Electron Correlation and Post-SCF Methods

As we have seen throughout this book, the Hartree-Fock method provides a reasonable model for a wide range of problems and molecular systems. However, Hartree-Fock theory also has limitations. They arise principally from the fact that Hartree-Fock theory does not include a full treatment of the effects of electron correlation the energy contributions arising from electrons interacting with one another. For systems and situations where such effects are important, Hartree-Fock results may not be satisfactory. The theory and methodology underlying electron correlation is discussed in Appendix A. 

A variety of theoretical methods have been developed which include some effects of electron correlation. Traditionally, such methods are referred to as post-SCF methods because they add correlation corrections to the basic Hartree-Fock model. As of this writing, there are many correlation methods available in Gaussian, including the following  

MoUer-Plesset perturbation theory energies through fifth-order (accessed via the keywords MP2, MP3, MP4, and MP5), optimizations via analytic gradients for second-order (MP2), third-order (MP3) and fourth-order (without triples MP4SDQ), and analytic frequencies for second-order (MP2). 

4- Quadratic Cl energies, optionally including triples and quadruples terms (QCISD, QCISDOl, and QCISDfTQI) and optimizations via analytic gradients for QCISD. 

4- Coupled Cluster methods, including doubles (energies and optimizations) or singles and doubles (energies only), and optional triples terms (CCD, CCSD, CCSD(T)). 

---

Density-Functional Theory. Transition metals pose a problem for classical quantum chemical methods like self-consistent field (SCF), perturbation theory, configuration interaction (Cl), and variations on these methods, because of the very large electron correlation. SCF underestimates binding substantially, and post-SCF methods are so expensive for transition metals that one can do a calculation only on models with few atoms. DFT on the other hand is relatively cheap it is about as expensive as SCF. Moreover, with the development of the generalized-gradient approximations it is also reasonably accurate. A large majority of quantum chemical... 

Methods based on Density Functional Theory also include some electron correlation effects (we ll consider them a bit later in this chapter). Of the traditional post-SCF methods, we ll be primarily using MP2, MP4, QCISD and QCISDfO in this work. 

The cc correlation energy is determined by the single and double amplitudes and the two-electron MO integrals. However, the recent progress in computational methods is largely influenced by DFT. One primary advantage of DFT over conventional HF-SCF procedures is that the former includes electron correlation fairly adequately (some times too much) at a fractional cost compared to a typical post-SCF (Cl, MBPT, or CC) calculation. 

Quantum mechanics (QM) can be further divided into ab initio and semiempiri-cal methods. The ab initio approach uses the Schrodinger equation as the starting point with post-perturbation calculation to solve electron correlation. Various approximations are made that the wave function can be described by some functional form. The functions used most often are a linear combination of Slater-type orbitals (STO), exp (-ax), or Gaussian-type orbitals (GTO), exp (-ax2). In general, ab initio calculations are iterative procedures based on self-consistent field (SCF) methods. Self-consistency is achieved by a procedure in which a set of orbitals is assumed and the electron-electron repulsion is calculated. This energy is then used to calculate a new set of orbitals, and these in turn are used to calculate a new repulsion energy. The process is continued until convergence occurs and self-consistency is achieved. 

In the HF method, the electron correlation is missing. For the post-HF methods, the correlation may be considered in different ways. In the configuration interaction (Cl) treatment, excited state determinants are constructed from the SCF solution, and the total wave function is written as a superposition of these determinants. The coefficients of the determinants in these... 

---