Quantum chemistry Ritz method

Klahn, B. and Bingel, W. A. [19 77] The Convergence of the Rayleigh-Ritz Method in Quantum Chemistry , Theoretica Chimica Acta, 44, p. 9. 

B. Klahn, W.A. Bingel, The convergence of the Rayleigh-Ritz method in quantum chemistry. I. The criteria for convergence, Theor. Chim. Acta 44 (1977) 9-26. 

In practice, the Ritz variational method is used most often. One of the technical problems to be solved is the size of the basis set. Enormous progress in computation and software development now facilitates investigations that 20 years ago were absolutely beyond the imagination. The world record in quantum chemistry means a few billion expansion functions. To accomplish this, quantum chemists have had to invent some powerful methods of applied mathematics. 

The available methods in molecular electronic structure theory are illustrated in Figure 1 with a family tree of quantum chemistry labeled with the acronyms of some of the most often used methods. The variety is a bit daunting to newcomers, who might be cautioned by a comment by Levine If you learn enough abbreviations you can convince some people that you know quantum chemistry. Flowever, as for most areas of science, electronic structure theory looks much worse from the outside than from the inside. The tree has three main branches density functional theory (DFT), quantum Monte Carlo (QMC), and Rayleigh-Ritz variational theory (RRV). Each of these leads to additional branches. In addition there are a number of interbranch connections indicated by dotted lines. We give a brief description below of the DFT and RRV branches and their relation to QMC, which is described in sections follow-... 

Figure 1 A family tree of quantum chemistry DFT, density functional theory QMC, quantum Monte Carlo RRV, Rayleigh-Ritz variational theory X-a, X-alpha method KS, Kohn-Sham approach LDA, BP, B3LYP, density functional approximations VQMC, variational QMC DQMC, diffusion QMC FNQMC, fixed-node QMC PIQMC, path integral QMC EQMC, exact QMC HF, Hartree-Fock EC, explicitly correlated functions P, perturbational MP2, MP4, Maller-Plesset perturbational Cl, configuration interaction MRCI, multireference Cl FCI, full Cl CC, CCSD(T), coupled-cluster approaches. Other acronyms are defined in the text.

Such a method of finding approximate eigenvalues and eigenvectors of the Hamiltonian is known as the Ritz method and is frequently applied in quantum chemistry. 

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