Tensor Product Multiscale Many-Particle Spaces with Finite-Order Weights for the Electronic Schrödinger Equation

We study tensor product multiscale many-particle spaces with finite-order weights and their application for the electronic Schrödinger equation. Any numerical solution of the electronic Schrödinger equation using conventional discretization schemes is impossible due to its high dimensionality. Therefore, typically Monte Carlo methods (VMC/DMC) or nonlinear model approximations like Hartree-Fock (HF), coupled cluster (CC) or density functional theory (DFT) are used. In this work we develop and implement in parallel a numerical method based on adaptive sparse grids and a particle-wise subspace splitting with respect to one-particle functions which stem from a nonlinear rank-1 approximation. Sparse grids allow to overcome the exponential complexity exhibited by conventional discretization procedures and deliver a convergent numerical approach with guaranteed convergence rates. In particular, the introduced weighted many-particle tensor product multiscale approximation spaces include the common configuration interaction (CI) spaces as a special case. To realize our new approach, we first introduce general many-particle Sobolev spaces, which particularly include the standard Sobolev spaces as well as Sobolev spaces of dominated mixed smoothness. For this novel variant of sparse grid spaces we show estimates for the approximation and complexity orders with respect to the smoothness and decay parameters. With known regularity properties of the electronic wave function it follows that, up to logarithmic terms, the convergence rate is independent of the number of electrons and almost the same as in the two-electron case. However, besides the rate, also the dependence of the complexity constants on the number of electrons plays an important role for a truly practical method. Based on a splitting of the one-particle space we construct a subspace splitting of the many-particle space, which particularly includes the known ANOVA decomposition, the HDMR decomposition and the CI decomposition as special cases. Additionally, we introduce weights for a restriction of this subspace splitting. In this way weights of finite order q lead to many-particle spaces in which the problem of an approximation of an N-particle function reduces to the problem of the approximation of q-particle functions. To obtain as small as possible constants with respect to the cost complexity, we introduce a heuristic adaptive scheme to build a sequence of finite-dimensional subspaces of a weighted tensor product multiscale many-particle approximation space. Furthermore, we construct a multiscale Gaussian frame and apply Gaussians and modulated Gaussians for the nonlinear rank-1 approximation. In this way, all matrix entries of the corresponding discrete eigenvalue problem can be computed in terms of analytic formulae for the one and two particle operator integrals. Finally, we apply our novel approach to small atomic and diatomic systems with up to 6 electrons (18 space dimensions). The numerical results demonstrate that our new method indeed allows for convergence with expected rates.Tensorprodukt-Multiskalen-Mehrteilchenräume mit Gewichten endlicher Ordnung für die elektronische Schrödingergleichung In der vorliegenden Arbeit beschäftigen wir uns mit gewichteten Tensorprodukt-Multiskalen-Mehrteilchen-Approximationsräumen und deren Anwendung zur numerischen Lösung der elektronischen Schrödinger-Gleichung. Aufgrund der hohen Problemdimension ist eine direkte numerische Lösung der elektronischen Schrödinger-Gleichung mit Standard-Diskretisierungsverfahren zur linearen Approximation unmöglich, weshalb üblicherweise Monte Carlo Methoden (VMC/DMC) oder nichtlineare Modellapproximationen wie Hartree-Fock (HF), Coupled Cluster (CC) oder Dichtefunktionaltheorie (DFT) verwendet werden. In dieser Arbeit wird eine numerische Methode auf Basis von adaptiven dünnen Gittern und einer teilchenweisen Unterraumzerlegung bezüglich Einteilchenfunktionen aus einer nichtlinearen Rang-1 Approximation entwickelt und für parallele Rechnersysteme implementiert. Dünne Gitter vermeiden die in der Dimension exponentielle Komplexität üblicher Diskretisierungsmethoden und führen zu einem konvergenten numerischen Ansatz mit garantierter Konvergenzrate. Zudem enthalten unsere zugrunde liegenden gewichteten Mehrteilchen Tensorprodukt-Multiskalen-Approximationsräume die bekannten Configuration Interaction (CI) Räume als Spezialfall. Zur Konstruktion unseres Verfahrens führen wir zunächst allgemeine Mehrteilchen-Sobolevräume ein, welche die Standard-Sobolevräume sowie Sobolevräume mit dominierender gemischter Glattheit beinhalten. Wir analysieren die Approximationseigenschaften und schätzen Konvergenzraten und Kostenkomplexitätsordnungen in Abhängigkeit der Glattheitsparameter und Abfalleigenschaften ab. Mit Hilfe bekannter Regularitätseigenschaften der elektronischen Wellenfunktion ergibt sich, dass die Konvergenzrate bis auf logarithmische Terme unabhängig von der Zahl der Elektronen und fast identisch mit der Konvergenzrate im Fall von zwei Elektronen ist. Neben der Rate spielt allerdings die Abhängigkeit der Konstanten in der Kostenkomplexität von der Teilchenzahl eine wichtige Rolle. Basierend auf Zerlegungen des Einteilchenraumes konstruieren wir eine Unterraumzerlegung des Mehrteilchenraumes, welche insbesondere die bekannte ANOVA-Zerlegung, die HDMR-Zerlegung sowie die CI-Zerlegung als Spezialfälle beinhaltet. Eine zusätzliche Gewichtung der entsprechenden Unterräume mit Gewichten von endlicher Ordnung q führt zu Mehrteilchenräumen, in denen sich das Approximationsproblem einer N-Teilchenfunktion zu Approximationsproblemen von q-Teilchenfunktionen reduziert. Mit dem Ziel, Konstanten möglichst kleiner Größe bezüglich der Kostenkomplexität zu erhalten, stellen wir ein heuristisches adaptives Verfahren zur Konstruktion einer Sequenz von endlich-dimensionalen Unterräumen eines gewichteten Mehrteilchen-Tensorprodukt-Multiskalen-Approximationsraumes vor. Außerdem konstruieren wir einen Frame aus Multiskalen-Gauss-Funktionen und verwenden Einteilchenfunktionen im Rahmen der Rang-1 Approximation in der Form von Gauss- und modulierten-Gauss-Funktionen. Somit können die zur Aufstellung der Matrizen des zugehörigen diskreten Eigenwertproblems benötigten Ein- und Zweiteilchenintegrale analytisch berechnet werden. Schließlich wenden wir unsere Methode auf kleine Atome und Moleküle mit bis zu sechs Elektronen (18 Raumdimensionen) an. Die numerischen Resultate zeigen, dass sich die aus der Theorie zu erwartenden Konvergenzraten auch praktisch ergeben

