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arXiv:2409.17033 · The Journal of Chemical Physics

Susceptibility formulation of density matrix perturbation theory

Anders M. N. Niklasson, Adela Habib, Joshua Finkelstein, Emanuel H. Rubensson

Published Dec 16, 2024• 0 citations
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Abstract

Density matrix perturbation theory based on recursive Fermi-operator expansions provides a computationally efficient framework for time-independent response calculations in quantum chemistry and materials science. From a perturbation in the Hamiltonian, we can calculate the first-order perturbation in the density matrix, which then gives us the linear response in the expectation values for some chosen set of observables. We present an alternative, dual formulation, where we instead calculate the static susceptibility of an observable, which then gives us the linear response in the expectation values for any number of different Hamiltonian perturbations. We show how the calculation of the susceptibility can be performed with the same expansion schemes used in recursive density matrix perturbation theory, including generalizations to fractional occupation numbers and self-consistent linear response calculations, i.e., similar to density functional perturbation theory. As with recursive density matrix perturbation theory, the dual susceptibility formulation is well suited for numerically thresholded sparse matrix algebra, which has linear scaling complexity for sufficiently large sparse systems. Similarly, the recursive computation of the susceptibility also seamlessly integrates with the computational framework of deep neural networks used in artificial intelligence (AI) applications. This integration enables the calculation of quantum response properties that can leverage cutting-edge AI-hardware, such as NVIDIA Tensor Cores or Google Tensor Processing Units. We demonstrate performance for recursive susceptibility calculations using NVIDIA Graphics Processing Units and Tensor Cores.

DOI
10.1063/5.0239961

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Density matrix perturbation theory based on recursive Fermi-operator expansions provides a computationally efficient framework for time-independent response calculations in quantum chemistry and materials science. From a perturbation in the Hamiltonian, we can calculate the first-order perturbation in the density matrix, which then gives us the linear response in the expectation values for some chosen set of observables. We present an alternative, dual formulation, where we instead calculate the static susceptibility of an observable, which then gives us the linear response in the expectation values for any number of different Hamiltonian perturbations. We show how the calculation of the susceptibility can be performed with the same expansion schemes used in recursive density matrix perturbation theory, including generalizations to fractional occupation numbers and self-consistent linear response calculations, i.e., similar to density functional perturbation theory. As with recursive density matrix perturbation theory, the dual susceptibility formulation is well suited for numerically thresholded sparse matrix algebra, which has linear scaling complexity for sufficiently large sparse systems. Similarly, the recursive computation of the susceptibility also seamlessly integrates with the computational framework of deep neural networks used in artificial intelligence (AI) applications. This integration enables the calculation of quantum response properties that can leverage cutting-edge AI-hardware, such as NVIDIA Tensor Cores or Google Tensor Processing Units. We demonstrate performance for recursive susceptibility calculations using NVIDIA Graphics Processing Units and Tensor Cores.
Time-independent quantum response calculations are performed using Tensor cores. This is achieved by mapping density matrix perturbation theory onto the computational structure of a deep neural network. The main computational cost of each deep layer is dominated by tensor contractions, i.e., dense matrix-matrix multiplications, in mixed-precision arithmetics, which achieves close to peak performance. Quantum response calculations are demonstrated and analyzed using self-consistent charge density-functional tight-binding theory as well as coupled-perturbed Hartree-Fock theory. For linear response calculations, a novel parameter-free convergence criterion is presented that is well-suited for numerically noisy low-precision floating point operations and we demonstrate a peak performance of almost 200 Tflops using the Tensor cores of two Nvidia A100 GPUs.
An orbital-free quantum perturbation theory is proposed. It gives the response of the density matrix upon variation of the Hamiltonian by quadratically convergent recursions based on perturbed projections. The technique allows treatment of embedded quantum subsystems with a computational cost scaling linearly with the size of the perturbed region, O(N(pert.)), and as O(1) with the total system size. The method allows efficient high order perturbation expansions, as demonstrated with an example involving a 10th order expansion. Density matrix analogs of Wigner's 2n+1 rule are also presented.
We present several finite-temperature recursive Fermi-operator expansion schemes based on the second-order spectral projection (SP2) method. Our approach builds on a previous observation that the electronic structure problem, as formulated through a recursive SP2 expansion, can be mapped onto the architecture of a deep neural network. Using this perspective, we generalize SP2 to finite electronic temperatures by constructing machine learning models that determine optimized recursive expansion coefficients. The same approach is also applied to the prediction of the electronic entropy for fractional occupation numbers. The coefficients are trained for a specified chemical potential and electronic temperature and are not available in closed analytical form. However, by employing an appropriate affine rescaling strategy to the Hamiltonian matrix, we eliminate the need to retrain the model for different temperatures and chemical potentials. Our approach avoids explicit diagonalization and relies solely on highly optimized matrix-matrix multiplication kernels. Compared to state-of-the-art diagonalization, we achieve an order-of-magnitude speedup in the single-particle finite-temperature density matrix calculation for small and moderately sized matrices on modern GPUs and dense matrix multiply units.
An implicit purification scheme is proposed for calculation of the temperature-dependent, grand canonical single-particle density matrix, given as a Fermi-Dirac operator expansion in terms of the Hamiltonian. The computational complexity is shown to scale with the logarithm of the polynomial order of the expansion, or equivalently, with the logarithm of the inverse temperature. The system of linear equations that arise in each implicit purification iteration is solved efficiently by a conjugate gradient solver. The scheme is particularly useful in connection with linear scaling electronic structure theory based on sparse matrix algebra. The efficiency of the implicit temperature expansion technique is analyzed and compared to some explicit purification methods for the zero temperature density matrix.
Abstract Dynamical correlation functions are essential for revealing emergent phenomena in quantum matter, since they measure the frequency-resolved response of a system at equilibrium to a perturbation and thereby probe its full spectrum. While neural quantum states provide an accurate and flexible description of correlated ground states, accessing real-frequency dynamical correlators from them remains a major challenge. Here we introduce a general and systematically improvable framework to compute real-frequency linear-response functions directly from neural quantum state ground states. Our method performs the Lanczos subspace construction through an operator recursion and controls operator growth with a suited truncation scheme. We benchmark it on demanding quantum impurity models, where high precision is required close to the Fermi energy, and show that rich spectral structure and the self-energy can be resolved over several orders of magnitude. The approach generalizes straightforwardly, for example providing a direct path toward more broadly applicable dynamical mean-field theory solvers.
A time-dependent density-functional tight-binding method in real time domain is developed to calculate the absorption spectra of very large systems. The time-dependent first-order response of the density matrix due to an external perturbation is solved using Chebyshev method with high efficiency and accuracy. Linear scaling of CPU time and memory usage with the system size is achieved by exploring the sparsity of the involving matrices as well as by introduction of a cutoff for the first-order density matrix. The compressed sparse row scheme is used to store the matrices, and SPARSEKIT is employed for sparse matrix multiplication. The absorption spectra of three-dimensional water clusters (H2O)216{({\mathrm{H}}_{2}\mathrm{O})}_{216}(H2​O)216​, (H2O)432{({\mathrm{H}}_{2}\mathrm{O})}_{432}(H2​O)432​, (H2O)648{({\mathrm{H}}_{2}\mathrm{O})}_{648}(H2​O)648​, and (H2O)864{({\mathrm{H}}_{2}\mathrm{O})}_{864}(H2​O)864​ are calculated using the present approach. The error due to the cutoff of density matrix is negligible. It is shown from these calculations that the presented method is very efficient and capable of calculating the absorption spectra for very large three-dimensional systems.