# Tensor product methods for strongly correlated molecular systems

### On the Ordering of Sites in the Density Matrix Renormalization Group using Quantum Mutual Information

Ali, Mazen

The density matrix renormalization group (DMRG) of White 1992 remains to this day an integral component of many state-of-the-art methods for efficiently simulating strongly correlated quantum systems. In quantum chemistry, QC-DMRG became a powerful tool for ab initio calculations with the non-relativistic Schrödinger equation. An important issue in QC-DMRG is the so-called ordering problem -- the optimal ordering of DMRG sites corresponding to electronic orbitals that produces the most accurate results. To this end, a commonly used heuristic is the grouping of strongly correlated orbitals as measured via quantum mutual information. In this work, we show how such heuristics can be directly related to minimizing the entanglement entropy of matrix product states and, consequently, to the truncation error of a fixed bond dimension approximation. Key to establishing this link is the strong subadditivity of entropy. This provides a rigorous theoretical justification for the orbital ordering methods and suggests alternate ordering criteria.

### The low-energy physics of bilinear-biquadratic spin-1 chains

Barthel, Thomas

We give a comprehensive analysis of the low-energy physics of the bilinear-biquadratic spin-1 chains using dynamic spin and quadrupolar structure factors. They are computed to high precision with an efficient infinite matrix product state algorithm. Comparing with Bethe ansatz solutions at integrable points and field-theoretical predictions, we explain the nature of the relevant low-lying excitations. The bulk of the Haldane phase is qualitatively described by the non-linear sigma model, which predicts a gapped single-magnon excitation and corresponding multi-magnon continua. For the vicinity of the critical Takhtajan-Babujian point, Tsvelik suggested a Majorana field theory. We discuss significant deviations from the field-theoretical predictions, and explain the prominent features in the structure factors using a non-interacting approximation. As one approaches the integrable Uimin-Lai-Sutherland point, the excitations deform into soliton continua. Within the extended critical phase, these continua contract, which we explain by comparison to a level-1 SU(3) Wess-Zumino-Witten field theory. We observe and characterize new elementary excitations at higher energies and find that their dispersion relations converge to simple cosine functions as we approach the purely biquadratic point. We discuss their relation to the Bethe ansatz solution for the Temperley-Lieb chain. The Temperley-Lieb chain can also be used to describe the physics at the opposite biquadratic point, which places the model in the gapped dimerized phase. Here, the excitation spectrum is related to that of an anisotropic spin-1/2 chain. In the ferromagnetic phase, the two-magnon excitations can be computed exactly and contain bound and resonant states in addition to two-particle continua. For the Temperley-Lieb and Uimin-Lai-Sutherland points, we characterize edge singularities and thresholds in the dynamic structure factors which may inform further Bethe ansatz and field-theoretical computations.

### Molecular Properties Using Multiresolution Analysis

Bischoff, Florian

We present the computation of molecular properties using multiresolution analysis. In the absence of basis-set incompleteness related artifacts a closer look at method errors and highly accurate calculations are possible. Examples are the computation of local exchange potentials or the structure determination of molecules in extreme magnetic fields. For correlated wave functions the coefficient tensors for the representation of the wave function must be decomposed into appropriate forms, e.g. SVD or tensor train. Computations in the decomposed tensor format are the time-determining step in the calculations.

### Concept of Orbital Entanglement and Correlation in Quantum Chemistry

Das, Sreetama

A recent development in quantum chemistry has established the quantum mutual information between orbitals as a major descriptor of electronic structure. This has already facilitated remarkable improvements in numerical methods and may lead to a more comprehensive foundation for chemical bonding theory. Building on this promising development, our work provides a refined discussion of quantum information theoretical concepts by introducing the physical correlation and its separation into classical and quantum parts as distinctive quantifiers of electronic structure. In particular, we succeed in quantifying the entanglement. Intriguingly, our results for different molecules reveal that the total correlation between orbitals is mainly classical, raising questions about the general significance of entanglement in chemical bonding. Our work also shows that implementing the fundamental particle number superselection rule, so far not accounted for in quantum chemistry, removes a major part of correlation and entanglement seen previously. In that respect, realizing quantum information processing tasks with molecular systems might be more challenging than anticipated.

### Entanglement between orbitals in many-electron systems

Ding, Lexin

Entanglement is one of the most fascinating concepts of modern physics. In striking contrast to its abstract, mathematical foundation, its practical side is, however, remarkably underdeveloped: Even for the simplest setting of just two orbitals or sites no faithful entanglement measure is known for generic quantum states. By exploiting the spin symmetries of realistic many-electron systems and implementing the crucial superselection rule, we eventually succeed in deriving a closed formula for the relative entropy of entanglement between any two electronic orbitals. The broad relevance of such a formula for quantum-many body physics is highlighted by its application to different systems: (i) an analytical description of the long range entanglement between two distant sites in free electron chains is found, (ii) the presence of the distinctive bond-order wave phase in the extended Hubbard model can be confirmed, and (iii) the orbital correlation in molecular systems is mainly classical, raising questions about the role of entanglement within chemical bonding theory.

