Expansion of the density matrix in

Expansion of the density matrix in terms of irreducible tensor operators 

T(k) 7hey are tensors of rank k and have (2k+l) different components labelled by the projection quantum number q, where -k s q s k. In terms of these operators we may write the density matrix in the form 

Now in order that the relaxation rates should be consistent with the requirements of spherical symmetry, it is 

Thus in an isotropic environment the number of different relaxation rates is reduced from (21 +1) to (2Jg+l), giving one relaxation rate for each multipole moment of the excited state. Experimentally the number of effective relaxation rates might still be embarrassingly large were it not for the fact that in resonance fluorescence we prepare and monitor the excited atoms through the absorption and emission of electric dipole radiation. The electric-dipole matrix elements have the properties associated with rank one tensors and consequently the observable multipole moments in these experiments are limited to those corresponding to tensors of rank 0, 1, and 2 respectively. 

The irreducible tensor operators are defined in terms of the Dirac bra and ket vectors, spectively, by the expression 

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Here n,(k) stands for the occupation number of the one-electron states, and pj j are the coefficients the expansion of the density matrix (in direct space),... 

The VS98 meta-GGA exchange-correlation functional (van Voorhis and Scuseria 1998) is the first exchange functional incorporating the kinetic energy density. Similarly to the PF exchange functional (see Sect. 5.2), this functional is derived from the analytical expansion of the density matrix in Eq.(5.7) (Negele and Vautherin 1972),... 

For further details concerning the using of chemical potential in spin thermodynamics the reader should refer to (Philippot, 1964). To calculate the expectation values of energies Hz) and Hss) at low temperatures one has to take into account the higher-order terms in the expansion of the density matrix in Equation 26. As a consequence the factorization condition 28 is violated and the Zeeman subsystem and the reservoir of spin-spin interactions carmot be considered as independent. So the advantage of the above-mentioned choice of thermodynamic coordinates is lost. Besides at low temperatures the entropy written in terms a and fi... 

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