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Calculation of vibronic coupling constant and vibronic coupling density analysis
Journal of Molecular Structure 838 (2007) 116–123 www.elsevier.com/locate/molstruc
Calculation of vibronic coupling constant and vibronic coupling density analysis Ken Tokunaga a, Tohru Sato a
a,b,*
, Kazuyoshi Tanaka
a,c
Department of Molecular Engineering, Graduate School of Engineering, Kyoto University, Kyoto-Daigaku-Katsura, Nishikyo-ku, Kyoto 615-8510, Japan b Fukui Institute for Fundamental Chemistry, Kyoto University, Takano-Nishihiraki-cho 34-4, Sakyo-ku, Kyoto 606-8103, Japan c Core Research for Evolutional Science and Technology, Japan Science and Technology Agency (JST-CREST), Japan Received 20 December 2006; accepted 29 December 2006 Available online 18 January 2007
Abstract Vibronic coupling, electron–phonon interaction, constants of Jahn–Teller molecules, C5H5 and C5D5, are computed as matrix elements of the electronic operator of the vibronic coupling operator using the electronic wave functions calculated by generalized restricted Hartree–Fock (GRHF) and state-averaged complete active space self-consistent-field (CASSCF) methods. The calculated values of vibronic coupling constants for C5H5 and C5D5 agree well with the experimental values. Vibronic coupling density analysis can explain the isotope effect on the vibronic coupling from view of the electronic and vibrational structures. Changes of the vibrational modes as well as the frequencies upon the deuteration can affect the vibronic coupling. Vibronic density analysis provides a local picture of the coupling in a molecule, and it enables us to control the coupling. This could open a way to engineering of the vibronic coupling, vibronics. 2007 Elsevier B.V. All rights reserved. Keywords: Jahn–Teller effect; Vibronic coupling; Electron–phonon coupling; Isotope effect; Cyclopentadienyl
1. Introduction
0 0 ½E002 1 ¼ a1 e 2 ;
Vibronic coupling, electron–phonon coupling, is one of the most investigated problems [1–4] since it plays an important role in many interesting phenomena. Cyclopentadienyl radical (C5H5, Fig. 1) is one of the most investigated Jahn–Teller (JT) molecules, since it has been a target of spectroscopy and quantum chemistry [5–11]. Since the strength of the coupling is measured by a vibronic (electron–phonon) coupling constant, the coupling constant has been calculated [12,13]. The electronic state of the radical with D5h symmetry is 2 00 E1 (see Fig. 2). The vibrational mode which couples to the electronic E001 state can be deduced as
as long as a linear vibronic coupling is considered. Therefore, the radical will give rise to a distortion along the e02 modes, and have a C2v geometrical structure. Four e02 modes are Jahn–Teller active among 3N 6 = 24 vibrational modes:
*
Corresponding author. Address: Department of Molecular Engineering, Graduate School of Engineering, Kyoto University, Kyoto-DaigakuKatsura, Nishikyo-ku, Kyoto 615-8510, Japan. Tel.: +81 75 383 2803; fax: +81 75 383 2556. E-mail address: [email protected] (T. Sato). 0022-2860/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.molstruc.2006.12.049
Cvib ¼ 2a01 a02 3e01 4e02 a002 e001 2e002 :
ð1Þ
ð2Þ
It is well known that one of the main isotope effects is red shift in frequency due to an increase of reduced mass. However, isotope substitution can affect not only the frequencies but also the direction and displacement of normal modes. Considering these two points, deuteration effect on vibronic coupling of C5H5 had been discussed by Applegate et al. [10]. We discuss the deuteration effect on the vibronic coupling using vibronic coupling density analysis proposed in our previous work [14]. Vibronic coupling density analysis enables us to explain the order of magnitude
K. Tokunaga et al. / Journal of Molecular Structure 838 (2007) 116–123
117
Table 2 Total energy (a.u.) and vibronic coupling constants V e02 ðjÞ (104 a.u.) of C5X5 Energy
GRHF
CAS(5,5)
192.1719
192.2357
C5H5
e02 ð1Þ e02 ð2Þ e02 ð3Þ e02 ð4Þ
8.76 13.60 14.61 7.83
4.78 11.63 14.58 3.20
C5D5
e02 ð1Þ e02 ð2Þ e02 ð3Þ e02 ð4Þ
0.27 12.54 17.73 6.44
1.21 8.66 17.09 2.23
Fig. 1. Structure of C5X5 and the coordinate axes. Because of linear vibronic coupling, symmetry of C5X5 is lowered from D5h to C2v.
E
"(1)
Basis set employed is 6-31G(d,p). The vibrational vectors employed are those obtained by RHF/6-31G(d,p) for C5 X 5.
e1"(1)
Table 3 Scaled dimensionless vibronic coupling constants D0e0 ðjÞ and experimental 2 values of C5X5
e2
(a2) ψε
ψθ (b1)
CAS(5,5)/6-31G(d,p)
Expt.
C5H5
e02 ð1Þ e02 ð2Þ e02 ð3Þ e02 ð4Þ
0.67 1.16 0.97 0.07
0.62 1.07 0.85 –
C5D5
e02 ð1Þ e02 ð2Þ e02 ð3Þ e02 ð4Þ
0.21 1.16 1.19 0.07
– 0.88 1.12 –
a2"(1) (b1) Fig. 2. Frontier p orbitals of C5X5.
of vibronic coupling constants [14] and the relation between the frontier electron density and the vibronic coupling [15]. In this paper, we present a calculation of the vibronic coupling constants of C5X5 (X = H, D) and compare them with experimental data by Applegate et al. [11]. Furthermore, using the vibronic coupling density, the isotope effect on the vibronic coupling is discussed from view of the electronic and vibrational structures. The paper is organized as follows: in Section 2, the model Hamiltonian is presented. The method of the calculation
The vibrational vectors and frequencies employed in these calculations were obtained with RHF/6-31G(d,p) for C5 X 5 . Negative signs are neglected. The experimental values are taken from Ref. [11].
