Multireference Electron Correlation Research Articles

Multireference Electron Correlation Methods: Journeys along Potential Energy Surfaces.

Multireference electron correlation methods describe static and dynamical electron correlation in a balanced way and, therefore, can yield accurate and predictive results even when single-reference methods or multiconfigurational self-consistent field theory fails. One of their most prominent applications in quantum chemistry is the exploration of potential energy surfaces. This includes the optimization of molecular geometries, such as equilibrium geometries and conical intersections and on-the-fly photodynamics simulations, both of which depend heavily on the ability of the method to properly explore the potential energy surface. Because such applications require nuclear gradients and derivative couplings, the availability of analytical nuclear gradients greatly enhances the scope of quantum chemical methods. This review focuses on the developments and advances made in the past two decades. A detailed account of the analytical nuclear gradient and derivative coupling theories is presented. Emphasis is given to the software infrastructure that allows one to make use of these methods. Notable applications of multireference electron correlation methods to chemistry, including geometry optimizations and on-the-fly dynamics, are summarized at the end followed by a discussion of future prospects.

Strong Electron Correlation in Nitrogenase Cofactor, FeMoco.

FeMoco, MoFe7S9C, has been shown to be the active catalytic site for the reduction of nitrogen to ammonia in the nitrogenase protein. An understanding of its electronic structure including strong electron correlation is key to designing mimic catalysts capable of ambient nitrogen fixation. Active spaces ranging from [54, 54] to [65, 57] have been predicted for a quantitative description of FeMoco's electronic structure. However, a wave function approach for a singlet state using a [54, 54] active space would require 1029 variables. In this work, we systematically explore the active-space size necessary to qualitatively capture strong correlation in FeMoco and two related moieties, MoFe3S7 and Fe4S7. Using CASSCF and 2-RDM methods, we consider active-space sizes up to [14, 14] and [30, 30], respectively, with STO-3G, 3-21G, and DZP basis sets and use fractional natural-orbital occupation numbers to assess the level of multireference electron correlation, an examination of which reveals a competition between single-reference and multireference solutions to the electronic Schrödinger equation for smaller active spaces and a consistent multireference solution for larger active spaces.

Electronic Structures of Anti-Ferromagnetic Tetraradicals: Ab Initio and Semi-Empirical Studies.

The energy relationships and electronic structures of the lowest-lying spin states in several anti-ferromagnetic tetraradical model systems are studied with high-level ab initio and semi-empirical methods. The Full-CI method (FCI), the complete active space second-order perturbation theory (CASPT2), and the n-electron valence state perturbation theory (NEVPT2) are employed to obtain reference results. By comparing the energy relationships predicted from the Heisenberg and Hubbard models with ab initio benchmarks, the accuracy of the widely used Heisenberg model for anti-ferromagnetic spin-coupling in low-spin polyradicals is cautiously tested in this work. It is found that the strength of electron correlation (|U/t|) concerning anti-ferromagnetically coupled radical centers could range widely from strong to moderate correlation regimes and could become another degree of freedom besides the spin multiplicity. Accordingly, the Heisenberg-type model works well in the regime of strong correlation, which reproduces well the energy relationships along with the wave functions of all the spin states. In moderately spin-correlated tetraradicals, the results of the prototype Heisenberg model deviate severely from those of multi-reference electron correlation ab initio methods, while the extended Heisenberg model, containing four-body terms, can introduce reasonable corrections and maintains its accuracy in this condition. In the weak correlation regime, both the prototype Heisenberg model and its extended forms containing higher-order correction terms will encounter difficulties. Meanwhile, the Hubbard model shows balanced accuracy from strong to weak correlation cases and can reproduce qualitatively correct electronic structures, which makes it more suitable for the study of anti-ferromagnetic coupling in polyradical systems.

Enhanced computational efficiency in the direct determination of the two-electron reduced density matrix from the anti-Hermitian contracted Schrödinger equation with application to ground and excited states of conjugated π-systems.

Determination of the two-electron reduced density matrix (2-RDM) from the solution of the anti-Hermitian contracted Schrödinger equation (ACSE) yields accurate energies and properties for both ground and excited states. Here, we develop a more efficient method to solving the ACSE that uses second-order information to select a more optimal step towards the solution. Calculations on the ground and excited states of water, hydrogen fluoride, and conjugated π systems show that the improved ACSE algorithm is 10-20 times faster than the previous ACSE algorithm. The ACSE can treat both single- and multi-reference electron correlation with the initial 2-RDM from a complete-active-space self-consistent-field (CASSCF) calculation. Using the improved algorithm, we explore the relationship between truncation of the active space in the CASSCF calculation and the accuracy of the energy and 2-RDM from the ACSE calculation. The accuracy of the ACSE, we find, is less sensitive to the size of the active space than the accuracy of other wavefunction methods, which is useful when large active space calculations are computationally infeasible.

