RPA Transition Moments

The excitation propagator (particle-hole propagator) at the RPA level of approximation can be expressed as 

For real orbitals, the dipole length matrix elements (0 r fc) are real, while dipole velocity matrix elements (0 fc) are purely imaginary. This means that 

For exact eigenstates of the many-electron hamiltonian, it holds that (0 i A ) (0 [r,ff] jfc) 

This means that the oscillator strengths of an absorption spectrum calculated within the RPA (in a complete orbital basis) will be identical in the dipole length form 

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In RPA, transition moments in the dipole length and in the velocity formulations are ... 

First consider the dipole operator (O = r). The matrix elements on rhs of eq. 17 are thus just the dipole transition moments, and the commutator becomes C = -ip. As the exact solution (complete basis set limit) to the RPA is under consideration, we may use eq. 10 to obtain... 

Even though the response approach represents a significant simplification, it is still computationally demanding for three-photon absorption, and the third order transition moments are therefore obtained at the RPA level, i.e. cubic response... 

We now wish to write Eq. (98) in a spectral form which readily allows identification of excitation energies and transition moments in the same way as we did for the exact propagator in Eq. (13), i.e. from the poles and residues. This is done by introducing the RPA eigenvector matrix (Oddershede et al, 1984, Section 3.2)... 

The excitation energies and transition moments of MCRPA can be determined as in RPA. Except for the dimensions of the A and B matrices and a different metric matrix (S, A) the propagators in MCRPA and RPA have the same form and we can thus express (P Q e in Eq. (116) in a spectral form using the transformations described in Eqs (102)-(105). The solution of the eigenvalue problem that arises has to be handled a little differently due to the non-imit metric matrix (see e.g. Oddershede et al, 1984, Appendix A). 

Another desirable aspect of using the TDA and RPA approaches is that they both use a common set of molecular orbitals, which aids both in developing qualitative interpretations of the excitation process and also in calculating properties such as transition moments. The latter depends on (i j r i i )p, where r = is the dipole operator. It is easy to evaluate such a one-electron property provided i / and are described in terms of the same orthonormal orbital set. When different orbitals are used in and l —typically to get the best possible solution for both states—the resultant nonorthogonality causes a number of complications. This is particularly true when an entire spectrum of electronic states is the objective and all transition moments are required. Nevertheless, all the methods discussed so far neglect electron correlation effects, and one must go beyond the single configuration approximation if quantitative accuracy is to be achieved. 

It can also be shown that an approximation of the transition moments with the physical solutions of the eigenvalue problem yields the RPA for the transition moments. The transition moments thus have all the RPA properties like, e. g., the exact equivalence of length and velocity form. 

An attractive aspect of CIS and RPA is that they both use a common set of MOs for the ground and excited states, which helps in developing qualitative interpretations of the excitation process and in calculation of transition moments. It is straightforward to evaluate (zlrla) and (z V a) provided and both belong to the same orthonormal MO set. When different MO sets are used for different electronic states (to get the best possible solution), the resultant nonorthogonality of [Pg.482]

Excitation energies and transition moments can in principle be obtained as poles and residua of polarization propagators as discussed in Section 7.4. However, only in the case that the set of operators hn in Eq. (7.77) is restricted to single excitation and de-excitation operators q i,qai is it computationally feasible to determine all excitation energies. This restricts this approach to single-excitation-based methods like the random phase approximation (RPA) discussed in Sections 10.3 and 11.1 or time-dependent density functional theory (TD-DFT). 

As has already been mentioned, the variational nature of the Hartree-Fock wave-function means that the CHF/TDHF equations are equivalent to the RPA equations. Unlike RPA and its correlated extensions, however, an atomic-orbital-based solution of the iterative CHF equations cannot give excitation energies and transition moments. 

There is only one other ab initio implementation of the theory of optical activity to calculate optical rotatory strengths, that due to Hansen and Bouman, based on the random-phase approximation (RPA) and implemented in the program package, RPAC. The RPA method is intended to include those first-order correlation effects that are important both for electronic transition intensities and for excitation energies. The electric and magnetic dipole transition moments in RPA are given by equations (14), (15), and (16) (analogous to equations 7, 8, and 9, above). 

For exact wave functions and those that fulfill the hyper-virial theorem by construction [e.g., time-dependent Hartree-Fock (TDFIF) or random phase approximation (RPA), TDDFT, see below] both forms are equivalent. Note that all virial theorems are exactly fulfilled only in a complete (i.e., usually infinitely large) AO basis. By a simultaneous computation of the transition dipole moments in the length and velocity forms and subsequent numerical... 

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