An ab initio study of excited states of the phthalocyanine magnesium complex and its cation radical

20 September 1996

CHEMICAL PHYSICS LETTERS ELSEVIER

Chemical Physics Letters 260 (1996) 178-185

An ab initio study of excited states of the phthalocyanine magnesium complex and its cation radical Naoto Ishikawa 1'*, David Maurice, Martin Head-Gordon * a Department of Chemistry, University of California, Berkeley, CA 94720, USA b Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA Received 6 May 1996; in final form 25 June 1996

Abstract

Excited states of the magnesium phthalocyanine complex [Mg(Pc)] and its cation radical are studied by ab initio configuration interaction with single substitutions (CIS), restricted open-shell CIS (ROCIS) and the extended CIS (XCIS) method. Moderately satisfactory agreement between calculated and observed transitions is obtained, and an assignment of the absorption bands of the radical species is proposed.

1. Introduction

Excited states of phthalocyanines (Fig. 1) have been long subjected to quantum chemical study. Particularly significant early work with the SCMOPPP-CI model predicted two absorption bands in the visible region, termed the Q and B bands, in excellent agreement with experimental data [1]. Subsequent CI studies [2,3] and extended Hiickel calculations [4,5] are still relied on by experimentalists to analyze their data. As a non-empirical treatment, DV-Xet [6-8] and DFT [9] methods have been used in assignments of photoelectron and optical spectra of metallophthalocyanines although their treatments of the excited states were not rigorous. For ground states, ab initio SCF, CI and MCSCF methods have * Corresponding authors. Present address: Department of Chemistry, Tokyo Institute of Technology, O-okayama, Meguro-ku, Tokyo 152, Japan.

been used to interpret X-ray photoemission spectroscopic data of the copper complex [10]. For the smaller closely related macrocycles, porphyrins, accurate, but computationally demanding multireference CASSCF/CASPT2 [11] and single reference SAC-CI [12] calculations of excited states were reported recently. The simpler CIS(D) method [13], which perturbatively approximates the linear response [14,15] (or equation of motion [16]) coupled-cluster theory with single and double substitutions (CCSD), has been shown to predict excitation energies of metalloporphyrins to roughly comparable accuracy [17]. Excitation energies calculated at the lower single excitation configuration interaction (CIS) level are qualitatively correct, but often have significant errors due to neglect of electron correlation. Unlike most general CI methods, CIS, CIS(D), excited state CCSD, and SAC-CI are strictly sizeconsistent (excitation energies of a system of infinitely separated molecules include those obtained

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N. Ishikawa et al. / Chemical Physics Letters 260 (1996) 178-185

179

report calculations on the excited states of the neutral closed-shell species, [Mg(Pc)].

2. Methods of calculation

Fig. 1. Molecular structure of metallophthalocyanine.

from calculations on the individual molecules alone); hence their quality has no artificial dependence on molecular size. Although there may be some corrections to the energy ordering of the states in the B band manifold and higher, the fundamental understanding of the low-lying electronic excited states of closed shell phthalocyanines appears to be solid. It is hence a suitable case with which to validate the performance of a theoretical method prior to subsequent applications to less well characterized phthalocyanines. For example, for the corresponding radical cation species, there are still fundamental arguments concerning the assignment of the absorption bands. The radical associates with closed-shell species to form dimer and trimer radicals such as [Lu(Pc) 2] [18] and [Lu2(Pc)3 ]+ [19] and radical salt crystals [20-22] which are well investigated organic conductors. To begin to understand their unique properties, a qualitative understanding of the electric structure of the ground and excited states of the monomer radical is desirable. The first extensive study of the absorption band assignments of phthalocyanine radical cation is the one reported by Stillman and co-workers as part of their extensive MCD and absorption spectrum analysis [23,24]. Subsequently, Ishikawa et al. have reported a semi-empirical calculation, which suggests a totally different assignment except for the lowest excited state (Q band) [25]. The purpose of this paper is to report the first ab initio molecular orbital calculations of the excited states of the magnesium phthalocyanine radical cation, [Mg(Pc)] + to attempt to clarify the assignment of the experimentally observed absorption spectrum. For comparison, we also

