Spin-adapted form

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). 

HyperChem uses single detenu in am rather than spin-adapted wave fn n ction s to form a basis set for th e wave Fin ciion sin a con -figuration interaction expansion. That is, HyperChem expands a Cl wave function, m a linear combination of single Slater deterniinants P,... 

The inherent spin-impurity problem is sometimes fixed by using the orbitals whieh are obtained in the UHF ealeulation to subsequently form a properly spin-adapted wavefunetion. For the above Li atom example, this amounts to forming a new wavefunetion (after the orbitals are obtained via the UHF proeess) using the teehniques detailed in Section 3 and Appendix G ... 

In order to get significant results, the initial data must be formed by a set of clearly non-A -representable second-order matrices, which would generate upon contraction a closely ensemble A -representable 1-RDM. It therefore seemed reasonable to choose as initial data the approximate 2-RDMs built by application of the independent pair model within the framework of the spin-adapted reduced Hamiltonian (SRH) theory [37 5]. This choice is adequate because these matrices, which are positive semidefinite, Hermitian, and antisymmetric with respect to the permutation of two row/column indices, are not A -representable, since the 2-HRDMs derived from them are not positive semidefinite. Moreover, the 1-RDMs derived from these 2-RDMs, although positive semidefinite, are neither ensemble A -representable nor 5-representable. That is, the correction of the N- and 5-representability defects of these sets of matrices (approximated 2-RDM, 2-HRDM, and 1-RDM) is a suitable test for the two purification procedures. Attention has been focused only on correcting the N- and 5-representability of the a S-block of these matrices, since the I-MZ purification procedure deals with a different decomposition of this block. 

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,... 

Eqs.(6) and (7) imply that the basis in the model space is formed by spin-adapted combinations of Slater determinants. This way of constructing the model space is specific for the unitary group approach to the theory of TV-electron systems [15, 16]. 

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... 

Laidig and Bartlett/li8(a)/ have implemented the theory in its linearized form. They applied it to the symmetric dissociating potential curve for the ground state of water and the symmetric abstraction from BeH2. Each T is truncated at the two-body level. The results were compared to the corresponding MR-CISD values. To our knowledge, a quadratic extension of the formalism has not been applied computationally. A spin-adapted version of the linearized model has also been recently formu1 ated/118[Pg.332]

When fhe lasf fwo indices are expanded from lower triangles to squares, symmetric and antisymmetric matrices are formed. The superscripts S and A denote these cases. Diagonal elements for the symmetric case have a different normalization factor. The rank of the spin-adapted H matrix for the closed-shell case is o- -v- -vo - -ov. ... 

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 ). 

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. ... 

Consider a system with one unpaired electron and its excited states of this type. Two examples are the sodium atom and states obtained by exciting the 3s electron into unoccupied orbitals and states of N2 obtained by removing one electron from N2. These can be studied in ACES II with several methods. These include the QRHF-CC schemes and the EA-EOM-CC and IP-EOM-CC schemes. One can conveniently study states of Na using EA-EOM-CC. One would use Na" " as the reference state and then obtain the energies of the states that can be formed by adding one electron. The states of N2 could be studied by the IP-EOM-CC method, starting from N2. Starting from closed-shell reference states, the doublet states obtained are spin adapted. 

Jeziorski et al, have formulated a first-quantization form of the CCD equations where the pair functions are not expressed in terms of double replacements but as expansions in Gaussian geminals, In the original derivation of the theory, they have employed a spin-adapted formulation in terms of singlet and triplet pairs, but a spin-orbital formalism will be used in the following for the sake of a compact presentation,... 

In the Hartree-Fock approximation, the electrc ic wave function is approximated by a single configuration of spin orbitals (i.e. by a single Slater determinant or by a single space- and spin-adapted CSF) and the energy is optimized with respect to variations of these spin orbitals. Thus, the wave function may be written in the form... 

The spin- and spatial- symmetry adapted N-eleetron funetions referred to as CSFs ean be formed from one or more Slater determinants. For example, to deseribe the singlet CSF eorresponding to the elosed-shell orbital oeeupaney, a single Slater determinant... 

Z (-1)) CSFs are, by no means, the true eleetronie eigenstates of the system they are simply spin and spatial angular momentum adapted antisymmetrie spin-orbital produets. In prineiple, the set of CSFs i of the same symmetry must be eombined to form the proper eleetronie eigenstates Fk of the system ... 

It is essential to realize that the energies (i H Oi> of the CSFs do n represent the energies of the true electronic states Ek the CSFs are simply spin- and spatial-symmetry adapted antisymmetric functions that form a basis in terms of which to expand the true electronic states. For R-values at which the CSF energies are separated widely, the true Ek are rather well approximated by individual (i H Oi> values such is the case near Rg. 

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