Methods of accounting for correlation effects

Correlation effects have already been mentioned a number of times. Let us discuss them here in more detail. In the Hartree-Fock approach (to be more exact, in its single-configuration version, which we have been considering so far) it is assumed that each electron in an atom is orbiting independently in the central field of the nucleus and of the remaining electrons (one-particle approach). However, this is not exactly so. There exist effects, usually fairly small, of their correlated, consistent movement, causing the so-called correlation energy. 

However, there are cases, especially for the configurations with dN and fN shells as well as for some excited configurations of multiply charged ions whose energy levels lie very close or even overlap, in which the conventional single-configuration approach is completely unfit (see also Chapter 31). 

As a rule, only the calculations that account for correlation effects can correctly predict the existence of stable negative ions, representing very weakly bound electronic systems [207-216]. 

There are a number of methods that take into account correlation effects perturbation theory, random phase approximation with exchange, method of incomplete separation of variables, so-called extended method of calculation, superposition-of-configurations, multi-configuration approach, etc. The first two methods have so far been applied only to comparatively simple systems (e.g. configurations having few electrons above the closed shells). 

A comprehensive review of non-relativistic and relativistic versions of the random phase approximation may be found in [217]. Some applications are described in [218]. A survey of various aspects of the modern theory of many-electron atoms is presented in [219]. 

In order to improve the theoretical description of a many-body system one has to take into consideration the so-called correlation effects, i.e. to deal with the problem of accounting for the departures from the simple independent particle model, in which the electrons are assumed to move independently of each other in an average field due to the atomic nucleus and the other electrons. Making an additional assumption that this average potential is spherically symmetric we arrive at the central field concept (Hartree-Fock model), which forms the basis of the atomic shell structure and the chemical regularity of the elements. Of course, relativistic effects must also be accounted for as corrections, if they are small, or already at the very beginning starting with the relativistic Hamiltonian and relativistic wave functions. 

In fact, electrons do not move independently, they feel each other, there is a certain correlation between the electrons in their mutual Coulomb field (many-body effects). 

The Cl method is often based on the use of analytic functions, which form the basis set. The parameters of a many-body wave function obtained in this way, as well as the mixing coefficients, are then found while 

In the MCHF approach a number of superposed configurations are chosen and the mixing coefficients (weights of the configurations) and also the radial parts of the wave functions are varied. This method does not depend on choice of the basis set and both analytical and numerical wave functions may be used. However, MCHF calculations for complex electronic configurations would require variation of a large number of parameters, which needs powerful computers. Problems may also occur with the convergence of the procedure [45]. 

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The parameters X of Eq. (9.3) and M of Eq. (9.5) are determined by the electronic structure of the molecule. They were calculated in 1997 for the ground 0+ state of TIE by Parpia [89] and by Laerdahl, Saue, Faegri Jr., and Quiney [88] using the Dirac-Fock method with gaussian basis sets of large sizes (see Table 4). Below we refer to paper [127] with the calculations presented in details and not to the brief communication [88] of the same authors. There was also a preliminary calculation of electronic structure in TIF performed by Wilson et al. in [128]. In paper [19] the first calculation of accounts for correlation effects was performed (see also [129]... 

The results obtained in post-HF methods for solids refer mainly to the energy of the ground state but do not provide the correlated density matrix. The latter is calculated for sohds in the one-determinant approximation. The density matrix calculated for crystals in RHF or ROHF one-determinant methods describes the many-electron state with the fixed total spin (zero in RHF or defined by the maximal possible spin projection in ROHF). Meanwhile, the UHF one-determinant approximation formally corresponds to the mixture of many-electron states with the different total spin allowed for the fixed total spin projection. Therefore, one can expect that the UHF approach partly takes into account the electron correlation. In particular, of interest is the question to what extent UHF method may account for correlation effects on the chemical bonding in transition-metal oxides. An answer to this question can be obtained in the framework of the molecular-crystalline approach, proposed in [577] to evaluate the correlation corrections in the study of chemical bonding in crystals. 

In this paper, the main features of the two-step method are presented and PNC calculations are discussed, both those without accounting for correlation effects (PbF and HgF) and those in which electron correlations are taken into account by a combined method of the second-order perturbation theory (PT2) and configuration interaction (Cl), or PT2/CI [100] (for BaF and YbF), by the relativistic coupled cluster (RCC) method [101, 102] (for TIF, PbO, and HI+), and by the spin-orbit direct-CI method [103, 104, 105] (for PbO). In the ab initio calculations discussed here, the best accuracy of any current method has been attained for the hyperfine constants and P,T-odd parameters regarding the molecules containing heavy atoms. 

