Interfaces to the Hamburg CheMPS2 Extension

The Hamburg CheMPS2 extension offers two interfaces to interact with users. First, it offers a C++ interface for compiled applications that make use of the Hamburg CheMPS2 library. Here all necessary classes and features can be used and controlled from external applications. Second, it has build-in executables for the most common tasks, such as:

• Thechemps2dynexecutable for time-dependent calculations using MPS.

• Thechemps2fciexecutable for time-dependent calculations using the com-plete many-body state.

• Thechemps2convexecutable to convert a real-valued MPS to a complex-valued MPS.

• Thechemps2ion executable for removing an electron from an orbital in a complex-valued MPS.

These executables come with every installed version of the Hamburg CheMPS2 exten-sion. They show a complete documentation of input parameters by calling them using the help command./chemps2xxx –help. All of the executables can be controlled by inputfiles that allow the user to specify the task. This is an adaption of the original CheMPS2 package that also offers an interface via input files, however, we extended the interface for time-dependent MPS calculations.

Listing3.1shows an example input file to perform a time-dependent MPS calculation using thechemps2dynexecutable. It starts with the definition of the system, for example, giving the file of integrals for the many-body Hamiltonian (FCIDUMP), a symmetry group identifier, the number of electrons in the system (lines1to8) and the order of orbitals used (here reordering based on the Fiedler minimization of the Hamiltonian band with [151]). It then continues with parameters to specify the time evolution in lines10to22, such as an identifier for the time evolution method (here Krylov), parameters as time step size and the time to propagate to, and the form of the initial state. It ends with the definition of parameters for the optimization algorithm, giving the number of sweeps and the bond dimension of the MPS tensors.

Thechemps2dynexecutable allows for input of three different types of initial states.

First and most basic option is to give the initial state in terms of occupation numbers of the orbitals, such as given in line 19 of Listing3.1. The program then forms an uncorrelated state using the given occupation numbers. Of course, the sum of the occupation numbers need to match to the number of electrons in the system that is given in line 5 of Listing3.1. Second option is to define a superposition of uncorrelated states as the initial state. This is done by writing

1 TIME_NINIT = 2 , 2 , 2 , 2 , 0 , 0 , 0 , 0 , 0 , 2 , 1 , 0 , 0 , 0 , 0 , 2 , 2 , 0 , 0 , 0 , 0 2 TIME_2_NINIT = 2 , 2 , 2 , 2 , 0 , 0 , 0 , 0 , 0 , 1 , 2 , 0 , 0 , 0 , 0 , 2 , 2 , 0 , 0 , 0 , 0 3 TIME_PREFACS = 0.7071067811865475 , 0.0 , 0.7071067811865475 , 0.0

in the input file. The program then forms a superposition of the two uncorrelated states with similar prefactors. The initial state is constructed as

|ψ(t0)iMPS = 1√

2|1iMPS+ 1√

2|2iMPS, (3.39)

where the states|1i_MPSand|2i_MPSare the uncorrelated states given in line 1 and line 2. Such an initial state has been used in Section5. Third option to specify the initial state is to give an MPS that has to match to the system and the chosen symmetry (size of the system, number of electrons, multiplicity). Here any MPS stored in the HDF5format can be used by giving

1 TIME_INIT = CheMPS2_CMPS_ION . h5

in the input file.

The Hamburg CheMPS2 extension offers two ways of providing the results. Some of the result are printed to the console. This includes quantities that serve more for debugging purposes such as the energy at any point in time (which is supposed to be constant in ideal situations), the norm of the time-evolved state (which should stay close to1). But also the occupation number of the molecular orbitals, CI weights, as

well as, the projection onto the initial state, which is a one-particle Green’s function are printed at every time step.

For more complex output parameters the Hamburg CheMPS2 extension also creates a binary output in the version 5 of the Hierarchical Data Format (HDF5). In this binary file the results are stored in a simple to parse format. Here the user also gets the one-particle reduced density matrix, the two-one-particle reduced density matrix if requested, full CI coefficients and more calculation depending numbers such as the size of the MPS tensors and the time since start of the calculation. The HDF5 file can be easily extend by arbitrary observables and correlators.

