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An approximation strategy on the Grassmann manifold to compute accurate SCF initial guesses for repeated calculations at different geometries

Étienne Polack, Aleksandr Mikhalev, Geneviève Dusson, Benjamin Stamm, Filippo Lipparini

Published May 15, 2020• 0 citations
OpenAlex

Abstract

Repeated computations on the same molecular system, but with different geometries, are often performed in quantum chemistry, for instance, in ab-initio molecular dynamics simulations or geometry optimizations. While many efficient strategies exist to provide a good guess for the self-consistent field procedure, which is usually the main computational task to be performed, little is known on how to efficiently exploit in this direction the abundance of information generated during the many computations. In this article, we present a strategy to provide an accurate initial guess for the density matrix, expanded in a set of localized basis functions, within the self-consistent field iterations for parametrized Hartree-Fock problems where the nuclear coordinates are changed along a few user-specified collective variables, such as the molecule's normal modes. Our approach is based on an offline-stage where the Hartree-Fock eigenvalue problem is solved for some particular parameter values and an online-stage where the initial guess is computed very efficiently for any new parameter value. The method allows non-linear approximations of density matrices, which belong to a non-linear manifold that is isomorphic to the Grassmann manifold. The so-called Grassmann exponential and logarithm map the manifold onto the tangent space and thus provides the correct geometrical setting accounting for the manifold structure when working with subspaces rather than functions itself. Numerical tests on different amino acids show promising initial results.

