Electronic Structure Theory has led to a variety of developments and applications. In the Nooijen group the focus is on the development and use of Coupled Cluster based approaches. Coupled Cluster is a very strong and accurate approach to the quantum mechanical problem. The research results presented in the thesis testify to the Similarity Transformed Equation of Motion Coupled Cluster (STEOM-CC) for being a very accurate and yet computationally inexpensive approach for excited states. This study reveals new features about STEOM and provides promise regarding future improvement in the methodology. STEOM can be used as the first step in the construction of the Vibronic model, which is a strong tool to move to paradigms beyond the Born-Oppenheimer approximation. Spin-Orbit Coupling (SOC) is a very important ingredient required to study relativistic phenomena and its quantum mechanical implementation for many body systems is not straightforward. The most widely used SOC operator in Chemical Physics is the Breit-Pauli operator, which requires employing non-trivial approximations to the Dirac equation to adapt the theory to many body systems. The integration of electronic structure approaches, Vibronic Coupling, and SOC is essential to study the phenomenon of intersystem crossing (transition between spin states) in fine detail. In this thesis a computational benchmark of STEOM is discussed, while the frameworks of Vibronic Coupling and Spin-Orbit Coupling (SOC) are considered on a theoretical level.

A novel method by which to calculate the near edge X-rayabsorption fine structure region of the X-ray absorption spectrum has been derived and implemented. By means of damped coupled cluster theory at coupled cluster levels CCS, CC2, CCSD and CCSDR(3), the spectra of neon and methane have been investigated. Using methods incorprating double excitations, the important relaxation effects maybe taken into account by simultaneous excitation of the core electron and relaxation of other electrons. An asymmetric Lanczos-chain driven approach has been utilized as a means to partially resolve the excitation space given by the coupled cluster Jacobian. The K-edge of the systems have been considered, and relativistic effects are estimated with use of the Douglas--Kroll scalar relativistic Hamiltonian. Comparisons have been made to results obtained with the four-component static-exchange approach and ionization potentials obtained by the {Delta}SCF-method. The appropriate basis sets by which to describe the core and excited states have been been determined.  The addition of core-polarizing functions and diffuse or Rydberg functions is important for this description. Scalar relativistic effects accounts for an increase in excitation energies due to the contraction of the 1s-orbital, and this increase is seen to be 0.88 eV for neon. The coupled cluster hierachy shows a trend of convergence towards the experimental spectrum, with an 1s -> 3p excitation energy for neon of an accuracy of 0.40 eV at a relativistic CCSDR(3) level of theory. Results obtained at the damped coupled cluster and STEX levels of theory, respectively, are seen to be in agreement, with a mere relative energy shift.

Electronic Structure Theory has led to a variety of developments and applications. In the Nooijen group the focus is on the development and use of Coupled Cluster based approaches. Coupled Cluster is a very strong and accurate approach to the quantum mechanical problem. The research results presented in the thesis testify to the Similarity Transformed Equation of Motion Coupled Cluster (STEOM-CC) for being a very accurate and yet computationally inexpensive approach for excited states. This study reveals new features about STEOM and provides promise regarding future improvement in the methodology. STEOM can be used as the first step in the construction of the Vibronic model, which is a strong tool to move to paradigms beyond the Born-Oppenheimer approximation. Spin-Orbit Coupling (SOC) is a very important ingredient required to study relativistic phenomena and its quantum mechanical implementation for many body systems is not straightforward. The most widely used SOC operator in Chemical Physics is the Breit-Pauli operator, which requires employing non-trivial approximations to the Dirac equation to adapt the theory to many body systems. The integration of electronic structure approaches, Vibronic Coupling, and SOC is essential to study the phenomenon of intersystem crossing (transition between spin states) in fine detail. In this thesis a computational benchmark of STEOM is discussed, while the frameworks of Vibronic Coupling and Spin-Orbit Coupling (SOC) are considered on a theoretical level.

Stuttgart, Univ., Diss., 2008.

University, Diss., 2003--Bonn.

