N-state system

Tang et al. [20] have analyzed a reduction procedure using the Schrodinger representation of the dynamics of the n-state system. The molecular wave function of the n-state system can be written as a superposition... 

If all states of the system are strongly coupled to each other, the system dynamics can only be described by completely solving the above equations. However, it is extremely unlikely that this is the case. Rather, it is commonly found that some pairs of states are strongly coupled and other pairs of states are weakly coupled. Then we expect that the population transfers among strongly coupled states dominate the system dynamics and that it should be possible to study the n-state system dynamics in the subspace of strongly coupled states with a correction from the influence of the weakly coupled states. 

The reduction schemes used by Tang et al. [20] to define the surrogate fewer state system follows the method proposed by Shore [62]. The scheme has a compact form when we introduce two orthogonal projection operators P and Q and work in the frequency domain instead of the time domain. The time evolution matrix for the n-state system dynamics, U(f). and its Fourier transform, G(w), satisfy the following equations ... 

We now generalize our previous development to obtain a diagrammatic representation of RS perturbation theory as applied to an N-state system. Consider the problem of finding the perturbation expansion for the lowest eigenvalue of such a system. Here we still have only one hole state, 1>, but there are now N - 1 particle states n>, n = 2, 3,..., AT. We draw the same set of diagrams as before. However, now we can label the particle lines with any index n. For example, the diagram... 

Exercise 6.2 Use diagrammatic techniques to obtain the fourth-order perturbation energy of a particular state (say, i) of an N-state system. That is, evaluate the diagrams... 

Let us generalize these results to the N-state system. The diagrams generated by adding dots to the second-order diagram are... 

We now consider an n-state system and generalize the expression given above for the 2-state case. In this case, we have... 

It is also instructive to start from the expression for entropy S = log(g(A( m)) for a specific energy partition between the two-state system and the reservoir. Using the result for g N, m) in section A2.2.2. and noting that E = one gets (using the Stirling approximation A (2kN)2N e ). 

Hence, in order to contract extended BO approximated equations for an N-state coupled BO system that takes into account the non-adiabatic coupling terms, we have to solve N uncoupled differential equations, all related to the electronic ground state but with different eigenvalues of the non-adiabatic coupling matrix. These uncoupled equations can yield meaningful physical... 

The total orbital wave function for this system is given by an electronically adiabatic n-state Bom-Huang expansion [2,3] in terms of this electronic basis set as... 

As is well known. Molecular Dynamics is used to simulate the motions in many-body systems. In a typical MD simulation one first starts with an initial state of an N particle system F = xi,..., Xf,pi,..., pf) where / = 3N is the number of degrees of freedom in the system. After sampling the initial state one numerically solves Hamilton s equations of motion ... 

The so-ealled Hohenberg-Kohn theorem states that the ground-state eleetron density p(r) deseribing an N-eleetron system uniquely determines the potential V(r) in the Hamiltonian... 

Irradiation of ethyleneimine (341,342) with light of short wavelength ia the gas phase has been carried out direcdy and with sensitization (343—349). Photolysis products found were hydrogen, nitrogen, ethylene, ammonium, saturated hydrocarbons (methane, ethane, propane, / -butane), and the dimer of the ethyleneimino radical. The nature and the amount of the reaction products is highly dependent on the conditions used. For example, the photoproducts identified ia a fast flow photoreactor iacluded hydrocyanic acid and acetonitrile (345), ia addition to those found ia a steady state system. The reaction of hydrogen radicals with ethyleneimine results ia the formation of hydrocyanic acid ia addition to methane (350). Important processes ia the photolysis of ethyleneimine are nitrene extmsion and homolysis of the N—H bond, as suggested and simulated by ab initio SCF calculations (351). The occurrence of ethyleneimine as an iatermediate ia the photolytic formation of hydrocyanic acid from acetylene and ammonia ia the atmosphere of the planet Jupiter has been postulated (352), but is disputed (353). 

For a correlated N-electron system with a non-degenerate ground state > the one-particle Green s function has the spectral representation (20,21) ... 

N-particle system in its ground state and of the N l-particle states. 

In the DC-biased structures considered here, the dynamics are dominated by electronic states in the conduction band [1]. A simplified version of the theory assumes that the excitation occurs only at zone center. This reduces the problem to an n-level system (where n is approximately equal to the number of wells in the structure), which can be solved using conventional first-order perturbation theory and wave-packet methods. A more advanced version of the theory includes all of the hole states and electron states subsumed by the bandwidth of the excitation laser, as well as the perpendicular k states. In this case, a density-matrix picture must be used, which requires a solution of the time-dependent Liouville equation. Substituting the Hamiltonian into the Liouville equation leads to a modified version of the optical Bloch equations [13,15]. These equations can be solved readily, if the k states are not coupled (i.e., in the absence of Coulomb interactions). 

Figs. 11 and 12 show typical mo diagrams for square planar and octahedral complexes. Inspection reveals that the metal orbital (z is the axial direction) in a square planar complex is involved in the n bonding system and available for a bonding in the transition state. This is a feature shared by nucleophilic substitution at square planar complexes with the spectacularly associative nucleophilic aromatic substitutions. The octahedral complexes discussed in this chapter... 

With regard to the results obtained for the ground state system N (ls ) -i- He, all the crossings are shifted towards shorter internuclear distances and have a lower height. The main remark concerns, however, the presence of an avoided crossing between the entry... 

We have written X = y = Cx just so that we can package this equation in matrix form in the next step. Substitution of Eq. (9-26) in the state model (9-13) and together with (9-27), we can write this (n + 1) system as... 

The a-n interaction in the excited-state n electron systems is also successfully treated. The 1,2-addition will take place with cis mode as is indicated in Fig. 7.39. This was predicated in reference B6>. Experimental evidence 64>149> is the photoinduced addition of IV-chlorourethane to olefins which gives mainly cis addition product, while thermal addition produces a dominantly trans adduct. 

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