Spin-adaptation

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,... 

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... 

It has been demonstrated that a given eleetronie eonfiguration ean yield several spaee- and spin- adapted determinental wavefunetions sueh funetions are referred to as eonfiguration state funetions (CSFs). These CSF wavefunetions are not the exaet eigenfunetions of the many-eleetron Hamiltonian, H they are simply funetions whieh possess the spaee, spin, and permutational symmetry of the exaet eigenstates. As sueh, they eomprise an aeeeptable set of funetions to use in, for example, a linear variational treatment of the true states. 

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 ... 

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. 

The most commonly used model space in quantum chemistry is the so-called full configuration interaction (FCI) space. It is the antisymmetric and spin-adapted N-fold tensorial power, of the 2A -dimensional spin-orbital one-electron space V. The... 

For the sake of practical simplicity (the use of Slater rules and a two-slope space geometry [2]) one may employ the larger, just antisymmetric but non-spin-adapted space spanned by ( ) Slater determinants. 

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...

Strength of positivity conditions Spin and spatial symmetry adaptation 1. Spin adaptation and S-representabiUty Open-shell molecules... 

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] ... 

By particle-hole duality, the same block structure appears in the spin-adapted two-electron RDM. The four blocks of the 2-RDM have the following traces [57] ... 

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

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... 

Similar to spin adaptation each 2-RDM spin block may further be divided upon considering the spatial symmetry of the basis functions. Here we assume that the 2-RDM has already been spin-adapted and consider only the spatial symmetry of the basis function for the 2-RDM. Denoting the irreducible representation of orbital i as T, the 2-RDM matrix elements are given by... 

To illustrate the advantage of spin- and spatial-symmetry adaptation, consider the BH molecule in a minimal basis set. If only is considered, the largest block of the two-electron RDM (i.e., is of dimension 36. Spin adaptation divides into two blocks, with sizes 15 and 21,... 

Using the definition of the spin-adapted 2-RDMs, however, we have the following relation between the a/1 block of the 2-RDM and the spin-adapted 2-RDMs ... 

E. Perez-Romero, E. M. Tel, and C. Valdemoro, Traces of spin-adapted reduced density matrices. Int. J. Quantum Chem. 61, 55 (1997). 

The main difference between the spin-adapted 2-CSE and the nonadapted one is that in the rhs of the spin-adapted 2-CSE, the 2-RDM only appears in the ajS block. 

C. Valdemoro, Theory and practice of the spin adapted reduced Hamiltonian, in Density Matrices and Density Functionals (R. Erdahl and V. Smith, eds.). Proceedings of the A. J. Coleman Symposium, Kingston, Ontario, 1985, Reidel, Dordrecht, 1987. 

C. Valdemoro, Spin-adapted reduced Hamiltonian. 1 Elementary excitations. Phys. Rev. A 31, 2114 (1985). 

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. 

As can be seen, the 2-CSE depends not only on the 2-RDM but also on the 3- and 4-RDMs. This fact lies at the root of the indeterminacy of this equation [63, 107]. As already mentioned, in the method proposed by Colmenero and Valdemoro [46 8] and in those further proposed by Nakatsuji and Yasuda [49, 51] and by Mazziotti [52, 111], a set of algorithms for approximating the higher-order ROMs in terms of the lower-order ones [46, 47, 108] allows this equation to be solved iteratively until converging to a self-consistent solution. In the approach considered in this work, the spin-adapted 2-CSE has been used. This equation is obtained by coupling the 2-CSE with the second-order contracted spin equation [50]. 

Once the 3- and 4-RDMs are evaluated, these trial matrices, jointly with the 2-RDM obtained by contracting them, are replaced in the rhs of the spin-adapted 2-CSE coupled equations, which become a matrix represented in the two-electron space, Raa - aa, such that... 

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