30
F. E. Harris
the earliest endeavor of this type was that of Hylleraas, whose study of the He atom
[1] used a wave function that included as a multiplicative factor the explicit appearance of the interelectron distance r 12 . Although it was many years before wave functions of this type came into widespread use (probably awaiting the availability of
digital computers), a few landmark studies using such wave functions (which we
identify as traditional Hylleraas functions) were soon carried out, including in particular a study of the hydrogen molecule by James and Coolidge [2], published in
1936. There followed in 1960 a further study of the hydrogen molecule ground state
by Kolos and Roothaan [3], in 1968 a detailed study of the lithium atom by Larsson
[4], and in 1994 an essentially quantitative computation of the ground state of the
hydrogen molecule by Kolos [5].
Attempts to apply Hylleraas methods to larger systems revealed that the occurrence of a wide variety of combinations of r ij factors (and higher powers thereof)
led to exceedingly complicated computations, and as early as 1971 it was proposed
by Sims and Hagstrom [6], and independently by Woźnicki [7], to consider configurations (wave function terms) that contained at most a single, linear r ij factor.
Methods based on wave functions of this type, now usually referred to as HylleraasCI (Hy-CI), were over time more fully developed and applied to a variety of atomic
problems. Representative work in this area is in [8–11].
An alternative to the Hy-CI development is the use of exponentially-correlated
wave functions. This type of wave function was proposed in the mid-1960s by Bonham [12, 13], but at that time calculations based on it seemed impractical. However,
it was practical to use exponentially-correlated Gaussian orbitals, and work in that
area has been pursued by Rychlewski et al. [14]. Calculations based on exponentiallycorrelated Slater-type orbitals finally became practical for small atomic systems after
publication of an extraordinary paper by Fromm and Hill [15].
The possibility of a practical extension to the Hy-CI method (identified by its
proposers as E-Hy-CI) has been examined by the Wang et al. [16], who developed
formulas for the “unlinked” integrals (defined in Sect. 4) that are encountered when
the single r ij of a Hy-CI wave function is generalized to a form of the generic type
r
p ij
ij
exp(−í µí»½ ij r ij ).
The present contribution reviews some aspects of electronic-structure computations by these Hylleraas-inspired methods, including a discussion of recentlydiscovered methods for simplifying the computation of the kinetic-energy matrix
elements in both Hy-CI and E-Hy-CI.
2 Wave Functions
Before incorporation into an antisymmetrized space-spin function, the spatial parts
of the exponentially-correlated three- and four-body spatial wave functions, written
in terms of their internal coordinates (in which one particle, often a nucleus, is at the
origin of the coordinate system), can take the form
F. E. Harris
the earliest endeavor of this type was that of Hylleraas, whose study of the He atom
[1] used a wave function that included as a multiplicative factor the explicit appearance of the interelectron distance r 12 . Although it was many years before wave functions of this type came into widespread use (probably awaiting the availability of
digital computers), a few landmark studies using such wave functions (which we
identify as traditional Hylleraas functions) were soon carried out, including in particular a study of the hydrogen molecule by James and Coolidge [2], published in
1936. There followed in 1960 a further study of the hydrogen molecule ground state
by Kolos and Roothaan [3], in 1968 a detailed study of the lithium atom by Larsson
[4], and in 1994 an essentially quantitative computation of the ground state of the
hydrogen molecule by Kolos [5].
Attempts to apply Hylleraas methods to larger systems revealed that the occurrence of a wide variety of combinations of r ij factors (and higher powers thereof)
led to exceedingly complicated computations, and as early as 1971 it was proposed
by Sims and Hagstrom [6], and independently by Woźnicki [7], to consider configurations (wave function terms) that contained at most a single, linear r ij factor.
Methods based on wave functions of this type, now usually referred to as HylleraasCI (Hy-CI), were over time more fully developed and applied to a variety of atomic
problems. Representative work in this area is in [8–11].
An alternative to the Hy-CI development is the use of exponentially-correlated
wave functions. This type of wave function was proposed in the mid-1960s by Bonham [12, 13], but at that time calculations based on it seemed impractical. However,
it was practical to use exponentially-correlated Gaussian orbitals, and work in that
area has been pursued by Rychlewski et al. [14]. Calculations based on exponentiallycorrelated Slater-type orbitals finally became practical for small atomic systems after
publication of an extraordinary paper by Fromm and Hill [15].
The possibility of a practical extension to the Hy-CI method (identified by its
proposers as E-Hy-CI) has been examined by the Wang et al. [16], who developed
formulas for the “unlinked” integrals (defined in Sect. 4) that are encountered when
the single r ij of a Hy-CI wave function is generalized to a form of the generic type
r
p ij
ij
exp(−í µí»½ ij r ij ).
The present contribution reviews some aspects of electronic-structure computations by these Hylleraas-inspired methods, including a discussion of recentlydiscovered methods for simplifying the computation of the kinetic-energy matrix
elements in both Hy-CI and E-Hy-CI.
2 Wave Functions
Before incorporation into an antisymmetrized space-spin function, the spatial parts
of the exponentially-correlated three- and four-body spatial wave functions, written
in terms of their internal coordinates (in which one particle, often a nucleus, is at the
origin of the coordinate system), can take the form