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Journal of Molecular Structure 798 (2006) 27–33

Calculation of kinetic energy functions for the ring-twistingand 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 2006Available online 1 September 2006

Abstract

Vector methods have been developed for the computation of the kinetic energy (reciprocal reduced mass) expressions for the ring-twisting and ring-bending vibrations of bicyclic molecules in the tetralin family. The definitions of the bond vectors in terms of thesecoordinates are presented. Both one- and two-dimensional kinetic energy surfaces have been calculated for tetralin and 1,4-benzodioxanand 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 determina-tions of potential energy surfaces utilizing spectroscopicmethods [1–5]. These calculations also require the kineticenergy expressions for use in the Schrodinger wave equa-tion. In 1982 we first described how to utilize vector meth-ods for carrying out kinetic energy (reciprocal reducedmass) calculations needed for the determination of poten-tial energy functions for the out-of-plane vibrations offour- and five-membered ring molecules such as cyclobu-tane and cyclopentene [6,7]. Later we also described similarcalculations for a variety of other molecules includingasymmetric five-membered ring molecules [8], bicyclic mol-ecules containing benzene rings [1,9–12], 1,4-cyclohexadi-ene and analogs including 9,10-dihydroanthracene [13],and 1,3-cyclohexadiene and analogs such as 1,2-dihyro-naphthalene [14]. The book by Fitts [15] very nicely showshow to make effective use of vectors for studying molecularsystems. In the present paper we present the methodologynecessary to calculate the kinetic energy expressions for

0022-2860/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.molstruc.2006.07.024

* Corresponding author. Tel.: +1 979 845 3352.E-mail address: laane@mail.chem.tamu.edu (J. Laane).

tetralin (TET) and related molecules such as 1,4-benzodio-xan (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 parame-ters for the bond distances (R1 to R6 for the C–C bonds andE1 to E4 for the C–H bonds) and bond angles (b1 and b2 inthe benzene ring, a and c in the saturated six-memberedring, and T1 and T2 for the HCH angles). Additional vec-tors u

*

23, u*

24, and u*

25, which are useful for defining thering-twisting (s) and ring-bending (h) motions, are alsoshown. u

*

23 is the vector connecting atoms 1 and 4 (stan-dard organic nomenclature) in the saturated ring, u

*

24 isthe same as u

*

4 when the saturated ring is not twisted,

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.

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

and u*

25 is the vector connecting the midpoints of u*

23 andu*

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 onlyfor TET and 14BZD but also for any molecule with a sym-metric saturated six-membered ring attached to a benzenering. 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 canbe obtained from the following vector equations:

ju*4j ¼ R4; ð1Þu*

4 � u*

24 ¼ R24 cos s; ð2Þ

u*

4 � u*

25 ¼ 0; ð3Þ

u*

3 and u*

5 can be solved from u*

4, u*

23, and u*

25 using the fol-lowing vector equations:

u*

3 þ u*

4 þ u*

5 þ u*

23 ¼ 0; ð4Þ

u*

3 þ �u*

5

� �¼ 2u

*

25; ð5Þ

ju*3j ¼ ju*

5j ¼ R3: ð6Þ

In order to calculate the kinetic energy expressions forthe ring-twisting and ring-bending modes, it is necessaryto define each of the vectors in Fig. 1 in terms of the hand s coordinates so that the derivatives d r

*

i=dh andd r*

i=ds can be calculated by numerical methods. The r*

i vec-tor for each atom i is the atom coordinate vector in the cen-ter-of-mass system [5–10]. For the calculation it is assumedthat all bond distances remain fixed during the vibrations,that the bisector of each CH2 group remains coincidentwith the corresponding C–C–C angle bisector, and thateach C–H bond lies along the corresponding C–C–C anglebisector of the benzene ring. Once the derivatives have been

calculated, they can be used to set up a 4 · 4 G�1 matrix fora one-dimensional problem or a 5 · 5 G�1 matrix for atwo-dimensional problem. Inversion of the matrix resultsin the G matrix containing the g44 terms for one-dimen-sional cases and the g44, g45, and g55 terms for two-dimen-sional cases [5–10]. Note that subscripts 1 to 3 correspondto the molecular rotations.

3. Results

3.1. One-dimensional kinetic energy functions

Whenever a vibrational problem is taken to be one-di-mensional, a 4 · 4 G�1 matrix is set up, and, after inversionof the matrix, the resulting g44 term is the kinetic energyterm (reciprocal reduced mass) for the molecular structureand conformation that was assumed. Typically this variesconsiderably with the coordinate and a polynomial expres-sion for g44(x) can be written, where x is the vibrationalcoordinate. These are of the form

g44ðxÞ ¼Xn

i¼0

gðiÞ44xi ¼ 1

lðxÞ : ð7Þ

The computed g44(s) expression for TET in its S0

ground state was determined to be

gS044ðsÞ ¼ 0:0306595� 0:00230104s2 � 0:0157768s4

þ 0:00458234s6: ð8Þ

The bending expansion g44(h) is

gS044ðhÞ ¼ 0:0333862� 0:00328062h2 � 0:00325037h4

þ 0:00148704h6: ð9Þ

In the S1(p,p*) excited state g44(s) for the twisting is

6.04.02.00.02.0-4.0-6.0-820.0

920.0

030.0

130.0

230.0

g 44 )τ(

τ )dar(

g S0

44(τ)

g S1

44(τ)

