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Perturbation theory is an important tool for describing real quantum systems, as it turns out to be very difficult to find exact solutions to the Schrödinger equation for Hamiltonians of even moderate complexity.
In this, the Schrieffer–Wolff transformation is an operator version of second-order perturbation theory. The Schrieffer–Wolff transformation is often used to project out the high energy excitations of a given quantum many-body Hamiltonian in order to obtain an effective low energy model. [1]
The standard exposition of perturbation theory is given in terms of the order to which the perturbation is carried out: first-order perturbation theory or second-order perturbation theory, and whether the perturbed states are degenerate, which requires singular perturbation. In the singular case extra care must be taken, and the theory is ...
Such degenerate states are often the case of atomic and molecular valence states. To counter the restrictions, there was an attempt to implement second-order perturbation theory in conjunction with complete active space self-consistent field (CASSCF) wave functions. [3]
The pseudo Jahn–Teller effect (PJTE), occasionally also known as second-order JTE, is a direct extension of the Jahn–Teller effect (JTE) where spontaneous symmetry breaking in polyatomic systems (molecules and solids) occurs even when the relevant electronic states are not degenerate. The PJTE can occur under the influence of sufficiently ...
We evaluate the derivatives and () given they are the coefficients of the following expansion in q where q is considered small with respect to k (+) = + + + Given (+) are eigenvalues of ^ + We can consider the following perturbation problem in q: ^ + = ^ + (+) + Perturbation theory of the second order states that = + ^ + ′ | ^ | ′ +...
It involves expanding the eigenvalues and eigenkets of the Hamiltonian H in a perturbation series. The degenerate eigenstates with a given energy eigenvalue form a vector subspace, but not every basis of eigenstates of this space is a good starting point for perturbation theory, because typically there would not be any eigenstates of the ...
Moreover, any perturbation to the system (which will be of the same form as the Hamiltonian) can be added to the system in the eigenbasis of the unperturbed Hamiltonian and analysed in the same way as above. Therefore, for any perturbation the new eigenvectors of the perturbed system can be solved for exactly, as mentioned in the introduction.