Obtaining the two-body density matrix in the density matrix renormalization group method

We present an approach that allows to produce the two-body density matrix during the density matrix renormalization group (DMRG) run without an additional increase in the current disk and memory requirements. The computational cost of producing the two-body density matrix is proportional to O(M3k2+M2k4). The method is based on the assumption that different elements of the two-body density matrix can be calculated during different steps of a sweep. Hence, it is desirable that the wave function at the convergence does not change during a sweep. We discuss the theoretical structure of the wave function ansatz used in DMRG, concluding that during the one-site DMRG procedure, the energy and the wave function are converging monotonically at every step of the sweep. Thus, the one-site algorithm provides an opportunity to obtain the two-body density matrix free from the N-representability problem. We explain the problem of local minima that may be encountered in the DMRG calculations. We discuss theoretically why and when the one- and two-site DMRG procedures may get stuck in a metastable solution, and we list practical solutions helping the minimization to avoid the local minima.     

Spin-adapted density matrix renormalization group algorithms for quantum chemistry

We extend the spin-adapted density matrix renormalization group (DMRG) algorithm of McCulloch and Gulacsi [Europhys. Lett. 57, 852 (2002)] to quantum chemical Hamiltonians. This involves using a quasi-density matrix, to ensure that the renormalized DMRG states are eigenfunctions of S?(2), and the Wigner-Eckart theorem, to reduce overall storage and computational costs. We argue that the spin-adapted DMRG algorithm is most advantageous for low spin states. Consequently, we also implement a singlet-embedding strategy due to Tatsuaki [Phys. Rev. E 61, 3199 (2000)] where we target high spin states as a component of a larger fictitious singlet system. Finally, we present an efficient algorithm to calculate one- and two-body reduced density matrices from the spin-adapted wavefunctions. We evaluate our developments with benchmark calculations on transition metal system active space models. These include the Fe(2)S(2), [Fe(2)S(2)(SCH(3))(4)](2-), and Cr(2) systems. In the case of Fe(2)S(2), the spin-ladder spacing is on the microHartree scale, and here we show that we can target such very closely spaced states. In [Fe(2)S(2)(SCH(3))(4)](2-), we calculate particle and spin correlation functions, to examine the role of sulfur bridging orbitals in the electronic structure. In Cr(2) we demonstrate that spin-adaptation with the Wigner-Eckart theorem and using singlet embedding can yield up to an order of magnitude increase in computational efficiency. Overall, these calculations demonstrate the potential of using spin-adaptation to extend the range of DMRG calculations in complex transition metal problems.     

Density matrix renormalisation group Lagrangians

We introduce a Lagrangian formulation of the density matrix renormalisation group (DMRG). We present Lagrangians which, when minimised, yield the optimal DMRG wavefunction in a variational sense, both within the general matrix product ansatz and within the canonical form of the matrix product that is constructed within the DMRG sweep algorithm. Some of the results obtained are similar to elementary expressions in Hartree-Fock theory, and we draw attention to such analogies. The Lagrangians introduced here will be useful in developing theories of analytic response and derivatives in the DMRG.     

State-of-the-art density matrix renormalization group and coupled cluster theory studies of the nitrogen binding curve

We study the nitrogen binding curve with the density matrix renormalization group (DMRG) and single-reference and multireference coupled cluster (CC) theory. Our DMRG calculations use up to 4000 states and our single-reference CC calculations include up to full connected hextuple excitations. Using the DMRG, we compute an all-electron benchmark nitrogen binding curve, at the polarized, valence double-zeta level (28 basis functions), with an estimated accuracy of 0.03 mEh. We also assess the performance of more approximate DMRG and CC theories across the nitrogen curve. We provide an analysis of the relative strengths and merits of the DMRG and CC theory under different correlation conditions.     

