The similarities between Hartree-Fock (HF) theory and the density matrix renormalization group (DMRG) are explored. Both methods can be formulated as the variational optimization of a wave-function Ansatz. Linearization of the time-dependent variational principle near a variational minimum allows to derive the random phase approximation (RPA). We show that the nonredundant parameterization of the matrix product state (MPS) tangent space [J. Haegeman, J. I. Cirac, T. J. Osborne, I. Pi\ifmmode \check{z}\else \v{z}\fi{}orn, H. Verschelde, and F. Verstraete, Phys. Rev. Lett. 107, 070601 (2011)] leads to the Thouless theorem for MPS, i.e., an explicit nonredundant parameterization of the entire MPS manifold, starting from a specific MPS reference. Excitation operators are identified, which extends the analogy between HF and DMRG to the Tamm-Dancoff approximation (TDA), the configuration interaction (CI) expansion, and coupled cluster theory. For a small one-dimensional Hubbard chain, we use a CI-MPS Ansatz with single and double excitations to improve on the ground state and to calculate low-lying excitation energies. For a symmetry-broken ground state of this model, we show that RPA-MPS allows to retrieve the Goldstone mode. We also discuss calculations of the RPA-MPS correlation energy. With the long-range quantum chemical Pariser-Parr-Pople Hamiltonian, low-lying TDA-MPS and RPA-MPS excitation energies for polyenes are obtained.
The similarities between Hartree-Fock (HF) theory and the density matrix renormalization group (DMRG) are explored. Both methods can be formulated as the variational optimization of a wave-function Ansatz. Linearization of the time-dependent variational principle near a variational minimum allows to derive the random phase approximation (RPA). We show that the nonredundant parameterization of the matrix product state (MPS) tangent space [J. Haegeman, J. I. Cirac, T. J. Osborne, I. Pi\ifmmode \check{z}\else \v{z}\fi{}orn, H. Verschelde, and F. Verstraete, Phys. Rev. Lett. 107, 070601 (2011)] leads to the Thouless theorem for MPS, i.e., an explicit nonredundant parameterization of the entire MPS manifold, starting from a specific MPS reference. Excitation operators are identified, which extends the analogy between HF and DMRG to the Tamm-Dancoff approximation (TDA), the configuration interaction (CI) expansion, and coupled cluster theory. For a small one-dimensional Hubbard chain, we use a CI-MPS Ansatz with single and double excitations to improve on the ground state and to calculate low-lying excitation energies. For a symmetry-broken ground state of this model, we show that RPA-MPS allows to retrieve the Goldstone mode. We also discuss calculations of the RPA-MPS correlation energy. With the long-range quantum chemical Pariser-Parr-Pople Hamiltonian, low-lying TDA-MPS and RPA-MPS excitation energies for polyenes are obtained.
The density matrix renormalization group (DMRG) has an underlying variational ansatz, the matrix product state (MPS). The accuracy of this ansatz is controlled by its virtual dimension. In most applications for quantum chemistry this parameter is very large, and the high-accuracy or FCI regime is studied. With a good orbital choice and ordering, and by exploiting the symmetry group of the Hamiltonian, DMRG then allows to retrieve highly accurate results in active spaces beyond the reach of FCI. However, DMRG can also be of use with smaller virtual dimensions. I will discuss how fundamental ideas from Hartree-Fock (HF) theory are transferable to DMRG. Both methods have a variational ansatz. The time-independent variational principle leads for HF and DMRG to a self-consistent mean-field theory in the particles and lattice sites, respectively. The time-dependent variational principle results in the random-phase approximation (RPA). RPA reveals the elementary excitations: particle and site excitations for HF and DMRG, respectively. Exponentiation of these excitations provides a nonredundant parameterization of the ansatz manifold, formulated by Thouless’ theorem. A Taylor expansion in the nonredundant parameters leads to the configuration interaction expansion. I will also discuss how auxiliary field quantum Monte Carlo for Slater determinants can be extended for matrix product states.
Linear response theory for the density matrix renormalization group (DMRG-LRT) was first presented in terms of the DMRG renormalization projectors [J. J. Dorando, J. Hachmann, and G. K.-L. Chan, J. Chem. Phys. 130, 184111 (2009)]. Later, with an understanding of the manifold structure of the matrix product state (MPS) ansatz, which lies at the basis of the DMRG algorithm, a way was found to construct the linear response space for general choices of the MPS gauge in terms of the tangent space vectors [J. Haegeman, J. I. Cirac, T. J. Osborne, I. Pižorn, H. Verschelde, and F. Verstraete, Phys. Rev. Lett. 107, 070601 (2011)]. These two developments led to the formulation of the Tamm-Dancoff and random phase approximations (TDA and RPA) for MPS. This work describes how these LRTs may be efficiently implemented through minor modifications of the DMRG sweep algorithm, at a computational cost which scales the same as the ground-state DMRG algorithm. In fact, the mixed canonical MPS form implicit to the DMRG sweep is essential for efficient implementation of the RPA, due to the structure of the second-order tangent space. We present ab initio DMRG-TDA results for excited states of polyenes, the water molecule, and a [2Fe-2S] iron-sulfur cluster.
