Numerical Evaluation of Energy Levels and Wave Functions for

Research: Science and Education

Numerical Evaluation of Energy Levels and Wave Functions for Hindered Internal Rotation Gianfranco Ercolani Dipartimento di Scienze e Tecnologie Chimiche, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 00133 Roma, Italy; [email protected]

The existence of barriers to rotation about chemical bonds is fundamental to the structural properties of molecules and conformational analysis (1–3). The three minimum-energy staggered conformations and three maximum-energy eclipsed structures of ethane are a classic example of the way in which the energy changes with a bond rotation. The origin of this energy barrier has aroused much discussion (4–6 ) and there appears to be a consensus that it arises from antibonding interactions between vicinal hydrogen atoms. In spite of the important consequences of torsional motion in molecular structure and dynamics, quantum mechanical treatments of restricted internal rotation are practically ignored by physical chemistry texts. No doubt this is because the Schrödinger equation for the internal rotation of even a simple molecule such as ethane is difficult to solve analytically. However, owing to the wide availability of computers, students should be prompted to become more familiar with numerical methods and to go beyond problems having analytic solutions. Numerical treatments of the one-dimensional Schrödinger equation based on either the linear variation method (7, 8) or the Noumerov method (9–11) have already been reported in this Journal. Here I wish to introduce a simple and efficient numerical method, the finite-difference boundary-value method, which is particularly convenient for the treatment of one-dimensional problems having periodic nature, such as internal rotation. This treatment has been adapted, with simplifications, from an article by ChungPhillips (12). Most of the mathematics is straightforward and should be accessible to teachers and students at the upper level associated with physical chemistry. Since the quantum mechanics of internal rotation is presented at an introductory level, even those who will not follow the math should find the results of interest. Method Consider two groups rotating about a common axis. In classical mechanics, the kinetic energy of internal rotation is

dϕ T = 1 Ir 2 dt

are symmetric tops, a degree of simplification is allowed; and thus, for low-resolution spectroscopy and for thermodynamic purposes, one may treat symmetric top rotation as independent from external rotation (15, 16 ). For this reason I confine my discussion principally to molecules such as CH3CH3 or CH3CCl3, which contain two coaxial symmetric tops, and suggest only an approximate procedure for the treatment of internal rotations involving asymmetric tops. In the case of two coaxial symmetric tops the reduced moment of inertia is

Ir =

(2)

I1 + I2

where I1 and I2 are the moments of inertia of the two groups about the axis of internal rotation. Let V(ϕ) represent the potential energy for the internal rotation. Then the Schrödinger equation for the torsional motion may be written as follows:

2 h2 d ψ + V ϕ ψ = Eψ 2Ir dϕ2

(3)

To obtain a numerical solution of this one-dimensional eigenvalue equation, consider a sequence of mesh points, denoted ϕ0, ϕ1, …, ϕn, which are spaced apart by a constant step s = 2π/n, over the range from π to π, so that

ϕ i = π + is

(4)

where i = 0, 1, …, n. Let us indicate with ψi the values of the wave function at the ϕi’s. Since the periodic boundary condition requires that ψ(ϕi) = ψ(ϕi + 2 π) = ψ(ϕi + ns), it follows that ψi = ψi+n. Consider now the three-point central-difference approximation (17) for the second-order derivative of the wave function at ϕi :

dψ 2

2

dϕ

(1)

where Ir is the reduced moment of inertia for the relative motion of the two groups about the axis around which the dihedral angle ϕ (torsion angle) is measured. If the groups have a threefold or higher symmetry axis (e.g., CH3), they are classified as symmetric tops. Such groups have the property that their rotation does not alter the principal moments of inertia of the molecule. In general, a rigorous treatment of internal rotation is complicated by the coupling with overall rotation of the entire molecule (13, 14). However, if the rotating groups

I 1I 2

2

≈

ψi+1 – 2ψi + ψi–1 (5)

s2

i

This is easily obtained considering a finite Taylor expansion of the wave function at ϕ i+1 ,

