QuTree: A tree tensor network package.

We present QuTree, a C++ library for tree tensor network approaches. QuTree provides class structures for tensors, tensor trees, and related linear algebra functions that facilitate the fast development of tree tensor network approaches such as the multilayer multiconfigurational time-dependent Hartree approach or the density matrix renormalization group approach and its various extensions. We investigate the efficiency of relevant tensor and tensor network operations and show that the overhead for managing the network structure is negligible, even in cases with a million leaves and small tensors. QuTree focuses on providing simple, high-level routines while retaining easy access to the backend to facilitate novel developments. We demonstrate the capabilities of the package by computing the eigenstates of coupled harmonic oscillator Hamiltonians and performing random circuit simulations on a virtual quantum computer.

A general method for the development of diabatic spin-orbit models for multi-electron systems.

Spin-orbit (SO) coupling can have significant effects on the quantum dynamics of molecular systems, but it is still difficult to account for accurately. One promising way to do this is to devise a diabatic SO model combined with the molecular potential energy. Few such models have been developed utilizing spatial and time-reversal symmetry. These models are tedious to derive and are specific for the molecular symmetry and included spin states. Here, we present a relatively simple approach to construct such models for various spin states with S≠12 from a basic one-electron SO case with S=12. The multi-electron fine structure states are expressed in terms of Slater determinants of single-electron spin functions (spinors). The properties of all single-electron matrix elements over the SO operator are derived and expressed as Taylor expansions in terms of symmetry-adapted nuclear coordinates. The SO matrix elements for the multi-electron case are then obtained from these single-electron matrix elements using the Slater-Condon rules. This yields the full SO matrix and symmetry properties of the multi-electron matrix elements in a straightforward way. The matrix elements are expressed as symmetry-adapted polynomials up to arbitrary order. This approach is demonstrated first for an abstract model of two electrons in a set of p orbitals in a C3v symmetric environment and then applied to set up a diabatic model for the photodissociation of methyl iodide (CH3I). The high accuracy of this new approach is demonstrated in comparison to an available analytic SO model for CH3I.

Symmetries in the multi-configurational time-dependent Hartree wavefunction representation and propagation.

In multi-configurational time-dependent Hartree (MCTDH) approaches, different multi-layered wavefunction representations can be used to represent the same physical wavefunction. Transformations between different equivalent representations of a physical wavefunction that alter the tree structure used in the multi-layer MCTDH wavefunction representation interchange the role of single-particle functions (SPFs) and single-hole functions (SHFs) in the MCTDH formalism. While the physical wavefunction is invariant under these transformations, this invariance does not hold for the standard multi-layer MCTDH equations of motion. Introducing transformed SPFs, which obey normalization conditions typically associated with SHFs, revised equations of motion are derived. These equations do not show the singularities resulting from the inverse single-particle density matrix and are invariant under tree transformations. Based on the revised equations of motion, a new integration scheme is introduced. The scheme combines the advantages of the constant mean-field approach of Beck and Meyer [Z. Phys. D 42, 113 (1997)] and the singularity-free integrator suggested by Lubich [Appl. Math. Res. Express 2015, 311]. Numerical calculations studying the spin boson model in high dimensionality confirm the favorable properties of the new integration scheme.

The multi-configurational time-dependent Hartree approach in optimized second quantization: Imaginary time propagation and particle number conservation.

The multilayer multiconfigurational time-dependent Hartree (MCTDH) in optimized second quantization representation (oSQR) approach combines the tensor contraction scheme of the multilayer MCTDH approach with the use of an optimized time-dependent orbital basis. Extending the original work on the subject [U. Manthe and T. Weike, J. Chem. Phys. 146, 064117 (2017)], here MCTDH-oSQR propagation in imaginary time and properties related to particle number conservation are studied. Differences between the orbital equation of motion in real and imaginary time are highlighted and a new gauge operator, which facilitates efficient imaginary time propagation, is introduced. Studying Bose-Hubbard models, particle number conservation in MCTDH-oSQR calculations is investigated in detail. Interesting properties of the single-particle functions used in the multilayer MCTDH representation are identified. Based on these results, a tensor contraction scheme, which explicitly utilizes particle number conservation, is suggested.

