Restricted Hartree-Fock method electron correlation methods

Such kind of calculations with a precise self-consistent account of crystal surrounding were performed by, at least, four scientific groups Baetzold [16], Das with coworkers [22,23], Winter etal. [28] and Ladik with coworkers [29]. Winter etal. [28] performed cluster calculation by restricted Hartree-Fock (RHF) method, so they did not take into account the electron correlation. The others groups used the umestricted Hartree-Fock (UHF) method for cluster calculations which allows to some extent the electron correlation. The strong covalent C-O bonding in planes and chains was revealed (in accordance with results obtained in Refs. [20,25,26]). For covalent systems,... 

The second approach to treating nondynamical correlation has an air of the ostrich about it ignore the spin symmetry of the wave function and use unrestricted Haxtree-Fock (UHF) theory as the single configuration description [7]. Since the UHF wave function comprises one spin-orbital for each electron, a molecular UHF wave function should dissociate to atomic UHF wave functions, for example. This is certainly not the case for spin-restricted Hartree-Fock (RHF) molecules and atoms in general. And there is an attractive simplicity about UHF — no active orbitals to identify, and so forth. However, where nondynamical correlation would be important in an RHF-based treatment, the UHF method will suffer from severe spin-contamination, while where nondynamical correlation is not important the RHF solution may be lower in energy than any broken-symmetry UHF solution, so potential curves and surfaces may have steps or kinks where the spin symmetry is broken in the UHF treatment. 

The many-electron wave function of a molecular system is taken as the antisymmetrized product of (pt, and for closed-shell systems it is convenient to represent it by a Slater determinant. Such an approach is known as the restricted Hartree-Fock (RHF) method and is the most widely used method in chemisorption calculations. Its principal drawback is the neglect of Coulomb electron correlation, which is of crucial importance for adequate treatment of chemical rearrangements with varying numbers of electron pairs. 

The few attempts at describing excited states in transition metal complexes within the Restricted Hartree Fock (RHF) formalism were rapidly abandoned due to the computational difficulties (convergence of the low-lying states in the open-shell formalism) and theoretical deficiencies (inherent lack of electronic correlation, inconsistent treatment of states of different multiplicities and d shell occupations). The simplest and most straightforward method to deal with correlation energy errors is the Configuration Interaction (Cl) approach where the single determinant HF wave function is extended to a wave function composed of a linear combination of many de-... 

In the field of quantum chemistry, theories on electronic motion states commencing with the Hartree-Fock method have intended to incorporate electron correlation more efficiently. However, due to the long computational times required, quantum chemistry calculations had been mostly restricted to trial applications by theoreticians until the 1980s. In the 1990s, density functional theory (DFT) appeared in quantum chemistry to resolve this situation. After this, DFT has become widespread, so that is now the main theory, which is used in more than 80% of quantum chemistry papers in 2014. 

In Tables 2 and 3, triplet doubly excited energies of 2s ns (n = 3,4,. .10) states and 3s ns (n = 4, 5,11) states of He, computed at the CSCF level, are presented. Calculations of Ref. [46] were restricted to only singly excited states. Therefore, we compare our CSCF calculations with accurate theoretical calculations based on a configuration interaction approach with the explicitly correlated HyUeraas basis set functions [48]. One can see that the accuracy of the CSCF calculations is improved when n increases. This observation is in agreement with Ref. [46]. whose authors pointed out that In those states where n 1, the electrons are spatially well separated and one might anticipate intuitively that they will be weakly correlated and that the Hartree-Fock method, which neglects such effects, may be an excellent approximation. ... 

If we except the Density Functional Theory and Coupled Clusters treatments (see, for example, reference [1] and references therein), the Configuration Interaction (Cl) and the Many-Body-Perturbation-Theory (MBPT) [2] approaches are the most widely-used methods to deal with the correlation problem in computational chemistry. The MBPT approach based on an HF-SCF (Hartree-Fock Self-Consistent Field) single reference taking RHF (Restricted Hartree-Fock) [3] or UHF (Unrestricted Hartree-Fock ) orbitals [4-6] has been particularly developed, at various order of perturbation n, leading to the widespread MPw or UMPw treatments when a Moller-Plesset (MP) partition of the electronic Hamiltonian is considered [7]. The implementation of such methods in various codes and the large distribution of some of them as black boxes make the MPn theories a common way for the non-specialist to tentatively include, with more or less relevancy, correlation effects in the calculations. 

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 electronic structure methods are based primarily on two basic approximations (1) Born-Oppenheimer approximation that separates the nuclear motion from the electronic motion, and (2) Independent Particle approximation that allows one to describe the total electronic wavefunction in the form of one electron wavefunc-tions i.e. a Slater determinant [26], Together with electron spin, this is known as the Hartree-Fock (HF) approximation. The HF method can be of three types restricted Hartree-Fock (RHF), unrestricted Hartree-Fock (UHF) and restricted open Hartree-Fock (ROHF). In the RHF method, which is used for the singlet spin system, the same orbital spatial function is used for both electronic spins (a and (3). In the UHF method, electrons with a and (3 spins have different orbital spatial functions. However, this kind of wavefunction treatment yields an error known as spin contamination. In the case of ROHF method, for an open shell system paired electron spins have the same orbital spatial function. One of the shortcomings of the HF method is neglect of explicit electron correlation. Electron correlation is mainly caused by the instantaneous interaction between electrons which is not treated in an explicit way in the HF method. Therefore, several physical phenomena can not be explained using the HF method, for example, the dissociation of molecules. The deficiency of the HF method (RHF) at the dissociation limit of molecules can be partly overcome in the UHF method. However, for a satisfactory result, a method with electron correlation is necessary. 

A somewhat modified MO LCAO scheme, without restriction on the identity of spin orbitals (p and

unrestricted Hartree-Fock (UHF) method and is usually used to treat open-shell systems (free radicals, triplet states, etc.). Electron correlation is partially taken into account in this method, and therfore it can be expected to be more efficient than the RHF method when applied to calculate potential energy surfaces of chemical rearrangements whose intermediate or final stages may involve the formation of free- or bi-radical structures. The potentialities of the UHF method are now under active study in organic reaction calculations. Also, it is successfully coming into use in chemisorption computations (6). 

The choice of reference space for MRCI calculations is a complex problem. First, a multieonfigurational Hartree-Fock (MCSCF) approach must be chosen. Common among these are the generalized valence-bond method (GVB) and the complete active space SCF (CASSCF) method. The latter actually involves a full Cl calculation in a subspace of the MO space—the active space. As a consequence of this full Cl, the number of CSFs can become large, and this can create very long Cl expansions if all the CASSCF CSFs are used as reference CSFs. This problem is exacerbated when it becomes necessary to correlate valence electrons in the Cl that were excluded from the CASSCF active space. It is very common to select reference CSFs, usually by their weight in the CASSCF wave function. Even more elaborate than the use of a CASSCF wave function as the reference space is the seeond-order Cl, in which the only restriction on the CSFs is that no more than two electrons occupy orbitals empty in the CASSCF wave function. Such expansions are usually too long for practical calculations, and they seldom produce results different from a CAS reference space MRCI. 

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