ψi+1 ≈ ψi +

dψ d2ψ s 2 s+ dϕ i dϕ2 i 2

ψi–1 ≈ ψi –

dψ d2ψ s 2 s+ dϕ i dϕ2 i 2

(6)

and at ϕi–1

JChemEd.chem.wisc.edu • Vol. 77 No. 11 November 2000 • Journal of Chemical Education

(7)

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Adding eqs 6 and 7 and rearranging the resulting equation, eq 5 is obtained. Of course the smaller the step size s, the greater the accuracy of eq 5. Substituting eq 5 into the Schrödinger equation (eq 3) evaluated at ϕi , the following equation is obtained: a ψi–1 + bi ψi + a ψi+1 = E ψi

(8)

where

a=

h2 2I r s 2

2

and

b i = Vi +

h Ir s 2

Note that a is a constant, whereas bi depends on the potential energy at the point ϕi . Let us consider a specific eigenvalue E j , and attach the subscript j to the corresponding wave function so that ψij = ψj ( π + is). Stepping through the points, from i = 1 to i = n, the following system of linear equations is obtained from eq 8:

a ψ1 j + b 2 ψ2 j + a ψ3 j = E j ψ2 j

(12)

2 s 1 ψ0j2 + ψ1j2 + ψ2j2 + … + ψn–1j + 1 ψn2j = 1 2 2

(13)

n

The periodic boundary condition requires that in the first equation of the system ψ0j = ψnj , and in the last equation ψn+1, j = ψ1 j . Considering all the eigenvalues in the range of j from 1 to n, n systems of equations, one for each Ej value, are obtained. All these systems are represented in matrix notation (18) by the following equation: (10)

where F is a real symmetric n × n matrix whose nonzero elements are Fii = bi (those along the principal diagonal), Fi, i+1 = Fi, i-1 = a (those along the diagonal above and below the main one, respectively), F1n = Fn1 = a (the right upper corner element and the left lower corner element, respectively); ⌿ is a n × n matrix whose elements are ψij , and thus the elements of each column represent the values of the autofunction ψj at ϕ = π + is for i ranging from 1 to n; E is a diagonal n × n matrix whose elements are zero except those on the diagonal that are Ej. Equation 10 is a matrix eigenvalue equation where ⌿ is the matrix of eigenvectors and E is the matrix of eigenvalues. Since F is a real symmetric matrix, its eigenvectors form a complete, orthogonal set (i.e., the scalar products of all pairs of eigenvectors are zero), its eigenvalues are all real, and the inverse ⌿ 1 is the same as the transpose, ⌿T (18). Then multiplying eq 10 from the left by ⌿T gives (11)

In eq 11 the matrix F is reduced to the diagonal form by a so-called unitary transformation with the matrix ⌿. Since the matrix F is easily set up, the problem is to seek the matrix ⌿ that diagonalizes F. This procedure is called matrix diagonalization. Because of the importance of eigenvalue equations in all areas of science and engineering, many methods have

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ψj2dϕ = 1

Introducing the periodic boundary condition ψ0j = ψnj, eq 13 can be rewritten in the more compact form

a ψn–1, j + b n j ψn j + a ψn+1, j = E j ψn j

⌿T F ⌿ = E

π

(9)

a ψn–2, j + b n–1, j ψn–1, j + a ψn j = E j ψn–1, j

F ⌿= ⌿E

π

The integral in eq 12 can be approximated by the trapezoidal rule (17, 19) as follows:

a ψ0 j + b 1 ψ1 j + a ψ2 j = E j ψ1 j a ψ2 j + b 3 ψ3 j + a ψ4 j = E j ψ3 j

been devised for diagonalizing matrices. A common and readily understood numerical method is that of Jacobi, which dates from the middle of the 19th century. However, modern eigenvalue/eigenvector algorithms are based on Householder reduction of the starting real symmetric matrix to tridiagonal form, followed by diagonalization by the QR method (17, 19). Diagonalization routines are available on most college and university mainframe computers and are also available for personal computers in a number of language versions (19). Matrix diagonalization can also be carried out by a number of easy-to-use numerical software packages such as Maple, MathCad, Mathematica, or MatLab (20). Diagonalization of the matrix F yields approximate energy levels E j and their wave functions ψj. In order to be normalized, the wave functions should satisfy the following condition:

s Σ ψij2 = 1 i=1

(14)

