15686
J. Phys. Chem. 1995, 99, 15686-15693
Polarization of Molecules Induced by Intense Nonresonant Laser Fields Bretislav Friedrich" and Dudley Herschbach* Department of Chemistry, Harvard University, I2 Oxford Street, Cambridge, Massachusetts 02138 Received: June 28, 1995; In Final Form: August 24, 1995@
The anisotropic interaction of the electric field vector of intense laser radiation with the dipole moment induced in apolarizuble molecule by the laser field creates aligned pendular states. These are directional superpositions of field-free states, governed by a cos2 8 potential (with 8 the angle between the molecular axis and the field vector). We show that the spatial alignment and other eigenproperties can be derived from spheroidal wave functions, give explicit expressions for low- and high-field limits, contrast the induced pendular states with those for permanent dipoles subject to static fields, and present calculations demonstrating the utility of the induced states for rotational spectroscopy, laser alignment, and spatial trapping of molecules.
1. Introduction When subject to an external electric field, the electronic distribution of any molecule (or atom) is distorted to some extent. This interaction, govemed by the molecular polarizability, results in an induced dipole moment. For experimentally feasible static fields, such induced moments are very weak, typically only on the order of D units.' However, far stronger induced moments, well above 1 D, can now be produced by intense laser fields, using either pulsed lasers or supermirror techniques to build up a CW laser mode. This makes it possible to study nonpolar molecules by means that otherwise require permanent dipole moments. The polarizability interaction with an intense nonresonant laser field can be exploited to extend rotational spectroscopy to nonpolar molecules, to suppress rotational tumbling and thereby align the internuclear axis (as in kindred work with permanent dipole^^.^), or to attain spatial trapping of molecule^.^ The polarizability interaction is govemed by a cos2 8 potential, in contrast to a cos 8 potential for a permanent dipole (with 8 the angle between the molecular axis and the electric field direction). If the field is sufficiently strong, these anisotropic interactions create pendular states, directional superpositions of the field-free rotational states in which the molecular axis librates about the field direction. In section 2, we evaluate the pendular energy levels and wave functions for the cos2 8 potential in terms of the well-known spheroidal function^.^,^ As this is a double-well potential, with end-forend symmetry, the levels exhibit characteristic tunnel-effect splittings. We also examine correlations with the field-free rotor states and the high-field harmonic librator limit. In section 3, we compare level patterns, torques, and angular momentum transfer with those for the permanent dipole case. In section 4, we determine the angular distribution of the molecular alignment, characterized by (cos2 6 ) and its ensemble average. Section 5 presents calculations for some representative molecules and prospective applications. 2. Pendular Eigenstates for an Induced Dipole For simplicity, we consider a linear rotor subject to planewave radiation with electric field strength E = EO cos(2nvt). The Schrodinger equation is
with
the squared angular momentum operator, B the rotational
constant, E the eigenenergy, and 8 the polar angle between the molecular axis and the electric field direction. For a permanent dipole moment ,u along the internuclear axis and polarizability components cq and ai parallel and perpendicular to the axis, the interaction potentials are
v,(e) = --p€ cos e
+
V,(O) = - 1 / 2 ~ 2 COS' ( q0, a, sin2 0)
(2b)
For nonresonant frequencies much greater than the reciprocal of the laser pulse duration, Y >> zp-', averaging over the pulse period tpquenches the V, interaction and converts c2 in Va to EO'/^. The time average of the Hamiltonian thus becomes
H = BJ'
+ V, = BJ'
- '/4~:[(~41- a,)cos2
e + a,]
(3)
The same form holds for a nonpolar molecule subject to a static field of strength ed2'l2. Spheroidal Wave Equation. The Schrodinger equation obtained from eq 3 reduces to a spheroidal wave equation,
where z = cos 8. This simple reduction apparently has only recently been noted.4 The quantity c2,a dimensionless anisotropy parameter, is given by
c2 = Aw = wl, - w, = ( q / a , - l)w,
(5)
where
=
@ 1 1 , 1 [oil,l€O2/(4B>l
(6)
When QI > aL,which always holds for linear molecules, A u > 0; this corresponds to the oblate spheroidal case. When ql al,which occurs for some planar molecules such as benzene, Am .c 0; this corresponds to the prolate spheroidal case, which we consider also because it provides instructive contrasts. As seen in Figure 1, for the oblate case V, is a double-well potential, with minima at 8 = 0" and 180", whereas for the prolate case Va has a single minimum at 8 = 90". The separation constant, A j , M , and eigenfunctions, S j , M , can be obtained from extensive tabulations or computed with arbitrary accuracy by standard methods.5.6 The separation constants are related to the eigenenergies by
(7)
@Abstractpublished in Advance ACS Abstracrs, October 1, 1995.
