The Origin of Aromaticity

Transcription

1 The Orgn of Aromatty Ths problem s one of the hstores of organ hemstry. Many researhers have propose ther own unerstanng methos amng at soluton. owever, t s not hear that the problem of the orgn why aromat property exsts was solve. In fat, f the bas onepts of quantum mehans are apple, omparatvely easly, almost all problems are unerstanable an thus seem to be alreay solve. But those eas have not arrve to organ hemsts. I wll explan as ntellgbly as possble.. Funamental Problem of Aromat Property Those who are famlar wth aromatty may skp ths part. Benzene In orer to larfy the man problems of aromatty, the features of benzene an ylobutaene are taken as typal examples of aromatty an antaromatty. Benzene s the most typal example of N systems whh have aromat propertes. Fgures an show ükel π-moleular orbtals wth energy-levels of benzene an referental hexatrene, respetvely. Fg.. The ükel moleular orbtals MOs an energes of benzene. Whte rles express the phase of, a blak ones express the phase of -, an a sze expresses the sze of the absolute value of the oeffent. The oupe orbtals of benzene are,, an an every par of π eletrons oupes them.

2 Clearly, benzene s hemally an thermoynamally stable ompare wth the referental han-lke lnear unsaturate hyroarbon: π eletron energy of benzene s 8. Let s look at orresponng hexatrene.,, an n hexatrene are oupe orbtals. The energy of beomes hgh about -0. above that of benzene. The energy of π eletrons s.99, an that s hgher than benzene by -.0 an are negatve onstants. Although the energy sum of of benzene an s, that of an of hexatrene s.9, an the energy of benzene beomes low about 0.. In aton to the lower, other energy-levels of other oupe orbtals n benzene are lower than those of hexatrene, the π eletron energy n benzene beomes very low as a whole Fg.. ükel MOs of hexatrene. Cylobutaene Let's see ylobutaene whh s a typal example of N systems whh have antaromat propertes. Fgures an show the moleular orbtals an ther energy-levels n ylobutaene an butaene, respetvely. Exept the lowest oupe orbtal, the orbtals are oubly egenerate n the yl system Fg.. The hghest oupe moleular orbtals OMOs are, an. Sne eletrons oupy spersely, an trplet state.e. braal, the stablzaton energy by the hemal bonng oes not take plae. The ase that two eletrons oupy or snglet state s onsere; there s an energy fall by a hemal bon. But s shoul be ponte out that sne the bon stane between the bonng arbons s shortene,.e., the geometry s hange, ths s not the same system as

3 ylobutaene. When two eletrons go nto the oubly egenerate orbtals the energy of the trplet state s generally lower. But sne there s an energy fall by a hemal bon n the snglet state, t annot be easly etermne whh s thermoynamally more stable as the groun state. Moreover, sne t s easy to beome the trplet state braal even f t takes the snglet state, hemal atvty seems to be hghly "lable" for hemsts. Fg.. The MOs of ylobutaene. See that the lowest an the hghest orbtals are not egenerate an the other orbtals are oubly egenerate. The energy when an eletron oupes the -th orbtal of MO n ylobutaene s gven by the followng formula. sr rs Q ere, s the ükel amltonan. Well, aorng to the supposton of the MO metho,, an other rs are zero. So,