On the importance of orbital localization in QC‐DMRG calculations

We investigate the importance of orbital localization in the application of the Density Matrix Renormalization Group (DMRG) technique to ab initio studies of molecular electronic structure. Our previous implementation of DMRG has been generalized to allow for the use of localized nonorthogonal orbitals. Simple cycles of equidistant hydrogen atoms, which are good examples of one dimensional metal‐like lattices with fully delocalized electronic structures, have been taken as test models. In this study, the efficiency of the DMRG method and the importance of orbital localization for the generation of the DMRG building blocks are confirmed. However, it is found that the convergence of the procedure based on nonorthogonal orbitals is slower and requires more DMRG components than the standard orthogonal formulation. Symmetrically orthonormalized atomic orbitals are shown to be a good compromise solution: they satisfy the requirement of orbital localization for the generation of the DMRG blocks and improve the convergence, reducing the number of components of the DMRG expansion. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

频域空间密度矩阵重正化群的研究进展

密度矩阵重正化群(DMRG)作为低维强关联体系中电子结构计算的强有力方法被广泛熟知, 并被迅速地应用于量子化学, 不仅在电子结构计算中发挥重要作用, 同时也在近几年迅速地成为复杂体系量子动力学计算的重要方法. 在DMRG框架中, 衍生出了一系列计算动态响应性质的有效方法, 并得到了广泛应用. 本文简述了DMRG的基本理论, 其矩阵乘积态(MPS)表示有效地扩展了该方法的应用范围. 重点介绍了基于线性响应理论的动态DMRG, 在频率空间求解系统在零温以及有限温度下响应性质的算法, 并介绍其在电子关联问题和电子-声子关联问题中的应用, 最后展望了该领域的未来发展方向.     

Decomposition of density matrix renormalization group states into a Slater determinant basis

The quantum chemical density matrix renormalization group (DMRG) algorithm is difficult to analyze because of the many numerical transformation steps involved. In particular, a decomposition of the intermediate and the converged DMRG states in terms of Slater determinants has not been accomplished yet. This, however, would allow one to better understand the convergence of the algorithm in terms of a configuration interaction expansion of the states. In this work, the authors fill this gap and provide a determinantal analysis of DMRG states upon convergence to the final states. The authors show that upon convergence, DMRG provides the same complete-active-space expansion for a given set of active orbitals as obtained from a corresponding configuration interaction calculation. Additional insight into DMRG convergence is provided, which cannot be obtained from the inspection of the total electronic energy alone. Indeed, we will show that the total energy can be misleading as a decrease of this observable during DMRG microiteration steps may not necessarily be taken as an indication for the pickup of essential configurations in the configuration interaction expansion. One result of this work is that a fine balance can be shown to exist between the chosen orbital ordering, the guess for the environment operators, and the choice of the number of renormalized states. This balance can be well understood in terms of the decomposition of total and system states in terms of Slater determinants.     

Construction of environment states in quantum-chemical density-matrix renormalization group calculations

The application of the quantum-chemical density-matrix renormalization group (DMRG) algorithm is cumbersome for complex electronic structures with many active orbitals. The high computational cost is mainly due to the poor convergence of standard DMRG calculations. A factor which affects the convergence behavior of the calculations is the choice of the start-up procedure. In this start-up step matrix representations of operators have to be calculated in a guessed many-electron basis of the DMRG environment block. Different possibilities for the construction of these basis states exist, and we first compare four procedures to approximate the environment states using Slater determinants explicitly. These start-up procedures are applied to DMRG calculations on a sophisticated test system: the chromium dimer. It is found that the converged energies and the rate of convergence depend significantly on the choice of the start-up procedure. However, since already the most simple start-up procedure, which uses only the Hartree-Fock determinant, is comparatively good, Slater determinants, in general, appear not to be a good choice as approximate environment basis states for convergence acceleration. Based on extensive test calculations it is demonstrated that the computational cost can be significantly reduced if the number of total states m is successively increased. This is done in such a way that the environment states are built up stepwise from system states of previous truncated DMRG sweeps for slowly increasing m values.     

