We describe a formulation of multi-reference perturbation theory that obtains a rigorous upper bound to the second order energy by minimizing the Hylleraas functional in the space of matrix product states (MPS). The first order wavefunctions so obtained can also be used to compute the third order energy with little overhead. Our formulation has several advantages including (i) flexibility with respect to the choice of zeroth order Hamiltonian, (ii) recovery of the exact uncontracted multi-reference perturbation theory energies in the limit of large MPS bond dimension, (iii) no requirement to compute high body density matrices, (iv) an embarrassingly parallel algorithm (scaling up to the number of virtual orbitals, squared, processors). Preliminary numerical examples show that the MPS bond dimension required for accurate first order wavefunctions scales sub-linearly with the size of the basis.
We describe a formulation of multi-reference perturbation theory that obtains a rigorous upper bound to the second order energy by minimizing the Hylleraas functional in the space of matrix product states (MPS). The first order wavefunctions so obtained can also be used to compute the third order energy with little overhead. Our formulation has several advantages including (i) flexibility with respect to the choice of zeroth order Hamiltonian, (ii) recovery of the exact uncontracted multi-reference perturbation theory energies in the limit of large MPS bond dimension, (iii) no requirement to compute high body density matrices, (iv) an embarrassingly parallel algorithm (scaling up to the number of virtual orbitals, squared, processors). Preliminary numerical examples show that the MPS bond dimension required for accurate first order wavefunctions scales sub-linearly with the size of the basis.
We present two efficient and intruder-free methods for treating dynamic correlation on top of general multiconfiguration reference wave functions — including such as obtained by the density matrix renormalization group (DMRG) with large active spaces. The new methods are the second order variant of the recently proposed multireference linearized coupled cluster method (MRLCC) [Sharma, S.; Alavi, A. J. Chem. Phys. 2015, 143, 102815] and of N-electron valence perturbation theory (NEVPT2), with expected accuracies similar to MRCI+Q and (at least) CASPT2, respectively. Great efficiency gains are realized by representing the first order wave function with a combination of internal contraction (IC) and matrix product state perturbation theory (MPSPT). With this combination, only third order reduced density matrices (RDMs) are required. Thus, we obviate the need for calculating (or estimating) RDMs of fourth or higher order; these had so far posed a severe bottleneck for dynamic correlation treatments involving the large active spaces accessible to DMRG. Using several benchmark systems, including first and second row containing small molecules, Cr 2, pentacene, and oxo-Mn(Salen), we show that active spaces containing at least 30 orbitals can be treated using this method. On a single node, MRLCC2 and NEVPT2 calculations can be performed with over 550 and 1100 virtual orbitals, respectively. We also critically examine the errors incurred due to the three sources of errors introduced in the present implementation – calculating second order instead of third order energy corrections, use of internal contraction, and approximations made in the reference wave function due to DMRG.
We propose a multireference linearized coupled cluster theory using matrix product states (MPSs-LCC) which provides remarkably accurate ground-state energies, at a computational cost that has the same scaling as multireference configuration interaction singles and doubles, for a wide variety of electronic Hamiltonians. These range from first-row dimers at equilibrium and stretched geometries to highly multireference systems such as the chromium dimer and lattice models such as periodic two-dimensional 1-band and 3-band Hubbard models. The MPS-LCC theory shows a speed up of several orders of magnitude over the usual Density Matrix Renormalization Group (DMRG) algorithm while delivering energies in excellent agreement with converged DMRG calculations. Also, in all the benchmark calculations presented here, MPS-LCC outperformed the commonly used multi-reference quantum chemistry methods in some cases giving energies in excess of an order of magnitude more accurate. As a size-extensive method that can treat large active spaces, MPS-LCC opens up the use of multireference quantum chemical techniques in strongly correlated ab initio Hamiltonians, including two- and three-dimensional solids.
In this work, we generalize the recently proposed matrix product state perturbation theory (MPSPT) for calculating energies of excited states using quasi-degenerate (QD) perturbation theory. Our formulation uses the Kirtman-Certain-Hirschfelder canonical Van Vleck perturbation theory, which gives Hermitian effective Hamiltonians at each order, and also allows one to make use of Wigner's 2n + 1 rule. Further, our formulation satisfies Granovsky's requirement of model space invariance which is important for obtaining smooth potential energy curves. Thus, when we use MPSPT with the Dyall Hamiltonian, we obtain a model space invariant version of quasi-degenerate n-electron valence state perturbation theory (NEVPT), a property that the usual formulation of QD-NEVPT2 based on a multipartitioning technique lacked. We use our method on the benchmark problems of bond breaking of LiF which shows ionic to covalent curve crossing and the twist around the double bond of ethylene where significant valence-Rydberg mixing occurs in the excited states. In accordance with our previous work, we find that multi-reference linearized coupled cluster theory is more accurate than other multi-reference theories of similar cost.
We propose a multireference linearized coupled cluster theory using matrix product states (MPSs-LCC) which provides remarkably accurate ground-state energies, at a computational cost that has the same scaling as multireference configuration interaction singles and doubles, for a wide variety of electronic Hamiltonians. These range from first-row dimers at equilibrium and stretched geometries to highly multireference systems such as the chromium dimer and lattice models such as periodic two-dimensional 1-band and 3-band Hubbard models. The MPS-LCC theory shows a speed up of several orders of magnitude over the usual Density Matrix Renormalization Group (DMRG) algorithm while delivering energies in excellent agreement with converged DMRG calculations. Also, in all the benchmark calculations presented here, MPS-LCC outperformed the commonly used multi-reference quantum chemistry methods in some cases giving energies in excess of an order of magnitude more accurate. As a size-extensive method that can treat large active spaces, MPS-LCC opens up the use of multireference quantum chemical techniques in strongly correlated ab initio Hamiltonians, including two- and three-dimensional solids.
The computationally expensive evaluation and storage of high-rank reduced density matrices (RDMs) has been the bottleneck in the calculation of dynamic correlation for multireference wave functions in large active spaces. We present a stochastic formulation of multireference configuration interaction and perturbation theory that avoids the need for these expensive RDMs. The algorithm presented here is flexible enough to incorporate a wide variety of active space reference wave functions, including selected configuration interaction, matrix product states, and symmetry-projected Jastrow mean field wave functions. It enjoys the usual attractive features of Monte Carlo methods, such as embarrassing parallelizability and low memory costs. We find that the stochastic algorithm is already competitive with the deterministic algorithm for small active spaces, containing as few as 14 orbitals. We illustrate the utility of our stochastic formulation using benchmark applications.
We explore various ways to group orbitals into clusters in a matrix product state (MPS). We explain how a generic cluster MPS can often lead to an increase in computational cost and instead propose a special cluster structure, involving only the first and last orbitals/sites, with a wider scope for computational advantage. This structure is a natural formalism to describe correlated multireference (MR) theories. We demonstrate the flexibility and usefulness of this approach by implementing various uncontracted MR configuration interaction, perturbation, and linearized coupled cluster theories using an MPS with large cluster sites. Applications to the nitrogen dimer, the chromium dimer, and benzene, including up to triple excitations in the external space, demonstrate the utility of an MPS with up to two large sites. We use our results to analyze the quality of different multireference approximations.