Excited states, perturbation theory formulation

There are several possible ways of deriving the equations for TDDFT. The most natural way departs from density-functional perturbation theory as outlined above. Initially it is assumed that an external perturbation is applied, which oscillates at a frequency co. The linear response of the system is then computed, which will be oscillating with the same imposed frequency co. In contrast with the standard static formulation of DFPT, there will be special frequencies cov for which the solutions of the perturbation theory equations will persist even when the external field vanishes. These particular solutions for orbitals and frequencies describe excited electronic states and energies with very good accuracy. 

Coalson, R.D. (1987). Time-domain formulation of photofragmentation involving nonra-diatively coupled excited states, and its implementation via wave packet perturbation theory, J. Chem. Phys. 86, 6823-6832. 

This relation shows how the action of the antisymmetrizer can mix different orders in perturbation theory. Secondly, the projected functions AglO ) 0 > do not form an orthogonal set in the antisymmetric subspace of the Hilbert space L2(r3N) if we take all excited states a > and b > in order to obtain a complete set a > b >, the projections As a > b > form a linearly dependent set. Expanding a given (antisymmetric) function in this overcomplete set is always possible, but the expansion coefficients are not uniquely defined. How the different symmetry adapted perturbation theories that have been formulated since the original treatment by Eisenschitz and London in 1930 , actually deal with these two problems can be read in the following reviews Usually, the first order interaction... 

Finite order perturbation theory (PT) for low-spin open-shell singlet states is formulated in terms of spin-free unitary group approach (UGA). Both M0ller Plesset and Bpstein-Nesbet partitionings of the Hamiltonian are considered. The method is tested on low lying excited states of CH2, NH2+ and H2O using various basis sets. 

Despite the fact that we may also formulate the perturbation theory for excited states, we will assume that we are dealing with the ground state (and denote it by subscript 0 ). In what is called the polarization approximation, the zeroth-order wave function will be taken as a product ... 

Response theory deviates in one crucial aspect from the formulation of time-dependent perturbation theory in most textbooks on quantum mechanics the response parameters are not explicitly expressed in terms of the excited states. As a consequence, knowledge of the excited state wave functions is not needed in response theory. Instead, we must solve the response equations (O Eq. 5.4) for each set of perturbation operators V (tu). This is a tremendous computational advantage as there are significantly fewer pertiubation operators, and hence response equations to solve, than excited states in virtually all cases of practical interest. 

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