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International Journal of Quantum Chemistry
Matrix‐covariant representation of high‐order configuration interaction and coupled cluster theories Anatoliy V. Luzanov
Published Nov 12, 2007 • 10 citations
Abstract Abstract We present the closed form of the reduced density matrices (RDMs) of arbitrary order for configuration interaction (CI) wave functions at any excitation level, up to the full CI. A special operator technique due to Bogoliubov is applied and extended. It focuses on constructions of matrix‐covariant expressions independent of the basis set used. The corresponding variational CI equations are given in an explicit form containing the matrices related to conventional excitation operators. A subsequent transformation of the latter to an irreducible form makes it possible to generate the matrix‐covariant representation for coupled cluster (CC) models. Here this transformation is performed for a simplified high‐order CC scheme somewhat reminiscent of the quadratic CI model. A generalized spin‐flip approximation closely related to high‐order CI and CC models is presented, stressing on a possible inclusion of nondynamical and dynamical correlation effects for multiple bond breaking. A derivation of the full CI and simple CC models for systems involving effective three‐electron interactions is also given, thereby demonstrating the capability of the proposed method to deal with complicated many‐body problem. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008
DOI 10.1002/qua.21551 Related papers Unitary group approach to spin‐adapted open‐shell coupled cluster theory
Bogumił Jeziorski, Josef Paldus, Piotr Jankowski
International Journal of Quantum Chemistry • Nov 5, 1995• 79 citations
Theory of strongly correlated electron systems: Hubbard–Anderson models from an exact Hamiltonian, and perturbation theory near the atomic limit within a nonorthogonal basis set
I. Sandalov, Börje Johansson, Olle Eriksson
arXiv:cond-mat/0011259 • International Journal of Quantum Chemistry • Jan 1, 2003• 28 citations
Spin‐free quantum chemistry: What one can gain from fock's cyclic symmetry
Anatoliy V. Luzanov
International Journal of Quantum Chemistry • Dec 7, 2010• 13 citations
Abstract We present the closed form of the reduced density matrices (RDMs) of arbitrary order for configuration interaction (CI) wave functions at any excitation level, up to the full CI. A special operator technique due to Bogoliubov is applied and extended. It focuses on constructions of matrix‐covariant expressions independent of the basis set used. The corresponding variational CI equations are given in an explicit form containing the matrices related to conventional excitation operators. A subsequent transformation of the latter to an irreducible form makes it possible to generate the matrix‐covariant representation for coupled cluster (CC) models. Here this transformation is performed for a simplified high‐order CC scheme somewhat reminiscent of the quadratic CI model. A generalized spin‐flip approximation closely related to high‐order CI and CC models is presented, stressing on a possible inclusion of nondynamical and dynamical correlation effects for multiple bond breaking. A derivation of the full CI and simple CC models for systems involving effective three‐electron interactions is also given, thereby demonstrating the capability of the proposed method to deal with complicated many‐body problem. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008
Abstract We show that the irreducible tensor operators of the unitary group provide a natural operator basis for the exponential Ansatz which preserves the spin symmetry of the reference state, requires a minimal number of independent cluster amplitudes for each substitution order, and guarantees the invariance of the correlation energy under unitary transformations of core, open‐shell, and virtual orbitals. When acting on the closed‐shell reference state with nc doubly occupied and nv unoccupied (virtual) orbitals, the irreducible tensor operators of the group U(nc) ⊗ U(nV) generate all Gelfand‐Tsetlin (GT) states corresponding to appropriate irreducible representation of U(nc + nv). The tensor operators generating the M‐tuply excited states are easily constructed by symmetrizing products of M unitary group generators with the Wigner operators of the symmetric group SM. This provides an alternative to the Nagel‐Moshinsky construction of the GT basis. Since the corresponding cluster amplitudes, which are also U(nc) ⊗ U(ns) tensors, can be shown to be connected, the irreducible tensor operators of U(nc) ⊗ U(nv) represent a convenient basis for a spin‐adapted full coupled cluster calculation for closed‐shell systems. For a high‐spin reference determinant with n, singly occupied open‐shell orbitals, the corresponding representation of U(n), n=nc + nv + ns is not simply reducible under the group U(nc) ⊗ U(ns) ⊗ U(nv). The multiplicity problem is resolved using the group chain U(n) ⊃ U(nc + nv) ⊗ U(ns) ⊃ U(nc) ⊗U(ns)⊗ U(nv) ⊗ U(nv). The labeling of the resulting configuration‐state functions (which, in general, are not GT states when nc > 1) by the irreducible representations of the intermediate group U(nc + nv) ⊗U(ns) turns out to be equivalent to the classification based on the order of interaction with the reference state. The irreducible tensor operators defined by the above chain and corresponding to single, double, and triple substitutions from the first‐, second‐, and third‐order interacting spaces are explicitly constructed from the U(n) generators. The connectedness of the corresponding cluster amplitudes and, consequently, the size extensivity of the resulting spin‐adapted open‐shell coupled cluster theory are proved using group theoretical arguments. The perturbation expansion of the resulting coupled cluster equations leads to an explicitly connected form of the spin‐restricted open‐shell many‐body perturbation theory. Approximation schemes leading to manageable computational procedures are proposed and their relation to perturbation theory is discussed. © 1995 John Wiley & Sons, Inc.
