Arrow diagram approach to nonorthogonal electron group functions in extended systems

Abstract

For nonorthogonal electron group functions, the arrow diagram (AD) method (Kantorovich and Zapol 1992 J. Chem. Phys. 96 8420; 1992 J. Chem. Phys. 96 8427) provides a convenient procedure for calculating matrix elements| O | of arbitrary symmetrical operators O. The total wavefunction of the system = A I I is represented as an antisymmetrized product of nonorthogonal many-electron group functions I of each group I in the system. For extended (e.g. infinite) systems the calculation of the mean value of an operator is ill defined, however, as it requires that each term of the diagram expansion be divided by the normalization integral S = | which is given by an AD expansion as well. In this work, we cast the mean value of a symmetrical operator in a form of an AD expansion which is a linear combination of linked ADs. By analysing an exactly solvable one-dimensional Hartree-Fock problem, we find that pre-factors, attached to every linked AD in the linear combination, can be expanded in a power series with respect to overlap. A general method of calculating these pre-factors in a form of a power series expansion with respect to overlap is suggested. This advance makes the AD theory applicable to extended systems, and allows one to calculate the mean value of an arbitrary symmetrical operator correct up to the desired order of overlap within the group function theory. In particular, we derive the effective Hamiltonian of a quantum cluster surrounded by overlapping group functions (e.g. bonds) in the environment region which is correct up to the second order with respect to overlap (an embedding problem).

Characterization of hydrogen dissociation over aluminium-doped zinc oxide using an efficient massively parallel framework for QM/MM calculations

“…This work along with a number of complementary studies (French et al 2001a(French et al ,b, 2003a(French et al ,b, 2004(French et al , 2008Bromley et al 2003;Catlow et al 2003Catlow et al , 2005aCatlow et al ,b, 2007Catlow et al , 2008Catlow et al , 2010Phala et al 2005;Sokol et al 2007) provided the first comprehensive computational survey of the atomistic and electronic mechanisms involved in methanol synthesis over the surfaces of pure ZnO and metal clusters supported on ZnO. Analogous methods by other groups have proved to be successful and accurate in the study of localized states and processes in a wide range of recent fundamental and applied studies (Nygren et al 1994;Mejias et al 1995;Llusar et al 1996;Rösch et al 1996;San Miguel et al 1998;de Carolis et al 1999;Kantorovich 1999Kantorovich , 2000aSushko et al 1999;Nasluzov et al 2001Nasluzov et al , 2010Karlsen et al 2003;Rivanenkov et al 2003;Danyliv & Kantorovich 2004;Mysovsky et al 2004;Seijo & Barandiarán 2004;Di Valentin et al 2006;Wang & Kantorovich 2006;Danyliv et al 2007;Giordano et al 2007;Sanz & Márquez 2007;Shor et al 2007Shor et al , 2009Kimmel et al 2008;Ramo et al 2008;Swerts et al 2008;…”

Arrow diagram theory for non-orthogonal electronic groups: the continued fractions method

“…It is also convenient to introduce a reciprocal manifold T r = [T ], i.e. an artificial system obtained by 'removing' from the entire system all groups comprising the set T [17,18]. Then, S(T r ) = S([T ]) ≡ S[T ] is obtained from equation ( 6) by retaining only ADs in which any of the groups belonging to T is not present.…”

“…It was argued in [17] that Ō can be represented as a sum of all linked (connected) ADs only, Ō = | Ô| c , thereby solving the problem. However, it has been found recently [18] that this representation for the mean value is only approximate, and one has to multiply each linked diagram by a numerical pre-factor. In general, these pre-factors differ from unity, and only when the overlap between different groups is insignificant (i.e.…”

“…To obtain the pre-factors in a general case of arbitrary overlap between the group functions, a method based on the power series expansion of the pre-factors with respect to overlap has been proposed recently [18]. The suggested approach is quite general and can be applied to any systems (including extended ones).…”

Treating periodic systems using embedding: Adams-Gilbert approach

“…[22][23][24] Although in this case a general formulation of the embedding method is still possible, 24 it is very complicated and has not yet been implemented in practice.…”