Table 1 Frequencies (cm1) of C5 X 5 calculated using RHF/6-31G(d,p) with the structure optimized by RHF/6-31G(d,p) and those of C5X5 with the D5h symmetry calculated using GRHF/6-31G(d,p) with the structure optimized by RHF/6-31G(d,p) Species
Method
e02 ð1Þ
e02 ð2Þ
e02 ð3Þ
e02 ð4Þ
C5 H 5 C5H5
RHF GRHF Expt.
914 918 872
1144 1198 1041
1497 1596 1320
3263 3313 –
C5 D 5 C5D5
RHF GRHF Expt.
780 814 –
939 939 861
1453 1556 1353
2401 2448 –
Experimental values of C5X5 are also represented. The experimental values are given in Ref. [11].
Fig. 3. Contour map on the plane z = 1.0 (a.u.) of frontier electron density qh(r) of C5X5.
118
K. Tokunaga et al. / Journal of Molecular Structure 838 (2007) 116–123
is described in Section 3. In Section 4, we show the results and compare them with the experimental values. We also discuss the isotope effect on the vibronic coupling using the vibronic coupling density in this section. Finally, we conclude this work in Section 5. 2. Vibronic Hamiltonian The model Hamiltonian employed in this work is [14] ! X X oU h2 o2 H¼ ðr; R Þ þ Q þ H e 0 oQi R0 i 2 oQ2i i i X1 x2i Q2i ; þ ð3Þ 2 i where r is a set of electronic coordinates, Qi a normal coordinate of the mode i, R0 a Jahn–Teller crossing geometry, He an electronic Hamiltonian, U a sum of an electron–electron, electron–nuclear, nuclear–nuclear potential operator, xi a frequency of the mode i. Only the linear vibronic coupling is considered. The operator involved in Eq. (3),
oU oQi
¼ Vi ¼
X
R0
vi ðaÞ;
ð4Þ
a
is called the electronic operator of the mode i, and Qi Vi describes the vibronic coupling. Vi is a sum of a oneelectron operator vi. The model Hamiltonian is treated within the space spanned by the electronic state jEhæ and jEæ; " ^ JT ¼ E0 þ H
X i
þ
(
h2 2
o2 oQ2i
!
1 þ x2i Q2i 2
X
hEhjVi jEhi
hEhjVi jEi
i
hEjVi jEhi
hEjVi jEi
)# ^0 r ! Qi ;
ð5Þ
where {|Ehæ, |Eæ} denotes the degenerate electronic state ^0 unit matrix. {j E001 hi; j E001 i}, E0 eigenenergy of He , and r Using Wigner–Eckart theorem, the vibronic coupling matrix of e02 ðjÞ modes can be reduced as
Fig. 4. Contour maps on the plane z = 1.0 (a.u.) of potential derivative ve02 ðjÞh ðrÞ of (a) e02 ð1Þh mode of C5H5 and (b) e02 ð2Þh mode of C5D5.
K. Tokunaga et al. / Journal of Molecular Structure 838 (2007) 116–123
V^ e02 ðjÞh ¼
hEhjVe02 ðjÞh jEhi
hEjV jEhi 1 0 ¼V e02 ðjÞ ; 0 1 e02 ðjÞh
hEhjVe02 ðjÞh jEi hEjV
e02 ðjÞh
!
jEi
hEhjVe02 ðjÞ jEhi hEhjVe02 ðjÞ jEi hEjVe02 ðjÞ jEhi hEjVe02 ðjÞ jEi 0 1 ¼V e02 ðjÞ ; 1 0
ð6Þ !
V^ e02 ðjÞ ¼
ð7Þ
where the integrals hEhjVe02 ðjÞh jEhi and so on are called vibronic coupling integrals (VCI). e02 ðjÞh and e02 ðjÞ denote a pair of degenerated normal mode e02 ðjÞ. The vibronic coupling constants (VCC) is defined as
119
Finally, we introduce here some dimensionless quantities. For a vibrational mode i, dimensionless normal coordinate qi, dimensionless coupling constant Di, and scaled dimensionless coupling constant D0i can be defined as [15] rffiffiffiffiffi xi ð9Þ qi ¼ Q; h i Vi Di ¼ pffiffiffiffiffiffiffiffi ; ð10Þ hx3i D0i ¼jDi ;
ð11Þ
where j is a scaling parameter which reproduces the Jahn– Teller stabilization energy calculation with the same level of theory.
V e02 ðjÞ ¼hEhjVe02 ðjÞh jEhi ¼ hEjVe02 ðjÞh jEi ¼hEhjVe02 ðjÞ jEi ¼ hEjVe02 ðjÞ jEhi;
ð8Þ
which is the quantity we will calculate in this article. The VCC can be obtained from a calculation of these integrals.
3. Method of calculation In the generalized restricted Hartree–Fock (GRHF) method, the wave functions for E states are expressed as a single Slater determinant:
Fig. 5. Contour maps on the plane z = 1.0 (a.u.) of potential derivative ve02 ðjÞh ðrÞ of (a) e02 ð3Þh mode of C5H5 and (b) e02 ð3Þh mode of C5D5.
