Extended-Lagrangian formalism

In case of LR-TDDFT, the forces on the nuclei are derived within the Tamm-DancofF approximation [36,37] from nuclear derivatives of the excited-state energies using the extended Lagrangian formalism introduced by Hutter [34], In general, LR-TDDFT MD simulations are about 70-90 times faster than P-TDDFT MD simulations. The LR-TDDFT scheme has also been combined with our QM/MM approach [38,39] in order to enable the calculation of excitation spectra [40-42] and excited-state dynamics in condensed-phase systems. 

Metadynamics was originally introduced as a coarse dynamics in collective variable space, much in the spirit of Ref. 39. This dynamics was then accelerated by adding a penalty to the already visited states, similarly to the taboo search method" and the local elevation method." Later, a version based on an extended Lagrangian formalism was introduced. Presently, the most adopted variant is the continuous one. In this chapter, we will only focus on this latter version. 

Notice that if s(t) = 1, then this extended Lagrangian reduces exactly to Eq. (9.11). Formally, we can determine the equations of motion associated with this extended Lagrangian using Eq. (9.12). These equations were written by Hoover in a convenient form using slightly different variables than the extended Lagrangian above ... 

Nos developed a deterministic approach to constant temperature simulations based on an extended Lagrangian which does not disturb the dynamic properties of the system. Since then, modifications of this approach have been developed to generate constant pressure and temperature simulations. These extended system methods have been known to suffer from stability problems, as well as occasional failures in ergodicity. Klein and co-workers introduced the concept of Nos6-Hoover chains to overcome these problems, and used a Liouville operator formalism to obtain reversible integrators generating these chains. 

The formalism can be extended for a quantum Jield with the TFD Lagrangian density given by t = — , where is a replica of for the tilde fields so leading to similar equations of motion. For the purpose of our applications, we shall restrict our analysis to free massless fields. Thus, considering the free-massless boson (Klein-Gordon) field, the two-point Green function in the doubled space is given by... 

Hong and Noolandi [74, 75] extended the approach taken by Helfand and Wasserman [8-11] to formulate a system of equations, via functional integral formalism, for multicomponent systems containing a solvent, and then solved the self-consistent system equations with the incompressibility constraint using a Lagrangian multiplier. Note that earlier Helfand and Wassermann used the incom-... 

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