Stark Effects on Rigid-Rotor Wavefunctions A Quantum Description of Dipolar Rotors Trapped in Electric Fields as Pendulum ~sciilators Giles Henderson and Brad Logsdon Eastern Illinois University, Charleston, IL61920 Undergraduate physical chemistry students are exposed to qualitative descriptions of the Stark effect a s a precise means of measuring the electric dipole moment of molecules ( I d ) . The Stark effect has also been a popular example for introducing students in quantum mechanics to the application of perturbation theory (6-11). Approximate eigenvalue expressions derived from perturbation theory are frequently used by authors of spectroscopy texts to calculate energy-level diagrams and to explain the details of the splitting and assignments of atomic and molecular spectra (12-15). These traditional discussions focus primarily on the effects of a n electric field on the energy levels of a n atom or molecule, but they fail to disclose the effects of the field on rotational wavefunctions, the molecular motion, and molecular orientations. For many years it was deemed (16,17) impractical to orient gas-phase molecules with electric fields because their rotational kinetic energies (&T)are large compared to the potential barrier due to the interaction energy of a rotating dipole with accessible laboratory fields. However, it is possible to cool gas molecules in supersonic expansions to rotational temperatures of less than 1K by using molecular beam techniques. I t has been shown (18,19) that with such dramatic cooling, significant molecular orientation is feasible with electric fields of ordinary magnitude. We can anticipate the importance of this technology in future experiments and processes designed to investigate or exploit the dependence of chemical reaction on the orientation of reactant molecules. The quantum mechanical description of how the Stark effect influences molecular dipole orientation is best understood by examining how the quantum probability funcis altered by a n applied tion of a rotating molecule, yr'*'v, electric field. Computer graphics are used in this project to illustrate orientational probabilities obtained from variational wavefunctions. These results will better enable students to correlate the evolution of free-rotor eigenstates with harmonic-oscillator (librator) states and to compare classical and quantum descriptions of a pendulum oscillator. Calculations We wish to consider a linear rigid rotor with a moment of inertia I and electric moment iiinteracting with a uniform electric field %. The full threk-dimensional description of this problem requires two spherical polar angles to describe the orientation of the rotor in our laboratory frame. The corresponding spherical harmonic wave functions of the rigid rotor in the zero-field limit can be correlated with two-dimensional librator wavefunctions of a n isotropic spherical pendulum oscillator in the high-field limit. The linear n-particle rotor is characterized by a moment of inertia. 1=zrnjr;
a.
D. PARTICLE IN A CIRCLE
LINEAR FREE ROTOR
c STARK PENDULUM
gravity / field Fiaure 1. The two-~articierotor (a) is inertiallv eauivalent to a sinale. rei~cea-mass ( C I bane e in a c ick co, s~ro&e w r c I e ds can Gap Ihe o polar molecL e as a Slark penaJ .m (c, !hat Jndergoes bralor mobon analogods lo an Inert al pendL L m n a grav ly 11elo10, Figure l a illustrates a diatomic rotating about its center of mass where mi and mz track about circular paths of radii ri and rz. This two-particle system is inertially equivalent to the simpler single particle in a circle (Fig. l b ) in which I=&
where
is the reduced mass and R = rl + rz is the bond length. Rotation Restricted to a Plane Strong electric fields can trap the dipolar molecule as a Stark pendulum (Fig. lc) that undergoes librator motion analogous to an inertial pendulum in a gravity field (Fig. Id). I t greatly simplifies the desired calculations to restrict the rotation to a plane in which the orientation of the rotor is specified by a single angle q (Fig. 1).Moreover, this constraint allows students to better relate the zero-field to high-field correlation to their previous experience with one-dimensional harmonic oscillators. Volume 72 Number 11 November 1995
1021
In the absence of a n electric field, we then have the familiar free-rotor problem (9)in which the allowed kinetic energies of the rotor are given as eigenvalues to the zeroorder Schrodinger equation.
where ti is Planck's constant divided by 2n. The eignefunctions are simple complex plane waves,
characterized by a quantum number m = 0, il,i2, ..., whose sign designates the direction of rotation. The allowed energy levels are given by the eigenvalues,
The rotor experiences an orienting torque if its dipole moment interacts with a uniform electric field 8. This interaction results i n a potential energy, which hinders the rotational motion and, if sufficiently large, can trap the rotor in a bound librator state. This interaction i s now included in the Hamiltonian.
