Calculation of kinetic energy functions for the ring-twisting and ring-bending vibrations of tetralin and related molecules

Journal of Molecular Structure 798 (2006) 27–33 www.elsevier.com/locate/molstruc

Calculation of kinetic energy functions for the ring-twisting and ring-bending vibrations of tetralin and related molecules Juan Yang, Jaan Laane

*

Department of Chemistry, Texas A&M University, College Station, TX 77843-3255, USA Received 21 July 2006; accepted 21 July 2006 Available online 1 September 2006

Abstract Vector methods have been developed for the computation of the kinetic energy (reciprocal reduced mass) expressions for the ringtwisting and ring-bending vibrations of bicyclic molecules in the tetralin family. The definitions of the bond vectors in terms of these coordinates are presented. Both one- and two-dimensional kinetic energy surfaces have been calculated for tetralin and 1,4-benzodioxan and both are significantly coordinate dependent. The results for the S0 electronic ground states and S1(p, p*) excited states are presented.  2006 Elsevier B.V. All rights reserved. Keywords: Kinetic energy functions; Vector methods; Tetralin; Ring-twisting; Ring-bending; 1,4-Benzodioxan

1. Introduction For many years we have been carrying out determinations of potential energy surfaces utilizing spectroscopic methods [1–5]. These calculations also require the kinetic energy expressions for use in the Schro¨dinger wave equation. In 1982 we first described how to utilize vector methods for carrying out kinetic energy (reciprocal reduced mass) calculations needed for the determination of potential energy functions for the out-of-plane vibrations of four- and five-membered ring molecules such as cyclobutane and cyclopentene [6,7]. Later we also described similar calculations for a variety of other molecules including asymmetric five-membered ring molecules [8], bicyclic molecules containing benzene rings [1,9–12], 1,4-cyclohexadiene and analogs including 9,10-dihydroanthracene [13], and 1,3-cyclohexadiene and analogs such as 1,2-dihyronaphthalene [14]. The book by Fitts [15] very nicely shows how to make effective use of vectors for studying molecular systems. In the present paper we present the methodology necessary to calculate the kinetic energy expressions for *

Corresponding author. Tel.: +1 979 845 3352. E-mail address: [email protected] (J. Laane).

0022-2860/$ - see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.molstruc.2006.07.024

tetralin (TET) and related molecules such as 1,4-benzodioxan (14BZD). O

O

TET

14BZD

2. Calculation methods Fig. 1 presents for TET the definition of the bond vec* * tors u 1 to u 22 along with the essential geometrical parameters for the bond distances (R1 to R6 for the C–C bonds and E1 to E4 for the C–H bonds) and bond angles (b1 and b2 in the benzene ring, a and c in the saturated six-membered ring, and T1 and T2 for the HCH angles). Additional vec* * * tors u 23 , u 24 , and u 25 , which are useful for defining the ring-twisting (s) and ring-bending (h) motions, are also * shown. u 23 is the vector connecting atoms 1 and 4 (stan* dard organic nomenclature) in the saturated ring, u 24 is * the same as u 4 when the saturated ring is not twisted,

28

J. Yang, J. Laane / Journal of Molecular Structure 798 (2006) 27–33

Fig. 1. Vectors for defining the atom positions of tetralin and the definition of the ring-twisting coordinate s and the ring-bending coordinate h. *

*

and u 25 is the vector connecting the midpoints of u 23 and * u 24 . Table * 1 presents * * the components of each bond vector * u i ¼ ai i þbi j þci k . These vectors can be used not only for TET and 14BZD but also for any molecule with a symmetric saturated six-membered ring attached to a benzene * * * ring. The vectors u 23 , u 24 , and u 25 can be solved easily, * and formulas for these are also shown in Table 1. u 4 can be obtained from the following vector equations: *

ju 4 j ¼ R4 ; * u4 * u4

*

 u 24 ¼

ð1Þ R24

ð2Þ

cos s;

*

 u 25 ¼ 0;

* u3

ð3Þ

*

*

*

*

and u 5 can be solved from u 4 , u 23 , and u 25 using the following vector equations:

