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.
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.
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.
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.
Density functional theory provides a powerful tool for computations of the quantum state of atoms, molecules and solids, and of ab-initio molecular dynamics. It was conceived in its initial naïve and approximative version by Thomas and Fermi immediately after the foundation of quantum mechanics, in 1927. Just thirty years ago, in the middle of the sixties, Hohenberg, Kohn and Sham on the one hand established a logically rigorous density functional theory of the quantum groundstate on the basis of quantum mechanics, and on the other hand, guided by this construction, introduced an approximative explicit theory called the local-density approximation, which for computations of the quantum groundstate of many-particle systems proved to be superior to both Thomas-Fermi and Hartree-Fock theories. From that time on, density functional theory has grown vastly in popularity, and a flood of computational work in molecular and solid state physics has been the result. Motivated by its success, there has been always a tendency to widen the fields of application of density functional theory, and in these developments, some points which were left somewhat obscure in the basic theory, were brought into focus from time to time. This led in the early eighties to a deepening of the logical basis, essentially by Levy and Lieb, and finally Lieb gave the basic theory a form of final mathematical rigour. Since that treatment, however, is based on the tools of modern convex functional analysis, its implications only gradually became known to the many people who apply density functional theory. A thorough treatment of the dependence on particle number on the basis of Lieb’s theory is given for the first time in the present text.
Despite the success of modern quantum chemistry in predicting properties of organic molecules, the treatment of inorganic systems, which have many close lying states, remains out of quantitative reach for current methods. To treat non-dynamic correlation, we take advantage of the density matrix renormalization group (DMRG) method that has become very successful in the field of solid state physics. We present a detailed study of the DMRG method, and we pay special attention to the evolution of the understanding behind the mathematical structure of the DMRG wave function. Our primary goal is to develop a density matrix renormalization group self--consistent--field (DMRG-SCF) approach, analogous to the complete active space self--consistent field (CASSCF) method, but dealing with large active spaces that are too demanding for the full configuration interaction (FCI) method. \nAs a first step towards such a DMRG-SCF procedure, we present a spin-adapted DMRG algorithm designed to target spin- and spatial-symmetry states that are hard to obtain while using an unrestricted algorithm. \nOur next step is a modification of the DMRG algorithm to obtain decreasing energy at every step during the sweep. This monotonically convergent DMRG scheme lets us obtain the two-body density matrix as a by--product of the existing procedure without any additional cost in storage. Additionally, the two-body density matrix produced at convergence using this scheme is free from the N-representability problem that is present when the two--body density matrix is produced with the two-site DMRG scheme without additional storage cost. Finally, taking advantage of the modifications developed herein, we present results obtained from our DMRG-SCF method. Lastly, we discuss possible ways of incorporating dynamical correlation into the DMRG scheme, in order to obtain a modern multireference approach.
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 have compared different strategies for ab initio quantum chemistry density matrix renormalization group treatments. The two starting orbital blocks include all valence and active orbitals of the reference complete active space self consistent field wave function. To generate the remaining blocks, we propose following the order of the contributions to the correlation energy: a posteriori using approximate occupation numbers or a priori, using a Møller–Plesset type of arguments, by explicit evaluation of second-order interactions. We have compared two different schemes for orbital localization to identify the important and less important orbital interactions and simplify the generation of the orbital blocks. To truncate the expansion we have compared two approaches, keeping constant the number m of components or the threshold λ to fix the residue of the expansion at each step. The extrapolation of the energies is found to provide accurate estimates of the full configuration interaction energy, making the expansion independent on the actual values of the two parameters m and λ. We propose to generate the factors for the two blocks from ground and excited eigenvectors of the Hamiltonian matrix.