Grid-based methods for chemistry simulations on a quantum computer

First-quantized, grid-based methods for chemistry modeling are a natural and elegant fit for quantum computers. However, it is infeasible to use today’s quantum prototypes to explore the power of this approach because it requires a substantial number of near-perfect qubits. Here, we use exactly emulated quantum computers with up to 36 qubits to execute deep yet resource-frugal algorithms that model 2D and 3D atoms with single and paired particles. A range of tasks is explored, from ground state preparation and energy estimation to the dynamics of scattering and ionization; we evaluate various methods within the split-operator QFT (SO-QFT) Hamiltonian simulation paradigm, including protocols previously described in theoretical papers and our own techniques. While we identify certain restrictions and caveats, generally, the grid-based method is found to perform very well; our results are consistent with the view that first-quantized paradigms will be dominant from the early fault-tolerant quantum computing era onward

Interactions between large molecules pose a puzzle for reference quantum mechanical methods

Quantum-mechanical methods are used for understanding molecular interactions throughout the natural sciences. Quantum diffusion Monte Carlo (DMC) and coupled cluster with single, double, and perturbative triple excitations [CCSD(T)] are state-of-the-art trusted wavefunction methods that have been shown to yield accurate interaction energies for small organic molecules. These methods provide valuable reference information for widely-used semi-empirical and machine learning potentials, especially where experimental information is scarce. However, agreement for systems beyond small molecules is a crucial remaining milestone for cementing the benchmark accuracy of these methods. We show that CCSD(T) and DMC interaction energies are not consistent for a set of polarizable supramolecules. Whilst there is agreement for some of the complexes, in a few key systems disagreements of up to 8 kcal mol−1 remain. These findings thus indicate that more caution is required when aiming at reproducible non-covalent interactions between extended molecules

Soft and accurate norm conserving pseudopotentials and their application for structure prediction