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Repeated computations on the same molecular system, but with different geometries, are often performed in quantum chemistry, for instance, in ab-initio molecular dynamics simulations or geometry optimizations. While many efficient strategies exist to provide a good guess for the self-consistent field procedure, which is usually the main computational task to be performed, little is known on how to efficiently exploit in this direction the abundance of information generated during the many computations. In this article, we present a strategy to provide an accurate initial guess for the density matrix, expanded in a set of localized basis functions, within the self-consistent field iterations for parametrized Hartree-Fock problems where the nuclear coordinates are changed along a few user-specified collective variables, such as the molecule's normal modes.
Our approach is based on an offline-stage where the Hartree-Fock eigenvalue problem is solved for some particular parameter values and an online-stage where the initial guess is computed very efficiently for any new parameter value.
The method allows non-linear approximations of density matrices, which belong to a non-linear manifold that is isomorphic to the Grassmann manifold.
The so-called Grassmann exponential and logarithm map the manifold onto the tangent space and thus provides the correct geometrical setting accounting for the manifold structure when working with subspaces rather than functions itself.
Numerical tests on different amino acids show promising initial results.
Repeated computations on the same molecular system, but with different geometries, are often performed in quantum chemistry, for instance, in ab-initio molecular dynamics simulations or geometry optimizations. While many efficient strategies exist to provide a good guess for the self-consistent field procedure, which is usually the main computational task to be performed, little is known on how to efficiently exploit in this direction the abundance of information generated during the many computations. In this article, we present a strategy to provide an accurate initial guess for the density matrix within the self-consistent field iterations for parametrized Hartree-Fock problems where the nuclear coordinates are parametrized by a few user-specified parameters such as normal modes or collective variables in general. Our approach is based on an offline-stage where the Hartree-Fock eigenvalue problem is solved for some particular parameter values and an online-stage where the initial guess is computed very efficiently for any new parameter value. The method is based on non-linear approximations of density matrices, which belong to a non-linear manifold that is isomorphic to the Grassmann manifold, by means of the so-called Grassmann exponential and logarithm in order to map the manifold onto the tangent space and thus providing the correct geometrical setting accounting for the manifold structure when working with subspaces rather than functions itself. Numerical tests on different amino acids show promising initial results.
Ab-initio molecular dynamics simulations are powerful but require expensive calculations. In general, the most costly part of the computation consists of the many resolutions of the Kohn-Sham equations arising at each time-step of the simulation. Proposing good initial guesses of the density matrix for the Self-Consistent Field procedure is therefore key to reduce the overall computational cost of the simulation. In this talk, I will first present a method to provide such guesses based on previous calculations, exploiting the geometrical structure of the manifold to which the density matrix belongs, in order to perform linear extrapolation in a vector space, while at the same time, retaining the correct physical properties of the extrapolated density. I will also show how the time-reversibility of the simulation can be obtained. I will then illustrate the performance of the method by showing numerical results on a few real-life, multiscale, polarizable QM/MM simulations. Third, I will provide a theoretical analysis of the method for a better understanding of the observed results.
The inherently high computational cost of iterative self-consistent field (SCF) methods proves to be a critical issue delaying visual and haptic feedback in real-time quantum chemistry. In this work, we introduce two schemes for SCF acceleration. They provide a guess for the initial density matrix of the SCF procedure generated by extrapolation techniques. SCF optimizations then converge in fewer iterations, which decreases the execution time of the SCF optimization procedure. To benchmark the proposed propagation schemes, we developed a test bed for performing quantum chemical calculations on sequences of molecular structures mimicking real-time quantum chemical explorations. Explorations of a set of six model reactions employing the semi-empirical methods PM6 and DFTB3 in this testing environment showed that the proposed propagation schemes achieved speedups of up to 30% as a consequence of a reduced number of SCF iterations.
We present a self-consistent field (SCF) approach within the adaptive derivative-assembled problem-tailored ansatz variational quantum eigensolver (ADAPT-VQE) framework for efficient quantum simulations of chemical systems on near-term quantum computers. To this end, our ADAPT-VQE-SCF approach combines the idea of generating an ansatz with a small number of parameters, resulting in shallow-depth quantum circuits with a direct minimization of an energy expression that is correct to second order with respect to changes in the molecular orbital basis. Our numerical analysis, including calculations for the transition-metal complex ferrocene [Fe (C 5 H 5 ) 2 ], indicates that convergence in the self-consistent orbital optimization loop can be reached without a considerable increase in the number of two-qubit gates in the quantum circuit by comparison to a VQE optimization in the initial molecular orbital basis. Moreover, the orbital optimization can be carried out simultaneously within each iteration of the ADAPT-VQE cycle. ADAPT-VQE-SCF thus allows us to implement a routine analogous to the complete active space SCF, a cornerstone of state-of-the-art computational chemistry, in a hardware-efficient manner on near-term quantum computers. Hence, ADAPT-VQE-SCF paves the way toward a paradigm shift for quantitative quantum-chemistry simulations on quantum computers by requiring fewer qubits and opening up for the use of large and flexible atomic orbital basis sets in contrast to earlier methods that are predominantly based on the idea of full active spaces with minimal basis sets.
High Resolution Image Download MS PowerPoint Slide We demonstrate a physics-informed program synthesis (PIPS) approach that can be used to identify entirely new algorithms that approximate the results of single-reference electronic structure approaches like Hartree–Fock (HF) and density-functional theory (DFT)─but without any self-consistent field iterations at all. Our PIPS strategy exploits the fact that the eigenvectors of the Fock matrix F (or Kohn–Sham matrix K ) are the same as the eigenvectors of a broad class of matrix functions, f ( F ). As a result, PIPS can be used to seek matrices M that yield the same molecular orbital coefficients as converged HF or DFT calculations. We demonstrate this approach by generating new algorithms that accurately predict total energies for a series of heterodiatomic molecules (LiCl, LiF, NaCl, NaF) and C 1 –C 4 hydrocarbons; further simulations of C 8 –C 20 alkane species demonstrate further transferability and efficiency of the resulting algorithms. We obtain novel algorithms that can reproduce HF or DFT energies to within 0.1 kcal/mol/atom while requiring only a single matrix-diagonalization operation, rather than an iterative self-consistent field convergence. The approach demonstrated here could be similarly applied to more complex wave function ansatze, opening an interesting optimization-based pathway to identifying accurate yet efficient algorithms for molecular quantum chemistry.
Abstract We present a fully numerical framework for the optimization of molecule-specific quantum chemical basis functions within the quantics tensor train format using a finite-difference scheme. The optimization is driven by solving the Hartree–Fock equations (HF) with the density-matrix renormalization group algorithm on Cartesian grids that are iteratively refined. In contrast to the standard way of tackling the mean-field problem by expressing the molecular orbitals as linear combinations of atomic orbitals (LCAO) our method only requires as much basis functions as there are electrons within the system. Benchmark calculations for atoms and molecules with up to ten electrons show excellent agreement with LCAO calculations with large basis sets supporting the validity of the tensor network approach. Our work therefore offers a promising alternative to well-established HF-solvers and could pave the way to define highly accurate, fully numerical, molecule-adaptive basis sets, which, in the future, could lead to benefits for post-HF calculations.