The concept of symmetry is of paramount importance in physics and chemistry. The exploitation of the inherent symmetries and corresponding conservation laws in electronic structure calculations not only reduces the degrees of freedom by block-diagonalization of the Hamiltonian into different symmetry sectors but also ensures the conservation of “good” quantum numbers and, thus, the physical correctness of calculated quantities. It also allows us to target a specific many-body subspace of the problem at hand. Commonly utilized symmetries in electronic structure calculations are discrete translational and point group symmetries, angular momentum, and z-projection of the total spin. Due to a non-straight-forward implementation and accompanying increased computational cost, one often ignored symmetry is the global SU(2) spin-rotation symmetry of spin-preserving, nonrelativistic Hamiltonians, common to many molecular systems studied.
In addition to the abovementioned Hilbert space size reduction and conservation of the total spin S, solving for the eigenstates of the Hamiltonian in a spin-eigenbasis allows targeting distinct—even (near-)degenerate—spin eigenstates, which allows the calculation of spin gaps between states inaccessible otherwise, and facilitates a correct physical interpretation of calculations and description of chemical processes governed by the intricate interplay between them. Moreover, by working in a specific spin sector, convergence of projective techniques that rely on the repeated application of a propagator to an evolving wavefunction is greatly improved, especially where there are near spin-degeneracies in the exact spectrum.
The Full Configuration Interaction Quantum Monte Carlo (FCIQMC) approach is a method that benefits from working in a spin-pure many-body basis. In my PhD, I implemented FCIQMC in a fully spin-adapted basis based on the unitary group approach. I applied the above methodology to obtain results not otherwise attainable with conventional approaches for the spin-gap of the high-spin cobalt atom ground- and low-spin excited state and the electron affinity of scandium within chemical accuracy to experiment. Furthermore, we establish the ordering of the scandium anion bound states, which has until now not been experimentally determined. Recently, my spin-adapted implementation of FCIQMC enabled us to study the magnetic properties and spin interactions of transition-metal complexes in the form of iron-sulfur cluster, Nagaoka ferromagnetism in the hole-doped Hubbard model and a combination of the unitary and symmetric group approach for low-dimensional spin systems, possibly extending our method to allow the study of spin-liquid phases.
Two-dimensional Hubbard lattices with two or three holes are investigated as a function of U in the large-U limit. In the so-called Nagaoka limit (one-hole system at infinite U), it is known that the Hubbard model exhibits a ferromagnetic ground state. Here, by means of exact full configuration interaction quantum Monte Carlo simulations applied to periodic lattices up to 24 sites, we compute spin-spin correlation functions as a function of increasing U. The correlation functions clearly demonstrate the onset of ferromagnetic domains, centered on individual holes. The overall total spin of the wave functions remains the lowest possible (0 or 12, depending on the number of holes). The ferromagnetic domains appear at interaction strengths comparable to the critical interaction strengths of the Nagaoka transition in finite systems with strictly one hole. The existence of such ferromagnetic domains is the signature of Nagaoka physics in Hubbard systems with a small (but greater than 1) number of holes.