Relativistic DMRG calculations on the curve crossing of cesium hydride

Over the past few years, it has been shown in various studies on small molecules with only a few electrons that the density-matrix renormalization group (DMRG) method converges to results close to the full configuration-interaction limit for the total electronic energy. In order to test the capabilities of the method for molecules with complex electronic structures, we performed a study on the potential-energy curves of the ground state and the first excited state of 1sigma+ symmetry of the cesium hydride molecule. For cesium relativistic effects cannot be neglected, therefore we have used the generalized arbitrary-order Douglas-Kroll-Hess protocol up to tenth order, which allows for a complete decoupling of the Dirac Hamiltonian. Scalar-relativistic effects are thus fully incorporated in the calculations. The potential curves of the cesium hydride molecule feature an avoided crossing between the ground state and the first excited state, which is shown to be very well described by the DMRG method. Compared to multireference configuration-interaction results, the potential curves hardly differ in shape, for both the ground state and the excited state, but the total energies from the DMRG calculations are in general consistently lower. However, the DMRG energies are as accurate as corresponding coupled cluster energies at the equilibrium distance, but convergence to the full configuration-interaction limit is not achieved.     

Quantification of electron correlation effects: Quantum Information Theory vs Method of Increments

Understanding electron correlation is crucial for developing new concepts in electronic structure theory, especially for strongly correlated electrons. We compare and apply two different approaches to quantify correlation contributions of orbitals: Quantum Information Theory (QIT) based on a Density Matrix Renormalization Group (DMRG) calculation and the Method of Increments (MoI). Although both approaches define very different correlation measures, we show that they exhibit very similar patterns when being applied to a polyacetelene model system. These results suggest one may deduce from one to the other, allowing the MoI to leverage from QIT results by screening correlation contributions with a cheap (“sloppy”) DMRG with a reduced number of block states. Or the other way around, one may select the active space in DMRG from cheap one-body MoI calculations.     

On the spin and symmetry adaptation of the density matrix renormalization group method

We present a spin-adapted density matrix renormalization group (DMRG) algorithm designed to target spin and spatial symmetry states that can be difficult to obtain while using a non-spin-adapted algorithm. The algorithmic modifications that have to be introduced into the usual density matrix renormalization group scheme in order to spin adapt it are discussed, and it is demonstrated that the introduced modifications do not change the overall scaling of the method. The new approach is tested on HNCO, a model system, that has a singlet-triplet curve crossing between states of the same symmetry. The advantages of the spin-adapted DMRG scheme are discussed, and it is concluded that the spin-adapted DMRG method converges better in almost all cases and gives more parallel curves to the full configuration interaction result than the non-spin-adapted method. It is shown that the spin-adapted DMRG energies can be lower than the ones obtained from the non-spin-adapted scheme. Such a counterintuitive result is explained by noting that the spin-adapted method is not a special case of the non-spin-adapted one; consequently, the spin-adapted result is not an upper bound for the non-spin-adapted energy.     

Orbital optimization in the density matrix renormalization group, with applications to polyenes and beta-carotene

In previous work we have shown that the density matrix renormalization group (DMRG) enables near-exact calculations in active spaces much larger than are possible with traditional complete active space algorithms. Here, we implement orbital optimization with the DMRG to further allow the self-consistent improvement of the active orbitals, as is done in the complete active space self-consistent field (CASSCF) method. We use our resulting DMRG-CASSCF method to study the low-lying excited states of the all-trans polyenes up to C24H26 as well as beta-carotene, correlating with near-exact accuracy the optimized complete pi-valence space with up to 24 active electrons and orbitals, and analyze our results in the light of the recent discovery from resonance Raman experiments of new optically dark states in the spectrum.     