The summation in eq 14 is given by the sum of the square of the elements of a column of the eigenvector matrix ⌿. Since the diagonalization procedure yields the eigenvectors as vectors of unit length, such summation is equal to 1. Thus, in order to normalize the wave functions so as to satisfy eq – 14, the matrix of eigenvectors must be divided by √ s. Free Internal Rotation To evaluate the accuracy of the numerical solution as a function of the step size s, and therefore of the number of mesh points n, I considered a problem that has a simple analytic solution (viz., the free internal rotation) and compared the approximate numerical results with the exact ones. If V(ϕ) = 0, the internal rotation is free, and eq 3 can be rewritten in the following form:

d2 ψ dϕ2

+ k2 ψ = 0

(15)

where

k2 =

2I r E h2

It is easy to show that the solution of eq 15 is1 ψ = A cos(kϕ) + B sin(kϕ)

(16)

The boundary condition is that ψ should be periodic with period of 2π, which requires that k = 0, ±1, ±2, …. The energy levels are therefore

Journal of Chemical Education • Vol. 77 No. 11 November 2000 • JChemEd.chem.wisc.edu


Ej =

h 2k 2 2I r

(17)

where k = 0, ±1, ±2, … and the j label assumes the values 1 when k = 0, 2k when k > 0, and 1 – 2k when k < 0. Apart from the case k = 0, for which after normalization the wave function is

ψ1 = 1 2π

k =0

(18)

in all the other cases the two degenerate wave functions corresponding to ± k have the following form

ψ± = A cos k ϕ ± B sin k ϕ

±k ≠ 0

(19)

Formulation of degenerate wave functions is inherently ambiguous because any linear combination of degenerate wave functions is an equally acceptable solution of the wave equation with the same eigenvalue. However, since the diagonalization procedure of a real symmetric matrix affords orthogonal eigenvectors, to compare the analytic solutions with the numeric ones, we must form orthogonal linear combinations of the degenerate wave functions. The wave functions shown in eqs 19 are not orthogonal; however, taking their sum and their difference, we obtain, after normalization, the wave functions shown in eqs 20 and 21, which, as could be easily demonstrated, are orthogonal.

ψ2k = 1 cos k ϕ π ψ1–2k = 1 sin k ϕ π

Figure 1. Plots of theoretical and approximate eigenvalues for a free rotator with Ir = 10 amu Å2 as a function of the absolute value of the quantum number k.

k >0 k 0 to eq 20 and k < 0 to eq 21 is arbitrary; the opposite choice would do as well. To compare the numerical results with the analytical ones I have considered a hypothetical free rotator having a reduced moment of inertia of, say, 10 amu Å2,2 and carried out the numerical integration of the Schrödinger equation by the method outlined above for n = 90, 180, 360, 720, 1440.3 In Figure 1 are reported the theoretical (from eq 17) and approximate Ej values, obtained numerically, as a function of the absolute value of the quantum number k, up to |k| = 100. The curve relative to n = 1440 is reported because it is practically indistinguishable from the theoretical curve, the relative error being less than 1% up to |k| = 80. One can see that the accuracy of the approximate eigenvalues increases, as expected, on increasing n, and that for a given n value, it decreases on increasing k. In fact, the energy of the first few levels can also be obtained with good accuracy with relatively low n values. This is not surprising because from eqs 20 and 21 it appears that k represents the number of cycles of the wave function in the range from π to π. Each wave function is thus sampled by n/k points per cycle. Since waves of different frequency would require the same number of points per cycle to be sampled with the same accuracy, it is intuitive that wave functions corresponding to lower k values are better sampled, and, consequently, their eigenvalues are better evaluated. In Figure 2 are reported plots of the first few free rotator wave functions, which illustrate the excellent accordance between selected points obtained by the numerical procedure and the theoretical curves. The Internal Rotation in the Ethane Molecule

Figure 2. Plots of the first five free-rotator wave functions. The points are a sample of the elements of the first five columns of the normalized eigenvector matrix, obtained by the numerical procedure with n = 1440. The theoretical curves are calculated by eqs 18, 20, and 21.

