Fig. 1. An example of resonance.

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1 Why Does Resonance Take Place? Conugaton and/or resonance are the most mportant and useful concepts n organc chemstry. But have you ever consdered the cause? The cause of resonance s not wrtten n the textbooks of organc chemstry at all and I thnk that t s a most complan for organc chemsts that they are unable to understand what s wrtten n the books of quantum chemstry. I would lke to explan ths problem here as planly as possble. But there s no tellng whether t s more ntellgble to read t snce I (tanuk ) ) myself thnk that t s freely ntellgble. If are unable to understand, please ask tanuk by e-mal. Although t may not be necessary to explan anew, resonance or conugaton s the followng phenomenon. A B A C D Fg.. An example of resonance. For example, n butadene (A), resonance s a phenomenon n whch a part of π electrons of the double bonds flows nto sngle bond (B). The textbook of organc chemstry explans t as the state where the electronc structures of A, B, and C are put together. Although ths explanaton borrows the results of the valence bond method (valence bond theory: VB), one theory of the quantum chemstry. A smple queston s the reason why such a phenomenon necessarly needs to happen. The textbooks of quantum chemstry explan t n the way merely that the energy s lowered and do not menton why t falls. NH A part of lone-par electrons of N n anlne flows nto the benzene rng (thus, the bascty of the amno group s decreased). Look, the electrons move from N wth a large electronegatvty to small C. It s clear nconsstency of the theory of organc chemstry. Although there s explanaton by conugated structures, snce the problem here s why conugaton happens, then, ths s not an answer.

2 If a concluson s sad prevously, the cause of resonance and/or conugaton orgnates n the most fundamental expresson of nature called the Hesenberg's uncertanty relaton. Only now, snce you probably do not understand, the route whch results n ths concluson s explaned systematcally.. When an electron s shut up n a narrow space, the electron has bg energy. It s a fact of nature although one may thnk what that s at all. If thngs are consdered classcally (n macroscopc way), such a fact s a very unacceptable phenomenon. In our world (macroscopc world), even f an obect s put nto a bg box or put nto a small box, the energy whch the obect has s same. However, these well-known deas n the macroscopc world sometmes completely dffer n the mcroscopc world. An electron confned n a narrow space has bg knetc energy. In other words, t would be sad that the electron moves volently. On the contrary, f restrcton of movement s not added, the knetc energy of the electronc s 0. Such an electron does not move. Ths surprsng phenomenon s known by the followng dscusson. When an electron s shut up n the cube of L, the behavor of the electron follows Eq. (Schrödnger equaton). (How to solve ths equaton s not explaned here.) In Eq., are an operator called Laplacan and m s the rest mass of electron and h s a constant called Planck's constant. E s knetc energy and ψ s the wavefuncton whch determnes the behavor of the electron. If you have the knowledge the frst grade of scentfc course of unversty, you may solve the Eq.. (See [partcles n a box] for knowng how to solve. ) The knetc energy of the electron shut up n the box (cube) s gven by Eq.. L L h Eψ ( x, y, z) = ψ ( x, y, z) 8π m L E( nx, n y, n y ) = x y h ( n n n ) z 8mV Fg.. Schrödnger equaton and ts soluton of an electron n a box. In Eq. V s the volume (L ) of the box. And, n x, n y, and n z are called quantum numbers, are values automatcally ntroduced n the solvng process, and take ndependently the natural number as,,, and... Now, let's consder only the case of n x =n y =n z =. Equaton shows that E becomes large, f volume V of the box s made small. Conversely, f V s enlarged and s made nfnte, the knetc energy of he electron wll approach 0. Snce any system tends to decrease ts energy as low as possble, t can be sad that an electron has the character of spreadng. Therefore, as for the case between the adacent π bonds, the π electrons tend to spread rather than stayng n one double bond. Smlarly, as for the amno group of anlne, the lone-par electrons tend to enter nto the benzene rng rather than stayng on N. Ths happens because the fall

