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1、The original Renner-Teller eff ect Paul E.S. Wormer Theoretical Chemistry, University of Nijmegen December 31, 2003 Abstract In his classic paper R. Renner Z. Phys. 92, 172 (1934) considered electronically excited CO2as an example of an open-shell linear triatomic molecule. Linear triatomic molecule
2、s have a doubly degenerate bending vibration. Renner was the fi rst to study the eff ect of the electronic motion on the bending vibrational wave functions. For a electronic (ground) state there is no coupling of the electronic motion with vibration, for a electronic state there is appreciable coupl
3、ing, while the magnitude of the couplings with ( = 2), ( = 3), ., electronic states, decreases steeply as function of electronic angular momentum (in the model of Renner). Renner did not consider electron spin or the rotation of the triatom. In this report Renners arguments will be summarized. Intro
4、duction In 1933 Gerhard Herzberg and Edward Teller Z. Phys. Chemie B21, 410 (1933) recognized that the potential of a triatomic linear molecule in a de- generate electronic state splits into two when the molecule is bent. A year later this eff ect was worked out in detail by Rudolf Renner, who gave
5、an ex- planation of this splitting and showed that the bending and electronic motion are coupled. He predicted that this coupling would give rise to anomalies in the vibrational side bands of electronic spectra. Herzberg refers to this as the “Renner-Teller” eff ect in one of his infl uential books
6、Molecular Spectra and Molecular Structure Vol. III, Reprint Edition, Krieger, Malabar (1991) and consequently the eff ect is now generally called after Renner and Teller, although it would be more appropriate to speak of the “Renner-Herzberg” or “Renner-Herzberg-Teller” eff ect as Herzberg was the f
7、i rst author on the 1933 paper. In its simplest form the Renner-Teller eff ect occurs in open-shell linear triatomic molecules. Let the electronic state | i be an eigenstate of Lel z, the projection of the electronic angular momentum operator on the molec- ular axis, with eigenvalue . The states | i
8、 are degenerate with en- ergy Eel |. Let | v,l i be a harmonic bending vibration function, which is an eigenfunction of Lvib z , the projection of the vibrational angular momen- tum operator on the molecular axis, with eigenvalue l. The v + 1 states 1 | v,l i, l = v,v + 2,.,v are degenerate with ene
9、rgy h(v + 1). The orthonormal product states | 1 i | i | v,K i and | 2 i | i | v,K + i are eigenfunctions of Ltot z Lel z 1 + 1 Lvib z (1) with eigenvalue K. In zeroth-order the product kets are degenerate with energy Eel | + h(v + 1). The bending of the molecule generates a coupling between the ele
10、ctronic and vibrational motion (a so-called vibronic coupling) that acts as a perturbation.Diagonalization of the 2 2 matrix of this perturbation on basis of | 1 i and | 2 i gives two new orthonormal zeroth-order eigenvectors (0) + = cos| 1 i + sin| 2 i and (0) = sin| 1 i + cos| 2 i with fi rst orde
11、r energies: E(1) .Clearly, the functions (0) are no longer simple products; this breakdown of the Born-Oppenheimer approximation is the Renner-Teller eff ect. Also belonging to the Renner-Teller eff ect is the observation that the functions (0) are not degenerate in fi rst order. The linear combinat
12、ions (0) are not eigenfunctions of Lel z or Lvib z . However, they are eigenfunctions of Ltot z . Much has been published about the Renner-Teller eff ect after its fi rst experimental observation in 1959. Nowadays, authors referring to this eff ect have usually broader physical phenomena and larger
