The Mechanism of the Propagation in the Anionic Polymerization of Polystyryllithium in Non-Polar Solvents Elucidated by Density Functional Theory Calculations. A Study of the Negligible Part Played by Dimeric Ion-Pairs under Usual Polymerization Conditions

The elementary processes occurring in the anionic polymerization of styrene with dimerically associated polystyryllithium (propagation during the anionic polymerization of dimeric polystyryllithium) in the gas phase and cyclohexane were studied using MX062X/6-31+G(d), a recently developed density functional theory (DFT) method and compared with the polymerization of styrene with non-associated polystyryllithium, which was described in a previous study. The most stable transition state in the reaction of styrene with dimeric polystyryllithium has a structure in which the side chains of styrene and the two chain end units of polystyryllithium are located in the same direction around the Li atom near the reactive site. The relative enthalpy for this transition state in cyclohexane is 28 kJ·mol−1, which is much lower than that for the reaction of non-associated polystyryllithium (51 kJ·mol−1). However, the relative free energy (which determines the rate constant) for the former is 93 kJ·mol−1, which is greater than that for the latter by 7 kJ·mol−1, indicating that the latter reaction (reaction with non-associated polystyryllithium) is advantageous over the former (reaction with dimeric polystyrylllithium). Their rates of reaction are also affected by initiator concentrations; in the case of reactions with low initiator concentrations, from which high molecular weight polymers are usually obtained, the rate of reaction corresponding to non-associated polystyryllithium is much larger than that corresponding to dimeric polystyryllithium.


Introduction
In the case of the anionic polymerization of styrene in non-polar solvents, it is generally accepted that polystyryllitium is mainly associated into dimeric species (PStLi) 2 in equilibrium with a small amount of non-associated PStLi chains [1][2][3][4][5]. A kinetic order of 0.5 with respect to [PStLi] for this reaction indicates that only non-associated PStLi ion-pairs are able to propagate [6,7]. However, the possibility of a reaction of styrene with dimeric (PStLi) 2 or higher aggregates was proposed based on experimental data, such as the addition of butadiene to freeze-dried polystyryllithium and the As described earlier, some researchers are working on the polymerization of styrene with dimeric or higher species. However, it is not clearly shown how and why the dimeric and higher species are more reactive than the non-associated species. The aim of this study is to clarify the mechanism of the anionic polymerization of styrene with dimeric polystyryllithium using DFT calculations in a manner similar to those performed for non-associated polystyryllithium, as described in our previous study [20]; further we intend to compare and contrast the differences between the reactions of the dimeric species and non-associated species.

Methods
Polystyryllithium species obtained by the addition of styrene to alkyllithium can be denoted as R(St) m Li, where R denotes the alkyl group of the initiator. We employed HStLi by setting m = 1 and substituting H for R. Using this structural model, the addition of styrene to dimerically associated polystyryllithium (a propagation reaction) was studied (in our previous study, we set m = 1 and 2 and compared their transition states in order to understand the effect of the penultimate unit and elucidate the reaction mechanism of styrene polymerization with non-associated polystyryllithium). Our purpose in this investigation is to compare the reactions of non-associated and dimerically associated polystyryllitium. Actual calculations on the transition state of St/(HSt 2 Li) 2 (m = 2) are rather complicated and difficult to perform, therefore we set m = 1 and compared the obtained results of St/(HStLi) 2 with those of St/HStLi (m = 1) performed in the previous study.
In this study, calculations were performed using the M062x/6-31+G(d)//M062x/6-31+G(d) method [21] with the Gaussian 09W program [25], in essentially the same manner as those performed in our previous study. The dimeric species ((HStLi) 2 ) and intermediate complexes (precursor complexes, transition states, etc.) and products of the reaction of (HStLi) 2 with styrene were optimized, and the obtained geometries, and enthalpy and Gibbs free energy values at 25 • C were used for discussion. The transition states were confirmed to have one imaginary frequency. The precursor complexes and products were obtained by first applying the intrinsic reaction coordinate (IRC) approach [26][27][28] to the transition states and then optimizing the obtained intermediate structures completely. For calculations in cyclohexane, the integral equation formalism (IEFPCM) variant module of the polarizable continuum model (PCM), a widely used method, was employed [29,30].