The Hamburg CheMPS2 extension is published under the GNU General License version 2 and can be freely copied and extended, however, with the condition that modified versions are published under the GPL2-2.0 or one of its successors.

4

Analysis of the Matrix Product State Approach to Study Ultrafast Dynamics in Molecules

At this point, we established the concepts to operate with many-body states in MPS representation and discussed the Hamburg CheMPS2 extension to perform numerical calculations on time-dependent problems in quantum chemical systems.

However, the MPS representation has never been used to describe electron dynamics in molecules before, therefore, a profound analysis of the performance of this novel method is due. In this chapter, we focus on simple systems that are represented by small orbital sets, such that the complete many-body state from Equation2.35is still manageable using our computing facilities. This gives us the opportunity to challenge the MPS representation with the complete many-body state. We can directly observe the impact of convergence parameters (MPS bond dimension, time evolution method, time step size) on the validity of the results and conclude regularities to operate with the MPS approach in the context of time-dependent quantum chemistry.

We begin the analysis by studying different types of hydrogen chains, starting from the simple hydrogen molecule H₂and continuing with longer chains H₁₀. For these molecules, we compare time-dependent reduced density matrices obtained from the different representations of the many-body state. Further, we study the one-body Green’s function in both, the time domain, as well as, in the frequency domain. Next, we continue with higher-dimensional molecules in extended basis sets. In particular, we study electron dynamics following a sudden (double) ionization of the molecules hydrogen fluoride FH, water H₂O, ammonia NH₃, and methane CH₄in the 6-31G Gaussian basis set [218]. To rate the accuracy of the MPS representation, we discuss the relative error of the one-body reduced density matrix originating from the MPS truncation. In the last part, we analyze different methods to perform time evolution of many-body states in MPS representation, which reveals an entanglement between the MPS approach and the time evolution method. A correctly chosen time evolution method allows us to improve the performance of the MPS representation in special situations.

Parts of this Chapter have been published in theJournal of Chemical Theory and Computation[219] and in theEuropean Physical Journal Web of Conferences[220].

Especially the analysis of the hydrogen chain, as well as, the results on the time evolution methods have found consideration in these journals.

4.1 Hydrogen Based Molecules

We start the analysis of the MPS representation by describing electron dynamics in one-dimensional chains of hydrogen atoms. In Section2.4.2, we highlighted the optimal performance of the MPS approach when representing gapped ground states of short-ranged one-dimensional systems. This optimal performance is a consequence of the limited electron entanglement in this type of states (remember the area law).

Therefore, MPS have an intrinsic benefit when representing states of one-dimensional systems (this is due to the factorization of the coefficient tensor in a one-dimensional chain of matrix products), which we rely upon to start the analysis of time-dependent MPS representation defensively.

The hydrogen molecule and the hydrogen chain have received much interest by physicists and chemists from the beginning of quantum mechanics [221]. The cationic hydrogen molecule H₂⁺is exactly solvable in the Born–Oppenheimer approximation, which started the field of molecular orbital theory a hundred years ago [222] . This model is still at the heart of our current understanding of electrons in molecules (see Chapter2). In recent years, hydrogen chains have been extensively used as simplified model systems for complex molecular chains [223–227]. Its actual simple structure in combination with the long-ranged Coulomb interaction allows the hydrogen chain to be simple, while representing most features of more complex systems. The hydrogen chain has also been subject to numerous benchmark studies of many-body methods.

For example, it was the system of interest for the benchmark studies ofdensity matrix renormalization group[205,228–230] anddensity-matrix embedding theory[231–233], and in an outstanding review benchmarking almost all latest many-body methods by Motta et al. [24]. The distance between the atoms here serves as a handle to tune the electronic correlation, which enables to measure how stable a particular many-body method handles correlations.