The characteristics of irregular algorithms make a parallel implementation difficult, especially for PC clusters or clusters of SMPs. These characteristics may include an unpredictable access behavior to dynamically changing data structures or strong irregular coupling of computations. Problems are an unknown load distribution and expensive irregular communication patterns for data accesses and redistributions. Thus the parallel implementation of irregular algorithms on distributed memory machines and clusters requires a special organizational mechanism for a dynamic load balance while keeping the communication and administration overhead low. We propose task pool teams for implementing irregular algorithms on clusters of PCs or SMPs. A task pool team combines multithreaded programming using task pools on single nodes with explicit message passing between different nodes. The dynamic load balance mechanism of task pools is generalized to a dynamic load balance scheme for all distributed nodes. We have implemented and compared several versions for task pool teams. As application example, we use the hierarchical radiosity algorithm, which is based on dynamically growing quadtree data structures annotated by varying interaction lists expressing the irregular coupling between the quadtrees. Experiments are performed on a PC cluster and a cluster of SMPs.

We investigate the quantum dynamics of the spins on different Heisenberg antiferromagnetic spin lattice systems. Firstly, we applied the coupled-cluster method to the spin-1/2 antiferromagnetic XXZ model on a square lattice by employing an approximation which contains two-body long-range correlations and high-order four-body local correlations. Improvement is found for the ground-state energy, sublattice magnetization, and the critical anisotropy when comparing with the approximation including the two-body correlations alone. We also obtain the full excitation spectrum which is in good agreement with the quantum Monte Carlo results and the high-order spin-wave theory. Secondly, we study the longitudinal excitations of quantum antiferromagnets on a triangular lattice by a recently proposed microscopic many-body approach based on magnon-density waves. We calculate the full longitudinal excitation spectra of the antiferromagnetic Heisenberg model for a general spin quantum number in the isotropic limit. Similar to the square lattice model, we find that, at the center of the first hexagonal Brillouin zone Γ(q=0) and at the magnetic ordering wavevectors ±[Q= (4π/3,0)], the excitation spectra become gapless in the thermodynamic limit, due to the slow, logarithmic divergence of the structure factor. However, these longitudinal modes on two-dimensional models may be considered as quasi-gapped, as any finite-size effect or small anisotropy will induce a large energy gap, when compared with the counterpart of the transverse spin-wave excitations. We have also investigated the excited states of the quasi-one-dimensional quantum antiferromagnets on hexagonal lattices, including the longitudinal modes based on the magnon-density waves. A model Hamiltonian with a uniaxial single-ion anisotropy is first studied by a spin-wave theory based on the one-boson method; the ground state thus obtained is employed for the study of the longitudinal modes. The full energy spectra of both the transverse modes (i.e., magnons) and the longitudinal modes are obtained as functions of the nearest-neighbor coupling and the anisotropy constants. We have found two longitudinal modes due to the non-collinear nature of the triangular antiferromagnetic order, similar to that of the phenomenological field theory approach by Affleck. The excitation energy gaps due to the anisotropy and the energy gaps of the longitudinal modes without anisotropy are then investigated. We then compares our results for the longitudinal energy gaps at the magnetic wavevectors with the experimental results for several antiferromagnetic compounds with both integer and non-integer spin quantum numbers, and we find good agreements after the higher-order contributions are included in our calculations.

Multireference coupled cluster (MRCC) methods are a highly accurate approach for sys- tems with quasi-degeneracies, where the static correlation plays an important role. How- ever, while canonical MRCC is successful for many systems, it can be used only for small sized systems. Nonetheless, it was shown that large systems can be described by the domain-based local pair natural orbital approach (DLPNO). In our group, we developed DLPNO-MkCCSD, DLPNO-TCCSD and DLPNO-MkCCSD(T) methods, which were able to recover more than 99.7% of the canonical correlation energy, while the computation of systems with more than 2000 basis functions took only a few hours on a single CPU core. Moreover, we also implemented a tailored variant of MRCC which successfully described excited states of cyclobutadiene, while the traditional MRCC under-performed.

