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Numerical Estimation of the Pseudo-Jahn-Teller Effect Using Non-Adiabatic Coupling Integrals in Monocyclic and Bicyclic Conjugated Molecules Shiro Koseki, Azumao Toyota, Takashi Muramatsu, Toshio Asada, and Nikita Matsunaga J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b09632 • Publication Date (Web): 29 Nov 2016 Downloaded from http://pubs.acs.org on December 2, 2016

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The Journal of Physical Chemistry November 29, 2016

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Submit to J. Phys. Chem. A

Numerical Estimation of the Pseudo-Jahn-Teller Effect Using Non-Adiabatic Coupling Integrals in Monocyclic and Bicyclic Conjugated Molecules

Shiro KOSEKI,†,‡* Azumao TOYOTA,§ Takashi MURAMATSU,§ Toshio ASADA†,‡ and Nikita MATSUNAGA‖ †

Department of Chemistry, Graduate School of Science, Osaka Prefecture University, 1-1 Gakuen-cho, Sakai, Osaka 599-8531, Japan

‡

The Research Institute for Molecular Electronic Devices (RIMED), Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai 599-8531, Japan

§

Environmental Education Center, Miyagi University of Education, Sendai 980-0845, Japan

‖

Department of Chemistry & Biochemistry, Long Island University, Brooklyn, NY

11201, U.S.A.

(Received on

, 2016)

*To whom correspondence should be addressed. e-mail: [email protected] (S) Supporting Information

1 ACS Paragon Plus Environment

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ABSTRACT: The pseudo-Jahn-Teller (pJT) effect in monocyclic and bicyclic conjugated molecules was investigated by using the state-averaged multi-configuration self-consistent field (MCSCF) method, together with the 6-31G(d,p) basis sets.

Following the perturbation theory, the force constant

along a normal mode Q is given by the sum of the classical force constant and the vibronic contribution (VC) resulting from the interaction of the ground state with excited states.

The latter

is given as the sum of individual contributions arising from vibronic interactions between the ground state and excited states.

In the present work, each VC was calculated on the basis of

non-adiabatic coupling (NAC) integrals.

Furthermore, the classical force constant was estimated

by taking advantage of the VC and the force constant obtained by vibrational analyses.

For

pentalene and heptalene, the present method seems to overestimate VC in absolute value because of the small energy gap between the ground state and the lowest excited state.

However, we are

confident that the VC and the classical force constant for the other molecules are reasonable in magnitude in comparison with available literature information.

Thus, it is proved that the present

method is applicable and useful for numerical estimation of pJT effect.


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INTRODUCTION Recent developments of computers and program codes have made it easy to theoretically search for the most stable geometrical structures of molecules.

If no imaginary frequencies are

obtained at the optimized structure of a molecule, the molecule successfully locates at an energy minimum on the multi-dimensional potential energy surface.

Even if it is not always the global

energy minimum on the surface, the structure can be considered to have a life time that is long enough to be detected experimentally. If one or more imaginary frequencies appear at an optimized symmetrical structure, the geometrical structure should be distorted along the corresponding vibrational modes into a more stable one in energy.

However, it is not always clear why the structure is stabilized energetically

along these vibrational modes.

It is well known that Jahn-Teller theory (Renner-Teller theory)1−9

can explain such structural deformations theoretically.

When an electronic state is degenerate, the

Jahn-Teller effect (first-order Jahn-Teller effect or Renner-Teller effect) breaks the degeneracy of the electronic state, and the geometrical structure is deformed into a less symmetrical nuclear arrangement.

In such a case, two different structures are generated by this distortion.

simplest example, let us consider the cyclopropenyl radical (C3H3).

As the

When the radical has a D3h

planar structure, the ground state should be 2E” and one of the e’ modes has an imaginary frequency. Structural distortion or Jahn-Teller distortion along the e’ mode provides ethylenic and allylic structures possessing C2v symmetry, where the ground states belong to A2 and B1 irreducible representations, respectively. These structures are connected by an in-plane distortion via a Cs path on the pseudo-rotation surface.10−14

The detailed results can be found in appropriate

10−14

reports.

Even though the ground state is not degenerate, the geometrical structures of some kinds of molecules are known to be distorted by the pseudo-Jahn-Teller (pJT) effect (or the second-order Jahn-Teller effect).

In the series of our research projects,15−24 energy component analyses were

performed to gain an insight into the nature of the pJT effect for various π -conjugated hydrocarbons.

Cyclobutadiene (CBD) and cyclooctatetraene (COT)18 are known to be typical pJT

molecules and have been investigated by many theoretical and experimental researchers in the past few decades.25–27

Bicyclic non-alternant hydrocarbons,17 such as pentalene and heptalene, are also

known to be good examples of pJT molecules.

In both molecules, analyses revealed that the pJT

distortion gives rise to totally symmetrical expansion of the molecular skeleton and hence leads to a reduction of the inter-electronic and inter-nuclear repulsion energies.

