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International Journal of Quantum Chemistry

Spin‐free quantum chemistry: What one can gain from fock's cyclic symmetry

Anatoliy V. Luzanov

Published Dec 7, 2010• 13 citations
Publisher

Abstract

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

DOI
10.1002/qua.22943

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•Oct 1, 1963•0 citations
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
Abstract The spin structure of the first order reduced density matrix (RDM‐1) for an arbitrary many‐electron state with zero z‐projection of the total spin is examined. It is well known that for the state ΨS0(r1σ1,…,rN σN), which is an eigenstate of operators S2^{\widehat {S^2}}S2 and S^z{\widehat S}_zSz​ with quantum numbers S and M = 0, the matrix elements for spins α and β are equal for any r and r′: ρ (r|r′) = ρ (r|r′). In the present article, it is shown that the same is true for any state ΦM = 0(r1σ1,…,rN,σN) with indefinite total spin if in the expansion ΦM = 0 = ∑S DS ΨS0 only spins S with the same parity are present. To prove the statement, it is shown that the wave function ΨS0 acquires the phase factor (−1)N/2−S when all spin functions α(σi) are changed for β(σi) and vice versa. In the developed proof, the Hamiltonian was not used at all and it was not even assumed that the wave function ΨS0 is an eigenfunction of some Hamiltonian. Therefore the obtained result is valid for the stationary and non‐stationary states, ground and excited states, with and without homogeneous magnetic field imposed, exact and approximate wave functions. From the result obtained it follows, in particular, that for the stationary state to be spin‐polarized (ρ (r|r) ≠ ρ (r|r)) it is necessary for the Hamiltonian to mix states with different parity spins. The consequences from the proved statement for the antiferromagnetic state are discussed. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008
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
Through the 3,5-contracted Schrödinger equation (3,5-CSchE) quantum energies and 3-particle reduced density matrices (3-RDMs) are determined directly without wave functions. Since the 3,5-CSchE involves the 5-RDM, its solution is indeterminate without N-representability conditions. However, the indeterminacy of the 3,5-CSchE may be removed through a reconstruction strategy for building the 4- and 5-RDMs from the 3-RDM. We present a systematic procedure for obtaining corrections for Valdemoro's reconstruction functionals from two complementary approaches, the particle–hole duality and the theory of cumulants. With the cumulants we are able to demonstrate that we have obtained all terms in the reconstruction functionals which may be written as antisymmetric products of the lower rdms. The cumulants allow us to understand the reconstruction functionals in terms of a renormalized many-body perturbation theory. The reconstruction functionals also lead to a natural generalization of Wick's theorem for evaluating expectation values of fermionic annihilation and creation operators with respect to correlated reference states. Previous work [Phys. Rev. A 57, 4219 (1998)] has explored the determination of correlation energy and 2-RDMs through the 2,4-CSchE, also known as the density equation. Because the reconstruction functionals employed with the 3,5-CSchE depend only on the antisymmetric products of lower RDMs in constrast to those used with the 2,4-CSchE, the 3,5-CSchE method presented here does not require the solution of systems of linear equations during reconstruction or the storage of the reconstructed RDMs. Application of the 3,5-CSchE technique to a quasi-spin model generates ground-state energies and 2-RDMs similar in accuracy to single–double configuration interaction (SDCI). We employ a simple iterative procedure for the solution of the 3,5-CSchE without traditional diagonalization. The CSchE techniques offer an approximate solution of the N-representability problem and a new approach to electron correlation. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 557–570, 1998
The traces of the p-order reduced density matrices (p-RDM) split into independent contributions associated to the subsets of p-electron eigenstates of the Ŝ2 and Ŝz operators. Here, we report the partial traces for the blocks of the low-order RDMs corresponding to pure spin states of an N-electron system. A systematic method for calculating those of higher order RDMs is described and some useful relations are also given. All these relations which must be fulfilled independently by a RDM can be considered as N- and S-representability conditions © 1997 John Wiley & Sons, Inc.
Abstract The ensemble N‐representability problem for the k‐th order reduced density matrix (k‐RDM) as well as the problem of reconstruction of the N‐particle system density matrices (N‐DM) from a given k‐RDM are studied. The spatial parts of the k‐RDM expansion in terms of spin tensorial operators Θ are represented using particular values (at specially chosen ) of the Radon transform of the N‐DM spatial parts (or their sums) 𝒟Nλ(x′ | x″) (here, is a d‐plane in the n‐space ℝn of x = (x′, x″)), with n = 6N, d = 3 (N − k), x′ ≡ (r′1, ⃛, r′N), x′ ≡ (r1″, ⃛, r″N ()). In this way, the problem is reduced to investigation of the properties of the functions . For a normalizable N – DM, it is proved that are bounded functions. The properties of implied by the N‐DM permutational symmetry, Hermiticity, and positive definiteness are found. A formal procedure of reconstruction of all N‐DM corresponding to a given k‐RDM is proposed. © 1995 John Wiley & Sons, Inc.
The density operator (density matrix) of a quantum mechenical system can be decomposed into operators which transform as irreducible representations of the symmetry group in coordinate and spin space. Each of these components has a physical meaning connected with the expectation values of certain operators. The reduced density matrices can be decomposed in a completely analogous way. The symmetry properties of the total wave function give rise to degeneracies of the eigenvalues of the reduced density matrices. These degeneracies can be removed by requiring that the natural spin orbitals (NSO, defined as the eigenfunctions of the first-order density matrix), as well as the natural spin geminals (NSG, the eigenfunctions of the second-order density matrix) and their spinless counterparts transform as irreducible representations of the symmetry group and are eigenfunctions of S/sup 2/ and S/sub z/. In many cases this requirement is compatible with the original definition of the NSO, the NSG, etc., e.g., when there is no spatial degeneracy of the total wave function and when the Z- component of the total spin vanishes. When these conditions are not fulfilled an alternative definition of the NSO and the NSG is proposed. (auth)