In molecules such as ethane, because of the interactions between the hydrogen atoms on each carbon, the internal rotation is not free, but is said to be restricted or hindered. As the two methyl groups rotate about the carbon–carbon bond, the hydrogen atoms become alternately eclipsed (directly opposite each other) and staggered. The potential energy associated with this rotation is shown in Figure 3. The maxima correspond to the eclipsed conformation, and the minima correspond to the staggered conformation.


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The potential energy curve can be approximately represented by the following equation: V=

V0 2

1 – cos zϕ

(22)

where V0 is the height of the barrier (V0 = 12.4 kJ mol1 for ethane [22]), z is the number of equivalent minima (z = 3 for ethane), and ϕ is the torsion angle, chosen so that ϕ = 0 corresponds to a minimum. In most cases, equivalent minima are a consequence of the symmetry of the rotating groups; then z coincides with the symmetry number for internal rotation (σint ), which is defined as the number of indistinguishable but nonidentical positions into which the molecule can be turned by internal rotation. Before discussing the results relative to the quantum mechanical treatment of the hindered internal rotation it is useful to consider the character of the energy levels near the potential minima (Ej > V0) (15). Near the potential minima, the wave function will be confined to small values of ϕ, and we can expand eq 22 in a Maclaurin series truncated after the second-order term:

V≈

V0 z2 ϕ 2 4

(23)

With this potential, the wave function is that for a simple harmonic oscillator with a force constant of V0 z 2/2, and the motion is best indicated as torsional vibration or libration. Accordingly the energy levels are given by with

4

Ev = (v + 1⁄2 )h ν

ν= z 2π

v = 0, 1, 2, …

V0 2I r

(24)

(25)

Therefore the restricted rotator energy levels will approach the harmonic oscillator levels at energies well below the top of the barrier. At high energies, well above the top of the barrier, one can replace the potential shown in eq 22 by its average value V0 /2. The resulting Schrödinger equation is that of a free rotator in which the zero of the energy scale has been shifted by V0 /2. Thus the energy levels are 2 2

Ej =

V h k + 0 2π 2

k = 0, ± 1, ± 2, …

(26)

The restricted rotator energy levels, therefore, will approach the shifted free rotator levels at energies well above the top of the barrier. Let us consider now the quantum mechanical treatment of the hindered internal rotation in the ethane molecule. The required parameters for the evaluation of the reduced moment of inertia of ethane are the mass of the 1H isotope, mH = 1.0078 amu (23a); the C–H bond length, l = 1.094 Å; and the C–C–H bond angle, θ = 111.2° (23b). It could be easily shown that the distance r of a hydrogen atom from the C–C rotation axis is r = l sin θ. Since the moment of inertia of a collection of masses m j, each located at some distance rj from 1498

an axis of rotation, is given by Σm j rj2, the moment of inertia of a methyl group is 3mH l 2 sin2 θ. From eq 2 it appears that the reduced moment of inertia of ethane is just half of that of a methyl group; thus Ir = 1.57 amu Å2. By substituting eq 22 into eq 3, the Schrödinger equation for the hindered rotation of two coaxial symmetric tops is obtained

2 h 2 d ψ V0 + 1 – cos zϕ ψ = E ψ 2 I r dϕ 2 2

(27)

I have carried out the numerical integration of eq 27 for ethane by the method outlined above, with n = 1440. The obtained energies of the first few levels are reported in Table 1. The accuracy of the results is probably of the same order of magnitude as that evaluated for the free rotator. In Table 1 are also reported the energy levels for the corresponding harmonic oscillator, calculated by eqs 24 and 25, and for the corresponding shifted potential free rotator, calculated by eq 26. Note that the three minima in the potential energy curve of ethane (Fig. 3) have been considered as three independent harmonic oscillators; therefore each energy level of the torsional vibration appears in Table 1 as threefold degenerate. Table 1 shows, as expected, that the hindered rotator energy levels tend to those of the harmonic oscillator in the lowenergy region and to those of the shifted potential free rotator at high energies. As for the splitting of energy levels, from Table 1 it appears that on going from the harmonic oscillator to the hindered rotator, each of the threefold degenerate energy levels is split into two sublevels, one of these being doubly Table 1. The First Few Energy Levels for the Hindered Rotation of Ethane and the Corresponding Torsional Vibration and Shifted Potential Free Rotation Hindered Rotation E/kJ mol 1

