Correlation perturbative treatment with

There is also a hierarchy of electron correlation procedures. The Hartree-Fock (HF) approximation neglects correlation of electrons with antiparallel spins. Increasing levels of accuracy of electron correlation treatment are achieved by Mpller-Plesset perturbation theory truncated at the second (MP2), third (MP3), or fourth (MP4) order. Further inclusion of electron correlation is achieved by methods such as quadratic configuration interaction with single, double, and (perturbatively calculated) triple excitations [QCISD(T)], and by the analogous coupled cluster theory [CCSD(T)] [8],... 

The Time Dependent Processes Section uses time-dependent perturbation theory, combined with the classical electric and magnetic fields that arise due to the interaction of photons with the nuclei and electrons of a molecule, to derive expressions for the rates of transitions among atomic or molecular electronic, vibrational, and rotational states induced by photon absorption or emission. Sources of line broadening and time correlation function treatments of absorption lineshapes are briefly introduced. Finally, transitions induced by collisions rather than by electromagnetic fields are briefly treated to provide an introduction to the subject of theoretical chemical dynamics. 

First let us review static and dynamic electron correlation. Dynamic (dynamical) electron correlation is easy to grasp, if not so easy to treat exhaustively. It is simply the adjustment by each electron, at each moment, of its motion in accordance with its interaction with each other electron in the system. Dynamic correlation and its treatment with perturbation (Mpller-Plesset), configuration interaction, and coupled cluster methods was covered in Section 5.4. 

As in all perturbational approaches, the Hamiltonian is divided into an unperturbed part and a perturbation V. The operator is a spin-free, one-component Hamiltonian and the spin-orbit coupling operator takes the role of the perturbation. There is no natural perturbation parameter X in this particular case. Instead, J4 so is assumed to represent a first-order perturbation The perturbational treatment of fine structure is an inherent two-step approach. It starts with the computation of correlated wave functions and energies for pure spin states—mostly at the Cl level. In a second step, spin-orbit perturbed energies and wavefunctions are determined. 

In a mainly experimental work, Fujihara et al performed photoelectron spectroscopy experiments on Na2 in small water clusters, i.e., Na2 (H20)K with w < 6. In addition, they performed electronic-structure calculations with Hartree-Fock plus 2nd order Moller-Plesset perturbation treatment of correlation effects. Further... 

SAPT avoids the subtraction of large energy values that is necessarily part of a supermolecule ab initio calculation. A supermolecule calculation obtains the interaction energy of monomers A and B, AVab, as Tab - Va - Vb. whereas SAPT finds AVab directly. The interaction evaluated in SAPT is defined so as to be free of BSSE however, the other requirements on basis-set quality and for correlation effects still hold. SAPT has yielded highly accurate interaction data, first for rare gas atoms interacting with small molecules [72 74] and more recently with molecule molecule clusters such as the CO2 dimer [75]. Further examples are the very accurate results achieved for Ne HCN [76] and a parr potential for water [77]. Another example study of perturbative treatment of the interaction potential has been a study of rare gas HCN clusters [78] which included vibrational analysis. 

An important issue with regard to any perturbation treatment is the convergence behavior of the perturbation series. This is considered in Section 4 where problematic cases are identified. Then a potentially viable treatment of such cases, based on vibrational SCF and post-SCF procedures, is elaborated in Section 5. In Section 6 we turn to tire practical Issues of basis set requirements and treatment of electron correlation. Here tire emphasis is on quasilinear pi-conjugated molecules and, for that case, we examine the difficulties encountered with tire use of density functional theory. 

Finally, we would like to mention Pople s (1986) recent work this treats the derivatives of the (MP) correlation energy as a double perturbation problem, with respect to a physical perturbation (e.g. nuclear coordinate change) and a non-physical perturbation (electron correlation). This provides a unified theory for the treatment of geometry and property derivatives at the correlated level. 

However, a reasonable quantitative treatment for TM systems seems to require a fairly high level of theory. One particularly promising approach has been developed by Roos [28] based on the Complete Active Space Self Consistent Field (CASSCF) method with a second order perturbation treatment of the remaining (dynamical) electron correlation effects, CASPT2. 

Along the ordinate, a sequence of correlation-consistent cc-pVXZ basis sets with X > 2 is depicted. Along the abscissa, the FCI limit is approached—beginning with Hartree-Fock theory and followed by the first correlated level, at which the single and double excitations are described by MP2 perturbation theory. The same excitations are subsequently treated by coupled-cluster theory at the CCSD level, which is then further improved upon by a perturbation treatment of the triple excitations at the CCSD(T) level. At the CCSDT level, the triple excitations are fully treated by coupled-cluster theory, and so on. In this manner, the hierarchy Hartree-Fock -> MP2 — CCSD -> CCSD(T) > CCSDT —---------------> FCI is established. 

Thus the double perturbation theory enables to discuss the effects of electron correlation on molecular properties systematically, but there are not many numerical calculations proceeding along these lines instead of calculating the effect of /I2 by veuriation methods and subsequently dealing with d 1 by ordinary perturbation theory. The former procedure has some advantages, for, in principle, it should include ZI2 to infinite order (in other words, A1 and A 2 might enter on different levels of the perturbation treatment). 

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