There are a lot of interesting problems in surface chemistry where quantum chemistry could give great insight, like reaction mechanisms in heterogeneous catalysis, the effect of surface functionalization on semiconductors, or the influence of defects on the reactivity of crystal surfaces. Plane wave based methods applied to crystals cannot handle problems that are localized in nature like surface defects and adsorbates. On the other hand, molecular electronic structure techniques, which describe these effects and the locality of the electronic correlation well, are too computationally expensive to use on these systems. In this work, we introduce translationally-transformed coupled-cluster (TT-CC) theory, a new electronic structure method that incorporates the periodicity of crystals and the locality of electronic correlation. This is accomplished by encoding the periodicity into the amplitudes, instead of using plane waves, in order to be able to use a local basis to reflect the decay of the electronic correlation at sufficiently large distances. This avoids the calculation of redundant amplitudes. Perfectly periodic surfaces are envisioned as reference wavefunctions for localized defects and chemical reactions. The working equations in one dimension are derived starting from the amplitude equations of conventional coupled cluster singles and doubles (CCSD) on an infinite system and rearranging them such that the distance to an anonymous cell is an explicit degree of freedom, L. The formally infinite summations can be truncated by systematically neglecting numerically insignificant amplitudes. The generalization of the amplitude equations to higher dimensions is straightforward, albeit laborious. We show a general strategy to incorporate defects. These will be subjects of future dissertations. We present a proof of principle for 1-dimensional chemical systems of increasing size (He, H2, Be, Ne and N2) using the 6-31G basis set. We compute the energies, with TT-CCSD, at different distances and compared them against the perfectly periodic intensive energy (PPIE) using conventional CCSD. All results, up to L=3, show that the energies of TT-CCSD converge to the PPIE. For neon, TT-CCSD shows an error of -6.2x10-6 Eh per cell against the PPIE at the bonding distance with the potential computational cost of 7 cells using CCSD, as an upper bound. For nitrogen, TT-CCSD shows an error of -2.2x10-9 Eh at 7.5 Å per cell with the same potential cost as upper bound.

