Diradical and Ionic Characters of Open-Shell Singlet Molecular

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Diradical and Ionic Characters of Open-Shell Singlet Molecular Systems Masayoshi Nakano, Kotaro Fukuda, Soichi Ito, Hiroshi Matsui, Takanori Nagami, Shota Takamuku, Yasutaka Kitagawa, and Benoît Champagne J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b11647 • Publication Date (Web): 01 Jan 2017 Downloaded from http://pubs.acs.org on January 2, 2017

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The Journal of Physical Chemistry

Diradical and Ionic Characters of Open-Shell Singlet Molecular Systems Masayoshi Nakano,*,†,¶ Kotaro Fukuda,† Soichi Ito,† Hiroshi Matsui,† Takanori Nagami,† Shota Takamuku,† Yasutaka Kitagawa,†,¶ Benoît Champagne§ †

Department of Materials Engineering Science, Graduate School of Engineering Science, Osaka University, Toyonaka,

Osaka 560-8531, Japan ¶

Center for Spintronics Research Network (CSRN), Graduate School of Engineering Science, Osaka University,

Toyonaka, Osaka 560-8531, Japan §

Laboratoire de Chimie Théorique, University of Namur, rue de Bruxelles, 61, 5000 Namur, Belgium

-1ACS Paragon Plus Environment


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ABSTRACT: The diradical and ionic natures of open-shell singlet systems have been investigated using new definitions of the diradical and ionic characters as well as of their densities within the valence configuration interaction (VCI) model with two electrons in two active orbitals. The two-site symmetric and asymmetric diradical models are examined by using these diradical/ionic characters. For realistic compounds, we investigate a diradicaloid diphenalenyl and a rectangular graphene nanoflake in the presence of an external static electric field, as well as π-stacked phenalenylderivative dimers with varying the intermonomer distance, where the central carbon atoms in the phenalenyl rings are substituted by boron (B) and nitrogen (N) atoms, respectively. It is found that the increase of charge asymmetricity induced by the static electric field decreases the diradical character and finally induces an ionic character in the ground state, while the first excited state is changed from pure ionic to diradical-dominant as the field amplitude increases. On the other hand, when increasing the intermonomer distance, the B/N substitution in the phenalenyl dimer changes the electronic state from open-shell singlet with small diradical character to closed-shell with large ionic character. These results indicate that the application of a static electric field to diradicaloids and the asymmetric substitution of a pancake bonded π-dimer combined with the variation of intermonomer distance could tune the diradical/ionic characters and therefore control the nonlinear optical responses.


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1. INTRODUCTION Recent developments in the synthesis, the experimental physicochemical characterizations, and the theoretical modeling of the electronic states of open-shell singlet molecular systems have opened up a new field towards highly efficient multifunctional molecular systems for optoelectronics, spintronics and so on.1-4 These functionalities are predicted to originate from the unique electronic structures of the open-shell singlet systems, which are characterized by intermediate/strong electron correlations.3 For example, the longer the polycyclic aromatic hydrocarbons (PAHs), the narrower the band gap with a singlet open-shell ground state, followed by a triplet state a few kcal/mol higher in energy.5, 6 Within the single-determinant broken-symmetry approximation, this feature leads to spin polarization between the zigzag edges.7-10 Although this spin polarization is an artifact of this approximation, it is not the case of the related odd electron density, which accounts for the high reactivity of the zigzag edges of long oligoacenes and thus for the difficulty of their synthesis.11-14 In the solid state, such open-shell singlet characteristics makes it possible to realize high charge carrier mobilities because the covalent-like spin-pairing intermolecular interactions are of comparable strength to the intramolecular interactions.15-19 Using quantum chemistry methods, several interpretation schemes have been elaborated to describe these structure-property relationships and to elucidate the variations of the excitation energies, transition properties, and nonlinear optical response (NLO) properties as a function of the degree of open-shell singlet nature.20-23

As evidenced by these facts, the quantitative estimation of the open-shell singlet

character as well as the clarification of the correlation between the open-shell singlet character, the molecular structures, and conventional chemical concepts are necessary for deepening the understanding of effective chemical bonds and for revealing novel structure-property relationships in open-shell singlet molecular systems. The theoretical understanding of these open-shell singlet electronic structures began with the introduction of the “diradical character” concept,24-26 which was originally defined as twice the



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weight of the doubly-excited configuration. The diradical and/or diradicaloid (= diradical-like) character appears in molecular systems possessing two weakly interacting electrons having similar energy,27,

28

which give rise to a pair of near-degenerate bonding and anti-bonding molecular

orbitals (MOs). Although the original studies were theoretical and conceptual and have been dealing with model systems, including unstable compounds like the stretched hydrogen molecule and twisted ethylene, recent developments in synthesis have led to a variety of thermally stable diradicaloids and multiradicaloids like PAHs, diphenalenyl compounds,15-19 transition metal compounds,29, 30 and main group compounds.31 Yamaguchi characterized the diradical character by referring to the chemical concept of “instability of chemical bonds”, and to estimate this, they proposed an approximate scheme based on the broken-symmetry (spin-unrestricted) Hartree-Fock (UHF) method with a spin projection correction.25 Later, Takatsuka and coworkers proposed related expressions for the odd (unpaired) electron density and for the number of odd (unpaired) electrons,32 which were later modified by Head-Gordon.33 Furthermore, there have been other definitions of the diradical character, which have been applied to describe a broad range of electronic structures including pancake bonds.34-38

Although these theoretical expressions are

useful for obtaining intuitive and pictorial descriptions of odd (unpaired) electron distributions as well as of radical and diradical characters in open-shell molecular systems, their applicability is limited to open-shell singlet systems with a dominating covalent (neutral) component rather than an ionic one. However, in general, arbitrary electronic states including time-dependent superpositions of the ground and excited states could display large ionic components. Although the ionic character was already defined from expanding the electronic populations,39 we consider here an alternative definition of the ionic as well as of the diradical character, both of which can be straightforwardly connected to each other, since we consider simple and useful chemical indices for characterizing functionalities of systems with a wide range of not only diradical but also of ionic characters. Indeed, the previously proposed diradical characters are found to be correlated to the excitation


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energies and properties.20-23 Therefore, the present study aims at defining alternative diradical and ionic characters and their densities on the basis of a two-site model with two electrons in two active orbitals and at applying these definitions to attractive singlet molecular systems. As it will be discussed in details, the present indices are related to the effective neutral and ionic chemical bond orders, and they describe the degree of electron correlation for arbitrary electronic states including ground and excited states and thus for any superposition of electronic states. Since the molecular energy and the properties are influenced by electron correlation,21 these indices will not only provide a useful tool for understanding the effective chemical bond nature and the electron correlation but they will also contribute to building structure-property relationships for diradical/ionic-component dominant systems.4, 21, 40-45 This paper is organized as follows. In Sec. 2.1, within the valence configuration interaction (VCI) approach, we summarize the theoretical framework for describing the ground and excited states of a two-site model with variable asymmetricity and we analyze the conventional ground state diradical character and its asymmetricity dependence. In Sec. 2.2, we introduce alternative definitions of the diradical and ionic characters based on the neutral and ionic components of the two-electron density, and we present the definitions of their densities as well as their effective bond orders. Then, we address the relationship between the present and previous diradical characters definitions in Sec. 2.3. In order to demonstrate the performance of the present definitions, we discuss in Sec. 3.1 a simple example of the two-site model and of the variations of its diradical and ionic characters as well as in their densities for the ground and excited states. In Sec. 3.2 and 3.3, this analysis scheme is applied to realistic compounds, that is, an open-shell singlet diphenalenyl diradicaloid, a rectangular graphene nanoflake, and boron(B)-/(nitrogen)N-doped phenalenyl dimers. These results are summarized in Sec. 4.



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2. THEORY 2.1. Valence Configuration Interaction (VCI) Two-Site Model. The VCI scheme for a twosite A•–B• model with two electrons in two magnetic orbitals46,

47

is adopted to describe the

diradical character dependences of the wavefunctions and energies.21-23

Details of the VCI

Hamiltonian matrix H and definition of each physical parameter are given in Supporting Information. This scheme employs the following dimensionless quantities:23

tab U

≡ rt (≥ 0) ,

t U 2K ab h ≡ rK (≥ 0) , ≡ rh (≥ 0) , a ≡ rU (≥ 0) , and ab(aa) ≡ rtab (≥ 0) , tab(bb) Ub U U

where t ab is the average transfer integral of tab(aa) ≡ ab Hˆ aa

(1)

and tab(bb) ≡ ab Hˆ bb ,

U [≡ (U a +U b ) / 2] is the average effective Coulomb repulsion, h is the one-electron core Hamiltonian difference h ≡ hbb − haa ( h ≥ 0 or haa ≤ hbb), and K ab (≥ 0) is a direct exchange integral. We define the “pseudo diradical character” yS as an alternative to rt by

yS = 1−

4rt 1+16rt2

,

(2)

which corresponds to the conventional diradical character for a symmetric two-site diradical system.21 For asymmetric two-site diradical system, the true diradical character is referred to as yA or just y in this study and it is a function of (rt, rK, rh, rU, rtab). Larger asymmetric electron distributions, corresponding to larger population on A than on B, are obtained by increasing rh (≥ 0) and/or by decreasing rU (≤ 1) and/or by increasing rtab (≥ 1) (eq 1). In this study, we investigate the first cause by varying rh between 0 and 2 while keeping (rU , rtab) = (1, 1) for simplicity, which corresponds to the situation where the asymmetricity is primarily governed by the difference of ionization energies between atoms A and B. Therefore, the eigenvalues and eigenvectors depend on the dimensionless quantities (rt, rK, rh, rU = 1, rtab = 1).


