Matrix Elements for Explicitly-Correlated Atomic Wave Functions

  • Frank E. HarrisEmail author
Conference paper
Part of the Progress in Theoretical Chemistry and Physics book series (PTCP, volume 31)


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 exponentially-correlated 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.



Completion of the numerical verifications referred to in this work involved significant consultations with Drs. María Belén Ruiz and James Sims. The author is pleased and grateful to acknowledge their assistance.


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of UtahSalt Lake CityUSA
  2. 2.Quantum Theory ProjectUniversity of FloridaGainesvilleUSA

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