Isolated Molecules

For a polyatomic, there are many questions on the role of the polyad number in energy flow from the molecule to the bath. Does polyad number conservation in the isolated molecule inlhbit energy flow to the batii Is polyad number breaking a facilitator or even a prerequisite for energy flow Finally, does the energy flow to the bath increase the polyad number breaking in the molecule One can only speculate until these questions become accessible to fiiture research. 

We consider an isolated molecule in field-free space with Hamiltonian //. We let Pbe the total angular momentum operator of the molecule, that is... 

We hope that by now the reader has it finnly in mind that the way molecular symmetry is defined and used is based on energy invariance and not on considerations of the geometry of molecular equilibrium structures. Synnnetry defined in this way leads to the idea of consenntion. For example, the total angular momentum of an isolated molecule m field-free space is a conserved quantity (like the total energy) since there are no tenns in the Hamiltonian that can mix states having different values of F. This point is discussed fiirther in section Al.4.3.1 and section Al.4.3.2. 

The translational linear momentum is conserved for an isolated molecule in field free space and, as we see below, this is closely related to the fact that the molecular Hamiltonian connmites with all... 

We can describe the conservation of linear momenUim by noting the analogy between tire time-dependent Schrodinger equation, (equation A1.4.1 OS ), and (equation A1.4.991. For an isolated molecule, //does not depend explicitly on t and we can repeat the arguments expressed in (equation Al.4.98), (equation A1.4.99), (equation A1.4.1 OOl. (equation A 1.4.1011 and (equation A1.4.1021 with X replaced by t and Py replaced by // to show that... 

Figure A3.13.1 illustrates our general understanding of intramolecular energy redistribution in isolated molecules and shows how these processes are related to intemiolecular processes, which may follow any of the mechanisms discussed in the previous section. The horizontal bars represent levels of nearly degenerate states of an isolated molecule.

Figure A3.13.1. Schematic energy level diagram and relationship between mtemiolecular (collisional or radiative) and intramolecular energy transfer between states of isolated molecules. The fat horizontal bars indicate diin energy shells of nearly degenerate states.

In view of the foregoing discussion, one might ask what is a typical time evolution of the wave packet for the isolated molecule, what are typical tune scales and, if initial conditions are such that an entire energy shell participates, does the wave packet resulting from the coherent dynamics look like a microcanonical... 

Newton M D 1999 Electron transfer from isolated molecules to biomolecules Advanced Chemicai Physics vol 106, ed J Jortner and M Bixon (New York Wiley) pp 303-75... 

Wynne K and Hochstrasser R M 1999 Coherence and adiabaticity in ultrafast electron transfer Adv. Chem. Phys. 107 (Electron transfer from isolated molecules to biomolecules) part 2, 263-309... 

Here t. is the intrinsic lifetime of tire excitation residing on molecule (i.e. tire fluorescence lifetime one would observe for tire isolated molecule), is tire pairwise energy transfer rate and F. is tire rate of excitation of tire molecule by the external source (tire photon flux multiplied by tire absorjDtion cross section). The master equation system (C3.4.4) allows one to calculate tire complete dynamics of energy migration between all molecules in an ensemble, but tire computation can become quite complicated if tire number of molecules is large. Moreover, it is commonly tire case that tire ensemble contains molecules of two, tliree or more spectral types, and experimentally it is practically impossible to distinguish tire contributions of individual molecules from each spectral pool. 

Chemical reaction dynamics is an attempt to understand chemical reactions at tire level of individual quantum states. Much work has been done on isolated molecules in molecular beams, but it is unlikely tliat tliis infonnation can be used to understand condensed phase chemistry at tire same level [8]. In a batli, tire reacting solute s potential energy surface is altered by botli dynamic and static effects. The static effect is characterized by a potential of mean force. The dynamical effects are characterized by tire force-correlation fimction or tire frequency-dependent friction [8]. 

S. Zilberg and Y. Haas, The photochemistry of 1,4-cyclohexadiene in solution and in the gas phase Conical intersections and the origin of the helicopter-type motion of H2 photogenerated in Che isolated molecule, PCCP 4, 34 (2002). 

In the strictest meaning, the total wave function cannot be separated since there are many kinds of interactions between the nuclear and electronic degrees of freedom (see later). However, for practical purposes, one can separate the total wave function partially or completely, depending on considerations relative to the magnitude of the various interactions. Owing to the uniformity and isotropy of space, the translational and rotational degrees of freedom of an isolated molecule can be described by cyclic coordinates, and can in principle be separated. Note that the separation of the rotational degrees of freedom is not trivial [37]. 

Atoms not explicitly included in the trajectory must be generated. The position at which an atom may be placed is in some sense arbitrary, the approach being analogous to the insertion of a test particle. Chemically meaningful end states may be generated by placing atoms based on internal coordinates. It is required, however, that an atom be sampled in the same relative location in every configuration. An isolated molecule can, for example, be inserted into... 

SISM for an Isolated Linear Molecule An efficient symplectic algorithm of second order for an isolated molecule was studied in details in ref. [6]. Assuming that bond stretching satisfactorily describes all vibrational motions for linear molecule, the partitioned parts of the Hamiltonian are... 

To reiterate a point that we made earlier, these problems of accurately calculating the free energy and entropy do not arise for isolated molecules that have a small number of well-characterised minima which can all be enumerated. The partition function for such systems can be obtained by standard statistical mechanical methods involving a summation over the mini mum energy states, taking care to include contributions from internal vibrational motion. 

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