Despite the fact that higher-order tensors (HOTs) plague electronic structure methods and severely limits the modeling of interesting chemistry problems, introduction and application of higher-order tensor (HOT) decompositions, specifically the canonical polyadic (CP) decomposition, is fairly limited. The CP decomposition is an incredibly useful sparse tensor factorization that has the ability to disentangle all correlated modes of a tensor. However the complexities associated with CP decomposition have made its application in electronic structure methods difficult. Some of the major issues related to CP decomposition are a product of the mathematics of computing the decomposition: determining the exact CP rank is a non-polynomially hard problem, finding stationary points for rank-R approximations require non-linear optimization techniques, and inexact CP approximations can introduce a large degree of error into tensor networks. While other issues are a result of the construction of computer architectures. For example, computer processing units (CPUs) are organized in a way to maximize the efficiency of dense linear algebra and, thus, the performance of routine tensor algebra kernels, like the Khatri-Rao product, is limited. In this work, we seek to reduce the complexities associated with the CP decomposition and create a route for others to develop reduced-scaling electronic structure theory methods using the CP decomposition. In Chapter 2, we introduce the robust tensor network approximation. This approximation is a way to, in general, eliminate the leading-order error associated with approximated tensors in a network. We utilize the robust network approximation to significantly increase the accuracy of approximating density fitting (DF) integral tensors using rank-deficient CP decompositions in the particle-particle ladder (PPL) diagram of the coupled cluster method with single and double substitutions (CCSD). We show that one can produce results with negligible error in chemically relevant energy differences using a CP rank roughly the same size as the DF fitting basis; which is a significantly smaller rank requirement than found using either a nonrobust approximation or similar grid initialized CP approximations (the pseudospectral (PS) and tensor hypercontraction (THC) approximations). Introduction of the CP approximation, formally, reduces the complexity of the PPL diagram from 𝓞(N⁶) to 𝓞(N⁵) and, using the robust approximation, we are able to observe a cost reduction in CCSD calculations for systems as small as a single water molecule. In Chapter 3, we further demonstrate the utility of the robust network approximation and, in addition, we construct a scheme to optimize a grid-free CP decomposition of the order-four Coulomb integral tensor in 𝓞(N⁴) time. Using these ideas, we reduce the complexity of ten bottleneck contractions from 𝓞(N⁶) to 𝓞(N⁵) in the Laplace transform (LT) formulation of the perturbative triple, (T), correction to CCSD. We show that introducing CP into the LT (T) method with a CP rank roughly the size of the DF fitting basis reduces the cost of computing medium size molecules by a factor of about 2.5 and introduces negligible error into chemically relevant energy differences. Furthermore, we implement these low-cost algorithms using newly developed, optimized tensor algebra kernels in the massively-parallel, block-sparse TiledArray [Calvin, et. al Chemical Reviews 2021 121 (3), 1203-1231] tensor framework. / Doctor of Philosophy / Electronic structure methods and accurate modeling of quantum chemistry have developed alongside the advancements in computer infrastructures. Increasingly large and efficient computers have allowed researchers to model remarkably large chemical systems. Sadly, for as fast as computer infrastructures grow (Moores law predicts that the number of transistors in a computer will double every 18 months) the cost of electronic structure methods grows more quickly. One of the least expensive electronic structure methods, Hartree Fock (HF), grows quartically with molecular size; this means that doubling the size of a molecule increase the number of computer operations by a factor of 16. However, it is known that when chemical systems become sufficiently large, the amount of physical information added to the system grows linearly with system size.[Goedecker, et. al. Comput. Sci. Eng., 2003, 5, (4), 14-21] Unfortunately, standard implementations of electronic structure methods will never achieve linear scaling; the disparity between actual cost and physical scaling of molecules is a result of storing and manipulating data using dense tensors and is known as the curse of dimensionality.[Bellman, Adaptive Control Processes, 1961, 2045, 276] Electronic structure theorists, in their desire to apply accurate methods to increasingly large systems, have known for some time that the cost of conventional algorithms is unreasonably high. These theorists have found that one can reveal sparsity and develop reduced-complexity algorithms using matrix decomposition techniques. However, higher-order tensors (HOTs), tensors with more than two modes, are routinely necessary in algorithm formulations. Matrix decompositions applied to HOTs are not necessarily straight-forward and can have no effect on the limiting behavior of an algorithm. For example, because of the positive definiteness of the Coulomb integral tensor, it is possible to perform a Cholesky decomposition (CD) to reduce the complexity of tensor from an order-4 tensor to a product of order-3 tensors.[Beebe, et. al. Int. J. Quantum Chem., 1977, 12, 683-705] However, using the CD approximated Coulomb integral tensors it is not possible to reduce the complexity of popular methods such as Hartree-Fock or coupled cluster theory. We believe that the next step to reducing the complexity of electronic structure methods is through the accurate application of HOT decompositions. In this work, we only consider a single HOT decomposition: the canonical polyadic (CP) decomposition which represents a tensor as a polyadic sum of products. The CP decomposition disentangles all modes of a tensor by representing an order-N tensor as N order-2 tensors. In this work, we construct the CP decomposition of tensors using algebraic optimization. Our goal, here, is to tackle one of the biggest issues associated with the CP decomposition: accurately approximating tensors and tensor networks. In Chapter 2, we develop a robust formulation to approximate tensor networks, a formulation which removes the leading-order error associated with tensor approximations in a network.[Pierce, et. al. J. Chem. Theory Comput., 2021 17 (4), 2217- 2230] We apply a robust CP approximation to the coupled cluster method with single and double substitutions (CCSD) to reduce the overall cost of the approach. Using this robust CP approximation we can compute CCSD, on average, 2.5-3 times faster and introduce negligibly small error in chemically relevant energy values. Furthermore in Chapter 3, we again use the robust CP network approximation in conjunction with a novel, low cost approach to compute order-four CP decompositions, to reduce the cost of 10 high cost computations in the the perturbative triple, (T), correction to CCSD. By removing these computations, we are able to reduce the cost of (T) by a factor of about 2.5 while introducing significantly small error.