13、molecules in mind than Renner had in 1934. For instance, Bunker and Jensen Molecular Symmetry and Spectroscopy, NRC Research Press, Ottawa (1998) very specifi cally state that the eff ect is not only due to the coupling of electrons and vibrations, but also to coupling with rotationsa rovibronic eff
14、 ect. Examples they quote are linear 5- and 6-atom molecules such as HC3N and HC4H+. It is of interest, historically and scientifi cally, to recall the content and scope of the original Renner paper. Renners paper is well accessible due to the English translation by H. Hettema Quantum Chemistry: Cla
15、ssic Sci- entifi c Papers, World Scientifi c, Singapore (2000). Renners work will be rephrased in modern language and condensed in this report. A knowledge of the theory of the 2-dimensional harmonic oscillator is presupposed in Ren- ners paper and since we do not want to assume this we start with a
16、 short exposition of the 2D oscillator. Then we will give a resum e of Renners work. In doing this, we will sometimes modify the original arguments somewhat, but we will indicate this, either in the text or in a footnote. 2 The 2-dimensional harmonic oscillator The Hamiltonian of the 2-D isotropic h
17、armonic oscillator is Hv= 2 2 ? 2 x2 + 2 y2 ? + 1 2(x 2 + y2)f, where is the (reduced) mass, f is the force constant and is Plancks con- stant divided by 2. Multiply left and right hand side of the corresponding eigenvalue equation by 1/p/f, substitute r (f/2)1/4r and it becomes Hv ? 1 2 ? 2 x2 + 2
18、y2 ? + 1 2(x 2 + y2) ? = E, where E is in units 1= pf/. As is well-known, the eigenvalue equation of Hvfactorizes into an x and a y part: 1 2 ? 2 x2 x2 ? nx(x)=(nx+ 1 2)n x(x) (2) 1 2 ? 2 y2 y2 ? ny(y)=(ny+ 1 2)n y(y), (3) where nxand nyare natural numbers and the total energy is v + 1, with v nx+ n
19、y. In polar coordinates ?x y ? = ?rcos r sin ? , Hvbecomes: Hv=Tv+ 1 2r 2 (4) with Tv=1 2 ? 2 r2 + 1 r r + 1 r2 2 2 ? .(5) Substitute v(r,) = eilR(r) into Hvv= (v + 1)vand we get for the r-dependent part ? 2 r2 + 1 r r l2 r2 r2+ 2(v + 1) ? R(r) = 0.(6) Recall that periodicity in requires l to be int
20、eger. By three substitutions the equation in r can be transformed into the associated Laguerre diff erential equation see, e.g., H. Margenau and G. M. Murphy, The Mathematics of 3 Physics and Chemistry, Van Nostrand, New York, 2nd Ed. (1964). First we substitute R(r) = rlexp(r2 /2)L(r) (l 0), which
21、gives a diff erential equation for L(r), 2L(r) r2 + L(r) r ? 2r + 2l + 1 r ? + L(r)(2v 2l) = 0. Next substitute x = r2, then x 2L(x) x2 + L(x) x (l + 1 x) + L(x)(v l) 2 = 0. Substitute fi nally n = (v + l)/2 and k = l, so that we get x 2L(x) x2 + L(x) x (k + 1 x) + L(x)(n k) = 0, which has as soluti
22、ons the associated Laguerre polynomials Lk n(x) = dk dxk Ln(x) with n = 0,1,2,. and k n. Here Ln(x) is an ordinary Laguerre polynomial of order n. In summary, v(r,) = eilr|l|exp(r2/2) L|l| (v+|l|)/2(r 2) is an (unnormalized) eigenfunction of Hvwith eigenvalue v+1, v = 0,1,2, Further |l| v and l = v
23、+ 2n, so that l = v,v + 2,.,v 2,v. The vibrational angular momentum quantum number l labels the v+1 degenerate states. Also for fi xed l: v = |l|,|l| + 2,|l| + 4, Renner uses the notation RNm(r) for the radial part of these solutions, with m |l| and RNm(r) Lm (N+m)/2,m(r 2), so that N v. Electron-vi
24、bration coupling according to Renner Of course, even an approximate solution of the electronic Schr odinger equa- tion for a molecule of the size of CO2was completely out of reach in 1934. Renner had to make many simplifi cations and to use intuitive arguments to obtain the required coupling terms.
