Mpller-Plesset approach

The difference between the Hartree-Fock energy and the exact solution of the Schrodinger equation (Figure 60), the so-called correlation energy, can be calculated approximately within the Hartree-Fock theory by the configuration interaction method (Cl) or by a perturbation theoretical approach (Mpller-Plesset perturbation calculation wth order, MPn). Within a Cl calculation the wave function is composed of a linear combination of different Slater determinants. Excited-state Slater determinants are then generated by exciting electrons from the filled SCF orbitals to the virtual ones ... 

The other area in which projection of an unrestricted result has received attention is projected Mpller-Plesset perturbation theory, the PUMPn methods [31], where n is the order of the perturbation theory. In cases in which the UHF approximation is a poor starting point (considerable spin contamination, for example), the convergence of the MP perturbation expansion can be slow and/or erratic. The PUMP methods apply projection operators to the perturbation expansion, although usually not full projection but simply annihilation of the leading contaminants. This approach has met with mixed success again, it represents a rather expensive modification to a technique that was originally chosen partly for its economy — seldom a recipe for success. 

The Mpller-Plesset (MP) treatment of electron correlation [84] is based on perturbation theory, a very general approach used in physics to treat complex systems [85] this particular approach was described by M0ller and Plesset in 1934 [86] and developed into a practical molecular computational method by Binkley and Pople [87] in 1975. The basic idea behind perturbation theory is that if we know how to treat a simple (often idealized) system then a more complex (and often more realistic) version of this system, if it is not too different, can be treated mathematically as an altered (perturbed) version of the simple one. Mpller-Plesset calculations are denoted as MP, MPPT (M0ller-Plesset perturbation theory) or MBPT (many-body perturbation theory) calculations. The derivation of the Mpller-Plesset method [88] is somewhat involved, and only the flavor of the approach will be given here. There is a hierarchy of MP energy levels MPO, MP1 (these first two designations are not actually used), MP2, etc., which successively account more thoroughly for interelectronic repulsion. 

The configuration interaction (Cl) treatment of electron correlation [83,95] is based on the simple idea that one can improve on the HF wavefunction, and hence energy, by adding on to the HF wavefunction terms that represent promotion of electrons from occupied to virtual MOs. The HF term and the additional terms each represent a particular electronic configuration, and the actual wavefunction and electronic structure of the system can be conceptualized as the result of the interaction of these configurations. This electron promotion, which makes it easier for electrons to avoid one another, is as we saw (Section 5.4.2) also the physical idea behind the Mpller-Plesset method the MP and Cl methods differ in their mathematical approaches. 

The coupled cluster (CC) method is actually related to both the perturbation (Section 5.4.2) and the Cl approaches (Section 5.4.3). Like perturbation theory, CC theory is connected to the linked cluster theorem (linked diagram theorem) [101], which proves that MP calculations are size-consistent (see below). Like standard Cl it expresses the correlated wavefunction as a sum of the HF ground state determinant and determinants representing the promotion of electrons from this into virtual MOs. As with the Mpller-Plesset equations, the derivation of the CC equations is complicated. The basic idea is to express the correlated wave-function Tasa sum of determinants by allowing a series of operators 7), 73,... to act on the HF wavefunction ... 

An ab initio version of the Mpller-Plesset perturbation theory within the DPCM solvation approach was introduced years ago by Olivares et al. [26] following the above intuitive considerations based on the fact that the electron correlation which modifies both the HF solute charge distribution and the solvent reaction potential depending on it can be back-modified by the solvent. To decouple these combined effects the authors introduced three alternative schemes ... 

In this section, we briefly discuss some of the electronic structure methods which have been used in the calculations of the PE functions which are discussed in the following sections. There are variety of ab initio electronic structure methods which can be used for the calculation of the PE surface of the electronic ground state. Most widely used are Hartree-Fock (HF) based methods. In this approach, the electronic wavefunction of a closed-shell system is described by a determinant composed of restricted one-electron spin orbitals. The unrestricted HF (UHF) method can handle also open-shell electronic systems. The limitation of HF based methods is that they do not account for electron correlation effects. For the electronic ground state of closed-shell systems, electron correlation effects can be accounted for relatively easily by second-order Mpller-Plesset perturbation theory (MP2). In modern implementations of MP2, linear scaling with the size of the system has been achieved. It is thus possible to treat quite large molecules and clusters at this level of theory. 

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.

The relation between the supermolecule coupled cluster approach and the perturbation theory of intermolecular forces in even less obvious than the case of the Mpller-Plesset theory, and no formal analysis has been reported in the literature thus far. Rode et al.68 analyzed the long-range behavior of the CCSD(T) method65, and showed that this method, although very popular and in principle accurate, may lead to wrong results for systems with the electrostatic term strongly depending on the electronic correlation, e.g. the CO dimer. 

We have above discussed the CASSCF method and how we can choose the active space. We noted that this choice was closely connected to the method we use to compute the effects of dynamic correlation, in this case the CASPT2 method. The development of this approach was inspired by the success of the Mpller-Plesset second order perturbation theory (MP2), which has been used for a long time to treat electron correlation for ground states, where the reference function is a single determinant. It was assumed that such an approach would be even more effective with the more accurate CASSCF reference function. A first attempt was made soon... 

The type of correlated method that has enjoyed the most widespread application to H-bonded systems is many-body perturbation theory, also commonly referred to as Mpller-Plesset (MP) perturbation theory This approach considers the true Hamiltonian as a sum of its Hartree-Fock part plus an operator corresponding to electron correlation. In other words, the unperturbed Hamiltonian consists of the interaction of the electrons with the nuclei, plus their kinetic energy, to which is added the Hartree-Fock potential the interaction of each electron with the time-averaged field generated by the others. The perturbation thus becomes the difference between the correct interelectronic repulsion operator, with its instantaneous correlation between electrons, and the latter Hartree-Fock potential. In this formalism, the Hartree-Fock energy is equed to the sum of the zeroth and first-order perturbation energy corrections. 

Two general groups of methodologies are used to solve the Schrodinger equation in combination with cluster models, the Hartree-Fock (HF) approach and related methods to include correlation effects like Mpller-Plesset perturbation theory (MP2) or configuration interaction (Cl) [58,59] and the Density Functional Theory (DFT) approach [59,60]. 

A similar approach with the following modifications was used in the present paper (1) The more rigorous Mpller—Plesset (MP) perturbation theory was selected instead of the Hartree—Fock method. (2) Clusters of two molecules were employed for the geometry... 

Second-order Mpller—Plesset perturbation theory (MP2) is the quantum mechanical approach selected for the calculations because the sizes of the clusters that were employed were too large to use more accurate methods. In addition, MP2 provides accurate results regarding the calculation of the interaction energies for both H-bonded pairs and van der Waals interacting pairs. 

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