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By diagonalizing the dimensionless hamiltonian matrix H ND ( ≡ H /U ), we obtain numerically the eigenvalues and eigenvectors for the four states, {j} = {T, g, k, f} (T: triplet state, and g, k, f: singlet states), each of which being described by a set of 4 coefficients { Cab , j , Cba, j ,

Caa, j , Cbb, j }:

Ψ j = Cab, j ab + Cba, j ba + Caa, j aa + Cbb, j bb , 2

2

2

2

with Cab , j + Cba, j + Caa, j + Cbb , j = 1 .

(3)

For the triplet state the Cab ,T = −Cba,T = 1/ 2 and

Caa,T = Cbb,T = 0 conditions are satisfied, while Cab , j = Cba, j is satisfied for the singlet states.23 For asymmetric singlets, Caa, j ≠ Cbb, j . Alternatively, these singlet electronic states can be represented in the { G , S , D } basis set, of which the functions are constructed from the MOs (g and u) (see eq s1 in Supporting Information), and which correspond to the ground, the singly-excited and the doubly-excited determinants, respectively:23

1 ( gu + ug ) , and D ≡ uu . 2

G ≡ gg , S ≡

(4)

In that basis, the singlet eigenstates are expressed under the form of linear combinations of the { G ,

S , D }. For instance, the ground state reads:

Ψ g = ξ G +η S −ζ D ,

(5)

with ξ 2 + η 2 + ζ 2 = 1 to satisfy the normalization condition. By comparing eq 3 with 4, we obtain the following relations:23

ξ = Cab ,g +

1 1 1 Caa,g − Cbb ,g ) , and ζ = Cab ,g − Caa,g + Cbb ,g . Caa,g + Cbb ,g , η = ( 2 2 2

(

)

(


)

(6)


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Here, the diradical character is defined as the occupation number (nLUNO) of the lowest unoccupied natural orbital (LUNO) of state Ψ g obtained from the diagonalization of its density matrix.23 Therefore, using the coefficients in eq 6 the diradical character yA in the ground state reads:

yA ≡ nLUNO = 1− ξ − ζ

2 − (ξ − ζ ) , 2

(7)

which holds for both symmetric and asymmetric systems. For symmetric systems, eq 7 reduces to the usual definition, i.e., twice the weight of the doubly-excited configuration:

yA = yS = 2ζ 2 ,

(8)

Using eqs 6 and 7, the diradical character yA can also be expressed as a function of the LNO coefficients:23

yA = 1− Caa,g + Cbb ,g 2 − (Caa,g + Cbb ,g )2 ,

(9)

It was found that yA is smaller than yS, in particular in the middle region of yS as increasing the asymmetricity rh.23 Note that if yS = 1 then yA = 1 for rh < 1 but yA = 0 for rh > 1, while yA is close to ~0.134 for rh = 1.23 This behavior indicates the exchange of the dominant configurations (neutral/ionic) in state g between rh < 1 and rh > 1 for yS > 0.

2.2. Diradical/ionic Characters and Their Densities for Arbitrary States within the Two-Site Model. For an arbitrary state j (eq 3), the neutral (covalent or diradical) ( PN, j ) and ionic ( PI, j ) populations are defined as: 2

2

2

2

PN, j = Cab , j + Cba, j , and PI, j = Caa, j + Cbb , j ,

(10)

so that eq 3 can be rewritten

Ψ j = PN, j N j + PI, j I j = Ψ N, j + Ψ I, j , where the orthonormalized neutral and ionic states are described, respectively, by


(11)

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Nj =

(

)

(

)

1 1 Cab , j ab + Cba, j ba , and I j = Caa, j aa + Cbb , j bb . PN, j PI, j

(12)

The neutral two-electron density is then defined using the neutral two-electron density operator ρˆ N, j (≡ Ψ N, j Ψ N, j ) :

ρ N, j (r1, r2 ; r1, r2 ) = r1, r2 ρˆ N, j r1, r2 = Ψ N, j (r1, r2 )Ψ N,* j (r1, r2 ) = PN, j r1, r2 N j N j r1, r2

(13)

(

= r1, r2 Cab , j ab + Cba, j ba

) (C

ab , j

ab + Cba, j ba

)

*

r1, r2 ,

where eqs 11 and 12 are used. Similarly, the ionic two-electron density is represented using the ionic two-electron density operator ρˆ I, j (≡ Ψ I, j Ψ I, j ) , as

ρ I, j (r1, r2 ; r1, r2 ) = r1, r2 ρˆ I, j r1, r2 = Ψ I, j (r1, r2 )Ψ I,* j (r1, r2 ) = PI, j r1, r2 I j I j r1, r2

(14)

(

= r1, r2 Caa, j aa + Cbb , j bb

) (C

aa, j

aa + Cbb , j bb

)

*

r1, r2 .

Note here that ρ j (r1, r2 ;r1, r2 ) ≠ ρ N, j (r1, r2 ;r1, r2 ) + ρI, j (r1, r2 ;r1, r2 ) in general because the total two-

(

electron density operator is defined as ρˆ j ≡ Ψ N, j + Ψ I, j

)( Ψ

N, j

)

+ Ψ I, j ≠ ρˆ N, j + ρˆ I, j .

Next, the neutral and ionic one-electron densities of state j are represented as

ρN, j (r1; r1 ) = ∫ dr2 ρN, j (r1, r2 ; r1, r2 ) =

∫ dr { C 2

2

ab, j

2

ρab,ab (r1, r2 ; r1, r2 ) + Cba, j ρba,ba (r1, r2 ; r1, r2 )

* * +Cab , j Cba, j ρ ab ,ba (r1, r2 ;r1, r2 ) + Cba, j Cab , j ρ ba,ab (r1, r2 ; r1, r2 )} 2

(15)

2

= Cab , j ρ ab ,ab (r1; r1 ) + Cba, j ρba,ba (r1; r1 ), and

ρI, j (r1; r1 ) = ∫ dr2 ρ I, j (r1, r2 ; r1, r2 ) =

∫ dr { C 2

2

aa, j

+Caa, j C

* bb , j

2

2

ρ aa,aa (r1, r2 ; r1, r2 ) + Cbb , j ρbb,bb (r1, r2 ; r1, r2 )

ρaa,bb (r1, r2 ; r1, r2 ) + Cbb , j C

* aa, j

ρbb ,aa (r1, r2 ; r1, r2 )}

2

= Caa, j ρaa,aa (r1; r1 ) + Cbb , j ρbb ,bb (r1; r1 ).


(16)


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In these transformations the one-electron reduced densities are expressed from the corresponding two-electron ones in the LNO basis representation, e.g.,

ρab ,ab (r1;r1 ) = ∫ dr2 r1, r2 ab ab r1, r2 = ∫ dr2 ρab ,ab (r1, r2 ; r1, r2 ) .

(17)

These one-electron reduced densities satisfy the following relations due to the orthonormalization condition of LNOs.

∫ dr ρ

1 ab ,ab

(r1;r1 ) = ∫ dr1 dr2 r1, r2 ab ab r1, r2 = ab ab = 1 .

(18)

Similar relationships are also satisfied for other one-electron reduced densities. Then, from using these relationships, integrals of the neutral and ionic one-electron reduced densities provide the neutral and ionic populations, respectively:

∫ dr ρ

2

1 N, j

2

(r1;r1 ) = Cab , j + Cba, j = PN, j , and

∫ dr ρ

2

1 I, j

2

(r1;r1 ) = Caa, j + Cbb , j = PI, j .

