The conservation of generalized parity: molecular symmetry and conservation rules in chemistry

Journal of Molecular Structure (Theochem) 586 (2002) 209±223

www.elsevier.com/locate/theochem

The conservation of generalized parity: molecular symmetry and conservation rules in chemistry Xuezhuang Zhao*, Zunsheng Cai, Guichang Wang, Yinming Pan, Benxiang Wu Department of Chemistry, Nankai University, Tianjin 300071, People's Republic of China Received 2 June 2001; accepted 25 February 2002

Abstract Last century, one of the most important achievements in theoretical chemistry was the conservation principle of orbital symmetry, i.e. the Woodward±Hoffmann rules (W±H rules). Owing to the greatness of the accomplishment, some defects in connection with the rules have been neglected. For more than 20 years ago, we have paid attention to these problems. Our early works were somewhat crude and all of the relevant papers and book were announced in China. This work could not be easily known by most scientists in the world. Now, we have improved the papers and rewrite this article from a more fundamental basis. The main points of this article are as follows. (1) According to quantum ®eld theory, there is an important theorem (Noether's theorem), which relates the selection rules of different processes with their symmetries. There are two different conceptsÐ symmetry and an invariant, but they are not distinguished in W±H rules. Using Noether's theorem to analyze the W±H rules, these two concepts may be described more clearly. (2) For some more important chemical systems given the point symmetry, we can analyze them using Noether's theorem, and get the corresponding conservation rule. The invariant is a kind of generalized parity. (3) In chemistry, there are other selection rules as for the spectral transition processes. All of the rules may be related to the corresponding conservation rules of generalized parity. (4) By means of a quantum ®eld method, we can analyze the rules according to operator methods. To different processes (such as a chemical reaction, a spectral transition, etc.), we can obtain the corresponding operators. If the process is allowed by symmetry, the corresponding operator may transform the initial state functional to the ®nal one. If the process is forbidden, the corresponding process operator will perish and no process operator can act on the initial state. (5) For photochemical reactions, there are two consecutive steps: excitation with the absorption of a photon and chemical reaction of the excited molecule. Both would be considered. The mechanism of photochemical dimerization of ethylene, according to the common view of W±H rule is uncertain. (6) For extended system with the space group symmetry, we can analyze them by means of the Noether's theorem, too. Selection rules in connection with physical processes in solids can be obtained. (7) For some chemical reaction systems, such as sigmatropic reactions, there is no symmetry in connection with the point group operation of the reaction system. However, there are some special symmetry properties contained in such reaction system. (8) To describe such special symmetry, we introduce a new transformation. It is the reaction-reversal transformation in the intrinsic reaction space. The corresponding special symmetry is the group from the union of the reaction-reversal and point symmetry transformation. Using Noether's theorem and such special symmetry, the selection rules of sigmatropic reactions may be obtained. (9) As for such symmetry, not only sigmatropic reaction, but also some other reactions, such as the disproportionation of an alkene, may be analyzed. (10) The molecular local ®eld method had been introduced. We may analyze a reaction involving chirality or open-shell molecules. q 2002 Elsevier Science B.V. All rights reserved. Keywords: Conservation; Point symmetry; Sigmatropic reaction

* Corresponding author. Tel.: 186-22-2350-4854; fax: 186-22-2350-2458. E-mail address: [email protected] (X. Zhao). 0166-1280/02/$ - see front matter q 2002 Elsevier Science B.V. All rights reserved. PII: S 0166-128 0(02)00089-1

210

X. Zhao et al. / Journal of Molecular Structure (Theochem) 586 (2002) 209±223

Table 1 Some important symmetry properties and relative invariants Number

Symmetry properties

Invariants

1 2 3 4

Homogeneity (time) Homogeneity (space) Isotropy (space) Invariance under the gauge transformation Isotropy (isobaric space) Invariance under inversion Invariance under the charge conjugate transformation Invariance under the timereversal

Energy Momentum Angular momentum Charge

5 6 7 8

Isobaric spin P-parity Charge parity ±a

a The time-reversal transformation is not unitary, but it is antiunitary. There is not any relative invariant.

1. Introduction In nature, there are many conservation rules to describe the invariants, of which certain symmetry properties are always included in the background. Seemingly, nature is seeking the wonderful harmony of symmetry. As for quantum ®eld theory, the main idea is the creation and/or annihilation of elemental particles from the vacuum. We may take this point of view to look at what happens during a chemical reaction. We see that in the environment a molecular orbital (MO) is created and/or destroyed, or sometimes, a chiral molecule appears or disappears and so on. Therefore, we may use the ideas of quantum ®eld theory to study chemical reaction processes and to refresh some old doctrines in chemical kinetics. An important theorem (Noether's theorem) [1,2] reveals the inherent relation between the symmetry and the invariant in general. However, in particle physics, we are dealing with free space (the 4D time-space), and in chemistry, we are dealing with the environment space instead (the so-called environment space is the ®eld of electrons and nuclei, which forms the chemical reaction environment). In chemistry, accompanying the development of Man's knowledge toward the microcosmos, the inherent internal structure of molecules have been manifested and described by means of group theory with many successes. However, until the conservation of orbital symmetry was established by Woodward

and Hoffmann, some 30 years ago [3,4], symmetry was not used to describe chemical reaction kinetics. These conservation rules have been adopted by most chemists with many successful applications. However, there are some important topics neglected in these rules. These conservation rules touched correctly upon the identity, but did not refer to the difference between the symmetry and the invariant. Moreover, in the description of symmetry of chemical reactions, the sigmatropic reaction was not described correctly. In 1996, there was a colloquium entitled `Symmetries Throughout the Sciences', organized by Henley [5]. Many colloquium papers were announced, some of them consider symmetry in chemistry. In this colloquium, symmetry and conservation rules, especially the Noether's theorem, were considered [5]. In Dunitz's paper [6], `The Conservation Principle of Orbital Symmetry' introduced in more detail. However, the defects in connection with the Woodward±Hoffmann's principle had not been noticed. Then, the basis nature of the Woodward±Hoffmann's rules (W±H's rules), and the relationship between these rules and Noether's theorem could not be expounded clearly. In fact, since about 20 years ago, when we tried to introduce the Noether's theorem to chemical kinetics, we found desirable symmetry groups with the corresponding invariants (point-parity and space-parity, see later) and conservation rules [7,8], that they obey. Meanwhile, a new symmetry, the union of point-parity and reaction-reversal, was discovered [9,10]. It can make the sigmatropic reactions possess symmetry. Although these works are some crude, in this paper, we improve them and explain how to employ more consistently the quantum ®eld theory in discussing symmetry in chemical kinetics. There are expanding on what we have said previously. In fundamental physics [1], there is an important theorem to relate the selection rules of different processes with their symmetry properties, which is Noether's theorem [2]. According to this theorem, if a system is invariant under a given group of unitary transformation, then there must be a conservation rule for a certain mechanical observable of the system. In other words, when the observable is constant, the transformation process will be allowed, otherwise it will be forbidden, and we call it the corresponding