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K. Tokunaga et al. / Journal of Molecular Structure 838 (2007) 116–123
jEhðGRHFÞi ¼j wm awm b we00 h awe00 awe00 bi 1
1
1
¼:jhi; jEðGRHFÞi ¼j wm awm b we00 h awe00 awe00 h bi 1
1
¼:ji;
ð12Þ
1
ð13Þ
and a, b are spin functions. Generally, the vibronic coupling matrix for the GRHF wave function can be written as hhjVi jhi hhjVi ji GRHF ^ ¼ Vi : ð14Þ hjVi jhi hjVi ji For the e02 modes, the vibronic coupling constant over a single Slater determinants jhæ and jæ can be decomposed into orbital vibronic coupling constants [4] for the h mode:
where m runs over the occupied molecular orbitals with e01 , e001 symmetries, and w(m)h and w(m) denote the degenerate pair of molecular orbitals. n(m)h and n(m) are occupation numbers of w(m)h and w(m). Note that the orbitals with the e01 , e001 symmetries can couple to the e02 modes since ½E01 ¼ ½E001 ¼ A01 E02 . Furthermore, from a symmetry of Clebsch–Gordan coefficients, hwðmÞh jve02 ðjÞh jwðmÞh i þ hwðmÞ jve02 ðjÞh jwðmÞ i ¼ 0:
Therefore, the vibronic coupling matrix in the GRHF methods is equal to the orbital vibronic coupling matrix: ! 0 hwe00 jve02 ðjÞh jwe00 i 1 1 GRHF V^ e0 ðjÞh ¼ ; ð17Þ 2 0 hwe00 h jve02 ðjÞh jwe00 h i 1 1 ! 0 0 hwe00 jve2 ðjÞ jwe00 h i 1 1 GRHF V^ e0 ðjÞ ¼ ; ð18Þ 2 hwe00 h jve02 ðjÞ jwe00 i 0 1
V GRHF ¼hhjVe02 ðjÞh jhi ¼ hjVe02 ðjÞh ji e02 ðjÞ X ¼ m2e0 e00 fnðmÞh hwðmÞh jve02 ðjÞh jwðmÞh i 1
1
þ nðmÞ hwðmÞ jve02 ðjÞh jwðmÞ ig;
ð15Þ
ð16Þ
1
In the state-averaged CASSCF method, the wave functions for E states are expressed as a linear combination of Slater determinants. The procedure of the calculation is the same as that expressed in Ref. [14].
Fig. 6. Contour maps on the plane z = 1.0 (a.u.) of potential derivative ve02 ðjÞh ðrÞ of (a) e02 ð4Þh mode of C5H5 and (b) e02 ð4Þh mode of C5D5.
K. Tokunaga et al. / Journal of Molecular Structure 838 (2007) 116–123
In order to obtain the Jahn–Teller crossing geometry R0 and vibrational structure, restricted Hartree–Fock (RHF) method is employed for C5 H 5 . At the geometry R0, we employed state-averaged CASSCF method using Gaussian 03 [16] and GRHF method using CADPAC [17] to determine the wave functions of C5X5. All calculations were performed using the 6-31G(d,p) basis set. The vibronic coupling constant was evaluated using a program coded by us. 4. Results and discussion 4.1. Geometrical and vibrational structure The optimized symmetry of C5 H 5 by RHF/6-31G(d,p) ˚ , rC–H = 1.0800 A ˚ ). We use the is D5h(rC–C = 1.4021 A geometry of C5 H5 as that of the JT crossing point R0 of the radical C5X5. Vibrational frequencies are shown in Table 1. Though the frequencies of e02 ð3Þ modes are a little different among
121
two methods, vibrational structures are almost the same. Therefore, we took the geometrical and vibrational structure of C5 H 5 as that of the JT crossing structure throughout this study [18]. 4.2. Vibronic coupling constant The calculated V e02 ðjÞ and D0e0 ðjÞ of C5X5 using GRHF and 2 CASSCF methods are tabulated in Tables 2 and 3, respectively. Our results agree well with experimental data, and the calculated constants V e02 ðjÞ of GRHF method qualitatively coincide with those of CASSCF method. Thus, results of GRHF method are worth analyzing by vibronic coupling density analysis. 4.3. Vibronic coupling density analysis 4.3.1. Vibronic coupling density Vibronic coupling density ge0 ðjÞ ðrÞ [14] for the e02 ðjÞ mode 2 is defined by
Fig. 7. Contour maps on the plane z = 1.0 (a.u.) of potential derivative ve02 ðjÞh ðrÞ of (a) e02 ð2Þh mode of C5H5 and (b) e02 ð1Þh mode of C5D5.
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K. Tokunaga et al. / Journal of Molecular Structure 838 (2007) 116–123
ge0 ðjÞ ðrÞ ¼wh ðrÞwh ðrÞve02 ðjÞh ðrÞ 2
¼qh ðrÞve02 ðjÞh ðrÞ;
ð19Þ
where qh(r) is the frontier electron density of the molecular orbital wh, and ve02 ðjÞh ðrÞ the one-electron operator. Using ge0 ðjÞ , the VCC is written as 2 Z 0 ð20Þ V e2 ðjÞ ¼ drge0 ðjÞ ðrÞ: 2
Using the vibronic coupling density, we can analyze the calculated VCC in terms of the electronic structure qh(r) and the derivative of the potential with respect to the normal coordinate ve02 ðjÞh ðrÞ. 4.3.2. Frontier electron density In Fig. 3, frontier electron density of C5X5 is shown. It should be noted that frontier electron density is localized on C1 atom and C3–C4 bond.