(19)

The neutral and ionic characters of the wavefunction of state j are then defined as the deviations from the neutral and ionic components in the non-correlated (nc), e.g., Hartree-Fock, ground state of a symmetric system. In the non-correlated limit, the two-electron densities are given by

ρ N nc (r1, r2 ; r1, r2 ) = PN nc r1, r2 N nc N nc r1, r2 =

1 r1, r2 N nc N nc r1, r2 , 2

(20)

and

ρ I nc (r1, r2 ; r1, r2 ) = PI nc r1, r2 Inc I nc r1, r2 =

1 r1, r2 Inc I nc r1, r2 , 2

(21)

)

(22)

where

N nc =

(

)

(

1 1 ab + ba , and I nc = aa + bb . 2 2

The corresponding one-electron reduced densities are given by

ρ N nc (r1; r1 ) = ∫ dr2

1 r1, r2 N nc N nc r1, r2 , 2

and - 10 ACS Paragon Plus Environment

(23)

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ρ I nc (r1; r1 ) = ∫ d r2

1 r1, r2 Inc Inc r1, r2 , 2

(24)

which satisfy the relationships:

∫ dr ρ 1

N nc

(r1; r1 ) =

1 1 N nc N nc = , and 2 2

∫ dr ρ 1

I nc

(r1; r1 ) =

1 1 Inc I nc = . 2 2

(25)

This means that the non-correlated ground-state wavefunction is equally composed of neutral and ionic components. Thus, the degrees of neutral and ionic characters, which also correspond to a weight of electron correlation, can be quantified by using the population deviations with respect to the non-correlated populations (1/2). On the basis of these quantities (see eqs 19 and 24), we define the diradical and ionic characters as well as their densities.

In the case of PN, j ≥ PI, j , where PN, j + PI, j = 1 and thus

1 ≥ PN, j ≥ 1 / 2 , the new diradical character y N, j of state j reads y N, j =

∫ dr ρ 1

y N, j

(r1; r1 ) = 2PN, j −1,

(26)

and the diradical character density is defined by

ρ y (r1; r1 ) ≡ 2 ρN, j (r1;r1 ) − ρN nc (r1;r1 ) .

(27)

N, j

From this definition, y N, j takes a value between 0 and 1, y N, j = 0 implies the non-correlated state (composed of equal-weighted neutral and ionic components), while y N, j = 1 corresponds to a pure On the other hand, in the case of PI, j ≥ PN, j , where PN, j + PI, j = 1 and thus

diradical state.

1 ≥ PI, j ≥ 1 / 2 , the ionic character of state j reads y I, j =

∫ dr ρ 1

y I, j

(r1; r1 ) = 1− 2PI, j ,

(28)

and the ionic character density is defined by

ρ y (r1, r1 ) ≡ 2 ρI nc (r1;r1 ) − ρI, j (r1;r1 ).

(29)

I, j

Similarly, y I, j takes a value between 0 and -1, with y I, j = 0 describing the non-correlated limit (composed of equal-weighted neutral and ionic components) and y I, j = -1 the pure ionic limit. - 11 ACS Paragon Plus Environment


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Therefore, the “diradical character density” ρ y N, j (r1;r1 ) of state j describes the deviation of the neutral density in the diradical system ( PN, j ≥ PI, j ) from that in the non-correlated symmetric system ( PN nc, j = PI nc, j = 1/ 2 ). A positive (negative) diradical character density represents the spatial distribution of the positive (negative) diradical correlation, where a positive value means that the density increases upon correlation, and vice versa. On the other hand, the “ionic character density”

ρ y (r1; r1 ) of state j indicates the deviation of ionic component density in the ionic system I, j

( PI, j ≥ PN, j ) from that in the non-correlated symmetric state ( PN nc, j = PI nc, j = 1/ 2 ). A negative or positive ionic character density represents the spatial distribution of the positive or negative pairelectron correlation relative to the case of a non-correlated symmetric state. The diradical and ionic characters are obtained by the spatial integration of these densities. As seen from the margins of the diradical ( 0 ≤ y N, j ≤1 ) and ionic ( −1 ≤ y I, j ≤ 0 ) characters, their amplitudes indicate the degree of localization of odd-/pair-electron density on each site. Therefore, the effective neutral (diradical or covalent)/ionic bond orders are defined, respectively, by

q N, j = 1− y N, j for PN, j ≥ PI, j and q I, j = 1− y I, j for PI, j ≥ PN, j .

(30)

Here, 0 ≤ q N(I), j ≤ 1 is satisfied. For PN, j = PI, j , we obtain q N, j = q I, j = 1 , which implies that, in the non-correlated or mean-field level of approximation, the chemical bond is composed of a superposition of equal-weighted covalent and ionic bond components. In that case, q N, j and q I, j , the covalent and ionic bond strengths of state j, take both the maximum value of 1. It is important to note that the present definitions of the diradical/ionic character and their densities are applicable to arbitrary electronic states including ionic-dominant states and time-dependent superposition states in contrast to the conventional diradical character, which is only applicable to the electronic state with dominant covalent (diradical) character, i.e., small ionic bond nature.

- 12 ACS Paragon Plus Environment

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2.3. Relationship between the New Diradical Character and the Conventional Ones. For the ground state of the two-site model (eq 5), eq 10 can be rewritten by taking advantage of eq 6 and the symmetry relationship ( Cab = Cba ) for singlet states: 2 2 1 PN,g = Cab ,g + Cba,g = (ξ + ζ )2 , 2

(31a)

2 2 1 PI,g = Caa,g + Cbb ,g = (ξ − ζ )2 + η 2 , 2

(31b)

and

Therefore, using eq 26, the expression of the present diradical character y N, j becomes:

y N,g = 2PN,g −1 = 2ξζ − η 2 ,

(32a)

where we use ξ 2 + η 2 + ζ 2 = 1 . Similarly, starting from eq 28, the ionic character y I,g reads

y I,g = 1− 2PI,g = 2ξζ − η 2 .

(32b)

We now compare the present diradical character y N,g with the conventional one, which is defined by the LUNO occupation number ( nLUNO ) and is given by eq 7 (referred to as y hereafter). After transformation of eq 7, one gets

(1− y)2 = (ξ − ζ )2 [2 − (ξ − ζ )2 ] .

(33)

Thus, we obtain

(ξ − ζ )2 = 1− 2ξζ − η 2 =1− 1− (1− y)2 = 1− y(2 − y) .

(34)

where we use the fact that ξ and ζ have the same sign in the ground state of a diradical. From eqs 32a and 33, we obtain

y N,g + 2η 2 = y(2 − y) = nLUNO (2 − nLUNO ) .

(35)

Thus, for symmetric diradical systems ( η = 0 ), the relationship between the present and the conventional diradical character expressions reads:

y N,g = y(2 − y) = nLUNO (2 − nLUNO ) .

(36) - 13 -

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It is known that there are several definitions of the number and density of odd (unpaired) electrons as well as of the diradical character based on the one-electron reduced density.32,33 Among the well-known definitions: the conventional y ( ≡ min(nLUNO,1− nLUNO ) = nLUNO ), where the odd electron density and odd electron number are defined by Head-Gordon,33 and y T ≡ nLUNO (2 − nLUNO ) defined originally by Takatsuka and coworkers.32 As seen from Figure 1, the differences between y T and

y N,g and the conventional y increase from 0 to a maximum and then decrease to 0, with a difference maximum for large and intermediate y regions, respectively. Although the present and Takatsuka’s definitions have disadvantages of yielding unpaired electrons numbers larger than the total electron number, they are useful for comparing the diradical characters of open-shell frontier orbitals such as magnetic orbitals. Moreover, the present definition has advantages of being applicable to arbitrary electronic states of symmetric/asymmetric systems as well as of providing the ionic characters, which can be used for describing ionic dominant states.

3. RESULTS AND DISCUSSION 3.1. Application to a Two-Site Diradical Model. This section illustrates the usefulness of the present definitions of diradical and ionic characters with their densities by using a simple onedimensional two-site model with two electrons in two active orbitals. Nevertheless, as shown in the next section, the present approach is directly applicable to arbitrary many-electron systems within the complete active space configuration interaction with two electrons in two active orbitals [CASCI(2,2)] or within the perfect-pairing-type VCI scheme using (HONO-i, LUNO+i) (i = 0, 1, …), that both constitute an approximate description of the electronic structures of multiradical systems. For convenience, we consider the case where the distributions of the LNO bases { aa ,

ab , ba , bb } are localized on the four points {(rA, rA), (rA, rB), (rB, rA), (rB, rB)}, respectively,


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in the (1α, 2β) plane (see Figure S1 in Supporting Information),48 so that the {(rA, rB), (rB, rA)} and {(rA, rA), (rB, rB) points correspond to the neutral (covalent or diradical) and ionic configurations, respectively. In the present case, the following relations are held for the two-electron density in the LNO basis representation (see eq 18):

ρ ab ,ab (rA , rB;rA , rB ) = ρba,ba (rB, rA ;rB, rA ) = ρ aa,aa (rA , rA ;rA , rA ) = ρ bb ,bb (rB, rB;rB, rB ) = 1,

(37)

where, for each density, the other coordinate combinations give zero values. The neutral and ionic one-electron densities of state j as well as their non-correlated densities are then obtained from eqs 15, 16, 20–24, and 37 as 2

ρ N, j (rA ;rA ) = ρN, j (rA , rA ;rA , rA ) + ρN, j (rA, rB;rA , rB ) = Cab , j , 2

ρ N, j (rB;rB ) = ρN, j (rB, rA ;rB, rA ) + ρN, j (rB, rB;rB, rB ) = Cba, j , 2

ρ I, j (rA ;rA ) = ρ I, j (rA , rA ;rA , rA )+ ρ I, j (rA , rB;rA , rB ) = Caa, j , 2

(38) (39) (40)

ρ I, j (rB;rB ) = ρI, j (rB, rA ;rB, rA ) + ρ I, j (rB, rB;rB, rB ) = Cbb, j ,

(41)

1 4

(42)

and

ρ N nc (rA ;rA ) = ρ N nc (rB;rB ) = ρ I nc (rA ;rA ) = ρ I nc (rB;rB ) = .

