Base state

The concept of two-state systems occupies a central role in quantum mechanics [16,26]. As discussed extensively by Feynmann et al. [16], benzene and ammonia are examples of simple two-state systems Their properties are best described by assuming that the wave function that represents them is a combination of two base states. In the cases of ammonia and benzene, the two base states are equivalent. The two base states necessarily give rise to two independent states, which we named twin states [27,28]. One of them is the ground state, the other an excited states. The twin states are the ones observed experimentally. 

Ammonia is a two-state system [16], in which the two base states lie at a minimum energy. They are connected by the inversion reaction with a small baiiier. The process proceeds upon the spin re-pairing of four electrons (Fig. 15) and has a very low barrier. The system is analogous to the tetrahedral carbon one... 

Let us consider the SHV mode of the TCP flow as the base state of a dynamical system described by three velocity components Vr, V0 and Vz relative to the fixed cylindrical co-ordinate system depicted in Figure 4.4.7(b). This dynamical system is described mathematically by the equations of motion of the particle trajectories in the 3D (r, 0, z) coordinate system ... 

The acid-base chemistry of nicotine is now well known and investigations have shown that nicotine in tobacco smoke or in smokeless tobacco prodncts can exist in pH-dependent protonated or nnprotonated free-base forms. In tobacco smoke, only the free-base form can volatilize readily from the smoke particnlate matter to the gas phase, with rapid deposition in the respiratory tract. Using volatility-based analytical measurements, the fraction of nicotine present as the free-base form can be quantitatively determined. For smokeless tobacco products, the situation differs because the tobacco is placed directly in the oral cavity. Hence, the pH of smokeless tobacco prodncts can be measured directly to yield information on the fraction of nicotine available in the nnprotonated free-base form. It is important to characterize the fraction of total nicotine in its conjugate acid-base states as this dramatically affects nicotine bioavailability, because the protonated form is hydrophilic while the nnprotonated free-base form is lipophilic and thus readily diffuses across membranes (Armitage and Turner 1970 Schievelbein et al. 1973). As drug delivery rate and addiction potential are linked (Henningfield and Keenan 1993), increases in delivery rate due to increased free-base levels affect the addiction potential. 

Schnitzler, G., Sif, S., and Kingston, R.E. (1998) Human SWI/SNF interconverts a nucleosome between its base state and a stable remodeled state. Cell 94, 17-27. 

Once the base states are found for a given material system (to be defined later on), it is the quantum state that is susceptible of (laboratory) preparation, manipulation and/or measurement. 

An eigenstate corresponds to a row vector with zero amplitude everywhere except at the base state function that is the case. It is then defined in Hilbert space and does not stand for an individual molecular state. The corresponding base state cannot be a dynamically unstable state as it is time independent. This is an important difference with standard approaches [15]. 

Let us focus on molecular systems for which we know molecular Hamiltonian models, H(q,Q). Electronic and nuclear configuration coordinates are designated with the vectors q and Q, respectively x = (q,Q) = (qi,..., qn, Qi,---, Qm- For an n-electron system, q has dimension 3n Q has dimension 3m for an m-nuclei system. The wave function is the projection in configuration space of a particular abstract quantum state, namely P(x) P(q,Q), and base state func-... 

The equation is defined in an I-ffame and no rotation and translation base states are present yet these latter appear after translating the origin to an I-ffame in uniform motion with (classical) velocity v, a v-frame. This aspect is not examined in this paper but observe that global angular momentum of the system is described with the invariances to rotation of the v-frame and will apply to all internal electronuclear state of the I-frame [10]. 

The Ho operator is mapped on to the well-known molecular Hamiltonian H(q,Q) because this one can be rigorously diagonalized in a particular separable base state set [7,12], cf. eq. (4). We briefly review this point. Now, we write ... 

In principle, the time evolution of a particular linear superposition on the molecular base states will reflect a chemical process via the changes shown by the amplitudes. This represents a complete quantum mechanical representation of the chemical processes in Hilbert space. The problem is that the separability cannot be achieved in a complete and exact manner. One way to introduce a model that is able to keep as much as possible of the linear superposition principle is to use generalized electronic diabatic base functions. 

The time evolution operator propagates an arbitrary quantum state thanks to the non-zero matrix elements (Ve0ph)j m ,jm- The set ( )j(q)Qm(Q) has all electronic base states corresponding to all possible chemical species in the sense discussed above only because the generalized electronic diabatic set is complete. 

Moreover, there are two diradical states of symmetry Cs constructed on the basis of 1-electron base states. To help visualizing the analysis we use the planes associated to the CH2 groups. At n/4 the planes defining each CH2 sigma base states at opposite sites have normal vectors making a nil angle. The local 7i-axis serve to identify new 1-e-base states Yj+ and y. The + state has two local NPs the minus (-) state increases the number of nodes by one unit. The... 

The concept of correlation enters via the inclusion of asymptotic base states. Translated into the particle model, one type of asymptotic state is given by a doubly occupancy of an anti-bonding state the fragments show a repulsive... 

If one had a complete basis set then, for the stationary geometry, the corresponding electronic waveftmctions obtained by actual calculation would be models for the diabatic base functions. The latter statement has to be understood in the sense that the set of nodal planes so obtained must be kept fixed. This was the procedure used to extract diabatic base states for cis and trans ethylene. 

Finally, the rules of angular momentum construction can be made as if the system had spherical symmetry. The reason is that the invariance to rotation of the I-frame leads to angular momentum conservation. Once all base states have been constructed, the dynamics is reflected on the quantum state that is a linear superposition on that base. As the amplitudes change in time, motion of different kinds result. 

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