Response to Comment on the paper "Restrcted Geometry Optmzaton: A Dfferent Way to Estmate Stablzaton Energes for Aromatc Molecules of Varous Types" Zhong-Heng Yu* and Peng Bao Supportng nformaton. Contents: pages. σ-π Energy Partton ~4. The Physcal Meanng of Destablzng Energy Dfferences E An 4~6. The Fundamental Dfference, n the Way to Change Nuclear Repulson, between Benzene and Hexatrene..6~7
. σ-π Energy Partton Full delocalzed State (FUD) of the GL Geometry E FUD (GL)= -.97 6 4 5. E V (GL) = 7.6 E V-π = -4.4 E V-σ = 50.0 ().5 4.447 5 =.6874 π = 0.8 σ =.4004 =.645 π = 0.707 σ =.4049 6 π Drvng force E N = -.64006 E = -0.00 (-0.8 ) E N π = -0.4090 E N σ = -.996 (energy unt n hartree) E N = -.64006 E = 0.000 (0.8 ) E N π = -0.4090 E N σ = -.996 (energy unt n hartree) σ E π = -0.0059 E σ = 0.007 E -π = -0.008 E -σ = 0.005 Ground state of hexatrene E(G) = -.9855 E N = 9.64 E e = -47.04087.4 E V (GL) = 6.0 E V-π = -4.6 E V-σ = 47.6 () the GL Geometry Full localzed State (DSI) E(GL) = -.4095 Drvng force of the G Geometry DSI: delocalzed sgma state.5.450 E DSI (G) = -.40805 E N = 9.64 E e = -47.0508 Fgure. The ground state of,,5-hexatrene, and the restrctedly optmzed GL geometres. Molecular energes E, the nuclear repulson E N, the total electronc energes E e, and the π and σ components of the molecular energy dfferences (Vertcal resonance energy) E V at the BLYP/6-G*. Where, the energes of the FUD state were obtaned from the sngle pont energy calculaton for the GL geometry, the energes of the DSI state were obtaned from the restrcted sngle pont energy calculaton for molecule hexatrene. E FUD (GL) = -.84 Ground state of benzene Full delocalzed State σ E(G) = -.4866.4 (FUD) E N = 0.87 of the GL Geometry Drvng force D h E e = -45.4598 = 0.7904 E N = -0.80096 E = -0.005 (-6.6 ) D 6h E π e = 0.4764 E π N = -0.48 E π = 0.06 E σ e = 0.6479 E σ N = -0.67667 E σ = -0.088 (energy unt n hartree).449.97 E V (GL) = -4. E V-π (GL) = -66.5 E V-σ (GL) = 6. E V-π (G) = -5.9 E V-σ (G) = 4.7 E V (G) = -..449.4 D h the GL Geometry E(GL) = -.44 (DSI: delocalzed sgma system) π Full localzed State Drvng force (DSI) of benzene = 0.8758 E π e = 0.584 E π N = -0.48 E σ e = 0.559 E σ N = -0.67667 (energy unt n hartree) E N = -0.80096 E = 0.066 (0.4 ) E π = 0.96 E σ = -0.74 D 6h.97 E DSI (G) = -.48
Fgure. The ground state of benzene, and the restrctedly optmzed GL geometres. Molecular energes E, the nuclear repulson E N, the total electronc energes E e, and the π and σ components of the molecular energy dfferences (Vertcal resonance energy) E V at the BLYP/6-G* level. Where, the energes of the FUD state were obtaned from the sngle pont energy calculaton for the GL geometry, the energes of the DSI state were obtaned from the restrcted sngle pont energy calculaton for molecule hexatrene. ============================================================== The followng resonance energes are from lterature Table. The Table was quoted from the reference (van Lenthe, J. H.; Haventh, R. W. A.; Djkstra, F.; Jenneskens, L.W. Chem. Phys. Lett. 00, 6, 0). In Table, three vertcal resonance energes (VRE) for benzene, as well as three theoretcal resonance energes (TRE), arose from the three dfferent ways to optmze D 6h and D h geometres usng VB program. One of three values of VRE s 5., and the correspondng TRE s.. These values for both the VRE and TRE are consderably lower than most prevously reported values (range 5 to 95 ) ((a) Mo, Y.; Wu, W.; Zhang, Q. J. Phys. Chem. 994, 98, 0048.; (b) Janoschek, R. J. Mol. Struct. (Theochem.). 99, 9, 97.; (c) Bernard, F.; Celan, P.; Olvucc, M.; Robb, M. A.;Suzz-Vall, G. J. Am. Chem. Soc. 995, 7, 05.; H. Kollmar, J. Am. Chem. Soc. 0 (979) 48.; (d) Shak, S.; Shurk, A.; Danovch, D.; Hberty, P. C. Chem. Rev. 00, 0, 50.) In our method, as shown by the data n Fgure, VRE for benzene s., and TRE, the energy dfference [E(G) E(GL)], s -0.8. -------------------------------------------------------------------------
