State labeling

Each and every electronic energy state, labelled k, has a set, labelled L, of vibration/rotation energy levels k,L and wavefiinctions... 

Atomic units, with h = 2m = 1, are used througout the text. The state label for the energy E. is omitted. The elaborated form of T/ k is [2]... 

Consider a deteriiiinistic local reversible CA i.o. start with an infinite array of sites, T, arranged in some regular fashion, and a.ssume each site can be any of N states labeled by 0 < cr x) < N. If the number of sites is Af, the Hilbert space spanned by the states <7-(x is N- dimensional. The state at time t + 1, cTf+i(a ) depends only on the values cri x ) that are in the immediate neighborhood of X. Because the cellular automata is reversible, the mapping ai x) crt+i x ) is assumed to have a unique inveuse and the evolution operator U t,t + 1) in this Hilbert space is unitary,... 

The semi-classical expression shown in Eq. (54) for the rate of ejection of electrons from a specified initial vibration-rotation state Xi (Q) induced by non BO coupling to all accessible neutral-molecule-plus-free-electron final states (labeled f) gives this rate as ... 

The total Hamiltonian is the sum of the two terms H = H + //osc- The way in which the rate constant is obtained from this Hamiltonian depends on whether the reaction is adiabatic or nonadiabatic, concepts that are explained in Fig. 2.2, which shows a simplified, one-dimensional potential energy surface for the reaction. In the absence of an electronic interaction between the reactant and the metal (i.e., all Vk = 0), there are two parabolic surfaces one for the initial state labeled A, and one for the final state B. In the presence of an electronic interaction, the two surfaces split at their intersection point. When a thermal fluctuation takes the system to the intersection, electron transfer can occur in this case, the system follows the path... 

An electron-transfer reaction is the transition between two electronic states, an initial state labeled i, which in our example corresponds to... 

A formerly unknown state, labeled Co-Mo-S in Fig. 9.19, which is most evident in sulfided Co-Mo catalysts of low cobalt content. 

If we start with states of tt-symmetry (dashed lines) we find three distinct peaks in the XES spectra reflecting the occupied states. The 1 a2u and lelgTr-like orbitals are essentially intact from the gas phase, while the third state, labeled e2u, is not seen for the free molecule. Based on symmetry-selection rules, it can be shown that this state is derived from the lowest unoccupied molecular orbital (LUMO) e2utt -orbital that becomes slightly occupied upon adsorption. We anticipate a similar bonding mechanism as discussed in the previous section for adsorbed ethylene with the exception of a weaker rehybridization due to the extra stability in the -system from the aromatic character. 

It is well known that, besides the electronic ground state,. Y3Zg, of the O2 molecule, there are two low lying excited electronic states labeled... 

A and B are W-matrices, C is square, and at least some elements of D and E are non-zero. A splitting matrix is reducible, with the additional property that when the rows and columns corresponding to the transient states are erased a decomposable matrix is left. There are three sets of states, labelled with a, h, c. The states c are transient and deplete into a and b. There are (at least) two linearly independent eigenvectors with eigenvalue zero, viz. 

The molecules of an ideal gas provide a concrete example for the following consideration, which, however, is of more general validity. Consider a system which can be in different states labelled n, and whose evolution is described by a master equation... 

Electronic States of a Diatomic Molecule. In the picture of orbitals a one-electron excited state is described by the initial and final orbitals and by the multiplicity of the excited state. In an unsaturated molecule like ethylene (ethene) there will therefore be excited states labelled 3 (tt-tt ), 1 (xr—zr8 6), 3(7T-cr ), etc. in order of increasing energy. Two- or more electron excitations would require two or more such labels, but in practice the number of accessible excited states is quite small. For the purposes of photophysics and photochemistry it is almost always sufficient to consider one-electron excitations. 

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