Approximate Spin Projection for Broken-Symmetry Method and Its Application

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Chapter 7
Approximate Spin Projection for Broken-Symmetry
Method and Its Application
Yasutaka Kitagawa, Toru Saito and
Kizashi Yamaguchi
Additional information is available at the end of the chapter
http://dx.doi.org/10.5772/intechopen.75726
Abstract
A broken-(spin) symmetry (BS) method is now widely used for systems that involve
(quasi) degenerated frontier orbitals because of their lower cost of computation. The BS
method splits up-spin and down-spin electrons into two different special orbitals, so that a
singlet spin state of the degenerate system is expressed as a singlet biradical. In the BS
solution, therefore, the spin symmetry is no longer retained. Due to such spin-symmetry
breaking, the BS method often suffers from a serious problem called a spin contamination
error, so that one must eliminate the error by some kind of projection method. An approx-
imate spin projection (AP) method, which is one of the spin projection procedures, can
eliminate the error from the BS solutions by assuming the Heisenberg model and can
recover the spin symmetry. In this chapter, we illustrate a theoretical background of the
BS and AP methods, followed by some examples of their applications, especially for
calculations of the exchange interaction and for the geometry optimizations.
Keywords: quantum chemistry, ab initio calculation, orbital degeneracy, electron
correlation, broken-(spin) symmetry (BS) method, approximate spin projection (AP)
method, spin polarization, spin contamination error, effective exchange integral (J
ab
)
values
1. Introduction
For the past few decades, many reports about “polynuclear metal complexes”have been
presented actively in the field of the coordination chemistry [1–19]. Those systems usually
have complicated electronic structures that are constructed by metal–metal (d-d) and metal–
ligand (d-p) interactions. Those electronic structures caused by their unique molecular
© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative
Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,
distribution, and reproduction in any medium, provided the original work is properly cited.

structures often bring many interesting and noble physical functionalities such as a magnetism
[8–17], a nonlinear optics [18], an electron conductivity [19], as well as their chemical functional-
ities, e.g., a catalyst and so on. For example, some three-dimensional (3D) metal complexes show
interesting magnetic behaviors and are expected to be possible candidates for a single molecule
magnet, a quantum dot, and so on [11–16]. On the other hand, one-dimensional (1D) metal
complexes are studied for the smallest electric wire, i.e., the nanowire [3–7, 17, 19]. In addition,
it has been elucidated that the polynuclear metal complexes play an important role in the
biosystems [20–24], e.g., Mn cluster [25, 26] in photosystem II and 4Fe-4S cluster [27–30] in
electron transfer proteins. In this way, the polynuclear metal complexes are widely noticed from
a viewpoint of fundamental studies on their peculiar characters and of applications to materials.
From those reasons, an elucidation of a relation among electronic structures, molecular struc-
tures, and physical properties is a quite important current subject.
Physical properties of molecules are sometimes discussed by using several parameters such as
an exchange integrals (J
ab
), on-site Coulomb repulsion, and transfer integrals of Heisenberg
and Hubbard Hamiltonians, respectively, in material physics [31–35]. In recent years, on the
other hand, direct predictions of such electronic structures, molecular structure, and physical
properties of those metal complexes are fairly realized by the recent progress in computers and
computational methods. In this sense, theoretical calculations are now one of the powerful
tools for understanding of such systems. However, those systems are, in a sense, still challeng-
ing subjects because they are usually large and orbitally degenerated systems with localized
electron spins (localized orbitals). The localized spins are caused by an electron correlation
effect called a static (or a non-dynamical) correlation [36]. In addition, a dynamical correlation
effect of core electrons also must be treated together with the static correlation in the case of the
metal complexes. A treatment of both the static correlation and the dynamical correlation in
large molecules is still a difficult task and a serious problem in this field. For those systems, a
standard method for the static and dynamical correlation corrections is a complete active space
(CAS) method [37–38] or a multi-reference (MR) method [39] that considers all configuration
interaction in active valence orbitals, together with the second-order perturbation correction,
e.g., CASPT2 or MPMP2 methods. In addition to these methods, recently, other multi-
configuration methods such as DDCI [40–42], CASDFT [43–45], MRCC [46–48], and DMRG-
CT [49–51] methods are also proposed for the same purpose. These newer methods are
developing and seem to be promising tools in terms of accuracy; however, real molecules such
as polynuclear metal complexes are still too large to treat computationally with those methods
at this state. An alternative way is a broken-symmetry (BS) method, which approximates the
static correlation with a lower cost of computation [52–55]. The BS method (or commonly
known as an unrestricted (U) method) splits up and down spins (electrons) into two different
spatial orbitals (it is sometimes called as different orbitals for different spins; DODS), so a
singlet spin state of the orbitally degenerated system is expressed as a singlet biradical, namely,
the BS singlet [55]. The BS method such as the unrestricted Hartree-Fock (UHF) and the
unrestricted DFT (UDFT) methods are now widely used for the first principle calculations of
such large degenerate systems. In this sense, the BS method seems to be the most possible
quantum chemical approach for the polynuclear metal complexes, although it has a serious
problem called the spin contamination error [56–65]. Therefore one must eliminate the error by
Symmetry (Group Theory) and Mathematical Treatment in Chemistry122

