Per-Olov Löwdin
Rev. Mod. Phys. 34, 80 – Published 1 January 1962
ABSTRACT
INTRODUCTION
the theory of the electronic structure of atoms,
~ - molecules, and solid state, the exchange and
correlation phenomena play a fundamental role in
studying cohesive and elastic properties or electric and
magnetic behavior. Conventionally it seems as if the
exchange effects would be of particular importance in
ferromagnetic and antiferromagnetic materials and the
correlation effects in conductors and semiconductors,
but actually the two phenomena are closely coupled
together in all electronic systems.
https://www.physik.uzh.ch/lectures/electroncorrelations/Loewdin_correlation.pdf
https://sci-hub.se/https://doi.org/10.1103/RevModPhys.34.80
The exchange integral J was introduced by
Heisenberg, ' but since this concept has been strongly
criticized by several authors and particularly by Slater, '
we start by a redehnition of this important quantity.
In addition to exchange, another phenomenon is of
fundamental importance in many-electron theory,
namely the interelectronic correlation. Because of their
mutual coulomb repulsion, two electrons try always
to avoid each other to keep the energy as low as possible
which leads to a certain "correlation" between their
motions. In this connection, there is actually an
increase in the kinetic energy of the two electrons
because of the more complicated motions they have to
perform, but this is compensated by a still larger
decrease in the coulomb repulsion energy; the balance
is regulated by the virial theorem, (T)= —rs(V).
In the Hartree-Fock scheme, the essential part
of the correlation error is hence associated with electron
pairs having antiparallel spins. The corretafioe energy is not a physical quantity but
the measure of the energy error due to the neglect of
correlation in a certain approximation. One is particularly interested in the correlation error in the HartreeFock (HF) scheme corresponding to the band theory
in solid-state physics, and it seems convenient to use
the definition:
Correlation energy= E, „,—EHF,
where E,„„fis the true eigenvalue for the Hamiltonian
under consideration.
For the H2 molecule, the correlation energy is —1.06
ev corresponding to errors of +1.06 ev in the kinetic
energy and —2.12 ev in the potential energy. Since
1 ev= 23.07 kcal/mole, these quantities are appreciable
even from a chemical point of view. For the series of
He-like ions, the correlation energy is approximately
constant —1.2 ev, varying from —1.142 for He to —1.197 for C4+. For Be, the correlation energy is —2.4 ev but, for the series of Be-like ions, it seems to
vary linearly with Z depending on the degeneracy of
the 2s and 2p orbitals.
"
For the ¹like ions, it is
probably fairly constant of the order of magnitude —11 ev."
For the alkali metals I i, Na, and K, signer
"A. Froman, Phys. Rev. 112, 870 (1958).
has given the values —1.89, —1.73, and —1.58 ev,
respectively.
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