1 3
Theor Chem Acc (2015) 134:86
DOI 10.1007/s00214-015-1682-y
REGULAR ARTICLE
Hermitian “chemical” Hamiltonian: an alternative ab initio
method
I. Mayer
1
Received: 21 April 2015 / Accepted: 6 June 2015 / Published online: 4 July 2015
© Springer-Verlag Berlin Heidelberg 2015
1 Introduction
Three decades ago the present author studied the apparent
contradiction that one has one- and two-electron integrals up
to four-center ones in the ab initio quantum chemical theory,
while the empirical chemical facts indicate that the intramolecular interactions are basically of atomic and diatomic
character [ 1 ]. In chemical practice, one needs not to assume
the existence of any primary three- and four-atom effects
in a molecule, while the presence of the three- and fourcenter integrals in the theory would indicate the opposite.
The problem was approached by introducing a “projected”
integral approximation scheme [ 1 ], permitting to present
each three- and four-center integral as a sum of a leading
“physical” term containing only one- and two-center integrals, and a fi nite basis correction to it (this integral approximation scheme has some resemblance with Ruedenberg
classical proposition [ 2 ]). Combined with a special “mixed”
second quantized formalism for non-orthogonal basis orbitals, this permitted to present the fi nite basis version of the
Born–Oppenheimer Hamiltonian as a sum of atomic and
diatomic “physical” terms and fi nite basis correction ones.
An interesting theoretical property of these atomic Hamiltonians was that—despite the interatomic overlap of the basis
functions—the antisymmetrized product of atomic full CI
solutions was an eigenfunction of the respective sum of the
atomic Hamiltonians, and the eigenvalues were the sums
of the atomic full CI energies [ 1 ] (no analogous property
could be proved, however, for the Hartree–Fock wave functions). The diatomic terms of the Hamiltonian have been
also decomposed into terms of different physical nature, like
electrostatic and overlap effects.
These properties motivated us to call this formalism as
“Chemical Hamiltonian Approach” (CHA). The disadvantage of the formalism was the non-Hermiticity of the
Abstract Some previous results of the present author
are combined in order to develop a Hermitian version of
the “Chemical Hamiltonian Approach.” In this framework
the second quantized Born–Oppenheimer Hamiltonian is
decomposed into one- and two-center components, if some
fi nite basis corrections are omitted. (No changes are introduced into the one- and two-center integrals, while projective expansions are used for the three- and four-center ones,
which become exact only in the limit of complete basis
sets.) The total molecular energy calculated with this Hamiltonian can then presented as a sum of the intraatomic and
diatomic energy terms which were introduced in our previous “chemical energy component analysis” scheme. The
corresponding modifi ed Hartree–Fock–Roothaan equations
are also derived; they do not contain any three- and fourcenter integrals, while the non-empirical character of the
theory is conserved. This scheme may be useful also as a
“layer” in approaches like ONIOM.
Keywords Chemical Hamiltonian Approach · Alternative
non-empirical SCF formalism · Second quantized
Hamiltonian · Excluding three- and four-center integrals ·
Projective integral approximation
Published as part of the special collection of articles “Festschrift
in honour of Péter R. Surjan.”
Dedicated to the 60th birthday of Professor Péter R. Surján.
* I. Mayer
mayer@chemres.hu
1
Research Centre for Natural Sciences , Hungarian Academy
of Sciences , P.O.Box 286 , Budapest 1519 , Hungary
31
Reprinted from the journal
Theor Chem Acc (2015) 134:86
DOI 10.1007/s00214-015-1682-y
REGULAR ARTICLE
Hermitian “chemical” Hamiltonian: an alternative ab initio
method
I. Mayer
1
Received: 21 April 2015 / Accepted: 6 June 2015 / Published online: 4 July 2015
© Springer-Verlag Berlin Heidelberg 2015
1 Introduction
Three decades ago the present author studied the apparent
contradiction that one has one- and two-electron integrals up
to four-center ones in the ab initio quantum chemical theory,
while the empirical chemical facts indicate that the intramolecular interactions are basically of atomic and diatomic
character [ 1 ]. In chemical practice, one needs not to assume
the existence of any primary three- and four-atom effects
in a molecule, while the presence of the three- and fourcenter integrals in the theory would indicate the opposite.
The problem was approached by introducing a “projected”
integral approximation scheme [ 1 ], permitting to present
each three- and four-center integral as a sum of a leading
“physical” term containing only one- and two-center integrals, and a fi nite basis correction to it (this integral approximation scheme has some resemblance with Ruedenberg
classical proposition [ 2 ]). Combined with a special “mixed”
second quantized formalism for non-orthogonal basis orbitals, this permitted to present the fi nite basis version of the
Born–Oppenheimer Hamiltonian as a sum of atomic and
diatomic “physical” terms and fi nite basis correction ones.
An interesting theoretical property of these atomic Hamiltonians was that—despite the interatomic overlap of the basis
functions—the antisymmetrized product of atomic full CI
solutions was an eigenfunction of the respective sum of the
atomic Hamiltonians, and the eigenvalues were the sums
of the atomic full CI energies [ 1 ] (no analogous property
could be proved, however, for the Hartree–Fock wave functions). The diatomic terms of the Hamiltonian have been
also decomposed into terms of different physical nature, like
electrostatic and overlap effects.
These properties motivated us to call this formalism as
“Chemical Hamiltonian Approach” (CHA). The disadvantage of the formalism was the non-Hermiticity of the
Abstract Some previous results of the present author
are combined in order to develop a Hermitian version of
the “Chemical Hamiltonian Approach.” In this framework
the second quantized Born–Oppenheimer Hamiltonian is
decomposed into one- and two-center components, if some
fi nite basis corrections are omitted. (No changes are introduced into the one- and two-center integrals, while projective expansions are used for the three- and four-center ones,
which become exact only in the limit of complete basis
sets.) The total molecular energy calculated with this Hamiltonian can then presented as a sum of the intraatomic and
diatomic energy terms which were introduced in our previous “chemical energy component analysis” scheme. The
corresponding modifi ed Hartree–Fock–Roothaan equations
are also derived; they do not contain any three- and fourcenter integrals, while the non-empirical character of the
theory is conserved. This scheme may be useful also as a
“layer” in approaches like ONIOM.
Keywords Chemical Hamiltonian Approach · Alternative
non-empirical SCF formalism · Second quantized
Hamiltonian · Excluding three- and four-center integrals ·
Projective integral approximation
Published as part of the special collection of articles “Festschrift
in honour of Péter R. Surjan.”
Dedicated to the 60th birthday of Professor Péter R. Surján.
* I. Mayer
mayer@chemres.hu
1
Research Centre for Natural Sciences , Hungarian Academy
of Sciences , P.O.Box 286 , Budapest 1519 , Hungary
31
Reprinted from the journal