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A Reparameterization of PM6 Applied to Organic Diradical Molecules Toru Saito, Yasutaka Kitagawa, and Yu Takano J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b08530 • Publication Date (Web): 11 Oct 2016 Downloaded from http://pubs.acs.org on October 16, 2016

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The Journal of Physical Chemistry

A Reparameterization of PM6 Applied to Organic Diradical Molecules Toru Saito,∗,† Yasutaka Kitagawa,‡,¶ and Yu Takano† †Department of Biomedical Information Sciences, Graduate School of Information Sciences, Hiroshima City University, 3-4-1 Ozuka-Higashi, Asa-Minami-Ku, Hiroshima 731-3194 Japan ‡Division of Chemical Engineering, Department of Materials Engineering Science, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan ¶Center for Spintronics Research Network (CSRN), Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan E-mail: [email protected]

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Abstract We have performed a reparameterization of PM6 (called rPM6) to compute openshell species, specifically organic diradical molecules, within a framework of the spin unrestricted semi-empirical molecular orbital (SE-UMO) method. The parameters for the basic elements (hydrogen, carbon, nitrogen, and oxygen) have been optimized simultaneously using the training set consisting of 740 reference data. Based on the GMTKN30 database, the mean absolute error of rPM6 is decreased from 16.1 to 14.2 kcal/mol, which reassures its accuracy for ground state properties. Applications of the spin unrestricted rPM6 (UrPM6) method to small diradicals and relatively large polycyclic aromatic hydrocarbons have provided substantial improvement over the standard SE-UMO methods like UAM1, UPM3, and the original UPM6. The UrPM6 calculation is much less susceptible to spin contamination, and therefore reproduces geometric parameters and adiabatic singlet-triplet energy gaps obtained by UDFT (UB3LYP and/or UBHandHLYP) at much lower computational cost.

1. Introduction Open-shell species play important roles as key intermediates in chemical reactions, organic and inorganic materials, and antitumor drugs due to peculiar electronic structures induced by the orbital degeneracy. 1–10 There needs to be a clear understanding of the complicated electronic structures to control their functions. On the theoretical side, considerable efforts have been made to cope with (quasi-)degenerate states. 11,12 In particular, spin unrestricted density functional theory (UDFT) calculations such as UB3LYP 13–15 have been widely applied to diradical and polyradical molecules composed of up to a few hundreds of atoms, due to the optimal balance of computational cost and accuracy. A combined quantum mechanics / molecular mechanics (QM/MM) method has increased the scope of systems and thereby is now routinely used for investigating large molecules. 16,17 The precise free energy differences for macromolecular complexes can be estimated by 2 ACS Paragon Plus Environment

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molecular dynamics (MD) simulations on the basis of QM/MM energies and forces (denoted as QM/MM MD). 16–20 For closed-shell systems, semi-empirical molecular orbital (SE-MO) methods based on the neglect of diatomic differential overlap (NDDO) approximation 21 including MNDO, 22,23 AM1, 24 PM3, 25,26 and their modified versions 27–35 can replace DFT in QM/MM MD simulations 36–38 because they have time-tested reliability and are roughly two or three orders of magnitude faster than DFT computations. For open-shell systems, however, the situation is more complicated. Most SE-MO approaches have been designed with strategies to reproduce experimental molecular properties such as heats of formation. 22–34 Although various open-shell molecules may be included in a training set of molecules, detailed verifications of these electronic and geometric structures have not seemed to be carried out. Some studies suggested that the spin unrestricted SE-MO (SE-UMO) calculations inherently suffer from significant spin contamination even for organic diradical molecules as well as spin unrestricted Hartree-Fock (UHF). 39–41 The extent of spin contamination may be measured using the expectation value of total spin angular momentum ⟨S2 ⟩. One must keep in mind that large amount of spin contamination can appear in more complicated π-conjugated organic and transition-metal-containing compounds, and the resultant inappropriate electronic structure would provide incorrect reaction mechanisms and/or energy splittings of spin states. Therefore, there is virtually no reliable SE-UMO method that can safely substitute for first-principles UDFT. In the present study, we introduce a reparameterization of PM6, mainly focusing on the parameters for the basic elements (H, C, N, and O), to improve its accuracy for organic diradical species without degrading the applicability to a wide range of chemical problems. Our training set includes geometric parameters and adiabatic singlet-triplet energy gap (∆ES-T = ES − ET ) for methylene (1) and the three isomers of benzyne (2) (see Figure 1), all of which are often used for benchmark calculations. 42–47 To show the merits of the reparameterization, we explore six complex applications involving cyclobutadiene (3), meta-benzoquinodimethane (meta-xylylene, 4), oxyallyl (5), tetrabenzo[a,f,j,o]perylene (6),



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a heptazethrenebis(dicarboximide) (7), and polyacenes (8), as shown in Figure 1. Here, we examine 6-8 on behalf of polycyclic aromatic hydrocarbons (PAHs) because some qualitative agreement between UB3LYP calculations and experimental observations have been demonstrated on these compounds. 48–56 ortho

meta

para D 2h

1

D 4h

3

2

4

D 2h

5

6

n–1

7

8

Figure 1: Calculated molecules: methylene (1), benzyne (2), cyclobutadiene (3), metabenzoquinodimethane (meta-xylylene, 4), oxyallyl (5), tetrabenzo[a,f,j,o]perylene (6), a heptazethrenebis(dicarboximide) (7), and polyacenes (8)