The viability of treating higher excitations in coupled-cluster theory is discussed. An algorithm is presented for solving coupled-cluster (CC) equations which can handle any excitation. Our method combines the formalism of diagrammatic many-body perturbation theory and string-based configuration interaction (CI). CC equations are explicitly put down in terms of antisymmetrized diagrams and a general method is proposed for the factorization of the corresponding algebraic expressions. Contractions between cluster amplitudes and intermediates are evaluated by a string-based algorithm. In contrast to our previous developments [J. Chem. Phys. 113, 1359 (2000)] the operation count of this new method scales roughly as the (2n+2)nd power of the basis set size where n is the highest excitation in the cluster operator. As a by-product we get a completely new CI formalism which is effective for solving both truncated and full CI problems. Generalization for approximate CC models as well as multireference cases is also discussed.
Abstract The theory of correlated electron systems is formulated in a form that allows the use as a reference point a density functional theory in the local density approximation (LDA DFT) for solids or molecules. The theory is constructed in two steps. As a first step, the total Hamiltonian is transformed into a correlated form. To elucidate the microscopic origin of the parameters of the periodic Hubbard–Anderson model (PHAM) the terms of the full Hamiltonian that have the operator structure of PHAM are separated. It is found that the matrix element of mixing interaction includes ion configuration‐ and number of particles‐dependent contributions from the Coulomb interaction. In a second step the diagram technique (DT) is developed by means of generalization of the Baym–Kadanoff method for correlated systems. The advantages of the method are that: (1) A nonorthogonal basis can be used, in particular the one generated by LDA DFT; (2) the equations for Green's functions (GFs) for the Fermi and Bose types of quasiparticles can be formulated in the form of a closed system of functional equations. The latter allows us to avoid the question of the nonunique decoupling procedure existing in previous versions of the DT and perform the expansion in terms of dressed GFs. Although the expressions for all interactions depend on the overlap matrix, it is shown that the theory is formally equivalent to one with orthogonal states with redefined interactions. When the PHAM is treated from the atomic‐limit side the vertexes are generated by kinematic interactions. The latter arise due to nontrivial commutation relations between X‐operators and come from the mixing, hopping, and overlap of states. The equations for GFs are derived within the nonorthogonal basis set in Hubbard‐I, one‐loop and random‐phase approximations with respect to kinematic interactions. The self‐consistent equation for “Hubbard Us” is derived. The technique developed is general, in particular its “bosonic” part can be used for description of spin systems with arbitrary anisotropy, systems with orbital ordering, or ordering of multipoles. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 94: 113–143, 2003
Abstract The advantages and possibilities of a unitary coupled‐cluster (CC) theory are examined. It is shown that using a unitary parameterization of the wave function guarantees agreement between a sum‐over‐states polarization propagator and response theory calculation of properties of arbitrary order, as opposed to the case in conventional CC theory. Then, using the Zassenhaus expansion for noncommuting exponential operators, explicit diagrams for an extensive and variational method based on unitary CC theory are derived. Possible extensions to the approximations developed are discussed as well. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006
We review the basic principles of the various coupled cluster (CC) methods based on an exponential form for the many-body wavefunction, and contrast them with the configuration–interaction (CI) method. Particular emphasis is placed on their applicability to problems in quantum chemistry. We prove that in all cases we can construct an energy functional which variationally determines both the ground–state wavefunction and the dynamic equations of motion for nonstationary states. As a result the equations of motion assume the familiar classical canonical Hamiltonian form in some well-defined (multibody) configuration space. We also thereby construct the expectation-value functional for an arbitrary operator in such a way that the Feynman–Hellmann theorem is preserved at all natural levels of truncation of the appropriate configuration space. We show in detail that only in the case of the recently introduced extended CC method (ECCM) is the expectation-value functional expressed fully in terms of linked (multilocal) amplitudes. The ECCM is thereby capable of describing such global phenomena as shape transitions and other stereochemical properties, and the large-scale behavior of the molecular energy surfaces. We illustrate our methodology on the one-body density matrix, which is now much more easily discussed than by conventional methods in quantum chemistry.
Abstract The spin‐free wave function due to Fock (Zh Eksp Teor Fiz, 1940, 10, 961) is re‐examined with a stress on the reduced density matrix (RDM) theory. The key notion of the Fock approach is the cyclic symmetry of wave functions. It is a specific algebraic identity involving transpositions of numbers taken from two different columns of the corresponding Young tableau. We show first how to construct symmetry adapted states by accounting for high‐order cyclic symmetry conditions. For Young's projectors, it gives a new expression including nothing but antisymmetrizers. Next, transforming the Fock spin‐free state by a duality operator (the star operator in exterior algebra), we arrive at the representation closely related to spin‐flip models. In such spin‐flip models, a coupling operator is the basic object for which we show that the cyclic symmetry is transformed into a tracelessness of the coupling operator. The main results are related to the spin‐free theory of spin properties. In particular, the theorem previously stated (Luzanov and Whyman, Int J Quantum Chem, 1981, 20, 1179) is refined by an explicit general representation of spin density operators through spin‐free (charge) RDMs. Some applications implicating high‐order RDMs (collectivity numbers, the unpaired electron problem, cumulant spin RDMs, spin correlators, etc.) are also considered. For spin‐free RDM components, a new projection procedure without constructing any symmetry adapted state is proposed. An unsolved problem of constructing orthogonal representation matrices within the Fock theory is raised. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2010