Relativistic internally contracted multireference electron correlation methods.

We report internally contracted relativistic multireference configuration interaction (ic-MRCI), complete active space second-order perturbation (CASPT2), and strongly contracted n-electron valence state perturbation theory (NEVPT2) on the basis of the four-component Dirac Hamiltonian, enabling accurate simulations of relativistic, quasi-degenerate electronic structure of molecules containing transition-metal and heavy elements. Our derivation and implementation of ic-MRCI and CASPT2 are based on an automatic code generator that translates second-quantized ansätze to tensor-based equations, and to efficient computer code. NEVPT2 is derived and implemented manually. The rovibrational transition energies and absorption spectra of HI and TlH are presented to demonstrate the accuracy of these methods.

Highly accurate excited-state energies from direct computation of the 2-electron reduced density matrix by the anti-Hermitian contracted Schrödinger equation

ABSTRACTDirectly solving for the 2-electron reduced density matrix (2-RDM) via the anti-Hermitian contracted Schrödinger equation (ACSE) enables computations for excited states energies without the N-electron wave function. Of particular interest are excitations and dissociation curves that exhibit strong multi-reference correlation effects. The ground and excited states of the molecules HF, H2O, and N2 are examined at both equilibrium and non-equilibrium geometries to compare the ability of the ACSE and widely used ab initio techniques to treat strong multi-reference electron correlation. Calculations are performed with double-ζ basis sets for calibration with full configuration interaction (FCI). Multi-reference second-order perturbation theory (MRPT2) and the ACSE both provide qualitative precision with respect to FCI data, although the ACSE's capability to include higher order correlation effects permits near-FCI accuracy for ground and excited states and excitation energies.

Multireference electron correlation methods with density matrix renormalisation group reference functions

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.

Multireference explicitly correlated F12 theories

We review our recent developments in multireference explicitly correlated F12 theories (explicitly correlated internally contracted multireference perturbation and multireference configuration interaction theories) that achieve near-basis-set-limit accuracy of the underlying multireference electron correlation methods with basis sets of medium size. The applicability of the multireference F12 theories is the same as that of their non-F12 counterpart, and therefore it is a computational tool with predictive accuracy for complicated electronic structures with strong correlation. A comparison with the earlier developments by others is also discussed.

Balancing single- and multi-reference correlation in the chemiluminescent reaction of dioxetanone using the anti-Hermitian contracted Schrödinger equation

Direct computation of energies and two-electron reduced density matrices (2-RDMs) from the anti-Hermitian contracted Schrödinger equation (ACSE) [D. A. Mazziotti, Phys. Rev. Lett. 97, 143002 (2006)], it is shown, recovers both single- and multi-reference electron correlation in the chemiluminescent reaction of dioxetanone especially in the vicinity of the conical intersection where strong correlation is important. Dioxetanone, the light-producing moiety of firefly luciferin, efficiently converts chemical energy into light by accessing its excited-state surface via a conical intersection. Our previous active-space 2-RDM study of dioxetanone [L. Greenman and D. A. Mazziotti, J. Chem. Phys. 133, 164110 (2010)] concluded that correlating 16 electrons in 13 (active) orbitals is required for realistic surfaces without correlating the remaining (inactive) orbitals. In this paper we pursue two complementary goals: (i) to correlate the inactive orbitals in 2-RDMs along dioxetanone's reaction coordinate and compare these results with those from multireference second-order perturbation theory (MRPT2) and (ii) to assess the size of the active space-the number of correlated electrons and orbitals-required by both MRPT2 and ACSE for accurate energies and surfaces. While MRPT2 recovers very different amounts of correlation with (4,4) and (16,13) active spaces, the ACSE obtains a similar amount of correlation energy with either active space. Nevertheless, subtle differences in excitation energies near the conical intersection suggest that the (16,13) active space is necessary to determine both energetic details and properties. Strong electron correlation is further assessed through several RDM-based metrics including (i) total and relative energies, (ii) the von Neumann entropy based on the 1-electron RDM, as well as the (iii) infinity and (iv) squared Frobenius norms based on the cumulant 2-RDM.

Benchmarking Electronic Structure Calculations on the Bare UO22+ Ion: How Different are Single and Multireference Electron Correlation Methods?

