By Feinsilver P., Schott R.
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Additional info for Algebraic structures and operator calculus. Vol.1
127) (Alkali metal). These effects are associated with the electric charge of the particles and their Coulomb interactions, the representation of quantum particles by a closed necklace, and the non-local exchange potential associated with the indistinguishability of the quantum particles. 1 Ewald Summation Method As we have seen MD simulations are often conducted with PBC. That is the system of interest is composed of a ﬁnite size simulation cell of speciﬁed shape and volume containing the particles of interest (here our electron plasma composed of N-electrons).
Therefore, an effective force calculation requires a satisfactory sample of such paths. Since the exchange potential offers a barrier to paths with negative determinants, it biases the sampling of phase space toward conﬁgurations with positive determinants. Although many conﬁgurations with negative determinants may exist and evolve, they do not contribute to the exchange forces. In the previous Hamiltonian, the Coulomb interactions between electrons are long range and we use the Ewald summation method (see subsequent sections) to calculate their contributions to the energy and forces.
The force on the particle i due to interactions with a particle j; fij ðtÞ, can be obtained from fij ¼ Àrj /ij . To solve the equations of motion in a numerical MD simulation, one discretizes in time the differential equations of motion. A variety of numerical schemes can then be used to integrate the equations of motion and march the system in time. For instance, the simplest method is based on straightforward ﬁnite differences. Since Alder and Wainwright  used MD for simulating an N -body system, this method has been applied to a large variety of systems [64–67].