@article{PhysRevB.107.064405,title={Ferromagnetic domains in the large-$U$ Hubbard model with a few holes: A full configuration interaction quantum Monte Carlo study},author={Yun, Sujun and Dobrautz, Werner and Luo, Hongjun and Katukuri, Vamshi and Liebermann, Niklas and Alavi, Ali},journal={Phys. Rev. B},volume={107},issue={6},pages={064405},numpages={9},year={2023},month=feb,publisher={American Physical Society},doi={10.1103/PhysRevB.107.064405},url={https://link.aps.org/doi/10.1103/PhysRevB.107.064405},dimensions={true},project_guga={true},project_hubbard={true}}
A novel combined unitary and symmetric group approach is used to study the spin-12 Heisenberg model and related Fermionic systems in a total spin-adapted representation, using a linearly-parameterised Ansatz for the many-body wave function. We show that a more compact ground-state wave function representation—indicated by a larger leading ground-state coefficient—is obtained when combining the symmetric group Sn, in the form of permutations of the underlying lattice site ordering, with the cumulative spin coupling based on the unitary group U(n). In one-dimensional systems the observed compression of the wave function is reminiscent of block-spin renormalization group approaches, and allows us to study larger lattices (here taken up to 80 sites) with the spin-adapted full configuration interaction quantum Monte Carlo method, which benefits from the sparsity of the Hamiltonian matrix and the corresponding sampled eigenstates that emerge from the reordering. We find that in an optimal lattice ordering the configuration state function with highest weight already captures with high accuracy the spin-spin correlation function of the exact ground-state wave function. This feature is found for more general lattice models, such as the Hubbard model, and ab initio quantum chemical models, exemplified by one-dimensional hydrogen chains. We also provide numerical evidence that the optimal lattice ordering for the unitary group approach is not generally equivalent to the optimal ordering obtained for methods based on matrix-product states, such as the density-matrix renormalization group approach.
@article{PhysRevB.105.195123,title={Combined unitary and symmetric group approach applied to low-dimensional Heisenberg spin systems},author={Dobrautz, Werner and Katukuri, Vamshi M. and Bogdanov, Nikolay A. and Kats, Daniel and Li Manni, Giovanni and Alavi, Ali},journal={Phys. Rev. B},volume={105},issue={19},pages={195123},numpages={17},year={2022},month=may,publisher={American Physical Society},doi={10.1103/PhysRevB.105.195123},url={https://link.aps.org/doi/10.1103/PhysRevB.105.195123},dimensions={true},project_guga={true}}
We investigate Nagaoka ferromagnetism in the two-dimensional Hubbard model with one hole using the spin-adapted [SU(2) conserving] full configuration interaction quantum Monte Carlo method. This methodology gives us access to the ground-state energies of all possible spin states S of finite Hubbard lattices, here obtained for lattices up to 26 sites for various interaction strengths (U). The critical interaction strength, Uc, at which the Nagaoka transition occurs is determined for each lattice and is found to be proportional to the lattice size for the larger lattices. Below Uc, the overall ground states are found to favour the minimal total spin (S=12), and no intermediate spin state is found to be the overall ground state on lattices larger than 16 sites. However, at Uc, the energies of all the spin states are found to be nearly degenerate, implying that large fluctuations in total spin can be expected in the vicinity of the Nagaoka transition.
@article{PhysRevB.104.235102,title={Benchmark study of Nagaoka ferromagnetism by spin-adapted full configuration interaction quantum Monte Carlo},author={Yun, Sujun and Dobrautz, Werner and Luo, Hongjun and Alavi, Ali},journal={Phys. Rev. B},volume={104},issue={23},pages={235102},numpages={6},year={2021},month=dec,publisher={American Physical Society},doi={10.1103/PhysRevB.104.235102},url={https://link.aps.org/doi/10.1103/PhysRevB.104.235102},dimensions={true},project_guga={true},project_hubbard={true}}
Polynuclear transition-metal (PNTM) clusters owe their catalytic activity to numerous energetically low-lying spin states and stable oxidation states. The characterization of their electronic structure represents one of the greatest challenges of modern chemistry. We propose a theoretical framework that enables the resolution of targeted electronic states with ease and apply it to two [Fe(III)4S4] cubanes. Through direct access to their many-body wave functions, we identify important correlation mechanisms and their interplay with the geometrical distortions observed in these clusters, which are core properties in understanding their catalytic activity. The simulated magnetic coupling constants predicted by our strategy allow us to make qualitative connections between spin interactions and geometrical distortions, demonstrating its predictive power. Moreover, despite its simplicity, the strategy provides magnetic coupling constants in good agreement with the available experimental ones. The complexes are intrinsically frustrated anti-ferromagnets, and the obtained spin structures together with the geometrical distortions represent two possible ways to release spin frustration (spin-driven Jahn–Teller distortion). Our paradigm provides a simple, yet rigorous, route to uncover the electronic structure of PNTM clusters and may be applied to a wide variety of such clusters.