Moreover, energy

component analyses were carried out in the electronically excited states of fulvalene systems as well as CBD and benzene.18,19

In our recent study of this series, the same analyses were performed for 3 ACS Paragon Plus Environment



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azides23 and nitrile imines24 in order to determine why most of the azido groups take bent structures at their energy minima: organic and inorganic azides are used as useful explosive substances28−30 and nitrile imines are useful reagents in azaheterocyclic syntheses.28−31

From these works, it was

shown that the dominant energy components leading to the pJT stabilization differ from molecule to molecule and that the stabilization arises mostly from the combined effects due to an expansion or contraction of the molecular skeleton, a redistribution of electron density, and the proximity among the nuclei and electron clouds through electrostatic interactions. In the present investigation, the pJT effect was examined from a viewpoint different from the energy component analyses. Since the vibronic contributions (VCs) can be evaluated by using non-adiabatic coupling (NAC) integrals

Ψm d / dQ Ψ0

(see below), we will make clear whether

or not the lowest electronically excited state contributes greatly to the pJT stabilization in the ground state in the monocyclic and bicyclic conjugated molecules shown in Scheme 1.

Combined

with the VCs and the corresponding force constant obtained by vibrational analyses, it becomes possible to calculate

(

)

Ψ0 d 2 Hˆ / dQ 2 0 Ψ0 , which has been referred to as a classical force constant

in Pearson’s book4 and as a primary force constant in the review written by Bersuker et al.6,7,9 the next section, the method for evaluating VCs by using NAC integrals will be explained. details of molecular orbital calculations will be described in the third section.

In

The

Then calculated

results for some monocyclic and bicyclic conjugated molecules will be discussed in the following sections.

PSEUDO-JAHN-TELLER EFFECT and NON-ADIABATIC COUPLING INTEGRAL Assuming the most symmetrical nuclear arrangement for a molecule as the unperturbed system and using the second-order perturbation theory, the ground-state energy after small nuclear displacement Q along a normal coordinate is given by ˆ  dH  Ψ0 • Q E (Q) = E 0 + Ψ0   dQ  0

(1)

2   ∞ ˆ ˆ  d 2H  dH 1 + −1   Ψ0 • Q (E m − E 0 )  +  + Q • Ψ0  2  Ψ0 • Q − 2∑ Ψm   2 m  dQ 0  dQ 0   

where Ψm and E m are the wave function and the corresponding energy of the m-th state at the unperturbed system, respectively.1−9

Note that since this theory is based on the perturbation theory,

Q must be small and the energy difference (E m − E 0 ) in the denominators should not be small for the perturbation theory to be valid.

The second term in the right hand side of Eq.(1) is referred to


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as a Jahn-Teller term.

When the electronic state Ψ0 is not degenerate and the geometrical

structure of the unperturbed system is fully optimized, the Jahn-Teller term must be zero. The third term in the right hand side of Eq.(3) induces the pJT distortion.

When this term is

rewritten as 2

∞ ˆ ˆ  d 2H  dH  Ψ0 • Q (E m − E 0 )−1 = α Q + • Q , (2) Q + • Ψ0  2  Ψ0 • Q − 2∑ Ψm   dQ m  dQ  0  0

the constant α is given as a force constant K divided by the corresponding reduced mass µ : α = K /µ.

If α < 0, an imaginary frequency must be obtained along the vibrational mode Q.

Currently, most of the popular quantum chemistry codes cannot analytically calculate the first term in the left hand side of Eq.(2), though Bersuker et al. estimated the integral and its related integrals using the RHF/STO-3G wave functions:5 ˆ   d 2H (3) Q + • Ψ0  2  Ψ0 • Q = α 0 Q + • Q .  dQ  0

The parameter of α 0 multiplied by the corresponding reduced mass µ has been referred to as a classical force constant K 0 = α 0 µ in Pearson’s book4 and as a primary force constant in the reviews by Bersuker et al.6,7,9 On the other hand, the second term in the left hand side of Eq.(2) was referred to as vibronic contributions (VCs) in Pearson’s book.4

Although Bersuker et al. have also investigated these

contributions using RHF/STO-3G wave functions,6,7,9 each VC can be modified as (4)

Ψm

ˆ dH d Ψ0 = −(E m − E 0 ) Ψm Ψ0 , dQ dQ

where the conjugate orthogonality of wave functions is used for the Hamiltonian and the first derivative d dQ should be anti-Hermitian.32,33 exact.

Eq.(4) is exact, as long as the wave functions are

However, since the present investigation employs the approximate wave functions obtained

by using state-averaged MCSCF (SA-MCSCF) methods34−39 as described below, Eq.(4) must be approximately correct in the following discussion.

The integral

Ψm d / dQ Ψ0

in the right hand

side of Eq.(4) is known to be a non-adiabatic coupling (NAC) integral and can be evaluated by using some of the popular quantum chemistry codes.

Since the ground state in most neutral

molecules (not radicals) is totally symmetrical, the electronically excited state expressed by Ψm should belong to the same symmetry as that of the vibrational mode Q. investigation, the NAC integral

Ψm d / dQ Ψ0

In the present

was calculated by using the GAMESS suite of




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quantum chemistry codes,40,41 and then the VCs were evaluated in order to clarify which electronically excited states contribute to the pJT distortion and how large the VC of each excited state is.