0022-365419512099-15686$09.0010
(24
0 1995 American Chemical Society
J. Phys. Chem., Vol, 99,No. 42, 1995 15687
Polarization of Molecules Induced by Intense Laser Fields
I
4
Oblate
pOBLA'TC
2
Ea 0
-2 -4
-....................... .. *."
..... 22 ............ . ........ . . . . . . . ....... ... JIM
20
......
...
II
oo
YO'
...........
....
.....................
-,
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............ ....... .............
.......
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;,!(
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........ -
)ROO
-8
8
-90" -60'
-30'
0'
90" -W" .60° -70"
00"
30"
Aligiiiiicii1 Aiiglr 0
Prolate
0'
3(y GO"
90'
AllgnmciiI h g l c 0
Figure 2. Potential energy (bold full lines), energy levels (full lines) in units of the rotational constant B, and squares of spheroidal wave functions (dashed lines) for [Awl= 4. The oblate case (at left) has QJ al = 2; the prolate case (at right) has qJal = 1/2. For both oblate and prolate cases, the interaction potentials have V(0")lB = 011 and V(9O")lB = W I . All levels are shifted toward negative energies with respect to the free rotor ground state by 01. Although the physical range of the angle 8 is 0" 180",here the -90" 90" range is plotted to better display the oblate state wave functions. See text, eqs 3-6.
-
90" 180' 9 Figure 1. Contrast of optimal alignments for oblate and prolate versions of anisotropic polarizability interaction,governed by eq 3. For the oblate case, the pair of potential minima in the polar regions, 0 = 0" and 180", are separated by an equatorial barrier that hinders end-for-end rotation via 0 n - 8. For the prolate case, the potential minimum at 0 = 90" extends around the equator, so does not restrain end-forend rotation via the azimuthal angle p p n.
14.01
0'
-
-
,
-2.0
-10.01 -I6
Prolate ' -12
'
-8
m
dJ,&w)YJ,1,1(e9Q) for Sodd
J=2n+ 1
+
+
+
Here 2n = m IMI and 2n 1 = m IMI with m either 0,2, 4, ... or 1,3,5, .... The expansion coefficients, dJ,M(AOJ), depend solely on the interaction parameter AOJ.For any fixed value of the good quantum number M,the range of J involved in this coherent superposition or hybrid wave function increases with the AOJparameter. Since the hybrid wave functions comprise either even or odd Ss, each state is of a definite parity, For AOJ= 0, the eigenproperties become those of a fieldfree rotor; the eigenfunctions then coincide with spherical harmonics, and the eigenvalues become l j , , ~ , EJ,IMI/B = J(J 1). The eigenstates can thus be labeled by [MI and the nominal value j , designating the angular momentum for the field-free rotor state that adiabatically correlates with the highfield hybrid function. In the high-field limit, AOJ 03, the range of 8 is confined near the potential minimum and eq 4 reduces to that for a twodimensional angular harmonic librator. Note that in the oblate
-
+
-
'
'
'
'
'
1
-4
0
4
8
12
1G
A0 Figure 3. Dependence of the spheroidal separation constant Lj,l,wi of eq 4 on the anisotropy parameter A o for the lowest six pendular states. See eq 7 for relation to energy levels. For the oblate case, A o is positive; for the prolate case, negative.