In order to compare the stabilities of the structures in the gas phase, the values of relative enthalpy in the gas phase (∆H r ) were calculated using the obtained values of enthalpy (H) shown in Table A1 of Appendix A.1, according to the following procedure: where H[(HStLi) 2 ] denotes the enthalpy of a particular (HStLi) 2 , and H[(HStLi) 2 ] 0 is the enthalpy of the most stable (HStLi) 2 that has the lowest free energy among the studied dimers. For the most stable dimer, ∆H r = 0, and for other dimers, ∆H r indicates the extent of instability (in a thermodynamic sense) of the particular dimer, (HStLI) 2 , with respect to the most stable dimer.  where H[St/(HStLi) 2 ] denotes the enthalpy of a particular St/(HStLi) 2 system and H(St) is the enthalpy of styrene. In this case, ∆H r actually indicates the extent of instability of the particular St/(HStLi) 2 system with respect to the starting material (styrene and dimeric (HStLi) 2 ). ∆H r for the transition state corresponds to the apparent activation energy of the reaction.
The values of relative free energy in the gas phase (∆G r ) were also calculated in the same manner as those of ∆H r , using the values of G shown in Table A1.
∆H rch and ∆G rch , the values of relative enthalpy and free energy, respectively, in cyclohexane, were calculated in a manner similar to that used to calculate ∆H r and ∆G r in the gas phase, but using the values of H and G in cyclohexane (Table A2, Appendix A.1).
In the geometries of all the studied structures, C-Li with distances less than 0.245 nm were marked with full or dotted lines as η-coordinated bonds.

Results and Discussion
3.1. Addition of Styrene to (HStLi) 2 in the Gas Phase (HStLi) 2 . To study the addition of styrene to dimeric polystyryllithium, several types of suitable dimeric species should be chosen. To this end, a series of dimeric (HStLi) 2 used in our previous study on the addition of styrene to non-associated polystyryllithium was employed (Figure 1, 1-a to 1-f).
(1) Structures 1-a to 1-d exhibited sandwich-type features, while 1-e and 1-f were six-membered structures like the four-membered structure of alkyllithium dimers. The former four (HStLi) 2 with sandwich-type structures have much lower ∆H r and ∆G r values than the latter two six-membered structures. (2) In 1-a and 1-b, the side chains of the HSt-groups (chain end units) were located near each end of Li−Li one after another, while in 1-c and 1-d, they were located near one end of Li−Li. 1-a and 1-b exhibited lower ∆H r and ∆G r when compared to 1-c and 1-d.  Optimized geometries and relative energies of (HStLi)2 in the gas phase originally shown in our previous paper. The small drawing on the top of each structure represents simplified overhead view of the lower drawing; carbon atoms in one of the HSt-groups (the upper HSt-group in the lower drawing) are colored in blue. In the lower drawing, C−Li distances less than 0.225 nm are shown. ∆Hr and ∆Gr, which represent the relative enthalpy and free energy, respectively, are expressed in kJ·mol −1 .
Transition state. There are several transition states for the addition of styrene to each (HStLi)2 shown in Figure 1. These transition states are determined depending on the arrangement of their groups, i.e., whether styrene is coordinated with two Li atoms or only one Li atom, whether the side chain of styrene approaches in the same direction or opposite direction with respect to the side chain of the reacting HSt-group, and from which part of the phenyl group styrene approaches Li away from the reaction site (from C2-C3 or C5-C6 as shown in the drawing in Table 1) if it is coordinated with two Li atoms. In the case of (HStLi)2(1-a), which has the lowest ∆Gr value, the possible transition states for the addition of styrene were optimized and the typical transition states are shown in Figure  2 and Table 1 (2-a to 2-d). In each structure, the drawing on the left side represents the side view of the drawing on the right side and the upper drawing is the overhead view of the lower drawing. The carbon atoms of styrene are colored in blue, while those of the reacting HSt-group are brown colored. The blue arrows in each drawing show the main displacement vectors corresponding to the imaginary frequency of the transition state. Accordingly, the blue arrows at the α-carbon of the side chain of the reacting HSt-group and the terminal carbon of the side chain of styrene indicate that these two carbon atoms react in the direction of the arrows to form the product. The relative enthalpies (∆Hr), which are expected to correspond with the apparent activation energies of the reaction, and the relative free energies (∆Gr), which determine the rate constants, of these transition Optimized geometries and relative energies of (HStLi) 2 in the gas phase originally shown in our previous paper. The small drawing on the top of each structure represents simplified overhead view of the lower drawing; carbon atoms in one of the HSt-groups (the upper HSt-group in the lower drawing) are colored in blue. In the lower drawing, C−Li distances less than 0.225 nm are shown. ∆H r and ∆G r , which represent the relative enthalpy and free energy, respectively, are expressed in kJ·mol −1 .
