Machine learning a molecular Hamiltonian for predicting electron dynamics
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Machine learning a molecular Hamiltonian for predicting electron dynamics Harish S. Bhat1
· Karnamohit Ranka2 · Christine M. Isborn2
Received: 16 July 2020 / Revised: 30 August 2020 / Accepted: 22 September 2020 / Published online: 6 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We develop a computational method to learn a molecular Hamiltonian matrix from matrix-valued time series of the electron density. As we demonstrate for three small molecules, the resulting Hamiltonians can be used for electron density evolution, producing highly accurate results even when propagating 1,000 time steps beyond the training data. As a more rigorous test, we use the learned Hamiltonians to simulate electron dynamics in the presence of an applied electric field, extrapolating to a problem that is beyond the field-free training data. We find that the resulting electron dynamics predicted by our learned Hamiltonian are in close quantitative agreement with the ground truth. Our method relies on combining a reduced-dimensional, linear statistical model of the Hamiltonian with a time-discretization of the quantum Liouville equation within time-dependent Hartree Fock theory. We train the model using a least-squares solver, avoiding numerous, CPU-intensive optimization steps. For both field-free and field-on problems, we quantify training and propagation errors, highlighting areas for future development. Keywords electron dynamics · Electron density · Hamiltonian · Machine learning · System identification
1 Introduction An intriguing new application of machine learning is to predict the dynamical electronic properties of a molecular system [1–3], which is essential to understanding phenomena such as charge transfer and response to an applied laser field. When discussing such electron dynamics predictions, we must start with the electronic time-dependent Schrödinger equation (TDSE): i
dΨ (r, t) = Hˆ (r, t)Ψ (r, t). dt
(1)
Here Hˆ (r, t) is the electronic Hamiltonian operator that operates on the time-dependent many-body electronic wave
B
Harish S. Bhat [email protected] Karnamohit Ranka [email protected] Christine M. Isborn [email protected]
1
Applied Mathematics Department, University of California, Merced, 5200 N. Lake Rd., Merced, CA 95343, USA
2
Chemistry Department, University of California, Merced, 5200 N. Lake Rd., Merced, CA 95343, USA
function Ψ (r, t), where r represents the spatial and spin coordinates of all electrons. One can derive from (1) an evolution equation for the time-dependent density operator. This operator equation can be represented in a finite-dimensional basis, yielding a matrix system of ordinary differential equations: i
dP (t) = H (t), P (t) . dt
(2)
We call this the quantum Liouville–von Neumann equation. Boldface capital letters denote matrices, representations of operators in particular bases. Primes denote representations of operators in an orthonormal basis. Here P (t) and H (t) are time-dependent density and Hamiltonian matrices, respe
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