DOI: 10.1002/chem.201404978

Full Paper

& Bond Theory

Two States Are Not Enough: Quantitative Evaluation of the Valence-Bond Intramolecular Charge-Transfer Model and Its Use in Predicting Bond Length Alternation Effects Peter D. Jarowski*[a] and Yirong Mo[b]

Abstract: The structural weights of the canonical resonance contributors used in the Two-state valence-bond chargetransfer model, neutral (N, R1) and ionic (VB-CT, R2), to the ground states and excited states of a series of linear dipolar intramolecular charge-transfer chromophores containing a buta-1,3-dien-1,4-diyl bridge have been computed by using the block-localized wavefunction (BLW) method at the B3LYP/6-311 + G(d) level to provide the first quantitative assessment of this simple model. Ground- and excited-state analysis reveals surprisingly low ground-state structural weights for the VB-CT resonance form using either this Twostate model or an expanded Ten-state model. The VB-CT

state is found to be more prominent in the excited state. Individual resonance forms were structurally optimized to understand the origins of the bond length alternation (BLA) of the bridging unit. Using a Wheland energy-based weighting scheme, the weighted average of the optimized bond lengths with the Two-state model was unable to reproduce the BLA features with values 0.04 to 0.02  too large compared to the fully delocalized (FD) structure (BLW: ca. 0.13 to 0.07 , FD: ca. 0.09 to 0.05 ). Instead, an expanded Ten-state model fit the BLA values of the FD structure to within only 0.001  of FD.

Introduction Intramolecular charge-transfer chromophores (ICTCs) benefit from intense optical transitions in the visible light spectrum useful for non-linear optics and photovoltaics.[1] This class of compounds represents an important research theme in chromophore technologies and a challenging area in molecular design. Linear ICTCs generally contain an unsaturated bridging moiety (Bridge) that serves as an electronic couple between an electron-rich donor (D) and an electron-poor acceptor (A) moiety (Scheme 1).[2] The origin of the prominent electronic absorption feature of ICTCs is most often interpreted by a valence bond theory (VBT)[3] approach involving two VB states, a neutral (N) and an ionic (VB-CT), where, for the latter, partial charge has been transferred from the donor to the acceptor moiety. This Two-state model, as it is called, by application of resonance theory, mixes the two states into a resonance hybrid leading to a lowered energy with the relative contribution of each state depending on mutual overlap and its individual energy. The N state is closely associated with the ground-state (GS) electronic structure of the molecule with [a] Dr. P. D. Jarowski Department of Physics, Advanced Technology Institute University of Surrey, Guildford, GU2 7XH (United Kingdom) E-mail: [email protected] [b] Prof. Y. Mo Department of Chemistry, University of Western Michigan Kalamazoo, MI 49008-3842 (USA) Supporting information for this article is available on the WWW under http://dx.doi.org/10.1002/chem.201404978. Chem. Eur. J. 2014, 20, 17214 – 17221

Scheme 1. The N and VB-CT resonance forms in the two-state model for intramolecular charge-transfer in a linear ICTC to give a dipolar resonance hybrid. D = donor and A = acceptor.

a lesser contribution of dipolar character from the VB-CT state. For the excited state (ES), the contribution of VB-CT is expected to increase leading to the charge-transfer aspect of the electronic absorption and a strong change in molecular dipole associated with an intense process with a large extinction coefficient. This Two-state model was originally offered by Weiss[4a] for intermolecular charge-transfer complexes, developed through semi-classical quantum mechanical treatment by Mulliken[4b] and then the concepts were applied to the intramolecular case by Brooker for organic chromophores.[4c] The Twostate concept was subsequently developed quantitatively in the work of Goddard[5a] and Blanchard-Desce in the understanding of molecular hyperpolarizabilities and non-linear optics.[5b] Beyond theory, experimentalist have interpreted experimentally measurable ground-state features using the Twostate model. The most noteworthy example of this is the analysis of the bond length alternation (BLA) effect,[7] a key experimental indicator for non-linear optical response. The BLA character of an unsaturated chain often correlates very well to mo-