4.3.3. e02 ð1Þ, e02 ð3Þ, and e02 ð4Þ modes of C5H5 Normal modes and potential derivatives are shown in Figs. 4–6. From Figs. 4–6, we can see that e02 ð1Þ, e02 ð3Þ, and e02 ð4Þ modes of C5H5 correspond to e02 ð2Þ, e02 ð3Þ, and e02 ð4Þ modes of C5D5, respectively [19]. In these e02 modes, direction and displacement of C5H5 are very similar to those of C5D5. Therefore, vibronic coupling density, which is a product of frontier electron density (Fig. 3) and potential derivative (Figs. 4–6), is not influenced by deuteration. The isotope effect on the vibronic coupling of these modes is small. This result agrees well with the experimental data [11] and the previous theoretical data [10]. 4.3.4. e02 ð2Þ mode of C5H5 Normal modes and potential derivatives are shown in Fig. 7. From Fig. 7, we can see that e02 ð2Þ mode of C5H5 corresponds to e02 ð1Þ mode of C5D5 [19]. In contrast to the other normal modes, the deuteration leads to the large reduction of the displacement of C3–C4 stretching motion. This gives rise to a reduction of distribution of potential derivative on C3–C4 bond. Since coincidence between q (Fig. 3) and v (Fig. 7) is decreased, the vibronic coupling density (Fig. 8) is also reduced. Therefore, the isotope effect on the vibronic coupling of this mode is very large, as also pointed out by Applegate et al. [10]. These results indicate that not only red shift of frequencies but also change of the direction and displacement of normal modes due to deuteration can affect the vibronic coupling constant. 5. Conclusion We present a calculation of vibronic coupling constants of C5H5 and C5D5. The vibronic coupling constants are calculated as matrix elements of the electronic operator of the vibronic coupling using GRHF and CASSCF methods. Our results agree well with the experimental and theoretical values by Applegate et al. Vibronic coupling density analysis is applied to analyze the isotope effect on the vibronic coupling from view of the electronic and vibrational structures. This analysis reveals that the coupling of the e02 ð2Þ mode of C5H5 is largely reduced by isotope substitution, because the displacement of C–C stretching motion is decreased. This result indicates that not only red shift of frequencies but also changes in the direction and magnitude of the displacement due to isotope substitution can strongly affect the vibronic coupling. Vibronic density analysis provides a local picture of the coupling in a molecule, and it enables us to control the coupling. This could open a way to engineering of the vibronic coupling, vibronics. Acknowledgements
Fig. 8. Contour maps on the plane z = 1.0 (a.u.) of vibronic coupling density ge0 ðjÞ with respect to (a) e02 ð2Þh mode of C5H5 and (b) e02 ð1Þh mode 2 of C5D5.
Numerical calculation was partly performed in the Supercomputer Laboratory of Kyoto University and Research Center for Computational Science, Okazaki, Japan.
K. Tokunaga et al. / Journal of Molecular Structure 838 (2007) 116–123
References [1] I.B. Bersuker, The Jahn–Teller Effect and Vibronic Interactions in Modern Chemistry, Plenum, New York, 1984. [2] G. Fischer, Vibronic Coupling: The interaction Between the Electronic and Nuclear Motions, Academic Press, London, 1984. [3] I.B. Bersuker, V.Z. Polinger, Vibronic Interaction in Molecules and Crystals, Springer, Berlin, 1989. [4] I.B. Bersuker, Chem. Rev. (Washington, D.C.) 101 (2001) 1067. [5] A.D. Liehr, Z. Phys. Chem., Neue Folge 9 (1956) 338. [6] L.C. Snyder, J. Chem. Phys. 33 (1960) 619. [7] W.D. Hobey, A.D. McLachlan, J. Chem. Phys. 33 (1960) 1695. [8] R. Meyer, F. Graf, T.-K. Ha, Hs.H. Gu¨nthard, Chem. Phys. Lett. 66 (1979) 65. [9] W.T. Borden, E.R. Davidson, J. Am. Chem. Soc. 101 (1979) 3771. [10] B.E. Applegate, T.A. Miller, T.A. Barckholtz, J. Chem. Phys. 114 (2001) 4855. [11] B.E. Applegate, A.J. Bezant, T.A. Miller, J. Chem. Phys. 114 (2001) 4869.
