Theor Chem Acc (2015) 134:125
1 3
nonetheless, one wishes the spin-orbitals to be adapted to
spatial symmetry).
However, it has been proposed by various authors to
relax some or all of the above-mentioned constraints, to
gain variational freedom. For example, the different orbitals for different spins method (DODS) of Refs. [ 14 , 15 ]
(which is usually just called “unrestricted Hartree–Fock”
(UHF), but in this paper we use “DODS” to avoid confusions) relax the spin-equivalence restriction, and hence,
the HF solution is no longer an eigenfunction of S
2 . Other
authors [ 16 – 18 ] have advocated the use of general spinorbitals, mixing α -spin and β -spin parts, in conjunction
with the use of projectors [ 19 ].
Along the same line of thought, the use of complex spinorbitals has been proposed [ 17 , 20 ] to increase variational
freedom in the case of real Hamiltonian. Prat and Lefebvre
went a step further with the so-called hypercomplex spinorbitals to construct Slater determinants of arbitrary accuracy [ 21 ]. However, the coeffi cients of their spin-orbitals
were elements of a Clifford algebra of dimension 2 2n , that
was not a normed division algebra, also known as Cayley
algebra, for arbitrary values of n . This was unfortunate,
since such a structure appears to be a minimal requirement
for a quantum formalism, if, for example, Born’s interpretation of the wave function is to hold fi rmly. For n = 1, the
Clifford algebra of Prat et al. was actually the noncommutative fi eld of quaternions, therefore, a fortiori, a normed
division algebra. The only larger normed division algebra is
the octonion algebra. It is a Clifford algebra of dimension
8, which has also been proposed in a quantum mechanical
context [ 22 ], but this algebra is neither commutative nor
associative. The lack of these properties rises diffi culties
for its use for multipartite quantum systems; nevertheless,
these diffi culties can be overcome by keeping the product
of octonion coeffi cients in the form of a tensor product. So,
octonion-unrestricted HF appears to be the largest Cilfford
algebra-unrestricted single determinantal method that can
be considered in the spirit of Prat and Lefebvre’s proposal.
However, octonions seem incompatible with the desirable
requirement that the algebra of quantum observables be
what is now called a formally real Jordan algebra [ 23 ] acting on a vector space of arbitrarily large dimension. Octonions are also ruled out by the requirement of orthomodularity in infi nite dimension according to Solèr’s theorem [ 24 ,
25 ], which restricts quantum Hilbert spaces to be real, complex or at most quaternionic.
The fi rst HF molecular calculations with general complex spin-orbitals, without projecting out the symmetrybreaking part of the wave function, are maybe those of Ref.
[ 26 ]. It was found on the BH molecule around its equilibrium geometry that the general complex Hartree–Fock
(GCHF) energy was indeed lower than the DODS one,
which itself was lower than the restricted Hartree–Fock
(RHF) solution. So necessarily, the corresponding GCHF
wave functions had S
2 -spin contamination and S z -spin contamination, that is to say, the expectation values of these
operators were different from 0, the value expected for a
singlet ground state (it is not clear whether complex numbers were used for this molecule, but the authors did mention that they performed complex calculations for two-electron systems).
Relaxing the “ S z -constraint” hence the “collinearity constraint” becomes perfectly legitimate when hyperfi ne or
spin–orbit couplings are considered, since the operator S z no
longer commutes with the Hamiltonian. As a matter of fact,
real physical systems do exhibit either light [ 27 , 28 ] or strong
[ 29 , 30 ] noncollinearity of their spin densities. Similarly, the
use of complex spin-orbitals is natural, when considering
relativistic corrections resulting in a complex Hamiltonian
operator. So, in such a context, one should use no less than
general complex spin-orbitals in HF calculations [ 31 ]. The
“spin-same-orbit” coupling term used in these calculations
does not commute with the S
2
-operator. Therefore, one cannot strictly speak of “ S
2
-spin contamination” in relativistic
GCHF wave functions. However, calculating the expectation
value of S
2
, a bona fi de quantum observable, can still provide
valuable physical information about the system.
A general expression for the expectation value of S
2 has
been obtained in the DODS case [ 8 ] and has served as a
measure of S
2 -spin contamination. However, as far as we
are aware, no such formula has been published in the case
of a GCHF wave function. This gap will be fi lled in the
next section.
Studying departure from collinearity is more diffi cult
because of arbitrariness in the quantifi cation axis. One possible way to overcome the diffi culty would be to apply an
external magnetic fi eld to fi x the z -axis but small enough
not to perturb the GCHF solution. However, an elegant
alternative has been proposed recently by Small et al. [ 32 ].
It is based on studying the lowest eigenvalue of a (3 × 3)
-matrix built from expectation values of spin operator components and their products. In the GCHF case, the authors
provided the expressions required to compute the matrix
elements in a compact form. In the third section, we give
a more extended formula in terms of molecular orbital
overlap matrix elements. We also illustrate the connections
between spin contamination, noncollinearity and its correlative: “perpendicularity” on the H 2 O
+ cation example. We
sum up our conclusions in the last section.
2 Spin contamination in GCHF
A general complex Hartree–Fock (GCHF) wave function,
(1)
Φ GCHF = φ 1 ∧ · · · ∧ φ N e ,
176
Reprinted from the journal
1 3
nonetheless, one wishes the spin-orbitals to be adapted to
spatial symmetry).