The Table was quoted from ref. (Yrong Mo, Y.; Schleyer, P. von R.; Chem. Eur. J. 006,, 009). In adddton, 74. value of VRE was reported by Mo s paper n 994 (Mo, Y.; WU, W.; Zhang, Q. J. Phys. Chem. 994, 98, 0048). ------------------------------------------------------------ ==============================================================. The Physcal Meanng of Destablzng Energy Dfferences E An. The energy dfference E An between the GE-m and GL geometres can be consdered as the energy effect assocated wth the local resonance nteracton between two double bonds n the GL geometry. Accordng to the classcal vewpont, resonance nteracton should be stablzaton, and the sngle bond r v,s between two nteractng double bonds C(u)=C(v) and C(s)=C(t) should be shortened. However, as shown by Fgure, such energy effect s always destablzng, and the correspondng sngle bond r v,s s lengthened due to the local π orbtal nteractons. 4
(f) Step I π sub-fock matrx f for GL geometry f π A f π B Set Equal to zero Restrcted geometry optmzaton E o (CH) = -77.009 hartree (BSSE) for each - CH=CH - fragment Set Equal to zero Frst way to form GE- (j) f π C LD state of GL geometry.449 E LD (GL) = -.5 hartree π sub-fock matrx f for GE- geometry f π AB Set Equal to zero.449 A + B C Restrcted sngle pont calculaton Set Equal to zero f π C Step II GL (D H ) 4 B 5 C A 6.4.4 (g) E T (GL) = -.4 hartree. Restrcted geometry optmzaton () Step III + + Far away from each other Second way to form GE- (b) π sub-fock matrx f for the LD state f π AB Set Equal to zero E T (GE) = -.64 hartree (k).474.9 A + B.097.45 E H = -0.088, E two = 0.899 C Set Equal to zero = E H + E two E T (CH) =*E o (CH) = -.606 hartree for reactant sysytems E T =E T (GE) - ET (CH) = -84.8 E H = -67.500, E two =.507 = -.98994, E N =.7677 E T =E T (GE) - ELD (GL) = -0.7 E H =.98064, E two = -0.97986 =.00075, E N = -.0086 GE- (C v ) E T,, E H, E two and E N are the dfferences n molecular energy, total electronc enenrgy, one electron energy,.9 f π C E T = E LD ( GL) - ET (GL) = 0. = 0.06, E N = 0.00000 Restrcted geometry optmzaton E A =E T (GE) - E T (GL) = 9.4 E A H =.87774, E A two = -0.86087 A =.0686, E N A = -.0086 E T = E T (GL) - ET (CH) = -94. E H = -69.7874, E two = 4.794 = -5.00680, E N = 4.786 E T = + E N two electron energy and nuclear repulson, resctvely. The unts of all the enenrgy dfferences, except for E T, are n hartree. (h) 8 o (d) (c) (e) E A-π = -58.0 E A-σ = 67.4 (a) Fgure. A Fcttous thermodynamc cycle for the formaton of the GE- geometry of benzene; ( a, b, c, d and e) Varous energes dfferences at BLYP/6-G*; (f and h) the settngs for the restrcted optmzaton of the GL and GE- geometres; (g) the settng for the sngle pont energy calculaton for the GL geometry; (, j and k) the optmzed geometres GL and GE-, where the thn lnes mean that the π systems were artfcally localzed on ther respectve double bonds and thck lnes n the GE- geometry and n the LD state of the GL geometry mean that one of the π systems was artfcally delocalzed on the group C=C-C=C4.. In order to understand the physcal meanng of the destablzng energy dfferences E An, molecular energes, denoted as E T (GE), E T (GL), for the GE- and GL geometry of benzene, as well as the sum E T (CH) of molecular energes for three fragments CH=CH, were parttoned nto total electron energy E e and nuclear repulson E N, where E e s the sum of one electron energy E H and two electron energy E two. The molecular energy for each fragment CH=CH was obtaned from geometry optmzaton usng unrestrcted BLYP/6-G* calculaton. Afterward, ths molecular 5
energy was corrected for the bass set superposton error (BSSE). As shown by the practcal calculatons for the CH=CH- fragment, the unrestrcted BLYP calculaton can ensure that each molecular orbtal has a correct electron occupancy, and t can also guarantee that the π and σ molecular orbtals are, thoroughly, separated out. In Fgure, there are two ways to form the GE- geometry, whch forms a thermodynamc cycle for the formaton of the GE- geometry. The frst way s a mult-step procedure, and t ncludes the steps I, II and III whch are denoted by the thck lnes wth arrowhead at the left sde of the Fgure. In ths way, the GL geometry was dealt as an ntermedate (a facttous ntermedate) between the reactant systems (the three CH=CH fragments) and the GE- geometry. As shown by Fgure a, the molecular energy dfference between the reactant systems and GL geometry s -94., and t resulted from the nteractons between the three fragments ( Fgure f). Of all the components of ths energy dfference, the absolute value of the one electron energy dfference E H (-69.78740 hartree) s the greatest, and E H + E two > E N, ndcatng that n the GL geometry, the bond energy (-94. ) between three CH=CH fragments manly resulted from state electronc nteractons between the dfferent fragments. In