some kind of projection method. An approximate spin projection (AP) method, which is one of
the spin projection procedures, can eliminate the error from the BS solutions and can recover
the spin symmetry. In this chapter, we illustrate a theoretical background of the BS and AP
methods, followed by some examples of their applications.
2. Theoretical background of AP method
In this section, the theoretical background of the BS and AP methods for the biradical systems is
explained with the simplest two-spin model (e.g., a dissociated H
2
)asillustratedinFigure 1(a).
2.1. Broken-symmetry (BS) solution and approximate spin projection (AP) methods for the
(two-spin) biradical state
In the BS method, the spin-polarized orbitals are obtained from HOMO-LUMO mixing [55–
56]. For example, HOMO orbitals for up-spin (ψHOMO) and down-spin (ψHOMO) electrons of the
simple H
2
molecule are expressed as follows (Figure 1(b)):
Figure 1. (a) Illustration of the two-spin states of the simplest two-spin model. (b) HOMO and LUMO of spin-adapted
(SA) and BS methods. (c) Illustration of spin-symmetry recovery of BS method by AP method.
Approximate Spin Projection for Broken-Symmetry Method and Its Application
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ψBS
HOMO ¼cosθψHOMO þsinθψLUMO,(1)
ψBS
HOMO ¼cosθψHOMO sinθψLUMO,(2)
where 0 ≤θ≤45and ψHOMO and ψLUMO express HOMO and LUMO orbitals of spin-adapted
(SA) (or spin-restricted (R)) calculations, respectively, as illustrated in Figure 1(b). And the
wavefunction of the BS singlet (e.g., unrestricted Hartree-Fock (UHF)) becomes
ΨSinglet
BS
E¼cos2θψ
HOMOψHOMO
þsin2θψ
LUMOψLUMO
ffiffiffi
2
pcosθsinθΨTriplet
,(3)
where ψHOMO and ψHOMO express up- and down-spin electrons in orbital ψHOMO , respectively.
If θ= 0, the BS wavefunction corresponds to the closed shell, i.e., SA wavefunctions, while if θ
is not zero, one can have spin-polarized, i.e., BS wavefunctions. In the BS solution, ψHOMO 6¼
ψHOMO (Figure 1(b)), so that a spin symmetry is broken. In addition, it gives nonzero b
S2
DE
Singlet
BS
value, and as described later, up- and down-spin densities appeared on the hydrogen atoms.
We often regard such spin densities as an existence of localized spins. An interaction between
localized spins can be expressed by using Heisenberg Hamiltonian:
b
H¼2Jabb
Sab
Sb,(4)
where b
Saand b
Sbare spin operators for spin sites a and b, respectively, and J
ab
is an effective
exchange integral. Using a total spin operator of the system b
S¼b
Saþb
Sb, Eq. (4) becomes
b
H¼2Jab b
S2þb
S2
aþb
S2
b

:(5)
Operating Eq. (5) to Eq. (3), the singlet state energy in Heisenberg Hamiltonian (ESinglet
HH )is
expressed as
ESinglet
HH ¼Jab b
S2
DE
Singlet
þb
S2
a
DE
Singlet
þb
S2
b
DE
Singlet