Structure prediction and discovery of new materials are essential for the advancement of new technologies. This have been possible due to the developments in Density Functional Theory (DFT) and increase in computational power of the supercomputers. One of the key aspect is the reliability of the structures predicted by the DFT codes. In this regard pseudopotentials are essential for both fast and accurate predictions. Through the addition of softness constraints on the pseudo valence orbitals along with the non-linear core correction and semicore states, new soft and accurate dual space Gaussian type pseudopotentials have been generated for the Perdew Burke Ernzerhof (PBE) and PBE0 functionals. Despite being soft, these pseudopotentials were able to achieve chemical accuracy necessary for the production runs. These pseudopotentials have been benchmarked against the most accurate all-electron (μHa accuracy) reference data of molecular systems till date which has been obtained using the Multi-Wavelets as implemented in the MRCHEM. In addition the pseudopotentials for the PBE functional show remarkable accuracy in the Delta tests. These new soft and accurate pseudopotentials have been used for structure prediction of large clusters

Sparse Compression of Expected Solution Operators

We show that the expected solution operator of prototypical linear elliptic partial differential operators with random coefficients is well approximated by a computable sparse matrix. This result is based on a random localized orthogonal multiresolution decomposition of the solution space that allows both the sparse approximate inversion of the random operator represented in this basis as well as its stochastic averaging. The approximate expected solution operator can be interpreted in terms of classical Haar wavelets. When combined with a suitable sampling approach for the expectation, this construction leads to an efficient method for computing a sparse representation of the expected solution operator

The High-Order Symplectic Finite-Difference Time-Domain Scheme

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Computational modeling of the electron momentum density

The properties and the functionality of materials are determined to a large extent by their electronic structure. The electronic structure can be examined through the electron momentum density, which is classically equivalent to the velocity distribution of the electrons. Changes in the structure of materials induce changes on their electronic structure, which in turn are reflected as changes in the electron momentum densities that can be routinely measured using, e.g., x-ray Compton scattering. The changes in the momentum density can be linked back to the structural changes the system has experienced through the extensive use of computational modeling. This procedure naturally requires using a model matching the accuracy of the experiment, which is constantly improving as the result of the ongoing development of synchrotron radiation sources and beam line instrumentation. However, the accuracies of the current computational methods have not been hitherto established. This thesis focuses on developing the methods used to compute the electron momentum density in order to achieve an accuracy comparable to that of the experiment. The accuracies of current quantum chemical methods that can be used to model the electron momentum density are established. The completeness-optimization scheme is used to develop computationally efficient basis sets for modeling the electron momentum density at the complete basis set limit. A novel, freely available software program that can be used to perform all of the necessary electronic structure calculations is also introduced.Materiaalien ominaisuudet sekä toiminnallisuus ovat pitkälti niiden elektronisen rakenteen määräämiä. Elektronirakennetta voidaan tutkia elektroniliikemäärätiheyden avulla, joka vastaa klassisesti elektronien nopeusjakaumaa. Materiaalien rakenteessa tapahtuvat muutokset muuttavat niiden elektronirakennetta, joka puolestaan heijastuu niiden liikemäärätiheyksiin joka voidaan rutiininomaisesti mitata käyttämällä esimerkiksi röntgen-Compton-sirontaa. Liikemäärätiheydessä tapahtuvat muutokset voidaan yhdistää systeemissä tapahtuneisiin rakennemuutoksiin käyttämällä laskennallista mallinnusta. Tämä luonnollisesti vaatii sellaisen mallin käyttämistä, jonka tarkkuus on verrattavissa mittaustuloksen tarkkuuteen, joka taas paranee jatkuvasti synkrotronisäteilylähteiden ja mittauslaitteistojen kehityksen vuoksi. Nykyisten mallinnusmenetelmien tarkkuutta ei ole kuitenkaan vielä määritetty. Tässä väitöskirjassa kehitetään elektroniliikemäärätiheyden mallinnusmenetelmiä, tarkoituksena saavuttaa kokeisiin verrattavissa oleva tarkkuus. Nykyaikaisten kvanttikemiallisten menetelmien tarkkuudet määritetään. Täydellisyysoptimointimenetelmää käytetään laskennallisesti tehokkaiden, elektroniliikemäärätiheyden mallintamiseen suunnattujen kantajoukkojen muodostamiseen, joiden tulokset ovat kantajoukkorajalla. Esittelemme myös uuden, vapaasti saatavilla oleva ohjelman, jolla voidaan suorittaa kaikki mallintamisessa tarvittavat elektronirakennelaskut