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In general, {.a(x;R)} may contain a discrete and a continuous part. However, in this manuscript, we will forget the continuous part in order to simplify the mathematics.
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In general, {.a(x;R)} may contain a discrete and a continuous part. However, in this manuscript, we will forget the continuous part in order to simplify the mathematics.
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In the development of the classical formalism of thermodynamics, this point is crucial.
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This mistake is made, for example, in ref 31 when going from their expression 8 to their expression 9. Having not realized the existence of the two distinct sets of independent variables, ψ,R} and {c,R′, they treat the |ca|2 as constants when taking the partial derivative with respect to q (our R) at the right hand side of (8, thus arriving at (9, where the crossterms f dx η*a (x;R'(t, ▽j Ĥe(r'R'(t, ηb(x;R'(t, with b ≠ a, are incorrectly missing see the correct derivation of EMD in the adiabatic basis below
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b(x;R'(t)), with b ≠ a, are incorrectly missing (see the correct derivation of EMD in the adiabatic basis below).
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This fact is not in contradiction with the above discussion about coordinates' independence. The difference between Ehrenfest MD and gsBOMD is that in the Lagrangian for the latter, the orbital velocities φ do not appear, thus generating eqs 22a and 22b, which can be regarded as constraints between the φ and the R
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We are exploring how to overcome this limitation by mapping the real Hamiltonian into another one that produces the same dynamics but not having contributions from the empty states
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In principle the forces acting over the ions are given by eq 20b; however, due to the derivatives of the ionic potential (that can have very high Fourier components), this expression is difficult to calculate accurately on real-space grids. Fortunately, an alternative expression in terms of the gradient of the wave functions can be obtained for both the local and nonlocal parts of the pseudopotential.56
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