:(6)
Similarly, for triplet state
ETriplet
HH ¼Jab b
S2
DE
Triplet
þb
S2
a
DE
Triplet
þb
S2
b
DE
Triplet

:(7)
The energy difference between singlet (ESinglet
HH ) and triplet (ETriplet
HH ) states (S-T gap) within
Heisenberg Hamiltonian should be equal to the S-T gap calculated by the difference in total
energies of ab initio calculations (here we denote ESinglet
BS and ETriplet for the BS singlet and
triplet states, respectively). And if we can assume that spin densities of the BS singlet state on
spin site i(i= a or b) are almost equal to ones of the triplet state, i.e., b
S2
i
DE
Triplet ffib
S2
i
DE
Singlet, then J
ab
can be derived as
Symmetry (Group Theory) and Mathematical Treatment in Chemistry124

Jab ¼ESinglet
HH ETriplet
HH
b
S2
DE
Triplet b
S2
DE
Singlet ¼ESinglet
BS ETriplet
b
S2
DE
Triplet b
S2
DE
Singlet
BS
:(8)
If the method is exact and the spin contamination error is not found in both singlet and triplet
states (i.e., b
S2
DE
Singlet
Exact ¼0and b
S2
DE
Triplet
Exact ¼2), the S-T gap between those states can be expressed as
ESinglet
Exact ETriplet
Exact ¼2Jab:(9)
The spin contamination in the triplet state is usually negligible (i.e., b
S2
DE
Triplet
Exact ffib
S2
DE
Triplet ffi2), and one
must consider the error only in the BS singlet state, so the S-T gap becomes
ESinglet
BS ETriplet ¼2Jab Jab b
S2
DE
Singlet
BS :(10)
A second term in a right side of Eq. (10) indicates the spin contamination error in the S-T gap,
and consequently, a second term in a denominator of Eq. (8) eliminates the spin contamination
in the BS singlet solution. In this way, Eq. (8) gives approximately spin-projected (AP) J
ab
values. Eq. (8) can be easily expanded into any spin dimers, namely, the lowest spin (LS) state
and the highest spin (HS) state, e.g., singlet-quintet for S
a
=S
b
= 2/2 pairs, singlet-sextet for
S
a
=S
b
= 3/2 pairs, and so on, as follows:
Jab ¼ELS
BS EHS
b
S2
DE
HS b
S2
DE
LS
BS
:(11)
Eq. (11) is the so-called Yamaguchi equation to calculate J
ab
values with the AP procedure,
which is simply denoted by J
ab
here. The calculated J
ab
value can explain an interaction
between two spins. If a sign of calculated J
ab
value is positive, the HS, i.e., ferromagnetic
coupling state, is stable, while if it is negative, the LS, i.e., antiferromagnetic coupling state is
stable. Therefore, one can discuss the magnetic interactions in a given system.
2.2. Approximate spin projection for BS energy and energy derivatives
Because J
ab
calculated by Eq. (11) is a value that the spin contamination error is approximately
eliminated, it should be equal to J
ab
value calculated by the approximately spin-projected LS
energy (ELS
AP)as
Jab ¼ELS
BS EHS
b
S2
DE
HS b
S2
DE
LS
BS
¼ELS
AP EHS
b
S2
DE
HS
exact b
S2
DE
LS
ecact
:(12)
Here, we assume b
S2
DE
HS
Exact ffib
S2
DE
HS; then one can obtain a spin-projected energy of the singlet state
without the spin contamination error as follows [62–65]:
Approximate Spin Projection for Broken-Symmetry Method and Its Application
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125

ELS
AP ¼αELS
BS βEHS,(13)
where
α¼b
S2
DE
HS b
S2
DE
LS
exact
b
S2
DE
HS b
S2
DE
LS
BS
(14)
and
β¼α1 (14)
Then, we explain about derivatives of this spin-projected energy (ELS
AP). In order to carry out the
geometry optimization using the AP method, an energy gradient of ELS
AP is necessary. ELS
AP can
be expanded by using Taylor expansion:
ELS
AP RLS
AP