Using these results, in the case of PN,j ≥ PI,j (diradical dominant case), the diradical character of state j is given by eq 26 with eq 27 reading as: 2

1 2

ρ y (rA ;rA ) = 2 ρ N, j (rA ;rA ) − ρN nc (rA ;rA ) = 2 Cab , j − , N, j

(43)

and

1 2

ρ y (rB;rB ) = 2 ρN, j (rB;rB ) − ρ N nc (rB;rB ) = 2 Cba, j − . 2

N, j

2

(44)

2

Since 1/ 4 ≤ Cab , j (= Cba, j ) ≤ 1/ 2 , each density is equivalent and increases from 0 to 1/2, i.e., the odd electron density increases on each site, with increasing y N, j from 0 to 1 or decreasing covalent - 15 ACS Paragon Plus Environment


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bond order from 1 to 0. Similarly, in the case of PI, j ≥ PN, j (ionic dominant case), the ionic character of state j is represented by eq 28, of which the two contributions in the localized approximation are given by using eq 29 as

1 2

ρ y (rA ;rA ) = 2 ρI nc (rA ;rA ) − ρ I, j (rA ;rA ) = − 2 Caa, j , I, j

2

(45)

and

1 2

2

ρ y (rB;rB ) = 2 ρ I nc (rB;rB ) − ρ I, j (rB;rB ) = − 2 Cbb , j . I, j

(46)

2

2

Since 0 ≤ { Caa, j , Cbb , j } ≤ 1, each density could be positive or negative and range from -3/2 to 1/2. If the asymmetricity leads to a larger population on A than B, y I, j goes from 0 to -1, the ionic bond order decreases from 1 to 0, and ρ y I, j (rA ;rA ) ranges from 0 to -3/2 (increase in pair-electron density on A) and that of ρ yI, j (rB;rB ) from 0 to 1/2 (decrease in pair-electron density on B). In addition, the asymmetric character AS, j of state j, representing the asymmetric distribution of the two electrons on A and B, can be defined as

AS, j =

2

2

2

2

Caa, j − Cbb , j Caa, j + Cbb , j

.

(47)

It takes a value between -1 (two electrons distributed only on B) and 1 (two electrons distributed only on A), where the positive (negative) value indicates that A(B) is more populated than on B(A). Let us consider the two-site system with asymmetricity rh ranging from 0.0 to 2.0 for pseudo diradical character yS = 0.02 and 0.6 while keeping (rK, rU, rtab) = (0, 1, 1). Figure 2a-f and 3a-f show the variations in neutral ( PN, j ) and ionic ( PI, j ) populations of state j (= g, k, and f) as well as in diradical ( y N, j ) and ionic ( y I, j ) characters as a function of rh for yS = 0.02 and 0.6, respectively. For rh = 0.0, these two cases (yS = 0.02 and 0.6) can be regarded as prototypes of symmetric nonsubstituted systems with nearly closed-shell and intermediate diradical characters, respectively,


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before donor/acceptor substitutions. Note that in the present model, the LNOs (a and b) are considered as “effective” orbitals taking into account the effects of their substituents/environment, which are described/determined by (yS, rh). It is known that for symmetric closed-shell systems states g and f are composed of equal-weighted diradical and ionic components, while state k is a pure ionic state. As seen from Figure 2a, the increase in rh causes the decrease in PN,g (the increase in PI,g) and leads to a crossing at a finite rh value ( > 1). On the other hand, in the first excited state k (Figure 2b), PI,k decreases from 1 (PN,k increases from 0) with rh, but PI,k and PN,k do not intersect each other in the region 0.0 ≤ rh ≤ 2.0. In state f, PI,f increases (PN,f decreases) with rh (Figure 2c) and then no intersection between PI,f and PN,f could occur. Moreover, with increasing rh, the present diradical character yN,g decreases from a small value toward 0 at rh ~ 1.4, where it connects to yI,g (Figure 2d). On the other hand, the yI,g amplitude (yI,g is negative) increases with rh (Figure 2d). In contrast, the conventional y value decreases toward 0 with rh starting from a smaller positive value (= 0.02) than that (~ 0.2) of yN,g. This feature coincides with the discussion in Sec. 2.2. In states k and f, it is found that only the ionic characters yI,k and yI,f exist and they increase from -1 (decrease in amplitude) and decrease from ~ -0.2, respectively (Figure 2e and f). This corresponds to the fact that states k and f have dominant ionic components. On the other hand, for intermediate yS = 0.6 (Figure 3a-c) PN,g strongly decreases (PI,g increases) with rh, since the difference between PN,g and PI,g for rh = 0 is larger for yS = 0.6 than for yS = 0.02, which reflects the PN = PI relation for yS = 0. The PN,g and PI,g curves intersect each other at rh ~ 1, which is smaller than for yS = 0.02. This difference follows the analytical expression,

(

)

−1

2  2  given in Ref. 23, of the intersection position for rK = 0, i.e., rh = 2 Caa,g − Cbb ,g  , which states  

2

2

that for small yS (~ 0), corresponding to Caa,g ~ Cbb ,g , a large rh (> 1) is necessary for achieving 2

2

PN,g = PI,g, while for finite yS (> 0), corresponding to ( Caa,g , Cbb,g ) ~ (1/2, 0), PN,g = PI,g is achieved at rh ~ 1. Then, in the first excited state k (Figure 3b), PI,k decreases from 1 (PN,k - 17 ACS Paragon Plus Environment


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increases from 0) with rh, and the PI,k and PN,k curves cross each other at rh ~ 1.0. This indicates that for intermediate yS values, increasing rh leads to the inversion of the dominant electronic configurations (neutral/ionic) in states g and k around rh = 1. This inversion behavior between states g and k at rh = 1 is represented by the fact that for state g, yN,g varies from a positive value – and this value is larger than the conventional y – toward 0 at rh ~ 1, followed by the decrease in yI,g from 0 toward -1 (Figure 3d), whereas for state k, such behavior is inversed with respect to rh ~ 1 (Figure 3e). In contrast to yS = 0.02, for yS = 0.6, almost pure ionic character is observed for state f at any rh (Figure 3c), which corresponds to yI,f nearly equal to -1 at any rh (Figure 3f). Finally, we investigate the diradical ( ρ yN, j (X) ) and ionic ( ρ y I, j (X) ) character densities, where ρ N, j (X) ≡ ρ N, j (rX ;rX ) (X = A, B), for yS = 0.6 (see Figure 4a–c). As shown in eqs 26 and 30,

ρ y (X) and ρ y (X) lead to y N, j for PN, j > PI, j and y I, j for PI, j > PN, j , respectively. ρ y (A) is N, j

I, j

N, j

equal to ρ yN, j (B) due to the symmetry, while ρ y I, j (A) is smaller than ρ y I, j (B) in the present asymmetric system. For state g, when increasing rh, ρ yN,g (X) (X =A, B) varies from a positive value around 1/2 to a negative value around -1/2 through 0 around rh ~ 1, the behavior of which corresponds to the variation from a positive to a negative diradical correlation on sites A and B through non-correlation in the vicinity of rh ~ 1. On the other hand, with increasing rh, ρ yI,g (A) decreases from a positive value around 1/2 to a negative value close to -3/2 through 0 value in the vicinity of rh ~ 1, while ρ y I,g (B) retains a value of 1/2 at any rh, whereas for the symmetric pure diradical case ρ yI,g (A) = ρ y I,g (B) = 1/2 (Figure 4a). This indicates that increasing rh causes the concentration of population on site A, with the limiting values given by ( ρ yI,g (A) , ρ y I,g (B) ) = (-3/2, 1/2). This is further supported by the variation of the asymmetric character AS,g from 0 to 1. For state k, the behavior of the diradical/ionic character densities and asymmetric character are just inverted with respect to those of state g, with the exception of the rapid variation of ρ y I,k (X) (X = A, - 18 ACS Paragon Plus Environment

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B) and AS,k around rh ~ 0 (Figure 4b). These match the inverted behaviors of the diradical and ionic populations as well as of the diradical and ionic characters shown in Figures 3b and e. Similarly, in state f (Figure 4c), according to the variations in diradical/ionic population and diradical/ionic character shown in Figures 3c and f, ρ y I,f (A) amounts to ~ 1/2 ( ρ y I,f (B) ~ -3/2), while ρ yN,f (X) (X = A, B) almost retains a value in the vicinity of -1/2 for any rh. As a result, these diradical and ionic character densities are found to give a spatial diradical (odd electron) and ionic (pair-electron) correlation features, respectively.

