Variational configuration interaction

Variational Configuration Interaction Methods and Comparison with Perturbation Theory. 

The complete description of hydrogen bond and van der Waals interactions requires of course the inclusion of electron correlation effects however, almost always, a very useful starting point for subsequent refinements is represented by a Hartree-Fock description, which serves as the basis for both perturbation theory and variational configuration interaction approaches to the treatment of electron correlation. 

Pople JA, Seeger R, Krishan R (1979) Variational configuration interaction methods and comparison with perturbation theory. Int J Quantum Chem Symp 11 149-163... 

Popic, J. A., Seeger, R., and Krishnan, R., Variational configuration interaction methods and... 

Within the variational configuration interaction a limited Cl expansion is written as follows... 

Several routes are active in the search to recover that missing piece of energy (correlation energy), such as many body perturbation theory (MBPT), density functional theory (DFT) and the variational configuration interaction (Cl) method, often at the cost of nontrivial computational efforts. 

A particularly valuable addition to the arsenal of methods has been density functional theory (DFT), which aims to predict molecular properties with greater accuracy than Hartree-Fock calculations. Rather than directly integrating the electronic Schrodinger equation to get the AT-electron wavefunction (where N is the number of electrons in the molecule), DFT methods solve instead for the overall electron density, po. The many-electron wavefunction is a function of 3N coordinates Xi,yi,Zj for each electron i), but po depends on just 3 coordinates x, y, and z. Only this density function is needed to calculate the energy, and many other molecular properties. By skirting the need for the explicit, many-electron wavefunction, DFT methods provide a fast alternative route to predicting properties of molecules. Other methods make use of perturbation theory, variational configuration interaction, and extensions of valence bond theory. 

In an ab initio approach, the first step is to solve the Hartree-Fock problem using a suitable basis set. In the Hartree-Fock model, each electron experiences only the average potential created by the other electrons. In reality, the instantaneous position of each electron, however, depends on the instantaneous position of the other electrons but the Hartree-Fock model cannot account for this electron correlation. In order to obtain quantitative results, electron correlation (also referred to as dynamical correlation) should be included in the model and there are many methods available for accomplishing this task based on either variational or perturbation principles. The easiest method to understand conceptually is variational configuration interaction (CI). In this method, the electronic wavefunction is expanded in terms of configurations that are formed from excitations of electrons from the occupied orbitals in the Hartree-Fock wavefunction to the virtual orbitals. The expansion can be written as... 

Wlien first proposed, density llinctional theory was not widely accepted in the chemistry conununity. The theory is not rigorous in the sense that it is not clear how to improve the estimates for the ground-state energies. For wavefiinction-based methods, one can include more Slater detenuinants as in a configuration interaction approach. As the wavellmctions improve via the variational theorem, the energy is lowered. In density fiinctional theory, there is no... 

This is perhaps the easiest method to understand. It is based on the variational principle (Appendix B), analogous to the HF method. The trial wave function is written as a linear combination of determinants with the expansion coefficients determined by requiring that the energy should be a minimum (or at least stationary), a procedure known as Configuration Interaction (Cl). The MOs used for building the excited Slater determinants are taken from a Hartree-Fock calculation and held fixed. Subscripts S, D, T etc. indicate determinants which are singly, doubly, triply etc. excited relative to the... 

The idea of coupling variational and perturbational methods is nowadays gaining wider and wider acceptance in the quantum chemistry community. The background philosophy is to realize the best blend of a well-defined theoretical plateau provided by the application of the variational principle coupled to the computational efficiency of the perturbation techniques. [29-34]. In that sense, the aim of these approaches is to improve a limited Configuration Interaction (Cl) wavefunction by a perturbation treatment. 

As usual, the Hartree-Fock model can be corrected with perturbation theory (e.g., the Mpller-Plesset [MP] method29) and/or variational techniques (e.g., the configuration-interaction [Cl] method30) to account for electron-correlation effects. The electron density p(r) = N f P 2 d3 2... d3r can generally be expressed as... 

Starting from the normal mode approximation, one can introduce anharmonicity in different ways. Anharmonic perturbation theory [206] and local mode models [204] may be useful in some cases, where anharmonic effects are small or mostly diagonal. Vibrational self-consistent-field and configuration-interaction treatments [207, 208] can also be powerful and offer a hierarchy of approximation levels. Even more rigorous multidimensional treatments include variational calculations [209], diffusion quantum Monte Carlo, and time-dependent Hartree approaches [210]. 

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