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Full Paper lecular first hyperpolarizabilities[1] and has been instrumental in the molecular design of new active chromophores with the Two-state model at its centre through the analysis of single crystal X-ray diffraction data as detailed by Marder and coworkers.[7] According to the Two-state model, because the two resonance forms exchange double for single CC bonds (considering, for example, a polyen-di-yl bridge), their averaging would result in a smaller bond length alternation. However, a significant contribution of the VB-CT state in the ground state is necessitated to explain the observed bond distortions (vide infra). Closer inspection of the Two-state model reveals that this significant contribution is unlikely to be present, in general, within the ICTC structural theme. The most damning point is that this simple model defies the basic chemical concept in valence bond theory and resonance originally codified by Wheland[8] and taught in all introductory chemistry textbooks; opposite charges should remain adjacent, where possible, in writing reasonable resonance structures to maintain electrostatic attraction and achieve a lower energy. The VB-CT resonance form is in stark defiance of this longstanding chemical intuition, but due to its apparent logic and correctness with regards to the ES, continues to be invoked as the primary model for ICTC behaviour interpreted by experimentalists who primarily investigate GS properties. Although the model was met with scepticism early on[6a] with noted deficiencies implicated in pre-existent work,[6b] a lack of proof from direct calculation of the relevant valence-bond structures in both the ground and excited state remains despite its widespread use. Recently,[9] issues with the VB-CT model were revisited in this journal, and the authors appealed to the readership for further quantitative work to be performed. Work by Mo and co-workers,[10] using the block-localized wavefunction (BLW) method,[11] in the context of modern valence bond theory,[12] has begun to address the validity of the VB-CT state for small test molecules, but the weights have not been tested against experimental metrics, such as the BLA effect, which is needed to properly establish the utility of the model to experimentalists. In the present paper we have applied the valence-bond BLW method to quantitatively measure the N and VB-CT states in model linear dipolar molecules of varying dipolarity. The series of linear ICTCs investigated is presented in Figure 1 where the donor is chosen as amino (NH2), the bridge is the linear diendi-yl unit and the acceptor (CH=X) varies (X=CH2 (1), NH (2), O (3) and C(CN)2 (4)) with increasing dipolar character in the order presented (see the Supporting Information for optimized geometries and energies). The aim here is to systematically assess, for the first time, the structural weights of the N and VB-CT resonance structures and to test the validity of the Twostate model in the ground and excited state, because both states are crucial to understanding and designing ICTCs. Part of this analysis demanded that the valence-bond space be expanded to include other resonance structures. Thus, in addition to the canonical neutral (N) and ionic (VB-CT) forms (R1 and R2, respectively) of 1–4, other resonance structures (R3– R10) that have a strong precedence in VBT, especially in the interpretation of resonance, are also included.[12a] R2–R10 all inChem. Eur. J. 2014, 20, 17214 – 17221

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Figure 1. Major canonical resonance forms R1–R10 of the intramolecular charge-transfer process in a linear ICTC. Acceptor X = CH2 (1), NH (2), O (3) and CHC(CN)2 (4).

volve a change in bonding arrangement that breaks one unsaturated bond and are expected to be higher in energy than R1 but lower than R2 and therefore secondary contributors to the total electronic structure.

Computational Methods In order to get quantitative information about the relevance of the N and VB-CT states and other possible resonance forms, the BLW method has been applied at the B3LYP/6-311 + G(d) level as implemented within GAMESS(US).[13] Further information can be found in the Experimental Section. BLW is a variant of ab initio valence bond theory that allows for the self-consistent optimization of the wavefunction of individual resonance contributors to a resonance hybrid through artificial limitation of orbital expansions to define non-penetrating (but interacting!) two-centre bonds or lone pairs,” for example. Thus, BLW is a molecular orbital valence bond hybrid approach and offers the computational efficiency of molecular orbital theory with the more ready interpretability of VBT simultaneously. BLW provides energies and gradients (optimized structures) from the block-localized wavefunction and has more recently been extended[11b] to density functional theory (DFT). The most crucial result in VBT is the structural weight of a given resonance structure.[14] The weight indicates the importance of a given structure to the electronic character of the molecule relative to other structures under consideration. The sum of the weights is normalized to unity and therefore the percent character of a given structure is the weight times one hundred. There are numerous schemes for calculating structural weights in VBT. Wheland set out rules for determining the importance of a resonance structure to the total hybrid electronic structure, which he based entirely on energy arguments. Using the BLW method to quantify individual resonance structure energies we apply the following weighting equation where the resultant weight (Wi) is the inverse of the energy gap for a given resonance structure normalized against the sum of these gaps for all resonance structures [Eq. (1)]: ðei  e0 Þ1 Wi ¼ P n ðej  e0 Þ1

ð1Þ

j¼1

in which Wi is the Wheland weight of the ith resonance form, ei is its calculated vertical energy of the VB wavefunction on the fully

