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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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123

ψBS

HOMO ¼cosθψHOMO þsinθψLUMO,(1)

ψBS

HOMO ¼cosθψHOMO sinθψLUMO,(2)

where 0 ≤θ≤45and 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

H¼2Jabb

Sab

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

H¼2Jab 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

http://dx.doi.org/10.5772/intechopen.75726

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

β¼α1 (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 RðÞβRðÞGHS RðÞ

þ∂αRðÞ

∂RELS

BS RðÞEHS 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 RðÞβRðÞFHS RðÞ

,

þ2∂αRðÞ

∂RGLS

BS RðÞGHS RðÞ

þ∂2αRðÞ

∂2RELS

BS RðÞEHS 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

¼cos2θsin2θ¼cos2θ:(20)

And because occupation number (n) of natural orbital (NO) for the corresponding orbital is

expressed as n¼2cos2θ, we get the relation:

T¼cos2θ¼n1 (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

1cos2θ

2ψLUMOψLUMO

¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2

1þT2

r1þT

2ψHOMOψHOMO

1T

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

s1T

2

()

2

¼1

212T

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

y¼2WD¼n24nþ4

n22nþ2:(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

*

):

b¼nn∗

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 ﬃ1T(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 ﬃ2n(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

Approximate Spin Projection for Broken-Symmetry Method and Its Application

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129

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 102.4and 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

.

Approximate Spin Projection for Broken-Symmetry Method and Its Application

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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

References

[1] Cotton FA, Walton RA. Multiple Bonds Between Metal Atoms. Oxford: Clarrendon Press; 1993

[2] Bera JK, Dunbar KR. Angewandte Chemie, International Edition. 2002;41:23

[3] Mashima K, Tanaka M, Tani T, Nakamura A, Takeda S, Mori W, Yamaguchi WK. Journal

of the American Chemical Society. 1997;119:4307

[4] Mashima K. Bulletin of the Chemical Society of Japan. 2010;83:299

[5] Murahashi T, Mochizuki E, Kai Y, Kurosawa H. Journal of the American Chemical

Society. 1999;121:10660

[6] Wang C-C, Lo W-C, Chou C-C, Lee G-H, Chen J-M, Peng S-M. Inorganic Chemistry. 1998;

37:4059

Approximate Spin Projection for Broken-Symmetry Method and Its Application

http://dx.doi.org/10.5772/intechopen.75726

135

[7] Lai S-Y, Lin T-W, Chen Y-H, Wang C-C, Lee G-H, Yang M-H, Leung M-K, Peng S-M.

Journal of the American Chemical Society. 1999;121:250

[8] Gatteschi D, Kahn O, Miller JS, Palacio F, editors. Magnetic Molecular Materials. Dor-

drecht: Kluwer Academic Publishers; 1991

[9] Kahn O, editor. Magnetism: A Supermolecular Function; NATO ASI Series C. Vol. 484.

Dordrecht: Kluwer Academic Publishers; 1996

[10] Coronado E, Dekhais P, Gatteschi D, Miller JS, editors. Molecular Magnetism: From

Molecular Assemblies to the Devices, NATO ASI Series E. Vol. 321. Dordrecht: Kluwer

Academic Publishers; 1996

[11] Müller A, Kögerler P, Dress AWM. Coordination Chemistry Reviews. 2001;222:193

[12] Sessoli R, Gatteschi D, Aneschi A, Novak MA. Nature. 1993;365:141

[13] Taft KL, Delfs CD, Papaefthymiou GC, Foner S, Gatteschi D, Lippard SJ. Journal of the

American Chemical Society. 1994;116:823

[14] Oshio H, Hoshino N, Ito T, Nakano M. Journal of the American Chemical Society. 2004;

126:8805

[15] Hoshino N, Nakano M, Nojiri H, Wernsdorfer W, Oshio H. Journal of the American

Chemical Society. 2009;131:15100

[16] Clerac R, Miyasaka H, Yamashita M, Coulon C. Journal of the American Chemical Society.

2002;124:12837

[17] Peng S-M, Wang C-C, Jang Y-L, Chen Y-H, Li F-Y, Mou C-Y, Leung M-K. Journal of

Magnetism and Magnetic Materials. 2000;209:80

[18] Kisida H, Matsuzaki H, Okamoto H, Manabe T, Yamashita M, Taguchi Y, Tokura Y.

Nature. 2000;405:929

[19] S-Y Lin, Chen I-WP, Chen C-H, Hsieh M-H, Yeh C-Y, Lin T-W, Chen Y-H, Peng S-M.

Journal of Physical Chemistry B. 2004;108:959

[20] Messerschmidt A, Huber R, Poulos T, Wieghardt K, editors. Handbook of Metalloproteins.