In a recent investigation by some of us on the spectrum of the uranyl (UO22+) ion [Ral, F.; Vallet, V.; Marian, C.; Wahlgren, U. J. Chem. Phys. 2007, 126, 214302], a sizable difference between CASPT2 and linear response coupled cluster (LRCC) was observed both with and without the perturbative inclusion of spin-orbit coupling. This poses a serious question as to which of the two would be more reliable for investigating molecules containing actinides. In this paper we address this question by comparing CASPT2 and LRCC to a method known to accurately describe the spectra of actinide-containing molecules: the four-component intermediate Hamiltonian Fock-space coupled cluster (IHFSCC) method, where electron correlation and spin-orbit coupling are treated on an equal footing. Our results indicate that for UO22+ there is little difference between treatments of spin-orbit coupling, making electron correlation the main cause of discrepancies. We have found IHFSCC and LRCC to be the most alike in the overall description of excited states, even though individual LRCC energies are blue-shifted in comparison to IHFSCC due to the difference in the parametrization of the excited states' wave functions. CASPT2, on the other hand, shows good agreement with IHFSCC for individual frequencies but significantly less so for the spectrum as a whole, due to the difference in the degree of correlation recovered in both cases.

A diagnostic for determining the quality of single-reference electron correlation methods

It was recently proposed that the Euclidian norm of the t1 vector of the coupled-cluster wave function (normalized by the number of electrons included in the correlation procedure) could be used to determine whether a single-reference-based electron correlation procedure is appropriate. This diagnostic T1 is defined for use with self-consistent-field molecular orbitals and is invariant to the same orbital rotations as the coupled-cluster energy. T1 is investigated for several different chemical systems which exhibit a range of multireference behavior and is shown to be an excellent measure of the importance of nondynamical electron correlation and is far superior to C0 from a singles and doubles configuration interaction wave function. It is further suggested that when the aim is to recover a large fraction of the dynamical electron correlation energy, a large T1 (i.e., >0.02) probably indicates the need for a multireference electron correlation procedure.

Mobility of the metastable C+ ion in cooled He gas at 77 and 4.3 K

The mobility of C+ in He gas has been measured using a very low temperature drift tube mass spectrometer at 77 and 4.3 K. Ion mobility separation between the ground-state C+ (2Po) and the metastable C+ (4P) ions has been observed at the effective temperature Teff from 550 to 1500 K. The calculation of the mobility based on the classical momentum transfer theory has been carried out using the interaction potential provided by the multi-reference electron correlation method. The calculated mobility of C+ qualitatively agrees with the measurement, confirming the accuracy of the interaction potential between the metastable ion and He.

The valence-excited states T1–T4 and S1–S2 of acetylene: A high-level MR-CISD and MR-AQCC investigation of stationary points, potential energy surfaces, and surface crossings

Valence-excited singlet (S1,S2) and triplet (T1–T4) states of acetylene have been studied by means of extended multireference electron correlation techniques (MR-CISD, MR-CISD+Q, and MR-AQCC). Extrapolations to the basis set limit have been performed. Minima and saddle points have been calculated using a recently developed analytic gradient method for excited states. Planar as well as nonplanar structures have been considered. In particular, the existence of an asymmetric, planar cis-type minimum on the S2 surface has been confirmed conclusively. Moreover, an intersection S1/S2 has been located close to this minimum. This situation will most probably affect the interpretation of the absorption bands attributed to the trans 1 1Bu state. In-plane and out-of-plane saddle points for cis–trans isomerization have been determined and characterized by harmonic vibrational analysis. Several interesting surface crossings for different electronic states (S1/S2, T2/T3, and S1/T3) have been characterized. Implications of the flatness of the T3 surface around linear structures and the location of the S1/T3 crossing seam on the anomalities observed in the ZAC spectrum of the à 1Au state are discussed.

Ab initio study of the carbon (C3+) cation using multireference methods

The energy difference between the linear {sup 2}{Sigma}{sub u}{sup +} and cyclic {sup 2}B{sub 2} structures of C{sub 3}{sup +} has been investigated by using large (5s3p2dlf) basis sets and multireference electron correlation treatments, including complete active space SCF (CASSCF), multireference CI(MRCI), and averaged coupled-pair functional (ACPF) methods, as well as the single-reference quadratic configuration interaction (QCISD(T)) method. The authors best estimate, including a correction for basis set incompleteness, is that the linear form lies above the cyclic form by 5.2{sub {minus}1.0}{sup +1.5} kcal/mol. The {sup 2}{Sigma}{sub u}{sup +} state is probably not a transition state but a local minimum. Reliable computation of the cyclic/linear energy difference in C{sub 3}{sup +} is extremely demanding of the electron correlation treatment used: of the single-reference methods previously considered, CCSD(T) and QCISD(T) perform best. The MRCI+Q-(0.01)/(4s2pld) energy separation of 1.68 kcal/mol should provide a comparison standard for other electron correlation methods applied to this system.