X. Zhao et al. / Journal of Molecular Structure (Theochem) 586 (2002) 209±223

selection rule for the process. There are two different, but connected important concepts here: the ®rst is the `symmetry property' and the second is the `conserved quantity (invariant)'. These two concepts are independent formally, but dependent essentially. Some important symmetry properties and the relative invariants are shown in Table 1. Although sometime all invariants connected with invariance under the discrete symmetry transformation may be called parity (such as the P-parity and C-parity in particle physics), to avoid the confusion, we now call it generalized parity to distinguish from the P-parity. Now, let us consider the conservation of orbital symmetry, i.e. the W±H rules [3,4]. The traditional statements may be expressed: orbital symmetry is conserved in concerted reaction. Considering Noether's theorem, two different concepts, i.e. symmetry and the invariant, are not distinguished in the W±H rules. However, these two concepts ought to be distinguished. In fact, here, the symmetry is a property of the molecular reaction system, i.e. the environment space in which the MO being (the invariance of molecular reaction system or environment space under the action of relative point transformation group) and the invariant is a characteristic of the MO (the eigenvalue of the MO under the corresponding point group symmetry operation). We may call such eigenvalue point group parity or `point-parity' for short. The `symmetry' is the `cause', and the `invariant' is the `result'. Consequently, the so-called conservation of orbital symmetry ought to be the conservation of point-parity [7,8]. Obviously, point-parity belongs to generalized parity. On the other hand, using the principle of `the conservation of point-parity', not only the W±H rules of chemical reactions, but also the selection rules for spectral transitions may be explained [10]. In addition, for some chemical reaction systems, such as sigmatropic reactions, there is not any symmetry connected with the point group operation of the reaction system. However, there are some special symmetry properties of such reaction system. Therefore, some special invariantsÐthe invariants of the union of reaction-reversal and point group operation were introduced by us [9]. Besides the above point group operation, corre-

211

sponding to the space group operation there is a relative invariantÐspace group parity, or for short, space-parity. This is the eigenvalue of the space group symmetry operation. Then, we can also obtain the relative conservation principleÐthe conservation of space-parity and use it to examine some solid state systems [10]. 2. Generalized parity in the environment space with point group symmetry An orbital is a single electron wave function, C…r; t†: The environment space is constructed by the ®eld of other electrons and the nuclei or cores and it is not always homogeneity and isotropy. During a chemical process, the environment space may vary, but it often may still keep certain point group symmetries throughout. The potential energy surface from the Born±Oppenheimer approximation [11,12] may comply with certain point group symmetries throughout the process. Certainly, there are some conservation rules, which are responsive to various point group symmetries. Point group symmetry transformations are discrete and the corresponding invariance must be termed as generalized parities, and we call them the point-parities. Now, we start to analyze these point-parities and their corresponding conservation rules. 2.1. Various point-parity and the corresponding conservation rules Firstly, let us establish the connection between quantum ®eld theory and a chemical reaction. In chemistry, the MO is the one electron wave function, and during the chemical reaction, accompanying the movement of an electron, a certain MO disappears, while another MO appears. We may now take another look at this picture of chemical reaction as follows: the electron which associates with the MO at the moment of being the reactant is annihilated from the environment space of the whole system during the reaction, while one electron which associates with the MO at the moment of being the product is created from the environment space. Therefore, we may use the quantum ®eld theory to discuss the chemical reaction and other similar processes (i.e. the electron changes its state), and

212

X. Zhao et al. / Journal of Molecular Structure (Theochem) 586 (2002) 209±223

Table 2 Some point group symmetry properties and their corresponding invariants Number

1 2 3

Symmetry properties relative transformations

Operators symbol

Point-parity transformation Parity (space inversion) transformation Axis-parity transformation (rotation by an angle 2pn 0 =n about an axis) Mirror-parity transformation (re¯ection in a mirror) Identity transformation

4 5

use the ®eld operator C^ …r; t† instead of the wave function C…r; t†: The transformation law of the ®eld operator C^ …r; t† ^ which is under a certain unitary transformation G; operating upon the environment space may be expressed as ^ 21 ˆ g0 C^ …G 21 r; t† ^ C^ …r; t†G G

…1†

in which r is the coordinate of space vector, t is the time coordinate, and g 0 is an intrinsic quantity. When ^ is the symmetry transformation, which generates G the point-parity, g 0 may be called the intrinsic pointparity. Expand the C^ …r; t† into spherical wave, and we obtain that

C^ …r; t† ˆ X  p mp a^ klm Zl …k; r†Ylm …u; f† 1 a^ 1 klm Zl …k; r†Yl …u; f† …klm†

…2† in which r, u , f are the space spherical coordinates, k is the wave-vector, and the l and m are the azimuthal and magnetic quantum numbers, respectively. The expansion coef®cient, a^ klm is the annihilation operator for the particle with the k, l, m state, and its adjoint quantity a^ 1 klm is the creation operator for the particle with the corresponding state. Zl …k; r† is the spherical Bessel function and Zlp …k; r† is the Hermitian conjugate of the Zl …k; r†: Under an arbitrary point-parity ^ the Zl(k,r) and Zlp …k; r† should be transformation G; invariant. The equations ^ l …k; r† ˆ Zl …k; r† GZ

…3a†

Invariants (eigenvalues) Point-parity

Symbol

^ G P^ 0 ^ nn C

Point-parity P-parity Axis-parity

g p c

^ M ^ E

Mirror-parity Identity-parity

m e;1

and ^ lp …k; r† ˆ Zlp …k; r† GZ

…3b†

must be true. As for the spherical function Ylm …u; f† and its Hermitian conjugate Ylmp …u; f†; they may be ^ changed under the point-parity transformation G: m mp m ^ ^ GYl …u; f† and GYl …u; f† may not equal Yl …u; f† and Ylmp …u; f†; i.e. ^ lm …u; f† ± Ylm …u; f† GY

…4a†

and ^ lmp …u; f† ± Ylmp …u; f† GY

…4b†

Since all of the point-parity transformations are unitary, according to the Noether's theorem, a oneto-one correspondence can be established between the relative symmetry properties and the invariants [2]. For the common point group symmetry properties (i.e. the invariance under the point-parity transformations), the relative transformations and their invariants as shown in Table 2. For example, considering the symmetry property of the invariance under the space inversion transforma^ the relative invariant is the parity p. According tion P; to the terminologies of ®eld theory P^ and p are called parity (space inversion) transformation and parity, respectively. Obviously, the parity is a special pointparity, and might be called the center-parity or Pparity, but we still call it parity for short. As for the invariance under the transformation of 0 ^ nn ; the rotation by an angle 2pn 0 =n about an axis, C relative0 invariant is called the axis-parity, c, and we ^ nn the axis-parity transformation. call C Similarly, we can introduce the mirror-parity