* u3

*

*

*

þ u 4 þ u 5 þ u 23 ¼ 0;  * * * u 3 þ u 5 ¼ 2u 25 ; *

*

ju 3 j ¼ ju 5 j ¼ R3 :

ð4Þ ð5Þ ð6Þ

In order to calculate the kinetic energy expressions for the ring-twisting and ring-bending modes, it is necessary to define each of the vectors in Fig. 1 in terms of the h * and s coordinates so that the derivatives d r i =dh and * * d r i =ds can be calculated by numerical methods. The r i vector for each atom i is the atom coordinate vector in the center-of-mass system [5–10]. For the calculation it is assumed that all bond distances remain fixed during the vibrations, that the bisector of each CH2 group remains coincident with the corresponding C–C–C angle bisector, and that each C–H bond lies along the corresponding C–C–C angle bisector of the benzene ring. Once the derivatives have been

calculated, they can be used to set up a 4 · 4 G1 matrix for a one-dimensional problem or a 5 · 5 G1 matrix for a two-dimensional problem. Inversion of the matrix results in the G matrix containing the g44 terms for one-dimensional cases and the g44, g45, and g55 terms for two-dimensional cases [5–10]. Note that subscripts 1 to 3 correspond to the molecular rotations. 3. Results 3.1. One-dimensional kinetic energy functions Whenever a vibrational problem is taken to be one-dimensional, a 4 · 4 G1 matrix is set up, and, after inversion of the matrix, the resulting g44 term is the kinetic energy term (reciprocal reduced mass) for the molecular structure and conformation that was assumed. Typically this varies considerably with the coordinate and a polynomial expression for g44(x) can be written, where x is the vibrational coordinate. These are of the form n X 1 ðiÞ : ð7Þ g44 xi ¼ g44 ðxÞ ¼ lðxÞ i¼0 The computed g44(s) expression for TET in its S0 ground state was determined to be gS440 ðsÞ ¼ 0:0306595  0:00230104s2  0:0157768s4 þ 0:00458234s6 :

ð8Þ

The bending expansion g44(h) is gS440 ðhÞ ¼ 0:0333862  0:00328062h2  0:00325037h4 þ 0:00148704h6 : In the S1(p, p*) excited state g44(s) for the twisting is

ð9Þ

J. Yang, J. Laane / Journal of Molecular Structure 798 (2006) 27–33 Table 1 * * * * Components of the bond vectors u i ¼ ai i þbi j þci k for the ring-twisting and ring-bending vibrations of tetralin *

Vector u i

ai

bi

ci

* u1 * u2 * u3 * u4 * u5 * u6 * u7 * u8 * u9 * u 10 * u 11 * u 12 * u 13 * u 14 * u 15 * u 16 * u 17 * u 18 * u 19 * u 20 * u 21 * u 22 * u 23 * u 24 * u 25

R1 X2 X3 X4 X5 X2 X7 X8 X7 X8 px + vx px  vx Px + Vx Px  Vx Px + Vx Px  Vx px + vx px  vx X19 X20 X20 X19 X23 R4 0

0 Y2 Y3 Y4 Y5 Y2 Y7 Y8 Y7 Y8 py + vy py  vy Py + Vy Py  Vy Py + Vy Py  Vy py + vy py  vy Y19 Y20 Y20 Y19 0 0 Y25

0 0 Z3 Z4 Z5 0 0 0 0 0 pz + vz pz  vz Pz + Vz Pz  Vz Pz + Vz Pz  Vz pz + vz pz  vz 0 0 0 0 0 0 Z25

*

p ¼ E1 cos

* u3

* u2

S0

g44(τ) S1

g44(τ)

g44(τ)

0.029

0.028 -0.6

*

0.4

0.6

S0

g44(θ) S1

g44(θ) 0.034

g4 4 ( θ)

0.033

0.032

-0.4

-0.2

0.0

0.2

0.4

0.6

tions were carried out for the 14BZD S0 and S1 states and the results are given in Eqs. (12)–(15) and shown in Fig. 3: gS440 ðsÞ ¼ 0:0336319  0:00772337s2  0:0139991s4 þ 0:00390668s6 ; gS440 ðhÞ

*

the components of the vector P, and Vx, Vy, and Vz are the components of * the vector V; where s and h are the ring-twisting and ring-bending coordinates, respectively, as shown in Fig. 1.

