(e,o) |
= | x(e+1, o+1) |
(tex2html_deferredtex2html_deferred6.tex2html_deferred10 Y_0(e,o) & = & x(e+1, o+1) ) |

as illustrated in Figure 1b. Any walk has the same index in Figures 1a and 1b. For instance, the walk has an index of 5+4+3+2+2=16 (equation 6.6) from Figure 1a, and an index of 15+1=16 from Figure 1b.

In the so-called ``reverse-lexical'' ordering
scheme, all upper paths
for a fixed lower path have contiguous indices.
Vertex weights are now determined as

(tex2html_deferredtex2html_deferred6.tex2html_deferred11x(e, o) = x(e, o-1) + x(e-1, o-1) ) |

where the overbar indicates reversed-lexical ordering. Figure 2a depicts a reversed-lexical graph with all non-occupied orbital arcs set to zero. The occupied orbital arcs are computed as

(tex2html_deferredtex2html_deferred6.tex2html_deferred12Y_1 (e, o) = x (e, o) ) |

Figure 2b is the same except that now all occupied arcs have weights of zero. The non-occupied arc weights are

(tex2html_deferredtex2html_deferred6.tex2html_deferred13Y_0 (e, o) = x (e, o) + x (e + 1, o + 1) + + x (N - 1, o + N - e - 1) ) |

Note that string indices for reverse-lexical ordering are not necessarily the same as indices for lexical ordering. For the string considered previously, the index is calculated as 1+1+1+1+6=10 from Figure 2a, or as 5+5=10 from Figure 2b.

The arc weights given in Figures 1 and
2 cause the rightmost path to have an
index
*I*(*R*_{m}) = 0. If we change the arc weights
so that the leftmost path has index
*I*(*L*_{m}) = 0, we obtain
four more addressing schemes. The two simplest schemes for
*I*(*L*_{m})=0 are

tex2html_{d}eferredtex2html_{d}eferred6.tex2html_{d}eferred14Y_{0}(e, o) = 0 |
Y_{1}(e, o) = x(e, o + 1) |
(tex2html_deferredtex2html_deferred6.tex2html_deferred14Y_0(e, o) = 0 & Y_1(e, o) = x(e, o + 1) ) | |

(tex2html_deferredtex2html_deferred6.tex2html_deferred15 Y_1(e, o) = 0 & Y_0(e, o) = x(e - 1, o) ) |

where the overbars indicate that reversed-lexical vertex weights have been used. Alpha strings for 5 electrons in 7 orbitals employing these addressing schemes are depicted in Figure 3.

If we add another coordinate to each vertex, we can extend these simple digraphs to include point-group symmetry.