By using these results and the force constant K obtained by the vibrational analysis, the

classical force constant K0 can be obtained as 2

∞

(5) K 0 Q + • Q = KQ + • Q + 2∑ µ

ˆ  dH  Ψ0 • Q (E µ − E 0 )−1 × µ . Ψµ    dQ  0

In concluding this section, it should be noted that a pJT effect takes place in a molecule, provided that the absolute value of vibronic contribution (VC) is larger than the value of classical force constant K0.

In such a case, the energy should be lowered by the nuclear displacement Q

from the symmetrical nuclear arrangement to a less-symmetrical one.

METHODS OF CALCULATION Geometry optimizations and vibrational analyses were performed at the full-optimized reaction space (FORS) multi-configuration self-consistent field (MCSCF) level of theory,42 together with the popular 6-31G(d,p) basis sets.43,44

For the present in-plane pJT distortions, the MCSCF

active space includes one set of valence π orbitals in each molecule, with the outer valence π orbitals being also included in the smallest molecules, CBD and propalene.

Second-order

45

configuration interaction (SOCI) calculations and multi-configuration quasi-degenerate perturbation theory (MCQDPT) calculations46,47 were carried out in order to refine the excitation energies from the ground state to low-lying excited states and in order to refine the energy differences among stationary structures.

The NAC integrals among these states were evaluated by

using state-averaged MCSCF (SA-MCSCF) methods.34−39

The numbers of states in the

SA-MCSCF calculations will be described for each molecule in the following sections.

All

calculations were carried out using the GAMESS suite of quantum chemistry program codes.40,41

RESULTS AND DISCUSSION 1. Traditional interests Cyclobutadiene (CBD) – It is well known that structural distortion occurs from a D4h structure to a D2h structure in this molecule.

Hund’s rule suggests that the ground state is 3A2g at a D4h

structure, but it has been proved that the effects of electron correlation make the ground state 1B1g instead of 3A2g.25–27

The present calculations provide the same results as those reported previously.

Namely, the 1B1g state is calculated to be lower in energy than the 3A2g state by 0.39 (MCSCF), 0.28 (SOCI), or 0.22 (MCQDPT) eV (Table 1). 1

Vibrational analysis at the optimized structure for the

B1g state provides one b1g mode possessing an imaginary frequency. 6 ACS Paragon Plus Environment

In these calculations, the

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MCSCF active space includes two sets of valence π orbitals.

That is to say, the active space

includes eight orbitals and four electrons (a doubly degenerate orbital being counted twice). Vibrational analysis at this MCSCF level of theory provides a b1g mode possessing ν = i1436 cm-1 and µ = 10.50 amu.

Then, K = –1276 [J / m 2 ] .

The energy difference between the optimized

D4h and D2h structures is calculated to be 0.27 (MCSCF), 0.32 (SOCI), or 0.21 (MCQDPT) eV (Table 2).

This structural distortion results in changes in the C−C bond lengths from 1.442 Å to

1.352 Å and to 1.544 Å (see Table S1).

These results are consistent with the results reported

previously.25–27 In order to obtain non-zero VCs, the ground state 1B1g needs to interact with some electronically excited A1g states via the b1g mode.

The main electron configuration of the lowest

( )

A1g state is (core) (a 2 u ) e g , which is the same as that of the ground state. 2

2

SA-MCSCF

( )

calculation, including the three singlet states A1g, B1g, and B2g possessing (core) (a 2 u ) e g 2

2

as their

main electron configuration, was performed in order to calculate the NAC integrals among these states.

The excitation energy to this state is calculated to be 2.15 eV.

When a unit vector e(b1g )

along the b1g vibrational displacement is used for an easy understanding, the NAC integral between the A1g and B1g states is calculated to be Ψ1 (A1g )

[

]

−1 / 2 d ⋅ bohr −1 . Ψ0 (B1g ) • e(b1g ) = 0.9156 amu dQ(b1g )

When Eq.(3) is applied, the VC can be obtained as 2

[

]

ˆ   dH  Ψ (B ) • e(b1g ) (E1 − E 0 )−1 = –0.1323 hartree ⋅ amu −1 ⋅ bohr −2 . – 2 Ψ1 (A1g )   dQ(b )  0 1g 1g  0 

Then, 2

[

]

ˆ   dH  Ψ (B ) • e(b1g ) (E1 − E 0 )−1 × µ = –2163 J / m 2 . – 2 Ψ1 (A1g )   dQ(b )  0 1g 1g  0 

If only the lowest A1g state is assumed to be effective in the summation in Eq.(2),

[

2 K0 = K +2163 = 887 J / m

]

can be obtained by using Eq.(5). Bersuker et al.5 reported classical force constants of 100 ~ 400

[J / m ] for small triatomic molecules. 2




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In order to estimate the contributions of higher excited states, 16 low-lying singlet states48 were included in the SA-MCSCF calculation. Since the NAC integrals are zero between the ground state and 12 non-A1g states by symmetry, Table 3 shows the NAC integrals and the VCs between the ground state and only the four A1g states.

SA-MCSCF calculation including 23

49

low-lying singlet states was also performed (Table 4). calculation, six A1g states are included.

In the state-averaging process of this

These results show that the VC is not greatly changed

when the number of states in the SA-MCSCF calculations increases: –2163  –2131  –2142

[J / m ]. 2

It can also be seen that interactions with higher excited states are apparently smaller than

that with the lowest A1g state (see Tables 3 and 4).