4 s obtane. If the oeffents of or are substtute, the oeffent of wll be 0 an unerstanably, the stablzaton by bon formaton wll not be gven. Those orbtals are nonbonng ones whh have nether ontrbuton nor rebounng n the π bon formaton an those energy levels are set to. The lowest MO energy of ylobutaene s. So, t omes out an 0.8 beomes lower ompare to that of butaene. In the ase of butaene, sne the movement range of the π eletrons s restrte wthn that from C to C, the energy goes up that muh ue to the unertanty prnple. or of ylobutaene are egenerate an have the energy value of. Contrary to ths, n ase of butaene sne regons between C an C an between C an C of are bonng wth energy of 0.8, 0.8 lower ompare to that of ylobutaene. Cylobutaene has the lowest energy level of, however, an have no ontrbuton to π bon formaton ausng a hgh π eletron energy Fg.. MO orbtals of butaene If ylobutaene s referre, t may turn out that the thermoynamal energy s unusually hgh as t may ome from the mage of the wor of an ant-aromat seres. But, t may not bet so. The MO energy s an ths s equvalent to the π eletron energy of two ethylene moleules. That s, t s the level that two ethylene unts o not only onjugate the π energy of ethylene s per pee. owever, sne the n an r π MOs are egenerate, ylobutaene beomes trplet state braal ausng hemal reatvty very hgh. Therefore, ylobutaene tens to gve hemsts

5 the mpresson of beng unusually unstable.. Problems Whh Shoul be Solve. The MO metho omputes the magntue of aromatty quanttatvely. Then, what oes the MO metho alulate?. The lowest an the hghest energy of MO of a rular onfguraton system o not egenerate, whle other orbtals oubly egenerate, an the reason why suh egeneraton s resolve n han struture.. The reason why the lowest energy of MO of a rular onfguraton takes a onstant value nepenently of the sze of the rular system, an why the hghest energy s set to be -.. Although t s onsere that t s the ause of aromat stablty that the lowest energy takes the steay value of n both N an N system, suh an effet oes not take plae n N system. What s the reason?. Are those problems an the answers generalzable?. The Conept of Stablty n Chemstry as Two Meanngs Cylobutaene s well known as very unstable. That a substane s "stable" means that the substane oes not hange. Namely, the state that a substane oes not hange s smply alle stable an f not so, unstable. owever, beng low n energy s also alle "stable". The thermoynamal low state of energy s alle stable thermoynamally stable, whereas a hgh state s alle unstable thermoynamally unstable: ths may ause onfuson. For example, the ompouns a an b whh are the somers of benzene hange wth heatng et. to a oes not neessarly result n through b. Although the total energy of b s lower than a, b hanges to easly. On the other han, although the total energy of a s hgher than b, hange to s not easy. If these fats are enote by a graph, t wll beome as t s shown n Fg.. a stable but thermoymamally unstable moleule b thermoymally stalbe but unstable moleule thermoynamally stable an stable moleule Reaton oornate Fg.. Moleular stablty an thermoynamal stablty

6 Among these three moleules, a s stable as a moleule sne t hemally s not easy to hange but thermoynamally unstable whereas b s thermoynamally more stable than a but thermoynamally unstable. an be sa to be "thermoynamally an also hemally stable. ere t shoul be realle that when argung about the stablty of aromat ompouns, only thermoynams stablty shoul be susse, beause, the problem of aromat property s relate wth one moleule, an f t nlues the hemal stable an unstable problem, the elements more than moleule-tself are nvolve. Agan, aromatty s a problem of a sngle moleule.. Stanar of Stablty Chemal stanar of aromatty may be lnear unsaturate hyroarbon wth the same number of ouble bons. In the ase of benzene, the referene ompoun s,, -hexatrene an for ylobutaene, t s butaene. When usng a moleular orbtal theory, n orer to make t strter, varous methos of efnng aromatty are propose. The methos may be effetve f one quanttatvely onsers the aromatty of an unknown aromat ompouns. But t may not be effetve to solve essental problems of aromatty. Namely, the problem here s why a fferene arses n the π eletron energy between the yl-onjugate an lnearly onjugate strutures n the same arbon number. In other wors, what we have to lear s to explan oneptually the fferene of the π eletron energy between the ases that the movement range of the π eletrons s lmte to be lnear an that s annular..what Does MO Calulate? Answer to the problem π Eletron energy of MO an explan aromatty quanttatvely. Ths means that MO wll lnk wth the eluaton of the ause of aromatty retly. Thus we nee to know what the MO metho alulates. The energy of MO s has alreay beome lear as t s "the knet energy of the π eletrons n the potentals average from nule an other eletrons." Refer to the followng URL for more nformaton. http: // Thus, t may be sa that the problem of aromat property s a problem of beng hgh or low of the knet energy of π eletrons.. Knet Energy of the Lowest-Level Eletrons n Annular Movement s n the State of 0 Answer to Problem The knet energy of the π eletrons n the lowest MO of any annular unsaturate hyroarbon serves as zero theoretally. Ths s beause n MO, t s assume that the potentals from nule an other eletrons are onstant. Moreover, sne knet energy s ertanly a postve value, 0 s the