West Sussex, England: John Wiley & Sons, Ltd; 2001

[21] Siegbahn PEM. Chemical Reviews. 2000;100:4211

[22] Inoue T, Shiota Y, Yoshizawa K. Journal of the American Chemical Society. 2008;130:16890

[23] Nakatani N, Nakao Y, Sato H, Sakaki S. The Journal of Physical Chemistry. B;113:4826

[24] Noodleman L, Lovell T, Han W-G, Li J, Himo F. Chemical Reviews. 2004;104:459

[25] Siegbahn PEM. Inorganic Chemistry. 2000;39:2923

[26] Yamanaka S, Takeda R, Yamaguchi K. Polyhedron. 2003;22:2013

[27] Torres RA, Lovell T, Noodleman L, Case DA. Journal of the American Chemical Society.

2003;125:1923

Symmetry (Group Theory) and Mathematical Treatment in Chemistry136

[28] Day A, Jenney J FE, Adams MWW, Babini E, Takahashi Y, Fukuyama K, Hodgson KO,

Hedman B, Solomon EI. Science. 2007;318:1464

[29] Shoji M, Koizumi K, Kitagawa Y, Kawakami T, Yamanaka S, Okumura M, Yamaguchi K.

Lecture Series on Computer and Computational Sciences. 2006;7:499

[30] Kitagawa Y, Shoji M, Saito T, Nakanishi Y, Koizumi K, Kawakami T, Okumura M,

Yamaguchi K. International Journal of Quantum Chemistry. 2008;108:2881

[31] Carbo R, Klobukowski Ed M. Self-Consistent Field Theory and Applications. Amster-

dam: Elsevier; 1990. p. 727

[32] Yamaguchi K, Kawakami T, Takano Y, Kitagawa Y, Yamashita Y, Fujita H. International

Journal of Quantum Chemistry. 2002;90:370

[33] Yamaguchi K, Yamanaka S, Kitagawa Y. The nature of effective exchange interactions. In:

Palacio F et al., editors. Carbon Magnet. Elsevier; 2006. pp. 201-228

[34] Yamaguchi K, Kitagawa Y, Yamanaka S, Yamaki D, Kawakami T, Okumura M, Nagao H,

Kruchinin SK. In: Scharnburg K, editor. First Principle Calculations of Effective Exchange

Integrals for Copper Oxides and Isoelectronic Species, in NATO Series. Elsevier; 2006

[35] Caneschi A, Gatteschi D, Sangregorio C, Sessoli R, Sorace L, Cornia A, Novak MA,

Paulsen C, Wernsdorfer W. Journal of Magnetism and Magnetic Materials. 1999;200:182

[36] Cremer D. Molecular Physics. 2001;99:1899

[37] Roos BO, Taylor PR, Siegbahn PEM. Chemical Physics. 1980;48:157

[38] Andersson K, Malmqvist P-Å, Roos BO, Sadlej SJ, Wolinski K. The Journal of Chemical

Physics. 1990;94:5483

[39] Hirao K. Chemical Physics Letters. 1992;190:374

[40] Miralles J, Castell O, Cabollol R, Malrieu JP. Chemical Physics. 1993;172:33

[41] Calzado CJ, Cabrero J, Malrieu JP, Caballol R. The Journal of Chemical Physics. 2002;116:

2728

[42] Calzado CJ, Cabrero J, Malrieu JP, Caballol R. The Journal of Chemical Physics. 2002;116:

3985

[43] Miehlich B, Stoll H, Savin A. Molecular Physics. 1997;91:527

[44] Grafenstein J, Cremer D. Chemical Physics Letters. 2000;316:569

[45] Takeda R, Yamanaka S, Yamaguchi K. Chemical Physics Letters. 2002;366:321

[46] Laidig WD, Bartlett RJ. Chemical Physics Letters. 1983;104:424

[47] Jeziorski B, Paldus JJ. Chemical Physics. 1988;88:5673

[48] Mahapatra US, Datta B, Mukherjee D. The Journal of Physical Chemistry. A. 1999;103:

1822

Approximate Spin Projection for Broken-Symmetry Method and Its Application

http://dx.doi.org/10.5772/intechopen.75726

137

[49] Yanai T, Chan GK-L. The Journal of Chemical Physics. 2006;124:19416

[50] Kurashige Y, Yanai T. The Journal of Chemical Physics. 2009;130:234114

[51] Yanai T, Kurashige Y, Neuscamman E, Chan GK-L. The Journal of Chemical Physics.

2010;132:024105

[52] Löwdin P-O. Physics Review. 1955;97:1509

[53] Lykos P, Pratt GW. Reviews of Modern Physics. 1963;35:496

[54] Yamaguchi K, Yamanaka S, Nishino M, Takano Y, Kitagawa Y, Nagao H, Yoshioka Y.