X. Zhao et al. / Journal of Molecular Structure (Theochem) 586 (2002) 209±223

^ and relative invariant mirror-parity transformation M ^ there m. As for the identity (unity) transformation E; is a de®ned invariant e, it is identically equal to one, and we call it the identity-parity. According to the transformation rules of Ylm …u; f† and Ylmp …u; f† under the different point-parity transformations and using Eqs. (1) and (2), we obtain the different relative invariants: the parity

p ˆ p0 …21†l

the mirror-parity the axis-parity

…5a†

m ˆ m0 …21†l1m

…5b†

c ˆ c0 exp…2pn 0 mi=n†

…5c†

where the p 0, m 0 and c 0 are the corresponding intrinsic point-parities, They may be called the intrinsic parity, intrinsic mirror-parity, and intrinsic axis-parity, respectively. It must be noted that according to Eq. (2), C^ …r; t† is a scalar ®eld operator, so the spin quantum number has been ignored. Supposing the spin quantum number (s) is included, then instead of Eqs. (5a)±(5c), we have [13,14]: the parity

p ˆ p 0 …21†l

the mirror-parity the axis-parity

m ˆ m0 …21†l1m1s

…6a† …6b†

213

Therefore, using Eqs. (6a)±(6c), the point-parities of orbital (electron) and photon may both be established. In fact, the point-parities are equal to the corresponding characters of the one-dimensional irreducible representations of the relative reaction system. According to Noether's theorem, we can conclude that so long as the symmetry of a given point group is maintained through the whole process of a chemical reaction, each point-parity corresponding to such a symmetry must be invariant. That is, the principle of `The Conservation of Point-parity' [8]. Considering that the character of the irreducible representation and point-parity are correlated, it is obvious that so long as the symmetry of a certain point group is retained throughout the whole process, the corresponding irreducible representation of this reaction must be invariable. Based on the MO approximation, the corresponding irreducible representation of the single electronic state, i.e. the orbital, must also be invariant. That is, the essence of the W±H's rule [16]. In short, we can obtain the point-parities for the orbital or the single electronic state by the same methods. These point-parities must be invariant so long as the corresponding point group symmetry can be retained throughout the whole chemical reaction. If that is so, this reaction will be allowed, otherwise it will be forbidden.

c ˆ c0 exp‰2pn 0 …m 1 s†i=n†Š …6c†

Eqs. (6a)±(6c) may be applied to the spinor ®eld (electron) and the electromagnetic ®eld (photon). Comparing Eqs. (5a)±(5c) and (6a)±(6c), it is not dif®cult to see that Eqs. (6a)±(6c) can be obtained so long as we substitute the m in Eqs. (5a)±(5c) for …m 1 s†: As for the intrinsic point-parities of electron, because the number of electrons is invariable throughout the chemical processes, we need not consider them, which may always be cancelled. For convenience, we can let all of the electronic intrinsic pointparities have equal unit value. However, the number of photons may be changed throughout processes of interest. As the intrinsic point-parities of photons must be considered, base on the ®eld theory [15], we know that the intrinsic parity of the photon is 21. Owing to the photon is a vector particle, we can get the intrinsic mirror-parity and axis-parity of photon as a vector.

2.2. The operators of some processes Various kinds of point-parity and intrinsic pointparity have been introduced above and the conservation law of point-parity has been discussed in accordance with Noether's theorem. Now, some important processes will be analyzed by the operator method [17]. In accordance with the particle number representation, the one particle-state functional may be expressed as

fp ˆ a^ 1 p f0

…7†

where fp and f 0 are the p-state and vacuum state functionals, respectively. Supposing a certain point group G is involved in an interacting system, the jth ^ j ; and the p-state is point-group-transformation is G

214

X. Zhao et al. / Journal of Molecular Structure (Theochem) 586 (2002) 209±223

^ j ; then we get the eigenstate of G ^ 21 ^ j a^ 1 …p†G ˆ gpj …a†^a1 …p† G j

…8a†

and ^ 21 ^ j a^ …p†G G ˆ gj …a†^a…p† j

…8b†

where gpj …a† and gj(a ) are equal to the character of the one-dimensional irreducible representation a , to ^ j: which the p-state belongs under the operation of G When gpj …a† and gj …a† are real, gpj …a† ˆ gj …a†; and it is ^ j : Then it the point-parity of the p-state in relation to G 21 p ^ j fp ˆ G ^ j a^ 1 ^ follows that G f ˆ g f : For differG 0 j p j p ^ j ; the corresponding point-parity may be deterent G mined from Eqs. (6a)±(6c). The many-particle state functional may be expressed as

F ˆ P a^ p1 f0

…9†

…p†

And the initial and ®nal state functionals of a chemical reaction system may be expressed as follows   1 F…int:† ˆ P a^ ki fs …10a† …ki†

and



 1 ^ F…fin:† ˆ P a kf fs

…10b†

…kf†

where f s may be the vacuum-state functional f 0, the ^1 a^ 1 ki and a kf are the relevant particle creation operators at the moment of being the reactant (initial state, ki) and product (®nal state, kf), respectively. The ki and kf in the continue multiplication should run over all relevant particles involved in the chemical reaction. For simpli®cation of calculation, they run only over the electrons, which change state throughout the chemical reaction. The creation operators of other electrons and relevant particles may be included in f s. So the f s is equivalent to a partial state functional for those parts, which are unchanged when passing throughout the reaction. Then, we can introduce the chemical elementary reaction operator as follows: 2 0 13 X X 21 ^ j a^ 1 ^ j A5 …11† ^ ˆ ^ ^ ki G Re Pu…kf; ki†4 P @ G kf a …Pu†

…kf†…ki†

j

^ where all of Pu…kf; ki† are the permutation operators

for all different combination ways of kf and ki, and the ^ j …j ˆ 1; 2; ¼† are all symmetry transformations G belonging to the point group G, which would be maintained throughout the whole process. In Eq. (11), the a^ kf1 a^ ki represents P the chemical change, the P…kf†…ki† P 21 ^ ^ ^ ¼ Pu…kf; ki† mean the operations and G G j j …Pu† j from the point group symmetry and from the permutation symmetry of the arbitrary combination of kf and ^ ki ; respectively. Using the orthogonality ki in a^ 1 kf and a condition [18] of the character and Eqs. (8a) and (8b), we can get (ignoring the normalization constants): ^ F…ini:† ˆ F…fin:† Re …for point-pairty allowed process†