Quantum dynamics and geometric phase in E⊗e Jahn-Teller systems with general Cnv symmetry.

E ⊗ e Jahn-Teller (JT) systems are considered the prototype of symmetry-induced conical intersections and of the corresponding geometric phase effect (GPE). For decades, this has been analyzed for the most common case originating from C3v symmetry and these results usually were generalized. In the present work, a thorough analysis of the JT effect, vibronic coupling Hamiltonians, GPE, and the effect on spectroscopic properties is carried out for general Cnv symmetric systems (and explicitly for n = 3-8). It turns out that the C3v case is much less general than often assumed. The GPE due to the vibronic Hamiltonian depends on the leading coupling term of a diabatic representation of the problem, which is a result of the explicit n, α, and ß values of a CnvEα ⊗ eß system. Furthermore, the general existence of n/m (m∈N depending on n, α, and ß) equivalent minima on the lower adiabatic sheet of the potential energy surface (PES) leads to tunneling multiplets of n/m states (state components). These sets can be understood as local vibrations of the atoms around their equilibrium positions within each of the local PES wells symmetrized over all equivalent wells. The local vibrations can be classified as tangential or radial vibrations, and the quanta in the tangential mode together with the GPE determine the level ordering within each of the vibronic multiplets. Our theoretical predictions derived analytically are tested and supported by numerical model simulations for all possible Eα ⊗ eß cases for Cnv symmetric systems with n = 3-8. The present interpretation allows for a full understanding of the complex JT spectra of real systems, at least for low excitation energies. This also opens a spectroscopic way to show the existence or absence of GPEs.

On the multi-layer multi-configurational time-dependent Hartree approach for bosons and fermions.

A multi-layer multi-configurational time-dependent Hartree (MCTDH) approach using a second quantization representation (SQR) based on optimized time-dependent orbitals is introduced. The approach combines elements of the multi-layer MCTDH-SQR approach of Wang and Thoss, which employs a preselected time-independent orbital basis, and the MCTDH for bosons and multi-configuration time-dependent Hartree-Fock approaches, which do not use multi-layering but employ time-dependent orbital bases. In contrast to existing MCTDH-type approaches, the results of the present approach for a given number of configurations are not invariant with respect to unitary transformations of the time-dependent orbital basis. Thus a natural orbital representation is chosen to achieve fast convergence with respect to the number of configurations employed. Equations of motion for the present ansatz, called (multi-layer) MCTDH in optimized second quantization representation, are derived. Furthermore, a scheme for the calculation of optimized unoccupied single-particle functions is given which can be used to avoid singularities in the equations of motion.

Development of multi-mode diabatic spin-orbit models at arbitrary order.

The derivation of diabatic spin-orbit (SO) Hamiltonians is presented, which are expanded in terms of nuclear coordinates to arbitrary order including the treatment of multi-mode systems, having more than one mode of the same symmetry. The derivation is based on the microscopic Breit-Pauli SO operator and the consequent utilization of time reversal and spatial symmetry transformation properties of basis functions and coordinates. The method is demonstrated for a set of (2)E and (2)A1 states in C(3v)* (double group) symmetry, once for a 3D case of one a1 and one e mode and once for a 9D case of three a1 and three e coordinates. It is shown that the general structure of the diabatic SO Hamiltonian only depends on the basis states and is strictly imposed by time reversal symmetry. The resulting matrix can be expressed easily by a power series using six parametrized structure matrices as expansion coefficients multiplied by the associated monomials in terms of symmetrized coordinates. The explicit example presented here provides a full-dimensional diabatic SO model for methyl halide cations, which will be studied in the future.