Optimized geometries of [Mg(Pc)] and [Mg(Pc)] +' were obtained using restricted and restricted openshell HF methods, respectively, with the 6-31G* basis set. These calculations were performed using the GAUSSIAN 92 program [26]. The excited states were calculated using singleexcitation configuration interaction (CIS) methods [27]. While CIS methods do not yield quantitative accuracy due to the neglect of electron correlation effects, all electron calculations with reasonably large basis sets are now feasible at this level of theory for molecules the size of phthalocyanines even on workstation-class computers. CIS excited states are generally qualitatively correct for the excited states observed in one-photon absorption, and the combination of excitation energies and oscillator strengths often permits assignment of at least the most intense features of the absorption spectrum. For the radical species, we first employ the simple restricted open-shell CIS (ROCIS) [28] method. ROCIS theory for an open-shell doublet system incorporates the following three types of spin-adapted excited configurations: I(s --, v ) 5 =

'/'~,

I(d --~ s)) = q~, I(d -~, v) s) = { ~ + ~dV}/¢~. Here, s, d and v are singly occupied, doubly occupied and virtual orbitals, respectively. However, ROCIS has recently been shown to be significantly poorer than CIS applied to molecules with closedshell ground states. This is primarily because omitted configurations, I(d ~ v)T} = {2q~; -- qbdv + ~dV}/¢6 play a significant role in many low-lying radical excited states. As well as being a pure doublet configuration, this can be viewed as an analogue of a triplet state of a closed-shell system. This is closely related to the 'trip-doublet' states [29] studied extensively in transition metal porphyrin complexes such

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N. lshikawa et al. / Chemical Physics Letters 260 (1996) 1 7 8 - 1 8 5

as [Cu(II)OEP]: an excited state which would be triplet in a closed-shell complex becomes a spin-allowed doublet due to the unpaired electron on the central metal ion. We hence employ an extended CIS method (XCIS) [30] which includes these configurations in the excited states for calculations on radicals. XCIS is a particular choice of CI expansion which like CIS itself ensures both size-consistency and variationality in the excited states. As such it can also be obtained as a special case within a general configuration interaction program. All excited state calculations were performed using the Q-Chem quantum chemistry program package [31]. The CIS, ROCIS and XCIS theories are implemented using direct methods, in which no two-electron repulsion integrals are stored on disk. Non-abelian point group symmetry was fully utilized in the implementation, and in the calculations reported here. For both the closed-shell and cation radical species, we used the frozen core approximation: the lowest 45 MOs, which were dominated by ls orbitals of the C and N atoms and ls, 2s, 2p orbitals of the Mg atom, were excluded from construction of the excited configurations.

3. Results

and

5-

---

eV

"----

0,

7a~, (n*)

__

7eg (Tg*)

20alg (Ms 4s) 4bz~ (re*) 6a2~ (n*) 3bl. (rt*)

6%(x*) - - - -

__

2alu (r~)

--

5a~u (~) 3b2u (x)

-10

__

5eg (~) 4eg (~) 2bl. (n)

--

la]u (re)

--

4az~(~) 13b2g (n) 3e s (n) ~29e. (n)~ 15b z (O') 19a1~ (n) 2b~, (~)

discussion

3.1. lMg(Pc)I 6-31G The MO energy diagrams of [Mg(Pc)] calculated at H F / 6 - 3 1 G and 6-31G * are shown in Fig. 2. The molecular geometry is of D~ point group symmetry. As found in earlier calculations, the 2a~u HOMO and the 6eg LUMO are energetically separated from the other orbitals, which results in a lowest excited state which is dominated by the H O M O - L U M O transition. By contrast with the metal porphyrins which have accidentally degenerate alu and a2u HOMOs, the next-HOMO of [Mg(Pc)], 5a2,, is well-separated from the HOMO and forms a group with other rr-orbitals in the range from ca. - 9 to - l 0 eV. Four n-orbitals characterized by the non-bonding orbitals of the bridging nitrogen atoms lie below -11 eV, which is much lower than the results of earlier extended Hiickel calculations for [Zn(Pc)]. In that work, 13b2g and 3e, were predicted to be close to the 2alu HOMO, and 19a~g was predicted to be

--

- - 30eu(Mg4p)- - -

__

-5

8% (~*) 3ai~ (~*)

-~-~-

--

_~

--

6-31G*

Fig. 2. Energy level diagram of Hartree-Fock molecular orbitals for D4h [Mg(Pc)] calculated with the 6-31G (left) and 6-31G* (right) basis sets.