The difficulties are mainly caused by two problems (1) the fact that even a qualitatively correct description of excited states often requires multiconfigurational wave functions and (2) that dynamic electron correlation effects in excited states are often significantly greater than in the electronic ground states of molecules, and may also vary greatly between different excited states. For explanations of the concepts invoked in this section, see Section 3.2.3 of Chapter 22 in this volume. Therefore, an accurate modeling of electronic spectra requires methods that account for both effects simultaneously. 

The NRO approach is very efficient when accounting for correlation effects in the framework of the so-called extended method of calculation (see also Chapter 29) applied to electronic configurations having several shells of equivalent electrons with the same values of orbital quantum numbers. Its general theory is described in [199-204]. 

Another possibility to account for correlation effects in a shell of equivalent electrons is to use the so-called extended method of calculation. In a conventional single-configuration approach, all electrons of shell lN are described by one and the same radial orbital. In an extended method each electron in shell lN has different radial functions, angular parts remaining... 

Let us also briefly discuss some developments in accounting for correlation effects while performing calculations of spectral properties of complex many-electron atoms and ions. The method of superposition of configurations based on the transformed radial orbitals, briefly described in Chapter 29, has demonstrated its effectiveness particularly for complex electronic configurations. Code for transformed radial orbitals is described in [11]. There the following transformed radial orbitals are recommended ... 

There exist a number of methods to account for correlation [17, 45, 48] and relativistic effects as corrections or in relativistic approximation [18]. There have been numerous attempts to account for leading radiative (quantum-electrodynamical) corrections, as well [49, 50]. However, as a rule, the methods developed are applicable only for light atoms with closed electronic shells plus or minus one electron, therefore, they are not sufficiently general. 

Another favorable finding was reported by Burton" that the exponents of bond functions optimized at the SCF level and with the inclusion of the correlation energy (by the CEPA method) are rather close in absolute value and that bond functions are very effective in accounting for correlation effects. 

Electron correlation effects are known to be impx>rtant in systems with weak interactions. Studies of van der Waals interetctions have established the importance of using methods which scale linearly with the number of electrons[28] [29]. Of these methods, low-order many-body perturbation theory, in particular, second order theory, oflfers computational tractability combined with the ability to recover a significant firaction of the electron correlation energy. In the present work, second order many-body perturbation theory is used to account for correlation effects. Low order many-body perturbation theory has been used in accurate studies of intermolecular hydrogen bonding (see, for example, the work of Xantheas and Dunning[30]). 

The principle of corresponding states provides a practical method for making use of the measured properties of one or more substances to predict the properties of other substances under conditions for which no data exist and no satisfactory theoretical treatments may be applied. This principle was originally formulated by van der Waals [ ] for classical systems and subsequently a method of accounting for deviations caused by quantum effects was suggested by Byk [2]. The further development of the quantum mechanical principle of corresponding states has been due primarily to de Boer and his collaborators Their early efforts were concerned with equation of state and vapor pressure correlations and predictions, the most successful of which was the prediction of the vapor pressure and the critical parameters of He prior to its liquefaction [ ]. 

The main difficulty here is to clearly separate effects that can hardly be separated, namely relativistic and electron-correlation effects. Nevertheless, pioneering studies of this effect date back to the mid 1970s [1140]. Four-component methods have been employed to determine the contribution which is solely due to relativity [1141]. The four-component approach, for which Dirac-Hartree-Fock and — to also account for correlation effects — relativistic MP2 calculations have been utilized, confirms results first obtained with relativistic effective core potential methods [1142,1143]. It has been found [1141] that between 10% and 30% of the lanthanide contraction and 40% to 50% of the actinide contraction are caused by relativity in monohydrides, trihydrides, and monofluorides of La, Lu and Ac, Lr, respectively. 

Six methods are recommended by Reid et al. (1987) for the estimation of thermal conductivity of nonpolar compounds. These include the corresponding-states methods of Chung et al. (1984, 1988), Ely Hanley (1983) and Hanley (1976). The third method (Roy Thodos 1968, 1970) is recommended for polar as well as nonpolar compounds. Topical errors for nonpolar compounds are 5 to 7%, with higher errors expected for polar compounds. Both the Chung and Ely-Hanley methods correlate the Eucken factor, fy, = XMlriCv, with other variables such as C (heat capacity), Tr and 0) (acentric factor). Thus the viscosity is required to use these correlations. The Roy-Thodos correlation requires only the critical temperature and the pressure, and employs a group contribution method to account for the effect of internal degrees of freedom. 

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