Thus, it is proved that the pJT effect caused by

the interaction of the ground state with only the lowest A1g state can explain the structural distortion of this molecule. Cyclooctatetraene (COT) – Like CBD, the ground state at the D8h structure is 1B1g for which

( ) (e ) , and the D8h structure is known to be

the main electron configuration is (core) (a 2 u ) e1g 2

4

2

2u

distorted into a D4h structure by the pJT effect (see Table S1).

Bond alternation of the

8-membered ring is derived by one of the b1g modes, and the stabilization energy by this motion is calculated to be 0.27 (MCSCF), 0.28 (SOCI), or 0.25 (MCQDPT) eV (Table 2).

In contrast to

CBD, the distorted D4h structure possessing bond alternation of the periphery ring has a new imaginary-frequency mode.

The distortion along this mode gives a well-known non-planar tub

structure.50–52 Such distortions to non-planar structures will be the next target of our investigation. The calculated results show that one of b1g modes has ν = i2360 cm-1 and µ = 11.46 amu.

[

2

]

Then, K = –3758 J / m .

When SA-MCSCF calculation includes three singlet states (A1g, B1g,

( ) (e ) , the excitation energies

and B2g) originating from the electron configuration (core) (a 2 u ) e1g 2

4

2

2u

from the ground state (B1g) to the lowest A1g and B2g states are 1.36 and 3.09 eV, respectively (Table 5).

Then, the VC caused by the interaction between the ground state and the lowest A1g

[

2

]

state is calculated to be −4378 J / m .

[

2

This value is somewhat smaller than that for CBD.

Benzene – For comparison, similar calculations were performed for benzene.

Benzene is

well known to maintain the D6h structure and never to be distorted into a lower symmetrical structure in the ground state.

The ground state is Ag and its main configuration is

(core) (a 2 u ) (e1g ) . The bond alternation mode belongs to b2u irreducible representation in this 2

]

These values provide K0 = –3758 +4378 = 620 J / m .

4


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The lowest frequency among the b2u modes is 1181 cm-1 and µ = 2.32 amu.

molecule.

[

Then, K

]

2 = +191 J / m .

SA-MCSCF calculation including three singlet states (B2u, B3u, and E2u) originating from the

( ) (e )

electron configuration (core) (a 2 u ) e1g 2

3

1

2u

[

]

2 provides VC of –147 J / m caused by the

interaction between the ground state and the lowest B2u state (Table 5).

The absolute value of this

VC is apparently smaller than those in the previous two molecules, partly because the excitation energy from the ground state and the lowest B2u state53–56 is larger than those in the previous two molecules.

[

]

2 Then, K0 = 191 + 147 = 338 J / m .

For confirmation, larger SA-MCSCF calculations were performed.

[

]

SA-MCSCF calculation

2 including 15 low-lying singlet states57 provides VC of –150 J / m caused by the interaction

between the ground state and the lowest B2u state.

Additionally, VC between the ground state and

]

[

2 the second lowest B2u state is calculated to be –15.5 J / m , with the excitation energy being

larger than 17 eV.

[

]

2 Then, K0 = 191 + (150+15.5) = 357 J / m .

This value is close to that

obtained above. Benzene has another b2u mode possessing ν = 1338 cm-1 and µ = 1.55 amu, which provides

[

]

2 K = 163 J / m .

[

]

2 The VCs along this mode are obtained as –51.2 and –5.42 J / m , when the

ground state interacts with the lowest and second lowest B2u states, respectively.

[

]

Then,

2 K0 = 163 + (51.2+5.42) = 220 J / m .

This is slightly smaller than the value for the first b2u mode but comparable.

Thus, although the

classical force constant K0 is estimated to be somewhat smaller than those for CBD and COT, the absolute values of the VCs are apparently smaller than those for CBD and COT, and these results are consistent with the fact that benzene maintains a D6h structure. Lowest triplet state in benzene – In contrast to the ground state, it is known that pJT distortion occurs in the lowest triplet state 3B1u.18

The triplet state has an e2g mode possessing ν = i1400

[

]

2 cm-1 and µ = 9.57 amu at the D6h structure, which provides K = –1105 J / m (Table 5). The

pJT distortion gives two nonequivalent D2h structures: quinoid and anti-quinoid structures (see Figure 1 and Table S1).

The stabilization energies of the distortions to the quinoid and

anti-quinoid structures are calculated to be 0.11 (MCSCF and SOCI) and 0.08 (MCSCF and SOCI) eV, respectively, being larger than those (0.02 and 0.03 eV) calculated by the MCQDPT method (Table 2). 9 ACS Paragon Plus Environment



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When SA-MCSCF calculation includes the lowest 3B1u and 3E1u states that have

[

]

(core) (a 2 u ) (e1g ) (e 2 u ) as their main configuration, VC is calculated to be –1804 J / m 2 . Even 2

3

1

when many low-lying excited states are included in the SA-MCSCF calculation, VC is only changed slightly. Additionally, the absolute values of VCs caused by the interaction of the lowest triplet state 3B1u state with the second and third lowest 3E1u states are apparently smaller than that caused by the interaction with the lowest 3E1u state (see Table 5).

