Beta 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... 

This section describes several determinant-based Cl algorithms. The alpha and beta string formalism of Handy44 is introduced, and the equations for a — He within this formalism are derived for the full Cl and RAS Cl cases.46 Practical considerations for implementation are also discussed. 

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... 

Direct Cl methods often require an index vector which points to a list of all allowed excitations from a given iV-electron basis function. Using alpha and beta strings, the index vector need not be the length of the Cl vector—its size is dictated by the number of alpha or beta strings, which (for a full Cl) is approximately the square root of the number of determinants. This results from the fact that in determinant-based Cl, electrons in alpha spin-orbitals can be excited only to other alpha spin-orbitals, and electrons in beta spin-orbitals can be excited only to other beta spin-orbitals (because of the restriction to a single value of Ms). 

Given equation (108), it is possible to write the one-electron coupling coefficients in terms of alpha and beta strings ... 

S2 = sign associated with I2 loop over 7 = 1, number of beta strings e (7,7) = c(72,7) S2 end loop over 7 end loop over 7... 

In this section we discuss some of the practical issues relevant to the actual implementation of the determinant-based Cl algorithms. We also describe our experience with our own fully direct Cl program, DETCI, which is capable of evaluating any Cl wavefunction which can be formulated as a RAS Cl, subject to memory and disk limitations. This program is based in part on the alpha and beta string formalism of Handy44 and the algorithms of Olsen et al.46,83 It has been modified to allow more complex Cl spaces, as described in sections 4.8.3 and 5.4. 

This form of the Hamiltonian shows explicitly the couplings between wave functions with different Mk values and makes possible to factorize occupation vectors in alpha- and beta-strings like done in non-relativistic Cl theory. The difference with non-relativistic theory is that calculations are not restricted to one value of Mk. Applied without further approximation the formalism gives therefore no dramatic reduction in operation count over the symmetry-adapted unrestricted scheme described in the previous section. An advantage of the formalism is, however, that it facilitates incorporation of the relativistic scheme in non-relativistic Cl or MCSCF implementations [35] and that the scheme gives a natural subdivision of the full Cl matrix. 

Slater determinants as products of alpha and beta strings... 

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