3.2. Application to Diphenalenyl Diradicaloid and Graphene Nanoflake under Static Electric Field.

The present scheme is now applied to the description of the diradical/ionic

characters of real open-shell singlet molecules in their ground and excited states, i.e., a diphenalenyl diradicaloid, s-indaceno[1,2,3-cd;5,6,7-c’d’]diphenalene (IDPL),15 and a rectangular graphene nanoflake, PAH[3,5],9 in the presence of an external static electric field F applied along the molecular axis (see Figure 5a and b) to induce asymmetric electron distribution.49-51 In this section, quantum chemistry calculations were performed by Gaussian 09.52 In the present case (rU, rtab) = (1, 1) and rh is given by rF where rF = FRBA/U (F: static external field amplitude, RBA = (b|r|b) – (a|r|a)). The geometries were optimized at F = 0 using the RB3LYP/6-31G* method. These geometries were employed in all calculations, including those where an external field is applied in order to highlight the electronic effects (in other words, the field-induced nuclear relaxation effects are neglected). A complete-active-space configuration-interaction treatment with two electrons in two molecular orbitals [CASCI(2,2)] was performed to calculate the wavefunctions, diradical/ionic characters, and their densities for the singlet ground (g), the first-excited (k) and the second-excited (f) states. The symmetry-adapted MOs, g(r) and u(r), in the CAS space were calculated using the tuned long-range-corrected (LC)-BLYP method53 with a range-separating parameter µ (= 0.190 bohr-1 for IDPL and 0.178 for PAH[3,5]) that has been optimized for each system so as to satisfy - 19 ACS Paragon Plus Environment


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the exact condition (minus HOMO energy = vertical ionization potential) of density functional theory (DFT).54-58 This CASCI(2,2) method qualitatively reproduces the first optically-allowed excitation energies for condensed-ring systems (see Supporting Information). We have calculated the diradical/ionic characters and their densities using eqs s2 (in Supporting Information), 15, 16, 23, 24 and 26-29. Figure 6a and b show the variations of diradical/ionic characters (yN,j and yI,j, respectively) for states g, k and f of IDPL and PAH[3,5] as a function of the field amplitude F. Both systems show similar variations in diradical/ionic characters for each state as the function of F. In absence of external field, the ground states belong to the intermediate diradical character region [yN,g = 0.669 (IDPL), 0.583 (PAH[3,5])]. The variations of yN,j and yI,j as a function of F are in qualitative agreement with those of the two-site diradical model with yS = 0.6 (see Figure 3d-f). The rh = 1.0 in the two-site diradical model corresponds to F ~ 0.008 a.u. for IDPL and ~ 0.007 a.u. for PAH[3,3], where the dominant population in the ground (first excited) state switches from neutral (ionic) to ionic (neutral). The diradical/ionic character densities of the {g, k, f} states of IDPL are shown for F = 0.0 a.u. (yN,g = 0.669 (Figure 7(a-g)), yI,k = -1.0 (Figure 7(a-k)), yI,f = -0.669 (Figure 7(a-f))) and 0.016 a.u. (yI,g = -0.814 (Figure 7(b-g)), yN,k = 0.783 (Figure 7(b-k)), yI,f = 0.969 (Figure 7(b-f))). The diradical character densities (in state g and f) and ionic character density (in state k) for F = 0 a.u. are slightly more distributed on the both-end phenalenyl rings. These maps indicate that the unpaired (odd) electrons and pair-electrons are dominantly distributed on the both-end phenalenyl rings, which is consistent with the diradical and zwitterionic resonance structures, respectively.

They also satisfy Clar’s sextet rule,59 which states that the Kekulé

resonance structures with larger number of disjoint aromatic π-sextets contribute more to the ground state electronic structures (see Figure 5a). In contrast, using a strong field (F = 0.016 a.u.) gives rise to asymmetric electron distributions in states g and f with dominant ionic components (though their polarization directions are opposite to each other), but to dominant diradical character in state k. Also, for F = 0.016 a.u., positive and negative ionic character density distributions are


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well separated into the right- (left-) and left- (right-) hand phenalenyl rings in state g (f). This implies that the pair-electron density distribution is increased in the negative ionic character density region on the left- (right-) hand phenalenyl ring in state g (f). PAH[3,5] is described by similar field-induced variations of the diradical and ionic character density distributions for each state (see Figure 8(a-g)-(a-f) for F = 0.0 a.u. (yN,g = 0.583, yI,k = -1.0, yI,f = -0.583) and Figure 8(b-g)-(b-f) for F = 0.016 a.u. (yI,g = -0.822, yN,k = 0.783, yI,f = -0.961)). The diradical character densities are mostly distributed on the zigzag edges in state g (F = 0.0 a.u.) and k (F = 0.016 a.u.), while the ionic character densities for F = 0.016 a.u. are distributed on wider spatial regions. For the latter, positive and negative densities are well separated into right (left) and (right) in state g (f), which indicates that the pair-electron density distribution is increased in the negative ionic character domain on the left (right) side of PAH[3,5] in its state g (f). These distribution features in the ground state are also in qualitative agreement with those predicted from the weighted resonance structures based on the Clar’s sextet rule59 (see Figure 5b).

3.3. Application to π-Stacked Dimer of Phenalenyl-Derivatives with Varying Intermonomer Distance. In this section, we examine the properties of a π-stacked phenalenylderivative dimer model as a function of the intermonomer distance R (see Figure 9). In this dimer, the central carbon (C) atoms of the phenalenyl rings were replaced by boron (B) [Phn(B)] and nitrogen (N) [Phn(N)] atoms, respectively. This molecule, referred to as (B, N), is isoelectronic to the neutral phenalenyl radical dimer (C, C) that was found to exhibit a gradual increase of the diradical character with increasing R,60 which indicates that the dimer having covalent-like interactions (pancake bonding) at R = 3.0 Å (y = 0.540 at R = 3.0 Å) undergoes a homolytic dissociation to two phenalenyl radicals as R increases, for example, y = 0.999 at R = 5.0 Å. Considering the fact that B- (N-)doped phenalenyl have zero (two) electron in the central p orbital, it is expected that the (B, N) system presents a somewhat covalent-like interaction between the



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monomers, which allows partial electron transfer from N to B around the equilibrium position while it undergoes a heterolytic dissociation to B-/N-doped phenalenyls as increasing R. To verify this prediction, the diradical/ionic character analysis was applied. The dimer model was constructed by π-π statcking of B-/N-doped phenalenyls, where the monomer geometries were optimized at the UB3LYP/6-311G* level of theory (see Supporting Information).

These quantum chemical

calculations were performed using Gaussian 09.52 The diradical/ionic characters were calculated by the CASCI(2,2)/6-31G* method using the localized MOs, which were obtained from Boys localization of the SOMOs calculated at the ROHF/6-31G* level.

These calculations were

performed by using MOLPRO program package.61 Note here that the Boys localization of SOMOs for asymmetric π-stacked phenalenyl-derivative dimer models was conducted to obtain LNO bases in the asymmetric two-site model as shown in Sec. 3.1. On the other hand, for the field-induced models in Sec. 3.2, we do not need such localization of SOMOs since we can use LNO bases obtained from the symmetric systems in the absence of external fields. The evolution of the electronic distributions was assessed first, using the Hirshfeld charges. Figure 10a shows the variation in the electronic charge difference between the monomers, ele ele , as a function of R. ∆ρ spans the [0, 2] interval, corresponding to the ∆ρ = ρPhn(B) − ρPhn(N)

homolytic [Phn(N•+) and Phn(B•–), ∆ρ = 0] and heterolytic [Phn(N:) and Phn(B), ∆ρ = 2] dissociation limits, respectively. The asymmetricity of the electronic distribution between the monomers increases as increasing ∆ρ . At 3.0 Å, ∆ρ = 1.35, which implies that ~0.175 electrons are transferred from Phn(B•–) to Phn(N•+) (or ~0.325 electrons are transferred from Phn(N) to Phn(B)), which suggests a covalent-like interaction. Then, as increasing R, ∆ρ increases toward 2, which corresponds to a heterolytic dissociation. Next, as seen from Figure 10b, the diradical/ionic character decreases from a slightly positive value (~0.18, nearly closed-shell) at 3.0 Å to negative values (which reaches -1 around R = 5.0 Å) through zero value around R = 3.5 Å. This implies that in the (B, N) dimer the neutral component is slightly larger than the ionic component with nearly - 22 ACS Paragon Plus Environment

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closed–shell for R < 3.5 Å, while the ionic character increases with R and the ionic component becomes dominant for R > 3.5 Å. It is also found that the first excited state k shows the inversed feature, that is, it varies from a slightly ionic to a neutral diradical nature as increasing R, while the second excited state f remains nearly purely ionic as increasing R.

