2.5 Time Evolution of Matrix Product States
2.5.4 Non-Orthogonalized Krylov Space Time Evolution
The Krylov space basis generated from the vectors in Equation2.111is not the only valid basis of the Krylov space. When using an MPS representation of the Krylov basis vectors, the Krylov vectors will loose orthogonality due to the MPS truncation. In the time-evolution, Equation2.112, we handled this by introducing the inverse of the overlap matrix in the exponential. Another approach is to form the Krylov space from
non-orthonormal basis vectors in the first place. When constructing the basis in a non-orthogonalized fashion, we can use
|φk+1iMPS = ˆH|φkiMPS
|MPShφk|ˆH ˆH|φkiMPS|2, (2.113) which spans the Krylov space just as given in the Krylov space definition in Equation 2.110. This will be the second method to form a Krylov space basis that we discuss in the following analysis of the MPS approach.
The time evolution is again done according to Equation2.112, which already takes care of possible non-orthogonality of the basis states. The resulting time evolution is, up to a MPS truncation error, norm and energy conserving. When using the non-orthonormalized Krylov basis vectors, and forming the time evolved state from the basis vectors, the resulting equation for the time evolved state is very similar to the NKry-th expansion of the exact time evolution operator in Equation2.102, however, in-corporating a complex pre-factor in a way that the resulting time evolution is unitary.
This is a special feature of the non-orthonormal Krylov basis vectors, that will improve the performance of the MPS time evolution approach in special circumstances (see Section4).
However, this improved performance is limited to special problems, as time-evolution based on the non-orthogonal Krylov basis vectors is more prone to errors. Especially if the state to time evolve is close to an eigenstate of the Hamiltonian, then the non-orthonormal Krylov basis vectors tend to become linearly dependent, which causes large numerical errors, when calculating the inverseN−1in Equation2.112. Therefore we have to apply the non-orthogonalized Krylov approach more carefully.
The complexity of the non-orthogonalized Krylov approach is slightly reduced com-pared to the orthonormalized approach. We have theNKry−1operator application and only one addition of the previous Krylov vectors, however, in most cases the application of the Hamiltonian is the most costly part of the calculation. Therefore, the formal complexity of the non-orthogonalized Krylov space method is similar compared to orthogonalized Krylov space, however, the reduction in MPS bond di-mension can be significant, which allows for more efficient calculations using the non-orthogonalized Krylov method in special situations (see Section4).
These are the time evolution algorithms we want to apply in this analysis of the MPS based time evolution for dynamics in molecules. There exist many more approaches that have been developed, ranging from Chebyshev time evolution [192], more ex-tensions to the time-dependent variational principle [58], up to Hylleraas functions [193] to calculate dynamic Green’s functions in frequency space directly. However,
for the first application of the MPS approach in time-dependent ab initio quantum chemistry presented in this thesis, we have a good variety of methods to access the performance of the MPS approach in this field.
3
Competitive Implementation of the Matrix Product State
Approach
Many numerical implementations of the DMRG method, the MPS approach, and more complex tensor network approaches have been developed in recent years [51, 139,141,142,194–197]. There are multipurpose implementations of the tensor network approach that can be adapted to many physical problems. One of such implementa-tions is the open source ITensor package [141], where the author contributed himself.
The ITensor package is able to implement any tensor network, however, most fea-tures focus on the MPS approach to represent many-body states and on the MPO approach to represent many-body operators. For this type of tensor networks, it offers procedures to perform most operations discussed in Section2.4. Implementing the time evolution methods discussed in Section2.5is straight forward. For exam-ple, ITensor was the package of choice in a master project that was initiated by the author of this thesis. Kothe [198] used the ITensor package and its MPS implementa-tion to study time-dependent transport mechanisms in tunneling juncimplementa-tions at finite temperature.
Highly flexible packages such as ITensor come with a significant problem when using them for calculations on a competitive level. They sacrifice performance in computa-tion time and memory for ease of adapcomputa-tion. Especially in ab initio quantum chemistry there is little chance to apply non-optimized code, as the complexity of the problem is enormous for non-trivial molecules. Computational power and memory is always the bottleneck in finding results of high accuracy. The modern quantum chemistry codes have been developed and optimized for decades and they are advanced in terms of computation efficiency, memory usage, and parallelism concepts. Therefore, we need an implementation of the MPS approach specifically optimized for quantum chemistry problems when proposing MPS as a promising approach to study time-dependent effects in molecules. Only then we will be able to describe dynamics in molecules with satisfying precision and international competitive efficiency.