Transition state. There are several transition states for the addition of styrene to each (HStLi) 2 shown in Figure 1. These transition states are determined depending on the arrangement of their groups, i.e., whether styrene is coordinated with two Li atoms or only one Li atom, whether the side chain of styrene approaches in the same direction or opposite direction with respect to the side chain of the reacting HSt-group, and from which part of the phenyl group styrene approaches Li away from the reaction site (from C2-C3 or C5-C6 as shown in the drawing in Table 1) if it is coordinated with two Li atoms. In the case of (HStLi) 2 (1-a), which has the lowest ∆G r value, the possible transition states for the addition of styrene were optimized and the typical transition states are shown in Figure 2 and Table 1 (2-a to 2-d). In each structure, the drawing on the left side represents the side view of the drawing on the right side and the upper drawing is the overhead view of the lower drawing. The carbon atoms of styrene are colored in blue, while those of the reacting HSt-group are brown colored. The blue arrows in each drawing show the main displacement vectors corresponding to the imaginary frequency of the transition state. Accordingly, the blue arrows at the α-carbon of the side chain of the reacting HSt-group and the terminal carbon of the side chain of styrene indicate that these two carbon atoms react in the direction of the arrows to form the product. The relative enthalpies (∆H r ), which are expected to correspond with the apparent activation energies of the reaction, and the relative free energies (∆G r ), which determine the rate constants, of these transition states are also shown in Figure 2 and Table 1. In each of the transition states 2-a and 2-b, styrene is coordinated with the two Li atoms, and the styrene side chain approaches from a direction opposite to that of the side chain of the reacting HSt-group. They are different in terms of the portion of styrene coordinating with the Li atom away from the reactive site (C2-C3 or C5-C6), as clearly shown in the left side drawings of 2-a and 2-b. In 2-c, styrene is coordinated with two Li atoms and the side chain of styrene approaches in the same direction as the side chain of the reacting HSt-group, while in 2-d, styrene is coordinated with only the Li atom near the reactive site and the side chain of styrene approaches in the same direction as that of the reacting HSt-group. Comparing the ∆H r and ∆G r values of these transition states, it can be observed that the transition states in which styrene is coordinated with two Li atoms in a direction opposite to that of the reacting HSt-group, i.e., 2-a and 2-b, have lower ∆H r and ∆G r values than other transition states. In 2-a and 2-b, the phenyl group of styrene approaches Li from a different portion and calculations were conducted to determine which of them results in a lower ∆G r for the transition state. Table 1. Arrangement of the transition states shown in Figure 2 for the addition of styrene to (HStLi) 2 (1−a).   Figure 3 for the addition of styrene to each (HStLi)2 in Figure 1.

Item Detail 2-a 3-b 3-c 3-d 3-e 3-f
(HStLi)2 Type of (HStLi)2  (2) is the Li atom away from the reactive site. b Opp.: Opposite c Numbering of carbon atoms is shown below. d Distance between the two carbon atoms participating in the reaction.
As described earlier, 3-c and 3-d possess lower ∆Hr and ∆Gr than other transition states. In these transition states all three side chains, i.e., those of styrene plus two HSt-groups, are located around the Li atom near the reaction site as clearly shown in 3-g and 3-h (a view from another point for 3-c and 3-d, respectively). This placement may be responsible for the low ∆Gr of these transition states. In 3-c, the three said side chains are located in the same direction to Li−Li as shown in 3-g, while in 3-d, the side chains of the two HSt-groups interact face to face with each other (3-h). The structure of 3-c may have caused the oblique positioning of the reacting HSt-group with respect to Li−Li, parallel sandwiching of the phenyl groups of styrene and the unreacting HSt-group, resulting in a low ∆Gr value.
