Basis functions interaction terms between

With respect to an otherwise unspecified operator H one defines the energy ajL of a basis function ry, and the interaction term bllv between two basis functions r) and r)v as... 

The research on nanocarbons dispersed in polymer matrices in recent years has shown that this route is very efficient at small volume fractions above electrical percolation, where it can be the basis for new composite functionalities in terms of processing and properties. It is also clear that there is an inherent difficulty in dispersing these nanoscopic objects at high volume fractions, which therefore limits composite absolute properties to a very small fraction of those of the filler. Independent of their absolute properties, composites based on dispersed nanocarbons have served as a test ground to understand better the basic interaction between nanocarbons and polymer matrices, often setting the foundation to study more complex composite structures, such as those discussed in the following sections. 

The indices k in the Ihs above denote a pair of basis operators, coupled by the element Rk. - The indices n and /i denote individual interactions (dipole-dipole, anisotropic shielding etc) the double sum over /x and /x indicates the possible occurrence of interference terms between different interactions [9]. The spectral density functions are in turn related to the time-correlation functions (TCFs), the fundamental quantities in non-equilibrium statistical mechanics. The time-correlation functions depend on the strength of the interactions involved and on their modulation by stochastic processes. The TCFs provide the fundamental link between the spin relaxation and molecular dynamics in condensed matter. In many common cases, the TCFs and the spectral density functions can, to a good approximation, be... 

The assumption of constant interaction terms B between conjugated basis orbitals or of between bonded 2p (0 ) orbitals assumes that all bonds in the polyene, both double and single, have the same length Rq both in the neutral molecule M and in the radical cation M+, independent of its different configurations However, this is not the case. In a first, crude approximation the interatomic distances of the neutral molecule are linear functions of the bond orders and those of the radical cation, y, of the bond orders p y = — CiijCvj, j being the quantum number of the vacated... 

The two types of vibration-rotation interaction terms are the PaPc + PcPa term and the Pb term (the b axis is perpendicular to the symmetry plane which is maintained throughout the ring-puckering.). Their coefficients are functions of the vibrational coordinate or the vibrational momentum or both. It is possible to choose the coordinate system so that the PaPc + PcPa term is zero and all of the coupling between rotational angular momentum and vibrational momentum is manifested by the Pb term. The matrix elements for the Hamiltonian in the basis of the solutions to the pure vibrational (J = 0) problem, are... 

For A = B = P (pyrene), the energies associated with the two basis functions are equal, H =H22 = E(P) + (P ), as are the interaction terms Hi2 = H2i, which in the simplest approximation are equal to the interaction between the transition moments M ) > i (Equation 2.18) of the two molecules. The secular determinant expands to the quadratic equation (Hu— s)2— H 2 = 0, which yields two exciton states (Figure 2.24), the lower of which is stabilized relative to the energy of the separated molecules by the interaction energy Hl2. 

The remainder of this section is devoted to a simplified two-level treatment of the Zeeman and Stark effects in the presence of zero-field Stark effect and field-dependent interactions between basis functions 1M) and 2M). In the presence of a static field directed along the space Z-axis, Mj remains a good quantum number. The Zeeman and Stark Hamiltonians involve the interaction between a magnetic field or electric dipole, /r, in the molecule-fixed axis system and the space-fixed magnetic or electric field, F, parallel to the laboratory direction K. The interaction can be expressed in terms of direction cosines... 

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