In the next chapter, we want to propose the Hamburg CheMPS2 extension as one of such highly optimized implementations of the MPS approach, which enables time-dependent studies in quantum chemistry. We will discuss its origins and the
properties that make it one of the state-of-the-art MPS implementations. Together with the insights in the MPS approach in context of time-dependent quantum chem-istry (that we discuss in Chapter4and Chapter5), the Hamburg CheMPS2 extension is the main asset developed in this Ph.D. project. The results discussed in this thesis are just the starting point of time-dependent MPS in quantum chemistry, where the Hamburg CheMPS2 extension is used in numerous ongoing and future projects.
3.1 Symmetry Adapted Tensor Networks
A crucial benefit in computational physics originates in the symmetries of the physical system. Symmetries allow to reduce the dimension of the considered many-body Hilbert space, which then results in a simplification of the many-body state. For example, let us suppose we have a molecule where the electrons can occupyL= 20 orbitals. If we span the many-body Hilbert space without using any symmetries, its dimension is given bydim(H) = 4L ≈ 1.099·1012. To represent the many-body state computationally we needdim(H)≈1.099·1012coefficients (see Section2.2.2), which requires≈70.37T Bof data when using real double precision numbers. This is computationally very challenging using today’s computational resources, however, using symmetries we can reduce the problem to a subspace of the many-body Hilbert space of manageable size. If utilizing the electron number conservation and the spin conservation and focus on the many-body states withN↑ = 10spin-up electrons andN↓ = 10spin-down electrons, we can reduce the problem to a subspace of dim HN↑=10,N↓=10 = NL
↑
· NL↓ = 3.4134·1010coefficients, which only requires 0.2731T Bof data to store the many-body state. Calculations for many-body Hilbert space dimensions of this magnitude have been performed successfully [93]. With the MPS we have a complementary approach to simplify the representation of the many-body state. We can combine the symmetries of the molecular system with the MPS approach to gain an additive computational advantage. In the following section, we discuss how to reduce the effective dimension of the many-body Hilbert space using symmetries and how these symmetries are implemented into the tensors of the MPS approach.
When taking advantage of symmetries, we use that the Hamiltonian is block-diagonal if written in the eigenbasis of the symmetry generating operator (see Figure3.1). This is a consequence of
hˆH,ˆOi
= 0, (3.1)
where ˆOis the symmetry generating operator. The blocks correspond to eigenvalues of the operator ˆO, that are also good quantum numbers (for example the electron number) of the system. The many-body states with fixed quantum number form
...
0
0
...
HN−1
HN
HN+1 HNI
HNISSz
Figure 3.1.: Schematic factorization of the Hamiltonian into symmetry blocks. The symme-tries used in this Section are particle numberN, irreducible representationI, and spin quantum numbersSandSz.
a subspace of the many-body Hilbert space. When focussing on states with fixed quantum numbers, we can reduce the total problem onto one of the subspaces with a fixed quantum number. All other states can be neglected, since their coefficient in the many-body state are zero. Therefore, the effective Hilbert space dimension reduces and we obtain a computational advantage. This also holds for time-dependent situations, where the good quantum number is a conserved quantity of the system.
As soon as the initial state is chosen from one of the subspaces, the time evolved many-body state remains part of this subspace forever.
Already the first DMRG study of chemical systems by White and Martin [105] made use of symmetries. They incorporated the abelian electron number and spin-projection symmetries to find the accurate ground state energy of water. Shortly following, studies additionally incorporating abelian point group symmetries [44,199] were pub-lished, promoting the role of DMRG in quantum chemistry. With these symmetries implemented into the DMRG method, a rich family of quantum chemistry studies evolved with prior unmatched precise results. However, advancing the DMRG method even further by implementation of non-abelian symmetries proved more challenging.
First implementation of non-abelian symmetry was done by McCulloch and Gulácsi [200–202] however, not with the aim to study quantum chemical systems. Sharma and Chan [203], as well as Wouters et al. [51] were the first to implement the non-abelian spin symmetry to study eigenstates of quantum chemical systems. Especially implementation of the non-abelian total spin symmetry enhanced the capabilities of the DMRG method and the MPS approach, enabling description of nearly degenerated states with different spin quantum numbers.
In the Hamburg CheMPS2 extension (which is an extension of the implementation by Wouters [51]) we exploit three different symmetries of the molecule. We make use of the electron number symmetry, the point group symmetry, and the spin symmetry, which we discuss in detail in the following.