order to search for the drvng force for dstortng the GL geometry to the GE- geometry, t s necessary to construct a LD ( locally delocalzed ) electronc state of the GL geometry. In the LD electronc state, as shown by the thck lnes n Fgure j, the π-electrons, orgnally localzed on two dfferent fragments C()=C() and C()=C(4) (Fgure ), become delocalzatng on the C()=C() C()=C(4) group, and meanwhle the molecular geometry (GL) was kept unchanged. In the second step of the frst way, as shown by the π sub-fock matrces n Fgure g and h, the sngle pont energy calculaton for the GL geometry (restrcted sngle pont energy calculaton) was performed under the condtons same as those used to obtan the GE- geometry, and t provded the GL geometry wth a LD electronc state. If the delocalzaton of π-electrons was so fast that the structure of the GL geometry was kept unchanged at the moment when delocalzaton of the π-electrons was fnshed. In ths case, as shown by the energy dfferences between the DL electronc state and GL geometry (Fgure b), two electron energy dfference [E DL two(gl) - E two (GL)] (0.899 hartree) s destablzng, and t s greater n the absolute value than the one electron energy dfference E H (-.088 hartree ), leadng to E T = [E LD (GL) - E T (GL)] ( 0. kcal) > 0. Therefore, the electron repulson ( E two > 0) s a drvng force for dstortng the GL geometry toward the GE- geometry. As a result, the bond length r, was lengthened from.4 Å n the GL geometry to.474 Å n the GE- geometry, and meanwhle the nuclear repulson decreased from 0.9944 hartree for the GL geometry to 0.9896 hartree for the GE- geometry. At last, the GE- geometry was formed, and the molecular energy dfference, [E T (GE-) E T (GL)] = 9.4 ( Fgure d). Emphatcally, the molecular energy dfference [E T (GE-) E T (GL)] s only.4 % of the molecular energy dfference ( -94. ) between the reactant systems and GL geometry, and t s so small that n the GL and GE- geometry, the lengths of the sngle bond C C between the double bonds C=C and C=C4 are both shorter than 6
the length (.54 Å) of a standard Carbon-carbon sngle bond although the length (.474 Å) n the GE- geometry s longer than that (.449 Å) n the GL geometry.. The Dfference, n the Way to Change Nuclear Repulson, between Benzene and Hexatrene. In order to search for potental correlaton between energetc and geometrcal crtera, the dfference n the way to change n the repulson energy between benzene and hexatrene s compared The nuclear repulson energy, E nu, between the bonded carbon atoms can be wrtten as equaton (): E nu n m nu nu r = E + E = [( q / R ) + ( q / )] () where q s the nuclear charges of carbon atom, and R and r are the lengths of the formal sngle and double bonds. In the case of the benzene molecule, as shown by comparson of the bond lengths n the G and GL geometres (Fgure ), d(r ) = [r (G) r (GL)] > 0 and d(r ) = [R (G) R (GL)] < 0. Accordngly, we have the followng frst and second order dervatves of the nuclear repulson energy when dr = dr = -dr : de = q nu / dr = ( E = ( r R nu )/( r / R ) + E R ) < 0 nu / r = q = [(/ R ) (/ r )] () de nu = q / dr when r = R n = / R + ( ( R + r )/( R r ) > 0 de nu / dr = 0 E nu E nu / r = q [(/ R ) + (/ r )] () Therefore, the nuclear repulson energy s mnmum when r = R On the other hand, as shown by Fgure, dr > 0, dr > 0, and dr > dr > 0 for hexatrene. We have: / / / [ [(/ ) (/ denu dr = Enu R + Enu r = q R + r )] < 0 (4) = In the case of hexatrene, the nuclear repulson monotoncally decreases as the bond length alternaton decreases, and the frst order dervatve for hexatrene s greater, n the absolute value, than that for benzene. At BLYP/6-G*, for example, the decrease E N = [E N (G) - E N FUD (GL)] ( 0.80096 hartree) n the nuclear repulson of the benzene s 0.5 tme as great as the decrease (.64006 hartree) n that of hexatrene, and meanwhle the gan E = [E(G) - E FUD (GL)] (-0.005 hartree = -6.6 ) n the molecular energy of benzene s about eght tmes of the gan ( -0.00 hartree = -0.8 ) n that of hexatrene. Correspondngly, the energy dfferences E A are -0.8 = ` 7
(benzene) and 6.8 (hextrene), where E A can also be wrtten as E A = E V (GL) + [E(G) - E FUD (GL)] (Fgure ) from the thermodynamc vewpont. It seems reasonable to say that aromatcty of benzene can be partly ascrbed to the ablty of the sx-membered to gan the extra stablzaton energy ( -9.0 ) va the way to mnmze the nuclear repulson energy. 8