Scattering state antisymmetric

Both formal analysis and computational developments associated with DFT can be carried over intact to nDFT. For example, the exact two-particle ground-state density, no(x), can be determined through a constrained search [34] for that many-particle, properly symmetrized or antisymmetrized wave function, with symmetry imposed with respect to ordinary particles, which yields n0 and also minimizes the many-particle energy, T + Vpp, where Vpp denotes the interparticle interaction in two-particle space. Essentially any method developed within a single-particle application of DFT for the study of electronic structure can, with appropriate technical modifications, be extended to two-, or rc-particle states. The use of multiple-scattering theory to calculate fully correlated two-particle densities in solids will be given in a future publication. 

Fig. 4 H+H2(v = 0,j = 0) —> H2(v, /)+H state-to-state Pauli-antisymmetrized ICS computed by excluding (NGP) and including the geometric phase explicitly (GPl), by artificially changing the sign of the reactive scattering amplitude, and implicitly (GP2) with the vector potential approach...

The occurrence of antisymmetric scattering contributions is invariably associated with processes in which the symmetry of particular electronic states becomes important. In the non-resonance electronic (or the vibro-electronic) Raman effect (ERE or VERE), the initial and final states are different electronic (or vibronic) states. Thus, if these states are of appropriate symmetry, an antisymmetric contribution may be present in the scattering 46). The same is true for the special case of the VERE represented by vibrational Raman scattering in systems possessing degenerate electronic ground states (see Section 2.11). In the resonance ERE and resonance VERE as well as the vibrational resonance-Raman effect, it is the symmetry of the intermediate state which becomes significant. 

Returning now to Eqs.(28), it can be seen that, because of the antisymmetry relations, the resonant and non-resonant parts of each tensor element tend to cancel in the off-resonance limit Eq Ej > El > hco. In addition, the contributions from le> and s> states to each term in Eqs.(28) are of comparable magnitude off resonance. These contributions tend to cancel because of the difference in sign of the HT perturbation energy denominators, viz. (Ee - E )" = - (Es - Ee) . The net result is the disappearance of antisymmetric scattering in the off-resonance region. 

The symmetries of the modes active in coupling the two states are ig - 2g ig 2g> however, theAjg modes are known to be ineffective. The Bjg (but apparently not the 52g) modes are effective, and yield depolarised bands. The A2g modes are also effective but, owing to the fact (Section 2.9.2) that their scattering tensors are antisymmetric, the resulting bands are not observed off resonance they appear only on resonance and are then inversely polarised (pi =°°). 

As stated in Sec. 1.6, the polarizability tensors are symmetric in normal Raman scattering. If the exciting frequency approaches that of an electronic absorption, some scattering tensors become antisymmetric, and resonance Raman scattering can occur (Sec. 1.22). In this case, Eq. 1.194 must be written in a more general form [98]... 

In scattering on a pair of identical particles in H2, D2, H2O, D2O and close pairs of protons or deuterons in molecules or metal hydrides, the quantum exchange effect must be taken into account by antisymmetrization of the initial state 4 i,... 

For projectile-nucleus scattering we will only be interested in matrix elements of T between initial and final projectile-nucleus wave functions, representing physical states of the system. The symmetry properties of (S,- u,) [Fe 71] result in intermediate states in the second term of eq. (2.5) which span only the physical, antisymmetric states of the nucleus and the physical states of the projectile. Likewise, from eq. (2.4), only the projections of If and (f) onto physical states of the system need be retained. Thus we consider and (f) from here on to be expanded in terms of all antisymmetric states of the target and all physical states of the projectile. Note that these states do not form complete sets [Ke 59]. Antisymmetrization between the projectile and target nucleon labels (in the case of nucleon projectiles) is, for the moment, neglected. TTie total wave function is therefore expanded according to... 

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