¼ELS
AP RðÞþXTGLS
AP RðÞþ
1
2XTFLS
AP RðÞX,(15)
where GLS
AP RðÞand FLS
AP RðÞare the first and second derivatives (i.e., gradient and Hessian) of
ELS
AP RðÞ, respectively [62–65]; RLS
AP and Rare a stationary point of ELS
AP RðÞand a present posi-
tion, respectively; and Xis a position vector (X¼RLS
AP R). The stationary point RLS
AP is a
position where GLS
AP RðÞ¼0; therefore one can obtain RLS
AP if GLS
AP RðÞcan be calculated. By
differentiating ELS
AP RðÞin Eq. (13), we obtain
GLS
AP RðÞ¼
∂ELS
AP RðÞ
∂R¼αRðÞGLS
BS RðÞβRðÞGHS RðÞ

þ∂αRðÞ
∂RELS
BS RðÞEHS RðÞ

,(16)
where GLS
BS and GHS are the first energy derivatives (energy gradients) of the BS and the HS
states, respectively. As mentioned above, the spin contamination in the HS state is negligible,
so that b
S2
DE
HS is usually a constant. Then ∂αRðÞ=∂Rcan be written as
∂αRðÞ
∂R¼b
S2
DE
HS b
S2
DE
LS
exact
b
S2
DE
HS b
S2
DE
LS
BS

2
∂b
S2
DE
LS
BS
∂R:(17)
By using Eqs. (16) and (17), the AP optimization can be carried out. In addition, one can also
calculate the spin-projected Hessian (AP Hessian; FLS
AP RðÞin Eq. (15)) as follows:
FLS
AP RðÞ¼
∂2ELS
AP RðÞ
∂2R¼αRðÞFLS
BS RðÞβRðÞFHS RðÞ

,
þ2∂αRðÞ
∂RGLS
BS RðÞGHS RðÞ

þ∂2αRðÞ
∂2RELS
BS RðÞEHS RðÞ

,(18)
Symmetry (Group Theory) and Mathematical Treatment in Chemistry126

where FLS
BS and FHS are the Hessians calculated by the BS and the HS states, respectively. And a
second derivative of αcan be expressed by
∂2αRðÞ
∂R2¼
2b
S2
DE
HS b
S2
DE
LS
exact

b
S2
DE
HS b
S2
DE
LS
BS

3
∂b
S2
DE
LS
BS
∂R
0
B
@1
C
A
2
þb
S2
DE
HS b
S2
DE
LS
exact
b
S2
DE
HS b
S2
DE
LS
BS

2
∂b
S2
DE
LS
BS
∂R:(19)
By using Eqs. (18) and (19), the spin-projected vibrational frequencies are also calculated. The
AP optimization can be carried out based on Eq. (16) with ∂b
S2
DE
LS
BS=∂Robtained by numerical
fitting or analytical ways.
2.3. Relationship between the BS and projected wavefunctions
As well as a calculated energy and its derivatives, the BS wavefunction itself has also vital
information. Here let us go back to Eq. (3). From the equation, an overlap between up-spin (so-
called alpha) and down-spin (so-called beta) orbitals (T) becomes
T¼ψBS
HOMOjψBS
HOMO
DE
¼cos2θsin2θ¼cos2θ:(20)
And because occupation number (n) of natural orbital (NO) for the corresponding orbital is
expressed as n¼2cos2θ, we get the relation:
T¼cos2θ¼n1 (21)
On the other hand, we can define projected wavefunction (PUHF) by eliminating triplet
species from BS singlet wavefunction from Eq. (3) as follows:
ΨSinglet
PUHF
E¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
1þcos2θðÞ
2
s1þcos2θ
2ψHOMOψHOMO
1cos2θ
2ψLUMOψLUMO


¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
1þT2
r1þT
2ψHOMOψHOMO
1T
2ψLUMOψLUMO


:
(22)
If we focus on the second term, which is related to double (two-electron) excitation, its weight
(W
D
) can be obtained from Eqs. (21) and (22) as follows:
WD¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
1þT2
s1T
2
()
2
¼1
212T
1þT2
 (23)
This is the weight of double excitation calculated by the BS wavefunction. By applying
Eq. (21)–Eq. (23), the W
D
is related to the occupation number of the corresponding NO as
follows:
Approximate Spin Projection for Broken-Symmetry Method and Its Application
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127