[12] [13] [14] [15] [16] [17]
123
T.A. Barckholtz, T.A. Miller, J. Phys. Chem. A 103 (1999) 2321. T. Kato, K. Hirao, Adv. Quantum Chem. 44 (2003) 257. T. Sato, K. Tokunaga, K. Tanaka, J. Chem. Phys. 124 (2006) 024314. K. Tokunaga, T. Sato, K. Tanaka, J. Chem. Phys. 124 (2006) 154303. M.J. Frisch et al., Gaussian 03 Revision C.02, Wallingford, CT, 2004. Cambridge Analytic Derivative Package (CADPAC) Issue 6.5, Cambridge UK, 2001. [18] Energy gap at R0 between the C5 H 5 geometry obtained by RHF method and the C5H5 geometry obtained by CASSCF method is due to the distortion along two totally symmetric modes, so that V e02 ðjÞ is little influenced by such a geometry difference. [19] We can see by eye the similarity of the e02 ð2Þh mode of C5D5 (Fig. 4(b)) to the e02 ð1Þh mode of C5H5 (Fig. 4(a)) in the upper part of the pentagon, but to the e02 ð2Þh mode of C5H5 (Fig. 7(a)) in the lower part of the pentagon. The criterion of the correspondence in Figs. 4 and 7 is the reduced mass l. In Fig. 4, le02 ð1Þh;C5 H5 is 4.36 (amu) which is a little smaller than le02 ð2Þh;C5 H5 , 4.81 (amu), due to the difference between D and H. In Fig. 7, le02 ð2Þh;C5 H5 is 1.31 (amu) which is half as large as le0 ð1Þh;C5 D5 , 2.38 (amu). 2
Calculation of vibronic coupling constant and vibronic coupling density analysis Ken Tokunaga a, Tohru Sato a
a,b,*
, Kazuyoshi Tanaka
a,c
Department of Molecular Engineering, Graduate School of Engineering, Kyoto University, Kyoto-Daigaku-Katsura, Nishikyo-ku, Kyoto 615-8510, Japan b Fukui Institute for Fundamental Chemistry, Kyoto University, Takano-Nishihiraki-cho 34-4, Sakyo-ku, Kyoto 606-8103, Japan c Core Research for Evolutional Science and Technology, Japan Science and Technology Agency (JST-CREST), Japan Received 20 December 2006; accepted 29 December 2006 Available online 18 January 2007
Abstract Vibronic coupling, electron–phonon interaction, constants of Jahn–Teller molecules, C5H5 and C5D5, are computed as matrix elements of the electronic operator of the vibronic coupling operator using the electronic wave functions calculated by generalized restricted Hartree–Fock (GRHF) and state-averaged complete active space self-consistent-field (CASSCF) methods. The calculated values of vibronic coupling constants for C5H5 and C5D5 agree well with the experimental values. Vibronic coupling density analysis can explain the isotope effect on the vibronic coupling from view of the electronic and vibrational structures. Changes of the vibrational modes as well as the frequencies upon the deuteration can affect the vibronic coupling. Vibronic density analysis provides a local picture of the coupling in a molecule, and it enables us to control the coupling. This could open a way to engineering of the vibronic coupling, vibronics. 2007 Elsevier B.V. All rights reserved. Keywords: Jahn–Teller effect; Vibronic coupling; Electron–phonon coupling; Isotope effect; Cyclopentadienyl
1. Introduction
0 0 ½E002 1 ¼ a1 e 2 ;
Vibronic coupling, electron–phonon coupling, is one of the most investigated problems [1–4] since it plays an important role in many interesting phenomena. Cyclopentadienyl radical (C5H5, Fig. 1) is one of the most investigated Jahn–Teller (JT) molecules, since it has been a target of spectroscopy and quantum chemistry [5–11]. Since the strength of the coupling is measured by a vibronic (electron–phonon) coupling constant, the coupling constant has been calculated [12,13]. The electronic state of the radical with D5h symmetry is 2 00 E1 (see Fig. 2). The vibrational mode which couples to the electronic E001 state can be deduced as
as long as a linear vibronic coupling is considered. Therefore, the radical will give rise to a distortion along the e02 modes, and have a C2v geometrical structure. Four e02 modes are Jahn–Teller active among 3N 6 = 24 vibrational modes:
*
Corresponding author. Address: Department of Molecular Engineering, Graduate School of Engineering, Kyoto University, Kyoto-DaigakuKatsura, Nishikyo-ku, Kyoto 615-8510, Japan. Tel.: +81 75 383 2803; fax: +81 75 383 2556. E-mail address: [email protected] (T. Sato). 0022-2860/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.molstruc.2006.12.049
Cvib ¼ 2a01 a02 3e01 4e02 a002 e001 2e002 :
ð1Þ
ð2Þ
It is well known that one of the main isotope effects is red shift in frequency due to an increase of reduced mass. However, isotope substitution can affect not only the frequencies but also the direction and displacement of normal modes. Considering these two points, deuteration effect on vibronic coupling of C5H5 had been discussed by Applegate et al. [10]. We discuss the deuteration effect on the vibronic coupling using vibronic coupling density analysis proposed in our previous work [14]. Vibronic coupling density analysis enables us to explain the order of magnitude
K. Tokunaga et al. / Journal of Molecular Structure 838 (2007) 116–123
117
Table 2 Total energy (a.u.) and vibronic coupling constants V e02 ðjÞ (104 a.u.) of C5X5 Energy
GRHF
CAS(5,5)
192.1719
192.2357
C5H5
e02 ð1Þ e02 ð2Þ e02 ð3Þ e02 ð4Þ
8.76 13.60 14.61 7.83
4.78 11.63 14.58 3.20
C5D5
e02 ð1Þ e02 ð2Þ e02 ð3Þ e02 ð4Þ
0.27 12.54 17.73 6.44
1.21 8.66 17.09 2.23
Fig. 1. Structure of C5X5 and the coordinate axes. Because of linear vibronic coupling, symmetry of C5X5 is lowered from D5h to C2v.
E
"(1)
Basis set employed is 6-31G(d,p). The vibrational vectors employed are those obtained by RHF/6-31G(d,p) for C5 X 5.
e1"(1)
Table 3 Scaled dimensionless vibronic coupling constants D0e0 ðjÞ and experimental 2 values of C5X5
e2
(a2) ψε
ψθ (b1)
CAS(5,5)/6-31G(d,p)
Expt.
C5H5
e02 ð1Þ e02 ð2Þ e02 ð3Þ e02 ð4Þ
0.67 1.16 0.97 0.07
0.62 1.07 0.85 –
C5D5
e02 ð1Þ e02 ð2Þ e02 ð3Þ e02 ð4Þ
0.21 1.16 1.19 0.07
– 0.88 1.12 –
a2"(1) (b1) Fig. 2. Frontier p orbitals of C5X5.
of vibronic coupling constants [14] and the relation between the frontier electron density and the vibronic coupling [15]. In this paper, we present a calculation of the vibronic coupling constants of C5X5 (X = H, D) and compare them with experimental data by Applegate et al. [11]. Furthermore, using the vibronic coupling density, the isotope effect on the vibronic coupling is discussed from view of the electronic and vibrational structures. The paper is organized as follows: in Section 2, the model Hamiltonian is presented. The method of the calculation
The vibrational vectors and frequencies employed in these calculations were obtained with RHF/6-31G(d,p) for C5 X 5 . Negative signs are neglected. The experimental values are taken from Ref. [11].