6.04.02.00.02.0-4.0-6.0-130.0

230.0

330.0

430.0

530.0

g 44 )θ(

θ )dar(

g S0

44(θ)

g S1

44(θ)

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

Table 1Components of the bond vectors u

*

i ¼ ai i*

þbi j*

þci k*

for the ring-twistingand ring-bending vibrations of tetralin

Vector u*

i ai bi ci

u*

1 R1 0 0u*

2 X2 Y2 0u*

3 X3 Y3 Z3

u*

4 X4 Y4 Z4

u*

5 X5 Y5 Z5

u*

6 X2 �Y2 0u*

7 X7 Y7 0u*

8 X8 Y8 0u*

9 �X7 Y7 0u*

10 �X8 Y8 0u*

11 px + vx py + vy pz + vz

u*

12 px � vx py � vy pz � vz

u*

13 Px + Vx Py + Vy Pz + Vz

u*

14 Px � Vx Py � Vy Pz � Vz

u*

15 �Px + Vx �Py + Vy �Pz + Vz

u*

16 �Px � Vx �Py � Vy �Pz � Vz

u*

17 �px + vx �py + vy �pz + vz

u*

18 �px � vx �py � vy �pz � vz

u*

19 X19 Y19 0u*

20 X20 Y20 0u*

21 �X20 Y20 0u*

22 �X19 Y19 0u*

23 X23 0 0u*

24 �R4 0 0u*

25 0 Y25 Z25

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;X 19 ¼ �E3 cos b2�b1

2 ; Y 19 ¼ �E3 sin b2�b1

2 ;X 20 ¼ �E4 cos b2

2 ; Y 20 ¼ �E4 sin b2

2 ;X23 = R1 � 2R2 cos a; Y25 = R3 sin c cos h; Z25 = R3 sin c sin h;X 3 ¼ X 5 ¼ � X 4þX 23

2 ;Y3, Z3, Y5, and Z5 can be solved from the following equations:

Y 3 þ Y 5 ¼ �Y 4;Z3 þ Z5 ¼ �Z4;Y 2

3 þ Z23 ¼ Y 2

5 þ Z25 ¼ R2

3 � X 23;

8<:

p* ¼ �E1 cos

T 1

2

u*

2

R2� u

*3

R3

2 cosðaþc2 Þ

; v* ¼ �E1 sin

T 1

2

u*

2 � u*

3

R2R3 sinðaþ cÞ, where px, py,

and pz are the components of the vector p*

, and vx, vy, and vz are thecomponents of the vector v

*;

P*

¼ E2 cosT 2

2

u*

3

R3� u

*4

R4

2 cos c2

; V*

¼ E2 sinT 2

2

u*

3 � u*

4

R3R4 sin c, where Px, Py, and Pz are

the components of the vector P*

, and Vx, Vy, and Vz are the components ofthe vector V

*

;where s and h are the ring-twisting and ring-bending coordinates,respectively, as shown in Fig. 1.

J. Yang, J. Laane / Journal of Molecular Structure 798 (2006) 27–33 29

gS144ðsÞ ¼ 0:0310023� 0:00232109s2 � 0:0156682s4

þ 0:00454698s6; ð10Þ

and for the bending g44(h) is

gS144ðhÞ ¼ 0:0333679� 0:00286035h2 � 0:00367870h4

þ 0:00162500h6: ð11Þ

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

tions were carried out for the 14BZD S0 and S1 statesand the results are given in Eqs. (12)–(15) and shown inFig. 3:

gS044ðsÞ ¼ 0:0336319� 0:00772337s2 � 0:0139991s4

þ 0:00390668s6; ð12ÞgS0

44ðhÞ ¼ 0:0443117� 0:0167421h2 þ 0:00704256h4

� 0:00146557h6; ð13ÞgS1

44ðsÞ ¼ 0:0353983� 0:00735565s2 � 0:0119540s4

þ 0:00325897s6; ð14ÞgS1

44ðhÞ ¼ 0:0435493� 0:0165764h2 þ 0:00696244h4

� 0:00145000h6: ð15Þ

3.2. Two-dimensional kinetic energy functions

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

6.04.02.00.02.0-4.0-6.0-030.0

130.0

230.0

330.0

430.0

530.0

630.0

g 44 )τ(

τ )dar(

g S0

44(τ)

g S1

44(τ)

6.04.02.00.02.0-4.0-6.0-830.0

930.0

040.0

140.0

240.0

340.0

440.0

540.0

064.0

g 44 )θ(

θ )dar(

g S0

44(θ)

g S1

44(θ)

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 inits S0 and S1 states.