The density matrix renormalization group (DMRG) has an underlying variational ansatz, the matrix product state (MPS). The accuracy of this ansatz is controlled by its virtual dimension. In most applications for quantum chemistry this parameter is very large, and the high-accuracy or FCI regime is studied. With a good orbital choice and ordering, and by exploiting the symmetry group of the Hamiltonian, DMRG then allows to retrieve highly accurate results in active spaces beyond the reach of FCI [1,2]. However, DMRG can also be of use with smaller virtual dimensions. I will discuss how fundamental ideas from Hartree-Fock (HF) theory are transferable to DMRG. Both methods have a variational ansatz. The time-independent variational principle leads for HF and DMRG to a self-consistent mean-field theory in the particles and lattice sites, respectively. The time-dependent variational principle results in the random-phase approximation (RPA). RPA reveals the elementary excitations: particle and site excitations for HF and DMRG, respectively. Exponentiation of these excitations provides a nonredundant parameterization of the ansatz manifold, formulated by Thouless’ theorem. A Taylor expansion in the nonredundant parameters leads to the configuration interaction expansion [3,4]. References [1] S. Wouters, W. Poelmans, P. W. Ayers and D. Van Neck, Computer Physics Communications 185, 1501-1514 (2014). [2] S. Wouters, T. Bogaerts, P. Van Der Voort, V. Van Speybroeck and D. Van Neck, Journal of Chemical Physics 140, 241103 (2014). [3] S. Wouters, N. Nakatani, D. Van Neck and G. K.-L. Chan, Physical Review B 88, 075122 (2013). [4] N. Nakatani, S. Wouters, D. Van Neck and G. K.-L. Chan, Journal of Chemical Physics 140, 024108 (2014).
During the past 15 years, the density matrix renormalization group (DMRG) has become increasingly important for ab initio quantum chemistry. The underlying matrix product state (MPS) ansatz is a low-rank decomposition of the full configuration interaction tensor. The virtual dimension of the MPS controls the size of the corner of the many-body Hilbert space that can be reached. Whereas the MPS ansatz will only yield an efficient description for noncritical one-dimensional systems, it can still be used as a variational ansatz for other finite-size systems. Rather large virtual dimensions are then required. The two most important aspects to reduce the corresponding computational cost are a proper choice and ordering of the active space orbitals, and the exploitation of the symmetry group of the Hamiltonian. By taking care of both aspects, DMRG becomes an efficient replacement for exact diagonalization in quantum chemistry. DMRG and Hartree-Fock theory have an analogous structure. The former can be interpreted as a self-consistent mean-field theory in the DMRG lattice sites, and the latter in the particles. It is possible to build upon this analogy to introduce post-DMRG methods. Based on an approximate MPS, these methods provide improved ansätze for the ground state, as well as for excitations. Exponentiation of the single-particle (single-site) excitations for a Slater determinant (an MPS with open boundary conditions) leads to the Thouless theorem for Hartree-Fock theory (DMRG), an explicit nonredundant parameterization of the entire manifold of Slater determinants (MPS wavefunctions). This gives rise to the configuration interaction expansion for DMRG. The Hubbard-Stratonovich transformation lies at the basis of auxiliary field quantum Monte Carlo for Slater determinants. An analogous transformation for spin-lattice Hamiltonians allows to formulate a promising variant for MPSs.
An implementation of the single-site density matrix renormalization group (DMRG) approach for the quantum-chemical application to calculate approximate as well as exact electronic ground-state energies and wave functions of atoms and molecules is presented.The algorithm is formulated in terms of the variational and sizeconsistent class of wave functions, matrix product states, and their operator analogues, matrix product operators.A procedure to construct quantum-chemical Hamiltonian operators with long-range Coulomb interactions as matrix product operators is developed.The many-electron Hamiltonian operators are represented as sums and products of one-electron operators.For the efficient matrix product operator construction of quantum-chemical Hamiltonians, different basis spaces including discrete position and momentum space as well as molecular and atomic orbitals are investigated.Discrete position space yields the best matrix product operator size scaling, but introduces artificial long-range couplings in the kinetic energy term of the Hamiltonian which are a consequence of the DMRG algorithm in terms of matrix product states.The huge amount of discrete space points needed for an accurate description poses a problem because of the electron-electron interaction.Therefore, a momentum space or Fourier series expansion of the electron-electron interaction is shown which makes it possible to compress its matrix product operator representation.DMRG applied to quantum-chemical Hamiltonians in matrix product operator form can serve as means to directly (without mean-field treatment in advance) investigate quantum-chemical problems and find the analytically exact ground-state solution of small atoms and molecules.The next basis space analyzed in detail is the atomic orbital basis.The matrix product operator construction and the direct application of the DMRG method in this basis is examined.A physical compression scheme based on a singular value decomposition of the electron-electron interaction in the atomic orbital basis is demonstrated.Furthermore, DMRG calculations in the atomic orbital basis for certain systems where mean-field theory yields qualitatively wrong results and where higher-order approximative correction methods like truncated configuration interaction and coupled cluster cannot fully capture strong electron correlation effects are compared to results obtained in the molecular orbital basis.The comparison shows that atomic orbitals describe the physical situation well and thus lead to a tremendous reduction of the matrix product state dimension and the computational effort.i
A matrix product state formulation of the multiconfiguration time-dependent Hartree (MPS-MCTDH) theory is presented. The Hilbert space that is spanned by the direct products of the phonon degree of freedoms, which is linearly parameterized in the MCTDH ansatz and thus results in an exponential increase in the computational cost, is parametrized by the MPS form. Equations of motion based on the Dirac-Frenkel time-dependent variational principle is derived by using the tangent space projection and the projector-splitting technique for the MPS, which have been recently developed. The mean-field operators, which appear in the equation of motion of the MCTDH single particle functions, are written in terms of the MPS form and efficiently evaluated by a sweep algorithm that is similar to the density-matrix renormalized group sweep. The efficiency and convergence of the MPS approximation to the MCTDH are demonstrated by quantum dynamics simulations of extended excitonic molecular systems.