Reaction path (comparison with the reaction of non-associated polystyryllithium). The pathway of the reaction system whose transition state is 3-c is shown in Figure 4; this system will be called system(dim-r) hereafter. The reaction proceeds in three steps. First, an initial complex is formed and it goes through the first and second steps to the final step (the precursor complex, transition state, and product). In Figure 4, only the initial complex and details of the final step are shown because the complete process is complicated and we can discuss the reaction path using the information in Figure For the other (HStLi) 2 moieties shown in Figure 1, the possible transition states were calculated in the same way as for those shown in Figure 2, and the transition states with the lowest ∆G r for each type of (HStLi) 2 are shown in Figure 3 and Table 2 as structures 3-b to 3-f, along with transition state 2-a that has the lowest ∆G r of St/[(HStLi) 2 (1-a)] system. In each of these transition states, styrene is coordinated with two Li atoms and the side chain of styrene is placed in direction opposite to that of the side chain of the reacting HSt-group, which is as expected. The ∆H r values of these transition states are low, around 22 (for 3-c) to 39 kJ·mol −1 (for 3-e), while the ∆G r values are fairly high, around 87 (for 3-c) to 113 kJ·mol −1 (for 3-e), and their Li−Li distances range from 0.25 to 0.35 nm. From the ∆G r values of these transition states, transition state 3-c was found to be the most stable, followed by 3-d, 3-b, 2-a, 3-f, and 3-e. In 3-c, the reacting HSt-group is situated oblique to the Li-Li line (red line in the upper drawing of structure 3-c); the planes of styrene and the unreacting HSt-group are situated parallel to each other (two red lines in the left drawing of structure 3-c) and the Li−Li distance is 0.35 nm, the longest of all transition states. The orders of magnitude of ∆H r and ∆G r of these transition states do not agree with the orders of magnitude of ∆H r and ∆G r values of the original (HStLi) 2 . For example, the ∆H r and ∆G r values of (HStLi) 2 (1-c) and (HStLi) 2 (1-d) are higher than those of (HStLi) 2 (1-a) and (HStLi) 2 (1-b) by about 10 kJ·mol −1 , as shown in Figure 1 Figure 3, by 3−11 kJ·mol −1 . However, the difference is not as high as that between ((HStLi) 2 (1-e) and (HStLi) 2 (1-f)) and ((HStLi) 2 (1-a) and (HStLi) 2 (1-b)), which is about 50 kJ·mol −1 , as shown in Figure 1. to that of the side chain of the reacting HSt-group. They are different in terms of the portion of styrene coordinating with the Li atom away from the reactive site (C2-C3 or C5-C6), as clearly shown in the left side drawings of 2-a and 2-b. In 2-c, styrene is coordinated with two Li atoms and the side chain of styrene approaches in the same direction as the side chain of the reacting HSt-group, while in 2d, styrene is coordinated with only the Li atom near the reactive site and the side chain of styrene approaches in the same direction as that of the reacting HSt-group. Comparing the ∆Hr and ∆Gr values of these transition states, it can be observed that the transition states in which styrene is coordinated with two Li atoms in a direction opposite to that of the reacting HSt-group, i.e., 2-a and 2-b, have lower ∆Hr and ∆Gr values than other transition states. In 2-a and 2-b, the phenyl group of styrene approaches Li from a different portion and calculations were conducted to determine which of them results in a lower ∆Gr for the transition state.  Table 1 and also in the relevant description in the main text. Hydrogen atoms are not shown. ∆Hr and ∆Gr, the relative enthalpy and free energy, respectively, are expressed in kJ·mol −1 .  Table 1 and also in the relevant description in the main text. Hydrogen atoms are not shown. ∆H r and ∆G r , the relative enthalpy and free energy, respectively, are expressed in kJ·mol −1 .
As described earlier, 3-c and 3-d possess lower ∆H r and ∆G r than other transition states. In these transition states all three side chains, i.e., those of styrene plus two HSt-groups, are located around the Li atom near the reaction site as clearly shown in 3-g and 3-h (a view from another point for 3-c and 3-d, respectively). This placement may be responsible for the low ∆G r of these transition states. In 3-c, the three said side chains are located in the same direction to Li−Li as shown in 3-g, while in 3-d, the side chains of the two HSt-groups interact face to face with each other (3-h). The structure of 3-c may have caused the oblique positioning of the reacting HSt-group with respect to Li−Li, parallel sandwiching of the phenyl groups of styrene and the unreacting HSt-group, resulting in a low ∆G r value.   Reaction path (comparison with the reaction of non-associated polystyryllithium). The pathway of the reaction system whose transition state is 3-c is shown in Figure 4; this system will be called system(dim-r) hereafter. The reaction proceeds in three steps. First, an initial complex is formed and it goes through the first and second steps to the final step (the precursor complex, transition state, and product). In Figure 4, only the initial complex and details of the final step are shown because the complete process is complicated and we can discuss the reaction path using the information in Figure 4 (the complete reaction path is shown in Figure A1 of Appendix A.2). It can be observed from Figure 4 that the values of ∆H r were very low, as (HStLi) 2 forms the initial complex, precursor complex, transition state, and product without dissociating into non-associated HStLi. The ∆H r value corresponding to the initial complex was −19 kJ·mol −1 , indicating an exothermic phenomenon. The ∆G r values were relatively high, which will be discussed in later sections. The distance between the two carbon atoms participating in the reaction becomes shorter as the reaction proceeds, from 0.37 nm for initial complex 4-a, through 0.24 nm for transition state 3-c, to the normal single bond distance for product 4-d (0.155 nm). Table 2. Arrangement of the transition states shown in Figure 3 for the addition of styrene to each (HStLi) 2 in Figure 1.
