# Tensor product methods for strongly correlated molecular systems

For each poster contribution there will be one poster wall (width: 97 cm, height: 250 cm) available. Please do not feel obliged to fill the whole space. Posters can be put up for the full duration of the event.

### Low-lying electronic states of heteronuclear di- and triatomic molecules built from 3d element and coinage metal atoms

Andrae, Dirk

Small molecules formed by atoms of the d-block elements are building blocks e.g. for catalytically and photochemically active systems, for nanostructured materials, and for electronic and magnetic devices that receive considerable attention in both fundamental and applied research. For low-lying terms of the diatomic molecules AB (A 3d element, B coinage metal), full potential energy curves, in both $\Lambda--S$ and $\Omega$ representation have been constructed using multireference configuration interaction (MRCI) techniques. Scalar relativistic effects were included by use of the spin-free Douglas-Kroll-Hess (DKH) Hamiltonian, with spin-orbit coupling subsequently incorporated by perturbation theory. For low-lying terms of the triatomic molecules AB$_2$, several $\Lambda--S$ potential energy surfaces were determined at MRCI/DKH level. We present a selection of results obtained for 48 systems included in this study.

### Quantum Dynamics with the Time-Dependent Density Matrix Renormalization Group

Baiardi, Alberto

Thanks to the advent of ultrafast spectroscopic techniques, the dynamics of a molecule can be resolved experimentally on the natural time scales of both its electronic and nuclear motions. Full Configuration Interaction (CI) methods enable one to simulate, in principle, the exact quantum dynamics of a molecular system [1,2] in a given basis. The exponential increase of the computational cost of CI with the basis size hinders, however, straightforward extensions to large molecular systems. In this contribution, we show how to limit this unfavorable scaling by expressing the wave function as a matrix product state, a parametrization employed in the Density Matrix Renormalization Group algorithm (DMRG) [3]. The different strategies that have been designed to integrate the resulting equation of motion are broadly defined as time-dependent DMRG (TD-DMRG). We generalize a recently developed tangent space-based TD-DMRG algorithm [4] to electronic- [5] and vibrational-structure [6-7] quantum chemical Hamiltonians. We apply the resulting theory to simulate the nuclear and electronic dynamics, possibly coupled together, of molecules with several dozens of degrees of freedom [8]. We assess the accuracy of the simulations by comparison with state-of-the-art CI results and experimental data. References: [1] Meyer H. D., WIRES Comp. Mol Sci. 2011, 2, 351. [2] Sato T., Ishikawa K. L., Phys. Rev. A. 2013, 88, 023402. [3] Baiardi A., Reiher M., J. Chem. Phys., 2020, 152, 040903. [4] Haegeman J., Lubich C., Oseledets I., Vandereycken F., Phys. Rev. B. 2016, 94, 1. [5] Keller S., Dolfi M., Troyer M., Reiher M., J. Chem. Phys. 2015, 143, 244118. [6] Baiardi A., Stein C.J., Barone V., Reiher M., J. Chem. Theory Comput. 2017, 13, 3764. [7] Baiardi A., Stein C.J., Barone V., Reiher M., J. Chem. Phys. 2019, 150, 094113. [8] Baiardi A., Reiher M., J. Chem. Theory Comput. 2019, 15, 3481.

### Accurate molecular properties using multi-resolution analysis

Bischoff, Florian

The formalism of Multiresolution Analysis (MRA) allows the computation of molecular properties at the limit of the complete basis. The real-space nature of MRA requires the reformulation of all working equations in first quantization. High-dimensional, many-particle methods, such as MP2 or Coupled-Cluster, which require the computation of the wave function in 6 or more dimensions, are only tractable by using appropriate tensor decomposition techniques. I will present the general method of computing standard quantum chemical properties, e.g. energies, gradients, hessians, excitation energies and give examples for such calculations. Describing electron correlation using methods such as perturbation theory or Coupled-Cluster require special care for the removal of the electronic singularities, whose presence would otherwise make computations on a 6-dimensional grid intractable. Diagrammatic methods which is the common way of deriving Coupled-Cluster equations may be used in MRA with only minor reinterpretations of the diagrams. The precision and the performance of MRA will be discussed, in particular with respect to ground and excited states

### Approximate Analytical Gradients and Nonadiabatic Couplings for the State-Average Density Matrix Renormalization Group Self-Consistent-Field Method

Freitag, Leon

Nonadiabatic surface-hopping dynamics is an established method of studying photophysical and photochemical processes in medium to large molecules. Its accuracy depends on the reliability of the underlying electronic structure method, and often, multiconfigurational electronic structure methods will be required, especially if the processes in question involve relaxation to the ground state or dissociation processes. The most commonly used multiconfigurational method is the state-average complete active space self-consistent field method (SA-CASSCF), whose exponential scaling with the number of active orbitals greatly limits the size of the molecules that can be studied. In contrast, state-average density matrix renormalization group-self consistent field method (SA-DMRG-SCF), i.e. an analogue to SA-CASSCF where the configuration interaction problem is solved with the density matrix renormalization group method (DMRG), shows only polynomial scaling with the number of active orbitals and thus can be employed for systems with larger active orbital spaces than the traditional SA-CASSCF. Here we present an approximate method for analytic state-average gradients and interstate non-adiabatic coupling vectors for state-average density matrix renormalization group-self consistent field (SA-DMRG-SCF) wavefunctions, which paves the way to nonadiabatic surface-hopping dynamics with DMRG-SCF as the electronic structure method.