Stork lnteroction Energy
Figure 2. The interaction of a molecular dipole with a uniform electric field results in a shifting and splitting of the rotor's degenerate energy levels (Stark effect). The wavefunction for each eigenstate i s specified i n terms of the original orthogonal basis @ by the corresponding eigenvector, W = @a or
Variational Method A
Because we are interested in circumstances where HI is not small compared with H, we will now use the variational method with exact matrix diagonalization rather than perturbation theory to find the eigenfunctions and eigenvales of this operator. I n order to construct the matrix representation of eq 3 we shall need integrals of the following type. 2n
277
lhlIm'i = J' 0:hl~,,,dq
(m
= +JO~COS
0
(q)amdq
0
form' m f 1
=0
(4)
The Hamiltonian matrix will then have the elements m W
(ml~lm)=-=m~~ 21
(5)
I t is convenient to construct this matrix in units of the rotational constant. B=-h2 21 The eigenvalues a r e obtained by diagonalizing t h e Hamiltonian matrix (20).
1022
Now that the wavefunction for each state is known, we can easily calculate the orientation's probability function.
Journal of Chemical Education
Using eqs 5 and 6, we can easily write computer codes to set u p the Harniltonian matrix for a given Stark interaction energy. The size of the free-rotor basis set and the order of @ will depend on the magnitude of the Stark interaction and the desired urecision. Abasis set of 0 < m < f13 will provide 1-ppm precision for the five lowest eigenstates with Stark energies 0.0 < p%< 4B. Standard subroutines (21) can be used to diagonalize fl to obtain the energy eigenvalues and the eigenvectors of the a i coefficients of eq 8. The a,j coefficients for a given j eigenstate are used i n a loop designed to sweep through the 9 domain and evaluate the position probability a t each orientation a s prescribed by eq 9. This procedure can be nestled inside an external loop that systematically increments the Stark interaction. The calculated 9(9)values are then written to disk files for subsequent plotting. Results and Discussion Effect on a Free Rotor's Energy Levels The effect of the Stark field on a free rotor's energy levels can be seen in Figure 2. At the zero-field limit we observe a pattern with familiar second-order m dependence. The two-fold degeneracies associated with -fm correspond to clockwise and counterclockwise rotations of the same energy As tht, t!lrctnc field is turned on, t h e m degeneracy is broken, and a s the field incrc~aseii,the lower-lving energies begin to assume harmonic pendulum oscill&or(libra~or) spacings: (1/2)hv, (312)hv, (5/2)hv, ..., . These lower-lying levels are trapped by the interaction of the rotating dipole moment with the applied electric field a s is evident for a Stark interaction p%= 4B in Figure 3.