These variations of the

neutral/ionic characters are in qualitative agreement with the other examples shown in Sec. 3.2. The diradical/ionic character densities for the g, k and f states at R = 3.0, 4.0 and 5.0 Å are shown in Figure 10c. For the ground state, we observe a small positive diradical character density (odd electron density) on each monomer at R = 3.0 Å, consistent with a nearly closed-shell system, while for R = 4.0 Å we observe positive and negative densities well separated on the Phn(B) and Phn(N) monomers, respectively, which indicates that the pair-electron density distribution is increased in the negative ionic character domain on the Phn(N) monomer. The amplitudes of ionic character densities at R = 5.0 Å are slightly larger than those at R = 4.0 Å. The diradical/ionic character density distributions are also shown for the other excited states to reflect the distribution of the π electron pair between the monomers and the diradical/ionic character amplitude of each state. As a result, the (B, N) dimer is found to vary from a covalent-like bonding (pancake) dimer with slight diradical character and slight electron transfer from Phn(B•–) to Phn(N•+) at R = 3.0 Å to a dimer with pure ionic character, which means that an electron is completely transferred from Phn(B•–) to Phn(N•+), resulting in Phn(B)–Phn(N) dimer at the dissociation limit.

4. SUMMARY The diradical and ionic natures of open-shell singlet molecules as well as their spatial contributions (densities) have been investigated using new definitions within the valence configuration interaction (VCI) two-site model with two electrons in two active orbitals. For the diradical character of ground states, the relationship between the present definition and the conventional one is clarified. However, the proposed chemical indices have the advantage of being applicable to arbitrary states,



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that is, not only to diradical-dominant states but also to ionic-dominant states. Moreover, they allow monitoring the variations in the diradical and ionic components of singlet ground and excited states as a function of the system asymmetricity rh. This asymmetricity is typically the difference of ionization energies of sites A and B but it can also be tuned by applying an external electric field. Then, by resorting to CASCI(2,2) calculations, this scheme has been applied to real compounds, that is, i) a diradicaloid diphenalenyl (IDPL) and a rectanglular graphene nanoflake PAH[3,5] in presence of an external static electric field, of which the magnitude is varied to induce asymmetric electron distributions, and ii) a π-stacked B-/N-doped phenalenyl dimer with varying intermonomer distance, which changes the ratio of ionic to diradical component. The analysis demonstrates the applicability of the present scheme to clarifying the variations of diradical/ionic characters/densities of the ground and excited states in a wide range of asymmetricity. These interpretation methods are expected to help controlling the properties, including NLO properties and singlet fission of symmetric/asymmetric open-shell singlet molecules.4,

62–65

Note that the

present examples use only the minimum active space, CAS(2,2), for diradical systems. Although the present diradicaloids are predicted to be well described with such a minimal active space judging from the previous BS DFT calculations,51,60 enlarging the active space may affect the diradical character quantitatively. In particular, the description of delocalized multiradicaloids is expected to require larger active spaces, which could be approximately treated by extending the present scheme to the perfect-pairing type CI with larger active spaces. Further applications to other di-/multi-radicaloids together with the evaluation of their NLO properties and an extension to their time-evolution are currently tackled in our laboratories.

ASSOCIATED CONTENT


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Supporting Information The Supporting Information is available free of charge on the ACS Publications website at DOI: Valence configuration interaction two-site model; Excitation energies of condensed-ring molecules: comparison between experimental and theoretical values; Cartesian coordinates for systems 1–11 shown in Figure S3.

AUTHOR INFORMATION Corresponding Author *E-mail: [email protected] Notes The authors declare no competing financial interests.

ACKNOWLEDGMENTS This work is a result of a collaboration between OsakaU and UNamur, supported by a Bilateral Program JSPS—F.R.S.-FNRS grant. It has been supported by JSPS KAKENHI Grant Numbers JP15J04949 and JPA2645050 in JSPS Research Fellowship for Young Scientists, Grant Number JP25248007 in Scientific Research (A), Grant Number JP24109002 in Scientific Research on Innovative Areas “Stimuli-Responsive Chemical Species”, Grant Number JP15H00999 in Scientific Research on Innovative Areas “π-System Figuration”, and Grant Number JP26107004 in Scientific Research on Innovative Areas “Photosynergetics”, the Belgian Government (IUAP N° P7/05 “Functional Supramolecular Systems”), and the Francqui Foundation. This is also partly supported



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by King Khalid University through a grant RCAMS/KKU/001-16 under the Research Center for Advanced Materials Science at King Khalid University, Kingdom of Saudi Arabia.


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REFERENCES (1) Lambert, C. Towards Polycyclic Aromatic Hydrocarbons with a Singlet Open-Shell Ground State. Angew. Chem., Int. Ed. 2011, 50, 1756–1758. (2) Sun, Z.; Wu, J. Open-Shell Polycyclic Aromatic Hydrocarbons. J. Mater. Chem. 2012, 22, 4151–4160. (3) Plasser, F.; Pašalić, H.; Gerzabek, M. H.; Libisch, F.; Reiter, R.; Burgdörfer, J.; Müller, T.; Shepard, R.; Lischka, H. The Multiradical Character of One- and Two-Dimensional Graphene Nanoribbons. Angew. Chem. Int. Ed. 2013, 52, 2581–2584. (4) Nakano, M. Excitation Energies and Properties of Open-Shell Singlet Molecules; Springer International Publishing, 2014. (5) Hachmann, J.; Dorando, J. J.; Avilés, M.; Chan, G. K.-L. The Radical Character of the Acenes: A Density Matrix Renormalization Group Study. J. Chem. Phys. 2007, 127, 134309–1–9. (6) Casanova, D.; Head-Gordon, M. Restricted Active Space Spin-Flip Configuration Interaction Approach: Theory, Implementation and Examples. Phys. Chem. Chem. Phys. 2009, 11, 9779– 9790. (7) Hod, O.; Barone, V.; Scuseria, G. E. Half-Metallic Graphene Nanodots: A Comprehensive First-Principles Theoretical Study. Phys. Rev. B 2008, 77, 035411–1–6. (8) Jiang, D. E.; Sumpter, B. G.; Dai, S. Unique Chemical Reactivity of a Graphene Nanoribbon’s Zigzag Edge. J. Chem. Phys. 2007, 126, 134701–1–13. (9) Nagai, H.; Nakano, M.; Yoneda, K.; Kishi, R.; Takahashi, H.; Shimizu, A.; Kubo, T.; Kamada, K.; Ohta, K.; Botek, E. et al. Signature of Multiradical Character in Second Hyperpolarizabilities of Rectangular Graphene Nanoflakes. Chem. Phys. Lett. 2010, 489, 212– 218. (10)

Jiang, D. E.; Dai, S. Circumacenes versus Periacenes: HOMO–LUMO Gap and Transition

from Nonmagnetic to Magnetic Ground State with Size. Chem. Phys. Lett. 2008, 466, 72–75.



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

(11)

Tönshoff, C.; Bettinger, H. F. Photogeneration of Octacene and Nonacene. Angew. Chem.

Int. Ed. 2010, 49, 4125–4128. (12)

Zade, S. S.; Bendikov, M. Heptacene and Beyond: The Longest Characterized Acenes.

Angew. Chem. Int. Ed. 2010, 49, 4012–4015. (13)

Kaur, I.; Jazdzyk, M.; Stein, N. N.; Prusevich, P.; Mille, G. P. Design, Synthesis, and

Characterization of a Persistent Nonacene Derivative. J. Am. Chem. Soc. 2010, 132, 1261–1263. (14)

Purushothaman, B.; Bruzek, M.; Parkin, S.; Miller, A.-F.; Anthony, J. E. Synthesis and

Structural Characterization of Crystalline Nonacene. Angew. Chem. Int. Ed. 2011, 50, 7013– 7917. (15)

Kubo, T.; Shimizu, A.; Sakamoto, M.; Uruichi, M.; Yakushi, K.; Nakano, M.; Shiomi, D.;

Sato, K.; Takui, T.; Morita, Y. et al. Synthesis, Intermolecular Interaction, and Semiconductive Behavior of a Delocalized Singlet Biradical Hydrocarbon. Angew. Chem. Int. Ed. 2005, 44, 6564–6568. (16)

Huang, J.; Kertesz, M. Intermolecular Covalent π−π Bonding Interaction Indicated by Bond

Distances, Energy Bands, and Magnetism in Biphenalenyl Biradicaloid Molecular Crystal. J. Am. Chem. Soc. 2007, 129, 1634–1643. (17)

Sun, Z.; Ye, Q.; Chi, C.; Wu, J. Low Band Gap Polycyclic Hydrocarbons: From Closed-

Shell Near Infrared Dyes and Semiconductors to Open-Shell Radicals. Chem. Soc. Rev. 2012, 41, 7857−7889 (2012). (18)

Abe, M. Diradicals. Chem. Rev. 2013, 113, 7011−7088.