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Full Paper delocalized (FD) geometry from molecular orbital theory, e0 is the FD energy on FD geometry, ej is the index of the vertical energies of all resonance forms and n is the number of states considered. This weighting scheme implicitly assumes that there is zero overlap among the different resonance structures and that the relevant information needed to weight the structures is incorporated sufficiently into the diagonal H matrix elements. In other words there is no interaction between the resonance structures. We have adopted this approach, in the first instance, since it is the most intuitive and most familiar to the general chemistry audience. An added benefit to this scheme is that it avoids unphysical negative structural weighting coefficients that arise when considering coefficients from multi-state analysis. To include overlap effects, multistate configuration interaction (CI), which has more recently become available[10a, 15] for BLW, has also been applied. Using the multi-state BLW analysis the CI total energy is computed (eCI0) and replaces the FD energy in [Eq. (1)] (e0) to produce [Eq. (2)], below: ðeCIi  eCI0 Þ1 Wi ðClÞ ¼ P n   eCIj  eCI0 1

The methodology applied in this work on relevant ICTCs has been arrived at through an analysis of formamide, a small dipolar molecule (homologous to 3) where conclusion about the proper choice of valence space and weighting scheme with respect to reproducing, with valence bond theory, key molecular properties derived readily from LCAO molecular orbital theory (MOT) can be made. The idea is that a proper space and weighting scheme from VBT, through weighted averaging of the VB structures and properties, should be able to return the same molecular properties as MOT. Thus, three valence spaces were considered including the Twostate space where the N and VB-CT structures are considered alone, a Three-state space with the inclusion of the remaining ionic structure with a 1,2-dipole oriented in the direction of the molecular dipole and a Five-state space where the 1,2-dipole has been reversed and the remaining ionic structure that satisfies the octet rule is added to more properly treat the ionic character of the bonds (Figure 2). The Five-state space can be considered a full ionic valence bond space for formamide. Using these three spaces,

ð2Þ

j¼1

in which the structural coefficients (eCI) now include overlap effects. Although these effects may be considerable, it should be noted that this method is only as good as the valence space under consideration and, at the limit of a full valence space, should be improved compared to [Eq. (1)]. However, where the valence space is lacking key structures the CI total energy will rise leading to an over weighting of low energy resonance structures at the cost of higher energy ones. The FD energy used in [Eq. (1)] can be considered as comprising a complete set of resonance structures implicitly treated in the molecular orbital approach and offers a more consistent standard energy. The cost/benefit of this outcome compared to the lack of overlap considerations will be assessed below. Moving away from energies, the BLW CI method produces valencebond structural coefficients in the ground state (GS) and first excited state (ES) that can be processed through numerous weighting schemes as well. However, negative structural coefficients (like the negative populations in the Mulliken population scheme) often arise and it is not clear what the physical significance of these is, if any. One tactic would be to simply ignore the sign of the coefficients (Ci ) by taking their modulus and normalizing against the sum of these moduli as is done in [Eq. (3)]: jCi j Mi ¼ P n   Cj 

ð3Þ

j¼1

The structural weight Mi is not dissimilar to the popular Hiberty scheme[3a] in which the square of a given coefficient (Ci ) is normalized with respect to the sum of the square of all coefficients in [Eq. (4)]: Ci2 Hi ¼ P n Cj2

ð4Þ

j¼1

This weighting (Hi) scheme is also used to avoid unphysical negative weights, a general problem in VBT.[3b, c] However, it should be noted that squaring of the coefficients in this way will tend to bias the result towards the more prominent resonance structures and perhaps obscure the more subtle effects. Chem. Eur. J. 2014, 20, 17214 – 17221

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Figure 2. Resonance forms for formamide comprising the Two-state, Threestate and Five-state valence space.

the four weighting schemes presented above were tested and the results are given in Table 1. Diradicaloid structures are not permissible within the BLW code and could not be tested. Starting with the ground-state weights there is good agreement between Wi and Mi, the former based on the energy and latter on the coefficients. As expected, the Wi(CI) weights are higher (Two-state) (R1: 0.822, R2: 0.178) than Wi (R1: 0.555, R2: 0.445) for the more significant structures and lower for the less significant ones due to the higher total energies compared to the FD values that fall closer to the lower energy structures. Likewise, the significant Hi weights are also higher, and less significant lower, than the Mi values since the Hiberty scheme relies on squaring the coefficients. We note that the results from Equations (2) and (4) for identical for a Two-state model and nearly identical for the Three- and Five-state models. As these weighting schemes treat larger valence spaces (three to five states) the weights tend to move closer to each other. The molecular dipole (m), Mulliken atomic charges (on N, C, O) and NC and CO bond lengths (L) were all obtained through weighted averaging using the derived weights of each resonance form and the standard deviation from MOT analysed. The lengths were taken from the optimized (diabatic) resonance structures using the BLW wavefunction, otherwise the properties are computed on the fully delocalized MOT structure. It was found, considering all weighting schemes together, that the Three-state space gave a good compromise achieving the best agreement in structure, comparable agreement with the Two-state space in reproducing the atomic charges and a close agreement in the molecular dipole compared to the Five-state space that performed best for that property. Considering the weighting schemes, the simple Wi weight was able to produce good agreement with atomic charges and dipole moment and performed comparably to Mi in returning good quality structural agreement in bond lengths. Thus, an ionic valence space considering only ionic structures with dipoles parallel to the molecular dipole and an energy-based scheme using the FD energy as a stan-