ð12Þ 2

¼ 0:0443117  0:0167421h þ 0:00704256h

gS441 ðsÞ

ð13Þ 2

¼ 0:0353983  0:00735565s  0:0119540s

4

þ 0:00325897s6 ;

ð14Þ 2

¼ 0:0435493  0:0165764h þ 0:00696244h  0:00145000h6 :

¼ 0:0310023  0:00232109s2  0:0156682s4

4

 0:00146557h6 ;

gS441 ðhÞ

4

ð15Þ

ð10Þ 3.2. Two-dimensional kinetic energy functions

and for the bending g44(h) is gS441 ðhÞ ¼ 0:0333679  0:00286035h2  0:00367870h4 þ 0:00162500h :

0.2

0.035

*

* * u u T 2 R33  R44 * T 2 u3  u4 , where Px, Py, and Pz are P ¼ E2 cos c ; V ¼ E2 sin 2 2 cos 2 2 R3 R4 sin c

6

0.0

τ (rad)

*

þ 0:00454698s ;

-0.2

Fig. 2. Coordinate dependence of the one-dimensional kinetic energy terms for the ring-twisting (s) and ring-bending (h) vibrations of tetralin in its S0 and S1 states.

* u3

6

-0.4

θ (rad)

and pz are the components of the vector p , and vx, vy, and vz are the * components of the vector v ;

gS441 ðsÞ

0.030

0.031 -0.6

T 1 R2  R3 * T1  ; v ¼ E1 sin , where px, py, 2 2 cosðaþc 2 R R sinða þ cÞ Þ 2 3 2

*

0.032

0.031

X2 = R2 cos a; Y2 = R2 sin a; X4 = R4 cos s; Y4 = R4 sin s sin h; Z4 = R4 sin s cos h; X7 = R5 cos b1; Y7 = R5 sin b1; X8 = R6 cos b2; Y8 = R6 sin b2; b2 b1 1 X 19 ¼ E3 cos b2 b 2 ; Y 19 ¼ E3 sin 2 ; X 20 ¼ E4 cos b22 ; Y 20 ¼ E4 sin b22 ; X23 = R1  2R2 cos a; Y25 = R3 sin c cos h; Z25 = R3 sin c sin h; 23 X 3 ¼ X 5 ¼  X 4 þX ; 2 Y3, Z3, Y5, and Z5 can be solved from the following equations: 8 < Y 3 þ Y 5 ¼ Y 4 ; Z 3 þ Z 5 ¼ Z 4 ; : 2 Y 3 þ Z 23 ¼ Y 25 þ Z 25 ¼ R23  X 23 ; * u2

29

ð11Þ

The coordinate dependence of g44(s) and g44(h) for TET in both S0 and S1 states is shown in Fig. 2. Similar calcula-

The one-dimensional calculations described above assume there is no vibrational interaction of the mode of interest with any other vibrations. For the two-dimensional calculations carried out here the assumption is that the twisting and bending modes interact with each other, but

30

J. Yang, J. Laane / Journal of Molecular Structure 798 (2006) 27–33 0.036

0.064 S0

S0

g44(τ) g44(τ)

0.035

g44(θ)

0.045

S1

S1

g44(θ)

0.044 0.034 0.043

g44(τ)

0.033

g 4 4( θ )

0.042 0.041

0.032 0.040 0.031 0.039 0.030 -0.6

-0.4

-0.2

0.0

0.2

0.4

0.038 -0.6

0.6

-0.4

τ (rad)

-0.2

0.0

0.2

0.4

0.6

θ (r ad )

Fig. 3. Coordinate dependence of the one-dimensional kinetic energy terms for the ring-twisting (s) and ring-bending (h) vibrations of 1,4-benzodioxan in its S0 and S1 states.

not with any other vibrations. A 5 · 5 G1 matrix is set up and the resulting G matrix after inversion provides g44 for the twisting, g55 for the bending, and g45 for the kinetic energy interaction. These values are calculated for a broad range of twisting (s) and bending (h) values, and from these

g44

the coordinate dependence of the gij expressions can be calculated. These are of the form m X n X ðk;lÞ gij sk hl ; i; j ¼ 4; 5 ð16Þ gij ðs; hÞ ¼ k¼0

l¼0

g55

g45

Fig. 4. The coordinate dependence of g44, g45, and g55 for tetralin S0 state. s and h are the ring-twisting and ring-bending coordinates, respectively.