(HStLi) 2 Type of (HStLi) 2 1-a  Table 2. Arrangement of the transition states shown in Figure 3 for the addition of styrene to each (HStLi)2 in Figure 1.  (2) is the Li atom away from the reactive site. b Opp.: Opposite c Numbering of carbon atoms is shown below. d Distance between the two carbon atoms participating in the reaction.
As described earlier, 3-c and 3-d possess lower ∆Hr and ∆Gr than other transition states. In these transition states all three side chains, i.e., those of styrene plus two HSt-groups, are located around the Li atom near the reaction site as clearly shown in 3-g and 3-h (a view from another point for 3-c and 3-d, respectively). This placement may be responsible for the low ∆Gr of these transition states. In 3-c, the three said side chains are located in the same direction to Li−Li as shown in 3-g, while in 3-d, the side chains of the two HSt-groups interact face to face with each other (3-h). The structure of 3-c may have caused the oblique positioning of the reacting HSt-group with respect to Li−Li, parallel sandwiching of the phenyl groups of styrene and the unreacting HSt-group, resulting in a low ∆Gr value.
Reaction path (comparison with the reaction of non-associated polystyryllithium). The pathway of the reaction system whose transition state is 3-c is shown in Figure 4; this system will be called system(dim-r) hereafter. The reaction proceeds in three steps. First, an initial complex is formed and it goes through the first and second steps to the final step (the precursor complex, transition state, and product). In Figure 4, only the initial complex and details of the final step are shown because the complete process is complicated and we can discuss the reaction path using the information in Figure  4 (the complete reaction path is shown in Figure A1 of Appendix A2). It can be observed from Figure  4 that the values of ∆Hr were very low, as (HStLi)2 forms the initial complex, precursor complex, transition state, and product without dissociating into non-associated HStLi. The ∆Hr value corresponding to the initial complex was −19 kJ·mol −1 , indicating an exothermic phenomenon. The ∆Gr values were relatively high, which will be discussed in later sections. The distance between the two carbon atoms participating in the reaction becomes shorter as the reaction proceeds, from 0.37 nm for initial complex 4-a, through 0.24 nm for transition state 3-c, to the normal single bond distance for product 4-d (0.155 nm).   Figure A1 of Appendix A2). The small drawing on the left side of each structure is the side view of the right-hand side drawing. Carbon atoms of styrene are colored in blue, while those of the reacting HSt-group are colored in brown. In each drawing the two carbon atoms participating in the reaction are connected by dotted or full line and the distance between them is shown in nm. Hydrogen atoms are not shown. ∆Hr and ∆Gr, the relative enthalpy and free energy, respectively, are expressed in kJ·mol −1 .
In this study, the reaction of (HStLi)2 (without the penultimate styrene unit) with styrene was used as discussed in the Methods section. Therefore, the reaction of non-associated HStLi (without the penultimate styrene unit) with styrene in the gas phase, which will be called system(mon-r) hereafter, was taken from our original paper and shown as   2 (1-c)] (system(dim-r)) in the gas phase. Although the reaction proceeds in three steps, only the first complex of the first step (4-a) (the initial complex) and the three structures of the final step (4-b, 3-c and 4-d) are shown here (the complete pathway is shown in Figure A1 of Appendix A.2). The small drawing on the left side of each structure is the side view of the right-hand side drawing. Carbon atoms of styrene are colored in blue, while those of the reacting HStgroup are colored in brown. In each drawing the two carbon atoms participating in the reaction are connected by dotted or full line and the distance between them is shown in nm. Hydrogen atoms are not shown. ∆H r and ∆G r , the relative enthalpy and free energy, respectively, are expressed in kJ·mol −1 . In this study, the reaction of (HStLi) 2 (without the penultimate styrene unit) with styrene was used as discussed in the Methods section. Therefore, the reaction of non-associated HStLi (without the penultimate styrene unit) with styrene in the gas phase, which will be called system(mon-r) hereafter, was taken from our original paper and shown as Figure 5. Comparing Figure 4 (system(dim-r)) with Figure 5 (system(mon-r)), it can be noted that the ∆H r value of transition state 3-c for system(dim-r) was 22 kJ·mol −1 , which is much lower than that of 5-b for system(mon-r), 50 kJ·mol −1 : this is because (HStLi) 2 does not undergo any preliminary dissociation in system(dim-r). However, the ∆G r of 3-c was 87 kJ·mol −1 , higher than that of 5-b by 5 kJ·mol −1 . As ∆G = ∆H − T∆S (T = absolute temperature and S = entropy), this difference is attributed to the difference in the -T∆S values of system(dim-r) and system(mon-r). The rate constant of the reaction is related to ∆G (which will be discussed in detail in the next section), and the reaction of non-associated polystyryllithium (system(mon-r)) is shown to be advantageous over that of dimer polystyryllithium (system(dim-r)).