7 lowest value. Generally, the lowest energy of MO of annular unsaturate hyroarbon s nepenently of the sze of annular system. Ths value orrespons to the zero knet energy. I wll explan somewhat n etal. Movement of a π eletron of a yl onjugate system turns nto rumferene movement n a lass mage. Generally the quantum mehanal energy of the lowest level n rumferene movement s ertanly zero. Ths s lear from the unertanty relaton: sne there s no restrant to rumferene movement, t turns out as qnfnty n the relatonshp, p q >h /, resultng n that the quantty of momentum tself approahes 0. Those who are aganst ths ea beause t s refute as " p0 s not that the quantty of momentum s 0 but the ambguty of quantty of momentum beng taken out" may refer to the omment of the unertanty relaton an energy shown below. Qorg/Tutoral/Elementary/Pf-/0.pf The followng omment whh explans the relatonshp between the regon of a partle movement an ts energy may also be useful. Qorg/Tutoral/Elementary/Pf-/0.pf What I want to emphasze s that the π eletrons n the lowest orbtal o not move n the annular onjugate system. Ths solves the rle why the lowest MO energy of annular unsaturate hyroarbon s rrespetve of the sze of annular system. In the ase of the even number atoms, there s no egeneraton also n the orbtal of the hghest energy level. In ths orbtal, the π eletrons are ompletely loalze on eah atom an t s thought that there s no rumferene movement sne the π eletrons are loalze, the knet energy beome large ue to the unertanty prnple. The energy when π eletrons loalze ompletely s -. See, s a negatve value. The resoluton for the orbtal of the hghest energy level omes out. It may be mentone that ths result s gven when the LCAO metho s use sne LCAO as a restrton on the soluton ths may not be worre muh. 7. General Conseraton: Meanng of Degeneraton an Non-Degeneraton Answer to the problem Let us onser generally the meanng of egeneraton an non-egeneraton of MOs n annular onjugate unsaturate systems. The MO energy of the j-th orbtal j s expresse as a funton of the number n of arbon atoms as Eq.. π j os j n n j 0, ±, ±, L, n j 0, ±, ±, L, ± for even n foro n The lowest energy level s j 0. Sne os of the rght-han se s set to whenever t substtutes 7

8 j 0 for Eq., t beomes. Ths s not relate to the number n of a rng. The orbtals other than the lowest an hghest energy-level n the ase of the even number are oubly egenerate. Sne they have two egrees of freeom, ths s a natural onsequene: lassally, beng lokwse movement an ounterlokwse movement. It shoul be realle that the same stuaton appears n the annular momentum n atom orbtals. The magnet quantum number m of an atom orbtal s a quantum number of rumferene movement about the z axs. Sne there s no rumferene movement when m s zero, a wave funton oes not egenerate takng a onstant therefore, knet energy s 0. Other orbtals take quantum number m as ±, ±,.. an the ± sgn orrespons to the same meanng of j n Eq.. ere, t must be mentone that an atual annular unsaturate hyroarbon system, sne nulear potental s not onstant, the knet energy of π eletrons n the lowest energy level oes not beome 0, but t was ponte out that the π eletron energy at that energy n benzene beomes low unusually f ompare wth those of other energy levels. 8. The Reason Why the Degenerate Orbtals n Annular Systems Resolve n Lnear Systems answer to the problem The MO metho obtans the knet energy an the orbtal of π eletrons by solvng the ükel Shrönger equaton base on the LCAO approxmaton. The Shrönger equaton of annular movement of eletrons an be solve analytally. If these results are ompare wth those of the MO metho, the reason why the egenerate orbtals n annular systems resolve n lnear systems may be unerstoo. The etals are as follows. O r ϕ m x Fg.. Crumferene movement by two-mensonal polar oornates. m s eletron mass, O s the enter of rumferene movement, r s the stane to the eletron, an ϕ s angle between x-axs an r. Let us onser the ase that an eletron moves rularly n a fxe potental U. To obtan the energy an orbtal of the eletron, the Shrönger equaton to be solve s, 8

