An auxiliary-field quantum Monte Carlo study of the chromium dimer

Abstract

The chromium dimer (Cr2) presents an outstanding challenge for many-body electronic structure methods. Its complicated nature of binding, with a formal sextuple bond and an unusual potential energy curve, is emblematic of the competing tendencies and delicate balance found in many strongly correlated materials. We present a nearexact calculation of the potential energy curve (PEC) and ground state properties of Cr2, using the auxiliary-field quantum Monte Carlo (AFQMC) method. Unconstrained, exact AFQMC calculations are first carried out for a medium-sized but realistic basis set. Elimination of the remaining finite-basis errors and extrapolation to the complete basis set (CBS) limit is then achieved with a combination of phaseless and exact AFQMC calculations. Final results for the PEC and spectroscopic constants are in excellent agreement with experiment. The chromium dimer is a strongly correlated molecule which poses a formidable challenge to even the most accurate many-body methods. It features a formal sextuple bond, with a weak binding energy (∼ 1.5 eV), a short equilibrium bond length (∼ 1.7Å), and an unusual "shoulder" structure in its potential energy curve (PEC). [1][2][3] The ground state of Cr 2 is highly multiconfigurational, and proper theoretical description requires an accurate treatment of the strong 3d electron correlations (both static and dynamic). The nature of the PEC in Cr 2 is representative of the competing tendencies separated by small energy differences seen in many strongly correlated materials. Because of the fundamental and technological significance of such materials, improving our abilities for accurate calculations in strongly correlated systems is one of the most pressing needs in condensed matter physics and quantum chemistry.Standard quantum chemistry methods, such as density functional theory (DFT), Hartree-Fock (HF), and post-HF methods such as single-reference second-order Møller-Plesset perturbation theory (MP2) and single-reference coupled cluster with singles, doubles, and perturbative triples [CCSD(T)], all fail to describe the correct binding of Cr 2 . Representative standard quantum chemistry results are shown in Fig. 1. As often is the case, the DFT results vary greatly, depending on the choice of exchange-correlation functional. There have also been numerous attempts to calculate the PEC of Cr 2 using sophisticated multireference quantum chemistry methods, 4-9 including the complete active space second-order perturbation theory (CASPT2) 10-12 and, more recently, CASPT2 based on a large density matrix renormalization group (DMRG) reference wave function (DMRG-CASPT2). 13 These calculations obtain qualitatively correct binding, but the results are sensitive to choice of active space and/or basis set. Standard quantum Monte Carlo (QMC) approaches 14,15 have also been severely challenged. A recent fixed-node diffusion Monte Carlo (DMC) study, which examined the use of a variety of single-and multi-determinant trial wave functions, did not obtain satisfactory binding (i...

Chemical Transformations Approaching Chemical Accuracy via Correlated Sampling in Auxiliary-Field Quantum Monte Carlo

“…This is crudely the case, yet the subtleties involved in such a claim warrant further discussion. Indeed, referring to the propagator in (11), while our correlated sampling method ensures that thev operators and the AFs, x, are the same for both the primary and secondary systems, the FBs,x, as defined in (9) and the expectation values with respect to the trial functions v will in general be different. In the limit that the primary and secondary systems are identical, the entire propagator in (11) is identical for both systems, and the statistical error in the energy difference will be exactly and trivially zero as a result of perfect walkerpair correlation.…”

“…Among the class of Quantum Monte Carlo (QMC) methods used in electronic structure theory, [26][27][28] Auxiliary-Field QMC, and in particular its "phaseless" variant (ph-AFQMC) [29,30] which controls the sign problem at the cost of introducing a bias which in principle can be systematically reduced, has produced stateof-the-art, benchmark-quality results for systems such as first-and second-row d elements, [31] the chromium dimer, [9] cobalt adatoms on graphene, [32] and various transition metal oxides, [33] while exhibiting lowpolynomial scaling (M 4 for Gaussian basis sets). However, the widespread use of ph-AFQMC in quantum chemistry has not yet taken place.…”

“…[1][2][3][4][5] Traditional methods such as full configuration interaction (FCI) [6] and single-reference coupled cluster with single, double, and perturbative triple excitations (CCSD(T)) [7,8] scale exponentially and with the seventh power of the system size, respectively, and the latter can break down for systems which exhibit sufficiently strong electron correlation. [9,10] Multi-reference methods such as CASSCF, [11][12][13] often supplemented with second order perturbation theory, [14,15] and FCI-QMC [16][17][18][19] have been shown to yield benchmark-quality results even for strongly correlated electronic systems, an import class of problems that includes many transition metal (TM)-containing systems. [20,21] However, while a judicious choice of the active space can make such calculations tractable for small systems, exponential scaling prohibits the use of these methods in studying most realistic systems of interest in biology, materials science, and chemical catalysis.…”

Bridging Single- and Multireference Domains for Electron Correlation: Spin-Extended Coupled Electron Pair Approximation

“…For this reason, a number of studies have attempted to elucidate this system using different methodologies. 76,[80][81][82][83][84][85][86][87][88][89][90] We used 18 core orbitals for SUHF (all of which are almost doubly occupied, even without constraints) and then correlated the 3p, 3d, and 4s electrons in the subsequent ECISD/ECEPA calculations, thus utilizing 12 frozen-core orbitals. Our calculations were carried out with the cc-pVnZ basis sets with n = D, T, and Q, and were extrapolated to the CBS limit using the two-point extrapolation formula.…”

Auxiliary-field Quantum Monte Carlo Methods in Nuclei

“…In condensed matter physics, AFMC has been used to study strongly correlated electron systems [12; 19; 20]. In quantum chemistry, it was applied to study the electronic structure of molecules, such as the recent study of the chromium dimer [21]. In cold atom physics, AFMC methods were used to study the thermodynamics of the two-species Fermi gas with contact interaction for both the uniform gas [22; 23] and the harmonically trapped gas [24], and the ground state of the Fermi gas in its unitary limit [25].…”