On the measure of electron correlation and entanglement in quantum chemistry based on the cumulant of the second-order reduced density matrix

Section: Theoretical Frameworkmentioning

confidence: 89%

Depicting electronic distributions from accurate computational first principles: On the relationship between the complex patterns of bonding interaction and the back‐donation phenomenon

Lobayan

Bochicchio

Valle

2016

“…Recently, quantum information theory has also appeared in quantum chemistry giving a fresh impetus to the development of methods in electronic structure theory . The amount of contribution of an orbital to the total correlation can be characterized, for example, by the single‐orbital entropy , and the the sum of all single‐orbital entropies gives the amount of total correlation encoded in the wave function .…”

Section: Introductionmentioning

confidence: 99%

Tensor product methods and entanglement optimization for ab initio quantum chemistry

Szalay

Pfeffer

Murg

et al. 2015

293

395

The treatment of high-dimensional problems such as the Schr€ odinger equation can be approached by concepts of tensor product approximation. We present general techniques that can be used for the treatment of high-dimensional optimization tasks and time-dependent equations, and connect them to concepts already used in many-body quantum physics. Based on achievements from the past decade, entanglement-based methods-developed from different perspectives for different purposes in distinct communities already matured to provide a variety of tools-can be combined to attack highly challenging problems in quantum chemistry. The aim of the present paper is to give a pedagogical introduction to the theoretical background of this novel field and demonstrate the underlying benefits through numerical applications on a text book example. Among the various optimization tasks, we will discuss only those which are connected to a controlled manipulation of the entanglement which is in fact the key ingredient of the methods considered in the paper. The selected topics will be covered according to a series of lectures given on the topic "New wavefunction methods and entanglement optimizations in quantum

“…Thus, regarding the fact that the first two terms in the rhs of Equation may be related to the net spin density contributions as shown above for the Hartree–Fock approach, the last term is clearly related to the correlation effects and states the difference among the cumulant crossed traces of the spin‐free 2‐RDM,

ε_{M} = \sum_{i, k} Γ_{k i}^{i k} (S_{z} = S)

. Consequently, as it has been shown, it represents a measure of the spin‐entanglement of the M ‐particle system and then, this term expressed by the

\pm (ε_{N \pm 1} - ε_{N})

means the contribution of the spin‐entanglement difference among the neutral and the ionic states of the system.…”

Section: Further Remarksmentioning

confidence: 90%

Structure of Fukui matrices

Bochicchio

2017

Fukui matrices considered as the generalization of the concept of Fukui densities are decomposed into their pairing and unpairing contributions within the theory of the reduced density matrices.Their algebraic structure become clear from this decomposition providing their relationships with the spin density matrices and the irreducible part of the second-order reduced density matrix cumulant, that is, the explicit contributions of the many-body or correlation effects. The uncorrelated state function approximation is a simple way to emphasize the physical meaning of these matrices and represents the appropriate starting point for the treatment of a quasi-analytical model to denote the occurrence of correlation effects. K E Y W O R D S Fukui matrices, pairing densities, reduced density matrices, unpairing densities 1 | T HE ORE TI CA L S CE NA R I O Fukui densities (or Fukui functions) are central to the reactivity concept in chemical physics. [1] The natural scenario in which they has beenproperly defined as the first derivative of the electron density respect to the number of particles at fixed external potential is conceptual density functional theory (DFT). [1,2] Its matrix formulation has been reported by use of the first-order reduced density matrices at the DFT level of approximation [3] and also it has been generalized for any type of molecular state functions. [4][5][6][7] The aim of this report is to use the rigorous version of PPLB proposal [8,9] based on the representation of the grand-canonical ensemble to properly express these magnitudes from which the concept of systems with a non-integer number of particles N (N 5N6m) is introduced in a natural manner. Thus, it allows the number of particles to be a mathematically continuous variable to derive the Fukui like density matrices [7] from the first-order reduced density matrices 1 D N6m (1-RDM), with N 2 N o ; m 2 R, and 0 < m < 1. [9] N o and R stand for the set of non-negative integer and real numbers, respectively. This type of systems may be identified with open systems, that is, an atom, a functional groups or a moiety, as a domain within a given molecular structure. [8] The convex structure of the energy in atomic or molecular systems driven by Coulomb interactions, [8] enables the ground state 1 D N6m , obtained by contraction mapping [10] of two integer M-particle system 1-RDMs, to be expressed by the simple convex expansion [8,9] 1 D N6m 5 m 1 D N61 1 ð12mÞ 1 D N (1)where 1 D N and 1 D N61 stand for the 1-RDMs of the systems with N and N 6 1 particles, that is, the neutral and ionic species, respectively. During the course of this work, the states of the neutral species will be considered as singlet closed shells. Equation 1 is the meaningful physical expression for the 1-RDM of a system possessing a non-integer number of electrons that cannot be described neither by a pure nor a canonical ensemble state. [9] Its trace N 6 m may be interpreted as an average of the number of electrons of the systems involved in the expansion. [8,9] The Fukui mat...