Graphical Methods Of Spin Algebras Research Articles

Matrix elements of unitary group generators in many-fermion correlation problem. I. tensorial approaches

The objective of this series of papers is to survey important techniques for the evaluation of matrix elements (MEs) of unitary group generators and their products in the electronic Gel’fand–Tsetlin basis of two-column irreps of U(n) that are essential to the unitary group approach (UGA) to the many-electron correlation problem as handled by configuration interaction and coupled cluster approaches. Attention is also paid to the MEs of one-body spin-dependent operators and of their relationship to a standard spin-independent UGA formalism. The principal goal is to outline basic principles, concepts, and ideas without getting buried in technical details and thus to help an interested reader to follow the detailed developments in the original literature. In this first instalment we focus on tensorial techniques, particularly those designed specifically for UGA purposes, which exploit the spin-adapted tensorial analogues of the standard creation and annihilation operators of the ubiquitous second-quantization formalism. Subsequent instalments will address techniques based on the graphical methods of spin-algebras and on the Green–Gould polynomial formalism. In the “Appendix A” we then provide a succinct historical outline of the origins of the Lie group and Lie algebra concepts.

Matrix elements of unitary group generators in many-fermion correlation problem. II. Graphical methods of spin algebras

This second instalment continues our survey inter-relating various approaches to the evaluation of matrix elements (MEs) of the unitary group generators, their products, and spin-orbital U(2n) generators, in the basis of the electronic Gel’fand–Tsetlin (G–T) [or Gel’fand–Paldus (G–P) or P-] states spanning the carrier spaces of the two-column irreducible representations (irreps) of the unitary group approach (UGA) to the many-electron correlation problem. Exploiting an intimate relationship between the U(n), $${\mathcal {S}}_N$$ , and SU(2) groups we rely here on the SU(2) formalism within the confines of UGA rather than on an earlier approach relying on the permutation group $${\mathcal {S}}_N$$ . The key is a close relationship between the P-states of UGA and the Yamanouchi–Kotani (Y–K) states of SU(2) that differ only in a phase. This enables one to exploit the powerful graphical techniques of spin (or angular momentum) algebras as first developed by Jucys. We show the usefulness and power of these techniques in the problem of UGA ME evaluation which, moreover, leads automatically to their segmentation, first derived for single generators by Shavitt and for generator products by Boyle and Paldus. We also show how the same technique can be successfully employed for MEs of spin-dependent U(2n) generators in the UGA P-basis.

On the representation matrices for the symmetric group adapted to electron‐pair and electron‐group wave functions using graphical methods of spin algebras

AbstractUsing graphical methods of spin algebras, we present a direct method for the determination of the representation matrices of the symmetric group in a basis adapted to electron‐pair and electron‐group wave functions. In particular, we consider nonstandard representations adapted to $S_{n_1 } \times S_{n_2 } \times \ldots \times S_{n_\ell }$ where n1 + n2 + … + nℓ = n. The Serber coupling scheme based on S2 × S2 × …× S2 is a special case suited for electron‐pair wave functions. Our approach exploiting graphical methods of spin algebras is contrasted with the use of Young diagrams and tableaux, Yamanouchi symbols and branching diagrams. © 2012 Wiley Periodicals, Inc.

Coupled‐Cluster approaches with an approximate account of triply and quadruply excited clusters: Implementation of the orthogonally spin‐adapted CCD + ST(CCD), CCSD + T(CCSD), and ACPQ + ST(ACPQ) formalisms

AbstractThe orthogonally spin‐adapted (OSA) coupled‐pair (CCD) methods with an approximate account of triply and quadruply excited clusters are considered. We focus on the CCD + ST(CCD) perturbative estimate of the singly and triply excited clusters due to Raghavachari [J. Chem. Phys. 82, 4607 (1985)] and its ACPQ + ST(ACPQ) analog proposed by Paldus and Piecuch [Int. J. Quantum Chem. 42, 135 (1992)]. The latter approach combines the perturbative treatment of singles and triples with an approximate CCD theory corrected for connected quadruply excited clusters (ACPQ). We also consider the OSA version of the CCSD + T(CCSD) method (coupled‐cluster [CC] approach with singly and doubly excited clusters and noniterative perturbative account of triply excited clusters) introduced by Urban et al. {J. Chem. Phys. 83, 4041 (1985)}. The explicit OSA expressions for the previously neglected {P. Piecuch and J. Paldus, Theor. Chim. Acta 78, 65 (1990)} S(CCD) and S(ACPQ) terms are derived using diagrammatic methods of many‐body perturbation theory and graphical methods of spin algebras. The CCD + ST(CCD), CCSD + T(CCSD), and ACPQ + ST(ACPQ) formalisms have been implemented and the general purpose ab initio programs have been written using a newly developed procedure for improving the convergence of the reduced linear equation method {P. Piecuch and L. Adamowicz, J. Chem. Phys. 100, 5857 (1994)}. Results of the pilot calculations for few nondegenerate and quasi‐degenerate systems are presented and compared with the full configuration interaction data. © 1995 John Wiley & Sons, Inc.