Table 1 Frequencies (cm1) of C5 X 5 calculated using RHF/6-31G(d,p) with the structure optimized by RHF/6-31G(d,p) and those of C5X5 with the D5h symmetry calculated using GRHF/6-31G(d,p) with the structure optimized by RHF/6-31G(d,p) Species
Method
e02 ð1Þ
e02 ð2Þ
e02 ð3Þ
e02 ð4Þ
C5 H 5 C5H5
RHF GRHF Expt.
914 918 872
1144 1198 1041
1497 1596 1320
3263 3313 –
C5 D 5 C5D5
RHF GRHF Expt.
780 814 –
939 939 861
1453 1556 1353
2401 2448 –
Experimental values of C5X5 are also represented. The experimental values are given in Ref. [11].
Fig. 3. Contour map on the plane z = 1.0 (a.u.) of frontier electron density qh(r) of C5X5.
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is described in Section 3. In Section 4, we show the results and compare them with the experimental values. We also discuss the isotope effect on the vibronic coupling using the vibronic coupling density in this section. Finally, we conclude this work in Section 5. 2. Vibronic Hamiltonian The model Hamiltonian employed in this work is [14] ! X X oU h2 o2 H¼ ðr; R Þ þ Q þ H e 0 oQi R0 i 2 oQ2i i i X1 x2i Q2i ; þ ð3Þ 2 i where r is a set of electronic coordinates, Qi a normal coordinate of the mode i, R0 a Jahn–Teller crossing geometry, He an electronic Hamiltonian, U a sum of an electron–electron, electron–nuclear, nuclear–nuclear potential operator, xi a frequency of the mode i. Only the linear vibronic coupling is considered. The operator involved in Eq. (3),
oU oQi
¼ Vi ¼
X
R0
vi ðaÞ;
ð4Þ
a
is called the electronic operator of the mode i, and Qi Vi describes the vibronic coupling. Vi is a sum of a oneelectron operator vi. The model Hamiltonian is treated within the space spanned by the electronic state jEhæ and jEæ; " ^ JT ¼ E0 þ H
X i
þ
(
h2 2
o2 oQ2i
!
1 þ x2i Q2i 2
X
hEhjVi jEhi
hEhjVi jEi
i
hEjVi jEhi
hEjVi jEi
)# ^0 r ! Qi ;
ð5Þ
where {|Ehæ, |Eæ} denotes the degenerate electronic state ^0 unit matrix. {j E001 hi; j E001 i}, E0 eigenenergy of He , and r Using Wigner–Eckart theorem, the vibronic coupling matrix of e02 ðjÞ modes can be reduced as
Fig. 4. Contour maps on the plane z = 1.0 (a.u.) of potential derivative ve02 ðjÞh ðrÞ of (a) e02 ð1Þh mode of C5H5 and (b) e02 ð2Þh mode of C5D5.
K. Tokunaga et al. / Journal of Molecular Structure 838 (2007) 116–123
V^ e02 ðjÞh ¼
hEhjVe02 ðjÞh jEhi
hEjV jEhi 1 0 ¼V e02 ðjÞ ; 0 1 e02 ðjÞh
hEhjVe02 ðjÞh jEi hEjV
e02 ðjÞh
!
jEi
hEhjVe02 ðjÞ jEhi hEhjVe02 ðjÞ jEi hEjVe02 ðjÞ jEhi hEjVe02 ðjÞ jEi 0 1 ¼V e02 ðjÞ ; 1 0
ð6Þ !
V^ e02 ðjÞ ¼
ð7Þ
where the integrals hEhjVe02 ðjÞh jEhi and so on are called vibronic coupling integrals (VCI). e02 ðjÞh and e02 ðjÞ denote a pair of degenerated normal mode e02 ðjÞ. The vibronic coupling constants (VCC) is defined as
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Finally, we introduce here some dimensionless quantities. For a vibrational mode i, dimensionless normal coordinate qi, dimensionless coupling constant Di, and scaled dimensionless coupling constant D0i can be defined as [15] rffiffiffiffiffi xi ð9Þ qi ¼ Q; h i Vi Di ¼ pffiffiffiffiffiffiffiffi ; ð10Þ hx3i D0i ¼jDi ;
ð11Þ
where j is a scaling parameter which reproduces the Jahn– Teller stabilization energy calculation with the same level of theory.
V e02 ðjÞ ¼hEhjVe02 ðjÞh jEhi ¼ hEjVe02 ðjÞh jEi ¼hEhjVe02 ðjÞ jEi ¼ hEjVe02 ðjÞ jEhi;
ð8Þ
which is the quantity we will calculate in this article. The VCC can be obtained from a calculation of these integrals.
3. Method of calculation In the generalized restricted Hartree–Fock (GRHF) method, the wave functions for E states are expressed as a single Slater determinant:
Fig. 5. Contour maps on the plane z = 1.0 (a.u.) of potential derivative ve02 ðjÞh ðrÞ of (a) e02 ð3Þh mode of C5H5 and (b) e02 ð3Þh mode of C5D5.