Torsional Vibration

Shifted Potential Free Rotation

v

E/kJ mol 1

E/kJ mol 1

k

1.8176 1.8177 1.8177

0

11.8931 11.8931 11.8931

16.2000 16.3284 16.3284

0 +1 1

5.2854 5.2854 5.2880

1

15.6792 1 5.6792 1 5.6792

16.7138 16.7138 17.3560

+2 2 +3

8.3543 8.3973 8.3973

2

1 9.4653 19.4653 1 9.4653

17.3560 18.2552 18.2552

3 +4 4

10.8703 10.8703 11.2309

3

13.2514 13.2514 13.2514

19.4112 19.4112 10.8241

+5 5 +6

12.4981 13.3765 13.3765

4

17.0375 17.0375 17.0375

10.8241 12.4939 12.4939

6 +7 7

15.0765 15.0765 17.0722

5

20.8237 20.8237 20.8237

14.4206 14.4206 16.6042

+8 8 +9

17.1209 19.4343 19.4343

6

24.6098 24.6098 24.6098

16.6042 19.0447 19.0447

9 +10 10


. . .

. . .

. . .

. . .

. . .

. . .

.


Figure 3. Potential energy curve for internal rotation of ethane.

degenerate. The degenerate sublevel is alternately the upper and lower one. The magnitude of the splitting is practically negligible for v = 0 and 1, but increases rapidly with increasing vibrational quantum number v of the torsion oscillation. On going from the free rotator to the hindered rotator, one can see that only the doubly degenerate energy levels corresponding to |k| = 3, 6, 9, …, etc. undergo a splitting, whose magnitude tends to reduce as soon as the hindered rotator energy levels go over into those of the free rotator. The splitting of the energy levels in ethane is readily understood; since hindered rotation is intermediate between the two limiting cases of torsional vibration and free rotation, the energy levels that are degenerate in both the harmonic oscillator and free rotator are also degenerate in the hindered rotator. It must be emphasized, however, that the illustrated splittings do not take into account interactions with external rotation, which produce additional splittings (16 ). In Table 2 are reported the torsional energy levels of ethane in reciprocal centimeters and a comparison of the calculated frequencies from this work with the experimental values; the accordance is very good. Figure 4 shows plots of the first three wave functions for the hindered rotation in ethane. Less Symmetrical Rotors If there is a single internal rotation in a molecule, it is convenient to consider it as the rotation of one top against the rest of the molecule thought of as a fixed frame. The reduction of the moment of inertia of the rotating top is required to uncouple the external and internal rotations. For the simplest case of a molecule made of two symmetrical coaxial tops, the reduced moment of inertia is given by eq 2. The next step, in terms of complexity, is a symmetric top attached to a rigid unsymmetrical framework—for example, CH3CH2Cl. For this case one may write

Figure 4. The first three wave functions for internal rotation of ethane. Note that ψ2 and ψ3 are degenerate wave functions.

Table 2. Torsional Energy Levels and Frequencies for Ethane v

Ev a/cm1

Transition

Frequency/cm1 Exptl b

Calcd c

ν (0 →1)

289

290

ν (1→2)

255 ( A ); 258 ( E ) 256 ( A ); 260 ( E )

0 152 ( A, E ) 1 442 ( E, A ) 2 698 ( A ); 702 ( E ) 3 909 ( E ); 939 ( A ) 4 1045 ( A ); 1118 ( E ) aCalculated from Table 1 considering 1 kJ mol1 = 83.5935 cm1 . A type levels are nondegenerate and E-type levels are doubly degenerate. bWeiss, S.; Loroi, G. E. J. Chem. Phys. 1968, 48 , 962. cThis work.