y¼2WD¼n24nþ4
n22nþ2:(24)
This yvalue is called an instability value of a chemical bond (or diradical character). In the case
of the spin-restricted (or spin-adapted (SA)) calculations, the yvalue is zero. However if a
couple of electrons tends to be separated and to be localized on each hydrogen atom, in other
words the chemical bond becomes unstable with the strong static correlation effect, the yvalue
becomes larger and finally becomes 1.0. So, the yvalue can be applied for the analyses of di- or
polyradical species, and it is often useful to discuss the stability (or instability) of chemical
bonds. The idea is also described by an effective bond order (b), which is defined by the
difference in occupation numbers of occupied NO (n) and unoccupied NO (n
*
):
b¼nn∗
2(25)
Different from the yvalue, the bvalue becomes smaller when the chemical bond becomes unsta-
ble. If we define the effective bond order with the spin projection b(AP), it is related to the yvalue:
bAPðÞ¼1–y(26)
Those indices show how the BS and AP wavefunctions are connected. In addition, one can
utilize the indices to estimate the contribution of double excitation for very large systems that
CAS and MR methods cannot be applied.
Finally, a relationship between the BS wavefunction and b
S2
DE
values are briefly explained. The
b
S2
DE
values of the BS singlet states do not show the exact value by the spin contamination error.
b
S2
DE
value of the SA calculation is.
b
S2
DE
SA ¼SSþ1ðÞ,where S ¼SaþSb(27)
However, in the case of the BS singlet state of H
2
molecule, it becomes
b
S2
DE
BS ¼b
S2
DE
exact þNdown X
ij
Tij ffi1T(28)
where N
down
and Tare number of down electrons and the overlap between spin-polarized up-
spin and down-spin orbitals in Eq. (21). Therefore b
S2
DE
is also closely related to a degree of
spin polarization. For the BS singlet state of the hydrogen molecule model, by substituting
Eq. (21) into Eq. (28), we can obtain
b
S2
DE
BS ffi2n(29)
Here we explain another aspect of the spin projection method. As depicted in Figure 1(c), the
BS wavefunction indicates only one spin-polarized configuration, e.g., BS1 in the figure.
Symmetry (Group Theory) and Mathematical Treatment in Chemistry128

However, in order to obtain a pure singlet wavefunction, which satisfies the spin symmetry,
the opposite spin-polarized state (BS2) must be included. The projection method can give a
linear combination of the both BS states, and therefore it can give an appropriate quantum
state for the singlet state.
3. Application of BS and AP methods to several biradical systems
3.1. Hydrogen molecule: comparison among SA, BS, and AP methods by simple biradical
system
In this section, we briefly illustrate how the BS and AP methods approximate a dissociation of
a hydrogen molecule. Figure 2(a) shows potential energy curves of Hartree-Fock and full CI
methods. In the case of the spin-adapted (SA) HF, i.e., the spin-restricted (R) HF method, the
curve does not converge to the dissociation limit. On the other hand, the BS HF, i.e., spin-
unrestricted (U) HF calculation, successfully reproduces the dissociation limit of full CI
method. This result indicates that the static correlation is included in the BS procedure.
Around 1.2 Å, there is a bifurcation point between RHF and UHF methods. Within the closed
shell (i.e., SA) region, where r
H-H
< 1.2 Å, the UHF solution does not appear, and the singlet
state is described by RHF (single slater determinant). In this region, the energy gap between
full CI and RHF that is known as correlation energy indicates a necessity of the dynamical
correlation correction as discussed later.
In order to elucidate how the double-excitation state is included in the BS solution, the
occupation numbers of the highest occupied natural orbital (HONO) are plotted along the H-
H distance in Figure 2(b). The figure indicates that the occupation number is 2.0 in the closed
shell region, while it suddenly decreases at the bifurcation point. And it finally closes to 1.0 at
the dissociation limit. In Figure 2(c), calculated y/2 values from the occupation numbers are
compared with the weight of the double excitation (W
D
) of CI double (CID) method. The
figure indicates that the BS method approximates the bond dissociation by taking the double
excitation into account. As frequently mentioned above, the BS wavefunction is not pure
singlet state by the contamination of the triplet wavefunction. In Figure 2(b),b
S2
DE
values of
the BS states are plotted. It suddenly increases at the bifurcation point and finally closes to the
1.0, which corresponds to occupation number nat the dissociation limit. And as mentioned
above, b
S2
DE
and 2-nvalues are closely related.
Next, we illustrate results of calculated effective exchange integral (J
ab
) values of the hydrogen
molecule by Eq. (11). The calculated Jvalues are shown in Figure 2(d). In a longer-distance
region (r
H-H
> 2.0 Å), the AP-UHF method reproduces the full CI result, indicating that the
inclusion of double excitation state and elimination of the triplet state work well within the BS
and AP framework. On the other hand, in a shorter-region (r
H-H
< 1.2 Å), a hybrid DFT
(B3LYP) method reproduces the full CI curve. In the region, the dynamical correlation that
the RHF method cannot include is a dominant. Therefore the dynamical correlation must be
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compensated by other approaches, such as MP, CC, and DFT methods. The hybrid DFT
methods are effective way in terms of the computational costs; however, one must be careful
in a ratio of the HF exchange. It is reported that a larger HF exchange ratio is preferable in the
intermediate region as well as the dissociation limit [69, 70].
3.2. Dichromium (II) complex: effectiveness of hybrid DFT method for calculation of J
value
Next, the BS and AP methods are applied for Cr
2
(O
2
CCH
3
)
4
(OH
2
)
2
(1) complex [1] as illus-
trated in Figure 3(a). This complex involves a quadruple Cr(II)-Cr(II) bond (σ,π
//
,π
⊥
, and δ
Figure 2. (a) Calculated potential energy surface of H
2
molecule by spin-restricted (R), spin-unrestricted (U), and approx-
imate spin-projected HF methods as well as full CI method. (b) Calculated b
S2
DE
, occupation number (n), and 2–nvalues
of H
2
molecule by UHF calculation. (c) a weight of double (two-electron) excitation (W
D
) by double CI (CID) calculation
and y/2 values in Eq. 24. (d) Calculated effective exchange integral (J) values of H
2
molecule with several H-H distances.
For all calculations, 6-31G** basis set was used.
Symmetry (Group Theory) and Mathematical Treatment in Chemistry130