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jEhðGRHFÞi ¼j wm awm b we00 h awe00 awe00 bi 1
1
1
¼:jhi; jEðGRHFÞi ¼j wm awm b we00 h awe00 awe00 h bi 1
1
¼:ji;
ð12Þ
1
ð13Þ
and a, b are spin functions. Generally, the vibronic coupling matrix for the GRHF wave function can be written as hhjVi jhi hhjVi ji GRHF ^ ¼ Vi : ð14Þ hjVi jhi hjVi ji For the e02 modes, the vibronic coupling constant over a single Slater determinants jhæ and jæ can be decomposed into orbital vibronic coupling constants [4] for the h mode:
where m runs over the occupied molecular orbitals with e01 , e001 symmetries, and w(m)h and w(m) denote the degenerate pair of molecular orbitals. n(m)h and n(m) are occupation numbers of w(m)h and w(m). Note that the orbitals with the e01 , e001 symmetries can couple to the e02 modes since ½E01 ¼ ½E001 ¼ A01 E02 . Furthermore, from a symmetry of Clebsch–Gordan coefficients, hwðmÞh jve02 ðjÞh jwðmÞh i þ hwðmÞ jve02 ðjÞh jwðmÞ i ¼ 0:
Therefore, the vibronic coupling matrix in the GRHF methods is equal to the orbital vibronic coupling matrix: ! 0 hwe00 jve02 ðjÞh jwe00 i 1 1 GRHF V^ e0 ðjÞh ¼ ; ð17Þ 2 0 hwe00 h jve02 ðjÞh jwe00 h i 1 1 ! 0 0 hwe00 jve2 ðjÞ jwe00 h i 1 1 GRHF V^ e0 ðjÞ ¼ ; ð18Þ 2 hwe00 h jve02 ðjÞ jwe00 i 0 1
V GRHF ¼hhjVe02 ðjÞh jhi ¼ hjVe02 ðjÞh ji e02 ðjÞ X ¼ m2e0 e00 fnðmÞh hwðmÞh jve02 ðjÞh jwðmÞh i 1
1
þ nðmÞ hwðmÞ jve02 ðjÞh jwðmÞ ig;
ð15Þ
ð16Þ
1
In the state-averaged CASSCF method, the wave functions for E states are expressed as a linear combination of Slater determinants. The procedure of the calculation is the same as that expressed in Ref. [14].
Fig. 6. Contour maps on the plane z = 1.0 (a.u.) of potential derivative ve02 ðjÞh ðrÞ of (a) e02 ð4Þh mode of C5H5 and (b) e02 ð4Þh mode of C5D5.
K. Tokunaga et al. / Journal of Molecular Structure 838 (2007) 116–123
In order to obtain the Jahn–Teller crossing geometry R0 and vibrational structure, restricted Hartree–Fock (RHF) method is employed for C5 H 5 . At the geometry R0, we employed state-averaged CASSCF method using Gaussian 03 [16] and GRHF method using CADPAC [17] to determine the wave functions of C5X5. All calculations were performed using the 6-31G(d,p) basis set. The vibronic coupling constant was evaluated using a program coded by us. 4. Results and discussion 4.1. Geometrical and vibrational structure The optimized symmetry of C5 H 5 by RHF/6-31G(d,p) ˚ , rC–H = 1.0800 A ˚ ). We use the is D5h(rC–C = 1.4021 A geometry of C5 H5 as that of the JT crossing point R0 of the radical C5X5. Vibrational frequencies are shown in Table 1. Though the frequencies of e02 ð3Þ modes are a little different among
121
two methods, vibrational structures are almost the same. Therefore, we took the geometrical and vibrational structure of C5 H 5 as that of the JT crossing structure throughout this study [18]. 4.2. Vibronic coupling constant The calculated V e02 ðjÞ and D0e0 ðjÞ of C5X5 using GRHF and 2 CASSCF methods are tabulated in Tables 2 and 3, respectively. Our results agree well with experimental data, and the calculated constants V e02 ðjÞ of GRHF method qualitatively coincide with those of CASSCF method. Thus, results of GRHF method are worth analyzing by vibronic coupling density analysis. 4.3. Vibronic coupling density analysis 4.3.1. Vibronic coupling density Vibronic coupling density ge0 ðjÞ ðrÞ [14] for the e02 ðjÞ mode 2 is defined by
Fig. 7. Contour maps on the plane z = 1.0 (a.u.) of potential derivative ve02 ðjÞh ðrÞ of (a) e02 ð2Þh mode of C5H5 and (b) e02 ð1Þh mode of C5D5.
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ge0 ðjÞ ðrÞ ¼wh ðrÞwh ðrÞve02 ðjÞh ðrÞ 2
¼qh ðrÞve02 ðjÞh ðrÞ;
ð19Þ
where qh(r) is the frontier electron density of the molecular orbital wh, and ve02 ðjÞh ðrÞ the one-electron operator. Using ge0 ðjÞ , the VCC is written as 2 Z 0 ð20Þ V e2 ðjÞ ¼ drge0 ðjÞ ðrÞ: 2
Using the vibronic coupling density, we can analyze the calculated VCC in terms of the electronic structure qh(r) and the derivative of the potential with respect to the normal coordinate ve02 ðjÞh ðrÞ. 4.3.2. Frontier electron density In Fig. 3, frontier electron density of C5X5 is shown. It should be noted that frontier electron density is localized on C1 atom and C3–C4 bond.
