Transition densities. Generalizations

So far we have been concerned with the properties of a single given state, and in particular with states represented by single determinants. Often, however, we need to discuss properties that depend jointly on two states. 

To this end, we introduce transition density matrices connecting pairs of states, and Wi say, by equations exactly analogous to (5.2.6) and (5.2.7)  

Clearly, for L = K we obtain the density functions for a single state, as used previously without any label p(KK xi xO is the density function Pixiix i) for state The transition-density functions determine 

Again we may integrate over spins (when dealing with spinless operators) to obtain transition densities in ordinary space  

It should be noted that, whereas the function P(KK r ) integrates to give the total number of electrons, P(KL (rj) K = L) integrates to zero 

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One of the primary features of the Gay-Berne potential is the presence of anisotropic attractive forces which should allow the observation of thermally driven phase transitions and this has proved to be the case. Thus using the parametrisation proposed by Gay and Berne, Adams et al. [9] showed that GB(3.0, 5.0, 2, 1) exhibits both nematic and isotropic phases on varying the temperature at constant density. This was chosen to be close to the transitional density for hard ellipsoids with the same ellipticity indeed it is generally the case that to observe a nematic-isotropic transition for Gay-Berne mesogens the density should be set in this way. The long range orientational order of the phase was established from the non-zero values of the orientational correlation coefficient, G2(r), at large separations and the translational disorder was apparent from the radial distribution function. 

The case of responses is more involved in the sense that matrix elements of the second operator in the commutator are transition densities which are generally complex. However, the first operator in the commutator still has real (for T-even A) or image (for T-odd A) matrix elements and so the averages can be finally reduced to... 

Equation (40) is a generalized eigenvalue equation. The eigenvalue j is interpreted as the excitation energy from the ground state to the Jth excited state. The vectors Xj and Yj are the first-order correction to the density matrix at an excitation and describe the transition density between the ground state and the excited state J. 

As was shown in chapter three we can compute the transition densities from the Cl coefficients of the two states and the Cl coupling coefficients. Matrix elements of two-electron operators can be obtained using similar expresssions involving the second order transition density matrix. This is the simple formalism we use when the two electronic states are given in terms of a common orthonormal MO basis. But what happens if the two states are represented in two different MO bases, which are then in general not oithonormal We can understand that if we realize that equation (5 8) can be derived from the Slater-Lowdin rules for matrix elements between Slater determinants. In order to be a little more specific we expand the states i and j ... 

Width of the transition. The a transitions are generally considerably sharper than secondary ones. The width of the a transition can be related to the degree of inhomogeneity of the spatial distribution of crosslink density (see Sec. 10.3.4). In contrast, secondary transitions can be considered intrinsically broad their width cannot be related to structural inhomogeneities. 

The effect of the medium denoted here as implicit reflects the influence of the solvent on the transition densities (i.e., spectral properties) of the D/A units, which determine the direct coupling Vs. The solvent explicitly enters into the definition of the coupling through the term T Xpiicit in (2.4), which describes an interaction between the two chromophores mediated by the medium, that generally leads to an overall reduction (i.e., a screening) of the D/A coupling. 

Different complicating factors led to the development of a more generalized approach [22, 23] in which the Coulomb interaction is now considered in terms of local interactions between donor and acceptor transition densities. This is... 

In general, the Coulombic interaction can be written in terms of the two-particle spinless transition density that connects the states K and L on the... 

As is well-known, density functional theories are very successful in describing ffuid-solid transitions in general [336, 337], and hence this approach is most promising for the related transition of mesophase ordering using the information on the disordered melt. Unlike the Landau-type expansion, Eq. (184X high-order nonlinear terms are included selfconsistently. 

Increasing the dipole coupling shifts the transition densities to lower densities. For instance, in dipolar soft spheres (256 particles) magnetic ordering sets in at a density p pa 0.85 for 7, 4 [103], p pa 0.65 for X 6.25 [102,103] and p pa 0.55 - 0.60 at A, 9 [102,103]. Transition densities are similar in DHS [113,135], but are much higher in Stockmayer fluids where a ferroelectric transition exists only for A, > 4 [127,136] at densities generally larger than the liquid densities at 1-g coexistence. MC simulations by Gao and... 

The antisymmetry of d (X) is a consequence of the orthonormality of the molecular orbitals, Eq. (43a). Here the adjective square has been emphasized in reference to the one-particle transition density matrix. The one particle transition density matrix is in general not symmetric, that is, the full or square matrix must be retained. However, in most electronic structure applications the associated one electron integrals, for example are symmetric, permitting the off-diagonal density matrix element to be stored in folded or triangular form. Since d is not symmetric, it is necessary to construct and store the transition density matrix in its unfolded or square form. 

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