…12a†

and ^ ˆ 0^ Re

…for point-pairty forbidden process† …12b†

If the reaction is point-parity allowed the F (ini.) should be changed into F (®n.) under the action of ^ On the other hand, if the reaction is point-parity Re: ^ will perish and no process operator forbidden, the Re will act on the F (ini.). It is obvious that these results are consistent with the principle of `the conservation of point-parity'. As for more complicated reactions, the overall reaction operator may be separated into some elementary reaction operators. So long as the corresponding elementary reaction operators are obtained, from the general chemical kinetics theory, it is not dif®cult to construct the overall reaction operator. Now, let us turn to analysis the operators for photon absorption and emission in the spectral transition processes. These two process operators are X ^ j a^ 1 ^ 21 ^ ˆ ^ pho a^ ki G Ja …13a† G kf a j j

and ^ ˆ Je

X j

^ j a^ 1 ^ 21 ^1 ^ ki G G j kf a pho a

…13b†

where the subscripts f and i denote the electron with ®nal and initial states, respectively, and the subscript pho denotes the photon. We can then ®nd the conservation rules of point-parity, which could serve as restrictions on the absorption and emission spectral

X. Zhao et al. / Journal of Molecular Structure (Theochem) 586 (2002) 209±223

215

and ^ Re…pho-che:† ˆ 0^ …for point-parity forbidden photochemical reaction† …16b† 1

Fig. 1. The SALC-BMOs and ABMOs of the ethylene dimerization reaction in connection with the variation of motion state (the solid and the open circle are the positive and negative lobes of the orbital, respectively).

transition processes may be expressed as follows: J^ F el …ini:† ˆ F el …fin:† …for point-pairty allowed transition†

…14a†

and J^ ˆ 0^

…for point-pairty forbidden transition† …14b†

In Eq. (14a), F el(ini.) and F el(®n.) indicate, respectively, the initial and ®nal state functionals, and J^ may ^ or Je: ^ Using Eqs. (14a) and (14b), we can be either Ja get the correct selection rules for the absorption and emission spectral transition processes. For photochemical reactions, there are two consecutive steps: excitation with the absorption of photon and chemical reaction of excited molecule. As a consequence the photochemical operator is [19] ^ ^ 1 Ja ^ Re…pho-che:† ˆ Re

…15†

1

^ where Re is a chemical reaction operator, whose ^ except including the excited form is the same as Re state. The photochemical reaction will be restricted by the conservation rule of point-parity as follows: ^ Re…pho-che:† F…ini:† ˆ F…fin:† …for point-parity allowed photochemical reaction† …16a†

^ does not exist, the Re…pho-che:† ^ ^ and/or Ja will If Re perish, and the photochemical reaction will be forbid^ 1 is den. Therefore, if only the contribution of Re considered, the conclusion may be not complete. Using the operators of different processes (such as the thermochemical reaction, spectral transition and photochemical reaction processes), we can get the relative selection rules for the corresponding processes easily. In additional, for more a complicated point group G, we may get the corresponding operators by means of G(m ), which is the subgroup of G. For the pointparity allowed process, the condition Eqs. (12a), (14a), and (16a) must be established in the cases of all subgroup G(m ). However, for the point-parity forbidden process, the corresponding operator perishes for any one subgroup G(m ), then the operator does for group G as well. 2.3. Special example Since the effectiveness of the symmetry selection rules of the chemical reaction ought to be the same as the conservation principle of orbital symmetry or the point-parity or by use of the process operator scheme. We may quote any one of the reactions that had been studied by W±H's rules, as examples of point-parity conservation. Many such examples can be found in our papers or books [10], published in the 1980s. Here, we cite one of them, i.e. the dimerization of ethylene [19], either by the themochemical or by the photochemical method. Some interesting topics may be analyzed, which have been neglected by usually with the traditional W±H rules. This reaction may be denoted as follows: 2C2 H4 …ethylene† ! C4 H8 …cyclobutane†

…17†

In this reaction, the primal SALC-BMOs and ABMOs (anti-bonding MOs) may be shown in Fig. 1. For thermochemical dimerization of ethylene, since the electronic con®guration must be transformation from p2A p2B into s A2 s C2 ; this reaction should be forbidden.

216

X. Zhao et al. / Journal of Molecular Structure (Theochem) 586 (2002) 209±223

According to the common view of the conservation of orbital symmetry, for photochemical dimerization, the reactant has absorbed a photon and in the pA2pBpCp excited con®guration can be transformed into the sA2sBp sC excited con®guration of the product. It is allowed [20]. However, since the intrinsic P-parity of photon is 21, the con®guration of reactant spices, after absorbtional a photon, cannot be transformed into the con®guration of pA2pBpCp . Meanwhile, this mechanism violates Einstein's law of photochemical equivalence [21±23]. According to this law, for the preliminary step of a photochemical reaction, a photon can be absorbed by only one molecule. However, it will be absorbed by two ethylene molecules simultaneously, in accord with this mechanism. If we check the photochemical Eq. (15) ^ P}; ^ based on the space inversion subgroup P ˆ {E; we can get the pB ! pCp spectral transition process
^ ˆ E^ ^ a 1 ppC a^ …pB †^apho E ^ 21 1 operator
as follows: Ja 1 21 p ^ a pC a^ …pB †^apho P^ : Then using Eqs. (8a) and P^ (8b) and considering all the eigenvalues of E^ are e ; 1 and all of the eigenvalues of P^ are p ˆ 21; it is not ^ ˆ 0: ^ Here, although Re ^ 1 is not dif®cult to obtain Ja 1 ^ ˆ 0^ and this ^ ^ Ja equal to null, the Re…pho-che† ˆ Re mechanism is point-parity forbidden. For this reason, we are certain think such mechanism for photochemical dimerization of ethylene is doubtful. The possible mechanism of this photochemical reaction may be as follows: with one photon absorbed by one and only one ethylene molecule, this molecule will be excited and react with another ground state ethylene molecule. The symmetry of the point group D2h for this bimolecular reaction system is no longer maintained. As the p-electron in this excited ethylene molecule is transformed into p p-ABMO, the CyC bond length of the excited ethylene molecule is longer than that of ground state one. The reaction system consisting of an excited ethylene molecule and a ground state one, now possesses only the symmetry of the point group C2v ; not D2h : The excited ethylene molecule with the p p con®guration can react with a normal ethylene molecule to form an excited cyclobutane molecule. According to such mechanism, the photochemical dimerization of ethylene is not a one, but multi-step reaction. Certainly, all steps of this mechanism are point-parity allowed in connection with the point group C2v : As for other reactions, we can often analyze them in

