Rayleigh—Schrodinger expansion

We can develop a series expansion for in powers of X. The coefficients in this expansion depend on the exact perturbed energy rather than, as is the case in the more familiar Rayleigh-Schrodinger expansion, on the unperturbed energy7 . Iterating the basic formula (2.21), we obtain... 

The current volume presents the compilation of splendid contributions distributed over 21 chapters. The very first chapter contributed by Istvan Hargittai presents the historical account of development of structural chemistry. It also depicts some historical memories of scientists presented in the form of their pictures. This historical description covers a vast period of time. Intruder states pose serious problem in the multireference formulation based on Rayleigh-Schrodinger expansion. Ivan Hubac and Stephen Wilson discuss the ciurent development and future prospects of Many-Body Brillouin-Wigner theories to avoid the problem of intruder states in the next chapter. The third chapter written by Vladimir Ivanov and collaborators reveals the development of multireference state-specific coupled cluster theory. The next chapter from Maria Barysz discusses the development and application of relativistic effects in chemical problems while the fifth chapter contributed by Manthos Papadopoulos and coworkers describes electronic, vibrational and relativistic contributions to the linear and nonlinear optical properties of molecules. 

For approach (i) the amplitude equations for the continuous transition between Brillouin-Wigner and Rayleigh-Schrodinger expansions have the... 

Substituting iteratively the full expression (3.29) for co in the denominators on the right hand side, we obtain the perturbed eigenfrequencies 2j (Rayleigh-Schrodinger expansion)... 

Let us discuss the relation between coupled cluster theories and the many-body perturbation expansion. Recall eq. (3.214), the Rayleigh-Schrodinger expansion in linked cluster form... 

It should be emphasised that the Brillouin-Wigner expansion contains only direct terms. Unlike the Rayleigh-Schrodinger expansion, it does not contain renormalization terms. In the Brillouin-Wigner case the subscript lc can be omitted. The matrix elements of the operator HiM are unknown, so we use a resolution of the identity operator to rewrite this equation as... 

The formulation of a multi-reference bwcc theory can now proceed in two distinct ways. In the first option, we can formulate a multi-root version of the multi-reference BWCC theory which yields all roots of the d-dimensional 9 space simultaneously. This is the approach employed in most multi-reference coupled cluster formulations which are based on the Rayleigh-Schrodinger expansion. In the second option, we can use the state-specific wave operator (4.59) and formulate a state-specific (or single root) version of multi-reference bwcc theory [10]. 

The energy coefficients E f in the Rayleigh-Schrodinger expansion can be written in sum-over-states form as follows... 

---