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2. Computational details 2.1 Diatomic core-core parameters in PM6 The nuclear-nuclear repulsion term in PM6 27 is basically given by ( ) 6 ) −αij (Rij +0.0003Rij En (i, j) = Zi Zj ⟨si si |sj sj ⟩ 1 + xij e ,

(1)

where Z, ⟨si si |sj sj ⟩, and Rij represent the charge of nucleus, two-electron two-center integral, and the distance between core i and j, respectively. The parameters xij and αij denote Voityuk’s diatomic term 57 for the atom pair i and j. For O-H and N-H interactions, Eq.(1) was replaced by Eq.(2) to improve the description of hydrogen bond interaction (e.g. O· · · H and N· · · H) energy, ( ) 2 −αij Rij En (i, j) = Zi Zj ⟨si si |sj sj ⟩ 1 + xij e .

(2)

The C-C core-core interaction is defined as Eq.(3) by adding a repulsive term to the original form Eq.(1), aiming at correcting the error found in computing the yne groups, ( ) 6 ) −αij (Rij +0.0003Rij ′ −α′ij Rij En (i, j) = Zi Zj ⟨si si |sj sj ⟩ 1 + xij e + xij e .

(3)

2.2 Parameter optimization for H, C, N, and O Our primary purpose is to improve the accuracy of PM6 in describing electronic and geometric structures of open-shell species, for which a slight modification was made. The electronic parameters consist of the one-electron one-center integrals, Uss and Upp , the resonance integrals, βs and βp , and the Slater orbital exponents, ζs and ζp , resulting in three parameters (Uss , βs , ζs ) for H atom and six parameters (Uss , Upp , βs , βp , ζs , ζp ) for C, N, O atoms (21 parameters in total). The remaining one-center two-electron integrals (Gss , Gsp , Gpp , Gp2 and Hsp ) were kept fixed at the original values in a similar way to the PDDG/PM3 approach. 31–33 5 ACS Paragon Plus Environment


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The diatomic core-core interaction term has four parameters for the C-C interaction defined in Eq.(3) and two parameters for other pairs defined in Eqs.(1) and (2), bringing the total to 22 parameters. Therefore, the number of the optimizable electronic and core-core parameters is added up to 43 (see Table 1 for more details). Here, we introduce a simple but essential training set to determine an optimal set of parameters instead of using many reference data. The training set used in the present study consist of three subsets as follows: I. Geometric parameters of 25 species consisting of 12 closed-shell and 13 open-shell molecules (595 reference data in total). II. Homolytic bond dissociation energies, reaction energies, and singlet-triplet energy gaps with 8 reference data in total. III. Part of the recently proposed GMTKN30 58,59 database and its variant GMTKN30CHNOF 35 was included by extracting cases associated with the H, C, N, O elements (called the GMTKN30-HCNO database). In the present study, we used 5 subsets in GMTKN30-HCNO database with 137 reference data which cover the atomization energies (W4-08), decomposition energies (M08-165), alkane bond separation energies (BSR36), reaction energies of selected G2/97 systems (G2RC), and binding energies of water (WATER27). The subset I includes both closed-shell and open-shell molecules, to name a few, imidazole, coronene, twisted ethylene, and phenyl nitrene, so that a modified set of parameters can keep describing structures of aliphatic and aromatic compounds well. All 25 species are presented in Table S1 in the Supporting Information. Although the number of variables is small, the subset II is composed of open-shell singlet diradicals, among which methylene and the three isomers of benzyne are particularly crucial for evaluating whether a computational method can be applicable to a wide range of diradical character (for details see Table S2 in the Supporting Information).


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A specific parameterization 60–62 is usually expected to deteriorate performance for generalpurpose applications, and thus we adopted above 5 subsets of the GMTKN30-HCNO database as the subset III, in an attempt to maintain the applicability to energetics of chemical reactions. Several subsets such as W4-08, G21IP (adiabatic ionization potentials), SIE11 (self-interaction error related problems), and RSE43 (radical stabilization energies) contain in open-shell doublets and/or triplets. We chose W4-08 from these subsets because it is the only one which involves open-shell singlets. The other 4 subsets mentioned above (M08-165, BSR36, G2RC, and WATER27) were also adopted, because the calculated mean absolute errors (MAEs) for the sets significantly vary with the standard AM1, PM3 and PM6 methods. 35 It may be advisable to change a set of parameters by regarding its behavior toward these subsets. See the related papers for details of the other subsets and the acronyms. 58,59,63 The total number of reference data, N , is 740. As with previous works, 22–27,30 given a set of parameters, P, a non-linear response function S is expressed as a sum of the square of the difference between reference data Qref (i) and calculated values Qcalc (i) with a weighting factor wi , S=

N ∑

(wi (Qcalc (i) − Qref (i)))2 .