between the HOMO and next-HOMO [4]. There is no metal-centered occupied orbital above - 15 eV. The calculated low-lying excited states of [Mg(Pc)] are shown in Fig. 3. Dipole-transition-allowed states are of E u (x, y polarized) and A2u (z polarized) symmetry. The calculation with the minimal STO-3G basis set does not give good agreement with experiment. Changing to a split-valence 6-31G basis improves the calculated excitation energies to some degree. The lowest singlet excited state, IE u, is lowered to the greatest extent by 0.7 eV. The effect of adding polarization functions (via the 6-

N. lshikawa et al./ Chemical Physics Letters 260 (1996) 178-185

31G* basis) leads to relatively small additional changes for the states shown in the figure. In a comparative study using smaller molecules, the CIS(D) method, which incorporates double excitations perturbatively, was shown to provide significant improvements relative to the CIS excitation energy (although other properties were less strongly improved) [32]. Recently we have shown that this is also the case for porphyrin and tetraazaporphyrin compounds. In particular, the T t - S t and S t - S 2 energy differences were significantly improved from 1.0 and 2.2 eV in CIS to 0.2 and 0.9 eV in CIS(D) for magnesium porphyrin [17] whereas experimental values for magnesium etioporphyrin are 0.4 and 1.0 eV [29,33]. The same is anticipated for [Mg(Pc)], but

181

we have not yet been able to complete such calculations because of computational limitations. Table 1 presents the CIS/6-31G energies and oscillator strengths, together with the main transitions characterizing each state. The 1E u state is predominantly a HOMO-LUMO transition (92% in the 6-31G basis). The Q band at 15000 cm -l corresponds to this state, consistent with all earlier assignments based on semiempirical calculations [1-5]. The 2E u state whose main character is 3b2, ~ 6eg lies below the 3E u state which has a large oscillator strength. This suggests the existence of a weak band at the foot of the broad B band. In this region, the second-derivative absorption spectrum of [AI(CI)(Pc)] in ethanol solution shows a weak band which ex-

Table 1 C I S / 6 - 3 1 G singlet excited states of [Mg(Pc)] Energy (eV)

f

Character

IE u lA2g IBlg IAlu lB2g

2.32 4.76 4.85 4.89 4.94

0.86 -

2E u

5.02

0.0054

2A2g

5.03

-

2B 2g

5.34

-

3Eu

5.36

2.40

2B ~g

5.58

-

4E u

5.74

0.85

5Eu

5.85

0.054

lA2u

5.90

0.029

6E u

6.14

0.18

7E u

6.68

0.0020

8Eu

7.24

0.060

2at~ ~ 6eg 2alu ~ 6a2u 2alu ~ 3blu 2atu ---, 20alg 5eg ~ 6eg 2alu ~ 462u 3b2, ~ 6eg 4a2u ~ 6eg 5eg ~ 3blu 5eg ~ 6eg 3b2u ~ 3blu 2alu ~ 462u 5eg ~ 6eg 5a2u ~ 6eg 4a2u ~ 6eg 2blu ~ 6eg 4eg ~ 6eg 5eg ~ 6eg 2al~ ~ 3btu 5a2u ~ 6eg 2blu ~ 6eg 2at~ ~ 7eg 2atu ~ 7eg 3b2u -'* 0eg 29% ~ 6eg 13b2g ~ 3blu 2blu ~ 6eg 4aEu ~ 6eg laju ~ 6eg 2bl~ ~ 6eg 2alu ~ 8eg lalu ~ 6eg

92% 91% 85% 98% 52% 25% 52% 21% 8% 74% 9% 65% 23% 43% 20% 11% 62% 10% 10% 33% 11% 10% 73% 11% 62% 17% 34% 31% 44% 23% 32% 27%

'Q'

t ~ Mg(4s)

"BI'

'B2'

(n, "~ * ) (n, ~r * ) 'N'

N. Ishikawa et aL / Chemical Physics Letters 260 (1996) 178-185

182

il

Table 2 CIS/6-31G triplet excited states of [Mg(Pc)] Energy (eV) Character

Abs. I

0

10

0

10

20

30

I

I

I

40

50

60

u ~ 50

6O

IE u 2Eu 3E u

0.71 3.61 4.10

2alu ~ 6eg 5a2u ~ 6eg 5a2u ---,6eg

93% 43% 29%

'Q' 'B'