9 h U 8π mr ϕ E The orbtal an ts energy E j s analytally gven as, j h E j U j 0, ±, ±, ± L ml' where L s the rumferene of the rng. The wave funtons are generally expresse as π exp±pϕ, but are transforme to more unerstanable forms, 0 for j 0, π j os jϕ for postve j, 7 π j sn jϕ for negatve j. 8 π From Eq., t turns out that the lowest orbtal j 0 s not egenerate n agreement wth the result of MO. The wavefunton s also a onstant an the eletrons n t have zero value n the knet energy. 7 Other orbtals are oubly egenerate an are expresse as os an sn funtons. They are n goo aorane wth the results of MO. The hange to lnear struture from a rular one s equvalent to nstallng an nfnte potental wall at zero or π. That s, n han struture, eletrons annot get out from the en of the han; ths means that there exsts a hgh potental wall. The wave of the partle must shrnk at 0 or π,.e., the wavefunton s neessarly zero there. If the wavefunton must take at 0 or π Fg. 7, the phase of one of energetally equvalent wavefuntons os funton nreases by one. The sgn wavefunton oes not hange. sn π ϕ 0 π π os π ϕ Cyl struture 9

10 l sn ϕ π sn π l ϕ 0 π π Chan struture Fg.7. Why the oubly egenerate states n yl system are resolve nto two energetally fferent states n lnear system. The lnear system orrespons to the state of the yl system n whh an mpenetrable barrer s plae at 0 or π. In ths stuaton the wave funton must shrnk to zero at 0 an π, formng a snusoal funton wth a hgher quantum number. Sne phase hange s ang restrton to the range of eletron's exstene, the energy of the wave funton beomes hgh. Ths s the reason why egenerate wavefuntons n a rng struture are resolve n a han struture. 9. Conernng the Infnte Lnearly Conjugate Systems Answer to Problems an Let's see the ase of the nfnte lnear unsaturate hyroarbon for omparson. The orbtal energy j of level j n the lnear unsaturate hyroarbon whh onssts of n numbers of arbon atoms s gven by Eq. 9 be autous of j begnnng from. π os j j,, L n j n, 9 The energy value of the lowest energy level j s hgher than. If n s nrease, the energy approahes to. Ths s beause the movement range of π eletrons beomes large, an s the result of the unertanty prnple also agreeng wth the prnple. That s, the orbtal energy of the lowest energy level of rumferene movement s n agreement wth the orbtal energy whh oes not have restrton n the movement range. Ths means that there s no restrant n movement of the π eletrons n the lowest-energy orbtal of a yl onjugaton system. 0. Eletron Confguratons Answers to Problems an 0