(4)

i

Two types of reference data were prepared depending on the subset of our training set. For the subset I and II, we used the standard UB3LYP/6-31G* model to deal with both the energetic and geometric features in a balanced manner, the latter of which are difficult to be obtained using highly correlated ab initio calculations at the moment. For the subset III, all reference data were taken from the GMTKN30 database available on the web site of Grimme’s group. 64 The contribution of each subset to S was controlled to have the same order of the magnitude by empirically adjusting the weighting factors appropriate to each subset, i.e., 0.10 (bohr−1 for bond lengths rad−1 for angles) for subset I, 1.00 hartree−1 for subset II, and 0.03 hartree−1 for subset III, respectively. All UB3LYP/6-31G* calculations were performed with the Gaussian 09 program package. 65 To obtain optimal parameter sets, the funciton S was minimized by employing the Newton-Raphson technique based on the 7 ACS Paragon Plus Environment


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first and second derivatives of S with respect to P.

2.3 Test calculations Once reoptimized parameter sets for PM6 (called rPM6 hereafter) were obtained, we examined rPM6 on the GMTKN30-HCNO database consisting of 24 subsets and 429 reference data to validate its accuracy for general use by comparing with the standard SE-MO (AM1, PM3 and PM6) calculations. 35,66 When estimating proton affinities with the use of the SE-MO methods, the experimental value of 367.171 kcal/mol 67 was used for the heat of formation of H+ in a similar way to the previous work. 35 These methods were also applied to 1 and 2 (see Figure 1), both of which are included in the training set, to see how much improvement can be gained by rPM6. Then we explored six applications that includes small but challenging disjoint diradicals 3-5 and relatively large PAHs 6-8 (see Figure 1). Apart from the calculations for the GMTKN30-HCNO database, geometry optimizations for both the singlet and triplet states were carried out at the UAM1, UPM3, UPM6, UrPM6, and UB3LYP/6-31G* levels. Cartesian coordinates for 1-8 optimized by UrPM6 are presented in the Supporting Information. For 3-8, we also examined UBHandHLYP (50% HF exchange), 15 which may yield more spin polarized electronic structures. The 6-31G* basis set was used for the UDFT calculations unless otherwise noted. Then, the ∆ES-T value was estimated. To remove spin contamination from ∆ES-T , an approximate spin projection (AP) method proposed by Yamaguchi 68,69 was used. The corrected ∆ES-T is expressed as

∆ES-T

⟨S2 ⟩S ⟨S2 ⟩T′ ES − 2 ET′ − ET , = 2 ⟨S ⟩T′ − ⟨S2 ⟩S ⟨S ⟩T′ − ⟨S2 ⟩S

(5)

where ES , ET , and ET′ denotes the energy of singlet (S), triplet (T) and triplet state at the singlet geometry(T′ ), respectively, and ⟨S2 ⟩S , ⟨S2 ⟩T , and ⟨S2 ⟩T′ are the corresponding expectation value of total spin angular momentum. In the AP method, the reference T′ state


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is assumed to be free from spin contamination.

3. Results and discussions 3.1 Reoptimized parameters The set of parameters for rPM6 are listed in Table 1 together with those for the original PM6 for comparison. Table 2 summarizes MAEs for each subset and the total set, relative to the reference values from the GMTKN30-HCNO database. Previous studies indicated that the original PM6 method gives rise to large deviations for the MB08-165 subset, with the MAE of more than 100 kcal/mol. 35,63 It is not surprising that the present rPM6 calculations substantially decrease the MAE to 40.0 kcal/mol because the MB08-165 subset was included in the training set. Concerning the remaining 4 subsets used in our training set, the reparameterization works well for the sets of G2RC and WATER27. It lowers the MAE for the G2RC (WATER27) subset from 30.9 (17.8) to 21.2 (7.5) kcal/mol and in particular rPM6 is the most accurate for the WATER27 subset. On the other hand, rPM6 deteriorates both the W4-08 and BSR36 subsets with the increase of MAEs by 10.6 and 15.0 kcal/mol, respectively. For the W4-08 subset, unlike the present rPM6 calculations, the MAEs of AM1, PM3, and PM6 were obtained with the removal of the zero-point vibrational energies (ZPVEs) and thermal correction. 35 We performed additional AM1, PM3, and PM6 calculations without such corrections in the same way as rPM6 using the Gaussian 09 program package, to see whether the present reparameterization causes a large deviation. The resultant MAE of PM6 (17.7 kcal/mol), as listed in Table 2, is comparable to that of rPM6 (18.0 kcal/mol). It is also found that the ZPVEs and thermal corrections reduce the error by ca. 50 % as reported previously, 35 suggesting that the inferior performance of rPM6 for the W4-08 subset is due not to the reparameterization but rather to lack of ZPVEs and thermal corrections. The AM1, PM3, and rPM6 are less accurate than PM6 for the BSR36 subset, whereas they 9 ACS Paragon Plus Environment


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Table 1: Optimized parametersa for rPM6 and those for the original PM6b rPM6 3.498705 2.274570 1.047843 0.210771 2.592208 0.834763 5.934270 9.286577 1.042810 0.179237 2.721866 0.886887 2.627044 0.651818 1.407931 0.200384 2.863882 1.020378 2.739991 0.780069 2.629206 0.547932