similar ways by the traditional method based on the principle of the conservation of orbital symmetry. Besides the bond formation and breaking during the chemical reaction, the other transitions of the electronic states are also transition though they are not the most important one. These transitions, however, cannot determine the reaction path. They ought to be constrained by the conservation law from Noether's theorem, and these transitions must be coordinated consistently with the reaction path predicted by the selection rules of the corresponding reaction. We had discussed the electrocyclic reaction as example [10], and will not analyze here. 3. Generalized parity in the intrinsic reaction space In Section 2.3, the invariant connected with point group symmetry and the related conservation rules were discussed according to Noether's theorem. The conservation rule of point-parity was established. Both the W±H rules and important spectral selection rules may be introduced in the same basis of point-parity conservation. On the other hand, as we known, for the sigmatropic reactions and some others, there is no any point group symmetry. Their symmetry conservation rules cannot be analyzed by means of the W±H rules based on the point group symmetry. For these reaction systems, however, there are some special symmetries other than the point group symmetry [24,25]. These symmetries are connected with the so-called reaction-reversal transformation, which ought to be a generalized parity transformation in the intrinsic reaction space. 3.1. Reaction-reversal transformation and the relevant symmetry Now, we introduce a new transformation, and its relevant symmetry and invariant. This transformation is called `reaction-reversal transformation' [9], which is connected with the time-reversal transformation. For the interaction system of a chemical reaction, the con®gurations of the relative position of all the atomic cores or nuclei can be described with an multi-dimensional (m-dimensional) con®guration space, which is the frame of the potential energy surface. The reaction coordinate refers to a curve in

X. Zhao et al. / Journal of Molecular Structure (Theochem) 586 (2002) 209±223

217

to the point a , it will be transformed into the state function C p(b ) under the reaction-reversal transfor^ that is: mation R; ^ C…a† ˆ C p …b† R

…18a†

^ corresponds For real wave functions C ˆ C p ; the R merely to the transition along the reaction coordinate from a to b , and ^ C…a† ˆ C…b† R Fig. 2. The potential energy surface and the reaction-reversal transformation.

this m-dimensional space. Across the center of this curve, there is an (m 2 1)-dimensional super-plane orthogonalized to the reaction coordinate curve. The operation of re¯ection at the (m 2 1)-dimensional super-plane is called the reaction-reversal transforma^ For a complex wave function, the tion, and denoted R: transformation of Hermitian conjugation should be ^ included in R: For example, there is the con®guration space with the coordinates r1 and r2 (where m ˆ 2), and L(rc) is the reaction coordinate (it may be the so-called intrinsic reaction coordinate and we may call such coordinate space the intrinsic reaction space), as shown in Fig. 2. Where point M is the center of L(rc), and B is the (m 2 1)-dimensional super-plane (it is a straight line here) orthogonalized to the reaction coordinate L(rc), across the point M. Chemists have an interest to the case, where the center of the reaction coordinate is the transition state of the relevant reaction. If C (a ) is a complex wave function corresponding

^ Fig. 3. The P^ R-symmetry of H3-system: (BMO) 2(NBMO) 1. (The 1 is the center of symmetry, the dot is the node, the solid and the open cycle are the positive lobe and negative one of the orbital, respectively).

…18b†

It may be noticed that by carrying out the correspond^ the molecular system of the chemical ing operation R; reaction may be transformed into a form, which may be distinguished from the original one. That means ^ C…a† ± cC…a† R

nor

cC p …a†

…19a†

where c is an arbitrary constant. However, many reaction molecular systems, after carrying out the reac^ and a certain point tion-reversal transformation R ^ group symmetry operation G successively, will be transformed back into the original form, ^ C…a† ˆ cC…a† ^G R

or

c C p … a†

…19b†

Obviously, there is a new symmetry in such reaction ^ ˆ ^G systems. Such a kind of symmetry operator R ^R ^ can be called the union transformation of point G ^ usually is the group and reaction-reversal, in which G ^ 2: ^ M ^ or C transformation of P; For example, considering the H3-system as shown in Fig. 3, the electronic con®guration of H3 is (BMO) 2(NBMO) 1, where BMO and NBMO are the bonding molecular orbital and non-bonding one, respectively. It can be seen that by performing the ^ P^ ˆ P^ R; ^ the corresponding symmetry operation, R BMO (occupied by a pair of electrons) and NBMO (occupied by only one electron) are transformed. Both are the eigenstates of the symmetry transformation ^ P: ^ However, the eigenvalues, i.e. the invariants, rp R ^ P: ^ of BMO and NBMO are different in relation to R They are 11 and 21, respectively. Thus, ^ P^ C…BMO† ˆ C…BMO†; and R ^ P^ C…NBMO† ˆ R 2C…NBMO†: We must notice that such a system is provided with ^ P; ^ but without that of the singular R ^ the symmetry of R ^ nor P: Notice that for this example (H3) under the P^ both the geometry con®guration and orbital phase are inverse through the symmetry center, while under the

218

X. Zhao et al. / Journal of Molecular Structure (Theochem) 586 (2002) 209±223

^ 2 transformation property of the BMO for the ^ u†C Fig. 4. The R… antarafacial [1,j]-sigmatropic reaction of allyl ((a) next HOMO rc ˆ 21; (b) HOMO rc ˆ 1). Both the dot (node) and solid circle (positive phase) are belong to the migrating group.

^ there is only translation of the middle hydrogen R; atom, but the orbital phase is unaltered. ^ and P^ are the generalized parity Evidently, both R transformation, and they are action on the intrinsic reaction space and orbital environment space, respectively. Of course, the H3-system is not the only system, which possesses the union symmetry of point-group and reaction-reversal. The more important reaction systems with such symmetry are sigmatropic reactions systems. In fact, we initially found such a new symmetry from research on sigmatropic reaction.

remains, and during the antarafacial (conrotatory) ^ 2 symmetry remains. ^ u†C reaction, the R… To start with, using the basis of real wave functions to discuss the [1,3]-sigmatropic reaction as an example, there is a pair of electrons occupying the BMO and an unpaired single electron occupying the NBMO in the allyl. The NBMO will be combined with the NBMO of migrating group to form the HOMO (highest occupying MO) and LUMO (lowest unoccupied MO). Thus, for the antarafacial reaction, both the occupied MOs, HOMO and next HOMO, are the ^ 2 as shown in Fig. 4. ^ u† C eigenstates of R… Their eigenvalues (rc) are conserved through the reaction process with 11 and 21, respectively. Therefore, it can be decided that this reaction is allowed antarafacially. However, as for the suprafacial shift, the next HOMO is still the eigenstate of ^ m †M ^ with eigenvalue rm equals 11, but its R… ^ m†M: ^ This [1,3]HOMO is not the eigenstate of R… sigmatropic reaction is forbidden suprafacially. This conclusion will be complicated, when a complex wave function is introduced. When a complex wave ^ it ^ G; function is used, according to the symmetry of R can be proven that the antarafacial mechanism of [1,3]-sigmatropic reaction is:

