Harris variational method

Nesbet, R.K. (1968). Analysis of the Harris variational method in scattering theory, Phys. Rev. 175, 134-142. 

Several other calculations of the first few partial-wave phase shifts for positron-helium scattering have been carried out using a variety of approximation methods in all cases, however, rather simple uncorrelated helium wave functions have been used. Drachman (1966a, 1968) and McEachran et al. (1977) used the polarized-orbital method, whereas Ho and Fraser (1976) used a formulation based on the static approximation, with the addition of several short-range correlation terms, to determine the s-wave phase shifts only. The only other elaborate variational calculations of the s-wave phase shift were made by Houston and Drachman (1971), who employed the Harris method with a trial wave function similar to that used by Humberston (1973, 1974), see equation (3.77), and with the same helium model HI. Their results were slightly less positive than Humberston s HI values, and are therefore probably less... 

The amphoteric nature of wool was demonstrated in the early studies of Speakman and Hirst (1933), Elod (1933), and in particular by the complete acid-base titration curve obtained by Speakman and Stott (1934). Even earlier attempts had been made to determine the isoelectric point of wool by the methods indicated in Table XXIII. Some variation in the isoelectric point is to be expected because the pH at which the net charge, including bound ions, is zero depends on the nature and concentrations of ions in the aqueous environment. For example, Sookne and Harris (1939) have shown that the early electrophoretic value of Harris (1932) was affected by the absorption of phthalate ions from the buffer solutions. With acetate buffer they obtained values of 4.2 and 4.5 for powdered wool and cortical cells, respectively. The isoelectric points listed in Table XXIII are... 

As noted above, these authors also proved the general validity, for the Rayleigh problem, of the principle of exchange of stabilities. Further, by formulating the problem in terms of a variational principle, Pellew and South-well devised a technique which led to a very rapid and accurate approximation for the critical Rayleigh number. Later, a second variational principle was presented by Chandrasekhar (C3). A review by Reid and Harris (R2) also includes other approximate methods for handling the Benard problem with solid boundaries. 

Second, canonical transformation methods may be employed to diagonalize the system-bath Hamiltonian partially by a transformation to new ( dressed ) coordinates. Such methods have been in wide use in solid-state physics for some time, and a large repertoire of transformations for different situations has been developed [101]. In the case of a linearly coupled harmonic bath, the natural transformation is to adopt coordinates in which the oscillators are displaced adiabatically as a function of the system coordinates. This approach, known in solid-state physics as the small-polaron transformation [102], has been used widely and successfully in many contexts. In particular, Harris and Silbey demonstrated that many important features of the spin-boson system can be captured analytically using a variationally optimized small-polaron transformation [45-47]. As we show below, the effectiveness of this technique can be broadened considerably when a collective bath coordinate is first included in the system directly. 

Isopiestic Method.— The isopiestic method can be used to determine provided only one of the components is volatile. The unknown solution and reference solution containing the same volatile component but different involatile components are placed in dishes on a metal block in an airtight enclosure. The volatile component distils between the two solutions until their compositions are such that they have the same chemical potential of volatile component. After equilibration the two solutions are analysed. As the isopiestic method is a comparative method it is necessary to know the variation of vapour pressure with composition for the reference solution. The technique has the advantage that no pressure measurements are required. The method has been frequently used in the study of electrolytes and involatile non-electrolytes in aqueous solutions. Corneliussen et a/. have used the method to study polymer solutions. An apparatus suitable for measurements with organic mixtures has been described by Harris and Dunlop. The method is suitable only when there is no possibility of distillation between the involatile component in the reference solution and the involatile component in the unknown solution. 

Approaches which consider one state at a time are often referred to as one-state or state-selective or single-root . They were first proposed in the late 1970s. A paper published by Harris [113] in 1977, entitled Coupled cluster methods for excited states, first introduced the state-selective approach. Four papers which were published in 1978 and 1979 advancing the state-selective approach parts 6 and 7 of a series of papers entitled Correlation problems in atomic and molecular systems part 6 entitled Coupled cluster approach to open-shell systems by Paldus et al. [114] and part 7 with the title Application of the open-shell coupled cluster approach to simple TT-electron model systems by Saute, Paldus and Cfzek [115], and two papers by Nakatsuji and Hirao on the Cluster expansion of wavefunction, the first paper [116] having the subtitle Symmetry-adapted-cluster expansion, its variational determination, and extension of open-shell theory and the second paper [117] having the subtitle Pseudo-orbital theory based on sac expansion and its application to spin-density of open-shell systems. 

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