@article{LiManni2022,doi={10.1021/acs.jpca.1c00397},url={https://doi.org/10.1021/acs.jpca.1c00397},year={2021},month=may,publisher={American Chemical Society ({ACS})},volume={125},number={22},pages={4727--4740},author={Manni, Giovanni Li and Dobrautz, Werner and Bogdanov, Nikolay A. and Guther, Kai and Alavi, Ali},title={Resolution of Low-Energy States in Spin-Exchange Transition-Metal Clusters: Case Study of Singlet States in [Fe({III})$_4$S$_4$] Cubanes},journal={The Journal of Physical Chemistry A},dimensions={true},project_guga={true},project_tm={true},chemrxiv={https://chemrxiv.org/engage/chemrxiv/article-details/60c75153bb8c1a47713dbc97}}
In this work, we demonstrate how to efficiently compute the one- and two-body reduced density matrices within the spin-adapted full configuration interaction quantum Monte Carlo (FCIQMC) method, which is based on the graphical unitary group approach (GUGA). This allows us to use GUGA-FCIQMC as a spin-pure configuration interaction (CI) eigensolver within the complete active space self-consistent field (CASSCF) procedure and hence to stochastically treat active spaces far larger than conventional CI solvers while variationally relaxing orbitals for specific spin-pure states. We apply the method to investigate the spin ladder in iron–sulfur dimer and tetramer model systems. We demonstrate the importance of the orbital relaxation by comparing the Heisenberg model magnetic coupling parameters from the CASSCF procedure to those from a CI-only (CASCI) procedure based on restricted open-shell Hartree–Fock orbitals. We show that the orbital relaxation differentially stabilizes the lower-spin states, thus enlarging the coupling parameters with respect to the values predicted by ignoring orbital relaxation effects. Moreover, we find that, while CASCI results are well fit by a simple bilinear Heisenberg Hamiltonian, the CASSCF eigenvalues exhibit deviations that necessitate the inclusion of biquadratic terms in the model Hamiltonian.
@article{Dobrautz2021,doi={10.1021/acs.jctc.1c00589},url={https://doi.org/10.1021/acs.jctc.1c00589},year={2021},month=sep,publisher={American Chemical Society ({ACS})},volume={17},number={9},pages={5684--5703},author={Dobrautz, Werner and Weser, Oskar and Bogdanov, Nikolay A. and Alavi, Ali and Manni, Giovanni Li},title={Spin-Pure Stochastic-{CASSCF} via {GUGA}-{FCIQMC} Applied to Iron{\textendash}Sulfur Clusters},journal={Journal of Chemical Theory and Computation},dimensions={true},project_guga={true},project_devel={true},project_tm={true},project_fciqmc={true}}
We present a protocol based on unitary transformations of molecular orbitals to reduce the number of nonvanishing coefficients of spin-adapted configuration interaction expansions. Methods that exploit the sparsity of the Hamiltonian matrix and compactness of its eigensolutions, such as the full configuration interaction quantum Monte Carlo (FCIQMC) algorithm in its spin-adapted implementation, are well suited to this protocol. The wave function compression resulting from this approach is particularly attractive for antiferromagnetically coupled polynuclear spin systems, such as transition-metal cubanes in biocatalysis, and Mott and charge-transfer insulators in solid-state physics. Active space configuration interaction calculations on N2 and CN– at various bond lengths, the stretched square N4 compounds, the chromium dimer, and a [Fe2S2]2– model system are presented as a proof-of-concept. For the Cr2 case, large and intermediate bond distances are discussed, showing that the approach is effective in cases where static and dynamic correlations are equally important. The [Fe2S2]2– case shows the general applicability of the method.