3.2. Special example As an example, considering the [1,j]-sigmatropic reaction with retention of con®guration at the shift site, the whole system of such a reaction can be ^ ^G shown to possess some type of symmetry of R [24]. Sometimes, complex wave functions would be needed and then reaction-reversal transformation of complex wave function would include the conjugate transformation. For a complex state function C in a certain reaction coordinate, its transition owing to the action of the reaction-reversal transformation can be written as ^ C…Dq ± † ˆ C p …2Dq± † R

…21† However, based on the suprafacial mechanism of the [1,3]-sigmatropic reaction, it will be found that the shift group X will split completely at the transition state:

…20†

…22†

where Dq ± denotes the interval in reaction coordinate with the origin in transition state. The wave function C and C p are conjugated each other. For the real ^ wave function, since C and C p are the same, R ± ± would only transform the Dq to 2Dq . For [1,j]sigmatropic reaction, we claim that during the supra^ m †M ^ symmetry facial (disrotatory) reaction, the R…

Therefore, the activation energy of the suprafacial mechanism ought to be much higher than that of the antarafacial one. The antarafacial [1,3]-sigmatropic shift is allowed and the suprafacial one is forbidden energetically. As for the [i, j]-sigmatropic reaction (both i and j are not equal one), the molecular fragment method could be introduced and the reaction

X. Zhao et al. / Journal of Molecular Structure (Theochem) 586 (2002) 209±223

symmetry selection rules can correctly be established, ^ symmetry [26±29]. ^G based on the R Other than sigmatropic reactions, the disproportio^ symme^ G; nation of alkene may also be analyzed in R try [9,30,31]. The disproportionations of an alkene are a very important catalytic reaction and in general the complex metal catalysts are used. For this reaction, there is a certain stereochemistry selection feature: the cis-alkene mainly yields cis-product and the trans-alkene mainly produces trans-product. For such a reaction, the concerted mechanism with a quasi-cyclobutane intermediate was proposed [32]. According to this mechanism, the stereochemical selection feature had been clari®ed from the aspect of steric hindrance and orbital symmetry. However, the further research made it clear that the mechanism of this reaction does not include a quasi-cyclobutane intermediate state, involved a chain mechanism instead. The reaction passes through a metal±carbene complex intermediate (I) and the metal inserts itself into their ortho-position of the other alkene molecule as follows [33±37]:

…23† There is no ordinary point group symmetry in this ^ symmetry must be ^G process, so for this reaction, R used and we may obtain the correct selection rules, too [9,30,31]. 4. Other special topics In previous sections, we consider the electronic wave function, MO, as a particle moving in the environment space with the symmetry of the point ^ or of the group union transgroup transformation G ^ ^ G: Now, we examine two other cases, formation R which do not connect with the W±H rules. In one case, the molecule itself is considered as a particle (by which molecule local ®eld is developed), and in the other case the environment space posses with the space group symmetry (by which the space-parity is developed).

219

4.1. The molecular local ®eld The local ®eld approximation is usually quoted in quantum ®eld theory. According to this approach, the corresponding particle is treated as a geometric point without any internal structure. The effect of internal motion will be described by some intrinsic parameters, which are not analyzed in detail. In the other words, we suppose that the states of the ®eld and their interaction at a given space-time point are completely determined by the ®eld functions and their derivatives evaluated at that particular point (i.e. do not depend on the neighborhood of the point). Structure of the particle will not be introduced. In the previous sections, our propositions are essentially based on a kind of MO (one electron) local ®eld. In general, it may be agreed that the molecule should be moved in a homogeneous and isotropic space, then the point-parity properties for the electronic moving (MO) within the molecule attribute to the molecular intrinsic point-parity. Considering that the electrons contained in a molecule often form pairs, the intrinsic symmetry for the whole molecule should be totally symmetric, i.e. all of the intrinsic point-parities equal one. Consequently, we usually give up the hope of directly predicting the selection rules of chemical reactions in connection with the conservation law of point-parity in the molecular local ®eld approach. However, we can sometimes apply the molecular local ®eld approach to speci®c chemical processes and obtain interesting information, which will be discussed here [38]. Now, let us develop the rules of the parity transformation in connection with the chirality of molecules during a reaction in which chiral molecule appear. It is well known that none of the second class of point symmetry operations, such as the inversion center and so on, can be found in a chiral molecule. On the other hand, the non-chiral molecule, including the mesomer is the parity transformation eigenstate. Although the chiral molecule cannot be the parity transformation eigenstate, but a molecular set with equal quantity of enantiomorphs, i.e. a racemic mixture will be the eigenstate of parity transformation. According to a similar procedure, as the analysis of the cross-sections of several strong interaction processes in the elementary particle theory using charge symmetry or the combined charge parity, we

220

X. Zhao et al. / Journal of Molecular Structure (Theochem) 586 (2002) 209±223

can conclude that the chiral product for non-chiral reactant must be a racemic mixture. Obviously, it is quite consistent with the experiment evidence. For instance, if reactant A is non-chiral and the product B is chiral, the enantiomorphs of B are B(d) and B(l), such reactions can be formulated as A ! B…d†

…24a†

A ! B…l†

…24b†

where A may represent more than one substance and B may also include several substances, but at least one of which must be chiral. The reaction cross-section of these two processes must be equal by the conservation law of parity, since both initial and ®nal states of the process (24b) arise from those of the process (24a) by ^ respectively. applying the P-parity transformation P; ^ denotes the interaction Hamiltonian, which Thus, if H would be invariant under the operation of P-parity, we can get that the respective transition amplitude of Eqs. (24a) and (24b) are equal. ^ ^ PuAl ^ ^ ^ PAl ^ kB…d†uHuAl ˆ kB…d†uP^ H ˆ kPB…d†u Hu ^ ˆ kB…l†uHuAl

…25†

where we introduce the A, B(d) and B(l) to denote the corresponding particle (i.e. the molecule) ®eld functional. The ®rst and last terms in Eq. (25) represent the transition moment of Eqs. (24a) and (24b), respectively. To understand this result, let us consider the following `Gedanken experiment'. Suppose we have an incident beam of A particles, which is `unpolarized' in the P-parity. Now, this beam undergoes some transmutation process after interactions. As a result of these interactions, we get a beam of outgoing product B particles. If the parity is conserved through the interactions, according to Eq. (25), we can come to the conclusion that the reaction cross-sections producing the enantiomorphs of B contained in the outgoing beam, which are unpolarized in P-parity, should be equal, i.e. the outgoing beam must be racemized. In addition to chiral molecules, the open-shell and excited molecules may be analyzed by the corresponding molecular local ®eld theorem [38]. However, the intrinsic point-parities of such molecules may not be equal to one. According to the molecular local ®eld theorem, we may study some elementary processes involved in the hydrocarbon