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Full Paper Table 1. Two-, Three- and Five-state analysis of formamide (5) considering weighting schemes from [Eqs. (1)–(4)]. Energies can be found in Supporting Information. Structural coefficients are provided (Ci) and the molecular orbital theory derived properties of formamide are provided in column 1 for the molecular dipole (m), Mulliken atomic charges (NCO) and bond lengths (L) for the NC and CO bonds. Resonance form/ Energy-based Multi-state property structural weights coefficients Wi(CI) Ci Wi GS GS GS ES Two-state R1 0.555 R2 0.445 Weighted properties m (4.100 Debye) 3.985 N (0.541) 0.484 C (0.037) 0.180 O (0.382) 0.476 1.385 LNC (1.361 ) 1.250 LCO (1.212 )

0.822 0.178

0.710 1.137 0.330 1.305

3.264 0.578 0.177 0.385 1.428 1.212

Three-state (dipole-oriented ionic) R1 0.417 0.624 R2 0.335 0.253 R3 0.248 0.123 Weighted properties m 4.151 3.697 N 0.517 0.548 C 0.249 0.213 O 0.527 0.454 1.390 1.409 LNC 1.257 1.234 LCO Five state (full ionic valence-space) R1 0.337 0.545 R2 0.271 0.247 R3 0.200 0.125 R4 0.096 0.041 R5 0.096 0.041 Weighted properties m 4.245 3.838 N 0.413 0.539 C 0.041 0.073 O 0.400 0.452 1.401 1.410 LNC 1.292 1.254 LCO

Coefficients-based structural weights Mi GS ES GS 0.683 0.466 0.317 0.534 4.124 0.529 0.179 0.432 1.405 1.232

0.802 1.041 0.506 1.436 0.371 0.287

Hi ES

0.822 0.431 0.178 0.569 3.264 0.578 0.177 0.385 1.428 1.212

0.478 0.376 0.301 0.520 0.221 0.104 4.011 0.529 0.241 0.506 1.397 1.250

0.620 0.335 0.247 0.639 0.133 0.026 3.699 0.550 0.216 0.456 1.410 1.234

mal number of resonance forms making their thorough study tractable. R2–R10 all adhere to the expected dipole orientation of the molecule but are not meant to form a comprehensive set as many alternative diradicaloid and ionic structures are absent. For the former, these can be assumed to be sufficiently high energy for exclusion. From the above analysis of formamide, the presentation for the weight of 1–4 has been simplified to include only weights from Wi and Mi and only a Twostate and Ten-state model are considered. After analysing the vertical structural weights in the ground and excited state, our methodology is verified by focusing on the BLA effects of the buta-1,3-dien-1,4diyl bridges with structural optimized (relaxed geometries) individual resonance forms. The changes in bond length parameters in the relaxed structures represent the “stress” contained within the fully delocalized (FD) structure from each contributing resonance form leading to the observed bond lengths (C1C2, C2C3, C3C4 and C4C5) of the linear carbon chain (see the Supporting Information for optimized bond lengths). These stresses are subjected to weighted averaging using the Wi weights and the resultant averaged structure compared to the FD structure as was done for formamide. The BLA is defined as the average difference in length between single and double bonds, [Eq. (5)]

