Single-Configuration Self-Consistent Field Methods

SINGLE-CONFIGURATION SELF-CONSISTENT FIELD METHODS 

Let us consider a situation in which we choose to work with a one-configuration wavefunction for which the orbitals arc allowed to vary. This single configuration 0 may still consist of a linear combination of determinants whose (fixed) coefficients are determined by the space and spin symmetry imposed on 0 . The orbital variations may be described by exp(/A) and an optimal set of orbitals determined as in the previous section [by simply neglecting terms involving exp(iS)]. The second-order Eq. (2.33) then reads 

The one-electron Fock potential V is thus far arbitrary. Different choices for V correspond to different choices of the spin-orbitals / , and their corresponding orbital energies ej, since we require the and to obey 

Since the one-electron density matrix is diagonal for the single-configuration case considered here, we have 

The part of the Fock potential not defined through the Brillouin condition is often chosen on physical ground [e.g., to have the resultant orbital energies represent ionization potentials and electron afiinities (via Koopmans theorem)] (MeWeeney and Sutcliffe, 1976). For a reference state containing a set of occupied spin-orbitals that we denote by a, / , y, S and a set of unoccupied spin-orbitals denoted ni, i, p, t/, the Fock potential in Eq. (2.89) is defined by the BT only between occupied and unoccupied orbitals. From Eq. (2.89) we get 

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C. SINGLE-CONFIGURATION SELF-CONSISTENT FIELD METHODS... 

C. Single-Configuration Self-Consistent Field Methods... 

Many of the principles and techniques for calculations on atoms, described in section 6.2 of this chapter, can be applied to molecules. In atoms the electronic wave function was written as a determinant of one-electron atomic orbitals which contain the electrons these atomic orbitals could be represented by a range of different analytical expressions. We showed how the Hartree-Fock self-consistent-field methods could be applied to calculate the single determinantal best energy, and how configuration interaction calculations of the mixing of different determinantal wave functions could be performed to calculate the correlation energy. We will now see that these technques can be applied to the calculation of molecular wave functions, the atomic orbitals of section 6.2 being replaced by one-electron molecular orbitals, constructed as linear combinations of atomic orbitals (l.c.a.o. method). 

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. 

Another class of methods uses more than one Slater determinant as the reference wave function. The methods used to describe electron correlation within these calculations are similar in some ways to the methods listed above. These methods include multiconfigurational self-consistent field (MCSCF), multireference single and double configuration interaction (MRDCI), and /V-clcctron valence state perturbation theory (NEVPT) methods.5... 

The calculations are not all at exactly the same bond length R. The basis set is indicated after the slash in the method. R, L, C, and T are basis sets of Slater-type functions. The aug-cc-pVDZ and aug-cc-pVTZ basis sets [360] are composed of Gaussian functions. SCF stands for self-consistent-field MC, for multiconfiguration FO, for first-order Cl, for configuration interaction MR, for multireference MPn, for nth-order Mpller-Plesset perturbation theory and SDQ, for singles, doubles, and quadruples. 

The simplest truncation of the iV-particle space is to use only one IV-electron basis function — a single configuration. If the energy is optimized with respect to the MO coefficients of Eq. 1.12 we then have the Hartree-Fock self-consistent field (SCF) method. This special case is sufficiently important that the difference... 

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