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

not with any other vibrations. A 5 · 5 G�1 matrix is set upand the resulting G matrix after inversion provides g44 forthe twisting, g55 for the bending, and g45 for the kineticenergy interaction. These values are calculated for a broadrange of twisting (s) and bending (h) values, and from these

g44

g45

Fig. 4. The coordinate dependence of g44, g45, and g55 for tetralin S0 state.

the coordinate dependence of the gij expressions can be cal-culated. These are of the form

gijðs; hÞ ¼Xm

k¼0

Xn

l¼0

gðk;lÞij skhl; i; j ¼ 4; 5 ð16Þ

g55

s and h are the ring-twisting and ring-bending coordinates, respectively.

g44 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

g45

g55

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.

J. Yang, J. Laane / Journal of Molecular Structure 798 (2006) 27–33 31

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 inEqs. (17)–(22) and shown in Figs. 4 and 5.

gS044ðs;hÞ¼ 0:030521883�0:002741035s2�0:014414611s4

þ0:003874489s6�0:000435594h2�0:000017005h4

þ0:000125230h6þ0:001867243s2h2�0:000479731s4h2

�0:000318589s2h4; ð17Þ

gS055ðs;hÞ¼ 0:033339463þ0:005145694s2�0:000419688s4

þ0:001137651s6�0:004483400h2�0:000918103h4

þ0:000485676h6�0:002098020s2h2þ0:001437924s4h2

þ0:000840282s2h4; ð18Þ

gS045ðs; hÞ ¼ �0:007049552shþ 0:002651319s3h

þ 0:001964390sh3 � 0:000687963s3h3; ð19ÞgS1

44ðs;hÞ¼ 0:030866163�0:002817808s2�0:014222202s4

þ0:003809592s6�0:000342597h2�0:000061248h4

þ0:000133963h6þ0:001669258s2h2�0:000392022s4h2

�0:000278588s2h4; ð20Þ

gS155ðs;hÞ¼ 0:033317277þ0:005271892s2�0:000494118s4

þ0:001163424s6�0:004160058h2�0:001156234h4

þ0:000538778h6�0:002471099s2h2þ0:001465732s4h2

þ0:000995707s2h4; ð21Þ

g44 g

g45

Fig. 7. The coordinate dependence of g44, g45, and g55 for 1,4-benzodioxanrespectively.

gS145ðs; hÞ ¼ �0:006859644shþ 0:002464505s3h

þ 0:001826140sh3 � 0:000566876s3h3: ð22Þ

Similarly, the kinetic energy expressions for the S0

ground state of 14BZD were obtained as

gS044ðs;hÞ¼ 0:033476535�0:006855096s2�0:014646045s4

þ0:003929556s6�0:003116382h2þ0:001133362h4

�0:000106793h6þ0:007942282s2h2�0:003060725s4h2

�0:001319295s2h4; ð23Þ

gS055ðs;hÞ¼ 0:044329620�0:001141322s2þ0:002428906s4

þ0:000121541s6�0:015598839h2þ0:005139534h4

�0:000686260h6þ0:009920440s2h2�0:000107406s4h2

�0:003277779s2h4; ð24Þ

gS045ðs; hÞ ¼ �0:014930208shþ 0:0083726413s3h

þ 0:006351398sh3 � 0:0040797066s3h3; ð25Þ

and the kinetic energy expressions for the 14BZD S1(p,p*)excited state are

gS144ðs;hÞ¼ 0:035277287�0:006634635s2�0:012450464s4

þ0:003244677s6�0:003583293h2þ0:001484587h4

�0:000173562h6þ0:007809779s2h2�0:002474722s4h2

�0:001570280s2h4; ð26Þ

55

S1 state. s and h are the ring-twisting and ring-bending coordinates,

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

gS155ðs;hÞ¼ 0:043567071�0:001398840s2þ0:001948076s4

þ0:000400976s6�0:015515522h2þ0:005131021h4

�0:000689175h6þ0:009406176s2h2�0:000573554s4h2

�0:003044145s2h4; ð27Þ

gS145ðs; hÞ ¼ �0:014264471shþ 0:007050807s3h

þ 0:0061752092sh3 � 0:003623517s3h3: ð28Þ

These gij (i, j = 4,5) expressions for both the S0 andS1(p,p*) states are presented graphically in Figs. 6 and 7,respectively.

For TET the two-dimensional expressions show thatg44(s,h) for the twisting depends almost only on thetwisting coordinate s. However, both g55(s,h) andg45(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 depen-dence on h.

4. Conclusions

In this paper we add to our arsenal of computer pro-grams for calculating the kinetic energy terms for thering-twisting and ring-bending of TET and other similarbicyclic molecules such as 14BZD. These are essential formeaningful potential energy calculations of the type wehave carried out before. Calculated results were presentedfor TET and 14BZD to show the nature of these kineticenergy expansions.

Acknowledgements

The authors thank the National Science Foundation(Grant CHE-0131935) and the Robert A. Welch Founda-tion (Grant A-0396) for financial assistance.

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