Orthogonally spin-adapted state-universal coupled-cluster formalism: Implementation of the complete two-reference theory including cubic and quartic coupling terms

The complete orthogonally spin-adapted Hilbert-space (or state-universal) coupled-cluster (CC) theory involving singly and doubly excited clusters for a model space spanned by two closed-shell configurations is considered. Explicit expressions for the previously neglected cubic and quartic coupling terms are derived using diagrammatic methods of many-body perturbation theory and graphical methods of spin algebras. The resulting formalism has been implemented and the general purpose ab initio program has been written using newly developed procedure for improving the convergence of the reduced linear equation method. Results of the pilot calculations for the two lowest singlet states of the minimum basis set and double zeta plus polarization basis H4 models as well as for the CH2 molecule at equilibrium and displaced geometries are presented and compared with the available single-reference CC and configuration interaction data. They indicate negligible role of the cubic and quartic coupling terms, which justifies the validity of approximations considered so far.

Orthogonally spin-adapted multi-reference Hilbert space coupled-cluster formalism: diagrammatic formulation

The problem of spin-adaptation of the multi-reference (MR) coupled-cluster (CC) formalism, employing Jeziorski-Monkhorst ansatz, is addressed. The diagrammatic technique based on graphical methods of spin algebras is generalized to the MR case, so that both direct and coupling terms can be determined. Usefulness of this fully diagrammatic spin-adaptation approach is illustrated on a derivation of explicit expressions for the linear and bilinear coupling terms that are required in the special two-reference MR-CC theory involving singly and doubly excited states (MR-CCSD formalism). Results obtained with the diagrammatic approach are compared with those derived earlier using the algebraic technique and relative advantages of both procedures are compared.

Coupled cluster approaches with an approximate account of triexcitations and the optimized inner projection technique

A general orthogonally spin-adapted formalism for coupled cluster (CC) approaches, with an approximate account of triexcited configurations, and for optimized inner projection (OIP) technique is described. Modifying the linear part of the CC equations for pair clusters (CCD) we obtain the orthogonally spin-adapted, non-iterative version of the CCDT-1 method of Bartlett et al. [J. Chem. Phys. 80, 4371 (1984), 81, 5906 (1984), 82, 5761 (1985)]. Similar modification of an approximate coupled pair theory corrected for connected quadruply excited clusters (ACPQ) yields a new approach called ACPTQ. Both the CCDT-1 and ACPTQ methods can be formulated in terms of effective interaction matrix elements between the orthogonally spin-adapted biexcited singlet configurations. The same matrix elements also appear in the orthogonally spin-adapted form of the CCD + T(CCD) perturbative estimate of triply excited contributions due to Raghavachari [J. Chem. Phys. 82, 4607 (1985)] and Urban et al. [J. Chem. Phys. 83, 4041 (1985)], and in the OIP method when applied to the Pariser-Parr-Pople (PPP) model Hamiltonians. We use the diagrammatic approach based on the graphical methods of spin algebras to derive the explicit form of these interaction matrix elements. Finally, the relationship between different diagrammatic spin-adaptation procedures and their relative advantages are discussed in detail.

Orthogonally spin‐adapted coupled‐cluster equations involving singly and doubly excited clusters. Comparison of different procedures for spin‐adaptation

AbstractA particularly compact form of the orthogonally spin‐adapted coupled‐cluster equations involving all singly and doubly excited clusters is derived for the general case of a non‐Hartree–Fock closed‐shell reference determinant. The diagrammatic approach based on the graphical methods of spin algebras is applied. The relationship of different diagrammatic procedures for spin‐adaptation, employing both bare and spin‐adapted two‐electron interaction vertices, is discussed. A comparison with the results obtained with algebraic spin‐adaption approaches is also given.

