Electron correlation Slater determinant

In this chapter we consider the method of configuration interaction (Cl) for obtaining the correlation energy. Of all the approaches considered in this book. Cl is conceptually (but not computationally ) the simplest The basic idea is to diagonalize the JV-electron Hamiltonian in a basis of N-electron functions (Slater determinants). In other words, we represent the... 

The remarkable thing is that the HF model is so reliable for the calculation of very many molecular properties, as 1 will discuss in Chapters 16 and 17. But for many simple applications, a more advanced treatment of electron correlation is essential and in any case there are very many examples of spectroscopic states that caimot be represented as a single Slater determinant (and so cannot be treated using the standard HF model). In addition, the HF model can only treat the lowest-energy state of any given symmetry. 

The parameterization of MNDO/AM1/PM3 is performed by adjusting the constants involved in the different methods so that the results of HF calculations fit experimental data as closely as possible. This is in a sense wrong. We know that the HF method cannot give the correct result, even in the limit of an infinite basis set and without approximations. The HF results lack electron correlation, as will be discussed in Chapter 4, but the experimental data of course include such effects. This may be viewed as an advantage, the electron correlation effects are implicitly taken into account in the parameterization, and we need not perform complicated calculations to improve deficiencies in fhe HF procedure. However, it becomes problematic when the HF wave function cannot describe the system even qualitatively correctly, as for example with biradicals and excited states. Additional flexibility can be introduced in the trial wave function by adding more Slater determinants, for example by means of a Cl procedure (see Chapter 4 for details). But electron cori elation is then taken into account twice, once in the parameterization at the HF level, and once explicitly by the Cl calculation. 

The HF method determines the best one-determinant trial wave function (within the given basis set). It is therefore clear that in order to improve on HF results, the starting point must be a trial wave function which contains more than one Slater Determinant (SD). This also means that the mental picture of electrons residing in orbitals has to be abandoned, and the more fundamental property, the electron density, should be considered. As the HF solution usually gives 99% of the correct answer, electron correlation methods normally use the HF wave function as a starting point for improvements. 

As mentioned in Chapter 5, one can think of the expansion of an unknown MO in terms of basis functions as describing the MO function in the coordinate system of the basis functions. The multi-determinant wave function (4.1) can similarly be considered as describing the total wave function in a coordinate system of Slater determinants. The basis set determines the size of the one-electron basis (and thus limits the description of the one-electron functions, the MOs), while the number of determinants included determines the size of the many-electron basis (and thus limits the description of electron correlation). 

There are three main methods for calculating electron correlation Configuration Interaction (Cl), Many Body Perturbation Theory (MBPT) and Coupled Cluster (CC). A word of caution before we describe these methods in more details. The Slater determinants are composed of spin-MOs, but since the Hamilton operator is independent of spin, the spin dependence can be factored out. Furthermore, to facilitate notation, it is often assumed that the HF determinant is of the RHF type. Finally, many of the expressions below involve double summations over identical sets of functions. To ensure only the unique terms are included, one of the summation indices must be restricted. Alternatively, both indices can be allowed to run over all values, and the overcounting corrected by a factor of 1/2. Various combinations of these assumptions result in final expressions which differ by factors of 1 /2, 1/4 etc. from those given here. In the present book the MOs are always spin-MOs, and conversion of a restricted summation to an unrestricted is always noted explicitly. 

The total wavefunction, , is an antisymmetrized product of the one-electron functions i/q (a Slater determinant). The i/tj are called one-electron functions since they depend on the coordinates of only one electron this approximation is embedded in all MO methods. The effects that are missing when this approximation is used go under the general name of electron correlation. 

Among the many ways to go beyond the usual Restricted Hartree-Fock model in order to introduce some electronic correlation effects into the ground state of an electronic system, the Half-Projected Hartree-Fock scheme, (HPHF) proposed by Smeyers [1,2], has the merit of preserving a conceptual simplicity together with a relatively straigthforward determination. The wave-function is written as a DODS Slater determinant projected on the spin space with S quantum number even or odd. As a result, it takes the form of two DODS Slater determinants, in which all the spin functions are interchanged. The spinorbitals have complete flexibility, and should be determined from applying the variational principle to the projected determinant. 

