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Advanced Theory and Simulations

Multireference Approaches to Spin‐State Energetics of Transition Metal Complexes Utilizing the Density Matrix Renormalization Group

Michael Roemelt, Dimitrios A. Pantazis

Published Feb 25, 2019• 65 citations
Publisher

Abstract

Abstract The accurate and reliable calculation of different electronic states in transition metal systems is a persistent challenge for theoretical chemistry. The widespread use of density functional theory for computing the relative energies of states with different spin in transition metal complexes not only has not discouraged, but often, owing to its limitations, has motivated the development and refinement of first‐principles wavefunction–based methods, including both single‐reference and multireference approaches. A significant boost for the latter has been the emergence of the density matrix renormalization group (DMRG) as a reliable tool in applied quantum chemistry. By enabling the use of larger active spaces than conventionally possible, DMRG has opened up areas of transition metal chemistry that were previously inaccessible to multireference approaches. The present perspective provides an overview of representative studies that make use of DMRG methods and discusses recent applications to spin‐state energetics of transition metal systems. These range from mononuclear to exchange‐coupled systems that define an important emerging field of DMRG applications. Major achievements are highlighted and potential pitfalls are identified with a view toward future methodological developments as well as extensions in the applicability of DMRG‐based approaches to problems of spin‐state energetics and exchange‐coupling in transition metal chemistry.

DOI
10.1002/adts.201800201

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Abstract The accurate and reliable calculation of different electronic states in transition metal systems is a persistent challenge for theoretical chemistry. The widespread use of density functional theory for computing the relative energies of states with different spin in transition metal complexes not only has not discouraged, but often, owing to its limitations, has motivated the development and refinement of first‐principles wavefunction–based methods, including both single‐reference and multireference approaches. A significant boost for the latter has been the emergence of the density matrix renormalization group (DMRG) as a reliable tool in applied quantum chemistry. By enabling the use of larger active spaces than conventionally possible, DMRG has opened up areas of transition metal chemistry that were previously inaccessible to multireference approaches. The present perspective provides an overview of representative studies that make use of DMRG methods and discusses recent applications to spin‐state energetics of transition metal systems. These range from mononuclear to exchange‐coupled systems that define an important emerging field of DMRG applications. Major achievements are highlighted and potential pitfalls are identified with a view toward future methodological developments as well as extensions in the applicability of DMRG‐based approaches to problems of spin‐state energetics and exchange‐coupling in transition metal chemistry.
A major challenge in molecular electronic structure theory arises from chemical systems where the mean-field picture based upon a single electronic configuration fails to properly describe mechanics of many electrons. Such complex electronic states should be accounted for using a correlated superposition of multiple determinants and are referred to as multireference in quantum chemistry. Application domain of traditional multireference methods is severely limited to systems containing a small number of correlated electrons because of their exponential complexity. In recent years, much attention has been drawn to the density matrix renormalization group (DMRG) theory as a promising alternative. It has been introduced in ab initio quantum chemistry calculations and shown to be a highly-scalable multireference method that can handle full quantum degrees of freedom for the strongly-correlated wavefunction with unprecedented large configuration space. In this article, we provide an explicative introduction to the underlying theory and formalism built into the DMRG method and give a brief overview of our methodological extensions that combine DMRG and active-space correlation models.
The accurate first-principles calculation of relative energies of transition metal complexes and clusters is still one of the great challenges for quantum chemistry. Dense lying electronic states and near degeneracies make accurate predictions difficult, and multireference methods with large active spaces are required. Often density functional theory calculations are employed for feasibility reasons, but their actual accuracy for a given system is usually difficult to assess (also because accurate ab initio reference data are lacking). In this work we study the performance of the density matrix renormalization group algorithm for the prediction of relative energies of transition metal complexes and clusters of different spin and molecular structure. In particular, the focus is on the relative energetical order of electronic states of different spin for mononuclear complexes and on the relative energy of different isomers of dinuclear oxo-bridged copper clusters.
In the past two decades, the density matrix renormalization group (DMRG) has emerged as an innovative new method in quantum chemistry relying on a theoretical framework very different from that of traditional electronic structure approaches. The development of the quantum chemical DMRG has been remarkably fast: it has already become one of the reference approaches for large-scale multiconfigurational calculations. This perspective discusses the major features of DMRG, highlighting its strengths and weaknesses also in comparison with other novel approaches. The method is presented following its historical development, starting from its original formulation up to its most recent applications. Possible routes to recover dynamical correlation are discussed in detail. Emerging new fields of applications of DMRG are explored, such as its time-dependent formulation and the application to vibrational spectroscopy.
This article presents an efficient and parallelized implementation of the density matrix renormalization group (DMRG) algorithm for quantum chemistry calculations. The DMRG method as a large-scale multireference electronic structure model is by nature particularly efficient for one-dimensionally correlated systems, while the present development is oriented toward applications for polynuclear transition metal compounds, in which the macroscopic one-dimensional structure of electron correlation is absent. A straightforward extension of the DMRG algorithm is proposed with further improvements and aggressive optimizations to allow its application with large multireference active space, which is often demanded for metal compound calculations. Special efficiency is achieved by making better use of sparsity and symmetry in the operator and wave function representations. By accomplishing computationally intensive DMRG calculations, the authors have found that a large number of renormalized basis states are required to represent high entanglement of the electron correlation for metal compound applications, and it is crucial to adopt auxiliary perturbative correction to the projected density matrix during the DMRG sweep optimization in order to attain proper convergence to the solution. Potential energy curve calculations for the Cr(2) molecule near the known equilibrium precisely predicted the full configuration interaction energies with a correlation space of 24 electrons in 30 orbitals [denoted by (24e,30o)]. The energies are demonstrated to be accurate to 0.6mE(h) (the error from the extrapolated best value) when as many as 10,000 renormalized basis states are employed for the left and right DMRG block representations. The relative energy curves for [Cu(2)O(2)](2+) along the isomerization coordinate were obtained from DMRG and other correlated calculations, for which a fairly large orbital space (32e,62o) is modeled as a full correlation space. The DMRG prediction nearly overlaps with the energy curve from the coupled cluster with singles, doubles, and perturbative triple [CCSD(T)] calculations, while the multireference complete active space self-consistent field (CASSCF) calculations with the small reference configuration (8e,8o) are found to overestimate the biradical character of the electronic state of [Cu(2)O(2)](2+) according to the one-electron density matrix analysis.
Recent advances in quantum chemical density matrix renormalisation group (DMRG) theory are presented. The DMRG, originally devised as an alternative to the exact diagonalisation in condensed matter physics, has become a powerful quantum chemical method for molecular systems that exhibit a multireference character, e.g., excited states, π-conjugated systems, transition metal complexes, and in particular for large systems by combining it with conventional multireference electron correlation methods. The capability of the current quantum chemical DMRG is demonstrated for an application involving the potential energy curve of the chromium dimer, which is one of the most demanding multireference systems and thus requires the best electronic structure treatment for non-dynamical and dynamical correlation as well as large basis sets.