Polymers 2019, 11, x FOR PEER REVIEW 10 of 18 Figure 5. Reaction pathway for St/HStLi (system(mon-r)) in the gas phase originally shown in our previous paper. In each drawing the two carbon atoms participating in the reaction are connected by dotted or full line and the distance between them is shown in nm. ∆Hr and ∆Gr, the relative enthalpy and free energy, respectively, are expressed in kJ·mol −1 .
The changes in the ∆Hr and ∆Gr values of system(dim-r) and (mon-r) are schematically shown in Figure 6 and 7, respectively. These figures clearly show that although ∆Hr of the transition state of system(dim-r) related to (HStLi)2 is low compared to that of system(mon-r) because of no dissociation in (HStLi)2, the ∆Gr value of the former is higher than that of the latter by 5 kJ·mol −1 , indicating that the route through the latter (system(mon-r)) is the predominant reaction path.  -r)) in the gas phase originally shown in our previous paper. In each drawing the two carbon atoms participating in the reaction are connected by dotted or full line and the distance between them is shown in nm. ∆H r and ∆G r , the relative enthalpy and free energy, respectively, are expressed in kJ·mol −1 .
The changes in the ∆H r and ∆G r values of system(dim-r) and (mon-r) are schematically shown in Figures 6 and 7, respectively. These figures clearly show that although ∆H r of the transition state of system(dim-r) related to (HStLi) 2 is low compared to that of system(mon-r) because of no dissociation in (HStLi) 2 , the ∆G r value of the former is higher than that of the latter by 5 kJ·mol −1 , indicating that the route through the latter (system(mon-r)) is the predominant reaction path.
The changes in the ∆Hr and ∆Gr values of system(dim-r) and (mon-r) are schematically shown in Figure 6 and 7, respectively. These figures clearly show that although ∆Hr of the transition state of system(dim-r) related to (HStLi)2 is low compared to that of system(mon-r) because of no dissociation in (HStLi)2, the ∆Gr value of the former is higher than that of the latter by 5 kJ·mol −1 , indicating that the route through the latter (system(mon-r)) is the predominant reaction path. Figure 6. Enthalpy changes in the gas phase for the addition of styrene to (HStLi)2(1-c) (system(dimr)) and HStLi (system(mon-r)).

Figure 6.
Enthalpy changes in the gas phase for the addition of styrene to (HStLi) 2 (1-c) (system(dim-r)) and HStLi (system(mon-r)).  Changes in the free energy in the gas phase for the addition of styrene to (HStLi)2(1-c) (system(dim-r)) and HStLi (system(mon-r)).

Addition of Styrene to (HStLi)2 in Cyclohexane
Anionic polymerization of styrene is generally performed in polar or non-polar solvents. SBR (styrene-butadiene rubber) and styrene-butadiene block copolymers have been produced in nonpolar solvents at an industrial scale. Therefore, it is important to study the propagation reaction of anionic polymerization of styrene in non-polar solvents.
The transition states of St/[(HStLi)2(1-c)] in cyclohexane were optimized using the IEFPCM method of PCM; the transition state with the lowest ∆Grch, which corresponds to 3-c in the gas phase and will be called the transition state of system(dim-r) in cyclohexane hereafter, is shown in Figure 8  as 8-a, along with 8-b that shows another view of 8-a from a different aspect. Structure of 8-a is nearly the same as that of 3-c (the C−C bond lengths of 8-a are essentially the same, while the C-Li bonds Figure 7. Changes in the free energy in the gas phase for the addition of styrene to (HStLi) 2 (1-c) (system(dim-r)) and HStLi (system(mon-r)).