3 of the knetc energy exceeds the rse due to opposng electronegatvty. Let us actually calculate how much the energy s, when an electron s confned n the box of Å. Into Eq., the constants, n x =n y =n z =,h= J s, m= kg, V=Å are put and the result s multpled by the Avogadro's number (6.0x0 mol - ) to gve about 0 4 kj/mol. In order to shut up an electron nto the box of Å, more than 0 4 kj/mol of energy s needed. Conversely, t may be sad that the electron shut up n the box of Å has the knetc energy of more than 0 4 kj/mol. Solvng the Schrödnger equaton makes us know that resonance and conugaton orgnate n the character that an electron tends to spread. However, even f the factor whch determnes the behavor of the electron s found, t wll not be the perfect explanaton why t becomes so. Namely, why does an electron have knetc the energy, f restrcton s added to the range of movement?. Concernng the Relatonshp Between Knetc Energy and the Uncertanty Relaton. The uncertanty relaton (prncple) s expressed by the followng formula. h h p x h = π Here, p and x are the ambgutes of momentum and locaton of the partcle. The uncertanty relaton was called uncertanty prncple, but uncertanty relaton seems to be recommended to use, snce ths s not a prncple but a relatonshp. Let us consder the relatonshp between the uncertanty relaton and the knetc energy quanttatvely. In order to smplfy the story, let us consder one-dmensonal coordnate. The poston of the electron confned n the range of length L on x s examned. Snce the electron exsts somewhere between 0 and L, the ambguty of poston s set to L (L= x). Snce the momentum (p: p=mv; m, electronc mass and v, ts velocty) has drectvty, the average ( p ) of the momentum by observaton becomes 0 (the probablty of rghtward movement and leftward movement s the same and when the average s taken t wll be offset mutually). The crossbar attached upwards shows the average value. The ambguty of momentum ( p) s the gap from the average value of momentum, p p. Therefore, ( p ) = ( p p) = p 4 Usng the uncertanty relatonshp, h p L, one may get ( p) = p h L and thus the average of observable knetc energy ( E ) s expressed by, p h E = 5 m 8mL

4 Equaton 5 s very smlar to Eq.. In Eq. 5, snce one dmenson s consdered, V / of Eq. s replaced by L. Look at Eq. 5. It s easly understood that E becomes nfntve at L approachng 0. That s, the more the electronc poston s made to decde small n ts ambguty (ths means L s made small), the more the average energy s large. In other words, an electron has a large knetc energy when t s packed n a small area. It orgnates n the uncertanty relaton,. e., a fundamental expresson of quantum theory. If t s rght, what wll be the true character of the uncertanty relaton? In advance, let us make a close look at wave.. Moton of Electron Is That of Wave Packet. Partcles are obects wth mass. Snce an electron has mass, t s a partcle. If we thnk a partcle macroscopcally, snce one can determne the speed of a partcle at a certan tme, t s easy to determne the momentum and the locaton of the partcle at the same tme as defnte values. But when t comes to a mcroscopc obect lke an electron, such measurement s mpossble. Next, let us consder the stuaton of wave. There are many knds of wave as sonc wave, the wave that moves on the surface of water, lght, and so on. A common character of wave s that centerng a pont, somethng vbrates up and down or forth and back. (Remember the shape of a sne wave.) The cases are that ths somethng has mass lke water wave or has not lke lght. And that somethng s not wave but a medum. What wave s characterzed are the dstance (wavelength: λ) of the peaks of vbraton, the strength (ampltude: A) and the wave velocty (u) of propagaton. The number of tmes of vbratng n second at a fxed poston (frequency: ν) s also added. x ut (The feature of a wave s denoted by mathematcal functon (for example, Ψ = Asn π ) π usng A, x, u, t, and Ψ s called a wavefuncton.) Thus, a wave s abstract whle a partcle s concrete and they are conflctng. But they are unted n the mcroscopc world n the form of a wave wth mass. What I emphasze here s that a partcle s a wave. x Fg.. The moton of a partcle s that of a wave. Snce t s confrmed by experments that a substance s a wave, ths fact cannot be dened. It s natural to thnk that the wave of a partcle naturally does not exst n the place where the 4