25、The simplifi cations he made are the following: the molecule does not rotate or translate. The CO symmetric and antisymmetric stretching vibrations are completely decoupled from the doubly degenerate bending. The electronic wave function is an orbital (a function of one electron). The molecule is be
26、nt as shown in Fig. 1. We write the spherical polar coordinates of the electron as q = (,), where is the 4 Figure 1: Bending coordinates in the molecule CO2 azimuthal (rotation around z-axis) angle and is the distance of the electron to the origin. In a linear molecule the orbitals I(q)=f(,)cos( )(7
27、) II(q)=f(,)sin( )(8) are degenerate, where is a natural number. The orbital I(q) is symmetric under refl ection in the plane of the distorted molecule in Fig. 1, while II(q) is antisymmetric under the same refl ection. Note that these electronic or- bitals contain explicitly the nuclear coordinate
28、. In the one-electron case Lel z = i/ and Lvib z = i/. Clearly,1the orbitals are eigenfunc- tions with K = 0 of Ltot z defi ned in Eq. (1). Renner now makes the crucial assumption that these orbitals are eigenfunctions of a clamped nuclei eff ec- tive one-electron Hamiltonian2He, even in the case of
29、 (small) bending, and writes HeI(q)=(E+ (r)I(q)(9) HeII(q)=(E (r)II(q),(10) here Eis the energy of the two orbitals in the linear (r = 0) case. The two orbital energies lie symmetrically with regard to E. Later in the paper it is assumed that (r) 0 for all r, so that the antisymmetric orbital is low
30、er in energy than the symmetric one. 1Renner did not see the angular momentum aspect of his work too clearly. His “Gesamt- drehimpuls” (total angular momentum) refers to vibrational coordinates only. 2Renner writes out a full N-electron Hamiltonian He and calls the single electron that he considers
31、consequently the “fi rst” electron 5 To evaluate (r), Renner assumes that the bending introduces a vibra- tional dipole moment nuc= er located at the origin and that the electron at position q interacts with this dipole. He approximates the usual charge- dipole interaction q nuc/3by U = ercos( )/2an
32、d sets up a matrix3 of this vibronic interaction on basis of the functions (7) and (8). The pertur- bation matrix decomposes into two blocks, one on basis of the sine functions and the other on basis of the cosine functions, because h I| cos( ) | IIi = 0. The matrix elements are taken to be proporti
33、onal to Z 2 0 coscoscosd= Z 2 0 sincossind= 2 , with , , 0, and = 1. There is a common proportionality constant, which is an integral over the remaining electron coordinates. Since the matrix elements vanish unless = 1, the two blocks are tridiagonal and its elements are all equal, except for the ca
34、se = 0 () and = 1 () in the cosine block, where the value is . The same matrix element is zero in the sine block. If the two blocks were identical, the perturbation would shift sine and cosine orbitals by the same amount and their degeneracy would not be lifted. However, the blocks diff er in the si
35、ngle matrix element h | cos( ) | i, which implies that the cosine orbitals shift diff erently than the sine orbitals. By applying second-order perturbation theory Renner shows that the energy of cos( ) shifts in second order in U, i.e., the splitting between the cosine and sine orbital is quadratic
36、in r. Similarly the orbitals split in fourth order (energy diff erence is quartic in r), etc. Renner then argues that only the case = 1 is of physical interest and considers subsequently only the second-order (in U) electronic energy splitting (r) = r2(11) as part of the vibronic coupling and restri