However, it has been proposed by various authors to
relax some or all of the above-mentioned constraints, to
gain variational freedom. For example, the different orbitals for different spins method (DODS) of Refs. [ 14 , 15 ]
(which is usually just called “unrestricted Hartree–Fock”
(UHF), but in this paper we use “DODS” to avoid confusions) relax the spin-equivalence restriction, and hence,
the HF solution is no longer an eigenfunction of S
2 . Other
authors [ 16 – 18 ] have advocated the use of general spinorbitals, mixing α -spin and β -spin parts, in conjunction
with the use of projectors [ 19 ].
Along the same line of thought, the use of complex spinorbitals has been proposed [ 17 , 20 ] to increase variational
freedom in the case of real Hamiltonian. Prat and Lefebvre
went a step further with the so-called hypercomplex spinorbitals to construct Slater determinants of arbitrary accuracy [ 21 ]. However, the coeffi cients of their spin-orbitals
were elements of a Clifford algebra of dimension 2 2n , that
was not a normed division algebra, also known as Cayley
algebra, for arbitrary values of n . This was unfortunate,
since such a structure appears to be a minimal requirement
for a quantum formalism, if, for example, Born’s interpretation of the wave function is to hold fi rmly. For n = 1, the
Clifford algebra of Prat et al. was actually the noncommutative fi eld of quaternions, therefore, a fortiori, a normed
division algebra. The only larger normed division algebra is
the octonion algebra. It is a Clifford algebra of dimension
8, which has also been proposed in a quantum mechanical
context [ 22 ], but this algebra is neither commutative nor
associative. The lack of these properties rises diffi culties
for its use for multipartite quantum systems; nevertheless,
these diffi culties can be overcome by keeping the product
of octonion coeffi cients in the form of a tensor product. So,
octonion-unrestricted HF appears to be the largest Cilfford
algebra-unrestricted single determinantal method that can
be considered in the spirit of Prat and Lefebvre’s proposal.
However, octonions seem incompatible with the desirable
requirement that the algebra of quantum observables be
what is now called a formally real Jordan algebra [ 23 ] acting on a vector space of arbitrarily large dimension. Octonions are also ruled out by the requirement of orthomodularity in infi nite dimension according to Solèr’s theorem [ 24 ,
25 ], which restricts quantum Hilbert spaces to be real, complex or at most quaternionic.
The fi rst HF molecular calculations with general complex spin-orbitals, without projecting out the symmetrybreaking part of the wave function, are maybe those of Ref.
[ 26 ]. It was found on the BH molecule around its equilibrium geometry that the general complex Hartree–Fock
(GCHF) energy was indeed lower than the DODS one,
which itself was lower than the restricted Hartree–Fock
(RHF) solution. So necessarily, the corresponding GCHF
wave functions had S
2 -spin contamination and S z -spin contamination, that is to say, the expectation values of these
operators were different from 0, the value expected for a
singlet ground state (it is not clear whether complex numbers were used for this molecule, but the authors did mention that they performed complex calculations for two-electron systems).
Relaxing the “ S z -constraint” hence the “collinearity constraint” becomes perfectly legitimate when hyperfi ne or
spin–orbit couplings are considered, since the operator S z no
longer commutes with the Hamiltonian. As a matter of fact,
real physical systems do exhibit either light [ 27 , 28 ] or strong
[ 29 , 30 ] noncollinearity of their spin densities. Similarly, the
use of complex spin-orbitals is natural, when considering
relativistic corrections resulting in a complex Hamiltonian
operator. So, in such a context, one should use no less than
general complex spin-orbitals in HF calculations [ 31 ]. The
“spin-same-orbit” coupling term used in these calculations
does not commute with the S
2
-operator. Therefore, one cannot strictly speak of “ S
2
-spin contamination” in relativistic
GCHF wave functions. However, calculating the expectation
value of S
2
, a bona fi de quantum observable, can still provide
valuable physical information about the system.
A general expression for the expectation value of S
2 has
been obtained in the DODS case [ 8 ] and has served as a
measure of S
2 -spin contamination. However, as far as we
are aware, no such formula has been published in the case
of a GCHF wave function. This gap will be fi lled in the
next section.
Studying departure from collinearity is more diffi cult
because of arbitrariness in the quantifi cation axis. One possible way to overcome the diffi culty would be to apply an
external magnetic fi eld to fi x the z -axis but small enough
not to perturb the GCHF solution. However, an elegant
alternative has been proposed recently by Small et al. [ 32 ].
It is based on studying the lowest eigenvalue of a (3 × 3)
-matrix built from expectation values of spin operator components and their products. In the GCHF case, the authors
provided the expressions required to compute the matrix
elements in a compact form. In the third section, we give
a more extended formula in terms of molecular orbital
overlap matrix elements. We also illustrate the connections
between spin contamination, noncollinearity and its correlative: “perpendicularity” on the H 2 O
+ cation example. We
sum up our conclusions in the last section.
2 Spin contamination in GCHF
A general complex Hartree–Fock (GCHF) wave function,
(1)
Φ GCHF = φ 1 ∧ · · · ∧ φ N e ,
176
Reprinted from the journal