Rayleigh-Schrddinger perturbation

Claverie P 1971 Theory of intermolecular forces. I. On the inadequacy of the usual Rayleigh-Schrddinger perturbation method for the treatment of intermolecular forces Int. J. Quantum Chem. 5 273... 

Her and Plesset proposed an alternative way to tackle the problem of electron correlation tiler and Plesset 1934], Their method is based upon Rayleigh-Schrddinger perturbation 3ty, in which the true Hamiltonian operator is expressed as the sum of a zeroth-er Hamiltonian (for which a set of molecular orbitals can be obtained) and a turbation, "V ... 

Up to this point we are still dealing with undetermined quantities, energy and wave funetion corrections at each order. The first-order equation is one equation with two unknowns. Since the solutions to the unperturbed Schrddinger equation generates a complete set of functions, the unknown first-order correction to the wave function can be expanded in these functions. This is known as Rayleigh-Schrddinger perturbation theory, and the equation in (4.32) becomes... 

Ho is the normal electronic Hamilton operator, and the perturbations are described by the operators Pi and P2, with A determining the strength. Based on an expansion in exact wave functions, Rayleigh-Schrddinger perturbation theory (section 4.8) gives the first- and second-order energy collections. 

Using the above partitioning into the Rayleigh-Schrddinger perturbation theory (RSPT) allows the perturbed reference function to be written as,... 

Rayleigh-Schrddinger perturbation theory provides the third-order energy. 

The new term in Ho approximates the actual interaction of an electron with the other electrons with a mean field U r), chosen to do so as accurately as possible. Applying standard Rayleigh-Schrddinger perturbation theory to Vc then gives the MBPT expansion. It is also possible to generalize to the relativistic case by introducing the instantaneous Breit interaction. 

In this expression is the electric dipole polarizability, is the mixed electric dipole-quadrupole polarizability, Xap is th magnetic susceptibility [56], rj p y and rj p ys can, respectively, be called electric dipole and electric quadrupole polarizability of the magnetic susceptibility. Quantum-mechanical definitions for these quantities within the framework of the Rayleigh-Schrddinger perturbation theory are given later. 

Let us go back to our problem we want to have Eq on the left side of the last equation, while - for the time being - Eq occurs on the right sides of both equations. To exit the situation, we will treat Eq occurring on the right side as a parameter manipulated in such a way as to obtain equality in both of these equations. We may do it in two ways. One leads to the BriUouin-Wigner perturbation theory, the other to the Rayleigh-Schrddinger perturbation theory. 

One of the basic computational methods for the correlation energy is the MP2 method, which gives the result correct through the second order of the Rayleigh-Schrddinger perturbation theory (with respect to energy). 

According to the Rayleigh-Schrddinger perturbation theory (discussed in Qiapter 5), the unperturbed Hamiltonian is a sum of the isolated molecules Hamiltonians ... 

The apparent garbage produced by tbe perturbational series for R = 3.0 a.u. represented for the Fade approximants precise information that the absurd perturbational corrections pertain the energy of the... 2 pou state of the hydrogen atom in the electric field of the proton. Why does this happen Visibly low-order perturbational corrections, even if absolutely crazy, somehow carry information about the physics of the problem. The convergence properties of the Rayleigh-Schrddinger perturbation theory depend critically on the poles of the function approximated (see the discussion on p. 250). A pole cannot be described by any power series (as happens in perturbation theories), whereas the Fade approximants have poles built in the very essence of their construction (the denominator as a polynomial). This is why they may fit so well to the nature of the problems under study. ... 

The Mpller-Plesset (MP) method [26] constitutes a relatively less expensive alternative and is conceptually related to Rayleigh-Schrddinger perturbation theory. The lowest term in the MPn series is MP2. The MP2 energy can be efficiently calculated as... 

Multi-reference Brillouin-Wigner theory overcomes the intruder state problem because the exact energy is contained in the denominator factors. Calculations are therefore state specific , that is they are performed for one state at a time. This is in contrast to multi-reference Rayleigh-Schrddinger perturbation theory which is applied to a manifold of states simultaneously. Multi-reference Brillouin-Wigner perturbation theory is applied to a single state. Wenzel and Steiner [105] write (see also [106]) ... 

The familiar Rayleigh-Schrddinger perturbation theory is obtained by making power series expansions in the perturbation parameter A for the exact energy... 

Further details of the Rayleigh-Schrddinger perturbation expansion can be found elsewhere [110]. 

The first-order wavefunction in Rayleigh-Schrddinger perturbation theory for a single perturbation is given by equation (24). Using equation (38), this may be written as the sum-over-states expression... 

For the second-order energy coefficient in Rayleigh-Schrddinger perturbation theory, the sum-over-states formula is... 

In 1934, Mailer and Plesset applied the Rayleigh-Schrddinger perturbation theory taken through second-order in the energy to the electronic structure problem in which the Hartree-Fock model is employed as a zero-order approximation. The Hartree-Fock wavefunction is a single determinant of the form... 

As an alternative to Brillouin-Wigner perturbation theory, we may consider Rayleigh-Schrddinger perturbation theory, which is size consistent. In this method the total energy is computed in a stepwise manner... 

According to the Rayleigh-Schrddinger perturbation theory (Chapter 5) the unperturbed Hamiltonian is a sum of the isolated molecules Hamiltonians + Hg. Following quantum theory tradition in the present chapter the symbol for the perturbation operator will be changed (when compared to Chapter 5) //(i) s V. 

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