Addition of Styrene to (HStLi) 2 in Cyclohexane
Anionic polymerization of styrene is generally performed in polar or non-polar solvents. SBR (styrene-butadiene rubber) and styrene-butadiene block copolymers have been produced in non-polar solvents at an industrial scale. Therefore, it is important to study the propagation reaction of anionic polymerization of styrene in non-polar solvents.
The transition states of St/[(HStLi) 2 (1-c)] in cyclohexane were optimized using the IEFPCM method of PCM; the transition state with the lowest ∆G rch , which corresponds to 3-c in the gas phase and will be called the transition state of system(dim-r) in cyclohexane hereafter, is shown in Figure 8 as 8-a, along with 8-b that shows another view of 8-a from a different aspect. Structure of 8-a is nearly the same as that of 3-c (the C−C bond lengths of 8-a are essentially the same, while the C-Li bonds are larger by a small proportion (up to 0.005 nm), and the Li−Li distance and distance between the two carbon atoms participating in the reaction are almost the same). The ∆H rch and ∆G rch values for 8-a are also shown in Table 3 together with those for St/HStLi (system(mon-r)) that was originally described in our previous paper. A tendency similar to that observed in the gas phase was observed in cyclohexane with respect to the ∆H rch and ∆G rch values. The ∆H rch value of the transition state of system(dim-r) was 28 kJ·mol −1 , which was much lower than that of system(mon-r), 51 kJ·mol −1 , owing to no preliminary dissociation of the dimeric species in system(dim-r). However, ∆G rch for system(dim-r) is 93 kJ·mol −1 , and higher than that for system(mon-r), 86 kJ·mol −1 , by 7 kJ·mol −1 .  Table 3. Relative enthalpies and free energies, respectively, in the gas phase and cyclohexane for the transition states of system(dim-r) and system(mon-r).

In the Gas Phase
Usually, high molecular weight polymers are produced using small amounts of initiator, as can be seen in Table 4, that shows conditions of the experiments preformed to obtain the apparent activation energies for the anionic polymerization of styrene as discussed in our previous study. At an initiator concentration of 10 -3 mol·l -1 which is approximately the average concentration for these experiments, the effect of initiator concentration is [Init] 1/2 /[Init] = 33, and Rm/Rd becomes much larger.   Table 3. Relative enthalpies and free energies, respectively, in the gas phase and cyclohexane for the transition states of system(dim-r) and system(mon-r). (1) The rate constant of system(mon-r) is larger than that of system(dim-r). (2) Usually, high molecular weight polymers are produced using small amounts of initiator, as can be seen in Table 4, that shows conditions of the experiments preformed to obtain the apparent activation energies for the anionic polymerization of styrene as discussed in our previous study.

Item
At an initiator concentration of 10 −3 mol·L −1 which is approximately the average concentration for these experiments, the effect of initiator concentration is [Init] 1/2 /[Init] = 33, and Rm/Rd becomes much larger. Table 4. Experimental conditions and results of the experiments performed to determine the apparent activation energies for the anionic polymerization of styrene (this was also discussed in our previous paper.). Thus, the advantage of the rate constant for the anionic polymerization of non-associated polystyryllithium over dimeric polystyryllithium has been proved, as shown in above (point (1), previous paragraph). Further, to obtain high molecular weight polymers which is often the desired outcome, the reactions are conducted at low catalyst concentrations; under these conditions, the difference in the rate of reaction becomes larger (point (2), previous paragraph).

Solvent
Some researchers, especially Fetters et al. [8][9][10][11][12][13] and Watanabe et al. [14], insist that polystyryllithium aggregates higher than dimeric aggregates coexist in the system and that the dimeric and/or higher aggregates participate in the polymerization reaction. In their investigations the existence of small amounts of higher aggregates was demonstrated using light and neutron scattering measurements; in addition, they studied the polymerization of butadiene with freeze-dried polystyryllithium. However, there is no decisive evidence for the advantages of polymerization of dimeric or higher species over that of non-associated species. Frischknecht et al. [32] reported the calculation result that star-like micelles and cylindrical micelles coexist in a polymeric system with butadienyllithium headgroups, based on the experimental results performed by Stellbrink et al. [33]. Calculations were carried out using the classical dipoles of point charges and they neglected the semi-empirical and DFT results of the binding energies shown in Figure 7 to 10 in their paper [32] due to inconsistency with the above Stellbrink et al.'s results, although these semi-empirical and DFT results agreed well with the generally accepted mechanism of the anionic polymerization of styrene [1][2][3][4][5].