I r = I 1 – I 12

λ x2 A

+

λ y2 B

+

λ z2 C

(28)

where I1 is the moment of inertia of the symmetric top; λ x , λ y , and λ z are, respectively, the direction cosines between the axis of internal rotation and the principal X, Y, Z axes of the molecule; and A, B, C are the principal moments of inertia of the molecule about the X, Y, Z axes, respectively (13a, 13b). If one considers the application of the formula shown in eq 28 rather laborious, he will find the evaluation of the reduced moment of inertia involving either two asymmetric tops or compound rotations (rotating groups attached in turn to other rotating groups) unduly complicated (13–15). Here I prefer to illustrate an interesting and useful approximate method to evaluate the reduced moment of inertia in complex problems that has been recommended for order-of-magnitude calculations (24). The method is based on eq 2; however the moment of inertia of each top is computed, not about the axis containing the twisting bond, but about the axis parallel to the bond and passing through the center of mass of the top. Let us consider, for example, 1,2-dichloroethane, a molecule containing two asymmetric tops. The geometry of the more stable, anti conformation of 1,2-dichloroethane as obtained by the ab initio MO program Gaussian 98 (25) at the HF/6-31G(d) level of theory, is as follows: C–C bond length,


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y

Cl x d1 rH

C

ξ x d2

H

H

Figure 5. Projection of one of the tops of 1,2-dichloroethane onto a plane perpendicular to the C–C rotation axis. The cross indicates the projected position of the center of mass of the top.

Figure 6. Potential energy curve for internal rotation of 1,2-dichloroethane.

l0 = 1.516 Å; C–Cl bond length, l1 = 1.792 Å; C–H bond length, l2 = 1.078 Å; C–C–Cl bond angle, θ1 = 109.4°; C–C–H bond angle, θ2 = 111.5°; improper Cl–C(C)–H dihedral angle (the angle between the two planes containing, respectively, the Cl–C1–C2 and the H–C1–C2 atoms), ξ = 118.6°. Merely for the purpose of evaluating the reduced moment of inertia, it is assumed that these parameters do not change upon internal rotation. A projection of one of the tops onto a plane perpendicular to the C–C rotation axis appears as depicted in Figure 5, where d 1 (= l1 sin θ1) and d 2

(= l 2 sin θ 2) are the distances of Cl and H atoms, respectively, from the rotation axis. The x coordinate of the center of mass of the top is equal to zero by symmetry considerations, whereas the y coordinate, given by

Table 3. Calculated Torsional Energy Levels of 1,2-Dichloroethane in cm –1 v

This Work anti

gauche a

Ref 12 anti

gauche a

10

62

64.1(733)

62

62.1(730)

11

186

192.1(861)

186

184.1(853)

12

307

316.1(985)

307

304.1(973)

13

426

437.(1106)

425

421.(1090)

14

543

554.(1223)

540

535.(1204)

15

657

668.(1337)

653

646.(1315)

16

768

777.(1446)

763

753.(1422)

17

877

881.(1550)

870

855.(1524)

18

982

979.(1648)

974

952.(1621)

19

1085

1064 (1733) 1067 (1735)

1075

1040 (1709) 1041 (1710)

10

1184

1123 (1792)

1172

1103 (1772)

11

1280

—

1266

—

12

1371

—

1356

—

13

1459

—

1442

—

14

1542

—

1523

—

15

1619

—

1600

—

16

1689

—

1669

—

17

1753

—

1731

—

NOTE: All levels are nondegenerate for the anti conformer and doubly degenerate for the gauche conformer with v < 9. aThe first value is the energy relative to the gauche minimum at 669 cm1; the value in parentheses is relative to the anti minimum at 0 cm1.