In principle, the deficiencies of HF theory can be overcome by so-called correlated wavefunction or post-HF methods. In the majority of the available methods, the wavefunction is expanded in terms of many Slater-determinants instead of just one. One systematic recipe to choose such determinants is to perform single-, double-, triple-, etc. substitutions of occupied HF orbitals by virtual orbitals. Pictorially speaking, the electron correlation is implemented in this way by allowing the electrons to jump out of the HF sea into the virtual space in order... 

This can only be true if p2 (xj, Xj) = 0. In other words, this result tells us that the probability of finding two electrons with the same spin at the same point in space is exactly zero. Hence, electrons of like spin do not move independently from each other. It is important to realize that this kind of correlation is in no way connected to the charge of the electrons but is a direct consequence of the Pauli principle. It applies equally well to neutral fermions and - also this is very important to keep in mind - does not hold if the two electrons have different spin. This effect is known as exchange or Fermi correlation. As we will show below, this kind of correlation is included in the Hartree-Fock approach due to the antisymmetry of a Slater determinant and therefore has nothing to do with the correlation energy E discussed in the previous chapter. 

Accounting for electron correlation in a second step, via the mixing of a limited number of Slater determinants in the total wave function. Electron correlation is very important for correct treatment of interelectronic interactions and for a quantitative description of covalence effects and of the structure of multielec-tronic states. Accounting completely for the total electronic correlation is computationally extremely difficult, and is only possible for very small molecules, within a limited basis set. Formally, electron correlation can be divided into static, when all Slater determinants corresponding to all possible electron populations of frontier orbitals are considered, and dynamic correlation, which takes into account the effects of dynamical screening of interelectron interaction. 

Most of the commonly used electronic-structure methods are based upon Hartree-Fock theory, with electron correlation sometimes included in various ways (Slater, 1974). Typically one begins with a many-electron wave function comprised of one or several Slater determinants and takes the one-electron wave functions to be molecular orbitals (MO s) in the form of linear combinations of atomic orbitals (LCAO s) (An alternative approach, the generalized valence-bond method (see, for example, Schultz and Messmer, 1986), has been used in a few cases but has not been widely applied to defect problems.)... 

The difference between the Hartree-Fock energy and the exact solution of the Schrodinger equation (Figure 60), the so-called correlation energy, can be calculated approximately within the Hartree-Fock theory by the configuration interaction method (Cl) or by a perturbation theoretical approach (Mpller-Plesset perturbation calculation wth order, MPn). Within a Cl calculation the wave function is composed of a linear combination of different Slater determinants. Excited-state Slater determinants are then generated by exciting electrons from the filled SCF orbitals to the virtual ones ... 

It is possible to divide electron correlation as dynamic and nondynamic correlations. Dynamic correlation is associated with instant correlation between electrons occupying the same spatial orbitals and the nondynamic correlation is associated with the electrons avoiding each other by occupying different spatial orbitals. Thus, the ground state electronic wave function cannot be described with a single Slater determinant (Figure 3.3) and multiconfiguration self-consistent field (MCSCF) procedures are necessary to include dynamic electron correlation. 

However, in a large number of closed shell molecules, a single Slater determinant describes the ground state wave function fairly accurately. Even in such cases inclusion of excited state configuration results in substantial lowering of total electronic energy, and this is referred to as nondynamic electron correlation. 

The treatment of systems where non-dynamic correlation is critical is quite more complicated from a methodological point of view. As mentioned above, non-dynamic correlation is associated to the presence of neardegeneracies in the electronic ground state of the system, which means that there are Slater determinants with a weight similar to that of the HF solution in equation 4. The problem of non-dynamic correlation is usually treated successfully by the CASSCF method [43] for organic systems. This method introduces with high accuracy the correlation in the orbitals involved in the near degeneracy, which constitute the so called active space. The problem in... 

J) Morokuma 85> defined (AE3 — AE 1) as the sum of polarization and dispersion energy. If Wx and Fb are single Slater determinants a3 is usual in SCF calculations, no correlation effects are taken into account and hence this contribution actually represents electron polarization alone. 

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