1 3
Theor Chem Acc (2015) 134:125
DOI 10.1007/s00214-015-1731-6
REGULAR ARTICLE
Spin contamination and noncollinearity in general complex
Hartree–Fock wave functions
Patrick Cassam-Chenaï
1
Received: 1 June 2015 / Accepted: 13 September 2015 / Published online: 12 October 2015
© Springer-Verlag Berlin Heidelberg 2015
operators S z and S
2 , a spin-symmetry respectful way of
using the Hartree–Fock method consists in:
(1) Using spin-orbitals of pure α - or β -spin, so that the HF
optimized Slater determinant is an eigenfunction of S z ;
(2) Imposing the spin-equivalence restriction [ 4 ], which
means that paired α - and β -spin-orbitals are formed
from the same set of linearly independent orbitals.
We have proved mathematically [ 5 – 7 ] that this additional constraint is a necessary and suffi cient condition to insure that a Slater determinantal wave function
is an eigenfunction of the spin operator S
2 . In other
words, we have shown that relaxing the S
2 -symmetry
constraint exactly amounts to allow different “paired
orbitals,” in the sense of Refs. [ 8 , 9 ], to have different
spins. This equivalence enabled us to characterize the
variational space explored by the restricted open-shell
Hartree–Fock (ROHF) method [ 10 ], which precisely
consists in optimizing a Slater determinant subject to
constraints (1) and (2) (plus spatial-symmetry constraints if any) [ 11 ]. The equivalence was also discovered independently [ 12 ] by optimizing a Slater determinant with a Lagrange multiplier, enforcing S 2 to be
arbitrarily close to the ROHF value, instead of applying the spin-equivalence restriction. Not surprisingly,
the determinant was approaching the ROHF solution.
Similar to the spin-free case is the “complex-free” one:
when a quantum system is described by a real Hamiltonian,
which obviously commutes with complex conjugation, one
can restrict oneself to the calculation of real eigenfunctions. Then, it is also possible to employ only real spinorbitals to construct the HF Slater determinant [ 13 ] (however, diffi culties may occur when the symmetry group of
the molecule cannot be represented over real numbers, and
Abstract An expression for the square of the spin operator expectation value, S 2 , is obtained for a general complex Hartree–Fock wave function and decomposed into
four contributions: the main one whose expression is formally identical to the restricted (open-shell) Hartree–Fock
expression. A spin contamination one formally analogous
to that found for spin unrestricted Hartree–Fock wave functions. A noncollinearity contribution related to the fact that
the wave function is not an eigenfunction of the spin- S z
operator. A perpendicularity contribution related to the fact
that the spin density is not constrained to be zero in the xy -
plane. All these contributions are evaluated and compared
for the H 2 O
+ system. The optimization of the collinearity
axis is also considered.
Keywords Spin contamination · Collinearity · General
complex Hartree–Fock
1 Introduction
Particle-independent models based on single Slater determinant wave functions have enjoyed considerable interest
in quantum chemistry, since the pioneering works of Hartree, Slater and Fock [ 1 – 3 ].
When a quantum system is described by a spin-free
Hamiltonian, which obviously commutes with the spin
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan”.
* Patrick Cassam-Chenaï
cassam@unice.fr
1
CNRS, LJAD, UMR 7351 , University of Nice Sophia
Antipolis , 06100 Nice , France
175
Reprinted from the journal
Theor Chem Acc (2015) 134:125
DOI 10.1007/s00214-015-1731-6
REGULAR ARTICLE
Spin contamination and noncollinearity in general complex
Hartree–Fock wave functions
Patrick Cassam-Chenaï
1
Received: 1 June 2015 / Accepted: 13 September 2015 / Published online: 12 October 2015
© Springer-Verlag Berlin Heidelberg 2015
operators S z and S
2 , a spin-symmetry respectful way of
using the Hartree–Fock method consists in:
(1) Using spin-orbitals of pure α - or β -spin, so that the HF
optimized Slater determinant is an eigenfunction of S z ;
(2) Imposing the spin-equivalence restriction [ 4 ], which
means that paired α - and β -spin-orbitals are formed
from the same set of linearly independent orbitals.
We have proved mathematically [ 5 – 7 ] that this additional constraint is a necessary and suffi cient condition to insure that a Slater determinantal wave function
is an eigenfunction of the spin operator S
2 . In other
words, we have shown that relaxing the S
2 -symmetry
constraint exactly amounts to allow different “paired
orbitals,” in the sense of Refs. [ 8 , 9 ], to have different
spins. This equivalence enabled us to characterize the
variational space explored by the restricted open-shell
Hartree–Fock (ROHF) method [ 10 ], which precisely
consists in optimizing a Slater determinant subject to
constraints (1) and (2) (plus spatial-symmetry constraints if any) [ 11 ]. The equivalence was also discovered independently [ 12 ] by optimizing a Slater determinant with a Lagrange multiplier, enforcing S 2 to be
arbitrarily close to the ROHF value, instead of applying the spin-equivalence restriction. Not surprisingly,
the determinant was approaching the ROHF solution.
Similar to the spin-free case is the “complex-free” one:
when a quantum system is described by a real Hamiltonian,
which obviously commutes with complex conjugation, one
can restrict oneself to the calculation of real eigenfunctions. Then, it is also possible to employ only real spinorbitals to construct the HF Slater determinant [ 13 ] (however, diffi culties may occur when the symmetry group of
the molecule cannot be represented over real numbers, and
Abstract An expression for the square of the spin operator expectation value, S 2 , is obtained for a general complex Hartree–Fock wave function and decomposed into
four contributions: the main one whose expression is formally identical to the restricted (open-shell) Hartree–Fock
expression. A spin contamination one formally analogous
to that found for spin unrestricted Hartree–Fock wave functions. A noncollinearity contribution related to the fact that
the wave function is not an eigenfunction of the spin- S z
operator. A perpendicularity contribution related to the fact
that the spin density is not constrained to be zero in the xy -
plane. All these contributions are evaluated and compared
for the H 2 O
+ system. The optimization of the collinearity
axis is also considered.
Keywords Spin contamination · Collinearity · General
complex Hartree–Fock
1 Introduction
Particle-independent models based on single Slater determinant wave functions have enjoyed considerable interest
in quantum chemistry, since the pioneering works of Hartree, Slater and Fock [ 1 – 3 ].
When a quantum system is described by a spin-free
Hamiltonian, which obviously commutes with the spin
Published as part of the special collection of articles “Festschrift
in honour of P. R. Surjan”.
* Patrick Cassam-Chenaï
cassam@unice.fr
1
CNRS, LJAD, UMR 7351 , University of Nice Sophia
Antipolis , 06100 Nice , France
175
Reprinted from the journal