Our study, including the results described in our previous study, shows that (1) Although the polymerization of styrene with non-associated polystyryllitnium requires the dissociation of dimeric polystyryllithium for the reaction, its true activation energy for the polymerization reaction is small. Therefore, the polymerization of non-associated polystyryllithium is very rapid. Especially at low catalyst concentrations where high molecular weight polymers are usually obtained, propagation reaction is very powerful. (2) Dimeric polystyryllithium can polymerize styrene. However, it is not as reactive as non-associated polystyryllithium, although its relative enthalpy is lower because there occurrs no preliminary dissociation in the dimeric species.
A reconstruction of their reports, taking our results also into account, is therefore recommended.

Conclusions
In the case of the anionic polymerization of styrene in non-polar solvents, it is generally accepted that polystyryllitium is mainly associated into dimeric species and only a small amount of non-associated polystyryllithium species can propagate. However, the possibility of the reaction of dimeric polystyryllithium and higher aggregates was proposed by some researchers based on experimental data such as the addition of butadiene to freeze-dried polystyryllithium and the existence of higher aggregates, which was demonstrated using high-performance analytical techniques.
In our previous study, the anionic polymerization of styrene with non-associated polystyryllithium in the gas phase and cyclohexane was studied using M062X/6-31+G(d), a DFT calculation method. It was shown that polystyryllithium mainly associated into dimeric species and a small amount of non-associated species reacted with styrene; its relative enthalpy of transition state in cyclohexane agreed with the apparent activation energies experimentally observed by Worsfold et al. and Ohlinger et al. Further, the most stable transition state was found to be the one with a new structure and the reason for the penultimate unit effect (slower addition of styrene to polystyryllithium with two or more styrene units when compared to that with one styrene unit) was described.
In this study, the polymerization of styrene with dimeric polystyryllithium was optimized in essentially the same manner in which styrene polymerization with non-associated polystyryllithium was optimized in our previous study and the following results were obtained. The most stable transition state of St/(HStLi) 2 in cyclohexane has a structure in which the side chains of styrene and two HStLi are situated in the same direction around Li near the reactive site (structure 8-a and 8-b). Comparing this transition state with the most stable transition state for the reaction of the non-associated polystyryllithium in cyclohexane, it was found that the relative enthalpy for the reaction of the dimeric species was 28 kJ·mol −1 , which is much lower than that of non-associated polystyryllithium, 51 kJ·mol −1 ; this result is attributed to no preliminary dissociation of dimeric polystyryllithium. However, the relative free energy of the transition state for the reaction of the dimeric polystyryllithium was 93 kJ·mol −1 , higher than that of non-associated polystyryllithium by 7 kJ·mol −1 . The reaction rate for this reaction, R, is expressed as k[St][Init] 1/n , where k is a rate constant expressed as e −Grch/RT and [St] and [Init] are the concentrations of styrene and the initiator, respectively, n is 1 for the reaction involving dimeric polystyryllithium and 2 for the reaction involving non-associated polystyryllithium. Therefore, these results demonstrate the advantage of a higher rate constant for the polymerization of styrene with non-associated polystyryllithium when compared to that with dimeric polystyryllithium (k m /k d = 21). At low initiator concentrations where high molecular weight polymers are usually obtained, the effect of initiator concentration can be described using the equation [init-m] 1/2 /[init-d] = 33 at an initiator concentration of 10 −3 mol·L −1 and the difference becomes much larger (R m /R d = (k m /k d ) ([init-m] 1/2 /[init-d]) = 690).
As described in the preceding section, some researchers proposed that the reaction involving dimeric and/or higher aggregates is highly reactive. However, there is no decisive evidence available on this point. In this study, it was demonstrated that dimeric polystyryllithium can react with styrene, but its reactivity is not as high as that of non-associated polystyryllithium, especially at low initiator concentrations where high molecular weight polymers are generally obtained.   Table A1. Calculated electronic energies, zero-point(ZP)-corrected electronic energies, enthalpies (H) and Gibbs free energies (G) at 25 • C in the gas phase for dimeric (HStLi) 2 , intermediates and products of their reaction with styrene, and intermediates and products of the reaction of non-associated HStLi with styrene.