We show that the recently developed phaseless auxiliary-field quantum Monte Carlo (AFQMC) method can be used to study excited states, providing an alternative to standard quantum chemistry methods. The phaseless AFQMC approach, whose computational cost scales as M^3-M^4 with system size M, has been shown to be among the most accurate many-body methods in ground state calculations. For excited states, prevention of collapse into the ground state and control of the Fermion sign/phase problem are accomplished by the approximate phaseless constraint with a trial wave function. Using the challenging C2 molecule as a test case, we calculate the potential energy curves of the ground and two low-lying singlet excited states. The trial wave function is obtained by truncating complete active space wave functions, with no further optimization. The phaseless AFQMC results using a small basis set are in good agreement with exact full configuration interaction calculations, while those using large basis sets are in good agreement with experimental spectroscopic constants.
The ability to calculate electronic excited states of molecules and extended systems is necessary to predict key phenomena and properties of technologically important systems. Compared to ground states, however, the accurate calculation of excitation energies is significantly more difficult. For molecules, a variety of many-body electronic structure quantum chemistry approaches have been developed, typically using a one-particle basis to represent the many-body wave function. For small molecules with modest basis sets, the full configuration-interaction (FCI) method is exact, but FCI is not practical for realistic calculations, since the computational cost scales exponentially as the system size is increased. For larger systems, approximate coupled cluster (CC) methods 1 are the standard, but these methods also have rather steep computational scaling with system size [e.g., O(M 7 ) for CCSD(T), CC with single and double excitations and perturbational triplets, where M is the number of basis functions]. For extended systems, less accurate approximations based on density functional theory (DFT) and timedependent DFT have been developed; GW and Bethe-Salpeter type methods have also been shown to be promising. 2 Correlated quantum chemistry methods have also been embedded in DFT calculations to treat extended systems. 3,4 The most commonly applied quantum Monte Carlo (QMC) method in electronic structure has been diffusion Monte Carlo (DMC), 5 which has also been used to compute excited states. 6,7 Compared to ground states, however, the accuracy of the results may depend on the symmetry 6 and show greater sensitivity 8 to the trial wave function used in the fixed-node approximation to control the sign problem and maintain orthogonality.
The recently developed phaseless auxiliary-field quantum Monte Carlo (AFQMC) method 9,10,11,12 is an orbital-based alternative many-body approach. AFQMC can be expressed with respect to any chosen single-particle basis (e.g., gaussians, planewaves, Wannier, etc.), and it exhibits favorable O(M 3 -M 4 ) scaling. For ground states, the new AFQMC method has been applied to close to 100 systems, including first-and second-row molecules, 11,12,13 transition metal oxide molecules, 10 simple solids, 9,14 post-d elements, 15 van der Waals systems, 16 and in molecules in which bonds are being stretched or broken. 17,18 In these calculations we have operated largely in an automated mode, inputting only the DFT or Hartree-Fock (HF) solutions as trial wave functions. The method demonstrated excellent accuracy, consistently able to correct errors in the mean-field trial wave function. In molecules, we have found that the accuracy of the phaseless AFQMC is comparable to CCSD(T) near equilibrium geometry and better when bonds are stretched. AFQMC thus provides new opportunities for the efficient and accurate manybody calculations of ground and excited states in both molecular and extended systems.
The seemingly simple C 2 molecule presents a significant challenge for many-body methods. 19,20 The C 2 molecule is difficult because of the strongly multireference nature of the ground state wave function [in which only ∼ 70% of the weight is the restricted Hartree-Fock (RHF) determinant] and the presence of nearby low-lying states. The shortcomings of standard quantum chemistry calculations for C 2 were shown by recent benchmark FCI calculations 19 of the potential energy curves (PECs) of its 1 Σ + g ground state and two lowlying singlet excited states. This benchmark shows that most correlated methods based on a single-determinant reference state wave function |Φ r exhibit large nonparallelity errors (NPE-defined as the difference between the maximum and minimum deviations from FCI along the PEC). Spin-restricted CCSD(T) [referred to as RCCSD(T) hereafter] was found to exhibit a large NPE of 98 mE h due to the poor behavior of RHF in the dissociation limit. Spin-unrestricted UCCSD(T), which is usually less accurate near equilibrium, has an NPE of 34 mE h . The excited state PECs are not accurately modeled by any of the commonly used single-reference methods, nor by CI including full quadruple substitutions. 19 Similarly, a recent DMC study 20 found that, even in its ground state at equilibrium geometry, the total energy of C 2 showed a large fixed-node error ∼ 40 mE h , if a single-determinant trial wave function is used.
As a new QMC method, the phaseless AFQMC provides an alternative route to the sign problem from fixed-node DMC. The random walks take place in a manifold of Slater determinants, in which fermion antisymmetry is automatically maintained in each walker. Applications have indicated that often this reduces the severity of the sign problem and, as a result, the phaseless approximation has weaker reliance on the trial wave function. It is interesting then to test the method for excited states, where QMC calculations depend more on the trial wave function and our knowledge of it is l
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