### Preconditioned Riemannian Optimization in Quantum Chemistry

Goette, Michael

We demonstrate that Riemannian optimization techniques also give comparable or in some sense even better results than DMRG. By applying ideas from differential geometry we set up a conjugated gradient scheme on the low-rank manifold induced by matrix product states of fixed rank. Key aspect of sufficiently fast convergence relies on the quality of preconditioning. We will present different preconditioning schemes and compare them to DMRG results.

### The role of vertex dressing in Multi-Reference Coupled Cluster

Margócsy, Ádám

Description of molecular systems having a predominantly Multi-Reference (MR) character has remained an open challenge for quantum chemistry to this day. The MR generalization of the Coupled Cluster method (MR-CC) might solve this problem; however, there is no unique way to define MR-CC, and various approaches have been proposed during the years. I shall present our approach to MR-CC: a state-specific, internally contracted (approximate) model which resembles the original, Single-Reference CC (SR-CC) as much as possible, MR effects (represented by density cumulants) being "hidden" in dressed 1- and 2-particle vertices. The model obtained this way retains the formal simplicity, low computational cost and other desirable properties of SR-CC, while the "implicit" MR effects make it suitable for treating strongly correlated systems. I shall also present pilot numerical applications to our method, including the description of multiple bond dissociation (e.g. N$_2$, H$_2$O), simple reaction pathways (insertion of Be into H$_2$) and dispersion interactions (the interaction energy of a He dimer).

### Variational two-site finite PEPS

Noack, Reinhard

We describe the implementation and application of the PEPS (Projected Entangled Pair States) algorithm on a finite lattice using two-site optimization with a variational diagonalization-based scheme. The algorithm is formulated in terms of PEPOs (PEPS-based operators), which are the generalization of MPOs (Matrix Product Operators) to PEPS. We describe the issues and approximations involved in the contraction of the PEPS state, including optimization strategies for tensor contractions and environment truncation schemes. We apply the algorithm to the two-dimensional Hubbard model and describe its performance and accuracy relative to other competitive algorithms.

### Calculating molecular properties of strongly correlated systems with the distinguishable cluster approximation

Schraivogel, Thomas

It is well known that standard single reference Coupled Cluster methods like CCSD break down for strongly correlated systems. The problem is due to the truncation of the excitation operator. Therefore for more than thirty years people try to find approximations to the standard methods like CCSD, which incorporate the relevant physics of the omitted higher excitations without hav- ing the cost of actually including them. One of such approximation was proposed in 2013 by Kats and Manby: The distinguishable cluster approximation (DCA). They showed that the distin- guishable cluster approximation with doubles (DCD) can accurately describe the potential energy surface of the uniform dissociation of a squared hydrogen lattice, which is considered to be a particular hard system to describe for most electronic structure methods due to the strong static correlation. For higher precision DCSD accounts for orbital relaxation with a singles similarity trans- formation. Calculations of molecular properties with DCSD for strongly correlated systems will be shown. Among other properties spin properties and ionisation potentials using the Extended Koopmans Theorem (EKT) with density matrices of DCSD will be presented. Additionally, we will present calculations with DC-CCSDT, which applies the DCA to the mixed triples equations of CCSDT.

### Multipartite entanglement and correlation and the chemical bond

Szalay, Szilárd

Correlations in quantum systems can be much stronger than in classical systems, an important manifestation of this is quantum entanglement. States of a bipartite system (pure or mixed) can be either uncorrelated or correlated, while for multipartite systems many different kinds of correlations arise. I will (i) show how to grasp this complicated structure efficiently, (ii) define proper correlation measures, (iii) formulate the multipartite correlation based clustering of the system, and (iv) give an algorithm for this clustering. The importance of the latter two points is that the existence of higher correlations makes the bipartite correlation based ("graph theoretical") clustering insufficient. I will also (v) illustrate the multipartite correlation theory by showing examples from molecular physics. This field provides an excellent playground for multipartite correlation theory, since here the ground states of the many-body interacting Hamiltonians are naturally factorized into approximate products of clusters of localized orbitals. Szilárd Szalay, Gergely Barcza, Tibor Szilvási, Libor Veis, Örs Legeza, The correlation theory of the chemical bond (research article), Sci. Rep. 7. 2237 (2017)

### MOLMPS: a new scalable and highly flexible quantum chemistry DMRG code

Veis, Libor

We present a new C++ implementation of the quantum chemistry density matrix renormalization group (QC-DMRG) method with the emphasis on scalability and high flexibility. We believe that our implementation has a potential to open the way for computations of challenging problems requiring very large active spaces not only in non-relativistic quantum chemistry, but due to its flexibility also for example in fully relativistic setting.