Fictitious kinetic energy

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. 

The scheme proposed by Car and Parinello1 in 1985 offers an attractive solution to this problem, by propagating the wave-function together with the nuclei. The ingenious idea of Car and Parinello was to include the fictitious kinetic energy term describing the wave-function motion into the classical Lagrangian ... 

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. 

The practical implication is the fact that in the CP MD simulation the molecular system does not evolve right on the Born-Oppenheimer PES, but stays close to it. A measure of deviations from the BO PES is the fictitious kinetic energy (wave-function temperature). Figure 4-2 demonstrates that this deviation is minor, as the electronic (fictitious) temperature is relatively low. The wave function stays cold (compared to the hot nuclei) in the MD terminology the term cold electrons is often used in this context. 

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)...

The Nose-Hoover thermostat, or chain of thermostats, can be used as well to control the wave function temperature, i.e. the fictitious kinetic energy. This prevents drifting of the wave function from the Born-Oppenheimer PES during long simulations. Wave function thermostats are introduced in a similar way to Eqs. 7-9. 

It should be pointed out that the use of a thermostat affects the energy conservation in MD. Namely, in thermostatted dynamics the conserved energy (kinetic and potential energy of nuclei plus the fictitious kinetic energy of the wave function) discussed in Section 2.1 is no longer conserved. Instead, the energy that includes additional terms due to the thermostats (nuclear and electronic ) is constant. For example, for a system thermostatted by a chain of n nuclear thermostats, controlled by variables J and QJ, the conserved energy takes the form ... 

To achieve an evenly distributed thermal excitation, the nuclei were brought to a temperature of 300 K by applying a sequence of 30 sinusoidal pulses, each of which was chosen to raise the temperature by 10 K. Each of the excitation vectors was chosen to be orthogonal to the already excited modes. The warmed-up systems were equilibrated for the 10 000 timesteps. The time step of 7 au. was used. Constraints were maintained by SHAKE algorithm.36 A temperature of 300 K was controlled by a Nose thermostat.23,24 The fictitious kinetic energy of the electrons was controlled in a similar fashion by a Nose thermostat.52... 

Secondly, the rate of change of the fictitious kinetic energy,... 

For the extended system of ions and particles C, the Lagrangian may be obtained by extension of the classical Lagrangian for ionic dynamics by means of a (fictitious) kinetic energy term due to the particles C ... 

We have stated earlier that the fictitious electronic mass /x should be such that the electronic d3mamics is fast enough to follow adiabatically the ionic motion. We will try to give here some conditions for choosing /x [111, 178]. First, /X should be small enough so that the fictitious kinetic energy satisfies... 

Otherwise stated, the fictitious kinetic energy should be a small term in the CP Hamiltonian. 

Fig. 3. Total energy and energy components for a system of 32 water molecules (simulations parameters see text). Top fictitious kinetic energy of the electrons (Kei), second from top instantaneous ionic temperature, Tions (proportional to the ions kinetic energy, Kiona), middle instantaneous Kohn-Sham energy Eks, second from bottom classical hamiltonian Eclass = Eks + Kions, bottom CP hamiltonian, Eham = Eclass + Kd- Note the change of scale of the vertical axis from one frame to the other...

Note that for simplicity, I have not written explicitly the k-point indices or the orbital occupancies.) In Eq. (21), the second term is the kinetic energy of the nuclei and the third term is the potential energy of the nuclei, which is also the electronic energy [Eq. (9) if calculated via DFT]. The first term is the fictitious kinetic energy of the orbitals, where // is the fictitious mass of the orbitals. The fourth term is a set of constraints which keep the orbitals orthonormal, where A,y are the undetermined multipliers. Note that additional constraints can be added into this Lagrangian (vide infra ). 

The development of plane-wave pseudopotential methods for electronic structure calculations of solids (e.g., Payne et al. 1992) has also opened the door to real first-principles molecular dynamics simulations using the algorithm of Car and Parinello (1985). Here, we let the wavefimctions become part of the dynamics of the system. To do this, we introduce a fictitious kinetic energy associated with a dynamical motion of the wavefunction ... 

The first term and second terms in Eq. (A45) correspond to the kinetic energy of the nuclei and the fictitious kinetic energy of the electrons in the system, respectively. The third term reports the overall electronic energy which corresponds to the potential energy of the nuclei. The last term represents the constraints that the orbitals must be orthonormal 1 . 

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