J. Yang, J. Laane / Journal of Molecular Structure 798 (2006) 27–33

g44

31

g55

g45

Fig. 5. The coordinate dependence of g44, g45, and g55 for tetralin S1 state. s and h are the ring-twisting and ring-bending coordinates, respectively.

g44

g55

g45

Fig. 6. The coordinate dependence of g44, g45, and g55 for 1,4-benzodioxan S0 state. s and h are the ring-twisting and ring-bending coordinates, respectively.

32

J. Yang, J. Laane / Journal of Molecular Structure 798 (2006) 27–33

The results for TET in both S0 and S1 states are given in Eqs. (17)–(22) and shown in Figs. 4 and 5. gS440 ðs; hÞ ¼ 0:030521883  0:002741035s2  0:014414611s4 2 6

þ 0:000125230h6 þ 0:001867243s2 h2  0:000479731s4 h2 ð17Þ

gS550 ðs; hÞ ¼ 0:033339463 þ 0:005145694s2  0:000419688s4 2 6

gS440 ðs; hÞ ¼ 0:033476535  0:006855096s2  0:014646045s4 þ 0:003929556s6  0:003116382h2 þ 0:001133362h4

4

2 2

 0:001319295s2 h4 ; 4 2

þ 0:000485676h  0:002098020s h þ 0:001437924s h þ 0:000840282s2 h4 ;

gS450 ðs; hÞ

ð18Þ

þ 0:000121541s6  0:015598839h2 þ 0:005139534h4

¼ 0:007049552sh þ 0:002651319s h

 0:000686260h6 þ 0:009920440s2 h2  0:000107406s4 h2

þ 0:001964390sh3  0:000687963s3 h3 ; þ 0:003809592s  0:000342597h  0:000061248h 6

þ 0:006351398sh3  0:0040797066s3 h3 ; ð20Þ

gS551 ðs; hÞ ¼ 0:033317277 þ 0:005271892s2  0:000494118s4 6

2 2

ð25Þ

4 2

 0:000278588s2 h4 ;

þ 0:001163424s  0:004160058h  0:001156234h

ð24Þ

gS450 ðs; hÞ ¼ 0:014930208sh þ 0:0083726413s3 h 4

þ 0:000133963h þ 0:001669258s h  0:000392022s h

2

 0:003277779s2 h4 ;

ð19Þ

2 2

6

ð23Þ

gS550 ðs; hÞ ¼ 0:044329620  0:001141322s2 þ 0:002428906s4

3

gS441 ðs; hÞ ¼ 0:030866163  0:002817808s2  0:014222202s4 2 6

ð22Þ

 0:000106793h6 þ 0:007942282s2 h2  0:003060725s4 h2

þ 0:001137651s  0:004483400h  0:000918103h 6

þ 0:001826140sh3  0:000566876s3 h3 :

Similarly, the kinetic energy expressions for the S0 ground state of 14BZD were obtained as

þ 0:003874489s  0:000435594h  0:000017005h4  0:000318589s2 h4 ;

gS451 ðs; hÞ ¼ 0:006859644sh þ 0:002464505s3 h

and the kinetic energy expressions for the 14BZD S1(p, p*) excited state are gS441 ðs; hÞ ¼ 0:035277287  0:006634635s2  0:012450464s4

4

þ 0:003244677s6  0:003583293h2 þ 0:001484587h4 4 2

þ 0:000538778h  0:002471099s h þ 0:001465732s h

 0:000173562h6 þ 0:007809779s2 h2  0:002474722s4 h2

þ 0:000995707s2 h4 ;

 0:001570280s2 h4 ;

ð21Þ

g44

ð26Þ

g55

g45

Fig. 7. The coordinate dependence of g44, g45, and g55 for 1,4-benzodioxan S1 state. s and h are the ring-twisting and ring-bending coordinates, respectively.

