Alpha strings

The occupation number vectors can be written as a product of alpha creation operators, an alpha string, times a product of beta creation operators, a beta string... 

A 1980 paper by Handy44 represented a major advance in determinant-based Cl, even though the paper was more concerned with how integrals and Cl coefficients are stored than with the computational advantages of determinants over CSFs. Handy realized that if determinants are used as IV-electron basis functions, and particularly if these determinants are expressed as alpha strings and beta strings, then the vector a can be evaluated very efficiently. 

Although Handy was the first to use alpha and beta strings, we will employ the subsequent notation of Olsen et al.46 An alpha string is defined as an ordered product of creation operators for spin orbitals with alpha spin. If Ia contains a list, j,... A of the Na occupied spin orbitals with alpha spin in determinant 7), then the alpha string a(Ia) is ajQaja... aj.a. A beta string is defined similarly. Thus a Slater determinant /) in terms of alpha and beta strings is... 

Figure 17 Alpha string graph for na = 5, n = 7. Vertex weights are determined according to lexical ordering, and arc weights are given so that the rightmost path has index zero, (a) All unoccupied arc weights Y0(e,o) are zero, (b) All occupied arc weights Vj (e, o) are zero.

Figure 18 Alpha string graph for na = 5, n = 7. Vertex weights are determined according to reverse-lexical ordering, and arc weights are given so that...

For Na alpha electrons distributed among n orbitals, the number of alpha strings (i.e. the number of rows in C) is given by the binomial coefficient... 

Another complication in RAS theory is the presence of blocks such as C that contain a large number of beta strings but few alpha strings - in our case one. For such blocks, the MOC algorithm of Section 11.8.4 becomes inefficient. Consider the contribution to fi-om C that involves two-electron integrals with four virtual indices... 

For this procedure to be efficient, each element must be used with many alpha strings, reducing the evaluation of (11.8.50) to a fraction of the total w k. A fith < ily one alpha string as above, the evaluation of G becomes the time-consuming step. 

The operators corresponding to the doubly occupied orbitals can be moved out in front of the other creation operators. The determinant then appears as a product of three strings - a core string, an alpha string and a beta string ... 

The effect of the lowering operator on the alpha string is to generate a linear combination of n strings, each obtained by lowering the spin projection of one cM bital in the alpha string. 

---