αH−H xH−H αC−H xC−H αC−C xC−C ′ αC−C ′ xC−C αN −H xN −H αN −C xN −C αN −N xN −N αO−H xO−H αO−C xO−C αO−N xO−N αO−O xO−O Uss Upp βs βp ζs ζp

H -10.575804

C -51.667769 -39.491401 -7.989550 -14.054866 -9.329734 1.266120 2.065862 1.673130 a

PM6 3.540942 2.243587 1.027806 0.216506 2.613713 0.813510 5.980000 9.280000 0.969406 0.175506 2.686108 0.859949 2.574502 0.675313 1.260942 0.192295 2.889607 0.990211 2.784292 0.764756 2.623998 0.535112

O N -91.440792 -58.051822 -70.095291 -48.928903 -64.976989 -17.514482 -21.076290 -15.323316 5.425583 2.384278 2.261714 1.988109

H -11.246958

C O N -51.089653 -91.678761 -57.784823 -39.937920 -70.460949 -49.893036 -8.352984 -15.385236 -65.635137 -17.979377 -7.471929 -21.622604 -15.055017 1.268641 2.047558 5.421751 2.380406 1.702841 2.270960 1.999246

In Å−1 for αij , eV for U and β, and bohr−1 for ζ;


b

From Ref. 27

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Table 2: Comparison of Mean Absolute Errors (MAEs) (in kcal/mol) for the GMTKN30HCNO database between the standard SE-MO (AM1, PM3, and PM6), and rPM6 calculations.a Subset

Method AM1 PM3 PM6 rPM6 overall 429 16.3 14.1 16.1 14.1 overall exc MB08-165 408 14.6 11.0 10.3 12.8 MB08-165 21 43.7 68.7 124.4 40.0 b c b c b c W4-08 41 9.5 (17.7 ) 9.7 (16.6 ) 8.2 (17.7 ) 18.0c G21IP 13 23.6 20.9 39.5 42.5 G21EA 11 22.5 12.9 22.4 28.0 PAd 8 12.9 16.1 18.5 10.6 SIE11 4 11.4 14.5 4.8 5.7 BHPERI 22 10.0 14.1 9.7 10.0 BH76 37 12.5 10.4 14.7 12.1 BH76RC 17 11.3 11.6 17.4 10.6 RSE43 28 3.9 4.1 5.7 5.7 O3ADD6 6 10.6 9.9 2.0 9.3 G2RC 13 12.4 21.9 30.9 21.2 ISO34 34 6.5 4.0 3.5 9.0 ISOL22 14 11.2 8.3 7.4 15.6 DC9 7 35.8 26.0 17.4 21.2 DARC 14 4.7 5.3 3.9 15.3 BSR36 36 39.6 16.7 7.4 22.4 IDISP 6 13.5 8.6 13.8 9.1 WATER27 27 48.6 31.6 17.8 7.5 S22 22 6.8 5.9 3.4 4.2 ADIM6 6 3.1 0.5 2.8 3.8 PCONF 10 5.4 3.7 2.3 3.7 ACONF 15 0.4 0.4 0.6 0.6 SCONF 17 2.4 3.1 2.6 3.0 a 35 MAEs for AM1, PM3, and PM6 were obtained in Ref. by applying Dewar’s half-electron approach 66 to open-shell species including singlets and doublets, while those for rPM6 were obtained using spin unrestricted calculations. ; b Energies are obtained by removing zero-point vibrational energies and thermal corrections 35 ; c Energies at 0 K estimated by using AM1, PM3, PM6, and rPM6 ; d The experimental value of 367.171 kcal/mol was used for the heat of formation of H+ . 67 N



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outperform PM6 for the MB08-165 subset, suggesting that there could be a trade-off between the two subsets.

Figure 2: Linear regression between the reference values of GMTKN30-HCNO and the calculated values obtained by PM6 and rPM6: (a) including and (b) excluding the M08-165 subset.

Overall, the MAE of rPM6 for the total set is calculated to be 14.1 kcal/mol, which is comparable to PM3 (14.1 kcal/mol) and more accurate than AM1 (16.3 kcal/mol) and the original PM6 (16.1 kcal/mol). An apparent improvement of R2 over PM6 obtained by a linear regression analysis can also be seen in Figure 2(a). However, one must recall that rPM6 is designed primarily for diradical molecules and it also has limitations. It significantly underperforms first-principles DFT calculations with the MAEs of 5-8 kcal/mol. 70 The overall superior performance of rPM6 is mainly due to the improvement in dealing with the MB08-165 subset, which was included in our training set as mentioned above. Excluding the M08-165 subset (overall exc MB08-165, see Table 2) leads the MAE of rPM6 to be higher by 2.5 kcal/mol than that of PM6. Looking at the accuracy of individual subsets, rPM6 underperforms the original PM6 in 13 subsets including DARC (reaction energies of Diels-Alder reactions) and ISOL22 (isomerization energies of large organic molecules). Nevertheless, the


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present reparameterization using part of the GMTKN30-HCNO database can demonstrate the ease of degradation (see Figure 2(b)).