4.3.3. e02 ð1Þ, e02 ð3Þ, and e02 ð4Þ modes of C5H5 Normal modes and potential derivatives are shown in Figs. 4–6. From Figs. 4–6, we can see that e02 ð1Þ, e02 ð3Þ, and e02 ð4Þ modes of C5H5 correspond to e02 ð2Þ, e02 ð3Þ, and e02 ð4Þ modes of C5D5, respectively [19]. In these e02 modes, direction and displacement of C5H5 are very similar to those of C5D5. Therefore, vibronic coupling density, which is a product of frontier electron density (Fig. 3) and potential derivative (Figs. 4–6), is not influenced by deuteration. The isotope effect on the vibronic coupling of these modes is small. This result agrees well with the experimental data [11] and the previous theoretical data [10]. 4.3.4. e02 ð2Þ mode of C5H5 Normal modes and potential derivatives are shown in Fig. 7. From Fig. 7, we can see that e02 ð2Þ mode of C5H5 corresponds to e02 ð1Þ mode of C5D5 [19]. In contrast to the other normal modes, the deuteration leads to the large reduction of the displacement of C3–C4 stretching motion. This gives rise to a reduction of distribution of potential derivative on C3–C4 bond. Since coincidence between q (Fig. 3) and v (Fig. 7) is decreased, the vibronic coupling density (Fig. 8) is also reduced. Therefore, the isotope effect on the vibronic coupling of this mode is very large, as also pointed out by Applegate et al. [10]. These results indicate that not only red shift of frequencies but also change of the direction and displacement of normal modes due to deuteration can affect the vibronic coupling constant. 5. Conclusion We present a calculation of vibronic coupling constants of C5H5 and C5D5. The vibronic coupling constants are calculated as matrix elements of the electronic operator of the vibronic coupling using GRHF and CASSCF methods. Our results agree well with the experimental and theoretical values by Applegate et al. Vibronic coupling density analysis is applied to analyze the isotope effect on the vibronic coupling from view of the electronic and vibrational structures. This analysis reveals that the coupling of the e02 ð2Þ mode of C5H5 is largely reduced by isotope substitution, because the displacement of C–C stretching motion is decreased. This result indicates that not only red shift of frequencies but also changes in the direction and magnitude of the displacement due to isotope substitution can strongly affect the vibronic coupling. Vibronic density analysis provides a local picture of the coupling in a molecule, and it enables us to control the coupling. This could open a way to engineering of the vibronic coupling, vibronics. Acknowledgements
Fig. 8. Contour maps on the plane z = 1.0 (a.u.) of vibronic coupling density ge0 ðjÞ with respect to (a) e02 ð2Þh mode of C5H5 and (b) e02 ð1Þh mode 2 of C5D5.
Numerical calculation was partly performed in the Supercomputer Laboratory of Kyoto University and Research Center for Computational Science, Okazaki, Japan.
K. Tokunaga et al. / Journal of Molecular Structure 838 (2007) 116–123
References [1] I.B. Bersuker, The Jahn–Teller Effect and Vibronic Interactions in Modern Chemistry, Plenum, New York, 1984. [2] G. Fischer, Vibronic Coupling: The interaction Between the Electronic and Nuclear Motions, Academic Press, London, 1984. [3] I.B. Bersuker, V.Z. Polinger, Vibronic Interaction in Molecules and Crystals, Springer, Berlin, 1989. [4] I.B. Bersuker, Chem. Rev. (Washington, D.C.) 101 (2001) 1067. [5] A.D. Liehr, Z. Phys. Chem., Neue Folge 9 (1956) 338. [6] L.C. Snyder, J. Chem. Phys. 33 (1960) 619. [7] W.D. Hobey, A.D. McLachlan, J. Chem. Phys. 33 (1960) 1695. [8] R. Meyer, F. Graf, T.-K. Ha, Hs.H. Gu¨nthard, Chem. Phys. Lett. 66 (1979) 65. [9] W.T. Borden, E.R. Davidson, J. Am. Chem. Soc. 101 (1979) 3771. [10] B.E. Applegate, T.A. Miller, T.A. Barckholtz, J. Chem. Phys. 114 (2001) 4855. [11] B.E. Applegate, A.J. Bezant, T.A. Miller, J. Chem. Phys. 114 (2001) 4869.
[12] [13] [14] [15] [16] [17]
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T.A. Barckholtz, T.A. Miller, J. Phys. Chem. A 103 (1999) 2321. T. Kato, K. Hirao, Adv. Quantum Chem. 44 (2003) 257. T. Sato, K. Tokunaga, K. Tanaka, J. Chem. Phys. 124 (2006) 024314. K. Tokunaga, T. Sato, K. Tanaka, J. Chem. Phys. 124 (2006) 154303. M.J. Frisch et al., Gaussian 03 Revision C.02, Wallingford, CT, 2004. Cambridge Analytic Derivative Package (CADPAC) Issue 6.5, Cambridge UK, 2001. [18] Energy gap at R0 between the C5 H 5 geometry obtained by RHF method and the C5H5 geometry obtained by CASSCF method is due to the distortion along two totally symmetric modes, so that V e02 ðjÞ is little influenced by such a geometry difference. [19] We can see by eye the similarity of the e02 ð2Þh mode of C5D5 (Fig. 4(b)) to the e02 ð1Þh mode of C5H5 (Fig. 4(a)) in the upper part of the pentagon, but to the e02 ð2Þh mode of C5H5 (Fig. 7(a)) in the lower part of the pentagon. The criterion of the correspondence in Figs. 4 and 7 is the reduced mass l. In Fig. 4, le02 ð1Þh;C5 H5 is 4.36 (amu) which is a little smaller than le02 ð2Þh;C5 H5 , 4.81 (amu), due to the difference between D and H. In Fig. 7, le02 ð2Þh;C5 H5 is 1.31 (amu) which is half as large as le0 ð1Þh;C5 D5 , 2.38 (amu). 2