2. Bicyclic non-alternant hydrocarbons Pentalene and heptalene are typical pJT molecules among non-alternant hydrocarbons and have been investigated by several research groups.58 We also investigated these molecules in the series of our investigations on pJT effect.17

In these molecules and their homologues “propalene”

and “nonalene”, the bond alternation of the periphery ring corresponds to the distortion from a D2h structure to a C2h structure.

In the following discussion, the molecular plane of the D2h structures

is set to the xy plane and the trans-annular bond is placed on the y axis.

Then, the motion of the

bond alternation belongs to b1g irreducible representation. Propalene –In a manner similar to that used in the calculations for CBD, the MCSCF active space includes two sets of valence π orbitals.

The energy difference between the D2h and C2h

structures is calculated to be 0.58 (MCSCF), 0.37 (SOCI), or 0.37 (MCQDPT) eV (Table 2), while it was reported to be 0.23 eV at the RHF/6-31G(d) level of theory in our previous paper.17

The

C−C bond lengths of the periphery ring are changed from 1.381 Å to 1.283 or 1.568 Å, with the trans-annular C−C bond length being changed from 1.499 Å to 1.683 Å (see Table S1). One of the b1g modes in the ground state at the D2h structure has ν = i204 cm-1 and µ = 8.82 amu.

[

]

2 These provide K = –20.2 J / m (Table 5).

When SA-MCSCF calculation includes only

the ground state and the lowest B1g state, the energy difference between these states is calculated to

[

]

2 be 1.47 eV and VC is obtained as –1801 J / m .

Furthermore, when 21 states are included in the

[

]

2 SA-MCSCF calculation, VC is calculated to be –2313 J / m and the fourth lowest B1g state has a

[

]

2 relatively large contribution (–29.4 J / m ) to the pJT effect, with the excitation energies to the

lowest and fourth lowest B1g states being 1.24 and 21.56 eV, respectively.

The excitation energy

from the ground state to the lowest B1g state is relatively small and is reduced from 1.47 eV to 1.24 eV when the number of states in the SA-MCSCF calculations increases.

These VCs are

comparable with those in CBD and COT and they can numerically explain the structural distortion 10 ACS Paragon Plus Environment

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in propalene.

Note that the C2h structure of propalene has two imaginary frequencies of i544 (cis

bent) and i367 (trans bent) cm-1 for out-of-plane motion of the C–H bonds at the current level of theory.

Structural distortion to such non-planar structures will be the next target of our

investigation. Pentalene – In our previous paper,17 it was reported that the ground state at the optimized D2h structure is Ag (closed shell) and that it is stabilized to a C2h structure by 0.70 eV at the RHF/6-31G(d) level of theory.

However, the present MCSCF calculations show that the ground

state at the optimized D2h structure is B1g and the adiabatic excitation energy from the ground state to the lowest Ag state is calculated to be 0.45 (MCSCF), 0.22 (SOCI), or 0.12 (MCQDPT) eV (Table 1).

Since the effects of dynamic electron correlation make the energy difference much

smaller, more sophisticated calculations would be needed to conclude that the ground state is B1g. Apart from the ground-state problem, the fact that the B1g and Ag states are very close in energy to each other suggests a large absolute value of VC to the ground state.

The present results show

that the stabilization energy caused by distortion from the D2h structure to a C2h structure is 0.43 (MCSCF), 0.44 (SOCI), or 0.25 (MCQDPT) eV (Table 2). the previous result (0.70 eV).

This energy is explicitly smaller than

This must be caused by the fact that the Ag state was considered to

be the ground state at the D2h structure in our previous study.

The geometrical parameters of the

stationary structures are shown in Table S1. Vibrational analysis for the ground state B1g at the D2h structure provides one b1g mode

[

possessing ν = i2754 cm-1 and µ = 11.58 amu.

]

2 These provide K = –5178 J / m (Table 5).

When the ground state and the first excited Ag state are included in the SA-MCSCF calculation,

]

[

2 VC between these states is obtained as –21958 J / m .

Mainly because the energy difference

between these states is very small (0.14 eV), such a large absolute value of VC is obtained for this molecule in comparison with those in CBD and COT. in Eq.(2) is very small, VC could be overestimated.

In other words, because the denominator The energy difference between these states

is calculated to be 0.45 (MCSCF), 0.22 (SOCI), or 0.12 (MCQDPT) eV (Table 1), even when the MCSCF calculations were performed individually for these states.

In such a case, a variation

61,62

method needs to be employed as reported by Liu et al.

In order to confirm these results, a larger SA-MCSCF calculation was performed. Twenty-two low-lying electronic states were included in the SA-MCSCF calculation because of the large energy gap between the 22nd and 23rd singlet states.

[

]

The VC between the ground state

2 and the lowest B1g state is calculated to be –13569 J / m (Table 5).

The absolute value of this

VC is about half of that obtained in the above-described SA-MCSCF calculation. 11 ACS Paragon Plus Environment

This would be



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

because the energy difference between the ground state and the first excited Ag state is calculated to be slightly larger in this calculation: 0.14  0.21 eV.

Thus, even though VC could be

overestimated when the energy difference between the interacting states is small, it is useful to apply the present method to studies of the pJT effect in molecules.

