Eigen state

Figure 41. Selective bond breaking of H2O by means of the quadratically chirped pulses with the initial wave packets described in the text. The dynamics of the wavepacket moving on the excited potential energy surface is illustrated by the density, (a) The initail wave packet is the ground vibrational eigen state at the equilibrium position, (b) The initial wave packet has the same shape as that of (a), but shifted to the right, (c) The initail wave packet is at the equilibrium position but with a directed momentum toward x direction. Taken from Ref. [37]. (See color insert.)...

According to relation (17), the p SRH matrix is a sum of terms and each term is the product of the energy and the RDM corresponding to an eigen-state of the system. Therefore for p > 2 the - SRH matrix has all the relevant information about the eigen-states of our system and is represented in a reduced space which renders it easy to handle. 

Let us start by assuming that in our 7V-electron state C there is a and eigen-state. A, having a coefficient of a much higher absolute value than all the rest i.e., A is the dominant configuration in C. Then the n/ are approximated... 

Note the importance of the off-diagonal electron-vibron coupling for the resonant effects. If the matrix M in the eigen-state representation is diagonal, there is no resonant coupling between different electronic states. 

V c r)- In the second case, HE as well as DFT-LDA eigen-states and eigen-energies may be used as starting points for the implementation of the GW approximation. 

There are several ways to define ionic charges. When the eigen-states of the effective one-electron Hamiltonian are expanded on a localised basis set, the probability of presence of the electrons contains a site contribution (square modulus of the projection of the wave function on the sites) and a bond term, related to the overlap of the basis functions. In standard Mulliken population analysis [16], each bond contribution is equally shared between the atom pair. The charges then depend sensitively upon the choice of the basis set and it is meaningless to compare absolute values obtained with different methods. Only charge variations within a given method bear significance. 

Figure 6.12 The decay rates of D2CO as a function of the internal energy. The points are the rate constants (in terms of peak widths in cm" ) for individual eigen states while the solid line is an RRKM calculation which represents the average rate. The average rate at 81 kcal/mol is about 2x10 sec". Taken with permission from Polik et al. (1990b).

Now, going to evaluate the atomic polarizability in terms of the quantum basic information contained within the atomic quantum numbers (e.g., n, k), one starts recognizing the general opeiatorial identity over the complete set of quantum (eigen) states (Putz, 2010c). 

THEOREM Two non-commutative hermitic operators cannot provide simultaneous observable measurements with the same precision on a given (prepared) eigen-state. 

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