### Nuclear-electronic all-particle density matrix renormalization group

Feldmann, Robin

We present the Nuclear-Electronic All-Particle Density Matrix Renormalization Group (NEAP-DMRG) to solve the time-independent Schrödinger equation for systems comprising more than one quantum species. NEAP relies on a generalized non-orthogonal configuration interaction wave function ansatz constructed from non-orthogonal Gaussian-type orbitals. We stochastically refine all non-linear basis set parameters iteratively to obtain the most compact wave function representation. Subsequently, we diagonalize the full molecular Hamiltonian in the space spanned by the optimized orbitals by parametrizing the wave function as a matrix product state and optimizing it with the DMRG. The efficient parameterization enables targeting molecular systems with more than three nuclei and 12 particles in total. We present the NEAP-DMRG results for two few-body systems, i.e., H$_2$ and H$^+_3$, and one larger system, namely, BH$_3$.

### Preconditioned Riemannian Optimization in Quantum Chemistry

Götte, Michael

In some sense DMRG is a restricted tangent space method. We were following the question if it is possible define a Riemannian optimization method on the whole tangent space, i.e. a precondionent projected gradient descent (PPGD). On this poster we present the method and give non competitive comparing results of DMRG and PPGD. The key idea is to use the inverse of the Fock operator as a preconditioner for the Hamiltonian. This is possible due to the special Laplace-like structure of the Fock operator.

### Minimal matrix product states

Simple wavefunctions of low computational cost but which can achieve qualitative accuracy across the whole potential energy surface (PES) are of relevance to many areas of electronic structure as well as to applications to dynamics. Here, we explore a class of simple wavefunctions, the minimal matrix product state (MMPS), that generalizes many simple wavefunctions in common use, such as projected mean-field wavefunctions, geminal wavefunctions, and generalized valence bond states. By examining the performance of MMPSs for PESs of some prototypical systems, we find that they yield good qualitative behavior across the whole PES, often significantly improving on the aforementioned ansätze.

### Tailoring CCSD with wave functions of polynomial cost

Leszczyk, Aleksandra

Tailored coupled-cluster theory represents a computationally inexpensive way to describe static and dynamic electron correlation effects. In this methodology, a subset of cluster amplitudes is extracted from an external model to describe the multireference nature of electronic structures. The subset of external amplitudes is incorporated into the CCSD ansatz. We focus on models that scale polynomially with system size: the matrix product state (MPS) ansatz optimized by the density matrix renormalization group (DMRG) algorithm and the orbital-optimized pair coupled cluster doubles (pCCD) model. Benchmark calculations are performed for molecules for which a single reference CCSD cannot produce accurate results.

### Functional Theory for Bose-Einstein Condensates and Origin of Quantum Depletion

Liebert, Julia

One-particle reduced density matrix functional theory would potentially be the ideal approach for describing Bose-Einstein condensates. It namely replaces the macroscopically complex wavefunction by the simple one-particle reduced density matrix, therefore provides direct access to the degree of condensation and still recovers quantum correlations in an exact manner. We eventually initiate and establish this novel theory by deriving the respective universal functional $\mathcal{F}$ for general homogeneous Bose-Einstein condensates with arbitrary pair interaction. Most importantly, the successful derivation necessitates a particle-number conserving modification of Bogoliubov theory and a solution of the common phase dilemma of functional theories. We then illustrate this novel approach in several bosonic systems such as homogeneous Bose gases and the Bose-Hubbard model. Remarkably, the general form of $\mathcal{F}$ reveals the existence of a universal Bose-Einstein condensation force. This generalization of the Fermi degeneracy pressure to interacting bosonic quantum systems of arbitrary size provides an alternative and more fundamental explanation for quantum depletion.

### Dissipation assisted matrix product factorization (DAMPF) method for open quantum systems

Lorenzoni, Nicola

We present a method capable of dealing with the simulation of spatially extended open quantum systems with an arbitrarily structured spectral density. In here, the introduction of an effective environment subjected to dissipation will make the matrix product state approximation feasible, enabling us to simulate the system in a numerically exact way.

