Electron correlation methods excited Slater determinants

Establishing a hierarchy of rapidly converging, generally applicable, systematic approximations of exact electronic wave functions is the holy grail of electronic structure theory [1]. The basis of these approximations is the Hartree-Fock (HF) method, which defines a simple noncorrelated reference wave function consisting of a single Slater determinant (an antisymmetrized product of orbitals). To introduce electron correlation into the description, the wave function is expanded as a combination of the reference and excited Slater determinants obtained by promotion of one, two, or more electrons into vacant virtual orbitals. The approximate wave functions thus defined are characterized by the manner of the expansion (linear, nonlinear), the maximum excitation rank, and by the size of one-electron basis used to represent the orbitals. 

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 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 ... 

Instead, practical methods involve a subset of possible Slater determinants, especially those in which two electrons are moved from the orbitals they occupy in the HF wavefunction into empty orbitals. These doubly excited determinants provide a description of the physical effect missing in HF theory, correlation between the motions of different electrons. Single and triple excitations are also included in some correlated ab initio methods. Different methods use different techniques to decide which determinants to include, and all these methods are computationally more expensive than HF theory, in some cases considerably more. Single-reference correlated methods start from the HF wavefunction and include various excited determinants. Important methods in inorganic chemistry include Mpller-Plesset perturbation theory (MP2), coupled cluster theory with single and double excitations (CCSD), and a modified form of CCSD that also accounts approximately for triple excitations, CCSD(T). 

Fig. 10.6. Symbolic illustration of the principle of the Cl method with one Slater determinant o dominant in the ground state (this is a problem of the many electron wave functions so the picture caimot be understood literally). The purpose of this diagram is to emphasize a relatively small role of electronic correlation (more exactly, of what is known as the dynamical comlation i.e., correlation of electronic motion). The function jrci is a linear combination (the c coefficients) of the determinantal functions of different shapes in the marry-electron Hilbert space. The shaded regions correspond to the negative sign of the function the rxxlal surfaces of the added functions allow for the effective deformation of V o to have lower and lower average energy, (a) Since C is small in comparison to cq, the result of the addition of the first two terms is a slightly deformed V o. (b) Similarly, the additional excitations just make cosmetic changes in the function (although they may substantially affect the quantities calculated with it).

As shown in the previous section, electron correlation can be essentially incorporated by a method that linearly combines excited CSFs with the ground CSF. This method is called the configuration interaction (Cl) method (McWeeny 1992). Slater indicated the lack of Cl in the Hartree-Fock method, before the Hartree-Fock SCF method was developed. In his paper published in 1929, in which he described how the formulation of the Hartree-Fock method is derived with the Slater determinant, he pointed out the problem of exchanging only occupied orbital electrons in the Slater determinants of wavefunctions and suggested a Cl method... 

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