@article{LiManni2020,doi={10.1021/acs.jctc.9b01013},url={https://doi.org/10.1021/acs.jctc.9b01013},year={2020},month=feb,publisher={American Chemical Society ({ACS})},volume={16},number={4},pages={2202--2215},author={Manni, Giovanni Li and Dobrautz, Werner and Alavi, Ali},title={Compression of Spin-Adapted Multiconfigurational Wave Functions in Exchange-Coupled Polynuclear Spin Systems},journal={Journal of Chemical Theory and Computation},dimensions={true},project_guga={true},project_tm={true},chemrxiv={https://chemrxiv.org/engage/chemrxiv/article-details/60c7451abb8c1a16363da600}}
We present NECI, a state-of-the-art implementation of the Full Configuration Interaction Quantum Monte Carlo (FCIQMC) algorithm, a method based on a stochastic application of the Hamiltonian matrix on a sparse sampling of the wave function. The program utilizes a very powerful parallelization and scales efficiently to more than 24 000 central processing unit cores. In this paper, we describe the core functionalities of NECI and its recent developments. This includes the capabilities to calculate ground and excited state energies, properties via the one- and two-body reduced density matrices, as well as spectral and Green’s functions for ab initio and model systems. A number of enhancements of the bare FCIQMC algorithm are available within NECI, allowing us to use a partially deterministic formulation of the algorithm, working in a spin-adapted basis or supporting transcorrelated Hamiltonians. NECI supports the FCIDUMP file format for integrals, supplying a convenient interface to numerous quantum chemistry programs, and it is licensed under GPL-3.0.
@article{Guther2020,doi={10.1063/5.0005754},url={https://doi.org/10.1063/5.0005754},year={2020},month=jul,publisher={{AIP} Publishing},volume={153},number={3},pages={034107},author={Guther, Kai and Anderson, Robert J. and Blunt, Nick S. and Bogdanov, Nikolay A. and Cleland, Deidre and Dattani, Nike and Dobrautz, Werner and Ghanem, Khaldoon and Jeszenszki, Peter and Liebermann, Niklas and Manni, Giovanni Li and Lozovoi, Alexander Y. and Luo, Hongjun and Ma, Dongxia and Merz, Florian and Overy, Catherine and Rampp, Markus and Samanta, Pradipta Kumar and Schwarz, Lauretta R. and Shepherd, James J. and Smart, Simon D. and Vitale, Eugenio and Weser, Oskar and Booth, George H. and Alavi, Ali},title={{NECI}: N-Electron Configuration Interaction with an emphasis on state-of-the-art stochastic methods},journal={The Journal of Chemical Physics},dimensions={true},project_guga={true},project_devel={true},project_fciqmc={true}}
After introducing the fundamental goals—solving the Schrödinger equation—and the associated problems of quantum chemistry, we describe the basics of multiconfigurational approaches to solve the latter. As an exact—or full configuration interaction (FCI)—solution, even in a finite basis set, comes with an exponential scaling cost, the importance of an efficient representation in either a Slater determinant or configuration state function basis is discussed. With the help of such an efficient representation it is possible to apply iterative techniques, like the Davidson method, to obtain the exact solution of the most important low-lying eigenstates of the Hamiltonian, describing a quantum chemical system. As the exponential scaling still restricts these direct approaches to rather modest system sizes, we discuss in depth the multi-configurational extension of the self-consistent field method (MCSCF), which captures the static correlation of a problem and serves as a starting point for many more elaborate techniques. In addition, we present the complete active space approach—and the generalized and restricted extensions thereof—, which allows an intuitive construction of the chemically important reference space and enables a much more compact description of the important degrees of freedom of a problem at hand. We explain the state-specific and state-averaged approaches to obtain excited states within the MCSCF method and conclude this chapter by presenting stochastic Monte-Carlo approaches to solve the FCI problem for unprecedented active space sizes.
