Abstract
Known force terms arising in the Ehrenfest dynamics of quantum electrons and classical nuclei, due to a moving basis set for the former, can be understood in terms of the curvature of the manifold hosting the quantum states of the electronic subsystem. Namely, the velocity-dependent terms appearing in the Ehrenfest forces on the nuclei acquire a geometrical meaning in terms of the intrinsic curvature of the manifold, while Pulay terms relate to its extrinsic curvature.
Highlights
In this paper we extend the formalism of Ref. [1] to the Ehrenfest forces in mixed, classicalnuclei / quantum-electrons dynamical calculations
For Ehrenfest dynamics of quantum electrons and classical nuclei, and for basis functions for the former that move with the latter, it has been shown how the extra terms appearing in the Ehrenfest forces acquire a natural geometric interpretation in the curved manifold given by the set of electronic Hilbert spaces defined at each set of nuclear positions
The velocity-dependent term, explicitly non-adiabatic, depends on the intrinsic curvature of the manifold
Summary
For Ehrenfest dynamics, considering a system of quantum electrons and classical nuclei, and following Todorov [14], we start from the Lagrangian. N j2 defined for the Ne wavefunctions ψn of independent electrons (we will disregard spin hereafter), and for the 3Nn position components of Nn nuclei in three dimensions, R j, as dynamical variables. N we express the electronic wavefunctions in a finite, non-orthogonal, and evolving basis set, {|eμ, t〉, μ = 1 . The symbol dt indicates the covariant time derivative in Ξt , defined as [1]. There is the possibility of orthonormalising the basis set at each time by, for instance, a time-dependent Löwdin transformation from the original non-orthogonal basis. The equations all stay as for the natural representation with no need of distinguishing upper/lower (contravariant/covariant) indices. It would be less efficient, so we keep the general non-orthogonal formalism for generality
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