5 partcle does not exst. The wave of a partcle locally exsts near the poston where the partcle exsts. Ths local wave s called wave. If the concept of a wave s used, movement of a partcle s nterpreted as that of a wave. By the way, a pure wave s determned wth a defnte wavelength, speed and ampltude, and does not change wth tme. A wave whch s decreased wth tme s not a pure wave but that of some knds of waves beng overlapped. Lkewse, a wave s a bunch of waves wth varous wavelengths. That s, the wave s made of superposton of many waves, and n order to express the wave functon of a wave, many wave functons are needed so that x becomes small. The example whch made the wave by the technque of the Fourer transformaton usng ten sne functons s shown n Fg. 4. Fg. 4. A wave conssts of many knds of waves 4. What the Uncertanty Relaton Means. Now we have suffcent knowledge to understand the reason why the uncertanty relaton exsts. We know that the wave ( Ψ ) correspondng to restrcted movement of an electron s expressed by superposton of many knds of wavefunctons. Here wthout losng generalty, we can assume that the ampltude of each wavefuncton (ψ ) s standardzed as, ψ dx = 6 Ths procedure s called normalzaton. A general promse s that unless mentoned n partcular, a wavefuncton s normalzed snce f not, dscussons became very complcated. Usng normalzed wavefunctons, the wavefuncton of a wave (Ψ ) s expressed n terms of ψ and ts rate of contrbuton (C ). Ψ = Cψ Cψ Cψ 7 5

6 In the above equaton, ψ, ψ, are the wavefunctons wth the frequences of ν, ν, or wth the momenta of p, p,. The square of the wavefuncton s nterpreted as the probablty of ts exstence. So, Eq. 7 s squared. ( Ψ ) = ( Cψ Cψ Cψ )( Cψ Cψ Cψ ) = C C ψ ψ Snce the electron must exst somewhere n space, summaton of ( Ψ ) dx wth respect to x becomes unty. If t s expressed by mathematcally, ( Ψ ) dx = CC ψ dx = ψ 9 s obtaned. And every wavefuncton can be assumed wthout losng generalty as the value of 8 ψ ψ dx s (beng normalzed) when =, otherwse 0 (beng orthogonalzed). Equaton 9 turns out to be, ( Ψ ) = dx C = C s nterpreted as the actual rate of contrbuton of ψ to the observable quantty of the wave ( Ψ ). So, the average momentum of the wave s, p C p C p C = p Equaton means that the average probablty of the wave takes the probablty of p s C, and that of p s C and so on. And, p s the dfference between the maxmum p max and mnmum p mn among p,p, p n. If one expresses the average knetc energy n terms of frequency ν usng the relatonshps of u=λν and ε=hν, the followng equaton s derved. ε ( C ν C ν C ) = ν h In order to decde the poston of an electron's exstence correctly, many knds of waves are needed, and the average value of momentum becomes large. Ths equals to the phenomenon that the average value of knetc energy becomes large. On the contrary, f the poston of an electron s not decded at all, the wave of such an electron s denoted by a sngle wavefuncton of a sngle wavelength. Snce the wavelength s constant, the momentum and therefore the knetc energy are also constant,.e., p=0. Ths s the essence of the uncertanty relaton. Snce a partcle s wave, t has a certan wdth around the poston of the partcles' exstence. If the wdth s made small, the waves of varous wavelengths (or frequences) must be added n. Ths 0 6

7 equals to takng many waves n. Ths means that the possblty of takng varous momenta ncreases so much,.e., the momentum becomes uncertan. 5. Summary An answer to the queston why resonance and conugaton take place s that an electron has the spreadng character. Why t spreads s because an electron has bg knetc energy when t s packed n a narrow area. The reason for havng bg knetc energy s due to a fundamental law of nature called the Hesenberg s uncertanty relaton (prncple). The reason why the uncertanty relaton exsts s that a substance s a wave. Don t you feel somethng uneasy was cleared up?. A tanuk s a raccoon-lke mountan dog lvng near-by human area n Japan. He s clever enough but he sometmes okes human-bengs. Be aware! 7