**Next-Generation, Explicitly-Correlated Electronic Structure Methods**

## Wesley D. Allen and Henry F. Schaefer III

#### Center for Computational Chemistry,

University of Georgia, Athens, GA 30622

Summary

***The quantum chemical wavefunction methods in current use for
electronic structure computations have accuracy limits arising from
their fundamental inability to properly describe electron-electron cusp
behavior. Our work is devoted to the development and dissemination of
rigorous, "explicitly-correlated" methods that solve the electron cusp
problem and deliver potential energy hypersurfaces for chemical systems
of exceptional accuracy (±0.1 kcal mol^--1^) in terascale computing
environments.***

The revolutions in physics in the early decades of the past century
provided essentially exact, predictive theories for all chemical
properties and processes. However, the mathematical intractability of
the Schrödinger and Dirac equations of molecular quantum mechanics
prevented the computation of accurate numerical solutions for chemical
systems until the past two decades. Carried by a synergy between
breathtaking improvements in computer hardware and groundbreaking
innovations in mathematical formulations and algorithms, computational
quantum chemistry has established a burgeoning presence in chemical
research, owing both to the elucidation of complex phenomena it provides
and its cost effectiveness vis-à-vis experimental methods.

Virtually all *ab initio* methods of quantum chemistry in some way
involve the orbital approximation, wherein the many-electron molecular
wavefunction (Ψ) is represented as superpositions of antisymmetrized
products of single-particle functions. Such methods, frequently aided by
empirical corrections, have been quite successful in approaching or
reaching "chemical accuracy" (1-2 kcal mol^--1^) for relative energies
of modestly-sized molecules. Moreover, the elementary bonding principles
of interpretive chemistry are built on longstanding orbital concepts.
However, order of magnitude improvements to "subchemical accuracy"
(0.1-0.2 kcal mol^--1^) are needed to expand the world's thermochemical
and spectroscopic database into inaccessible realms, establish its
building blocks beyond dispute, and supersede more expensive
experimental approaches.

Theoretical methods built on the orbital approximation have fundamental
accuracy limits and fall short of the subchemical accuracy target for
all but the simplest molecules. The essential flaw in these methods is
their inability to correctly describe the mathematical cusp behavior of
many-body wavefunctions in regions of close electron proximity, leading
to incomplete accounting of instantaneous, short-range correlation among
electrons. Achieving the subchemical accuracy dream will require
*next-generation* methodologies that solve the electron cusp problem by
explicitly incorporating interelectronic variables (*r*~12~) into
molecular wavefunctions, or equivalently by constructing Ψ from
two-particle components (geminals). Figure 1 is a scheme depicting the
merits of such "explicitly correlated" methods over conventional
orbital-based descriptions.

![](media/image1.png){width="2.998611111111111in"
height="2.6479166666666667in"}

*Figure 1. Superiority of explicitly correlated (R12) methods in the
cusp regions of many-electron wavefunctions.*

Our SciDAC research has focused specifically on the development of
"linear R12" methods and competing schemes for achieving complete basis
set limits of electronic structure theory. The computer code developed
in our group for evaluating the nonstandard two-electron integrals \[1\]
required by R12 theories employs a sophisticated architecture effective
for functions of arbitrarily high angular momentum. This
state-of-the-art integrals software has become part of an existing
massively parallel quantum chemical suite to take full advantage of
terascale computing facilities. Recently, we have executed novel
computational partial wave analyses \[2,3\] of molecular correlation
energies to unprecedented levels to better reveal the asymptotic
structure of both R12 and conventional electronic wavefunctions. This
work entailed the construction of new, generally applicable one-particle
basis sets with angular momentum through *l* = 7\
(*k* functions)! Recent chemical applications of our R12 research have
included studies to pinpoint the problematic barriers to linearity of
H~2~O and SiC~2~, and the heats of formation of NCO and \[H,N,C,O\]
isomers.\[2-4\]

Continuing work within our group will pursue the development of new
types of basis sets for explicitly-correlated theories, particularly
non-Gaussian forms especially designed for the evaluation of three-and
four-electron integrals, a daunting task which has been an obstacle to
achieving the full promise of next-generation methods. We also plan
continued work on R12 theories for *open-shell* species, both within
less complicated unrestricted (UHF) formalisms and more difficult
spin-adapted (ROHF) approaches. To reach the *ab initio* limit of
molecular quantum mechanics most efficiently, we conjoin lower-order,
explicitly-correlated methods with high-order conventional correlation
treatments via the focal-point scheme\[5\] of Allen and co-workers. Here
we are involved in SciDAC collaborations with the Piecuch group, experts
in high-order, renormalized coupled-cluster approaches for closely
approximating exact correlation (FCI) limits.

\[1\] E. F. Valeev and H. F. Schaefer, *J. Chem. Phys.* **113**, 3990
(2000).\
\[2\] J. P. Kenny, W. D. Allen, and H. F. Schaefer, *J. Chem. Phys*.
**118**, 7353 (2003). \[3\] E. F. Valeev, W. D. Allen, R. Hernandez, C.
D. Sherrill, and H. F. Schaefer, *J. Chem. Phys.* **118**, 8594 (2003).\
\[4\] M. Schuurman, S. Muir, W. D. Allen, and H. F. Schaefer, *J. Chem.
Phys*. **120**, in press (2004).\
\[5\] See, for example, A. G. Császár, W. D. Allen, and H. F. Schaefer,
*J. Chem. Phys.* **108**, 9751(1998).

**For further information on this subject contact:**

Dr. Wesley D. Allen, Center for Computational Chemistry, University of
Georgia; wdallen \@ccqc.uga.edu or www.ccc.uga.edu.