Nuclear dynamics electronic wave function

The idea of a classical treatment of the nuclear motion within the molecular dynamics (MD) scheme with ab initio determined, quantum-mechanical forces acing on nuclei is as old as quantum mechanics.11,12 The commonly used Born-Oppenheimer approximation12 introduces the concept of potential energy surface (PES). Different time-scales for nuclear and electronic motion allows for the adiabatic separation of the nuclear and electronic wave-function. In the Born-Oppenheimer molecular dynamics (BO-MD) the nuclei move according to Newton laws, while the quantum mechanics is required to determine the potential for this motion ... 

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... 

Electron Nuclear Dynamics (48) departs from a variational form where the state vector is both explicitly and implicitly time-dependent. A coherent state formulation for electron and nuclear motion is given and the relevant parameters are determined as functions of time from the Euler equations that define the stationary point of the functional. Yngve and his group have currently implemented the method for a determinantal electronic wave function and products of wave packets for the nuclei in the limit of zero width, a "classical" limit. Results are coming forth protons on methane (49), diatoms in laser fields (50), protons on water (51), and charge transfer (52) between oxygen and protons. 

Wigner rotation/adiabatic-to-diabatic transformation matrices, 92 Electronic structure theory, electron nuclear dynamics (END) structure and properties, 326-327 theoretical background, 324-325 time-dependent variational principle (TDVP), general nuclear dynamics, 334-337 Electronic wave function, permutational symmetry, 680-682 Electron nuclear dynamics (END) degenerate states chemistry, xii-xiii direct molecular dynamics, structure and properties, 327 molecular systems, 337-351 final-state analysis, 342-349 intramolecular electron transfer,... 

Once we have calculated the dynamics of the collision, we obtain the final electronic wave function, as well as the final nuclear momenta and positions. We observe that even for relatively large impact parameters, the projectile suffers an appreciable deflection. 

A curious effect, prone to appear in near degeneracy situations, is the artifactual symmetry breaking of the electronic wave function [27]. This effect happens when the electronic wave function is unable to reflect the nuclear framework symmetry of the molecule. In principle, an approximate electronic wave function will break symmetry due to the lack of some kind of non-dynamical correlation. A typical example of this case is the allyl radical, which has C2v point group symmetry. If one removes the spatial and spin constraints of its ROHF wave function, a lower energy symmetry broken (Cs) solution is obtained. However, if one performs a simple CASSCF or a SCVB [28] calculation in the valence pi space, the symmetry breaking disappears. On the other hand, from the classical VB point of view, the bonding of the allyl radical is represented as a superposition of two resonant structures. 

The particular iterative technique chosen by Car and Parrinello to iteratively solve the electronic structure problem in concert with nuclear motion was simulated annealing [11]. Specifically, variational parameters for the electronic wave function, in addition to nuclear positions, were treated like dynamical variables in a molecular dynamics simulation. When electronic parameters are kept near absolute zero in temperature, they describe the Bom-Oppenheimer electronic wave function. One advantage of the Car-Parrinello procedure is rather subtle. Taking the parameters as dynamical variables leads to robust prediction of values at a new time step from previous values, and cancellation in errors in the value of the nuclear forces. Another advantage is that the procedure, as is generally true of simulated annealing techniques, is equally suited to both linear and non-linear optimization. If desired, both linear coefficients of basis functions and non-linear functional parameters can be optimized, and arbitrary electronic models employed, so long as derivatives with respect to electronic wave function parameters can be calculated. 

There is also a drawback to treating electronic parameters as dynamical variables. Energy flow between the physically meaningful dynamical variables, the nuclear positions, and auxiliary dynamical variables introduced for computational reasons, the electronic wave function parameters, must be kept to a minimum. The arbitrary masses assigned to the wave function parameters as additional dynamical variables are adjusted so that the characteristic frequency of their motion is sufficiently high in comparison to nuclear... 

This is an eigenvalue equation for the nuclei degrees of freedom submitted to the model potential generated by the exact electronic wave function Yj(p aoi). Depending upon the particular mass vector, the electronic potential may sustain bound nuclear dynamical states of diverse frequencies. The nuclear stationary states Xik(R) in the inertial frame are obtained as functions relative to the electronic attractor potential Ej(R). 

Note that the electronic wave functions in the R-BO scheme have the parameter set aoi only as labels to remind us of the existence of an attractor related to the electronic wave function. It is the attractor (trapping potential) which acts on the nuclear dynamics. The aoi s are not dynamic coordinates in themselves. Thus, eq.(8) simply says that the set of all electronic wave functions in the R-BO scheme are orthogonal. The electronic Hamiltonian being diagonal for all R means that eq.(4) is consistent with eq.(8) and (11). 

In the R-BO scheme, the stationary electronic wave function drives the nuclear dynamics via the setup of a fundamental attractor acting on the sources of Coulomb field [11]. The nuclei do not have an equilibrium configuration as they are described as quantum systems and not as classical particles. The concept of molecular form (shape) is related to the existence of stationary nuclear state setup by the electronic attractor and their interactions with external electromagnetic fields. 

Two methods, identified as Car-Parinello [113] and Born-Oppenheimer [114], have been advanced for performing direct dynamics simulations. For the former, the motions of the electrons are determined simultaneously as the nuclear classical equations of motion are integrated, to determine the change in the electronic wave function as the nuclei move. For the second method the electronic wave function is optimized during the numerical integration of the classical trajectory. 

In mean-field or Ehrenfest methods, " the forces result from the contribution of two terms the first is related to the nonadiabatic coupling, the second is an average of the gradients of the potentials of the populated electronic states. Therefore, the forces acting on the nuclei depend directly on the population of the electronic states. The electronic problem and the nuclear dynamics have to be solved simultaneously. The time step must be sufficiently small to account for the time variation of the electronic wave functions. In this case, the solution of the TDSE can be propagated as ... 

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