Integral equation theories many coupled

As we stated above, SAPT is formulated in a top down manner. Eq. (6) then forms the top going down to workable equations, one is forced to introduce a multitude of approximations. In practice, i is restricted to the values 1 and 2 interactions of first and second order in Different truncation levels for j + k are applied, depending on the importance of the term (and the degree of complexity of the formula). Working out the equations to the level of one- and two-electron integrals is a far from trivial job. This has been done in a long series of papers that use techniques from coupled cluster theory and many-body PT see Refs. [147,148] for references to this work and a concise summary of the formulas resulting from it. 

Using all the above, Cizek presents the explicit, spin orbital and spin-adapted CC doubles equations (CCD) i.e., T = T2 (then called coupled-pair many-electron theory) in terms of one- and two-electron integrals over an orthogonal basis set. Assisted by Joe Paldus with some computations, he also reports some CCD results for N2, which though limited to only ti to Ug excitations, uses ab initio integrals in an Slater-type-orbital basis. He also does the full Cl calculation to assess convergence, a tool widely used in Cizek s and Paldus work and by most of us, today. He also reports results for the minimum-basis TT-electron approximation to benzene. 

Theoretical treatments of PCET reactions typically have equation (1.2) as a conceptual starting point. In Hammes-Schiffer s multistate continuum theory for PCET, the pre-exponential factor includes both electronic coupling and vibrational overlaps, and the rate is a sum over initial and final vibrational states integrated over a range of proton-donor acceptor distances. This theory has been elegantly applied to understand the intimate details of a variety of PCET reactions, but many of its parameters are essentially unattainable experimentally. 

The most simple and instructive way to study the main elements of F12 theory is to start with second-order Moller-Plesset perturbation theory, MP2. We will show in the following how the conventional, orbital-expansion based theory is supplemented by additional geminal functions and how the working equations change. We will then discuss how expensive many-electron integrals are avoided by reducing them to products of two-electron integrals, and finally, how F12 theory is transferred to second quantization, which in particular is required for the formulation of coupled-cluster theory with F12 terms. 

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