Electron nuclear dynamics , molecular function

Density functional theory, direct molecular dynamics, complete active space self-consistent field (CASSCF) technique, non-adiabatic systems, 404-411 Density operator, direct molecular dynamics, adiabatic systems, 375-377 Derivative couplings conical intersections, 569-570 direct molecular dynamics, vibronic coupling, conical intersections, 386-389 Determinantal wave function, electron nuclear dynamics (END), molecular systems, final-state analysis, 342-349 Diabatic representation ... 

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,... 

In this chapter, we discussed the permutational symmetry properties of the total molecular wave function and its various components under the exchange of identical particles. We started by noting that most nuclear dynamics treatments carried out so far neglect the interactions between the nuclear spin and the other nuclear and electronic degrees of freedom in the system Hamiltonian. Due to... 

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. 

The quantum number v embodies the set of nuclear dynamic states with their labels (see below) and /c stands for the electronic quantum state. Thus, the nuclear wave function is always determined relatively to particular electronic states which, in turn, must be correlated to the (point) symmetries of the system. This stationary wave function may define, for particular cases, a class of geometric elements having an invariant center of mass. Actually, the (equivalence) class of configurations are those for which symmetry operations leave invariant this center of mass. This framework shares the discrete symmetries, such as permutation and space reflection invariances that are properties of the molecular eigenstates. There exists, then, a specific geometric framework pok- At this point, the expectation value ofH ,. taken with respect to the universal wave function is stationary to any geometric variation. 

When the field is absent and the nuclei are viewed as quantum particles, a complete description requires inclusion of electron-phonon and spin-orbit coupling operators. Now, the By( )-coefficients couple electronic states whose diabatic potential energy surfaces jrield degenerate functions for the nuclear dynamics. Finally, these geometry-dependent coefficients couple all diabatic electronic states whenever a fuUy quantum-mechanical molecular system is embedded in an external field. 

The time-independent Schrodinger equation (SE) for a molecular system derives from Hamiltonian classical dynamics and includes atomic nuclei as well as electrons. Eigenfunctions are therefore functions of both electronic and nuclear coordinates. Very often, however, the nuclear and electronic variables can be separated. The motion of the heavy particles may be treated using classical mechanics. Particularly at high temperatures, the Heisenberg uncertainty relation Ap Ax > /i/2 is easy to satisfy for atomic nuclei, which have a particle mass at least 1836 times the electron mass. The immediate problem for us is to obtain a time-independent SE including not only the electrons but also the nuclei and subsequently solve the separation problem. 

The electronic energy thus computed at each molecular shape serves as a potential function working on nuclei, called (adiabatic) potential energy surface (PES), which drives nuclear wavepackets on it, and only in this stage time-variable is retrieved, to the time scale of nuclear dynamics mostly of the order of femtosecond. This is the standard theoretical framework for the study of the dynamics of molecules [59]. Very well structured and fast computer codes for quantum chemistry are now available, which can serve even as an alternative for experimental apparatus. 

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