Gaussian dynamics

Haliloglu T, Bahar I, Erman B (1997) Gaussian dynamics of folded proteins. Phys. Rev. Lett. 79 3090-3093... 

The semi-classical equations of motion obtained above involve only the transverse adiabatic vector potential which is, by definition, independent of the choice of gauge functions/(q) and g(q). The (Aj -f A2)/2M term in the potential is also independent of those two arbitrary functions. The locally quadratic approach to Gaussian dynamics therefore gives physically equivalent results for any choice of /(q) and g(q). The finding that the locally quadratic Hamiltonian approach developed here is strictly invariant with respect to choice of phases of the adiabatic electronic eigenstates supersedes the approximate discussion of gauge invariance given earlier by Romero-Rochin and Cina [25] (see also [40]). 

Like the exact QDT counterpart [cf. Eq. (4.6)], the POP-CS-QDT preserves both the reduced Gaussian dynamics and the effective local field pictinre for the DBO system. Its TZg [Eq. (4.11a)] has the same dissipation superoperator terms as those in ]Zf [Eq. (4.6b)]. The first and the last terms in the right-hand-side of Eq. (4.11a) for TZg or Eq. (4.6b) for are mainly responsible for the energy renormalization (or self-energy) contribution [38] and their dynamics implications are often neglected in phenomenological quantum master equations such as the optical Bloch-Redfield theory [36]. Note that the bath response function relates to the spectral density as [cf. Eq. (2.8)]... 

The calculation of the time evolution operator in multidimensional systems is a fomiidable task and some results will be discussed in this section. An alternative approach is the calculation of semi-classical dynamics as demonstrated, among others, by Heller [86, 87 and 88], Marcus [89, 90], Taylor [91, 92], Metiu [93, 94] and coworkers (see also [83] as well as the review by Miller [95] for more general aspects of semiclassical dynamics). This method basically consists of replacing the 5-fimction distribution in the true classical calculation by a Gaussian distribution in coordinate space. It allows for a simulation of the vibrational... 

To remedy this diflSculty, several approaches have been developed. In some metliods, the phase of the wavefunction is specified after hopping [178]. In other approaches, one expands the nuclear wavefunction in temis of a limited number of basis-set fiinctions and works out the quantum dynamical probability for jumping. For example, the quantum dynamical basis fiinctions could be a set of Gaussian wavepackets which move forward in time [147]. This approach is very powerfLil for short and intemiediate time processes, where the number of required Gaussians is not too large. 

To add non-adiabatic effects to semiclassical methods, it is necessary to allow the trajectories to sample the different surfaces in a way that simulates the population transfer between electronic states. This sampling is most commonly done by using surface hopping techniques or Ehrenfest dynamics. Recent reviews of these methods are found in [30-32]. Gaussian wavepacket methods have also been extended to include non-adiabatic effects [33,34]. Of particular interest here is the spawning method of Martinez, Ben-Nun, and Levine [35,36], which has been used already in a number of direct dynamics studies. 

The Gaussian wavepacket based spawning method, mentioned above, has also been used in direct dynamics where it is called ab initio multiple spawning... 

The method will, however, fail badly if the Gaussian form is not a good approximation. For example, looking at the dynamics shown in Figure 2, a problem arises when a barrier causes the wavepacket to bifurcate. Under these circumstances it is necessary to use a superposition of functions. As will be seen later, this is always the case when non-adiabatic effects are present. 

The standard semiclassical methods are surface hopping and Ehrenfest dynamics (also known as the classical path (CP) method [197]), and they will be outlined below. More details and comparisons can be found in [30-32]. The multiple spawning method, based on Gaussian wavepacket propagation, is also outlined below. See [1] for further infomiation on both quantum and semiclassical non-adiabatic dynamics methods. 

Both the BO dynamics and Gaussian wavepacket methods described above in Section n separate the nuclear and electronic motion at the outset, and use the concept of potential energy surfaces. In what is generally known as the Ehrenfest dynamics method, the picture is still of semiclassical nuclei and quantum mechanical electrons, but in a fundamentally different approach the electronic wave function is propagated at the same time as the pseudoparticles. These are driven by standard classical equations of motion, with the force provided by an instantaneous potential energy function... 

A more comprehensive Gaussian wavepacket based method has been introduced by Martinez et al. [35,36,218]. Called the multiple spawning method, it has already been used in direct dynamics studies (see Section V.B), and shows much promise. It has also been applied to adiabatic problems in which tunneling plays a role [219], as well as the interaction of a... 

Quantum chemical methods, exemplified by CASSCF and other MCSCF methods, have now evolved to an extent where it is possible to routinely treat accurately the excited electronic states of molecules containing a number of atoms. Mixed nuclear dynamics, such as swarm of trajectory based surface hopping or Ehrenfest dynamics, or the Gaussian wavepacket based multiple spawning method, use an approximate representation of the nuclear wavepacket based on classical trajectories. They are thus able to use the infoiination from quantum chemistry calculations required for the propagation of the nuclei in the form of forces. These methods seem able to reproduce, at least qualitatively, the dynamics of non-adiabatic systems. Test calculations have now been run using duect dynamics, and these show that even a small number of trajectories is able to produce useful mechanistic infomiation about the photochemistry of a system. In some cases it is even possible to extract some quantitative information. 

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