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Volumn 5, Issue 4, 2009, Pages 728-742

Modified ehrenfest formalism for efficient large-scale ab initio molecular dynamics

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EID: 65249161145     PISSN: 15499618     EISSN: None     Source Type: Journal    
DOI: 10.1021/ct800518j     Document Type: Article
Times cited : (146)

References (86)
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    • Computer simulations of the dynamics of systems of interacting molecules based on the Monte Carlo methods were presented some years before.70 Also, before the work of Alder and Wainwright, some previous "computations" were reported that did not utilize modern computers, but rather real physical models of the system, i.e. rubber balls linked by rods.71 The rapid improvement of digital computer machines discouraged this cumbersome, yet entertaining, methodology.
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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.
    • 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
    • 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
    • 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
    • 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
    • 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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    • For a one to one transformation U, we may define unitarity as the property of preserving the scalar product, i.e. given two arbitrary vectors f and g, we have f,g f, U. One can easily see that any reversible transformation U that enjoys the previous property is necessarily linear: , , , , , , As U is reversible, Ug is an arbitrary vector, and therefore, we must have U(f1, f2), Uf1, Uf2 Hence as the equations of motion (20a) and (20b) are clearly nonlinear, the evolution is nonlinear too, and consequently, it cannot be a unitary transformation as defined above. If we fix the value of the density for all times, then the operat
    • 2 Hence as the equations of motion (20a) and (20b) are clearly nonlinear, the evolution is nonlinear too, and consequently, it cannot be a unitary transformation as defined above. If we fix the value of the density for all times, then the operator will become linear and it will be unitary.
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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
    • 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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