+
c
1
,
I
-
-6.0-
and the Schrodinger eigenfunctions are given by
=
,
Oblate
+
with Q, the azimuthal angle. The eigenfunctions can be conveniently expanded in spherical harmonics,
,
-
case, states with = N 1 for j - IMI odd and = IN1 for j - (MIeven, with the same N = 2(k IM1/2) and k = 0, 1, 2, .... have equal energies in the harmonic librator limit. In the prolate case, states with the same 1 - IMI have the same harmonic energies. Eigenproperties. Figure 2 shows the interaction potential, -(wl Aocos2 e), the energy levels, and the squares of the spheroidal wave functions SJJMIfor typical values of the polarizability anisotropy: qdal = 2 (oblate), udal = 1/2 (prolate), but for simplicity, a low value for the reduced parameter lA.01 = 4. The potential is purely attractive. The w l term contributes a uniform shift toward negative energies; the AOJterm governs the amplitude and location of the potential wells. The number of bound states thus increases with /Awl, Le. grows quadratically with field strength. Figures 3 and 4 display, for the lowest six pendular states, the dependence on AOJ of the eigenvalues, &,MI, and the expectation values of the squared alignment cosine, (cos2 8)j.lMI. The latter is readily evaluated via the Hellmann-Feynman theorem,
+
+
(COS'
e),,,,
= -a(Ej,lMI/B)/a(A(U)
(10)
The energy levels decrease with increasing field strength for the oblate case, whereas they increase for the prolate case. This
Friedrich and Herschbach
15688 J. Phys. Chem., Vol. 99, No. 42, 1995 0.8 Prolate 0.6 A
a
N
8
v
0.4
0.2 0
.
0
~
" -8
"
"
"
0 4 X 12 Ih Am Figure 4. Dependence of the alignment parameter (cos2 6)j.IMl of eq 10 on Aw for the lowest six pendular states; c t Figure 3. -12
-16
-4
reflects directly the opposite sense of the polarizability anisotropy, depicted in Figure 1. Likewise, (cos2 e) increases with Am for the oblate case because the induced dipole moment is parallel to the molecular axis, whereas (cos2 e) decreases with increasing IAml for the prolate case because the induced dipole is perpendicular to the figure axis. Table 1 gives the leading terms in perturbation expansions for the eigenenergy Ej.1~1and for (cos2 e)j,,Ml applicable when IAol is small or large, corresponding to the low- and highfield limits. Effective Potentials. For any system treated in curvilinear coordinates, the nonuniform spatial weighting introduced by the Jacobian factor must be taken into account. This may be done in the usual way7 by transforming to a probability amplitude (D with unit Jacobian, I@I2 = lWI2 sin 8. The corresponding Hamiltonian then takes the form
H = -B- d2
de2
+ v,,Xe)
which describes one-dimensional motion in the polar angle 8, subject to an effective potential
This displays explicitly the role of the M-dependent centrifugal term, which for IMI > 0 provides a repulsive contribution competing with the attractive polarizability interaction. Figures 5 and 6 plot for the oblate and prolate case, respectively, the effective potentials and energy levels, for [MI = 0-5 and IAml = 49. For the oblate case, if M2 - I14 < Am, the effective potential exhibits a double well with an equatorial barrier. At the barrier maximum, VefX9O")IB = M2 - '12 w l . The two equivalent potential minima occur at e,, given by
and there V(6,)lB = -Am cos 28, - I14 - WL. The barrier height is thus AVIB = Am( 1 - sin2 e,)2. For the prolate-case, Vef has a single minimum flanked by baniers in the polar regions. This determines the angular ranges accessible to the libratingpinwheeling molecular axis. For both the oblate and prolate cases, the energy levels conform to E ~ J M
