Matrix Elements for Explicitly-Correlated
Atomic Wave Functions
Frank E. Harris
Abstract We refer to atomic wave functions that contain the interelectron distances as “explicitly correlated”; we consider here situations in which an explicit
correlation factor r ij can occur as a power multiplying an orbital functional form
(a Hylleraas function) and/or in an exponent (producing exponential correlation).
Hylleraas functions in which each wave-function term contains at most one linear
r ij factor define a method known as Hylleraas-CI. This paper reviews the analytical
methods available for evaluating matrix elements involving exponentially-correlated
and Hylleraas wave functions; attention is then focused on computation of integrals needed for the kinetic energy. In contrast to orbital-product and exponentiallycorrelated wave functions, no general formulas have been developed by others to
relate the kinetic-energy integrals in Hylleraas-CI (or its recent extension by the
Nakatsuji group) to contiguous potential-energy matrix elements. The present paper
provides these missing formulas, obtaining them by using relevant properties of vector spherical harmonics. Validity of the formulas is confirmed by comparisons with
kinetic-energy integrals obtained in other ways.
1 Introduction
Ever since the first days of quantum mechanics investigators have sought methods for describing the electronic structures of atoms and molecules that are more
rapidly convergent than superposition-of-configurations (also called configurationinteraction) expansions of the electronic wave function in orbital products. Probably
It is a pleasure to present this work as part of a tribute to Professor Josef Paldus in celebration
of his eightieth birthday.
F. E. Harris ( ✉ )
Department of Physics, University of Utah, Salt Lake City, UT, USA
e-mail: harris@qtp.ufl.edu
F. E. Harris
Quantum Theory Project, University of Florida, Gainesville, FL, USA
© Springer International Publishing AG, part of Springer Nature 2018
Y. A. Wang et al. (eds.), Concepts, Methods and Applications of Quantum Systems
in Chemistry and Physics, Progress in Theoretical Chemistry and Physics 31,
https://doi.org/10.1007/978-3-319-74582-4_2
29
Atomic Wave Functions
Frank E. Harris
Abstract We refer to atomic wave functions that contain the interelectron distances as “explicitly correlated”; we consider here situations in which an explicit
correlation factor r ij can occur as a power multiplying an orbital functional form
(a Hylleraas function) and/or in an exponent (producing exponential correlation).
Hylleraas functions in which each wave-function term contains at most one linear
r ij factor define a method known as Hylleraas-CI. This paper reviews the analytical
methods available for evaluating matrix elements involving exponentially-correlated
and Hylleraas wave functions; attention is then focused on computation of integrals needed for the kinetic energy. In contrast to orbital-product and exponentiallycorrelated wave functions, no general formulas have been developed by others to
relate the kinetic-energy integrals in Hylleraas-CI (or its recent extension by the
Nakatsuji group) to contiguous potential-energy matrix elements. The present paper
provides these missing formulas, obtaining them by using relevant properties of vector spherical harmonics. Validity of the formulas is confirmed by comparisons with
kinetic-energy integrals obtained in other ways.
1 Introduction
Ever since the first days of quantum mechanics investigators have sought methods for describing the electronic structures of atoms and molecules that are more
rapidly convergent than superposition-of-configurations (also called configurationinteraction) expansions of the electronic wave function in orbital products. Probably
It is a pleasure to present this work as part of a tribute to Professor Josef Paldus in celebration
of his eightieth birthday.
F. E. Harris ( ✉ )
Department of Physics, University of Utah, Salt Lake City, UT, USA
e-mail: harris@qtp.ufl.edu
F. E. Harris
Quantum Theory Project, University of Florida, Gainesville, FL, USA
© Springer International Publishing AG, part of Springer Nature 2018
Y. A. Wang et al. (eds.), Concepts, Methods and Applications of Quantum Systems
in Chemistry and Physics, Progress in Theoretical Chemistry and Physics 31,
https://doi.org/10.1007/978-3-319-74582-4_2
29