1500

Y=

Σm j y j Σm j

is 1.187 Å. This has been computed by the masses of the isotopes 12C (12 amu), 35Cl (34.9689 amu), and 1H (1.0078 amu) (23a), and by the y coordinates of the atoms: yC = 0, y Cl = d1, y H = d 2 cos ξ. The distances of the atoms from the axis parallel to the C–C–bond and passing through the center of mass – – – of the top are rC = Y, rCl = d1 – Y, rH = (Y 2 + d 22 – 2Y d2 cos ξ)1/2, the latter being obtained from the triangle shown in Figure 5. The moment of inertia of a chloromethyl group is then readily evaluated as 32.9 amu Å2, and the reduced moment of inertia of 1,2-dichloroethane as 16.45 amu Å2. The adiabatic or relaxed potential energy curve for internal rotation of 1,2-dichloroethane,5 obtained at the HF/ 6-31G(d) level of theory (12), is plotted in Figure 6. It can be represented by a six-term Fourier-series expansion z=6

V = 1 Σ V z 1 – cos zϕ 2 z=1

(29)

where the Fourier coefficients, expressed in kJ mol1, are V1 = 20.6706, V2 = 8.2119, V3 = 21.5321, V4 = 0.9372, V5 = 0.4422, V6 = 0.0117. The potential energy curve shows three minima. The lower one at ϕ = 0° corresponds to the anti conformation, whereas the two equivalent minima at ϕ = ±109.5° correspond to the gauche(+) and gauche() conformations, respectively. These have an energy of 8.00 kJ mol1 relative to the anti minimum. Substituting eq 29 into eq 3 and integrating the resulting Schrödinger equation by the finite-difference boundary-value method with n = 1440, one obtains the torsional energy levels shown in Table 3. The levels in the anti conformer are nondegenerate, whereas those in the gauche conformer are doubly degenerate for v < 9. For comparison, Table 3 also reports the corresponding levels obtained by the more elaborate method of



copy or conformational analysis, and in statistical treatments of thermodynamics and reaction rates. However, students should be aware that, owing to the approximations involved, especially in the treatment of asymmetric top rotations, caution should be exerted for the correct appraisal of results. Acknowledgments I wish to thank Mauro Di Mario for stimulating discussions and for an early implementation of a diagonalization routine in Basic. Notes

Figure 7. The first four wave functions for internal rotation of 1,2dichloroethane localized in the potential well of the anti conformer.

1. Solutions in terms of exp(± ik ϕ) are of course also permissible. 2. 1 amu Å2 = 1.66053873 × 1047 kg m2 (21). 3. Carrying out such calculations is a matter of few minutes on modern PCs. 4. Note that in eq 25 the barrier height is expressed in joules per molecule. 5. While a rigid geometry is assumed for the evaluation of the reduced moment of inertia, as to the evaluation of the potential energy curve, bond lengths and angles are allowed to relax to their equilibrium value on changing the internal rotation angle. 6. A small part of the deviation can be ascribed to the trivial fact that we used 1440 mesh points in the calculations, against the 720 used by Chung-Phillips.

Literature Cited Chung-Phillips (12), in which the reduced moment of inertia of 1,2-dichloroethane is accurately evaluated as a function of the torsional angle ϕ. The average relative deviation of our results with respect to those of Chung-Phillips, for the energy levels up to v = 10, is 0.5% for the anti conformer and 3.2% for the gauche conformers.6 The good performance of our simplified approach appears further supported by a comparison of the experimental torsional frequencies with the calculated ones: anti ν (0→1): exptl: 129 cm1 (26 ) calcd: 124 cm1 (this work); 123 cm1 (12) gauche ν (0→1): exptl: 126 cm1 (26 ) calcd: 127 cm1 (this work); 122 cm1 (12) The use of the simplified approach here presented appears justified, especially in view of the fact that treating torsion of asymmetric tops apart from other molecular motions is in itself an approximation that holds great uncertainty. In the gauche conformation of 1,2-dichloroethane, for instance, the torsion interacts strongly with C–Cl deformation (12). In Figure 7 are reported plots of the first few wave functions for the hindered rotation in 1,2-dichloroethane, which are localized in the potential well of anti conformation. Conclusions Internal rotations of other molecules may be easily investigated with the methods illustrated here. Calculated energy levels and wave functions so obtained can be of great value in many kinds of torsional studies in infrared spectros-