¯ame. For this problem, Shuler [39] made a discussion based on group theory. The symmetry selection rules of state-to-state reactions can be discussed with the similar method, in principle. 4.2. Space-parity and the corresponding conservation rule Similar to the point group transformation, if a system is invariant under certain space group transformation, according to Noether's theorem, from this symmetry property there follows the conservation of an observable quantity of this system [9,10]. We call this quantity the space group parity, or for short, the space-parity, which is the eigenvalue of the space group transformation. Obviously, the space-parity is the generalized parity corresponding to the discrete space group transformation. For example, correspond^ ing to the translation transformation, T…na†; the eigenvalue (wave-parity or translation-parity) of the eigenstate functional F k should be exp…knai† according to the Bolch's theorem [40,41], under the translation of the integral number (n) folds of the length of potential ®eld, a, so we get: ^ T…na† F k ˆ exp…knai†F k

…26†

where a is the periodic length of the potential ®eld and k is the wave-vector. As mentioned earlier, we can be sure that so long as the symmetry of a certain transla^ tion transformation T…na† is maintained through the whole process, the wave-parity tp …na† ˆ exp…knai† must be invariant during this process. If a approaches zero, in such a case, the k will be the ordinary real number (the module in®nite number) and means the momentum ought to be invariant. The conservation of wave-parity will then become the conservation of momentum. This situation is similar to what happens, when the axis-parity conservation becomes angular momentum conservation in an isotopic environment space. In the ®nite periodic potential energy ®eld, the conservation principle of wave-parity may illustrate the electron transition process of semi-conductor from the occupied band to the conduct band [42]. For such a process, according to the conservation principle of wave-parity, we get

tp …o:b:† £ tp …hn† ˆ tp …c:b:†

…27†

where t p(o.b.) and t p(c.b.) represent the wave-parities

X. Zhao et al. / Journal of Molecular Structure (Theochem) 586 (2002) 209±223

of the electrons in the occupied and conduct bands (before and after the transition), respectively. tp …hn† denotes the wave-parity of the absorbed photon. Using Eq. (26), we get: kk…o:b:†l 1 kk…hn†l ˆ kk…c:b:†l

…28†

where kk(o.b.)l, kk(c.b.)l and kk(hn )l represent the wavevectors of the electrons in occupied band, conduction band and of the absorbed photon, respectively. Since kk…hn†l is quite small in general, it may be proposed approximately that the kk(o.b.)l and kk(c.b.)l are nearly the same. It means that it is vertical transition. When the phonon participates such transition, since the wavevector of phonon may be change, then kk(o.b.)l and kk(c.b.)l may be different, i.e. and it is so called the non-vertical transition. Certainly, both the vertical and the non-vertical transitions ought to be constrained by the conservation law of the wave-parity. In addition to translation, the space symmetry transformations include the screw translation and glid re¯ection. These space transformations can be analyzed in a similar manner. They may be also considered as the combination of translation and the corresponding point transformation. The corresponding eigenvalues (spaceparities) may be obtained easily. Based on the statement above, we come to the conclusion that so long as the symmetry of a certain space group is maintained, during the whole process of a varied system, each space-parity corresponding to the symmetry is an invariant. That is, the principle of the `conservation of space-parity' and it is consistent with Noether's theorem. Consequently, the corresponding selection rules after one photon is adsorbed can be established just like the rules for the electron transition in a semi-conductor. Although the surface of a catalyst can be treated in connection with a two-dimensional space group, part of the symmetry in this kind of space group symmetry will be destroyed by the reagent system, when the catalytic reaction is in progress. Therefore, for a catalytic reaction, the conservation rules of space-parity are not as important as those of point-parity. 5. Conclusions (summary and outlook) Undoubtedly, last century, one of the most important achievements in theoretical chemistry is the

221

conservation principle of orbital symmetry, i.e. W± H rules established about 35 years ago. However, owing to the importance of their accomplishment, some defects in connection with the W±H rule have been neglected. In 1996, there was a colloquium entitled `Symmetries Throughout the Sciences', organized by Henley. Many of the colloquium papers were announced and published in Ref. [5]. Some of them are in relate to symmetry in chemistry. In this colloquium, symmetry and conservation rules, especially Noether's theorem has been studied. However, in this colloquium, the defects of W±H rule were not examined. Since 20 years ago, we have paid attention to these problems. However, these works were somewhat crude and all of the relevant papers and a book were published in China. The works could not be easily known by most scientists. Through this review, we introduce these works to more scientists. Now, we have improved our earlier works and write this review. The main points of this review are as follows. (1) According to Noether's theorem in quantum ®eld theory, which relates the process selection rules with their symmetries. There are two different concepts (symmetry and an invariant) they are not distinguished in W±H rules. However, they ought to be distinguished. Using Noether's theorem, we can make these two concepts more clearly. (2) Using Noether's theorem, we can analyze the more important chemical systems given the point symmetry, and get the corresponding conservation rule. The invariant is a kind of generalized parity. (3) In chemistry, there are some other selection-rules, such as for the spectral transition processes. All of them may be related to the corresponding conservation rules of generalized parity. (4) By means of the operator methods, for different processes (such as a chemical reaction, a spectral transition, etc.), we obtain the corresponding operators. If the process is allowed by symmetry, the corresponding operator may transform the initial state functional to the ®nal one. If the process is forbidden, the corresponding process operator will perish and no process operator can act on the initial state. (5) For photochemical reactions, there are two consecutive steps: excitation with the absorption of a photon and chemical reaction of the excited molecule. Both would be considered. The mechanism of

222

X. Zhao et al. / Journal of Molecular Structure (Theochem) 586 (2002) 209±223

photochemical dimerization of ethylene, according to the common view of W±H rule is uncertain. (6) For extended system with the space group symmetry, we can analyze them by means of the Noether's theorem, too. Selection rules in connection with physical processes in solids can be obtained. (7) For some chemical reaction systems, such as sigmatropic reactions, there is no point group symmetry, but there are some special symmetry properties contained. (8) To describe such special symmetry, we introduce a new reaction-reversal transformation in the intrinsic reaction space. The corresponding special symmetry is the group from the union of the reaction-reversal and point symmetry transformation. Using Noether's theorem and such special symmetry, the selection rules of sigmatropic reactions may be obtained. (9) As for such symmetry, not only sigmatropic reaction, but also some other reactions, such as the disproportionation of an alkene, may be analyzed. (10) The molecular local ®eld method had been introduced. We may analyze a reaction involving chirality or open-shell molecules. 5.1. Outlook Before our work in this ®eld, Noether's theorem had been used mainly to analyze some conservation rules in fundamental physics. In this paper, Noether's theorem has been used to explain conservation rules in more complex chemical processes and to obtain new information. Of course, the speci®c expressions are somewhat different. On the other hand, it ought to be possible, that we use this theorem to analyze more complex processes. Such as the genetic and replication processes in life-scienti®c ®eld, obviously, Noether's theorem would be a useful tool to analyze these processes. What is the invariant here? What is the symmetry in the background? Although it is not in the free 4D time-space (as in the particle physics) nor in the environment space with point group symmetry (as in the chemistry), using the Noether's theorem to analyze some conservation rules, ought to be bene®cial. As for the chemical processes, owing to the discovery of new processes, sometime Noether's theorem may be useful, such as some non-linear chemical kinetics processes, etc. which we are going to analyze.