3.2 Applications to diradical molecules 3.2.1 methylene Methylene has the triplet (3 B1 ) ground state, situated ca. 9.0 kcal/mol below the singlet (1 A1 ) state with weak diradical character. The evaluation of ∆ES-T value has been widely investigated from both the theoretical and experimental sides. 42–46,71 Table 3 summarizes the structural parameters (C-H bond length and H-C-H angle), ⟨S2 ⟩ value for each spin state, and the adiabatic singlet-triplet energy gaps (∆ES-T = ES − ET ) calculated at the UAM1, UPM3, UPM6, UrPM6, and UB3LYP levels of theory. The negative ∆ES-T value means that the singlet state lies below the triplet state. The available experimental data 71 are also listed for comparison. Table 3: Optimized structural parameters (in Å for length and degree for angles) for singlet and triplet methylene, and calculated adiabatic singlet-triplet energy gap (∆ES-T ) values (in kcal/mol)a Method

1

1 r(C-H) ̸ H-C-H ⟨S2 ⟩ UAM1 1.068 140.6 0.964 UPM3 1.067 131.3 0.945 UPM6 1.018 180.0 1.000 UrPM6 1.081 111.9 0.487 UB3LYP 1.100 113.1 0.733 Expt.b 1.107 102.3 a Values in parentheses are obtained

3

1 ∆ES-T r(C-H) ̸ H-C-H ⟨S2 ⟩ 1.062 151.7 2.016 18.1 (34.1) 1.063 149.9 2.024 23.5 (42.7) 1.019 180.0 2.007 15.8 (31.5) 1.039 152.6 2.005 10.7 (11.2) 1.082 133.1 2.005 6.0 (6.3) 1.077 134.0 9.0 b using the AP method; From Ref. 71

The shape of singlet methylene varies with its diradical characater, the degree of which can be measured by the calculated ⟨S2 ⟩S value. The UPM6 method predicts the singlet state to be a complete diradical with the ⟨S2 ⟩S value of 1.000 and therefore causes the incorrect linear structures with the H-C-H angle of 180◦ . The other SE-UMO methods reproduce the bent structures appropriately, and above all the H-C-H angle for the singlet state optimized 13 ACS Paragon Plus Environment


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by UrPM6 are quite similar to that optimized by UB3LYP (111.9 vs. 113.1◦ ). It is also found that the SE-UMO calculations systematically overestimate the H-C-H angle for the triplet state. The diradical character of the singlet state also reflects a calculated ∆ES-T value especially when the AP method 68,69 is used. As can be seen in Table 3, the strong diradical characters of the singlet state calculated by the standard SE-UMO methods (UAM1, UPM3, and UPM6) give rise to significant increases in the ∆ES-T values after the AP correction. The UrPM6 calculation yields the singlet state with weak diradical character and can circumvent this problem in line with UB3LYP. The resultant ∆ES-T value of 11.2 kcal/mol are relatively close to both the reference (6.3 kcal/mol) and the experimental value (9.0 kcal/mol). 71

3.2.2 three isomers of benzyne Photoelectron spectroscopy measurements revealed that all three isomers of benzyne have a singlet ground state. The ∆ES-T values proceed in the order ortho > meta > para, which confirms that ortho-benzyne has the smallest diradical character, followed by meta-benzyne and para-benzyne with strong diradical character. The three isomers of benzyne are good benchmarks to check whether a computational method can describe various electronic structures, and that is the reason why we included them in our training set. Table 4 summarizes the ∆ES-T values with and without the AP method obtained at the UAM1, UPM3, UPM6, UrPM6, and UB3LYP levels of theory. Table 4: Calculated adiabatic singlet-triplet energy gap (∆ES-T ) values (kcal/mol)a for the three isomers of benzyne and Mean Absolute Errors (in kcal/mol) for the MkMRCCSD/ccpVDZ resultsb

a

Method UAM1 UPM3 UPM6 UrPM6 UB3LYP MkMRCCSD Expt.c Values in parentheses

ortho meta para -11.6 (-41.5) 2.0 (3.1) -7.5 (-34.9) -15.4 (-49.4) 4.3 (6.7) -7.6 (-38.1) -10.5 (-23.4) -1.7 (-3.1) -4.0 (-12.1) -8.7(-14.7) -9.2 (-15.5) -3.5 (-7.3) -29.5 (-29.5) -11.7 (-18.0) -2.5 (-5.1) -35.1 -18.7 -4.5 -37.7 ± 0.7 -21.1 ± 0.4 -3.8 ± 0.5 are obtained using the AP method; b From 14 ACS Paragon Plus Environment

MAE 13.7 (19.5) 13.2 (24.4) 14.0 (12.5) 12.3 (8.8) 4.9 (2.3)