Targeted excited state algorithms

To overcome the limitations of the traditional state-averaging approaches in excited state calculations, where one solves and represents all states between the ground state and excited state of interest, we have investigated a number of new excited state algorithms. Building on the work of van der Vorst and Sleijpen [SIAM J. Matrix Anal. Appl. 17, 401 (1996)], we have implemented harmonic Davidson and state-averaged harmonic Davidson algorithms within the context of the density matrix renormalization group (DMRG). We have assessed their accuracy and stability of convergence in complete-active-space DMRG calculations on the low-lying excited states in the acenes ranging from naphthalene to pentacene. We find that both algorithms offer increased accuracy over the traditional state-averaged Davidson approach, and, in particular, the state-averaged harmonic Davidson algorithm offers an optimal combination of accuracy and stability in convergence.     

An algorithm for large scale density matrix renormalization group calculations

We describe in detail our high-performance density matrix renormalization group (DMRG) algorithm for solving the electronic Schrodinger equation. We illustrate the linear scalability of our algorithm with calculations on up to 64 processors. The use of massively parallel machines in conjunction with our algorithm considerably extends the range of applicability of the DMRG in quantum chemistry.     

UNO DMRG CAS CI calculations of binuclear manganese complex Mn(IV)2O2(NHCHCO2)4: Scope and applicability of Heisenberg model

Both direct exchange and super-exchange interactions cooperate to realize inter-spin magnetic interaction in binuclear manganese complex Mn(IV)₂O₂(NHCHCO₂)₄ with a di-μ-oxo path. We revisited this spin system using DMRG CAS methods and CAS selection procedures. Our results indicate that our previous “dynamically extended spin polarization” (DE-SP) procedure for organic polyradicals and so forth does not work well. Thus, we have examined another selection procedure, the “dynamically extended super-exchange” (DE-SE) procedure. DMRG CASCI [18,18] by UB3LYP(HS)-UNO(DE-SE) can realize antiferromagnetic J values similar to experimental ones (−87 cm⁻¹). In addition, all J values between all spin states (HS[septet],IS[quintet],IS[triplet],LS[singlet])were also shown to be correct under sufficiently large M values. © 2018 Wiley Periodicals, Inc.     

Investigation of challenging spin systems using Monte Carlo configuration interaction and the density matrix renormalization group

We investigate if a range of challenging spin systems can be described sufficiently well using Monte Carlo configuration interaction (MCCI) and the density matrix renormalization group (DMRG) in a way that heads toward a more “black box” approach. Experimental results and other computational methods are used for comparison. The gap between the lowest doublet and quartet state of methylidyne (CH) is first considered. We then look at a range of first‐row transition metal monocarbonyls: MCO when M is titanium, vanadium, chromium, or manganese. For these MCO systems we also employ partially spin restricted open‐shell coupled‐cluster (RCCSD). We finally investigate the high‐spin low‐lying states of the iron dimer, its cation and its anion. The multireference character of these molecules is also considered. We find that these systems can be computationally challenging with close low‐lying states and often multireference character. For this more straightforward application and for the basis sets considered, we generally find qualitative agreement between DMRG and MCCI. © 2017 Wiley Periodicals, Inc.     

含时密度矩阵重正化群的理论与应用

密度矩阵重正化群(DMRG)作为计算低维强关联体系强有力的方法为人熟知, 在量子化学电子结构计算中得到广泛应用. 最近几年, 含时密度矩阵重正化群(TD-DMRG)的理论取得较快发展, TD-DMRG逐渐成为复杂体系量子动力学理论模拟的重要新兴方法之一. 本文综述了基于矩阵乘积态(MPS) 和矩阵乘积算符(MPO)的DMRG基本理论, 并重点介绍了若干最常见的TD-DMRG时间演化算法, 包括基于演化再压缩(P&C) 的算法、 基于含时变分原理(TDVP)的算法和时间步瞄准(TST)算法; 还对利用TD-DMRG模拟有限温体系的纯化(Purification)算法和最小纠缠典型量子热态(METTS)算法进行了介绍. 最后, 对近年TD-DMRG在复杂体系量子动力学中的应用进行了总结.