orbitals). Due to the strong static correlation effect, it requires the multi-reference approach.
Within the BS procedure, as a consequence, the electronic structure of the complex is expressed
by the spin localization on each Cr(II) ions. First, let us examine the nature of the metal–metal
bond between Cr(II) ions. For the purpose, natural orbitals and their occupation numbers are
obtained from the BS wavefunctions using an experimental geometry.
As depicted in Figure 3(b), there are eight magnetic orbitals, i.e., bonding and antibonding σ,π
//
,
π
⊥
,andδorbitals that concern about the direct bond between Cr(II) ions. The NO analysis
clarifies the nature of the Cr-Cr bond. If d-orbitals of two Cr(II) ions have sufficient overlap to
form the stable covalent bond, the occupation numbers of each occupied orbital will be almost
2.0 (i.e., Tis close to 1.0). As summarized in Tab l e 1 , however, those bonds show much smaller
values. The occupation numbers of all of occupied σ,π,andδorbitals are close to 1.0, indicating
that electronic structure of the complex 1is described by a spin-polarized spin structure like the
biradical singlet state.
By substituting the obtained energies and b
S2
DE
values into Eq. (11), J
ab
values of the complex 1are
calculated as summarized in Tab l e 2 . In comparison with the experimental value, HF method
underestimates the effective exchange interaction, while B3LYP method overestimates it. This
result is quite similar to a tendency of the J
ab
curve of H
2
molecule at the intermediation region in
Figure 2(d). In that region, BH and HLYP method, which involves 50% HF exchange, gives
better value in comparison with B3LYP. The results also suggest an importance of the effect on
the ratio of the HF/DFTexchange for estimation of the effective exchange interaction [71, 72].
3.3. Singlet methylene molecule: Spin contamination error in optimized geometry by BS
method and its elimination by AP method
Finally, we examine the spin contamination error in the optimized structure. Here we focus on a
singlet methylene (CH
2
). As illustrated in Figure 4(a), the methylene molecule has two valence
Figure 3. (a) Illustration of Cr
2
(O
2
CCH
3
)
4
(OH
2
)
2
(1) complex. (b) Calculated natural orbitals of complex 1by UB3LYP/
basis set I(basis set I: Cr, MIDI+p; others, 6-31G*).
Approximate Spin Projection for Broken-Symmetry Method and Its Application
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131