(19)

Sun, Z.; Zeng, Z.; Wu, J. Zethrenes, Extended p-Quinodimethanes, and Periacenes with a

Singlet Biradical Ground State. Acc. Chem. Res. 2014, 47, 2582−2591. (20)

Nakano, M.; Kishi, R.; Nitta, T.; Kubo, T.; Nakasuji, K.; Kamada, K.; Ohta, S.; Botek, E.;

Champagne, B. Second Hyperpolarizability (γ) of Singlet Diradical Systems: Dependence of γ on the Diradical Character. J. Phys. Chem. A 2005, 109, 885−891. - 28 ACS Paragon Plus Environment

Page 28 of 57

Page 29 of 57

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


(21)

Nakano, M.; Kishi, R.; Ohta, S.; Takahashi, H.; Kubo, T.; Kamada, K.; Ohta, K.; Botek, E.;

Champagne, B. Relationship between Third-Order Nonlinear Optical Properties and Magnetic Interactions in Open-Shell Systems: A New Paradigm for Nonlinear Optics. Phys. Rev. Lett. 2007, 99, 033001−1−4. (22)

Nakano, M.; Yoneda, K.; Kishi, R.; Takahashi, H.; Kubo, T.; Kamada, K.; Ohta, K.; Botek,

E.; Champagne, B. Remarkable Two-Photon Absorption in Open-Shell Singlet Systems. J. Chem. Phys. 2009, 131, 114316–1–7. (23)

Nakano, M.; Champagne, B. Diradical Character Dependences of the First and Second

Hyperpolarizabilities of Asymmetric Open-Shell Singlet Systems. J. Chem. Phys. 2013, 138, 244306–1–13. (24)

Hayes, E. F.; Siu, A. K. Q. Electronic Structure of the Open Forms of Three-Membered

Rings. J. Am. Chem. Soc. 1971, 93 2090−2091. (25)

Yamaguchi, K. In Self-Consistent Field: Theory and Applications; Carbo, R., Klobukowski,

M., Eds.; Elsevier: Amsterdam, The Netherlands; 1990; pp 727−828. (26)

Nakano, M.; Fukui, H.; Minami, T.; Yoneda, K.; Shigeta, Y.; Kishi, R.; Champagne, B.;

Botek, E.; Kubo, T.; Ohta, K.; et al. Theoret. Chem. Acc. 2011, 130, 711–724; erratum 2011, 130, 725. (27)

Salem, L.; Rowland, C. The Electronic Properties of Diradicals. Angew. Chem., Int. Ed.

1972, 11, 92–111. (28)

Bonačiý-Koutecký, V.; Koutecký, J.; Michl, J. Neutral and Charged Biradicals, Zwitterions,

Funnels in S1, and Proton Translocation: Their Role in Photochemistry, Photophysics, and Vision. Angew. Chem., Int. Ed. 1987, 26, 170–189. (29)

Cotton, F. A.; Murillo, C. A.; Walton, R. A. Multiple Bonds between Metal Atoms; 3rd ed.;

Springer: New York, 2005.



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

(30)

Metal-Metal Bonding; Parkin, G., Ed.; In Structure and Bonding; Mingos, D. M. P., Ed.;

Vol. 136; Springer: Heidelberg, 2010. (31)

Breher, F. Stretching Bonds in Main Group Element Compounds—Borderlines between

Biradicals and Closed-Shell Species. Coord. Chem. Rev. 2007, 251, 1007–1043. (32)

Takatsuka, K.; Fueno, T.; Yamaguchi, K. Distribution of Odd Electrons in Ground-State

Molecules. Theor. Chim. Acta. 1978, 48, 175–183. (33)

Head-Gordon, M. Characterizing Unpaired Electrons from the One-Particle Density Matrix.

Chem. Phys. Lett. 2003, 372, 508–511. (34)

Machado, F. B. C.; Aquino, A. J. A.; Lischka, H. The Diverse Manifold of Electronic States

Generated by a Single Carbon Defect in a Graphene Sheet: Multireference Calculations Using a Pyrene Defect Model. ChemPhysChem 2014, 15, 3334–3341. (35)

Cui, Z.-h.; Lischka, H.; Mueller, T.; Plasser, F.; Kertesz, M. Study of the Diradicaloid

Character in a Prototypical Pancake-Bonded Dimer: The Stacked Tetracyanoethylene (TCNE) Anion Dimer and the Neutral K2TCNE2 Complex. ChemPhysChem 2014, 15, 165–176. (36)

Cui, Z.-h.; Lischka, H.; Beneberu, H. Z.; Kertesz, M. Double Pancake Bonds: Pushing the

Limits of Strong π−π Stacking Interactions. J. Am. Chem. Soc. 2014, 136, 12958−12965. (37)

Luzanov, A. V.; Prezhdo, O. V. The Spin-Polarized Extended Brueckner Orbitals. J. Chem.

Phys. 2011, 135, 094107-1-14. (38)

Feng, X.; Luzanov, A. V.;

Krylov, A. I., Fission of Entangled Spins: An Electronic

Structure Perspective, J. Phys. Chem. Lett. 2013, 4, 3845−3852. (39)

Papanikolaou, P.; Karafiloglou, P. Extracting Covalent and Ionic Structures from Usual

Delocalized Wave Functions: The Electron-Expansion Methodology. J. Phys. Chem. A 2008, 112, 8839–8848.


Page 30 of 57

Page 31 of 57

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


(40)

Minami, T.; Ito, S.; Nakano, M. Signature of Singlet Open-Shell Character on the Optically

Allowed Singlet Excitation Energy and Singlet–Triplet Energy Gap. J. Phys. Chem. A 2013, 117, 2000–2006. (41)

Smith, M. J.; Michl. J. Singlet Fission. Chem. Rev. 2010, 110, 6891–6936.

(42)

Son, Y. -W.; Cohen, M. L.; Louie, S. G. Half-Metallic Graphene Nanoribbons. Nature 2006,

444, 347–349. (43)

Minami, T.; Nakano, M. Diradical Character View of Singlet Fission. J. Phys. Chem. Lett.

2012, 3, 145–150. (44)

Minami, T.; Ito, S.; Nakano, M. Theoretical Study of Singlet Fission in Oligorylenes. J.

Phys. Chem. Lett. 2012, 3, 2719–2723. (45)

Minami, T.; Ito, S.; Nakano, M. Fundamental of Diradical-Character-Based Molecular

Design for Singlet Fission. J. Phys. Chem. Lett. 2013, 4, 2133–2137. (46)

Michl, J. Singlet and Triplet States of an Electron Pair in a Molecule—a Simple Model. J.

Molec. Struct. (THEOCHEM) 1992, 260, 299–311. (47)

Calzado, C. J.; Cabrero, J.; Malrieu, J. P.; Caballol, R. Analysis of the Magnetic Coupling in

Binuclear Complexes. I. Physics of the Coupling. J. Chem. Phys. 2002, 116, 2728–2747. (48)

Nakano, M.; Kishi, R.; Ohta, S.; Takebe, A.; Takahashi, H.; Furukawa, S.; Kubo, T.; Morita,

Y.; Nakasuji, K.; Yamaguchi, K.; et al. Origin of the Enhancement of the Second Hyperpolarizability of Singlet Diradical Systems with Intermediate Diradical Character. J. Chem. Phys. 2006, 125, 074113–1–9. (49)

Meyers, F.; Marder, S. R.; Pierce, B. M.; Brédas, J. L. Electric Field Modulated Nonlinear

Optical Properties of Donor-Acceptor Polyenes: Sum-Over-States Investigation of the Relationship between Molecular Polarizabilities (α, β and γ) and Bond Length Alternation. J. Am. Chem. Soc. 1994, 116, 10703–10714.



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

(50)

Kirtman, B.; Champagne, B.; Bishop, D. M. Electric Field Simulation of Substituents in

Donor−Acceptor Polyenes: A Comparison with Ab Initio Predictions for Dipole Moments, Polarizabilities, and Hyperpolarizabilities. J. Am. Chem. Soc. 2000, 122, 8007–8012.. (51)

Nakano, M.; Minami, T.; Yoneda, K.; Muhammad, S.; Kishi, R.; Shigeta, Y.; Kubo, T.;

Rougier, L.; Champagne, B.; Kamada, K. et al. Giant Enhancement of the Second Hyperpolarizabilities of Open-Shell Singlet Polyaromatic Diphenalenyl Diradicaloids by an External Electric Field and Donor–Acceptor Substitution. J. Phys. Chem. Lett. 2011, 2, 1094– 1098. (52)

Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J.