1. Internal Rotation in Molecules; Orville-Thomas, W. J., Ed.; Wiley: New York, 1974. 2. Lister, D. G.; MacDonald, J. N.; Owen, N. L. Internal Rotation and Inversion; Academic: London, 1978. 3. Mizushima, S. Structure of Molecules and Internal Rotation; Academic: New York, 1954 4. Smith, D. W. J. Chem. Educ. 1998, 75, 907–909, and references therein. 5. Sacks, L. J. J. Chem. Educ. 1986, 63, 487–489. 6. Eyring, H.; Grant, D. M.; Hecht, H. J. Chem. Educ. 1962, 39, 466. 7. Sims, J. S.; Ewing, G. E. J. Chem. Educ. 1979, 56, 546–550. 8. Dunbrack, R. L. Jr. J. Chem. Educ. 1986, 63, 953–955. 9. Blukis, U.; Howell, J. M. J. Chem. Educ. 1983, 60, 207–212. 10. Kubach, C. J. Chem. Educ. 1983, 60, 212–213. 11. Tellinghuisen, J. J. Chem. Educ. 1989, 66, 51. 12. Chung-Phillips, A. J. Comp. Chem. 1992, 13, 874–882. 13. (a) Pitzer, K. S.; Gwinn J. Chem. Phys. 1942, 10, 428–440. (b) Pitzer, K. S. J. Chem. Phys. 1946, 14, 239–243. (c) Kilpatrick, J. E.; Pitzer, K. S. J. Chem. Phys. 1949, 17, 1064– 1075. (d) Li, J. C. M.; Pitzer, K. S. J. Phys. Chem. 1956, 60, 466–474. 14. Gang, J.; Pilling, M. J.; Robertson, S. H. J. Chem. Soc., Faraday Trans. 1996, 92, 3509–3518. 15. Pitzer, K. S. Quantum Chemistry; Prentice-Hall: New York, 1953; pp 239–243, 494–500. 16. For a discussion about the coupling of internal and external rotation in ethane and methanol, see: Wilson, E. B. Jr. J. Chem. Phys. 1938, 6, 740–745. Koehler, J. S.; Dennison, D. M. Phys. Rev. 1940, 57, 1006. Herzberg, G. Molecular Spectra and Molecular Structure II. Infrared and Raman Spectra of Polyatomic


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22. Chao, R.; Wilhoit, R. C.; Zwolinski, B. J. J. Phys. Chem. Ref. Data 1973, 2, 427–438. 23. (a) Handbook of Chemistry and Physics, 79th ed.; Lide, D. R., Ed.; CRC Press: Boca Raton, FL, 1998; p 1-10. (b) Ibid. p 9-31. 24. Herschbach, D. R.; Johnston, H. S.; Pitzer, K. S.; Powell, K. S. J. Chem. Phys. 1956, 25, 736–741. 25. Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Zakrzewski, V. G.; Montgomery, J. A. Jr.; Stratmann, R. E.; Burant, J. C.; Dapprich, S.; Millam, J. M.; Daniels, A. D.; Kudin, K. N.; Strain, M. C.; Farkas, O.; Tomasi, J.; Barone, V.; Cossi, M.; Cammi, R.; Mennucci, B.; Pomelli, C.; Adamo, C.; Clifford, S.; Ochterski, J.; Petersson, G. A.; Ayala, P. Y.; Cui, Q.; Morokuma, K. D.; Malick, K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Cioslowski, J.; Ortiz, J. V.; Baboul, A. G.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Gomperts, R.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Gonzalez, C.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Andres, J. L.; Gonzalez, C.; Head-Gordon, M.; Replogle, E. S.; Pople, J. A. Gaussian 98, Revision A.7; Gaussian, Inc.: Pittsburgh, PA, 1998. 26. Lane, G. Ph.D. thesis, University of Manchester, 1966. Dew, G. Ph.D. thesis, University of Manchester, 1969. (Sources cited by Allen, G.; Fewster, F. In Internal Rotation in Molecules; OrvilleThomas, W. J., Ed.; Wiley: New York, 1974: Chapter 8.)


Numerical Evaluation of Energy Levels and Wave Functions for

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