Theoretical Chemistry Accounts. 1999;102:328

[55] Szabo A, Ostlund NS. Modern Quantum Chemistry. New York: Dover Publications, Inc;

1996. ch. 3. pp. 205-230

[56] Sonnenberg JL, Schlegel HB, Hratchian HP. In: Solomon IE, Scott RA, King RB, editors.

Computational Inorganic and Bioinorganic Chemistry (EIC Books). UK: John Wiley &

Sons Ltd; 2009. p. 173

[57] Mayer I. In: Löwdin P-O, editor. Advances in Quantum Chemistry. Vol. vol. 12. New

York: Academic Press, Inc; 1980. p. 189

[58] Löwdin P-O. Reviews of Modern Physics. 1964;36:966

[59] (a) Yamaguchi K, Takahara Y, Fueno T, Houk KN. Theoretica Chimica Acta. 1988;73:337;

(b) Takahara Y, Yamaguchi, K, Fueno T, Chemical Physics. Letters. 1989;157:211; (c)

Yamaguchi K, Toyoda Y, Fueno T. Chemical Physics. Letters. 1989;159:459

[60] Yamaguchi K, Okumura M, Mori W. Chemical Physics Letters. 1993;210:201

[61] Yamanaka S, Okumura M, Nakano M, Yamaguchi K. Journal of Molecular Structure:

THEOCHEM. 1994;310:205

[62] Kitagawa Y, Saito T, Ito M, Shoji M, Koizumi K, Yamanaka S, Kawakami T, Okumura M,

Yamaguchi K. Chemical Physics Letters. 2007;442:445

[63] Saito T, Nishihara S, Kataoka Y, Nakanishi Y, Matsui T, Kitagawa Y, Kawakami T,

Okumura M, Yamaguchi K. Chemical Physics Letters. 2009;483:168

[64] Kitagawa Y, Saito T, Nakanishi Y, Kataoka Y, Shoji M, Koizumi K, Kawakami T,

Okumura M, Yamaguchi K. International Journal of Quantum Chemistry. 2009;109:3641

[65] Kitagawa Y, Saito T, Nakanishi Y, Kataoka Y, Matsui T, Kawakami T, Okumura M,

Yamaguchi K. The Journal of Physical Chemistry. A. 2009;113:15041

[66] Petek H, Nesbitt DJ, Darwin DC, Ogilby PR, Moore CB, Ramsay DA. The Journal of

Chemical Physics. 1989;91:6566

[67] Bunker PR, Jensen P, Kraemer WP, Beardsworth R. The Journal of Chemical Physics.

1986;85:3724

Symmetry (Group Theory) and Mathematical Treatment in Chemistry138

[68] Hargittai M, Schults G, Hargittai I. Russian Chemical Bulletin, International Edition.

2001;50:1903

[69] Kitagawa Y, Soda T, Shigeta Y, Yamanaka S, Yoshioka Y, Yamaguchi K. International

Journal of Quantum Chemistry. 2001;84:592

[70] Kitagawa Y, Kawakami T, Yamaguchi K. Molecular Physics. 2002;100:1829

[71] Kitagawa Y, Kawakami T, Yoshioka Y, Yamaguchi K. Polyhedron. 2001;20:1189

[72] Kitagawa Y, Yasuda N, Hatake H, Saito T, Kataoka Y, Matsui T, Kawakami T, Yamanaka

S, Okumura M, Yamaguchi K. International Journal of Quantum Chemistry. 2013;113:290

[73] Saito T, Nishihara S, Kataoka Y, Nakanishi Y, Kitagawa Y, Kawakami T, Yamanaka S,

Okumura M, Yamaguchi K. The Journal of Physical Chemistry. A. 2010;114:12116

[74] Kitagawa Y, Yasuda N, Hatake H, Saito T, Kataoka Y, Matsui T, Kawakami T, Yamanaka

S, Okumura M, Yamaguchi K. International Journal of Quantum Chemistry. 2013;113:290

[75] Saito T, Kitagawa Y, Takano Y. The Journal of Physical Chemistry. A. 2016;120:8750

[76] Saito T, Kitagawa Y, Kawakami T, Yamanaka S, Okumura M, Takano Y. Polyedeon. 2017;

136:52

[77] Nakanishi Y, Matsui T, Kitagawa Y, Shigeta Y, Saito T, Kataoka Y, Kawakami T, Okumura

M, Yamaguchi K. Bulletin of the Chemical Society of Japan. 2011;84:366

[78] Kitagawa Y, Matsui T, Nakanishi Y, Shigeta Y, Kawakami T, Okumura M, Yamaguchi K.

Dalton Transactions. 2013;42:16200

[79] Kitagawa Y, Asaoka M, Natori Y, Miyagi K, Teramoto R, Matsui T, Shigeta Y, Okumura

M, Nakano M. Polyhedron. 2017;136:125

Approximate Spin Projection for Broken-Symmetry Method and Its Application

http://dx.doi.org/10.5772/intechopen.75726

139