3.2. [Mg(Pc)] + ~C

in

~

;C

j

I

30

4O

T, 1nO

20

r

,,, x =o.T,; ° . , 30

40

'it 631 0

'-

6-31G

,, 0

i

20

1'o

50

60

~o

60 '

T 20 '

30

40

Wavenumber/103cm-~ Fig. 3. Calculated excitation energies and oscillator strengths of Dab [Mg(Pc)] with the CIS method using various basis sets. Electric-dipole-transition-allowed (Eu and A2u) singlet, forbidden singlet, and triplet states are marked by a solid rhombus, triangle, and cross, respectively. The top of the figure is an absorption spectrum of a typical metallophthalocyanine, [AI(CIXPc)], in ethanol solution.

The ground state is Alu, where an unpaired electron is present in the 2a], orbital (see Fig. 2). Electric-dipole transitions to Eg and A2g states are allowed. Fig. 4 (middle) shows the result of ROCIS calculations of doublet excited states of [Mg(Pc)] +'. The characters of the states are summarized in Table 3. The lEg state is again dominated by the transition from the (half-filled) H O M O to LUMO. The 5Eg state can be assigned to the radical analog of the B band because of the intensity and its primary component, which is the n e x t - H O M O - L U M O excited configuration (although the configuration is spread out among higher levels of the same symmetry due to

Abs.

i

0

hibits an MCD A term indicating that it corresponds to an Eo state [34]. This band may be tentatively assigned to the 2E u state. The pair of 3Eu and 4E u states can be assigned to the two bands under the B band envelope which were elucidated by the band deconvolution study by Stillman and co-workers [35]. Both of the states include large contributions from the 5azu ~ 6 e g transition. The lowest A2u state, which is of the other dipole-allowed symmetry, is predicted at 5.90 eV (47600 c m - J ). The main contributions are from 29e~ --~ 6eg (62%) and 13bEg --~ 3blu (17%), which corresponds to the lowest (n, w * ) transitions. This contrasts with the earlier semiempirical calculations where (n, ~r * ) transitions were predicted below the B bands. The triplet states which correspond to the B band are predicted to lie between the singlet Q band and B band, as summarized in Table 2.

it 0

[

10

20

30

210

30

CIS/6-31G

110

40

T J 40

50

XCIS/6-31G 1

o, 0

,, 10

,; 20

,

...:, 30

Wavenumber/ 103cm-~

TT, 40

510

Fig. 4. Comparison of restricted open-shell CIS and XCIS excitation energies and oscillator strengths of [Mg(Pc)]+'. The basis set used is 6-31G. Only electric-dipole-transition-allowed (Eg and A2g) doublet states are shown. The top of the figure is an absorption spectrum of [Mg(Pe)]+" in methylene chloride reproduced from the data in Ref. [24].

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N. Ishikawa et a l . / Chemical Physics Letters 260 (1996) 178-185

Table 3 ROCIS/6-31G doublet excited states of [Mg(Pc)] ÷ Energy (eV)

f

Character

leg 2Eg 3Eg 4Eg 5Eg

1.53 4.03 4.33 5.77 5.84

0.29 0.0013 1.08 0.059 2.01

IAzg

6.48

0.032

2 a , u - , 6eg 4eg ~ 2alu 5eg ~ 2alu 2alu ~ 7eg 5a2u ~ 6eg 3eg~2alu 29% ~ 6eg 13b2g~3blu

92% 63% 74% 63% 24% 8% 27% 9%

'Q' (d ~ s) (d--* s) (s ~ v) (s ~ v) (d~s) (n, "tr * ) (n, rr *)

the close occupied orbital spacing). Particularly interesting is the 3Eg state which is located between the Q and B bands and has a large oscillator strength.