J. Yang, J. Laane / Journal of Molecular Structure 798 (2006) 27–33

gS551 ðs; hÞ ¼ 0:043567071  0:001398840s2 þ 0:001948076s4 6

2

Acknowledgements 4

þ 0:000400976s  0:015515522h þ 0:005131021h

 0:000689175h6 þ 0:009406176s2 h2  0:000573554s4 h2  0:003044145s2 h4 ;

ð27Þ

gS451 ðs; hÞ ¼ 0:014264471sh þ 0:007050807s3 h þ 0:0061752092sh3  0:003623517s3 h3 :

33

ð28Þ

These gij (i, j = 4,5) expressions for both the S0 and S1(p, p*) states are presented graphically in Figs. 6 and 7, respectively. For TET the two-dimensional expressions show that g44(s, h) for the twisting depends almost only on the twisting coordinate s. However, both g55(s, h) and g45(s, h) depend on both coordinates to a large degree. The situation for 14BZD is similar except that g55(s, h) depends almost only on h and g44(s, h) for the twisting, which depends mostly on s, has only a moderate dependence on h. 4. Conclusions In this paper we add to our arsenal of computer programs for calculating the kinetic energy terms for the ring-twisting and ring-bending of TET and other similar bicyclic molecules such as 14BZD. These are essential for meaningful potential energy calculations of the type we have carried out before. Calculated results were presented for TET and 14BZD to show the nature of these kinetic energy expansions.

The authors thank the National Science Foundation (Grant CHE-0131935) and the Robert A. Welch Foundation (Grant A-0396) for financial assistance. References [1] J. Laane, J. Phys. Chem. 104A (2000) 7715. [2] J. Laane, Intl. Rev. Phys. Chem. 18 (1999) 301. [3] J. Laane, Structure and dynamics of electronic excited states, in: J. Laane, H. Takahasi, A. Bandrauk (Eds.), Honolulu Symposium, Springer, Berlin, Germany, 1999, pp. 3–35. [4] J. Laane, Annu. Rev. Phys. Chem. 45 (1994) 179. [5] J. Laane, in: J. Laane, M. Dakkouri (Eds.), Structures and Conformations of Non-Rigid Molecules, Kluwer Publishing, Amsterdam, 1993, pp. 65–98. [6] J. Laane, M.A. Harthcock, P.M. Killough, L.E. Bauman, J.M. Cooke, J. Mol. Spectrosc. 91 (1982) 286. [7] M.A. Harthcock, J. Laane, J. Mol. Spectrosc. 91 (1982) 300. [8] R.W. Schmude, M.A. Harthcock, M.B. Kelly, J. Laane, J. Mol. Spectrosc. 124 (1987) 369. [9] T. Klots, S. Sakurai, J. Laane, J. Chem. Phys. 108 (1998) 3531. [10] S. Sakurai, N. Meinander, K. Morris, J. Laane, J. Amer. Chem. Soc. 121 (1999) 5056. [11] S. Sakurai, E. Bondoc, J. Laane, K. Morris, N. Meinander, J. Choo, J. Amer. Chem. Soc. 122 (2000) 2628. [12] J. Laane, Z. Arp, S. Sakurai, K. Morris, N. Meinander, T. Klots, E. Bondoc, K. Haller, J. Choo, Low-lying potential energy surfaces, in: M. Hoffman, K. Dyall (Eds.), ACS Symposium Series 828, Washington, D.C., 2002, pp. 380–399. [13] M.M. Strube, J. Laane, J. Mol. Spectrosc. 129 (1988) 126. [14] D. Autrey, Z. Arp, J. Choo, J. Laane, J. Chem. Phys. 119 (5) (2003) 2557. [15] D.D. Fitts, Vector Analysis in Chemistry, McGraw-Hill, New York, NY, 1974.