Spectral configuration, integrated

Starting with the method described above, extensive tables of the numerical values of mean energies, integrals of electrostatic and constant of spin-orbit interactions are presented in [137] for the ground and a large number of excited configurations, for atoms of boron up to nobelium and their positive ions. They are obtained by approximation of the corresponding Hartree-Fock values by polynomials (21.20) and (21.22). Such data can be directly utilized for the calculation of spectral characteristics of the above-mentioned elements or they can serve as the initial parameters for semi-empirical calculations [138]. 

Here, the integration over X was performed in Eq. (63) to define W%a (X, ) which is the integrated value of the combination of the spectral density function with the time independent operator. This spectral density function contains the quantum equilibrium structure of the system. (X, t) is the time evolved matrix element of the number operator for the product state B. Thus, to calculate the rate, one samples initial configurations from the quantum equilibrium distribution, and then computes the evolution of the number operator for product state B. The QCL evolution of the species operator is accomplished using one of the algorithms discussed in Sec. 3.2. Alternative approaches to the dynamics may also be used such as the further approximations to the QCLE discussed in Sec. 4. 

In order to prove this statement we assume the second functional derivative of Stot evaluated in the reference configuration to equal the integral over Q of a suitable quadratic form multiplied by a scalar quantity a and determine under which conditions the corresponding spectral problem admits positive eigenvalues. In particular the following quadratic form is assumed for the second functional derivative of Stot... 

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