We have implemented the single-site density matrix renormalization group algorithm for the variational optimization of SU(2) \times U(1) (spin and particle number) invariant matrix product states for general spin and particle number symmetric fermionic Hamiltonians. This class also includes non-relativistic quantum chemical systems within the Born-Oppenheimer approximation. High-accuracy ab-initio finite field results of the longitudinal static polarizabilities and second hyperpolarizabilities of one-dimensional hydrogen chains are obtained with the algorithm. A comparison with other methods is made.
We have implemented the single-site density matrix renormalization group algorithm for the variational optimization of SU(2) \times U(1) (spin and particle number) invariant matrix product states for general spin and particle number symmetric fermionic Hamiltonians. This class also includes non-relativistic quantum chemical systems within the Born-Oppenheimer approximation. High-accuracy ab-initio finite field results of the longitudinal static polarizabilities and second hyperpolarizabilities of one-dimensional hydrogen chains are obtained with the algorithm. A comparison with other methods is made.
We have implemented the sweep algorithm for the variational optimization of SU(2) U(1) (spin and particle number) invariant matrix product states (MPS) for general spin and particle number invariant fermionic Hamiltonians. This class includes non-relativistic quantum chemical systems within the Born-Oppenheimer approximation. High-accuracy ab initio finite field results of the longitudinal static polarizabilities and second hyperpolarizabilities of one-dimensional hydrogen chains are presented. This allows to assess the performance of other quantum chemical methods. For small basis sets, MPS calculations in the saturation regime of the optical response properties can be performed. These results are extrapolated to the thermodynamic limit.
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.
We describe how to efficiently construct the quantum chemical Hamiltonian operator in matrix product form. We present its implementation as a density matrix renormalization group (DMRG) algorithm for quantum chemical applications. Existing implementations of DMRG for quantum chemistry are based on the traditional formulation of the method, which was developed from the point of view of Hilbert space decimation and attained higher performance compared to straightforward implementations of matrix product based DMRG. The latter variationally optimizes a class of ansatz states known as matrix product states, where operators are correspondingly represented as matrix product operators (MPOs). The MPO construction scheme presented here eliminates the previous performance disadvantages while retaining the additional flexibility provided by a matrix product approach, for example, the specification of expectation values becomes an input parameter. In this way, MPOs for different symmetries - abelian and non-abelian - and different relativistic and non-relativistic models may be solved by an otherwise unmodified program.
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).
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 novel combined unitary and symmetric group approach is used to study the spin-21 Heisenberg model and related Fermionic systems in a total spin-adapted representation, using a linearly-parameterised Ansatz for the many-body wave function. We show that a more compact ground-state wave function representation---indicated by a larger leading ground-state coefficient---is obtained when combining the symmetric group Sn, in the form of permutations of the underlying lattice site ordering, with the cumulative spin coupling based on the unitary group U(n). In one-dimensional systems the observed compression of the wave function is reminiscent of block-spin renormalization group approaches, and allows us to study larger lattices (here taken up to 80 sites) with the spin-adapted full configuration interaction quantum Monte Carlo method, which benefits from the sparsity of the Hamiltonian matrix and the corresponding sampled eigenstates that emerge from the reordering. We find that in an optimal lattice ordering the configuration state function with highest weight already captures with high accuracy the spin-spin correlation function of the exact ground-state wave function. This feature is found for more general lattice models, such as the Hubbard model, and ab initio quantum chemical models, exemplified by one-dimensional hydrogen chains. We also provide numerical evidence that the optimal lattice ordering for the unitary group approach is not generally equivalent to the optimal ordering obtained for methods based on matrix-product states, such as the density-matrix renormalization group approach.