11 The energy of the j-th orbtal n annular or lnear unsaturate hyroarbons s expresse by Eq. or Eq. 9, respetvely. The number of π eletrons s n the same as N. N system J-value for OMO s If j±n s substtute n Eq., π homo os ± N N N N an the number of the rle s N. ere, j nreases as 0,±,±. 0 s obtane. OMOs serves as nonbonng orbtals whh are egenerate, an sne the oeffent of s 0, two eletrons n these orbtals have no ontrbuton to the hemal bon. Therefore, the π-eletron energy beomes hgh ompare wth the lnear unsaturate ompoun of the same arbon number as shown next. N Contrarly to ths, j-value for OMO n the lnear struture s N. The number of the han s N. It shoul be note that j starts as,,, If jn an nn are substtute n Eq.9, N π homo os N s obtane. The value of the os funton takes some value more than 0 an less than ; the oeffent of s not zero. Namely, the OMO s a bonng orbtal. N system J-value for OMO s gve, N N an the number of the rle s N. They are substtute n Eq. to π Nπ homo os ± N os N N Sne the os funton never gves 0, the oubly egenerate OMOs are bonng ausng the π eletron energy low. Eletrons oupy the OMOs lowerng the π eletron energy n aton to oupyng the lowest orbtal wth. Ths lowers the energy of the yl system more than orresponng lnear system. If N an nn are substtute n Eq. 9 for lnear system, N π homo os N s obtane. Ths orbtal also s bonng one. If N N an N N are ompare, the value of the former s always less than the latter. So, The π eletron energy of the annular system s always

12 lower than that of the lnear system. obey obtane the N/N rule mathematally. 8 The physal meanngs for hs results were gven. Those who are ntereste n may refer to the lteratures alreay gven.. What s Aromatty The knet energy of the lowest π moleular orbtal of a yl onjugaton system s n the state of 0. Ths s the man ause of the stablty of an annular unsaturate system. owever, OMOs n the N system, are nevtably nonbonng an oubly egenerate where two eletrons oupy them spersely ausng the state of braal. Sne they are nonbonng orbtals, eletrons n OMOs o not ontrbute to the energy fall of π eletrons, makng the energy of the system hgh ompare to the referene system. Moreover, sne the moleule has raal harater, hemal reatvty beomes hgh. Ths s the reason why N systems are thermoynamally an hemally unstable. On the other han, the ause of the aromat stablty of N systems s that there s an orbtal of the zero knet energy ue to yl onjugaton. Contrary to N systems, OMOs of N systems are bonng orbtals. Although they are oubly egenerate, four eletrons may oupy them wth parng an ontrbutng to energy stablzaton of π eletrons. If those mentone above are mae smpler, t turns out as the aromat property of an unsaturate hyroarbon of rular onfguraton s brought forth by the zero knet energy n the lowest π eletron orbtal. But n N systems, two eletrons oupy nonbonng OMO orbtals proung braal harater as well as no bnng energy. Ths anels the effet of the zero knet energy. Referenes. Mmkn, V. I.; Glukhovtsev, M. N.; Smkn, Ya. B., Aromatty an Antaromatty, John Wley & Sons, In., New York, 99.. When the number of a rng s N N,,.. n the yl onjugate ompouns, the ompouns are hemally more stable than those of lnear wth the same number of unsaturate bons an are alle to be aromat, whereas the ompouns of the seres of N s alle antaromat beause of beng unstable. Ths rule s the ükel rule or the N/N rule. Ths s one of the most mportant onepts n organ hemstry.. There are many books on MO. We reommen you the followng URL explane nlung the physal meanng of MO urrently n Japanese. Qorg/Tutoral/Elementary/EO-.html. In hemstry, even f a moleule has some strutural hanges, t s the same moleule, but when applyng a moleular orbtal metho, the moleule whh hanges struturally must be onsere as a fferent system. In ths ase, ylobutaene wth regular tetragon D h

13 struture an that of retangle D h struture are fferent moleules n quantum hemstry.. Emerson, G. F.; Watts, L.; Pettt, R., J. Am. Chem. So. 9, 87,.. Ihkawa,.; Sakata, K., Int. J. Quam. Chem., 00, 87, The reason why the knet energy of the wavefunton of a onstant s zero: f the wave-funton r,ϕk onstant s apple to the general metho of obtanng the expetaton value n quantum mehans, * h * E r, ϕ K U K K KU 8π mr ϕ s obtane to show that the ontrbuton of the fferental par s null. 8. obey, W. D. J. Org. Chem.97, 7, 7.