[

]

2 In addition, relatively large absolute values for VCs, –456 and –108 J / m , were obtained

between the ground state and the second lowest Ag state and between the ground state and the fourth lowest Ag state, with the excitation energies from the ground state to these Ag states being 3.27 and 7.25 eV, respectively.

Although these VCs are remarkably smaller than the VC

between the ground state and the lowest Ag state, it was found that they also contribute to the pJT distortion to the C2h structure. Heptalene – Like pentalene, we have reported that the ground state at the D2h structure is Ag on the basis of RHF/6-31G(d)) results.17

The present MCSCF calculations show that the ground

state at the D2h structure is B1g and that the lowest excited state is Ag.

The energy difference

between these two states is calculated to be only 0.37 eV (MCSCF), 0.27 (SOCI), or 0.04 (MCQDPT) eV (Table 1), with the active space of the SOCI calculations including only 97 orbitals that have the lowest eigenvalues of the standard MCSCF Fock operator due to our computer resource limitation.

It seems possible that the energetic order of these states is reversed

at higher levels of theory because of a very small difference in the MCQDPT calculations.

The

stabilization energy given by the structural distortion to a C2h structure is calculated to be 0.49 (MCSCF), 0.48 (SOCI), or 0.18 (MCQDPT) eV (Table 2).

As compared with pentalene, the

stabilization energy is smaller than the difference (0.74 eV) reported in our previous paper17 because the ground state was assumed to be Ag at the optimized D2h structure in our previous investigation. The geometrical parameters of the stationary structures are listed in Table S1. One of the b1g modes at the optimized D2h structure has ν = i4143 cm-1 and µ = 11.70 amu.

[

]

2 These provide K = –11830 J / m (Table 5).

When the ground state (B1g) and the lowest

[

]

2 excited Ag state are included in the SA-MCSCF calculation, VC is obtained as –29560 J / m .

As mentioned above for pentalene, VC could be overestimated because of the small energy difference of 0.13 eV between the ground state (B1g) and the first excited Ag state, although the absolute value of the force constant K is also large in this molecule.

Nevertheless, the present

method is useful to explain how strong the pJT distortion is in this molecule.

Nonalene – This molecule is the largest homologue of the bicyclic non-alternant hydrocarbons in this investigation, and it is too large to be investigated at higher levels of theory. 12 ACS Paragon Plus Environment

Page 13 of 26


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Additionally, since this molecule has 9-membered rings, it is apparent that structural distortion to a non-planar structure occurs.

Therefore, our attention was paid only to the in-plane structural

distortion from the D2h structure to the C2h structure in the present section. Individual MCSCF calculations show that the lowest Ag state is lower in energy than the lowest B1g state and is the ground state at the D2h structure, though the energy difference between these states is calculated to be 0.65 eV in the SA-MCSCF calculation (Table 5).

As described in

the paragraph on heptalene, pJT distortion to a C2h structure apparently occurs in this molecule and the VC between these states could be overestimated.

Unfortunately, this molecule is too big to

carry out vibrational analysis at the MCSCF level of theory in our computer systems, and it was therefore decided to use the results of RHF/6-31G(d,p) vibrational analysis in the present analyses. One of the b1g modes has ν = i2148 cm-1 and µ = 11.29 amu.

[

]

2 Then, K = –3070 J / m for this

mode (Table 5). When only the ground state (Ag) and the lowest B1g state are included in the SA-MCSCF

[

]

2 calculations, VC is calculated to be –4940 J / m , with the energy difference between these states

being 0.65 eV. Thus, although the absolute value of VC is somewhat large, this result is acceptable for explaining the pJT distortion in this molecule.

Naphthalene – Finally, the calculated results for naphthalene are described briefly in order to confirm the usefulness of the present method. One of the b3u modes corresponding to the bond alternation of the periphery ring has ν = 665 cm-1 and µ = 6.53 amu.

[J / m ] (Table 5). 2

These provide K = 170

When the ground state and the lowest B3u state are included in the

SA-MCSCF calculation, the excitation energy from the ground state to the lowest B3u state is calculated to be 4.26 eV.

Hashimoto et al. have reported that it is 4.34 (CASSCF), 4.09 (MRMP),

59

or 4.03 (CASPT2) eV and the experimental value is 3.97 eV.60

[

]

2 VC of –9.66 J / m .

Then, this calculation provides

This contribution is apparently smaller than those for the above molecules,

[

]

2 and the classical force constant is obtained as K0 = 170 + 9.66 = 180 J / m .

Thus, we are

confident that the present method is applicable to numerical investigations of the pJT effect on the in-plane distortion.

SUMMARY The pseudo-Jahn-Teller (pJT) effect in monocyclic and bicyclic conjugated molecules was studied by using the state-averaged multi-configuration self-consistent field (MCSCF) method, 13 ACS Paragon Plus Environment



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

together with the 6-31G(d,p) basis sets.

The vibronic contributions (VCs) of the pJT effect were

calculated on the basis of non-adiabatic coupling (NAC) integrals between the ground state and low-lying excited states.

The present analyses confirm that the absolute value of the VC between

the ground state and the lowest excited state is generally overwhelmingly large in comparison with those between the ground state and other excited states, indicating that the lowest excited state plays a dominant role in the pJT effect in the molecules examined here.