@misc{LiManni2021,doi={10.1002/9781119417774.ch6},url={https://doi.org/10.1002/9781119417774.ch6},year={2020},month=nov,publisher={Wiley},pages={133--203},author={Manni, Giovanni Li and Guther, Kai and Ma, Dongxia and Dobrautz, Werner},title={Foundation of Multi-Configurational Quantum Chemistry},dimensions={true},project_guga={true}}
2019
PhD Thesis
Development of Full Configuration Interaction Quantum Monte Carlo Methods for Strongly Correlated Electron Systems
Full Configuration Interaction Quantum Monte Carlo (FCIQMC) is a prominent method to calculate the exact solution of the Schrödinger equation in a finite antisymmetric basis and gives access to physical observables through an efficient stochastic sampling of the wavefunction that describes a quantum mechanical system. Although system-agnostic (black-box-like) and numerically exact, its effectiveness depends crucially on the compactness of the wavefunction: a property that gradually decreases as correlation effects become stronger. In this work, we present two -conceptually distinct- approaches to extend the applicability of FCIQMC towards larger and more strongly correlated systems. In the first part, we investigate a spin-adapted formulation of the FCIQMC algorithm, based on the Unitary Group Approach. Exploiting the inherent symmetries of the nonrelativistic molecular Hamiltonian results in a dramatic reduction of the effective Hilbert space size of the problem. The use of a spin-pure basis explicitly resolves the different spin-sectors, even when degenerate, and the absence of spin-contamination ensures the sampled wavefunction is an eigenfunction of the total spin operator. Moreover, targeting specific many-body states with conserved total spin allows an accurate description of chemical processes governed by the intricate interplay of them. We apply the above methodology to obtain results, not otherwise attainable with conventional approaches, for the spin-gap of the high-spin cobalt atom ground- and low-spin excited state and the electron affinity of scandium within chemical accuracy to experiment. Furthermore we establish the ordering of the scandium anion bound states, which has until now not been experimentally determined. In the second part, we investigate a methodology to explicitly incorporate electron correlation into the initial Ansatz of the ground state wavefunction. Such an Ansatz induces a compact description of the wavefunction, which ameliorates the sampling of the configuration space of a system with FCIQMC. Within this approach, we investigate the two-dimensional Hubbard model near half-filling in the intermediate interaction regime, where such an Ansatz can be exactly incorporated by a nonunitary similarity transformation of the Hamiltonian based on a Gutzwiller correlator. This transformation generates novel three-body interactions, tractable due to the stochastic nature of FCIQMC, and leads to a non-Hermitian effective Hamiltonian with extremely compact right eigenvectors. The latter fact allows application of FCIQMC to larger lattice sizes, well beyond the reach of the method applied to the original Hubbard Hamiltonian.
@phdthesis{dobrautz-phd,title={Development of Full Configuration Interaction Quantum Monte Carlo Methods for Strongly Correlated Electron Systems},author={Dobrautz, Werner},school={University of Stuttgart},year={2019},month=mar,url={http://dx.doi.org/10.18419/opus-10593},doi={10.18419/opus-10593},project_devel={true},project_fciqmc={true},project_guga={true},project_tc={true},project_thesis={true},owner={dobrautz},timestamp={2019.04.16}}
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 operator Ŝ 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 2F − 4F spin gap of the cobalt atom in large basis sets, achieving chemical accuracy to experiment, and the Σg+1, Σg+3, Σg+5, and Σg+7 spin-gaps of the stretched N2 molecule, an archetypal strongly correlated system.
@article{Dobrautz2019,doi={10.1063/1.5108908},url={https://doi.org/10.1063/1.5108908},year={2019},month=sep,publisher={{AIP} Publishing},volume={151},number={9},author={Dobrautz, Werner and Smart, Simon D. and Alavi, Ali},title={Efficient formulation of full configuration interaction quantum Monte Carlo in a spin eigenbasis via the graphical unitary group approach},journal={The Journal of Chemical Physics},dimensions={true},project_guga={true},project_devel={true},project_fciqmc={true}}
