By I. Prigogine, Stuart A. Rice
Advances in Chemical Physics covers contemporary advances on the leading edge of analysis relative to chemical physics. The sequence, Advances in Chemical Physics, offers a discussion board for serious, authoritative reviews of advances in each region of the self-discipline.
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Extra info for Advances in Chemical Physics, Vol.125 (Wiley 2003)
The present basis set used for the finite-size scaling calculations is built up on the perimetric coordinates that are only suitable for 50 sabre kais and pablo serra the three-body systems, thus restricting the basis set’s extension to treat larger molecules. Several types of Gaussian basis sets have been tested. The first one has the following general form: É ¼ ð1 þ P12 Þ M X 2 2 2 eÀam r1 Àbm r2 Àgm r12 ð95Þ m¼1 where M is the expansion length, P12 is the exchange operator and am , bm and gm are the variational parameters.
34 sabre kais and pablo serra To carry out the FSS procedure, one has to choose a convenient basis set to obtain the two lowest eigenvalues and eigenvectors of the finite Hamiltonian matrix. As basis functions for the FSS procedure, we choose the following basis set functions [104–106]: 1 k x1 ;~ x2 Þ ¼ pﬃﬃﬃ r1i r2j eÀðgr1 þdr2 Þ þ r1j r2i eÀðdr1 þgr2 Þ r12 F‘ ðy12 ; XÞ Èijk;‘ ð~ 2 ð78Þ where g and d are fixed parameters, we have found numerically that g ¼ 2 and d ¼ 0:15 are a good choice for the ground state , r12 is the interelectronic distance, and F‘ ðy12 ; XÞ is a suitable function of the angle between the positions of the two electrons y12 and the Euler angles X ¼ ðÂ; È; ÉÞ.
10 1/N Figure 5. aðNÞ as a function of 1=N for the ground state of the Yukawa potential. The extrapolated values are shown by dots. Because the same argument of regularity holds for the derivatives of the truncated expectation values, we have that qm hOiðNÞ qlm $ N ÀðmO ÀmÞ=n ð73Þ l¼lc hOiðNÞ is analytical in l; then using Eq. (73), the Taylor expansion could be written as  hOiðNÞ ðlÞ $ N ÀmO =n GO ðN 1=n ðl À lc ÞÞ ð74Þ where GO is an analytical function of its argument. This equivalent expression for the scaling of a given expectation value has a correct form to study the data collapse in order to test FSS hypothesis in quantum few-body Hamiltonians.
Advances in Chemical Physics, Vol.125 (Wiley 2003) by I. Prigogine, Stuart A. Rice