32
F. E. Harris
process was greatly simplified by the development of recurrence formulas enabling
all four-body integrals to be obtained from a single starting value. This objective
was achieved, first for traditional Hylleraas integrals only, by Pachucki et al. [21],
and later, for general exponential correlation, by the present author [22].
4 Hylleraas Integrals
Integrals containing various combinations of r ij as factors (we call these potentialenergy integrals) will occur in atomic Hylleraas calculations; each integral can be
identified with a diagram that is formed by
(1) Introducing a vertex corresponding to each particle not at the origin of the
coordinate system (usually the nucleus);
(2) For each factor r ij or 1∕r ij in the integral, drawing a line in the diagram that
connects vertices i and j.
Any vertex not connected to other vertices by a line corresponds to a one-particle
integral that is easily evaluated. Diagrams containing closed loops (with three or
more vertices) are termed linked, and any diagram or part of a diagram that is not
contained in a closed loop is called unlinked. Unlinked integrals and parts can be
evaluated in closed form after introducing the Laplace expansion of each 1∕r ij [23]
and/or its generalization to the related quantity r ij [24]. An alternative to the Laplacetype expansion for unlinked integrations is to use a coordinate system in which the
unlinked r ij are coordinates. That approach has been followed by Ruiz in [25–28] and
in other papers. Integrations over the particles in closed loops can also be treated
using Laplace-type expansions, but such linked integrations lead to infinite series
that are usually evaluated numerically.
For atomic Hy-CI, the limitations in the occurrence of r ij factors cause the
potential-energy integrals to consist only of completely unlinked integrals involving four or fewer vertices, except for one three-vertex integral containing a linked
product of the form r ij r ik ∕r jk . This type of integral, often called a “triangle” integral,
was first discussed by Szasz [29].
While the general development for exponentially correlated wave functions in
principle provided closed analytic formulas for the Hylleraas triangle integrals,
those formulas were often more laborious to evaluate than expansions based on
Laplace-type formulas. For triangle integrals with general spherical harmonics, the
most utilized current approach is probably the Levin u-transformation convergenceacceleration scheme [30] used by Sims and Hagstrom [9].
The diagrams denoting integrals arising in Hy-CI are the same as those occurring
in its extension E-Hy-CI, the only difference being that each diagram line refers to a
factor of type r
p ij
ij
exp(−í µí»½ ij r ij ) instead of simply r ij .
F. E. Harris
process was greatly simplified by the development of recurrence formulas enabling
all four-body integrals to be obtained from a single starting value. This objective
was achieved, first for traditional Hylleraas integrals only, by Pachucki et al. [21],
and later, for general exponential correlation, by the present author [22].
4 Hylleraas Integrals
Integrals containing various combinations of r ij as factors (we call these potentialenergy integrals) will occur in atomic Hylleraas calculations; each integral can be
identified with a diagram that is formed by
(1) Introducing a vertex corresponding to each particle not at the origin of the
coordinate system (usually the nucleus);
(2) For each factor r ij or 1∕r ij in the integral, drawing a line in the diagram that
connects vertices i and j.
Any vertex not connected to other vertices by a line corresponds to a one-particle
integral that is easily evaluated. Diagrams containing closed loops (with three or
more vertices) are termed linked, and any diagram or part of a diagram that is not
contained in a closed loop is called unlinked. Unlinked integrals and parts can be
evaluated in closed form after introducing the Laplace expansion of each 1∕r ij [23]
and/or its generalization to the related quantity r ij [24]. An alternative to the Laplacetype expansion for unlinked integrations is to use a coordinate system in which the
unlinked r ij are coordinates. That approach has been followed by Ruiz in [25–28] and
in other papers. Integrations over the particles in closed loops can also be treated
using Laplace-type expansions, but such linked integrations lead to infinite series
that are usually evaluated numerically.
For atomic Hy-CI, the limitations in the occurrence of r ij factors cause the
potential-energy integrals to consist only of completely unlinked integrals involving four or fewer vertices, except for one three-vertex integral containing a linked
product of the form r ij r ik ∕r jk . This type of integral, often called a “triangle” integral,
was first discussed by Szasz [29].
While the general development for exponentially correlated wave functions in
principle provided closed analytic formulas for the Hylleraas triangle integrals,
those formulas were often more laborious to evaluate than expansions based on
Laplace-type formulas. For triangle integrals with general spherical harmonics, the
most utilized current approach is probably the Levin u-transformation convergenceacceleration scheme [30] used by Sims and Hagstrom [9].
The diagrams denoting integrals arising in Hy-CI are the same as those occurring
in its extension E-Hy-CI, the only difference being that each diagram line refers to a
factor of type r
p ij
ij
exp(−í µí»½ ij r ij ) instead of simply r ij .