Ab initio electron correlation methods

Full ab initio electron-correlation methods, from MP2 to CCSD(T) (the acronyms refer to increasing complexity in the treatment of correlation, with increasing computational cost) include polarization and dispersion contributions and apply to any molecular system. Accuracy depends on the size of the basis set, but so-called complete set limit calculations can nowadays be carried out. 

In the last section of this review, we elaborated on the relevance and consequences of these concepts for transition-metal cluster chemistry on the basis of new results. We discussed problems and pitfalls that may arise in present-day quantum chemical DFT calculations on open-shell clusters. Clearly, these obstacles point to the necessity of developing improved density functionals and also new ab initio electron correlation methods, like, for example, the density renormalization group algorithm (151). 

Due to its excellent balance between accuracy and computational cost, Kohn-Sham density functional theory (KS-DFT) [13,14] is usually the method of choice to investigate electronic ground states and their properties in chemistry and solid-state physics [15,16]. Hartree-Fock (HF) wavefunctions, on the other hand, are the starting point for ab initio electron correlation methods [4,15] which are discussed in Section 4. 

Second-order Moller-Plesset perturbation theory (MP2) is the computationally least expensive and most popular ab initio electron correlation method [4,15]. Except for transition metal compounds, MP2 equilibrium geometries are of comparable accuracy to DFT. However, MP2 captures long-range correlation effects (like dispersion) which are lacking in present-day density functionals. The computational cost of MP2 calculations is dominated by the integral transformation from the atomic orbital (AO) to the molecular orbital (MO) basis which scales as 0(N5) with the system size. This four-index transformation can be avoided by introduction of the RI integral approximation which requires just the transformation of three-index quantities and reduces the prefactor without significant loss in accuracy [36,37]. This makes RI-MP2 the most efficient alternative for small- to medium-sized molecular systems for which DFT fails. 

The difficulty with traditional ab initio electron correlation methods is the scaling of the calculations with system size. Canonical second order Mpller-Plesset (MP2) methods, the least expensive approach of this type, scales formally as N, and cutoffs are significantly less effective here than in SCF-type calculations, particularly for large basis sets. QCISD (T) and related methods. scale as making them intractable for routine large molecule applications now and in the forseeable future. 

The electronic structure method used to provide the energies and gradients of the states is crucial in photochemistry and photophysics. Ab initio electronic structure methods have been used for many years. Treating closed shell systems in their ground state is a problem that, in many cases, can now be solved routinely by chemists using standardized methods and computer packages. In order to obtain quantitative results, electron correlation (also referred to as dynamical correlation) should be included in the model and there are many methods available for doing this based on either variational or perturbation principles [41],... 

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. 

The mean-field potential and the need to improve it to aohieve reasonably aeourate solutions to the true eleotronio Selirodinger equation introduoe three oonstniots that oharaoterize essentially all ab initio quantum ohemioal methods orbitals, configurations and electron correlation. 

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