Spin-adapted Configurations

Table 1 Calculations on HF with DZ+P basis. The total energies are reported as -(E + 99) hartree. Dimension should be understood as number of determinants (number of spin-adapted configurations in italics). NPE (non-parallelity error) is the difference between the maximal and minimal deviation from FCI...

Thus, when we wish to employ a model space Mo that is spanned by M suitable configurations i), f = 1,2, , M (represented by either the Slater determinants or spin-adapted configurations), so that dimAlo = M, we have essentially two options how to proceed (for an overview, see, e.g.. Refs. [16,18,21,24]). 

Most of the formalism to be developed in the coming sections of these lecture notes will be independent of the specific definition of the configurational basis, in which we expand the wave function. We therefore do not have to be very explicit about the exact nature of the basis states hn>. They can be either Slater determinants or spin-adapted Configuration State Functions (CSF s). For a long time it was assumed that CSF s were to be preferred for MCSCF calculations, since it gives a much shorter Cl expansion. Efficient methods like GUGA had also been developed for the solution of the Cl problem. Recent... 

In order to be able to write out all the terms of the direct Cl equations explicitly, the Hamiltonian operator is needed in a form where the integrals appear. This is done using the language of second quantization, which has been reviewed in the mathematical lectures. Since, in the MR-CI method, we will generally work with spin-adapted configurations a particularly useful form of the Hamiltonian is obtained in terms of the generators of the unitary group. The Hamiltonian in terms of these operators is written,... 

Although the investigation of Segal and coworkers gives a first basis for the understanding of the ultraviolet absorption spectrum of 1, it leaves a number of open questions, which can only be answered by more extended calculations. Such calculations are possible and have been done for derivatives of 1 (see, e.g., the MR-CI calculations for fluoro- and methyl-cyclopropanone with more than 50,000 spin adapted configurations " ) but not for 1 itself. Therefore, further calculations are needed to get additional information on the excited states of 1. 

The excited-state wave functions are written as linear combinations of spin-adapted configurational functions... 

Based on the work of Thiel, such a semiempirical MR-CI was developed some time ago. This MNDOC-CI method may be characterized as follows. For each state under consideration, one or a few reference configurations are used, and from these reference configurations single and double excitations within a properly chosen active space (AS) are considered in constructing spin-adapted configuration state functions (CSFs). To specify the active space and the refer-... 

The spin-adapted configurational state functions (CSF s) y(l,...iV) may in general be expressed in terms of Slater determinants... 

This problem has been investigated at the at initio level by Borden et alP using two different wavefunctions a two-configuration wavefunction (TC-SCF) and a wavefunction consisting of 52 Ag spin-adapted configurations (MC-SCF). Calculations have been carried out using both STO-3G and 3-2IG basis sets. Because the MC-SCF computations with the 3-21G basis required too much computer memory, the calculations were simulated by using ALIS to find the optimal orbitals. These optimized orbitals have been used to perform a full six-electron, six-orbital Cl. 

Section 2.S discusses electron spin and spin operators in many-electron systems and contains a description of restricted and unrestricted spin orbitals and spin-adapted configurations. Spin-adapted configurations, unlike many single determinants derived from restricted spin orbitals, are correct eigenfunctions of the total electron spin operator. Singlet, doublet, and triplet spin-adapted configurations as well as unrestricted wave functions, which are not eigenfunctions of the total electron spin operator, are described. 

We have described the spin of a single electron by the two spin functions a(this section we will discuss spin in more detail and consider the spin states of many-electron systems. We will describe restricted Slater determinants that are formed from spin orbitals whose spatial parts are restricted to be the same for a and p spins (i.e., xi = il iP ), Restricted determinants, except in special cases, are not eigenfunctions of the total electron spin operator. However, by taking appropriate linear combinations of such determinants we can form spin-adapted configurations which are proper eigenfunctions. Finally, we will describe unrestricted determinants, which are formed from spin orbitals that have different spatial parts for different spins (i.e., fjS ). 

Hamiltonian between, say, singlet and triplet spin-adapted configurations is zero. 

Let us generalize the above results for minimal basis H2. In Chapters 4 and 5 we use singlet spin-adapted configurations that arise as a result of single and double excitations from a closed-shell Hartree-Fock ground state,... 

The procedure for finding the appropriate linear combinations of singly and doubly excited determinants to form spin-adapted configurations is beyond the scope of this book we shall merely quote the results. A variety of methods are available for constructing spin eigenfunctions. An authoritative and clear description of many of these methods has been given by Paunz. ... 

The singlet spin-adapted configuration corresponding to the single excitation in which an electron has been promoted from spatial orbital to spatial orbital [1/ is... 

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