37、cts his attention to orbitals Eqs. (7) and (8) with = 1 only. The positive constant enters the theory as a free parameter. Often the splitting of the orbital energy upon bending is referred to as the Renner eff ect, but the description of this eff ect is only a prelude to the study of the eff ect of
38、 the splitting on the bending vibrational energies. 3Here and in the calculation of the perturbation matrix Renner sets = 0 6 The Renner equations Since we have here a case of two close lying potential energy surfaces, coin- ciding for r = 0, the usual single-product Born-Oppenheimer Ansatz will not
39、 do. Instead Renner makes the inner product Ansatz tot(q,r,) = ?I (q),II(q)? ? I(r,) II(r,) ? (12) where I(q) and II(q) are the eigenfunctions of He , the eff ective one- electron Hamiltonian appearing in Eqs. (9) and (10).In this subsection we will derive coupled equations from which I(r,) and II(r
40、,) may be solved. We refer to these equations as the “Renner equations”. The total vibronic Hamiltonian is Htot= He+ Hv(13) with Hv= Tv+ r2and Tv= 2 2 ? 2 r2 + 1 r r + 1 r2 2 2 ? ,(14) and where = 1 2f. Following Renner, we consider only the case = 1 and write E E=1.Substitute Eq. (12) into the eige
41、nvalue equation Htottot= Etottotcf. also Eqs. (7) and (8): Htottot=f(,)(E+ r2)cos( )I(r,) +f(,)(E r2)sin( )II(r,) +f(,)Hvcos( )I(r,) +f(,)Hvsin( )II(r,) =Etotf(,) ?cos( )I (r,) + sin( )II(r,)?. Here we assumed that Hvf(,) = f(,)Hv, i.e., that the parametric de- pendence of f(,) on the nuclear coordi
42、nates r and is weak. Defi ne e Tv 2 2 ? 2 r2 + 1 r r + 1 r2 2 2 1 r2 ? p 2 r2 then Tvcos( )I(r,)=cos( )eTvI(r,) + sin( )pI(r,) Tvsin( )II(r,)=sin( )eTvII(r,) cos( )pII(r,). 7 The eigenvalue equation of Htotcan be written as cos( ) h? e Tv+ ( + )r2 ? I(r,) pII(r,) i +sin( ) h? e Tv+ ( )r2 ? II(r,) +
43、pI(r,) i =E cos( )I(r,) + E sin( )II(r,), where we defi ned E = Etot Eand divided out f(,).Equating the factors of cos( ) and of sin( ), we get the coupled equations for the vibrational functions Iand II: e Tv+ ( + )r2p +p e Tv+ ( )r2 ! |z b H ? I(r,) II(r,) ? = E ? I(r,) II(r,) ? .(15) These equati
44、ons may be transformed to exponential basis; the total vibronic wave function is invariant under such a transformation, tot(q,r,) = 1 2 ?I (q),II(q)? ?1 1 ii ?1 i 1i ? I(r,) II(r,) ? . If = 0 the orbitals I(q) and II(q) are degenerate and then we may as well depart from e I(q)=f(,)ei()(16) e II(q)=f
45、(,)ei()(17) instead of from Eqs. (7) and (8). This has the advantage that the off -diagonal elements in Eq. (15) are moved to the diagonal. These off -diagonal elements are due to the fact that the fi rst derivative of a sine (cosine) is a cosine (sine), while the fi rst derivative of an exponential
46、 is an exponential. The coupled equations uncouple (for = 0) and both become a harmonic oscillator equa- tion in polar coordinates. Hence, we consider the following transformation of Eq. (15) 1 2 ?1 i 1i ? b H ?1 1 ii ?1 i 1i ? I II ? = E ?1 i 1i ? I II ? . Or, e Tv+ r2 ipr2 r2 e Tv+ r2+ ip ! e I(r,
47、) e II(r,) ! = E e I(r,) e II(r,) ! (18) 8 with e I(r,) e II(r,) ! = 1 2 ?1 i 1i ? I(r,) II(r,) ? .(19) Although Renner does not give Eqs. (18) explicitly, they may be justifi ably called the Renner equations; they are completely equivalent with Eqs. (15), which are given by Renner. We reiterate that the two parameters in this equation: (twice the force constant) and (twice the splitting of the orbitals due to bending) are free parameters. Approximate solution of the Renner equations Renner solves his equat
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