BLA ¼

ðC1C2 þ C3C4Þ  ðC2C3 þ C4C5Þ 2

ð5Þ

in which a value of 0  signifies no alternation and a value of roughly 0.12  (for polyenes, as example) signifies full alternation tending to the standard values of isolated double and single bonds (see Figure 1 for atom numbering). The BLAs of the FD structures (used as standards here) for these compounds decrease with increasing dipolarity from (in ) 1 (0.088), 2 (0.088), 3 4.088 3.863 0.506 0.544 (0.083) and 4 (0.053). Assuming, for a hypothetical 0.188 0.231 molecule, that the VB-CT state exactly exchanges the 0.472 0.480 bond length values with N along the bridge, a 50 % con1.400 1.405 tribution of the VB-CT state would be required to ach1.265 1.244 ieve a BLA of 0 . Experimentally measured values are more typically in the range of 0.04 to 0.08  (close to the FD calculated values above) and would still require 40 % to 30 % VB-CT contribution to the ground-state, redard will provide meaningful ground-state structural weights. For spectively. These contributions would represent nearly degenerate the excited states, no energies are available for the individual resoresonance forms as the weighting follows the reciprocal of the nance structures and thus the Mi weights will be used for their energy gap between the FD energy and the resonance structure analysis. energy. Thus, for the Two-state model, which treats asymmetric molecules with divergent electronic character from one end to the The ICTCs chosen for this work provide a number of features that other, a problem may lay in the VB-CT state itself, which is prima are important for initial conclusions to be drawn and are chosen facie a poor contributor in the ground state according to the stanfor their closer relation to experimental target systems with longer dard Wheland rule.[8] The non-adjacency of opposite charges genmolecular lengths and associated visible absorptions and colour. erally results in high-energy resonance structures, a factor that is First, they are the smallest molecules possible where the important compounded especially as the length of the bridge extends in experimental metric of BLA can be analysed (at least two conjugatICTCs. ed C=C double bonds). Due to the short conjugation length between the donor and acceptor, it is anticipated that the VB-CT state will have a significant structural weight compared to other possible longer systems and thus the present systems are robust Results and Discussion test cases. Second, polyene bridges are well-studied in this reMultistate structural weights are presented in Table 2 derived search theme and are where much of the seminal non-linear optifrom the modulus scheme [Eq. (3), Mi]. This is done for both cal molecular design research began and where the Two-state the ground and excited states of 1–4. The GS weights of the model has been heavily applied.[16] Third, the systems have a miniChem. Eur. J. 2014, 20, 17214 – 17221

1.005 1.097 0.693 1.496 0.635 0.290 0.100 0.054 0.057 0.056

0.404 0.278 0.255 0.040 0.023

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0.367 0.500 0.097 0.018 0.019

0.530 0.252 0.211 0.005 0.002

0.341 0.634 0.018 0.001 0.001

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Full Paper Table 2. Modulus structural weights (Mi) for 1–4 in resonance forms R1– R10 using a Two-state (R1–R2) and a Ten-state (R1–R10) model in the ground and first excited singlet state. The weights are derived using [Eq. (3)] from multi-state BLW CI. Resonance form

1

Modulus structural weights [Mi] 2 3 GS ES GS ES GS

4

GS

ES

Two-state R1 R2

ES

0.946 0.054

0.135 0.865

0.929 0.071

0.165 0.835

0.908 0.092

0.231 0.769

0.892 0.108

0.231 0.769

Ten-state R1 R2 R3 R4 R5 R6 R7 R8 R9 R10

0.483 0.009 0.006 0.168 0.006 0.030 0.056 0.030 0.156 0.056

0.312 0.099 0.133 0.259 0.015 0.018 0.050 0.041 0.011 0.062

0.472 0.013 0.010 0.169 0.008 0.036 0.052 0.030 0.153 0.057

0.320 0.107 0.166 0.210 0.016 0.015 0.026 0.044 0.022 0.075

0.428 0.018 0.004 0.149 0.011 0.045 0.058 0.036 0.169 0.082

0.287 0.166 0.164 0.151 0.039 0.020 0.016 0.060 0.025 0.073

0.421 0.023 0.006 0.152 0.013 0.046 0.056 0.037 0.169 0.078

0.270 0.243 0.128 0.086 0.056 0.033 0.045 0.046 0.034 0.059

VB-CT resonance form (R2) in all cases using the Two-state model are strikingly low and do not exceed approximately 0.11 even in the case of the most dipolar example (4) the rest of the weight belonging to R1 (N). There is a marginal increase from approximately 0.05 to 0.09 in the VB-CT weight in proceeding along the series 1 to 3 with increasing dipolarity (see the Experimental Section for molecular dipoles), but from these data it must be concluded that the R2 resonance form has an insignificant contribution to the GS electronic structure in these model dipolar molecules based on the valence-bond space including only R1 and R2 and that these molecules can be well-described from the single resonance form, N, with weights decreasing correspondingly from approximately 0.95 to 0.89 from 1–4. Upon increasing the valence space to include the ten resonance forms of Figure 1, the contribution of R2 is decreased to less than approximately 0.02 for all derivatives. However, other resonance forms now show more significant contributions with some of the weight of R1 shared with R4 and R9, both of which maintain a close proximity of the charged positions (1,3dipole and 1,2-dipole, respectively). The weights of these structures are not strongly correlated to the acceptor strength as the ionic character sits closer to the donor side of the molecules and the effect of the acceptor is not felt. R4 and R9 maintain structural weights between approximately 0.15 and 0.17 throughout. The rest of the resonance forms, with longerrange dipolar arrangements, do not exceed 0.06 in structural weight. R10 reaches approximately 0.08 for 3 and 4. The weightings follow the Wheland rule closely, where, in general, opposite charges are as close together as possible. There is a dramatic change in the modulus weights for the excited states of 1–4. Considering just two states (R1 and R2) there is an almost complete exchange of structural weights in proceeding from the GS to the ES with R2 taking the majority of the weight and only a residual weight for R1 of approxiChem. Eur. J. 2014, 20, 17214 – 17221