Ref. 47 ;

c

From Ref. 72

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The results of Mukherjee’s multireference coupled cluster singles and doubles (MkMRCCSD) 47 and experiments 72 are also listed for comparison. The accuracy of the methods can be evaluated in terms of the mean absolute errors (MAEs) of ∆ES-T relative to the corresponding MkMRCCSD result. The reference UB3LYP calculations combined with the AP method provide good agreement with the MkMRCCSD results for all isomers, which confirms that the AP method works well for this system. Concerning the standard UAM1, UPM3, and UPM6 approaches, we can see that the poor description of electronic structure brings about unsystematic results. The application of the AP method to UAM1 and UPM3 significantly stabilize not only the singlet state of ortho-benzyne but also that of para-benzyne, resulting in large MAEs as presented in Table 4. These methods predict that meta-benzyne has a triplet ground state with positive ∆ES-T values. The UPM6 method gives a correct ground state for meta-benzyne, but its magnitude is substantially small. The UrPM6 method represents improvement in describing the singlet state for each isomer. As listed in Table S3 in the Supporting Information, the ⟨S2 ⟩S values calculated by UrPM6 as well as UB3LYP proceed in the correct order para > meta > ortho, in contrast to para > ortho > meta obtained with the conventional SE-UMO methods. Besides, it decreases the spin contamination effect on the triplet state for meta-benzyne. Looking at ⟨S2 ⟩T′ values provided in Table S1, the smallest value obtained with UB3LYP exceeds the exact value by only 0.02, followed by UrPM6 (0.05), UPM6 (0.29), UPM3 (0.56), and UAM1 (0.59). The extent of spin contamination for the triplet state clearly reflects the magnitude of error in ∆ES-T . From the results, we can conclude that UrPM6 shows improvement over the conventional SE-UMO methods, though it ends up underestimating the ∆ES-T value for ortho-benzyne.

3.2.3 cyclobutadiene The automerization reaction of cyclobutadiene 3 involves the interconversion between two equivalent D2h structures passing through the transition state (D4h ) as shown in Figure 1.



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The ground state of D4h cyclobutadiene and the barrier height for the reaction (∆E ‡ ) have been of interest for many years. It is a typical case of the violation of Hund’s rule, as rigorous and robust theoretical calculations assigned the ground state of the D4h structure to be a singlet. 73,74 Table 5: Calculated adiabatic singlet-triplet energy gap (∆ES-T ) valuesa (in kcal/mol) for cyclobutadiene (3), meta-xylylene (4), and oxyallyl (5), and barrier height for automerization of cyclobutadienea (∆E ‡ , in kcal/mol). Method

3 ∆ES-T N.A. N.A. N.A. -9.7 (-22.0) N.A. -4.3 (-9.0) -13.2

a d

‡

∆E N.A. N.A. N.A. 1.0 (-11.3) N.A. 0.8 (-4.0) 6.3

4 ∆ES-T 14.0 (24.9) 13.2 (22.8) 12.0 (20.6) 10.5 (18.1) 10.9 (19.6) 6.7 (12.7)

5 ∆ES-T 3.9 (6.7) 3.3 (5.8) 1.6 (2.9) 4.1 (6.4) 3.0 (4.0) 1.3 (1.4)

UAM1 UPM3 UPM6 UrPM6 UBHandHLYP UB3LYP MR-AQCCb MRPT2c 12.0 c EOM-SF-CCSD 11.3 EOM-SF-CCSD(dT)d -1.5 e ULO-MkMRCCSD -1.6 Expt. -12.0f 1.6-10g 9.6 ± 0.2h -1.3i N.A.: not available. Values in parentheses are obtained using the AP method; b From Ref. 74 ; c From Ref. 75 ; From Ref. 76 ; e From Ref. 77 ; f From Ref. 78 ; g From Ref. 79 and Ref. 80 ; h From Ref. 81 ; i From Ref. 82 To our surprise, UBHandHLYP as well as the SE-UMO methods other than UrPM6

fails to locate a singlet transition state corresponding to the automerization reaction. As listed in Table 5, comparison of ∆ES-T value and the barrier height (∆E ‡ ) between UrPM6 and UB3LYP clearly shows a similar trend albeit with quantitative differences. The hybrid UDFT calculations combined with the AP method can yield a negative barrier height as suggested by the previous work. 83 Such situation can be avoided by choosing a specific functional suitable for this system, but it is out of scope of the present study. We would like to emphasize here that the UrPM6 method outperforms the standard SE-UMO methods significantly, being capable of locating the D4h structure as transition state and reproducing 16 ACS Paragon Plus Environment

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the singlet ground state.

3.2.4 meta-xylylene M eta-xylylene (4) has the triplet (3 B2 ) ground state and the first excited singlet (1 A1 ) lies 9.6 ± 0.2 kcal/mol above it. Robust approaches such as Multireference second-order Møller-Plesset (MRMP2) and equation of motion spin-flip coupled cluster singles and doubles (EOM-SF-CCSD) usually overestimates the experimental ∆ES-T value by a few kcal/mol (see Table 5). 75,84,85 The use of AP method on UB3LYP (12.7 kcal/mol) seems to slightly overshoot the experimental value of 9.6 ± 0.2 kcal/mol. The increases in error are more apparent for the SE-UMO calculations. As in the case of benzyne discussed above, the use of the AP method may be problematic when the methods other than UB3LYP are used because the reference triplet state is not free from spin contamination (see Table S4 in the Supporting Information). The extent of spin contamination for the triplet state clearly reflects the magnitude of error in ∆ES-T and UrPM6 is found to be comparable to UBHandHLYP.

