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Kluwer Academic Publishers eBooks

Reduced Density-matrix Treatment of Spin-orbit Interaction Terms in Many-electron Systems

Rossen L. Pavlov, Alexander I. Kuleff, P. Tz. Yotov, J. Maruani, Ya. I. Delchev

Published Dec 21, 2005• 0 citations
Publisher

Abstract

In the formalism of reduced density matrices and functions, using the irreducible tensor-operator technique and the space-spin separation scheme, the matrix elements of one of the main spin-relativistic corrections of the Breit-Pauli Hamiltonian, the spin-orbit interactions, are expressed in a form suitable for numeric implementation. A comparison with other methods is made and the advantages of such an approach are discussed. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/0-306-46951-0_5

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In the formalism of reduced density matrices and functions, using the irreducible tensor-operator technique and the space-spin separation scheme, the matrix elements of one of the main spin-relativistic corrections of the Breit-Pauli Hamiltonian, the spin-orbit interactions, are expressed in a form suitable for numeric implementation. A comparison with other methods is made and the advantages of such an approach are discussed. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
A formalism based on the unitary- and symmetric-group approaches to configuration-interaction methods has been applied to an evaluation of matrix elements of a spin-adapted reduced Hamiltonian (SRH). It has been shown that there is a simple closed-form expression for the matrix elements of the SRH matrices. This formula has been expressed as a linear combination of two-electron integrals with coefficients given as explicit functions of the parameters defining the finite-dimensional Hilbert space (number of orbitals, number of electrons, and the total spin quantum number).
A fully spin-adapted approach to many-electron density matrices is developed in the context of the unitary group approach to many-electron systems. An explicit expression for the single-electron spin-density operator, as a polynomial of degree two in the orbital U(n) generators, is derived for the case of spin-independent systems. Extensions to spin-dependent systems are also considered, leading to the appearance of total-spin transition densities, whose general properties are investigated. A corresponding formalism for the two-electron density matrix, which is capable of further generalization, is also developed. The results of this paper, together with recent developments on the matrix elements of the U(2n) generators in the electronic Gel’fand basis, afford a versatile method for the direct calculation of one- and two-body density matrices in the unitary group approach framework.
Spin Orbit Coupling (SOC) is introduced to molecular ab initio density matrix renormalization group (DMRG) calculations. In the presented scheme, one first approximates the electronic ground state and a number of excited states of the Born-Oppenheimer (BO) Hamiltonian with the aid of the DMRG algorithm. Owing to the spin-adaptation of the algorithm, the total spin S is a good quantum number for these states. After the non-relativistic DMRG calculation is finished, all magnetic sublevels of the calculated states are constructed explicitly, and the SOC operator is expanded in the resulting basis. To this end, spin orbit coupled energies and wavefunctions are obtained as eigenvalues and eigenfunctions of the full Hamiltonian matrix which is composed of the SOC operator matrix and the BO Hamiltonian matrix. This treatment corresponds to a quasi-degenerate perturbation theory approach and can be regarded as the molecular equivalent to atomic Russell-Saunders coupling. For the evaluation of SOC matrix elements, the full Breit-Pauli SOC Hamiltonian is approximated by the widely used spin-orbit mean field operator. This operator allows for an efficient use of the second quantized triplet replacement operators that are readily generated during the non-relativistic DMRG algorithm, together with the Wigner-Eckart theorem. With a set of spin-orbit coupled wavefunctions at hand, the molecular g-tensors are calculated following the scheme proposed by Gerloch and McMeeking. It interprets the effective molecular g-values as the slope of the energy difference between the lowest Kramers pair with respect to the strength of the applied magnetic field. Test calculations on a chemically relevant Mo complex demonstrate the capabilities of the presented method.
We investigate the structure of the one-body reduced density matrix of three electron systems, i.e., doublet and quadruplet spin configurations, corresponding to the smallest interacting system with an open-shell ground state. To this end, we use configuration interaction (CI) expansions of the exact wave function in Slater determinants built from natural orbitals in a finite dimensional Hilbert space. With the exception of maximally polarized systems, the natural orbitals of spin eigenstates are generally spin dependent, i.e., the spatial parts of the up and down natural orbitals form two different sets. A measure to quantify this spin dependence is introduced and it is shown that it varies by several orders of magnitude depending on the system. We also study the ordering issue of the spin-dependent occupation numbers which has practical implications in reduced density matrix functional theory minimization schemes, when generalized Pauli constraints (GPCs) are imposed and in the form of the CI expansion in terms of the natural orbitals. Finally, we discuss the aforementioned CI expansion when there are GPCs that are almost "pinned."
As was repeatedly seen in the last chapter, for computing most of the usual matrix elements only a few of the many coordinates of an N -particle wavefunction are used, where due to the symmetry properties it is irrelevant which ones. Density matrices are a tool to extract the relevant information out of such monstrous constructs as are N -particle wavefunctions. (For reviews, see [Erdahl and Smith, 1987, Coleman, 1963, McWeeny, 1960]; in the text below we try to keep the notation as canonical as possible.) These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
We investigate the structure of the one-body reduced density matrix of three electron systems, i.e., doublet and quadruplet spin configurations, corresponding to the smallest interacting system with an open-shell ground state. To this end, we use configuration interaction (CI) expansions of the exact wave function in Slater determinants built from natural orbitals in a finite dimensional Hilbert space. With the exception of maximally polarized systems, the natural orbitals of spin eigenstates are generally spin dependent, i.e., the spatial parts of the up and down natural orbitals form two different sets. A measure to quantify this spin dependence is introduced and it is shown that it varies by several orders of magnitude depending on the system. We also study the ordering issue of the spin-dependent occupation numbers which has practical implications in reduced density matrix functional theory minimization schemes, when generalized Pauli constraints (GPCs) are imposed and in the form of the CI expansion in terms of the natural orbitals. Finally, we discuss the aforementioned CI expansion when there are GPCs that are almost “pinned.”