orbitals (ψ
1
and ψ
2
) and two spins in those orbitals. Those two orbitals are orthogonal and
energetically quasi-degenerate each other. The ground state of the molecule is
3
B
1
(triplet) state,
and
1
A
1
(singlet) state is the first excited state. Components of the wavefunction of
1
A
1
state
obtained by BS method as illustrated in Figure 4(b) have been graphically explained [36]. The
spin-restricted method such as RHF considers only single component (the first term of Figure 4(b))
although the BS wavefunction involves three components as illustrated in Figure 4(b).Theexis-
tence of the triplet component is the origin of the spin contamination error in this system.
Both
1
A
1
and
3
B
1
methylene molecules have bent structures, but the experimental data indi-
cates a large structural difference between them. For example, as summarized in Table 3,
experimental HCH angles (θ
HCH
)of
1
A
1
and
3
B
1
states are 102.4and 134.0, respectively [66,
67]. There have also been many reports of the SA results as summarized in Ref. [68]. On the
other hand, the BS method is a convenient substitute for CI and CAS method, so here we
examined the optimized geometry of the
1
A
1
methylene by SA and BS methods. In order to
elucidate a dependency of the spin contamination error on the calculation methods, HF,
configuration interaction method with all double substitutions (CID), coupled-cluster method
with double substitutions (CCD), several levels of Møller-Plesset energy correction methods
(MP2, MP3, and MP4(SDQ)), and a hybrid DFT (B3LYP) method are also examined. In the case
of
1
A
1
state, all SA results are in good agreement with the experimental values; however, it is
reported that energy gap between the singlet and triplet (S-T gap) value is too much
underestimated [65]. On the other hand, all BS results overestimate the HCH angle. The
difference in HCH angle between the BS values and experimental one is about 10–20. The
HCH angles of UCI and UCC methods are especially larger than MP and DFT methods,
Orbital Occupation number (n) Overlap (T)
δ1.148 0.148
π
ave2
1.242 0.242
σ1.625 0.625
1
Cr, MIDI+p, and others, 6-31G*
2
Averaged value of π
⊥
and π
//
Table 1. nand Tvalues of complex 1calculated by UB3LYP/basis set I
1
.
Method J
ab
values
B3LYP 734
BH and HLYP 520
HF 264
Expt 490
1
In cm
1
2
Basis set Iwas used.
Table 2. Calculated J
ab
values
1
of complex 1by several functional sets
2
.
Symmetry (Group Theory) and Mathematical Treatment in Chemistry132

indicating that the post-HF methods even require some correction for such systems if the BS
procedure is utilized. Therefore it is difficult to use the BS solution for
1
A
1
state without some
corrections. On the other hand, by applying the AP method to the BS solution, the error is
drastically improved, and the optimized structural parameters became in good agreement
with experimental ones. The difference in the optimized θ
HCH
values between the BS and the
AP method, i.e., the spin contamination error in the optimized geometry, is about 10–20.
Figure 4. Illustrations of (a) a methylene molecule and (b) components of BS wavefunctions.
Method r
CHa
θ
HCHb
SA BS AP (
3
B
1
) SABSAP(
3
B
1
)
HF 1.097 1.083 1.098 1.071 103.1 115.5 102.9 130.7
CID 1.114 1.091 1.112 1.081 101.6 119.7 101.9 131.8
CCD 1.116 1.087 1.113 1.082 101.7 125.1 102.4 132.0
MP2 1.109 1.091 1.109 1.077 102.0 114.7 100.9 131.6
MP3 1.109 1.094 1.112 1.080 102.0 114.9 101.0 131.8
MP4(SDQ) 1.117 1.096 1.114 1.081 101.2 115.0 101.0 131.9
B3LYP 1.120 1.100 1.113 1.082 100.3 112.9 103.2 133.1
CASSCF(2,2) 1.097 102.9
CASSCF(6,6) 1.124 100.9
MRMP2(2,2) 1.109 102.0
MRMP2(6,6) 1.122 101.1
Expt.
d
1.107 1.077 102.4 134.0
a
In Å
b
In degree
c
6-31G* basis set was used
d
In Refs. [66, 67] for singlet and triplet states, respectively
Table 3. Optimized C-H bond lengths (r
CH
)
a
and H-C-H angle (θ
HCH
)
b
by SA, BS, and AP approaches with several methods
c
.
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133