R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.; et al. Gaussian 09, Revision B.01; Gaussian, Inc.: Wallingford, CT, 2010. (53)

Tawada, Y.; Tsuneda, T.; Yanagisawa, S.; Yanai, T.; Hirao, K. A Long-Range-Corrected

Time-Dependent Density Functional Theory. J. Chem. Phys. 2004, 120, 8425–8433. (54)

Stein, T.; Kronik, L.; Baer, R. Reliable Prediction of Charge Transfer Excitations in

Molecular Complexes Using Time-Dependent Density Functional Theory. J. Am. Chem. Soc. 2009, 131, 2818–2820. (55)

Stein, T.; Eisenberg, H.; Kronik, L.; Baer, R. Fundamental Gaps in Finite Systems from

Eigenvalues of a Generalized Kohn-Sham Method. Phys. Rev. Lett. 2010, 105, 266802–1–4. (56)

Refaely-Abramson, S.; Baer, R.; Kronik, L. Fundamental and Excitation Gaps in Molecules

of Relevance for Organic Photovoltaics from an Optimally Tuned Range-Separated Hybrid Functional. Phys. Rev. B 2011, 84, 075144–1–8. (57)

Minami, T.; Nakano, M.; Castet, F. Nonempirically Tuned Long-Range Corrected Density

Functional Theory Study on Local and Charge-Transfer Excitation Energies in a Pentacene/C60 Model Complex. J. Phys. Chem. Lett. 2011, 2, 1725–1730.


Page 32 of 57

Page 33 of 57

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


(58)

Kronik, L.; Stein, T.; Refaely-Abramson, S.; Baer, R. Excitation Gaps of Finite-Sized

Systems from Optimally Tuned Range-Separated Hybrid Functionals. J. Chem. Theor. Comput. 2012, 8, 1515–1531. (59)

Clar, E. The Aromatic Sextet; Wiley: New York, 1972.

(60)

Yoneda, K.; Nakano, M.; Fukuda, K.; Matsui, H.; Takamuku, S.; Hirosaki, Y.; Kubo, T.;

Kamada, K.; Champagne, B. Third-Order Nonlinear Optical Properties of One-Dimensional Open-Shell Molecular Aggregates Composed of Phenalenyl Radicals. Chem. Eur. J. 2014, 20, 11129–11136. (61)

MOLPRO, version 2015.1, a package of ab initio programs, Werner, H.-J.; Knowles, P. J.;

Knizia, G.; Manby, F. R.; Schütz, M. ; others, see http://www.molpro.net. (62)

Nakano, M.; Champagne, B. Theoretical Design of Open-Shell Singlet Molecular Systems

for Nonlinear Optics. J. Phys. Chem. Lett. 2015, 6, 3236−3256. (63)

Nakano, M.; Champagne, B. Nonlinear Optical Properties in Open-Shell Molecular Systems.

WIREs Comput. Mol. Sci., 2016, 6, 198–210. (64)

Nakano, M. Open-Shell-Character-Based Molecular Design Principles: Applications to

Nonlinear Optics and Singlet Fission. Chem. Rec. 2016. DOI: 10.1002/tcr.201600094. (65)

Fukuda, K.; Suzuki, Y.; Matsui, H.; Nagami, T.; Kitagawa, Y.; Champagne, B.; Kamada,

K.; Yamamoto, Y.; Nakano, M. Tuning Nonlinear Optical Properties by Diradical and Charge Transfer Characters of Chichibabin's Hydrocarbon Derivatives. ChemPhysChem., 2016, DOI: 10.1002/cphc.201601226.



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Figure captions

Figure 1.

Diradical characters (yN, yT, y) versus conventional y.

Figure 2.

rh dependences of PN,j and PI,j (a-c) as well as of diradical (yN,j) and ionic (yI,j) characters (d-f) of state j (= g, k and f) for yS = 0.02.

The evolution of the

conventional diradical character (y) is also shown for state g (d). The vertical dotted lines in (a) and (d) indicate the rh for PN,g = PI,g.

Figure 3.

rh dependences of PN,j and PI,j (a-c) as well as of diradical (yN,j) and ionic (yI,j) characters (d-f) of state j (= g, k and f) for yS = 0.6. The vertical dotted lines in (a) and (d) ((b) and (e)) indicate the rh for PN,g = PI,g (PN,k = PI,k).

Figure 4.

rh dependences of the diradical and ionic character densities, ρN, j (X) and ρI, j (X) , where X = A and B, of state j as well as of the asymmetric character AS, j (j = g, k and f). The vertical dotted line in (a) ((b)) indicates the rh for PN,g = PI,g (PN,k = PI,k).

Figure 5.

Diphenalenyl diradicaloid, s-indaceno[1,2,3-cd;5,6,7-c’d’]diphenalene (IDPL) (a) and a rectangular graphene nanoflake, PAH[3,5] (b), as well as their several resonance structures. The solid ring indicates Clar’s sextet. The direction of a static external electric field F along the molecular axis is also shown.


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Figure 6.

Variations of the diradical/ionic characters (yN,j and yI,j, respectively) for the ground (g), first (k), and second excited (f) states of IDPL (a) and PAH[3,5] (b) as the function of the amplitude [F = 0.0 – 0.016 a.u.] of the external static electric field.

Figure 7.

Diradical/ionic character densities and corresponding diradical/ionic characters (yN,j and yI,j, respectively) of the j = {g, k, f}states of IDPL for F = 0.0 a.u. (a-j) and 0.016 a.u. (b-j). The white and blue surfaces represent positive and negative diradical/ionic character densities with iso-surfaces of ±0.0009 a.u., respectively.

Figure 8.

Diradical/ionic character densities and corresponding diradical/ionic characters (yN,j and yI,j, respectively) of the j = {g, k, f}states of PAH[3,5] for F = 0.0 a.u. (a-j) and 0.016 a.u. (b-j).

The white and blue surfaces represent positive and negative

diradical/ionic character densities with iso-surfaces of ±0.0009 a.u., respectively.

Figure 9.

π-stacked B-/N-doped phenalenyl dimer model with intermonomer distance R.

Figure 10.

ele ele Variations of the electronic charge difference ∆ρ = ρPhn(B) for the ground − ρPhn(N)

state (g) (a) and diradical/ionic characters (yN,j and yI,j, respectively) for the ground (g) and excited (k, f) states (b) as a function of intermonomer distance R. The diradical/ionic character densities of each electronic state at R = 3.0, 4.0 and 5.0 Å are also shown (c). The vertical dotted line in (b) indicates the R satisfying yN/I,g = yN/I,k = 0. In (c), the yellow and blue surfaces represent positive and negative diradical/ionic character densities with iso-surfaces of ±0.001 a.u., respectively.



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M. Nakano et al., Fig. 1


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F = 0 .0 0 a .u .to 0 .0 1 6 a .u .

F

Closed-shell (Neutral) Closed-shell (Ionic)

(b)

F = 0 .0 0 a .u .to 0 .0 1 6 a .u .

F

D ir a d ic a l( N e u tr a l) C lo s e d s h e ll( N e u tr a l) C lo s e d s h e ll( Io n ic )

(a)

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Diradical (Neutral)




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(a)

(b)




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(a)

(b)

(c) State g

State k

State f

yN,g = 0.183

yI,k = –0.341

yI,f = –0.841

yI,g = –0.514

yN,k = 0.509

yI,f = –0.995

yI,g = –0.992

yN,k = 0.992

yI,f = –1.000

R = 3.0 Å

R = 4.0 Å

R = 5.0 Å




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TOC graphic


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Diradical character [-]

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1.0

0.8

0.6

yN 0.4

yT y

0.2

0.0 0.0

0.2

0.4

0.6

0.8

y (conventional) [-]


1.0


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1 2 3 4 5 6 7 (a) 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


(b) Closed-shell (Ionic)

Closed-shell (Neutral)

Diradical (Neutral)

Closed-shell (Ionic)

F

Closed-shell (Neutral)

F

F = 0.00 a.u. to 0.016 a.u.

F = 0.00 a.u. to 0.016 a.u.


Diradical (Neutral)


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

(a)

(b)


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(a-g) yN,g = 0.669

(b-k) yN,k = 0.783

(b-g) yI,g = -0.814

(b-f) yI,f = -0.969

(a-k) yI,k = -1.0

(a-f) yI,f = -0.669

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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

(a-g) yN,g = 0.583

(b-g) yI,g = -0.822

(a-k) yI,k = -1.0

(b-k) yN,k = 0.783

(a-f) yI,f = -0.583

(b-f) yI,f = -0.961


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N

B

R



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(a)

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(b)

(c) State g

State k

State f

yN,g = 0.183

yI,k = –0.341

yI,f = –0.841

yI,g = –0.514

yN,k = 0.509

yI,f = –0.995

yI,g = –0.992

yN,k = 0.992

yI,f = –1.000

R = 3.0 Å

R = 4.0 Å

R = 5.0 Å


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⎧⎪ 2P −1 ≥ 0 for P ≥ P N N I y N/I ≡ ⎨ ⎪⎩ 1− 2PI ≤ 0 for PN ≤ PI R = 3.0 Å

R = 4.0 Å B

N R

yN/I = 0.183

yN/I = –0.514

Diradical Character

Ionic Character


Diradical and Ionic Characters of Open-Shell Singlet Molecular

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