### 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) can tackle this problem. Various MR-CC approaches have been proposed over the years; however, a method optimal in all respects is yet to be found. I shall present an approximation within the internally contracted MR-CC formalism which can be thought of as a dressing of vertices appearing in ordinary Single-Reference CC (SR-CC). These dressed vertices are functions of density cumulants coming from the Generalized Wick Theorem[1]; diagrams incorporated in the dressed vertex approximation provide a generalization of ladder- and ring-type Goldstone diagrams[2] among others. Formal simplicity, moderate computational cost and other desirable properties of SR-CC are retained, while the ”implicit” MR effects (hidden in the dressed vertices) make the model more suitable for treating strongly correlated systems. The Generalized Valence Bond (GVB) wave function is used as a starting point for our MR-CC correction. I shall present the theory and pilot numerical applications (including the description of multiple bond dissociation on the example of N$_2$ and H$_2$O). References: [1] Kutzelnigg, Mukherjee: JCP 107 432 (1997); [2] Margócsy, Szabados: JCP 152 204114 (2020).

### Realizing Bose--Einstein condensation in a Mexican-hat-like potential

Máté, Mihály

Mihály Máté, Örs Legeza, Rolf Schilling, Mason Yousif, Christian Schilling We propose and study a model for $N$ hard-core bosons which allows for the interpolation between one- and high-dimensional behavior by variation of just a single external control parameter $s/t$. It consists of a ring-lattice of $d$ sites with a hopping rate $t$ and an extra site at its center. Increasing the hopping rate $s$ between the central site and the ring sites induces a transition from the regime with a quasi-condensed number $N_0$ of bosons proportional to $\sqrt{N}$ to complete condensation with $N_0 \simeq N$. In the limit $s/t \to 0, d \to \infty$ with $\tilde{s}=(s/t)\sqrt{d}$ fixed the low-lying excitations follow from an effective ring-Hamiltonian. An excitation gap makes the condensate robust against thermal fluctuations at low temperatures. These findings are supported and extended to the full parameter regime by large scale density matrix renormalization group computations. We show that ultracold bosonic atoms in a Mexican-hat-like potential represent an experimental realization allowing one to observe the transition from quasi to complete condensation by creating a well at the hat's center. The contribution is based on our recent paper [1]. [1] Mihály Máté, Örs Legeza, Rolf Schilling, Mason Yousif, Christian Schilling, Commun. Phys., DOI 10.1038/s42005-021-00533-3

### Tailored coupled cluster theory in varying correlation regimes

Mörchen, Maximilian

The tailored coupled cluster (TCC) approach is a promising ansatz that preserves the simplicity of single-reference coupled cluster theory while incorporating a multi-reference wave function through amplitudes obtained from a preceding multi-configurational calculation. Here, we present a detailed analysis of the TCC wave function based on model systems, which require an accurate description of both static and dynamic correlation. We investigate the reliability of the TCC approach with respect to the exact wave function. In addition to the error in the electronic energy and standard coupled cluster diagnostics, we exploit the overlap of TCC and full configuration interaction wave functions as a quality measure. We critically review issues, such as the required size of the active space, size-consistency, symmetry breaking in the wave function, and the dependence of TCC on the reference wave function. We observe that possible errors caused by symmetry breaking can be mitigated by employing the determinant with the largest weight in the active space as reference for the TCC calculation. We find the TCC model to be promising in calculations with active orbital spaces which include all orbitals with a large single-orbital entropy, even if the active spaces become very large and then may require modern active-space approaches that are not restricted to comparatively small numbers of orbitals. Furthermore, utilizing large active spaces can improve on the TCC wave function approximation and reduce the size-consistency error because the presence of highly excited determinants affects the accuracy of the coefficients of low-excited determinants in the active space.

### 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, Sci. Rep. 7. 2237 (2017)

### MOLMPS: the massively parallel quantum chemical density matrix renormalization group program

Veis, Libor

In the past decade, the quantum chemical version of the density matrix renormalization group method (QC-DMRG) has established itself as the method of choice for accurate calculations of strongly correlated molecular systems requiring large active spaces. Several groups have presented their own QC-DMRG implementations, but to the best of our knowledge, none of them has been presented as trully massively parallel. We present MOLMPS, the new C++ QC-DMRG implementation with the emphasis on scalability and high flexibility. Our parallel approach employs MPI and is based on low level lightweight tensor library allowing global memory storage. By combining operator and symmetry sector parallelisms, sufficient number of tasks to achieve massive parallelization is generated. We will present the scaling tests on typical candidates for DMRG calculations, namely Fe(II)-porphyrine model, extended $\pi$-conjugated system, and FeMoCo cofactor with the active spaces containing up to 76 orbitals. Due to its capabilities, our implementation is e.g. an ideal tool for machine learning. We believe that it 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.