Moreover, the VC and the

classical force constant are found to be reasonable in magnitude in comparison with available literature information, though the present method tends to overestimate VC in pentalene and heptalene because of the small energy gap between the ground state and the lowest excited state. Accordingly, we are confident that the present method is applicable and useful for numerical estimation of pJT effect.

Appropriate and reliable methods for calculation of pJT distortion from

planar structures to non-planar structures in COT and propalene are now being searched for in our laboratory.

In order to quantitatively explain such distortion in the molecules, it is necessary to

correctly estimate the interaction between σ and π orbitals.

We have also started investigating

the pJT distortion from linear structures to bent structures in azides and nitrile imines by using the present methods of calculation. ASSOCIATED CONTENT Supporting Information The supporting information is available free of charge on the ACS Publications website.

AUTHOR INFORMATION Corresponding Author *E-mail: [email protected]

Tel.: +81-72-254-9702.

Notes The authors declare no competing financial interest.


Page 15 of 26


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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Table Captions Table 1. Adiabatic excitation energies (eV) from the ground states to low-lying excited states at the highest symmetrical structures Table 2. Stabilization energies (eV) of the ground states by pseudo-Jahn-Teller effect Table 3. Non-adiabatic coupling (NAC) integrals and the vibronic contributions (VCs) in cyclobutadiene (CBD), where the state-averaged MCSCF (SA-MCSCF) calculation includes low-lying 16 singlet states Table 4. Non-adiabatic coupling (NAC) integrals and the vibronic contributions (VCs) in cyclobutadiene (CBD), where the state-averaged MCSCF (SA-MCSCF) calculation includes low-lying 23 singlet statesa

[

]

2 Table 5. Vibrational frequency ν [cm-1], reduced mass µ [amu], force constant K J / m ,

[

]

[

]

2 2 vibronic contribution VC J / m , classical force constant K0 J / m , and the energy

difference ∆E [eV] between the ground state and low-lying excited states


Page 21 of 26


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Table 1. Adiabatic excitation energies (eV) from the ground states to low-lying excited states at the highest symmetrical structures Mol. cyclobutadiene (CBD)

cyclooctatetraene (COT)

benzenea

Geom. D4h

D8h

D6h

State 1

pentalene heptalene nonalene

D2h D2h D2h D2h

SOCI

MCQDPT

∆E (SAMC) d

B1g

0

0

0

3

A2g

0.39

0.28

0.22

1

A1g

2.19

2.05

1.58

2.15

B1g

0

0

0

0

3

A2g

0.68

0.46

0.29

1

A1g

1.41

1.16

0.86

1.36

0

0

0

0

1.18

0.97

0.68

1.17

Ag

0

0

0

0

1

B1g

0.63

0.93

0.36

1.47

1

B1g

0

0

0

0

1

Ag

0.45

0.22

0.12

0.14

B1g

0

0

0

0

1

Ag

0.37

0.27b

0.04

0.13

1

Ag

0

c

0

0

B1g

0.06

0.33

0.65

1

3

B1u

3

propalene

MCSCF

E1u

1

1

1

a

0

The target state is not the ground state but the lowest triplet state. 97 orbitals were used. c It is impractical to perform effective SOCI calculations because of the limitation of our computer resource. d ∆E (SAMC) [eV] is the energy difference or the vertical excitation energy between the ground state and the excited states obtained by using SA-MCSCF method. b




1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Table 2. Stabilization energies (eV) of the ground states by pseudo-Jahn-Teller effect MCSCF

SOCI

MCQDPT

Previousa

Mol.

Distortion

cyclobutadiene (CBD) cyclooctatetraene (COT) benzene (3B1u)c

D4h  D2h

0.27

0.32

0.21

0.29

D8h  D4h

0.27

0.28

0.25

0.27

Liter.b 0.17 ~ 0.39 0.13 ~ 0.33

D6h  D2h quinoid

0.11

0.11

0.02

0.11

anti-quinoid

0.08

0.08

0.03

0.08

propalene

D2h  C2h

0.58

0.37

0.37

0.23

pentalene

D2h  C2h

0.43

0.44

0.25

0.70

heptalene

D2h  C2h

0.49

0.48d

0.18

0.74

nonalene

D2h  C2h

0.61

e

–0.05

0.49

a

MCSCF/6-31G(d) method was used for CBD, COT, and benzene [ref.18], while RHF/6-31G(d) method was used for propalene, pentalene, heptalene, and nonalene [ref.17]. b See refs. 25–27 (CBD), refs. 50–52 (COT), and refs. 53–56 (benzene). c The target state is not the ground state but the lowest triplet state. d 72 orbitals were used in the SOCI calculations. e It is impractical to perform effective SOCI calculations because of the limitation of our computer resource.