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mately 0.14 to 0.23 in order of increasing dipolarity. Using ten states the ES weights of R2 for methylene (1) and imine acceptor (2) maintain relatively small weights (< 0.11) but there is a strong increase for the formyl and dicyanovinyl derivatives bringing R2 to a comparable weight to R1. For these R2 is weighted as 0.17 (3) and 0.24 (4) compared to 0.29 and 0.27 for R1, respectively. It is noteworthy that R1 maintains its position as major resonance form throughout. In addition, R4 shows an important contribution, which decreases with increasing dipolarity from as much as 0.26 (1) to 0.09 (4). This result indicates that the ES is localized on the donor side of the molecule in the case of weak acceptor moieties and is drawn out to the acceptor side as the acceptor strength is increased and the localized R4 replaced in importance by the VB-CT (R2) resonance form. In summary, the VB-CT (R2) state is of little electronic importance in the GS of the dipolar molecules studied here when considering a Two-state or Ten-state model. Resonance structures that adhere to the Wheland rule are favoured, but the electronic structure is dominated by the N (R1) resonance structure irrespective of the dipolarity of the molecule. The resonance structure weights according to [Eq. (3)] adhere to the expected trends and offer insight into the importance of R2 and related ionic structure; thus, R2 is not of special importance to the GS of these dipolar molecules having a near zero weight from the multi-state configuration interaction analysis or a weight less than or equal to other resonance forms not generally considered. In the ES the weight of R2 is strongly increased and with sufficient acceptor strength can be comparable in the weight to R1. The ES is dominated by R1 and R4 for weak acceptors (1 and 2) and by R1 and R2 for strong acceptor moieties (3 and 4). Although these results are in line with trend expectations based on molecular dipoles, the values of the weights of R2–R10 are substantially low in the GS and do not agree with calculated bond length alternation effects for these systems. There is no single competing resonance form that could account for a decrease in single and an increase in double bond lengths along the carbon chain (see below). These effects most likely arise from R1 being over weighted using a coefficient scheme that relies on a limited number of resonance forms in the valence space. Using the Wheland weights [Wi, Eq. (1), Table 3] the Twostate model also expresses an increasing weighting of R2 proceeding from 1 to 4 and a corresponding diminishing contribution from R1 from approximately 0.79 to 0.67 in Table 2. R2 ranges from approximately 0.21 in 1 to 0.33 in 4. These are substantially higher in weight compared to the Mi weights above. In 3, the structural weight of R2 is approximately 0.24, which is comparable to the analogous resonance form of formamide (above) signifying a decrease in structural weight with increased length and separation of charge (R2: 1,3-dipole in formamide vs. 1,7-dipole in 3). The 0.33 contributions in 4 demonstrate again the effect of increasing acceptor strength as these values sit close to formamide despite the 1,7-dipolar arrangements. In 2, the weight is only 0.22 as the electronegativity has decreased relative to 3 (N vs. O). This, again, is related to the Wheland rules where the negative charge is progres-

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Full Paper Table 3. Wheland structural weights for 1–4 in resonance forms R1–R10 using a Two-state (R1–R2) and a Ten-state (R1–R10) model with weighting from [Eq. (1)]. Resonance form

1

Wheland structural weights [Wi] 2 3

Two-state R1 R2

0.794 0.206

0.780 0.220

0.756 0.244

0.672 0.328

Ten-state R1 R2 R3 R4 R5 R6 R7 R8 R9 R10

0.271 0.070 0.084 0.119 0.057 0.065 0.089 0.067 0.092 0.087

0.259 0.073 0.087 0.119 0.058 0.066 0.090 0.068 0.091 0.088

0.243 0.079 0.089 0.117 0.062 0.070 0.095 0.068 0.089 0.088

0.191 0.093 0.094 0.116 0.073 0.079 0.096 0.075 0.092 0.091

4

Figure 3. Bond length differences (DL in []) between resonance forms R1– R10 (grey circles) for 4 and the FD structure (abscissa) sized according to Wi. The weighted average (horizontal bar) is also shown.