Symmetry-adapted coupled-pair approach to the many-electron correlation problem. I.LS-adapted theory for closed-shell atoms

The orthogonally spin- and orbital-symmetry-adapted ($\mathrm{LS}$-coupled) coupled-pair many-electron theory and its extended version involving monoexcited and biexcited cluster coefficients is derived in a form suitable for application to closed-shell atomic systems. A diagrammatic approach, based on second quantization, the time-independent Wick theorem, and the graphical methods of spin algebras, is employed.

Unitary Group Approach to the Many-Electron Correlation Problem via Graphical Methods of Spin Algebras

The various existing approaches for the evaluation of matrix elements of unitary group generators and their products with respect to the basis of electronic Gelfand states or the corresponding Yamanouchi-Kotani states are interrelated, and their desirable features combined, yielding a direct algorithm for the evaluation of matrix elements of products of two generators and, consequently, a simple and efficient algorithm for the calculation of two-electron matrix elements of spin-independent Hamiltonians needed in the unitary group configuration interaction (shell model) approach. Moreover, this algorithm is compatible with the efficient generation and representation scheme for electronic Gelfand states based on the distinct row table concept. Diagrammatic techniques based on the time-independent Wick theorem and graphical methods of spin algebras are used to derive the required factors for both one and two-generator (or electron) matrix elements for three different phase conventions and several possible simplifications in the evaluation of the two-electron part of the Hamiltonian matrix are outlined.

Configuration interaction matrix elements. II. Graphical approach to the relationship between unitary group generators and permutations

AbstractThe explicit expressions for the matrix elements of unitary group generators between geminally antisymmetric spin‐adapted N‐electron configurations in terms of the orbital occupancies and spin factors, given as spin function matrix elements of appropriate orbital permutations, are derived using the many‐body time‐independent diagrammatic techniques. It is also shown how this approach can be conveniently combined with graphical methods of spin algebras to obtain explicit expressions for the spin factors, once a definite coupling scheme is chosen. This method yields explicit expressions for the orbital permutations defining the spin factors. However, if desired, the explicit determination of line‐up permutations can be avoided in this approach, since they are implicitly contained in the orbital diagrams. It also clearly indicates why the geminally antisymmetric spin functions have to be used when a simple formalism is desired.

Orthogonally-spin-adapted coupled-cluster theory for closed-shell systems including triexcited clusters

The orthogonally-spin-adapted form of the nonlinear system of equations for the extended coupled-pair many-electron theory, which involves the monoexcited, biexcited, and triexcited states and cluster components, is derived. A diagrammatic approach, based on second quantization, the time-independent Wick theorem, and the graphical methods of spin algebras, is employed. The advantages of using the orthogonally-spin-adapted states and cluster components, rather than the nonorthogonally-spin-adapted ones which have been previously employed, particularly in the triexcited case, are also discussed. It is also shown that the formulas for the direct configuration-interaction method can easily be obtained in compact form from the linear part of the system of extended coupled-pair equations.

Direct evaluation of elements of the representation matrices of the spin permutation group for transpositions

Graphical methods of spin algebras are used to evaluate elements of the representation matrices of the spin permutation group for arbitrary transpositions. The matrix elements are evaluated directly; no matrix multiplication is required. The method is discussed for the Young-Yamanouchi basis and the Jahn-Serber basis.

Application of graphical methods of spin algebras to limited CI approaches. II. A simple open shell case

AbstractA time independent diagrammatic technique based on the Wick theorem and graphical methods of spin algebras, as outlined in Part I, is applied to a simple open shell case having one unpaired electron in addition to the pure singlet closed shell. Compact explicit expressions for the matrix elements of a spin independent Hamiltonian between conveniently chosen spin symmetry adapted states are given for the ground, mono‐ and bi‐excited configurations.

Application of graphical methods of spin algebras to limited CI approaches. I. Closed shell case

AbstractThe time independent diagrammatic technique based on the mathematical methods of quantum electrodynamics (second quantization, Wick's theorem, Feynman‐like diagrams) is combined with graphical techniques of spin algebras to derive general expressions for the matrix elements of spin independent one‐ and two‐particle operators between spin symmetry adapted ground, mono‐ and bi‐excited configurations of a closed shell system. Two coupling schemes are considered for bi‐excited states and their relative merits are discussed. Finally, the results are used to derive compact expressions for the coupling coefficients of the direct configuration interaction from molecular integrals (CIMI) method.