Beyond the Single-Configuration Approximation

Until this point, the single-configuration approximation for each electronic state has been assumed/ The C2 b3E X E+ interaction is one example where it is necessary to represent one of the interacting states by a mixture of two configurations. Another example involving predissociation is discussed in Section 7.11.1. 

For many years, the ground state of the C2 molecule had been assumed to be a 3n state, in part because the relative energies of the manifolds of singlet and triplet states were unknown. The discovery of perturbations between the X E+ and b3E states (Ballik and Ramsay, 1963) allowed the relative energies of singlet and triplet states to be determined. 

Configuration mixing is often necessary to account for spin-orbit interactions between states with configurations which differ by more than one orbital (for 

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Another desirable aspect of using the TDA and RPA approaches is that they both use a common set of molecular orbitals, which aids both in developing qualitative interpretations of the excitation process and also in calculating properties such as transition moments. The latter depends on (i j r i i )p, where r = is the dipole operator. It is easy to evaluate such a one-electron property provided i / and are described in terms of the same orthonormal orbital set. When different orbitals are used in and l —typically to get the best possible solution for both states—the resultant nonorthogonality causes a number of complications. This is particularly true when an entire spectrum of electronic states is the objective and all transition moments are required. Nevertheless, all the methods discussed so far neglect electron correlation effects, and one must go beyond the single configuration approximation if quantitative accuracy is to be achieved. 

L. Smentek and B. G. Wybourne, Relativistic/ <—y f transitions in crystal fields It. Beyond the single-configuration approximation. Journal of Physics B, 34, 625-630 (2001). 

HF is the simplest of the ab initio methods, named after the fact that they provide approximate solutions to the electronic Schrodinger equation without the use of empirical parameters. More accurate, correlated, ab initio methods use an approximate form for the wavefunction that goes beyond the single Slater determinant used in HF theory, in that the wavefunction is approximated instead as a combination or mixture of several Slater determinants corresponding to different occupation patterns (or configurations) of the electrons in the molecular orbitals. When an optimum mixture of all possible configurations of the electrons is used, one obtains an exact solution to the electronic Schrodinger equation. This is, however, not computationally tractable. 

One of the original approximate methods is the wavefunction-theory-based Hartree-Fock (HF) method [40]. The HF method is a single determinant method that does not include any correlation interactions between the electrons, and as such has limited accuracy [41, 42]. Higher level wavefunction-based methods such as coupled cluster [43 5], configuration interaction [40,46,47], and complete active space [48-50] methods include multiple determinants to incorporate some of the electron-electron correlation. Methods based on perturbation theory, such as second order Mpller-Plesset perturbation theory [51], go beyond the HF method by perturbatively adding electron correlation. These correlated wavefunction-based methods have well-defined ways in which they approach the exact solution to the Schrodinger equation and thus have the potential to be extremely accurate, but this accuracy comes at a price [52]. 

Contents Experimental Basis of Quantum Theory. -Vector Spaces and Linear Transformations. - Matrix Theory. -- Postulates of Quantum Mechanics and Initial Considerations. - One-Dimensional Model Problems. - Angular Momentum. - The Hydrogen Atom, Rigid, Rotor, and the H2 Molecule. - The Molecular Hamiltonian. - Approximation Methods for Stationary States. - General Considerations for Many-Electron Systems. - Calculational Techniques for Many-Electron Systems Using Single Configurations. - Beyond Hartree-Fock Theory. 

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