Those results strongly indicate that the spin contamination sometimes becomes a serious
problem in the structural optimization of spin-polarized systems and the AP method can work
well for its elimination. On the other hand, the optimized structure with the AP-UHF method
almost corresponds to CASSCF(2,2) result. This means that the AP method approximates two-
electron excitation in the (2,2) active space well. The θ
HCH
values become smaller by including
higher electron correlation with the larger CAS space such as CASSCF(6,6) or with the dynam-
ical correlation correction such as MRMP2(2,2) and MRMP2(6,6). The result of the spin-
projected MP4 (AP MP4(SDQ)) successfully reproduced the MRMP2(6,6) result, indicating
that the AP method plus dynamical correlation correction is a promising approach.
By calculating Hessian, one can also obtain frequencies of the normal modes. In Table 4, the
calculated frequencies of the normal mode singlet methylene are summarized. The significant
difference between the BS and AP methods can be found in a bending mode. The BS result
underestimates the binding mode frequency by the contamination of the triplet state. On the
other hand, the AP result gives close to the experimental result of
1
A
1
species. In this way, the
AP method is also effective for the normal mode analysis as well as the geometry optimization.
4. Summary
In this chapter, we explain how the BS method breaks the spin symmetry and AP method
recover it. In addition, we also demonstrate how those methods work the biradical systems.
The theoretical studies of the large biradical and polyradical systems such as polynuclear
metal complexes have been fairly realized by the BS HDFT methods in this decade. The BS
method is quite powerful for the large degenerate systems, but one must be careful about the
spin contamination error. Therefore the AP method would be important for those studies. For
example, it is suggested that the spin contamination error misleads a reaction path that
involves biradical transition states (TS) or intermediate state (IM) [73]. In addition, in the case
of the more larger systems, e.g., metalloproteins, some kind of semiempirical approach com-
bined with the AP hybrid DFT method by ONIOM method will be effective [74]. By using the
method, the mechanisms of the long-distance electron transfers and so on will be elucidated. In
Method θ
HCH

Mode
Symmetry Bent Antisymmetry
BS 114.1 3008 1069 3152
AP 104.5 2959 1252 3054
Expt.
c
(
1
A
1
) 102.4 2806 1353 2865
(
3
B
1
) 134.0 2992 963 3190
a
In cm
1
b
B3LYP/6–31++G(2d,2p) was used
c
In Refs. [66, 67] for singlet and triplet states, respectively.
Table 4. Calculated vibrational frequencies
a
of singlet methylene by SA, BS, and AP approaches with several methods
b
.
Symmetry (Group Theory) and Mathematical Treatment in Chemistry134

such cases, one also must be careful about the parameter of the semiempirical approach to fit
the spin-polarized systems. Recently, some improvements for PM6 method have been pro-
posed [75, 76]. Because the PM6 calculation can be utilized for the outer region in ONIOM
approach, therefore the AP method is also the effective method for the larger systems. In
addition, the BS wavefunction can be applied for other molecular properties by combining
with other theoretical procedures. For example, it was reported that the electron conductivity
of spin-polarized systems could be simulated by using the BS wavefunction together with
elastic Green’s function method [77], and some applications for one-dimensional complexes
have reported [78, 79]. The results indicate that the BS wavefunctions can be applied for
calculations of the physical properties of the strong electron correlation systems as well as their
electronic structures. The spin-projected wavefunctions seem to be effective for such simula-
tions of the physical properties. From those points of view, the BS and AP methods have a
great potential to clarify chemical and physical phenomena that are still open questions.
Author details
Yasutaka Kitagawa
1,2
*, Toru Saito
3
and Kizashi Yamaguchi
4
*Address all correspondence to: kitagawa@cheng.es.osaka-u.ac.jp
1 Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka, Japan
2 Center for Spintronics Research Network (CSRN), Graduate School of Engineering Science,
Osaka University, Toyonaka, Osaka, Japan
3 Department of Biomedical Information Sciences, Graduate School of Information Sciences,
Hiroshima City University, Hiroshima, Japan
4 Graduate School of Science, Osaka University, Toyonaka, Osaka, Japan
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