3.2.5 oxyallyl The ground state of oxyallyl (5) had been controversial until the recent study using negative ion photoelectron spectroscopy 82 revealed that the singlet (1 A1 ) is the ground state with the ∆ES-T value of -1.3 kcal/mol. This finding was in contrast to the prior theoretical calculations which favored a triplet ground state. 86,87 Saito et al. 77 assessed the applicability of both single-reference and multireference methods systematically and pointed out that the sophisticated methods including EOM-SF-CCSD with perturbative triples corrections (EOM-SF-CCSD(dT)) 76 and MkMRCCSD with the size-consistent correction with the UHF localized natural orbitals (ULO-MkMRCCSD) are mandatory to achieve good agreement with experiment. They also found that hybrid UDFT calculations do not reproduce the singlet ground state. This is true for the present SE-UMO methods as none of the methods



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examined here gives the singlet ground state as listed in Table 5. We are reluctant to modify the parameter sets further for 5 in case errors increase elsewhere.

3.2.6 bistetracene Liu et al. 48 synthesized 6, a zigzag-edged PAH, which demonstrates the diradicaloid nature based on NMR and EPR analysis. The UB3LYP calculation supported that 6 has a singlet diradical ground state with the ∆ES-T value of -6.7 kcal/mol. We compare the performance of SE-UMO methods for first-principles UDFT calculations, especially focusing on optimized geometries and estimated ∆ES-T values. The calculated data are presented in Table 6. Table 6: Root-mean-square deviations (in Å) from the reference geometries optimized at the UB3LYP/6-31G* level and calculated adiabatic singlet-triplet energy gap (∆ES-T ) values (in kcal/mol)a for tetrabenzo[a,f,j,o]perylene (6) and a heptazethrenebis(dicarboximide) (7)b Method 6

7

⟨S2 ⟩

RMSD T 0.090 0.249 0.144 0.075 0.024

S CSb UAM1 0.095 UPM3 0.250 UPM6 0.121 UrPM6 0.078 UBHandHLYP 0.028 UB3LYP UAM1 0.148 0.149 0.147 UPM3 0.047 0.048 0.047 UPM6 0.180 0.272 0.180 UrPM6 0.185 0.187 0.184 UBHandHLYP 0.034 0.032 0.030 UB3LYP

S T 4.994 4.929 4.680 4.543 3.752 3.719 1.907 2.401 1.441 2.223 0.429 2.059 3.767 4.027 3.399 3.604 2.708 3.190 1.591 2.436 1.273 2.254 0.355 2.061

∆ES-T

∆ES-CS b

-18.5 -16.9 -13.6 -6.9 -7.3 -6.2 (-7.9) -14.7 -34.0 -13.6 -30.7 -11.5 -19.5 -8.1 -6.4 -8.7 -7.5 -8.0 (-10.2) -0.3 (-2.5)

a

Values in parentheses are obtained using the AP method; b For 7, the RMSD for the closed-shell singlet (CS) state and calculated adiabatic S-CS energy gap (∆ES-CS ) values (in kcal/mol) are presented. For both singlet and triplet states, UrPM6 gives the smallest root-mean-squared deviations (RMSDs) of the coordinates from the corresponding UB3LYP-optimized geometries, followed by UAM1, UPM6, and UPM3. It is obvious, when looking at the calculated ⟨S2 ⟩S and ⟨S2 ⟩T′ values, that the standard UAM1, UPM3, and UPM6 suffer from significant spin contamination. The UPM6 calculations overestimate the reference values obtained with 18 ACS Paragon Plus Environment

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UB3LYP by 3.324 and 1.660 for the singlet and triplet state, respectively, even though it provides improvement over UAM1 and UPM3. The UrPM6 method can further reduce these values from the original UPM6 method by ca. 50 and 35 %. The triplet state obtained with UB3LYP seems to have relatively small spin contamination (⟨S2 ⟩T′ = 2.059) and thus the AP method can be used only for this case (see Table 6. The SE-UMO calculations other than UrPM6 favor more spin polarized (spin-contaminated) solutions and systematically overestimate the stability of the singlet state with respect to UB3LYP. In contrast, the uncorrected ∆ES-T value calculated by UrPM6 (-6.9 kcal/mol) is quite close to that calculated by UBHandHLYP (-7.3 kcal/mol) and the UB3LYP calculated data with the AP correction (-7.9 kcal/mol).

3.2.7 heptazethrenebis(dicarboximide) Sun et al. synthesized the stable singlet diradical 7 and performed the UCAM-B3LYP 88 calculations to confirm its stability. 49 They found that the open-shell singlet state lies 5.8 and 7.5 kcal/mol below the closed-shell singlet and triplet state, respectively. In this case, the closed-shell singlet (CS) state is also considered as well. As in the case of 6, UrPM6 provides improvement over the standard SE-UMO methods in terms of the ∆ES-T and ∆ES-CS values albeit with slightly large RMSDs of the coordinates. While UPM3 gives the smallest RMSDs with less than 0.05 Å for both spin states, it leads to spurious energy lowering of the singlet state in line with UAM1 and UPM6, mainly because of large spin contamination. The magnitude of the ∆ES-CS values calculated by these three methods (ranging from -34.0 to -19.5 kcal/mol) seem to be particularly large as compared to those calculated by the UDFT calculations ranging from -7.5 to -0.3 kcal/mol. Concerning UrPM6, both singlet and triplet wave functions are more resistant to spin contamination with 42 and 24 % reduction of ⟨S2 ⟩S and ⟨S2 ⟩T′ values, respectively, from the original UPM6 method. The better description of the electronic structure makes the ∆ES-T and ∆ES-CS values comparable to the UDFT calculations (see Table 6).



