Spin adaptation operators

While previous variational 2-RDM calculations for electronic systems have employed the above formulation [20-31], the size of the largest block diagonal matrices in the 2-RDMs may be further reduced by using spin-adapted operators Ci in Eq. (9). Spin-adapted operators are defined to satisfy the following mathematical relations [54, 55] ... 

These operators satisfy the formal definition for spin-adapted operators in Eqs. (79) and (78). Inserting these four operators into Eq. (8), we can generate four... 

In summary, proper spin eigenfunetions must be eonstmeted from antisymmetrie (i.e., determinental) wavefunetions as demonstrated above beeause the total and total Sz remain valid symmetry operators for many-eleetron systems. Doing so results in the spin-adapted wavefunetions being expressed as eombinations of determinants with eoeffieients determined via spin angular momentum teehniques as demonstrated above. In... 

The Spin adapted Reduced Hamiltonian SRH) is the contraetion to a p-electron space of the matrix representation of the Hamiltonian Operator, 2 , in the N-electron space for a given Spin Symmetry [17,18,25,28], The basis for the matrix representation are the eigenfunctions of the operator. The block matrix which is contracted is that which corresponds to the spin symmetry selected In this way, the spin adaptation of the contracted matrix is insnred. 

To generate the spin-adapted matrix, we spin-adapt the products of one creation operator and one annihilation operator ... 

One particular advantage of Slater determinants constructed from orthonormsd spin-orbitals is that matrix elements between determinants over operators such as H sure very simple. Only three distinct cases arise, as is well known and treated elsewhere. It is perhaps not surprising that the simplest matrix element formulas should be obtained from the treatment that exploits symmetry the least, as only the fermion antisymmetry has been accounted for in the determinants. As more symmetry is introduced, the formulas become more complicated. On the other hand, the symmetry reduces the dimension of the problem more and more, because selection rules eliminate more terms. We consider here the spin adaptation of Slater determinants. 

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,... 

The eigenvalue problem of the Hamiltonian operator (1) is defined in an infinite-dimensional Hilbert space Q and may be solved directly only for very few simple models. In order to find its bound-state solutions with energies not too distant from the ground-state it is reduced to the corresponding eigenvalue problem of a matrix representing H in a properly constructed finite-dimensional model space, a subspace of Q. Usually the model space is chosen to be spanned by TV-electron antisymmetrized and spin-adapted products of orthonormal spinorbitals. In such a case it is known as the full configuration interaction (FCI) space [8, 15]. The model space Hk N, K, S, M) may be defined as the antisymmetric part of the TV-fold tensorial product of a one-electron space... 

Extensive numerical investigations of this formalism were undertaken by Nakatsuji/52/ and Hirao/53/ for IP and EE computations. For IP calculations, the operator manifold taken by them were W and W2, and product excitations of the form W T2 were also included. A similar approximation scheme for EE was also used, although all the spin-adapted W operators for the triplet EE calculations were not included. This should be contrasted with the scheme of... 

Although a spin-orbital formulation is conceptually simple, desirable properties such as spin-adaptation may be lost when the electronic state of interest is open shell, for example. A rigorously spin-adapted theory must include spin-free definitions of the cluster operators, T, and an appropriate (perhaps multideterminant) reference wavefunction (Refs. 39, 41, 42, 156-158). Such general coupled cluster derivations are beyond the scope of this chapter, though some of the issues associated with difficult open-shell problems are discussed in the next section. 

We find that it is convenient to work with the spin-adapted form of the coupled-cluster doubles (CCD) equations. The spin-adapted double excitation operators S (i),i = 1,2, are given, for example, in Oddershede et al. (1984, Appendix C). 

For closed-shell reference states, spin adaptation of the operator manifolds reduces the rank of H. Amplitudes for doublet final states have the following... 

In the spin-representation the two, three and four electron functions of the basis are simple products of fermion operators. Therefore, the upper limit of their occupation number is one. This upper bound value has also been adopted for the elements in a spin-adapted basis of representation. We know also that the diagonal elements must be positive. Finally, we know the value not only of the trace but also of the partial traces of the spin-adapted matrices (18, 19, 20). 

The general problem of spin-adaptation using multiple vacuua depending upon the model function the component of the wave operator exp(T ) acts upon, is a nontrivial and rather involved exercise. Here we will consider the simplest yet physically the most natural tmncation scheme in the rank of cluster operators T, where each such operator is truncated at the excitation rank two. For generating the working equations for the spin-adapted theory in this case, it is useful to classify the various types of excitation operators leading to various virtual CSFs as ... 

To derive the spin-adapted SS-MRCC equations in the SD truncation scheme of the cluster operators, we rewrite the Schrodinger equation for ip as follows ... 

---