Mpller-Plesset perturbation theory higher orders

A series of single-point energy calculations is carried out at higher levels of theory. The first higher-level calculation is the complete fourth-order Mpller-Plesset perturbation theory [13] with the 6-31G(d) basis set, i.e. MP4/6-31G(d). For convenience of notation, we represent this as MP4/d. This energy is then modified by a series of corrections from additional calculations ... 

All of the systems were initially optimized using a much higher level of theory, in order to ensure that the OM2 method provides a realistic description of the structure. The method employed was the second-order Mpller-Plesset perturbation theory (MP2) [50] using the cc-pVDZ basis set [51]. The resolution-of-identity (RI) approximation for the evaluation of the electron-repulsion integrals implemented in Turbomole was utilized [52]. 

In the G2 and G3 [10,11] theories, the Mpller-Plesset perturbation theories of the 2-nd and 4-th orders are used to estimate the consequences of extending orbital basis sets by including the diffuse and polarization functions. These attempts, however, do not allow one to eliminate a systematic error of about 6 millihartree per electronic pair, which, in the frame of the G2 and G3 theories, bears the pompous name of higher level correlation of unknown nature. These latter are parametrized in the form ... 

Table 3 Optimized regular interatomic distance (in A) and gain in energy per C2H2 unit (in eV) of regular polyacetylene by report to the alternating ground state, with the Neel state based higher order and the RVB ansatze. Results from Local-density-functional approach of Mintmire and White [48], Hartree-Fock and Mpller-Plesset perturbation Theory of Suhai [50], or Ashkenazi [49] are included.

One of the original approximate methods is the wavefunction-theory-based Hartree-Fock (HF) method [40]. The HF method is a single determinant method that does not include any correlation interactions between the electrons, and as such has limited accuracy [41, 42]. Higher level wavefunction-based methods such as coupled cluster [43 5], configuration interaction [40,46,47], and complete active space [48-50] methods include multiple determinants to incorporate some of the electron-electron correlation. Methods based on perturbation theory, such as second order Mpller-Plesset perturbation theory [51], go beyond the HF method by perturbatively adding electron correlation. These correlated wavefunction-based methods have well-defined ways in which they approach the exact solution to the Schrodinger equation and thus have the potential to be extremely accurate, but this accuracy comes at a price [52]. 

The least expensive and conceptually simplest correlation treatment that can be applied to medium-size molecules is the second-order Mpller-Plesset perturbation theory (MP2), which is the most popular single-reference approach for the low-level treatment of electron correlation [21]. Higher order MP perturbation theory such as MP3 and MP4 are typically less useful in particular, results coming from MP3 level are inferior to MP2 because of the characteristically oscillatory convergence of perturbation theory. MP4, MP5, and MP6 offer some improvements, but the high computational costs required prevent their routine application [22]. 

The formulation of approximate response theory based on an exponential parame-trization of the time-dependent wave function, Eq. (11.36), and the Ehrenfest theorem, Eq. (11.40), can also be used to derive SOPPA and higher-order Mpller-Plesset perturbation theory polarization propagator approximations (Olsen et al., 2005). Contrary to the approach employed in Chapter 10, which is based on the superoperator formalism from Section 3.12 and that could not yet be extended to higher than linear response functions, the Ehrenfest-theorem-based approach can be used to derive expressions also for quadratic and higher-order response functions. In the following, it will briefly be shown how the SOPPA linear response equations, Eq. (10.29), can be derived with this approach. 

Mpller-Plesset perturbation theory represents a useful approach to the calculation of size-extensive correlation energies for systems dominated by a single electronic configuration. The MP2 model, in particular, represents a successful compromise between computational cost and accuracy. Higher-order corrections may also be calculated, but it should be emphasized that the M0ller-Plesset series does not converge unconditionally. 

Perturbative approximation methods are usually based on the Mpller-Plesset (MP) perturbation theory for correcting the HF wavefunction. Energetic corrections may be calculated to second (MP2), third (MP3), or higher order. As usual, the open- versus closed-shell character of the wavefunction can be specified by an appropriate prefix, such as ROMP2 or UMP2 for restricted open-shell or unrestricted MP2, respectively. 

During the 1960s, Kelly [37-43] pioneered the application of what is today the most widely used approach to the description of correlation effects in atomic and molecular systems namely, the many-body perturbation theory [1,2,43 8]. The second-order theory using the Hartree-Fock model to provide a reference Hamiltonian is particularly widely used. This Mpller-Plesset (mp2) formalism combines an accuracy, which is adequate for many purposes, with computational efficiency allowing both the use of basis sets of the quality required for correlated studies and applications to larger molecules than higher order methods. 

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