sively placed on a more electronegative position thus favouring the VB-CT resonance form. When R2 is included in a more comprehensive basis of resonance forms (R1–R10) its structural weighting is significantly reduced, as is that of R1. The same effect was seen for the modulus weights. R2 ranges now from 0.07 to 0.09 from 1 to 4 using the ten-state model, whereas R1 ranges from 0.27 to 0.19 in the same order. Again, R2 is not the most energetically accessible resonance form. R4 has a consistent structural weighting of approximately 0.12 where the 1,3-dipolar arrangement is kept for 1 to 4, 0.05 to 0.03 higher than R2. R3 and R5–R10 are also roughly equivalent in weighting to R2. R9 is also seen as one of the more important resonance structures in accord with the modulus weights as well. Thus results from [Eq. (3)] and [Eq. (1)] are generally in qualitative agreement. The main difference is in the greater distribution of weights towards R3–R10 and away from R1 for [Eq. (1)] compared with [Eq. (3)] resulting from the use of the FD energy. Using the BLW gradients the self-consistent optimization of the individual resonance structures R1–R10 was performed for 1–4. To restate, the bond length parameters using the block-localized wavefunctions are expected to relate to the influence of a given resonance structure on the FD optimized structure and therefore the weighted average of the bond length parameters in R1–R10, according to Wi (with the best structure performance for formamide), should return the FD values. As an example, the optimized geometries for 4 are given in Figure 3 where the difference between the diabatic (relaxed) bond length parameters in R1–R10 compared to the FD structure (DL) is shown for each bond in the buta-1,3-dien-1,4-diyl bridge. The abscissa is the FD parameter and the data points are sized according to their adiabatic structural weights (Tenstate model, Wi) with the weighted average of each bond length given as a horizontal bar. The structural bond length parameters agree well with the resonance structures pictured in Figure 1 reflecting contracted and extended bond lengths where bond orders have changed and oscillate about the abChem. Eur. J. 2014, 20, 17214 – 17221

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scissa accordingly. R2 swaps all of the bond orders compared to R1 and both result in the largest DL values. Bond C1C2 in R1 and R2 has values of 0.04 and 0.15 , respectively. The former is compressed and the latter elongated reflecting a loss of delocalization in both cases compared to the FD structure. For C2C3, DL changes from 0.10  in R1 and 0.10  in R2 for the corresponding reason. Electrostatic interactions are also prevalent. For example, the C2C3 bond length in 4 for R6 is 0.07  longer than the FD parameter, but less extended than in R1 due to electrostatic charge–bond interaction between the C1=C2 bond and the C3 + centre. Similarly, DL for C3C4 in R6 is 0.10  compared to 0.15  in R2. In this way, R3–R10 all pull the final weighted structure towards the FD bond length values. Using the relaxed resonance structures for 1–4, the bond length alternation (BLA) has been calculated from their weighted averages. The predicted BLA values in Table 4 are taken

Table 4. Bond length alternation parameters [] for 1–4 obtained through structural averaging of the optimized geometries of R1–R2 (Twostate) and R1–R10 (Ten-state) based on the weights in Table 3.

1 2 3 4

FD

BLA [] Two-state

Ten-state

SD []

0.088 0.088 0.083 0.053

0.125 0.122 0.115 0.074

0.087 0.084 0.080 0.058

0.038 0.038 0.037 0.042

from the weighted averages of the bond length parameters over R1–R2 (Two-state) or R1–R10 (Ten-state) weighted according to the vertical structural weights of Table 3. Using just two states the BLA value is strongly overestimated, as the contribution of R2 is too low to provide the needed structural adjustments, even though the bonds are significantly different in the optimized structure R2 compared to R1. The BLA is between 0.04 and 0.02  too large compared to the FD values. Structures 1 to 3, in fact, are predicted to have roughly the maximum value of BLA. On the other hand, inclusion of ten states in the averaging leads to excellent agreement in BLA in all

