Fictitious mass

First, we imagine that there are fictitious masses at points of the surface S and they are distributed with surface density a in such way that the potential of their field is equal T. 

The second term in Eq. (10-14) represents the kinetic energy of 0j, which has a fictitious mass similar to that of a heavy atom. The last term represents the sum of the biasing potential for each titrating group, defined as... 

One may consider the above equation as a generalization of Born-Oppenheimer dynamics in which electrons always stay on the Born-Oppenheimer surface. For a given conformation of nuclei, the numerical value of the fictitious mass associated with electronic degrees of freedom determines how far the electron density is allowed to deviate from the Born-Oppenheimer one. Each consecutive step along the trajectory, which involves electronic and nuclear degrees of freedom, can be obtained without determining the exact Born-Oppenheimer electron density. 

The second term of Eq. (12) describes the fictitious kinetic energy of the electrons. The term contains an arbitrary parameter (fictitious mass parameter) p with appropriate units of energy times a squared time. 

Marchi and co-workers [27,28] have applied Equation (1.79) in the context of classical MD by using a Fourier pseudo-spectral approximation of the polarization vector field. This approach provides a convenient way to evaluate the required integrals over all volume at the price of introducing in the extended Lagrangian a set of polarization field variables all with the same fictitious mass. They also recognized the cmcial requirement that both the atomic charge distribution and the position-dependent dielectric constant be continuous functions of the atomic positions and they devised suitable expressions for both. 

The first term in this Lagrangian contains the fictitious mass of the wave-function, /1. This fictitious kinetic energy term should not be confused with the real kinetic energy of electrons included in the electronic Hamiltonian. 

Figure 4-2. Energy conservation in CP-MD the potential energy (Ee, main axis), temperature (kinetic energy, T, auxiliary, right-hand side axis), physical energy (T + Ee, auxiliary axis), and conserved energy (Econs). The difference between Ec0 s and T + Ee is the fictitious kinetic energy of the wavefunction. The data from the simulation for the ethylene molecule with the CPMD program13 (Troullier-Martins pseudopotentials1415, time step of 4 a.u., fictitious mass 400 a.u., cut-off energy 70 Ry, unit cell 12 Ax 12 A xl2 A)...

Parameters originating from equations of motion time steps and the fictitious mass... 

The Car-Parinello equations of motion (Eq. 3) contain the basic parameter of the method, i.e the fictitious mass of the wave function, p. 

The fictitious mass controls the magnitude of the drift from the BO PES the smaller its value is, the smaller is deviation from the BO PES. However, lowering pu implies that one has to decrease the time step. The maximum time step is related to the fictitious mass and the cutoff energy according to the following rule 2... 

Figure 4-6. Plane wave convergence of the carbon-carbon bond length in the ethylene and butadiene molecules, from the simulations with the CPMD program13 (Troullier-Martins pseudopotentials,1415 time step 4 a.u., fictitious mass 400 a.u., unit cell 12 A x 12 A x 12 A)...

The Hamiltonian in eqn.(10) can be viewed as representing the internal motions of a three-particle system or as the total energy of a two-particle system with fictitious masses mi, m2 and charges 91, q2 interacting with a charge 90 at the origin together with their mutual Coulomb interaction and a momentum dependent mass polarization potential . 

The dynamic treatment of the charges is quite similar to the extended Lagrangian approach for predicting the values of the polarizable point dipoles, as discussed in the previous section. One noteworthy difference between these approaches, however, is that the positions of the shell charges are ordinary physical degrees of freedom. Thus the Lagrangian does not have to be extended with fictitious masses and kinetic energies to encompass their dynamics. 

In the extended Lagrangian method, as applied to a fluctuating charge system,the charges are given a fictitious mass, M, and evolved in time according to Newton s equation of motion, analogous to Eq. [23],... 

Like CP, ADMP represents fictitious dynamics where the density matrix is propagated instead of being converged. The accuracy and efficiency is governed by the choice of the fictitious mass tensor, /i hence one must be aware of the limits on this quantity. We have derived two independent criteria [152,154] that place... 

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