Page 23 of 26


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Table 3. Non-adiabatic coupling (NAC) integrals and the vibronic contributions (VCs) in cyclobutadiene (CBD), where the state-averaged MCSCF (SA-MCSCF) calculation includes low-lying 16 singlet states State 1 B1g 1 A1g 1 A1g 1 A1g 1 A1g a

order 1 2 6 10 16

dEa

NACb

2.13 11.22 16.30 21.71

0.071372 0.033101 0.007364 0.017295

VCc −0.130380 −0.005313 −0.000181 −0.000750

−2131.42 −86.85 −2.96 −12.26

dE = En(A1g) – E0(B1g) [eV].

b

  d  Ψ (B ) • e(b1g ) [hartree/bohr]. Ψ1 (A1g )   dQ(b )  0 1g 1g  0 

NAC =

2

c

VCd

ˆ   dH  Ψ (B ) • e(b1g ) (E1 − E 0 )−1 VC = – 2 Ψ1 (A1g )   dQ(b )  0 1g 1g  0  2

d

[hartree ⋅ amu

ˆ   dH  Ψ (B ) • e(b1g ) (E1 − E 0 )−1 × µ VC = – 2 Ψ1 (A1g )   dQ(b )  0 1g 1g  0 

−1

]

⋅ bohr −2 .

[J / m ]. 2

Table 4. Non-adiabatic coupling (NAC) integrals and the vibronic contributions (VCs) in cyclobutadiene (CBD), where the state-averaged MCSCF (SA-MCSCF) calculation includes low-lying 23 singlet statesa State 1 B1g 1 A1g 1 A1g 1 A1g 1 A1g 1 A1g 1 A1g a

dE

NAC

2.12 11.21 18.29 21.69 24.31 28.08

0.071480 0.033077 0.007394 0.017256 0.013332 0.003495

VCb −0.131038 −0.005310 −0.000183 −0.000747 −0.000398 −0.000024

See the footnote in Table 3. VC in hartree ⋅ amu −1 ⋅ bohr −2 .

b c

order 1 2 6 10 16 23 30

VC in

[ [J / m ].

]

2


VCc −2142.18 −86.81 −2.99 −12.22 −6.51 −0.39



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

[

]

[

]

Table 5. Vibrational frequency ν [cm-1], reduced mass µ [amu], force constant K J / m 2 , vibronic contribution VC J / m 2 , classical force

[

]

constant K0 J / m 2 , and the energy difference ∆E [eV] between the ground state and low-lying excited states Mol. cyclobutadiene (CBD)

ν i1436 (b1g)

µ

10.50

K –1276.0

SAa 3 16 (4)

23 (7)

VC –2163.4 –2131.4 –86.9 –3.0 –12.3 –2142.2 –86.8 –3.0 –12.2 –6.5 –0.4

K0 +887.4 +957.4

+975.1

∆E (SAMC) b

∆E (SOCI) b

2.15 2.13 11.22 16.30 21.71 2.12 11.21 16.29 21.69 24.31 28.08

2.05


i2360 (b1g)

11.46

–3757.8

3 13 (7)

–4378.3 –4297.1 –108.5 –49.4

+620.5 +697.2

1.36 1.34 8.29 1021

1.16

benzene (1A1g)

1181 (b2u)

2.32

+191.0

4 (1) 15 (7) 15 (7)

–146.7 –47.1 –150.3 –15.5 –51.2 –5.4

+337.7

4.89 8.26 4.90 17.23 4.90 17.23

4.93

2 (1) 13 (7)

−1803.9

+699.0

1.17

0.97

–1852.7 –0.4 –0.2

+748.4

1.11 8.13 10.32

benzene (3B1u)c

1338 (b2u)

1.55

+163.4

i1400 (e2g)

9.57

–1104.9


+356.8 +220.0

Page 25 of 26


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

Table 5 (continued) Mol. propalene

ν i204 (b1g)

pentalene

heptalene nonalene naphthalene a

µ

8.82

K –20.2

SAa 2 21

VC –1801.2 –2312.8 –0.1 –2.4 –29.4 –2.6

K0 +1781.0 +2327.1

i2754 (b1g)

11.58

–5178.0

2 22

–21957.9 –13568.5 –456.1 –0.1 –108.4 –23.2 –36.9 –2.4

i4143 (b1g) i2148 (b1g) 665 (b3u)

11.70

–11829.6

2

11.29

–3070.0

6.53

+169.8

∆E (SAMC) b

∆E (SOCI) b

1.47 1.24 16.77 20.51 21.56 23.43

0.93

+16779.9 +9017.5

0.14 0.21 3.27 5.35 7.25 8.57 9.26 10.32

0.22

–29559.5

+17730.0

0.13

0.27

2

–4940.3

+1870.2

0.65

4

–9.7

+179.5

4.26

d

3.63

Number of states included in the SA-MCSCF calculations. The numbers of doubly degenerate states are shown in parentheses. ∆E (SAMC) [eV] is the energy difference or the vertical excitation energy between the ground state and the excited states obtained by using

b

SA-MCSCF method.

∆E (SOCI) [eV] is the energy difference or the adiabatic excitation energy between the ground state and the first excited

state obtained by the second-order configuration interaction (SOCI) methods at the geometrical structures optimized for the individual states. The target state is not the ground state but the lowest triplet state. d It is impractical to perform effective SOCI and MCQDPT2 calculations because of the limitation of our computer resource. c



Page 26 of 26

October 21, 2016 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Scheme 1

cyclobutadiene (CBD)


benzene

naphthalene

propalene

pentalene

heptalene

nonalene

Graphical Abstract Pseudo-Jahn-Teller Effect Vibronic Contribution

m

ˆ dH d 0  E m  E0  m 0 dQ dQ Non-Adiabatic Coupling Integral

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