-
5
1
-180
"
'
-120
'
"
"
-60
"
"
PHI
teg)60
120
180
Figure 3. The iowest-lying energy levels are described by a nearly harmonic librator in which the rotor is trapped by the Stark field and undergoes pendulumlike motions. The potential energy for this interaction is characterized by a cos (cp) dependence on the dipole orientation (eq 3). If we expand this potential in a power series,
for small cp, the potential is harmonic with a torque constant K, = @. The dashed curve i n Figure 3 allows u s to compare this harmonic lihrator potential with the actual Stark potential. Given this description, we can anticipate that the corresponding wavefunctions for these lower levels must begin to assume the character of the familiar harmonic oscillator (librator). Also, these new wavefunctions should he composed of a mixture of im basis functions because the trapped rotor spends half of its time rotating clockwise and the other half rotating counterclockwise as i t librates about its center of mass. This feature is evident in the ground-state eigenvedor given i n the table for V =
Figure 4. Polar plots of y*yi at various Stark fields. The zero-field rotor exhibits familiar circular harmonic wavefunctionsand m degeneracies. The Stark field causes an anisotropy in the angular distributions with maxima consistent with a nearly harmonic librator. transformations induced in these angular functions by the laboratory field are indeed similar to the corresponding hybridization of atomic orbitals induced by the fields of a bonding neighboring atom. Although these functions assume harmonic-lihrator qualities, to students this feature may seem disguised in the polar space. v*v plots for the lower-lying states are compared in both polar and Cartesian space i n Figure 5. Figure 3 shows t h a t these energy levels are below the Stark potential barrier. Classically, the respective rotors behave as librators with turning points given by
The classical turning-point angles are depicted i n Figure 4 a s dashed lines. Because the lihrator spends most of its
CARTES IAN
4B.
POLAR 90
Evolution of Free-Rotor Wavefunctions
I
'
I
We now examine how t h e free-rotor wavefunctions evolve into nearly harmonic-librator wavefunctions as the electric field is increased. Polar plots of the wavefunctions of the first five states are depicted in Figure 4 for interaction energies V = p% = 0.0,2.OB, and 4.OB. When the electric field is off (V = 0.0) the quantum description of t h e m = 0 ground-state rotor provides a n exact value of the angular momentum, L=nE = O
V
= 4B Eigenvector for the
u = 0 Ground State Librator. However, all orientations a r e equally probable. As the field is increased, the probability of the dipole being oriented i n the direction of the laboratory field (cp = 0.0) is dramatically increased a t the expense of o t h e r orientations. This new distribution characterizes t h e s m a l l amplitude zero-point motion of the ground-state (u = 0 ) librator. The orientational probability function of the higher-lying V = 4B excited maxima a t progressively l a r g e r +cp values. T h e
13 1 -1 2 -2 -3
4 -4 5 5
0.712436 0.472288 0.472288 0.149888 0.149888 0.026214 0.026214 0.002833 0.002833 0.000206 0.000206 0.000011
... ...
0.000011
' '
Fioure 5. The w*w function can be viewed in either a Dolar or ~Sesian soace: s his correlation allows the student to recodnize the expected nearly narmonc-Iorator characler lhal mtgnt otn& se be obsc-red in !he ess lam ar polar pots C assca tJrn nq po nts, given in both sets of figures as dashed lines, may be compared with gW maxima. Volume 72 Number 11 November 1995
1023
time a t q = *rp*, these angles have the maximum classical orientational probability. It is instructive to note that Pdq), the quantum probability functions, also exhibit maxima a t similar q values. Because P d q ) is a function of the Stark interaction, these functions can be plotted a s surfaces in the (q,V) space. These plots (Fig. 6) allow us to observe the free-rotor (V = 0.0) functions evolve into approximate harmonic-librator functions a s the Stark field is increased. I n the absence of a Stark field the m > 0 levels are two-fold degene r a t e . Therefore a n y l i n e a r combination of t h e corresponding eigenfunctions (eq 2) are also solutions to the Schrodinger equation. In this study, it is convenient to choose Euler combinations
-
mv +&4 ' =+ f '
and
'f-= v + m - Wrn so that these functions can he conveniently correlated with the V > 0 eigenstates of Figures 3 and 5. The graphics and computational methods described here can provide students with a better understanding of how quantum mechanics describes the effects of a n electric field on a dipolar rotor. Moreover, these descriptions suggest (18)that it is possible to orient rotationally cold molecules produced by supersonic expansions in a molecular beam. These circumstances will undoubtedly be exploited to control the orientation of polar molecules colliding with neutral atoms or nonpolar molecules. I n addition, spectroscopy and electron-diffraction studies might also be carried out on oriented gas-phase molecules rather than the usual random orientations of a static gas. Acknowledgement The author wishes to express his gratitude to Scott Martin for pointing out the recent work by B. Friedrich and D. R. Herschbach (19)in this area and to hoth Scott Martin and a reviewer for other useful suggestions.
