Supporting Information on
“Finite-Field Method with Unbiased Polarizable Continuum Model
for Evaluation of the Second Hyperpolarizability of an Open-Shell
Singlet Molecule in Solvents”
Tomoya Inui, Yasuteru Shigeta,* Katsuki Okuno, Takeshi Baba, Ryohei Kishi, and Masayoshi
Nakano*
Department of Materials Engineering Science, Graduate School of Engineering Science, Osaka
University, Toyonaka, Osaka 560-8531, Japan
S.1. Correlation between  and y for open-shell singlet molecules.
The diradical character yi, which is related to the occupation numbers of (HOMO − i) and (LUMO
+ i), is defined by twice the weight of doubly excited configuration in the multi-configuration
self-consistent-field (MCSCF) wave function.[15]
Under the broken symmetry approaches, yi is
approximately expressed using the overlap Ti between the corresponding orbital pair (HOMO–i and
LUMO+i) as[13-15]
yi = 1-
2Ti
,
1+ Ti 2
Ti = ò c HOMO-i ( r)hHOMO-i ( r) dr =
(S.1)
nHOMO-i - nLUMO+i
,
2
1
(S.2)
ì
æ qi ö
æ qi ö
ï c HOMO-i ( r ) = cos çè 2 ÷ø j HOMO-i ( r ) + sin çè 2 ÷ø j LUMO+i ( r )
ï
í
q
q
ïh
( r) = cos æçè i ö÷ø j HOMO-i ( r ) - sin æçè i ö÷ø j LUMO+i ( r)
ïî HOMO-i
2
2
(S.3)
where nj is the occupation number of the unrestricted Hartree-Fock (UHF) natural orbital (UNO) x
( x = (HOMO - i) and (LUMO + i)), and i expresses UNO and orbital mixing parameters,
respectively. The diradical character yi runs from 0 to 1, where yi = 0 and 1 mean the closed-shell
and the pure diradical states, respectively.
Our previous studies on both model and realistic systems
demonstrated that the second hyperpolarizability  in the intermediate y region is larger than those of
the closed-shell and the pure open-shell states, and has a maximum in the intermediate y region.
These works reveal that the diraical character is a reasonable indicator taken from its ground state
electronic structure calculation.
For example, we here demonstrate the y- relationship for the
compound A (Figure 1b) in gas obtained from FF calculations with the LC-UBLYP/6-31G(d) and
CCSD(T)/6-31G(d) methods of 
-31G(d) calculations of y.
The diradical
character y and the second hyperpolarizability  are shown in Figure S.1 as a function of an
intra-complex distance, R.
In the short distance region, where the diradical character y is relatively
large, the  value is found to be small to some extent.
the decreases of y and the increases of 
On the other hand, as the distance R increases,

Rmax = 2.2
Å and 2.6 Å at the LC-UBLYP and CCSD(T) levels of approximation, respectively. This figure
clearly shows the y- relationship, where the 
Although the 
rmediate y region.
Rmax depend on the choice of the calculation methods as well as on the
given system, this relationship is universal for all open-shell singlet systems [see details for refs.
16-30, 38 in text].
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Figure S.1. B-O distance (R) [Å] dependences of  [a.u.] and diradical character y [-] of compound A.
S.2 Dependence of  on the calculation methods
The variation of  values of the complex in gas and in water (using the FF-PCM) as the function of
B-O distance (R) are shown in Figures S.2 (at the UMP2, UMP4SDQ, PUMP2, UCCSD, and
UCCSD(T) level of approximations) and S.3 (at the UB3LYP, UM06, and UPW91 level of
approximations). The post-HF PCM calculation at R = 1.5 Å was not converged.
As seen from
all the results, although the  amplitude obtained by each method is different from each other, the 
values vary with R and have the maximum values at certain R values depending on the calculation
methods. As seen in the previous paper [ref. 38 in text], such variations are found to be associated
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with the variation in the diradical character with the increase in R (see Supporting Information S.1).
Similar to the previous results, it is found that all the PCM methods show the remarkable
enhancement of the  values and that the R value giving the maximum  in every PCM method is
larger than that in gas.
In case of using the post-HF methods, the  obtained by each method is shown to be
similar to each other. As rising the approximation levels, the results are shown to step-by-step
approach the UCCSD(T) results.
In contrast, the DFT results are found to strongly depend on the
choice of the functionals. The UPW91 and UB3LYP results are shown to be quite different from
the UCCSD(T) results, while the LC-UBLYP and UM06 methods are found to give similar results
to the UCCSD(T) results. Figure S.4 shows the variation in the ratio between  values in gas and
in solvent with the increase in R.
It is found that the UM06 method gives a bit large ratio as
compared to the UCCSD(T) method, while that the LC-UBLYP method well reproduces the
UCCSD(T) results. Thus, the LC-UBLYP method is employed to evaluate the optical properties
in this study.
1
K. Kubota, H. Takahashi, H. Fukui, S. Bonness, K. Yoneda, R. Kishi, T. Kubo, K. Kamada,
K. Ohta, B. Champagne, E. Botek, M. Nakano, Chem. Phys. Lett. 2009, 477, 309-314.
4
(a) In gas
(b) In solvent by the PCM method
Figure S.2. B-O distance (R) [Å] dependences of  [a.u.] of the complex (composed of compound A
and a water molecule) in gas and in water (FF-PCM) using the pots-HF methods, i.e.,
the UMP2, UMP4SDQ, PUMP2, UCCSD and UCCSD(T) methods.
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(a) In gas
(b) In solvent by the PCM method
Figure S.3. B-O distance (R) [Å] dependences of  [a.u.] of the complex (composed of compound A
and a water molecule) in gas and in water (FF-PCM) using the density functional theory
methods, i.e., the UB3LYP, UM06, UPW91 and UCCSD(T) methods.
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Figure S.4. Ratio between  [a.u.] of the complex (composed of compound A and a water molecule)
in gas and that in water (FF-PCM) calculated using the UB3LYP, UM06, UPW91 and
UCCSD(T) methods.
7