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3.2.8 polyacenes Finally, we conducted benchmark calculations of the ∆ES-T values of n-acenes 8 (n = 2-6, 8, 10, and 12) as with the previous work. 50 Figure 3(a) plots ∆ES-T value vs. length of polyacenes obtained with the SE-UMO calculations and the experimental observations.

Figure 3: Calculated adiabatic singlet-triplet energy gap (∆ES-T ) values with different length of acenes: (a) Comparison between UrPM6 and the standard UAM1, UPM3, and UPM6 results, and (b) comparison between UrPM6, UDFT, and DMRG results. The results of UB3LYP/6-31G* and DMRG/STO-3G were take from Ref. 50

Unlike the standard SE-UMO methods, UrPM6 performs well for small polyacenes up to hexacene (n = 6), showing an exponential decay of ∆ES-T value with respect to n similar to the experimental observations. 52–56 However, it does not hold true as the length of polyacene increases (n ≥ 8). Note that near degeneracies in the π-orbitals cause more complicated electronic structures with increasing the number of benzene rings. The natural orbital analysis based on the wave function obtained at the DMRG/STO-3G level 50 suggested that the ground state of the longer acenes has a singlet polyradical (multiconfigurational) character that should be treated using genuine multireference approaches. In fact, the UBHandHLYP and UB3LYP 20 ACS Paragon Plus Environment

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become less accurate with increasing conjugated length (see Figure 3(b)). Leaving aside the difficulties of dealing with polyradical character within a single-reference framework, we can see that UrPM6 obviously outperforms the standard SE-UMO methods and display similar behavior to UBHandHLYP rather than UB3LYP.

4. Conclusions In the present article, we report a study of organic diradical molecules with SE-UMO methods. To compute organic diradical molecules with accuracy similar to that of the widely used UB3LYP/6-31G* method, we have performed a reparameterization of PM6 (called rPM6). The parameters for the basic elements (hydrogen, carbon, nitrogen, and oxygen) have been optimized simultaneously on the basis of the training set, which contains both the electronic and geometric variables of diradical molecules of importance. Although inclusion of part of the GMTKN30 database in the training set contributes to the ease of degradation, rPM6 can show worse performance than the original PM6 for general-purpose applications. For diradical systems, the use of the spin unrestricted rPM6 (UrPM6) method to openshell compounds such as small diradicals (1-4) and PAHs (6-8) indicate that it shows substantial improvement over the standard SE-UMO (UAM1, UPM3, and the original UPM6) methods. While the standard SE-UMO computations as well as UHF inherently suffer from significant spin contamination, both the singlet and triplet UrPM6 wave functions are more resistant to spin contamination and can thus lead to the better geometric parameters and adiabatic singlet-triplet energy gaps. The performance is found to be comparable to the UBHandHLYP/6-31G* calculations for PAHs. It can cover systems with a wide range of diradical character except for difficult cases such as oxyallyl (5) and works poorly for polyradical (multiconfigurational) character. It should be emphasized that the advantage of UrPM6 over first-principles UDFT is obviously the computational speed. Given the largest molecule presented here (7) consisting



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of 118 atoms, it is about 7,000 times faster than the UB3LYP/6-31G* method per SCF iteration (30 ms vs. 209 s) with a serial mode on a quad-core Intel Xeon E3-1241v3 at 3.50 GHz CPU with 8 GB RAM. For large systems, optimized geometries generated by UrPM6 may be a good initial guess for a stationary point, which is to be optimized by UDFT. In that case, the number of iterations could be reduced. Parameters for another element can be optimized one by one as needed, with the fixed set of the parameters determined in the present work. It will increase the scope of the SE-MO methods for applications in QM/MM MD studies, for example on metalloenzymes. This work is in progress.

Author Information Corresponding Author E-mail: [email protected] Phone: +81 82 830 1617

Notes The authors declare no competing financial interest.

Acknowledgement This work was supported by JSPS KAKENHI Grant-in-Aid for Young Scientists (B) (No. JP16K17856) (to T. S.), by JSPS KAKENHI Grant-in-Aid for Scientific Research (C) (No. JP15KT0143) (to Y. K. and T. S.), by JSPS KAKENHI Grant-in-Aid for Scientific Research (C) (No. JP16K07325) (to Y. T.), by CREST, JST (to Y. T.), and by MEXT Grants-in-Aid for Scientific Research on Innovative Areas“ 3D Active-Site Science ”(No. JP26105012) (to Y. T. and T. S.).


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Supporting Information Available Information about the training set (Table S1 and Table S2); Cartesian coordinates for 1-8 optimized by UrPM6; Table S3; Table S4; full citation for the reference 65. This material is available free of charge via the Internet at http://pubs.acs.org/.

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