Wavefunction-based electron correlation calculations

Localized Orbital Methods for Wavefunction-based Electron Correlation Calculations... 

One of the major advances in recent years in attempts to calculate more exact wavefunctions, including electron correlation, has been the implementation by Boys and Handy,46 47 of a computational scheme based on the method of moments, called the transcorrelated wavefunction method. In this method, a correlation factor is built into the wavefunction, which is written in the form (1), where 0 is a Slater... 

Local electron-correlation methods are ab-initio wavefunction-based electronic-structure methods that exploit the short-range nature of dynamic correlation effects and in this way allow linear scaling 0 N) in the electron-correlation calculations [128,129,131-135] to be attained. 0 N) methods are applied to the treatment of extended molecular systems at a very high level of accuracy and rehabihty as CPU time, memory and disk requirements scale hneaily with increasing molecular size N. 

How does a rigorously calculated electrostatic potential depend upon the computational level at which was obtained p(r) Most ab initio calculations of V(r) for reasonably sized molecules are based on self-consistent field (SCF) or near Hartree-Fock wavefunctions and therefore do not reflect electron correlation in the computation of p(r). It is true that the availability of supercomputers and high-powered work stations has made post-Hartree-Fock calculations of V(r) (which include electron correlation) a realistic possibility even for molecules with 5 to 10 first-row atoms however, there is reason to believe that such computational levels are usually not necessary and not warranted. The Mpller-Plesset theorem states that properties computed from Hartree-Fock wave functions using one-electron operators, as is T(r), are correct through first order (Mpller and Plesset 1934) any errors are no more than second-order effects. 

The discussion presented in the subsequent parts of this chapter is based on the results of ab initio calculations of the electronic energy of molecular systems. Details about this kind of calculation are described in reference 11. In connection with this procedure, two major questions have to be addressed. The first is the choice of the wavefunction (basis set) to be used in the calculation, and the second whether or not to include electron correlation. 

As we just saw, MP2 calculations utilize the Hartree-Fock MOs (their coefficients c and energies e). The HF method gives the best occupied MOs obtainable from a given basis set and a one-determinant total wavefunction i(i, but it does not optimize the virtual orbitals (after all, in the HF procedure we start with a determinant consisting of only the occupied MOs - Sections 5.2.3.1-5.2.3.4). To get a reasonable description of the virtual orbitals and to obtain a reasonable number of them into which to promote electrons, we need a basis set that is not too small. The use of the STO-1G basis in the above example was purely illustrative the smallest basis set generally considered acceptable for correlated calculations is the 6-31G, and in fact this is perhaps the one most frequently used for MP2 calculations. The 6-311G basis set is also widely used for MP2 and MP4 calculations. Both bases... 

The mathematical term functional, which is akin to function, is explained in Section 7.2.3.1. To the chemist, the main advantage of DFT is that in about the same time needed for an HF calculation one can often obtain results of about the same quality as from MP2 calculations (cf. e.g. Sections 5.5.1 and 5.5.2). Chemical applications of DFT are but one aspect of an ambitious project to recast conventional quantum mechanics, i.e. wave mechanics, in a form in which the electron density, and only the electron density, plays the key role [5]. It is noteworthy that the 1998 Nobel Prize in chemistry was awarded to John Pople (Section 5.3.3), largely for his role in developing practical wavefunction-based methods, and Walter Kohn,1 for the development of density functional methods [6]. The wave-function is the quantum mechanical analogue of the analytically intractable multibody problem (n-body problem) in astronomy [7], and indeed electron-electron interaction, electron correlation, is at the heart of the major problems encountered in... 

Therefore the scaling transformation of the quantum-mechanical force field is an empirical way to account for the electronic correlation effects. As far as the conditions listed above are not always satisfied (e.g. in the presence of delocalized 7r-electron wavefunctions) the real transformation is not exactly homogeneous but rather of Puley s type, involving n different scale constants. The need of inhomogeneous Puley s scaling also arises due to the fact that the quantum-mechanical calculations are never performed in the perfect Hartree-Fock level. The realistic calculations employ incomplete basis sets and often are based on different calculation schemes, e.g. semiempirical hamiltonians or methods which account for the electronic correlations like Cl and density-functional techniques. In this context we want to stress that the set of scale factors for the molecule under consideration is specific for a given set of internal coordinates and a given quantum-mechanical method. 

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. 

Hyperpolarizabilities can be calculated in a number of different ways. The quantum chemical calculations may be based on a perturbation approach that directly evaluates sum-over-states (SOS) expressions such as Eq. (14), or on differentiation of the energy or induced moments for which (electric field) perturbed wavefunctions and/or electron densities are explicitly calculated. These techniques may be implemented at different levels of approximation ranging from semi-empirical to density functional methods that account for electron correlation through approximations to the exact exchange-correlation functionals to high-level ab initio calculations which systematically include electron correlation effects. 

In this section two approaches to the selection of CSF expansion spaces for MCSCF wavefunctions have been described. Most modern MCSCF methods are best suited to the a priori selection of CSFs based on orbital occupation and spin-coupling restrictions. These a priori selection approaches should be accompanied with empirical evidence (usually more extensive MCSCF or Cl calculations at selected geometries) that a balanced description of the molecular electron correlation is achieved. A reasonable approach to use for molecular systems is to begin with an RCI expansion of all the valence electrons. The less important electrons may then optionally be described with the more restrictive, and more economical, PPMC expansion. The description of the more important electrons in the RCI expansion may be generalized, if necessary, to other direct product type expansions, thereby allowing a more thorough treatment of the correlation of these electrons. 

Ab initio MP2/6-311++G(d,p) calculations were performed to obtain geometries and electron densities. The perturbative MP2 approach is one of the wavefunction-based methods most frequently used to include electron correlation and its well-gained reputation needs no further comments. The basis set... 

The Coupled-clusters (CC) method[7] based on the cluster expansion of the wavefunction has been established as a highly reliable method for calculations of ground state properties of small molecules with the spectroscopic accuracy. When this method is used together with a flexible basis set it recovers the dominant part of the electron correlation. Typically, CC variant explicitly considering single and double excitations (CCSD) is used. In order to save computer time the contributions from triple excitations are often calculated at the perturbation theory level (notation CCSD(T) is used in this case). CCSD(T) method can be routinely used only for systems with about 10 atoms at present. Therefore, it cannot be used directly in zeolite modeling, however, results obtained at CCSD(T) level for small model systems can serve as an important benchmark when discussing the reliability of more approximate methods. 

As one moves towards the realm of condensed matter physics, the hope of a wavefunction-based theory involving an exact treatment of spin-jK>larisation and exchange fades. Most modem work in this field is carried out within the density functional theory (DFT) introduced by Hohenbei, Kohn and Sham [1,2], in which the electron density tak i on the role of the primary variable. This allows scope for any number of s roximate treatments of electronic exchange and correlation, so that calculations for even the largest s> tems become tractable. Insofar as spin-polarisation is included, it is tinted in a parametric maimer, making use of exact results obtained for the homogeneous electron gas. 

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