Acknowledgements This work is supported by the National Natural Science Foundation of China. References [1] D.J. Gross, Proc. Natl Acad. Sci. USA 93 (1996) 14256± 14259. [2] E. Noether, Nach d Kel. Ges. D. Wiss. Gottingen (1918) 235. [3] R.B. Woodward, R. Hoffmann, Angew. Chem. 8 (1969) 781± 853. [4] R.B. Woodward, R. Hoffmann, The Conservation of Orbital Symmetry, Weinheim, New York, 1970. [5] E.M. Henley, Proc. Natl Acad. Sci. USA 93 (1996) 14215, and 14301. [6] J.D. Dunitz, Proc. Natl Acad. Sci. USA 93 (1996) 14260± 14266. [7] X. Zhao, Scientarum Naturlium Universitaties Nankaiensis (Nankai Daxue Xuebao) 1 (1978) 1±18 in Chinese. [8] X. Zhao, Chin. J. Mol. Sci. 1 (2) (1981) 99±107 Chem. Abstr. 96: 129928v. [9] X. Zhao, Chin. J. Mol. Sci. 2 (1) (1982) 43±48 Chem. Abstr. 97: 181339p. [10] X. Zhao, Application of Symmetry Principle in Field Theory to Chemistry, Science Press, Beijing, 1986 in Chinese; Chem. Abstr. 106: B38793f. [11] M. Born, J.P. Oppenheimer, Ann. Physik. 84 (1917) 457. [12] M. Weissbluth, Atoms and Molecules, Academic Press, New York, 1987 pp. 551±555. [13] X. Yi, J. Liu, X. Zhao, Chin. J. Mol. Sci. (Ed. Eng.) 1 (2) (1983) 57±64. [14] X. Yi, J. Liu, X. Zhao, Fenzi Kexue Yu Huaxue Yanjiu 3 (2) (1983) 41±48 in Chinese; Chem. Abstr. 99: 147426w. [15] P. Roman, Theory of Elementary Particles, North-Holland, Amsterdam, 1964 pp. 246±248. [16] X. Zhao, Sci. Bull. (K'o Hsueh T'ung Pao) 24 (2) (1979) 76± 78 in Chinese; Chem. Abstr. 91: 174952k. [17] X. Zhao, Mol. Sci. Chem. Res. (Fenzi Kexue Xuebao) 2 (4) (1982) 177±184 Chem. Abstr. 98: 222597n. [18] D.M. Bishop, Group Theory and Chemistry, Clarendon Press, Oxford, 1973 pp. 138±143. [19] X. Zhao, Sci. Bull. (K'o Hsueh T'ung Pao) 24 (19) (1979) 882±884 in Chinese; Chem. Abstr. 92: 172382s. [20] H. Eyring, S.H. Lin, S.M. Lin, Basic Chemical Kinetics, Wiley, New York, 1980 pp. 58±61. [21] A. Einstein, Ann. Phys. 37 (1912) 832. [22] A. Einstein, Ann. Phys. 38 (1912) 881. [23] J.G. Calvart, J.N. Pitts Jr., Photochemistry, Wiley, New York, 1966 pp. 19±21. [24] X. Zhao, Acta Chim. Sin. (Hua Hsueh Hsueh Pao) 37 (1979) 247±254 in Chinese; Chem. Abstr. 92: 117175b. [25] Y. Pan, Z. Cai, H. Song, X. Zhao, Chin. J. Chem. Phys. (Huaxue Wuli Xuebao) 9 (5) (1996) 401±405 in Chinese; Chem. Abstr. 126: 171143y.

X. Zhao et al. / Journal of Molecular Structure (Theochem) 586 (2002) 209±223 [26] X. Yi, J. Liu, X. Zhao, Chin. J. Mol. Sci. (Ed. Eng.) 3 (1) (1985) 47±56. [27] X. Zhao, M. Yuan, Chin. J. Mol. Sci. (Ed. Eng.) 2 (1) (1984) 57±65. [28] X. Zhao, M. Yuan, Fenzi Kexue Yu Huaxue Yanjiu 4 (1) (1984) 53±60 in Chinese; Chem. Abstr. 101, 54119t. [29] B. Wu, H. Sun, X. Zhao, Chem. J. Chin. Univ. (1990) 27±32 Special of Academic Theses in celebration of Professor Tang Auqing's teaching and researching for ®ve decades; in Chinese. [30] X. Zhao, Chin. Sci. Bull. (Ed. Eng.) 26 (7) (1981) 601±604. [31] X. Zhao, Kexue Tongbao 26 (5) (1981) 276±278 in Chinese. [32] J. Wang, H.R. Menapace, J. Org. Chem. 33 (1968) 3794± 3796. [33] J. Bilhou, J.L. Basset, W.F. Graydon, J. Am. Chem. Soc. 99 (1977) 4083±4090. [34] C.P. Casey, T.J. Burhardt, J. Am. Chem. Soc. 94 (1972) 6543± 6544.

223

[35] C.P. Casey, T.J. Burhardt, J. Am. Chem. Soc. 95 (1973) 5833± 5834. [36] C.P. Casey, T.J. Burhardt, J. Am. Chem. Soc. 96 (1974) 7808± 7809. [37] C.P. Casey, T.J. Burhardt, J. Am. Chem. Soc. 98 (1976) 608± 609. [38] X. Zhao, Sci. Bull. (K'o Hsueh T'ung Pao) (1980) 215±216 A Special Issue of Mathematics, Physics and Chemistry; in Chinese; Chem. Abstr. 94: 127458b. [39] K.E. Shuler, J. Chem. Phys. 21 (1953) 624±632. [40] J. Callaway, Quantum Theory of Solid State, Second ed, Academic Press, New York, 1991 pp. 2±4. [41] F. Bloch, Z. Physik. 52 (1928) 555. [42] K.W. Boer, Survey of Semiconductor Physics, Electrons and Other Particles in Bulk Semiconductors, Van Nostrand Reinhold, New York, 1990 pp. 697±749.