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Physical Review A

Spin-adapted reduced Hamiltonians. II. Total energy and reduced density matrices

Carmela Valdemoro

Published Apr 1, 1985• 31 citations
Publisher

Abstract

In the preceding paper (paper I of this series) the spin-adapted reduced Hamiltonian theory was developed and applied to the study of the elementary excitations of a test example; the Be atom. Here this theory is extended in order to obtain other observables: the total energy and the reduced density matrices for an m-electron system, always working within the two-body space. Preliminary applications of this formalism produce very satisfactory results.

DOI
10.1103/physreva.31.2123

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In the preceding paper (paper I of this series) the spin-adapted reduced Hamiltonian theory was developed and applied to the study of the elementary excitations of a test example; the Be atom. Here this theory is extended in order to obtain other observables: the total energy and the reduced density matrices for an m-electron system, always working within the two-body space. Preliminary applications of this formalism produce very satisfactory results.
The spin-adapted pth-order reduced Hamiltonians (SRH) are obtained for an m-electron system, within the formalism of second quantization. This transformation is applied to the usual many-body Hamiltonian and involves an augmenting mapping, a spin projection, and a reducing mapping into the p-body reduced space. These three operations are combined into a single algorithm, allowing the work to be carried out within the two-body space. The SRH matrices thus obtained (1) have the same trace as that of the corresponding full configuration-interaction (CI) matrix, and (2) can be written as expansions involving the p eigenvalues and the corresponding pth-order reduced density matrices (RDM) of the m-body system in the chosen spin symmetry. Both of these properties guarantee that all and only the relevant information about our system is contained in the SRH matrices (and the SRH operators). The excitations of the eigenstates of the 1-SRH and the 2-SRH can be interpreted as the normal modes (or elementary excitations) of our electronic system in a given symmetry. A preliminary calculation on the Be atom yields encouraging results when compared with experiment. In the following paper (paper II of this series) the theory is applied to obtain the total energy and the reduced density matrices.
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.
The formalism developed in the first paper of the series [arXiv:0901.1060] is applied to two thermodynamic systems: (i) of three global observables (the energy, the total electron number and the spin number), (ii) of one global observable (the internal electron energy) and two local (position-dependent) observables (the total electron density and the spin density). The two-component potential of the many-electron system of interest is constructed of a scalar external potential and a collinear magnetic field (coupled only with the spin operator). Various equilibrium characteristics of two systems are defined and investigated. Conditions for the equivalence between two systems (the same equilibrium density matrix demanded) are derived and thoroughly discussed. The applicability of the Hohenberg-Kohn theorem is extended to the thermodynamic spin-density functional theory. Obtained results provide a rigorous mathematical foundation for future derivation of the zero-temperature limit of this theory and determination of its chemical reactivity indices.
It is proposed that by evaluating pth-order spin-adapted reduced Hamiltonians and applying a model of independent groups of p electrons we can approximate the pth-order reduced density matrix (p-RDM). Subsequently this pth-order matrix is contracted to the corresponding 2-RDM. As a test example we calculate the total energy of the beryllium-atom isoelectronic sequence using an independent-trio model. The results, as expected, have a higher accuracy than those obtained with the independent-pair model.
We extend the spin-adapted density matrix renormalization group (DMRG) algorithm of McCulloch and Gulacsi [Europhys. Lett. 57, 852 (2002)] to quantum chemical Hamiltonians. This involves using a quasi-density matrix, to ensure that the renormalized DMRG states are eigenfunctions of Ŝ(2), and the Wigner-Eckart theorem, to reduce overall storage and computational costs. We argue that the spin-adapted DMRG algorithm is most advantageous for low spin states. Consequently, we also implement a singlet-embedding strategy due to Tatsuaki [Phys. Rev. E 61, 3199 (2000)] where we target high spin states as a component of a larger fictitious singlet system. Finally, we present an efficient algorithm to calculate one- and two-body reduced density matrices from the spin-adapted wavefunctions. We evaluate our developments with benchmark calculations on transition metal system active space models. These include the Fe(2)S(2), [Fe(2)S(2)(SCH(3))(4)](2-), and Cr(2) systems. In the case of Fe(2)S(2), the spin-ladder spacing is on the microHartree scale, and here we show that we can target such very closely spaced states. In [Fe(2)S(2)(SCH(3))(4)](2-), we calculate particle and spin correlation functions, to examine the role of sulfur bridging orbitals in the electronic structure. In Cr(2) we demonstrate that spin-adaptation with the Wigner-Eckart theorem and using singlet embedding can yield up to an order of magnitude increase in computational efficiency. Overall, these calculations demonstrate the potential of using spin-adaptation to extend the range of DMRG calculations in complex transition metal problems.