This suggests that the 'fingerprint' band of the cation radical observed at 20000 cm-1 should be assigned to this electronic transition from the highest occupied eg orbital to the half-filled HOMO. However, as discussed in Section 2, ROCIS results for radicals have to be viewed with some caution because of the neglect of the triplet-like configurations which contain partial double excitation character. In the present -rr-radical cation, such configurations involve coupling of the unpaired electron in the 2alg orbital with triplet states of neutral [Mg(Pc)] which involve transitions from MOs below the HOMO (e.g. 3b2u ~ 6eg and 5a2u -0 6eg from Table 2) to yield doublet states. As seen in the previous section for the closed-shell species, triplet

Table 4 XCIS/6-31G doublet excited states of [Mg(Pc)] + Energy (eV)

f

(d ~ v)T contribution

Character

lEg 2Eg

1.08 2.59

0.18 0.0020

5.7% 55%

3Eg

3.28

0.29

31%

4Eg

4.00

0.025

50%

2alu ~ 6eg 4eg --, 2aiu 3b2u ~ 6eg 5eg ~ 2a,u 4eg ~ 2a,u 2blu ~ 6eg 2alu ~ 7eg 4eg ~ 2alu 2b,u --* 7eg 5eg ~ 7a2u 3eg ~ 2alu 4a2u ~ 6eg 5a2u --~ 6eg 3b2u ~ 6eg 4a2u -~ 6eg 5aEu ~ 6eg 362u --~ 6eg 4a2u ~ 6eg 2btu -~ 6eg 5azu ~ 6eg 3b2u ~ 0eg 3b2u ~ 6eg 5a2u ~ 6eg 4eg ~ 4b2u 2blu ~ 6eg 3eg --* 2a~u 5a2u --~ 6eg 29% ~ 6eg 29% - , 6eg 13b2g ~ 3blu 13b2g ~ 3blu 2alu - , 7eg 2blu ~ 6eg 2alu --~ 8eg

5Eg

4.46

0.015

55%

6Eg

4.83

0.032

24%

7Eg

5.15

1.30

22%

8Eg

5.43

0.60

58o/0

9Eg

5.71

0.55

24%

1A2u

5.74

0.014

48%

10Eg

5.88

0.22

41%

86% 35% 9% 50% 9% 9% 16% 9% 8% 7% 28% 15% 15% 35% 9% 9% 9% 19% 19% 21% 9% 9% 8% 8% 24% 12% 8% 32% 24% 8% 9% 17% 9% 6%

,Q, (d (d (d (d (d

~ ~ ~ ~ ~

s) v)T s) s) v)T

(s --, v) (d (d (d (d (d (d

~ s) ~ v)T ~ v)T ~ s) ~ v)T --* v)T

(d --' v)s (d ~ v) s (d ~ v) s (d ~ v)* (d ~ v) s

(d ~ (d ~ (d ~ (d ~

v) s v) s v) s V)T

(d --, v)T (d -'} V)T

(d ~ v) s (d ~ S) (d ~ v) s (n, ~r* )s (n, 'rr *)T (n, "n" ")S (n, ~ ,)T (s ~ v) (d ~ v)s (d ~ v) s

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N. lshikawa et a l . / Chemical Physics Letters 260 (1996) 178-185

states are predicted below the B band of singlet [Mg(Pc)], and therefore for the cation radical the corresponding (d ~ v) T states are expected to be at similar energy. Although these configurations do not contribute to the transition dipole moment, through configuration interaction with other dipole-allowed configurations, allowed excited states can appear in this region. Indeed, a previous semiempirical calculation has predicted a state whose main character is a 'triplet-like' configuration below the (5a2u ~ 6eg) s state [25]. The results of XCIS calculations of the excitation energies of [Mg(Pc)] ÷" are presented in Fig. 4 (bottom) and Table 4. The three lowest excited states derive from the three lowest ROCIS excited states in Table 3, although there is substantial mixing of (d ~ v) T configurations into the latter two states. The lEg state corresponds to the Q band, while the 3Eg state appears to correspond to the 'fingerprint' band at 20000 cm -1, as in the ROCIS case. This assignment, however, disagrees with the previous proposal of Stillman and co-workers which assigned the 'fingerprint' band to the transition from a nondegenerate w-orbital to the half-filled HOMO [24]. The non-degeneracy was deduced from MCD data showing a B term maximum at this band. However, such a state was not found anywhere near this region in our calculations. Another candidate for the 'fingerprint' band would be the lowest non-degenerate allowed 1A2g state. In the XCIS calculations, it is located at relatively high energy, 5.74 eV (46300 c m - l ) , with an oscillator strength of 0.014. This lA2g state is predominantly described by (n, ~r*) transitions such as 29e u --->6eg (56%) and 13b2g 3blu (17%), which correspond to the 1A2u state in the closed-shell species. Although this state gives only an MCD B term, the calculated excitation energy seems too high to make it a strong candidate for the 20000 cm-1 fingerprint band. The next major question is the identity of the most intense feature in the spectrum, centered at roughly 30000 cm -1. Based on our calculations, this band appears to correspond to the manifold of states 7Eg, 8Eg, 9Eg and 10Eg. In particular, the 7Eg and 8Eg states can be viewed as being derived from the B band of neutral [Mg(Pc)], based on the primary transitions shown in Table 4. This assignment of the 30000 cm-1 band as the radical analog of the B