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Full Paper cases to within a 0.001 (1) to 0.005  (4) range of differences. The larger discrepancy for 4 may be related to the exclusion of resonance structures involving the pi system of the nitrile groups in our present analysis, which we have done for consistency among the compounds studied. It should be noted also that there is consistent overestimation of bond length (SD value is the standard deviation in bond length, Table 4), which most likely arises from a too strong ionicity in the bonds since the oppositely oriented dipoles are not considered in the basis of resonance structures included here (not a full ionic valence space). Various combinations of resonance structures were tested, but only a basis of the ten resonance structures gave an acceptable agreement compared to the FD structure, indicating that the proper description of the BLA effect in ICTCs requires consideration of a number of resonance structures that have roughly equivalent structural weighting to that of R2. The Two-state model is indeed too simple, as originally stated.[6a] It can now be understood that the increase in BLA along 1 to 4, and generally for dipolar molecules, is not related to an increase in the importance of the VB-CT state, but rather to an increasing distribution of the structural weights to a number of other ionic forms. Moreover, it is shown that a simple energy-weighting scheme gives absolute weights that are appropriate for comparison to the experimental determinable of bond length and to the bond length alternation. The excited state has a strong contribution from both resonance structures based on the Mi structural weights presented earlier. Optimized BLA effects (in ) for the first singlet ES from TD DFT (same level as above) analysis for 1 (0.004), 2 (0.005), 3 (0.024) and 4 (0.000) are considerably changed and in two cases the bond lengths switched in relative length (change in sign). However, it is not the ES that is being indirectly measured by the proxy measure of the BLA effects. Unfortunately, at this time the methodology to weight the excited state energies of the resonance forms does not exist (a multi-reference block-localized wavefunction configuration interaction) and this analysis will have to wait for further developments in the extension of the application of BLW. In general, there has been renewed interest in the quantitative energetic evaluation of theoretical constructs in organic chemistry[17, 18] that relate specific structural situations to molecular properties. Quantities such as conjugation and hyperconjugation,[17] aromaticity,[18a] strain,[18a] and 1,3-alkyl-alkyl interactions (now termed protobranching)[18a–c] are being re-evaluated in the context of highly accurate computational methods that have allowed access to studies formerly forbidden by experimental difficulties. A revival in valence bond theory,[19] especially through the use of the BLW method, has supported much of this work by allowing a systematic way of separating, electronically, parts of a molecule and a direct evaluation of internal interaction energies within a molecule. The present manuscript is an extension of this theme. The predictive utility of the Two-state model has been demonstrated over many years, especially in the analysis of first and second hyperpolarizabilities in the context of non-linear optics. Mulliken’s model and its application in Goddard’s treatment rely upon VB-CT as an approximate representation of the first excited state’s elecChem. Eur. J. 2014, 20, 17214 – 17221

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tronic structure to predict changes in non-linear optical response. This is entirely appropriate, as the optical transitions and corresponding hyperpolarizabilities are related directly to the excited-state wavefunction. The valence-bond landscape of the ES has been quantified here and indeed verifies the importance of the VB-CT state but also indicates important contributions from other states that should not be overlooked from a design perspective. For the GS, the VB-CT state is problematic since, as a contributing structure, it does not adhere to the important Wheland selection rules for writing resonance forms that represent a core part of the education of all chemists. The VB-CT form is a high-energy resonance form for the GS of linear dipolar molecules and plays a minor role in the groundstate wavefunction. Thus, the Two-state model cannot adequately rationalize bond length alternation effects alone through mixing of the N and VB-CT form. Instead, a more complex model is necessary. Thus, the focus in ICTCs on the termini of the linear molecule has been adversely set, where a more complete focus would consider ionic structures along the chain as predicted early on by Pauling.[6b] With this knowledge, experimentalists can better design ICTCs with larger BLA effects in the GS and better control over ES electronic structure to achieve a more refined control of the hyperpolarizabilities towards improved molecular components in optical devices by modifying the bridge[20] to accommodate specific ionic structures. The VBT BLW approach is efficient and versatile enough to address larger molecules than previously addressable and offers a means to apply this methodology for other interested research groups on more experimentally relevant molecular systems beyond the model systems treated in this work.

Experimental Section Computational work Compounds 1 to 4 were optimized at the B3LYP/6–311 + G(d) level using the standard molecular orbital approach to give fully delocalized (FD) wavefunctions and optimized GS structures (see the Supporting Information for Cartesian coordinates and SCF energies). TDDFT was used for the optimization of the first excited state using the root = 1 option. The fully delocalized molecular orbital theory results are used as standards where we have assumed that this level gives accurate data as can be reasonably expected from this well-tested functional and basis set. The structures have been optimized in the Ci point group to maintain planarity, which is useful for the description of the block-localized wavefunction, but we note that small imaginary frequencies associated with the pyramidalization of the amine nitrogen remain as this level, although the frequencies have little effect on the calculated energies. The calculated dipole moments (in Debye, long molecular axis) are 1 (3.97), 2 (7.42), 3 (8.66) and 4 (13.14) increasing in the order listed as expected from the nature of the accepting moieties.

Acknowledgements P.D.J. would like to thank the EPSRC National Service for Computational Chemistry Software (NSCCS) at Imperial College London for use of their computing facility and also Dr. Jiabo Li

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