Efficient formulation of full configuration interaction quantum Monte Carlo in a spin eigenbasis via the graphical unitary group approach

Abstract

We provide a spin-adapted formulation of the Full Configuration Interaction Quantum Monte Carlo (FCIQMC) algorithm, based on the Graphical Unitary Group Approach (GUGA), which enables the exploitation of SU(2) symmetry within this stochastic framework. Random excitation generation and matrix element calculation on the Shavitt graph of GUGA can be efficiently implemented via a biasing procedure on the branching diagram. The use of a spin-pure basis explicitly resolves the different spin-sectors and ensures that the stochastically sampled wavefunction is an eigenfunction of the total spin operatorŜ 2 . The method allows for the calculation of states with low or intermediate spin in systems dominated by Hund's first rule, which are otherwise generally inaccessible. Furthermore, in systems with small spin gaps, the new methodology enables much more rapid convergence with respect to walker number and simulation time. Some illustrative applications of the GUGA-FCIQMC method are provided: computation of the 2 F − 4 F spin gap of the cobalt atom in large basis sets, achieving chemical accuracy to experiment, and the 1 Σ + g , 3 Σ + g , 5 Σ + g , 7 Σ + g spin-gaps of the stretched N 2 molecule, an archetypal strongly correlated system.

Resolution of Low-Energy States in Spin-Exchange Transition-Metal Clusters: Case Study of Singlet States in [Fe(III)4S4] Cubanes

“…A detailed analysis on the role of these terms within CI wave functions can be found in the literature. 71,74,75 In this work we adopt the GUGA formalism, [66][67][68][69][70][71] which uses spin-adapted basis functions known as configuration state functions (CSFs), denoted here as |µ . The total number of CSFs, f (N, n, S), dependent on the A genealogical branching diagram describes the spin coupling of a given unpaired electron with all the previous ones in a cumulative manner.…”

“…e ij,kl =Ê ijÊkl − δ jkÊil (5) are the spin-free excitation operators. The advantage of using this formulation in FCIQMC has been discussed in 70,71 , namely full spin symmetry is dynamically preserved in the QMC simulation, allowing to target specific spin states.…”

“…54 We demonstrate that our paradigm challenges this assumption, offering a viable theoretical route to solve the problem on classical computers with modest computational resources. Our discovery is highly advantageous for methods that exploit the sparsity of the CI Hamiltonian matrix and its eigensolutions, such as the spin-adapted Full Configuration Interaction Quantum Monte Carlo (FCIQMC) [55][56][57][58][59][60][61][62][63][64][65] within the Graphical Unitary Group Approach (GUGA) [66][67][68][69][70][71] , used in this work. This paradigm provides a direct understanding of the fundamental mechanisms that govern the electron interactions and are responsible for the electronic structure of PNTM clusters.…”

Comparison of many‐particle representations for selected‐CI I: A tree based approach

“…Stochastic and semi-stochastic approaches such as the full-CI quantum Monte Carlo (FCIQMC) of Alavi et al [20][21][22] The FCIQMC method has also been recently formulated using a spin-adapted basis based on the GUGA. 23 The heat-bath CI approach, which is a semi-stochastic determinant based approximate FCI method [24][25][26] and is closely related to the selected-CI approach of the Scemama et al [27][28][29] The FCIQMC has also been combined with the coupledcluster method by Piecuch et al 30,31 In this hybrid approach a stochastic estimation of the Coupled-Cluster (CC) amplitudes via an FCIQMC run is performed in order to use these approximate amplitudes to perform a deterministic coupled-cluster (FCIQMC + CC) calculation.…”

“…Significant variations in the core idea as well as in the algorithm have been proposed, partially for algorithmic reasons and partially in order to adapt the approach to the powerful computational hardware of current times. The new variations to the approximate FCI problem can be separated into three broad categories as follows: Stochastic and semi‐stochastic approaches such as the full‐CI quantum Monte Carlo (FCIQMC) of Alavi et al 20–22 The FCIQMC method has also been recently formulated using a spin‐adapted basis based on the GUGA 23 . The heat‐bath CI approach, which is a semi‐stochastic determinant based approximate FCI method 24–26 and is closely related to the selected‐CI approach of the Scemama et al 27–29 The FCIQMC has also been combined with the coupled‐cluster method by Piecuch et al 30,31 In this hybrid approach a stochastic estimation of the Coupled‐Cluster (CC) amplitudes via an FCIQMC run is performed in order to use these approximate amplitudes to perform a deterministic coupled‐cluster (FCIQMC + CC) calculation. Purely variational approaches such as the iterative CI 32 and SDS CI of Liu et al where the static and dynamic correlation are treated at the same footing by systematically revising the static correlation part of the wavefunction under the response of the dynamical correlation part 33 …”