0
Literature Cited 1. Adamson, A. W A lkzlhook of Physical Chemistry, 3rd ed.; Academic: New York. 1986; P 829. 2. Levine, I. Physical Chemistry, 2nd ed.; McOraw-Hill: New York. 1983: p 700. 3. Moore, W JPhyslcol ChemiLry.4th ed.;Renti*Hall: En@ewood Cliffa. NJ. 1972; p 761. 4. Noggle, J. H . Physical Chemistry. 2nd ed.; k o t t and Forasman: Glennew IL. 1989: pp 748,822. 5. Vemulapalli, C. K Ph~.sicolChemistry; Prentice-Hall: Englewood Cliffs. NJ, 1993; p 423. 6. Atkins, E W. Mokculor Quantum Mechanics, 2nd ed.; Oxford University: Oxford, 1983; p 246. 7 . Dykstra, C. E. Quantum Chemistry and Mokulor Spcfmsmpy: Prentice-Hall: Englewood CliKs. NJ. 1992: p 386. 8. F'lygare. W. H. Molecvlar StrucfurrondDymmiii;P~~tiiiH~l Englewood Cliffs, NJ, 1978; p 191. 9. Psuling L.; Wilson, E. B. lnfmdvcfionto Quantum Mechanics: Mecraw-Hill: New York, 1 9 8 ; p 177. 10. SchiK L. I. Quantum Mechanics; MeCraw-Hill: New York. 1955: pp 252.268. 11. Ref5. p 496. 12. Barmw C. M. Infmduefionlo Molecular Swctroseonv: .. McOraw-Hill: New York. 1963; p 89. 13. oraybeal, J.D ~ ~ l e c u lspectmsmp~; ar Mecraw- ill: ~ e ~wo r k1988; , p 386.
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Author's Note: The authors have developed an electronic
abstract of this article for the World Wide Web. This multimedia environment allows the reader to view computer animations of the angular motions of a dipolar molecule in various levels, wavefunctionsand position probabilitiesare simultaneously depicted on a multi- panel screen. Hypertext links are used to provide cross-platformaccess to the text, images and animations. these materials are scheduled to be linked to the Journal of Chemical Education home page and the Journal of Chemical Education: Soffwarehome page in December 1995.
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Figure 6. The evolution of yrXV spherical harmonics of the free-rotor to nearly harrnonic-librator functions with an increasing Stark field can be easily visualized on a Cartesian surface. 14. Gufion, W A. lnlroduellan l o Molecular Sprdroscopy: AIlyn and Bacon: Boston. 1972;pp 79, 117. 15. steinfeld, J. I. M ~ I Z C U I ~and S ~ . d i ~ t H~~~~~ i ~ ~ ; and ROW:N ~ WY O F ~ ,1974: pp 54. 14s. 16. Brooks, P R.; Jones, E. M. J Chem. Phys. 1969,51,3073. 17. Bematein, R. B.;Hersehhaeh, D. R.: Levine,R. D. J. Phys Chsm. 1987,91,5365. 18. ~riedrich,B.; ~ ~ ~ s ~ h b0. a R. c h2., ~ h y D. s 1991. 18, 153.161. 19. ~ r i e d r i c h , ~~ersehhach, .; D. R. J.~ h y chsm. r 1881,95,811r8129. 20. &f 8 , P 98. 21. Algorithms for matrix diagonalizatian are well~known.Explicit FORTRAN code for the Jaeobi mothod is @"en by Johnson. K J. Nzrnaricoi Methods in Chemistry: Mareel Dekker:New York. 1980: ~ ~ 4 0 7 4 2 9 .