band of neutral [Mg(Pc)] differs from the conclusion of Stillman and co-workers, who assigned the pair of peaks in the vicinity of 25000 c m - l to B1 and B2 transitions [24]. However, it is consistent with the intensity of the B band being largest in the absorption spectrum for both the neutral and cation radical. Two states whose triplet-like (d ~ v) T contribution exceeds 50% lie above the 2Eg and 3Eg (d ~ s) states, and below the 7Eg and 8Eg states which we have identified as the B band manifold. The 4Eg and 5Eg states can be tentatively assigned to the two absorption peaks at around 25000 cm-1, although the calculated oscillator strength appears somewhat small. These assignments are generally consistent with those made by Ishikawa et al. in a semiempirical SCMO-CI calculation of the cation radical [25]. They calculated two kinds of allowed transitions, of (d ~ s) and (d ~ v) T character in the present notation, between the Q band and the B band. They assigned the 20000 cm -1 band to be eg(d) ~ alo(s) and the bands at 25000 cm-1 were assigned to a state which is dominated by (a2u(d)~ eg(V))T, in other words the trip-doublet analog of the B band.

4. Conclusions

We have reported ab initio calculations of the excited singlet and triplet states of the magnesium phthalocyanine complex, and doublet excited states of the corresponding cation radical. The principal objectives of the study were to examine the suitability of single-excitation-based excited state methods for studying large molecules of this type, and to clarify the assignment of the main absorption bands in the cation radical. Our conclusions are as follows: (1) Single excitation configuration interaction (CIS) calculations on [Mg(Pc)] lead to singlet excited states whose relative intensities and energy differences are in moderately satisfactory agreement with established experimental and theoretical data for this well-characterized molecule. Excitation energies are not predicted with quantitative accuracy, but a clear assignment of calculated states to experimental absorption bands is still possible. Additionally the calculations predict a triplet state which corresponds to the B band, located in energy between the singlet Q band and singlet B band.

N. Ishikawa et al./ Chemical Physics Letters 260 (1996) 178-185

(2) E x t e n d e d CIS ( X C I S ) calculations on the radical species, [Mg(Pc)] ÷" lead to an a s s i g n m e n t o f the principal absorption bands seen experimentally. The m o s t intense b a n d at 30000 c m - ~ is assigned to the doublet analog o f the B band, the m o s t intense transition in the neutral c o m p l e x . The l o w - l y i n g transition at 12000 c m - 1 is assigned to the radical analog o f the Q band in the neutral c o m p l e x . The ' f i n g e r p r i n t ' band at 20000 c m - 1 is assigned to a (5eg ~ 2a~o) band, based on its intensity and relative energy. T h e band at 25000 c m - ~ is assigned to a doublet analog o f the triplet-like B band excitation, w h i c h b e c o m e s a l l o w e d due to m i x i n g with other c o n f i g u r a t i o n s w h i c h h a v e a n o n - z e r o transition m o ment. Further calculations which properly account for d y n a m i c e l e c t r o n correlation are desirable to further test these conclusions.

Acknowledgements NI w i s h e s to a c k n o w l e d g e a f e l l o w s h i p f r o m the Japan S o c i e t y for the P r o m o t i o n of Science. T h i s w o r k was supported by the Director, O f f i c e o f Ene r g y R e s e a r c h , O f f i c e o f Basic E n e r g y Sciences, C h e m i c a l S c i e n c e s D i v i s i o n o f the U.S. D e p a r t m e n t of E n e r g y u n d e r Contract D E - A C 0 3 - 7 6 S F 0 0 0 9 8 . W e a c k n o w l e d g e the National E n e r g y Research Superc o m p u t e r C e n t e r for a grant o f c o m p u t e r time.

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