Directional states of symmetric-top molecules produced by combined static and radiative electric fields

Directional states of molecules are at the core of all methods to manipulate molecular trajectories. This is because only in directional states are the body-fixed multipole moments “available” in the laboratory frame where they can be acted upon by space-fixed fields.

In the case of polar molecules, the body-fixed permanent dipole moment is put to such a full use in the laboratory by creating oriented states characterized by as complete a projection of the body-fixed dipole moment on the space-fixed axis as the uncertainty principle allows.

The classic technique of hexapole focusing selects precessing states of symmetric top molecules (or equivalent) which are inherently oriented by the first-order Stark effect,¹ independent of the strength of the electric field applied (at low fields). For the lowest precessing state, only half of the body-fixed dipole moment can be projected on the space-fixed axis and utilized to manipulate the molecule. Pendular orientation is more versatile,^2,3 applicable to linear molecules as well as to symmetric and asymmetric tops, but works well only for molecules with a large value of the ratio of the body-fixed electric dipole moment to the rotational constant. However, for favorable values of this ratio, on the order of $100D∕cm−1$⁠, the ground pendular state of a $Σ1$ molecule utilizes up to 90% percent of the body-fixed dipole moment in the space-fixed frame at feasible electric field strength, of the order of $100kV∕cm$⁠.

Pendular states of a different kind can be produced by the induced-dipole interaction of a nonresonant laser field with the anisotropic molecular polarizability.^4–7 The resulting directional states exhibit alignment rather than orientation and can be used to extend molecular spectroscopy,⁸ suppress rotational tumbling,^9–11 focus molecules,¹² or to decelerate and trap molecules.¹³ 

The reliance on special properties of particular molecules has been done away with by the development of techniques that combine a static electric field with a nonresonant radiative field. The combined fields give rise to an amplification effect which occurs for any polar molecule, as only an anisotropic polarizability, along with a permanent dipole moment, is required. This is always available in polar molecules. Thus, for a number of molecules in their rotational ground state, a very weak static electric field can convert second-order alignment by a laser into a strong first-order orientation that projects about 90% of the body-fixed dipole moment on the static field direction. If the polar molecule is also paramagnetic, combined static electric and magnetic fields yield similar amplification effects. So far, the combined-field effects have been worked out for linear polar molecules and corroborated in a number of experiments by Buck and co-workers^14–16 and Poterya et al.¹⁷ on linear rare-gas clusters, as well as by Minemoto and co-workers who undertook a detailed femtosecond-photoionization study on oriented OCS.^18,19

Here, we analyze the amplification effects that combined electrostatic and nonresonant radiative fields bring out in symmetric-top molecules. In our analysis, we draw on our previous work on linear molecules^20,21 as well as on the work of Kim and Felker,²² who have treated symmetric-top molecules in pure nonresonant radiative fields.

For linear molecules, the amplification of the orientation arises from the coupling by the electrostatic field of the members of quasidegenerate opposite-parity tunneling doublets created by the laser field. Thereby, the permanent electric dipole interaction creates oriented states of indefinite parity.¹⁶ We find that this mechanism is also in place for a class of precessing symmetric top states in the combined fields. However, another class of precessing symmetric top states is prone to undergo a sharp orientation via a different mechanism: The laser field alone couples states with opposite parity, thereby creating indefinite parity states that can then be coupled by a weak electrostatic field to the field’s direction.

In Sec. II, we set the stage by writing down the Hamiltonian (in reduced, dimensionless form) for a symmetric-top molecule in the combined electrostatic and linearly polarized nonresonant radiative fields and consider all four symmetry combinations of the polarizability and inertia tensors as well as the collinear and perpendicular (tilted) geometries of the two fields. We derive the elements of the Hamiltonian matrix in the symmetric-top basis set and discuss their symmetry properties. We limit ourselves to the regime when the radiative field stays on long enough for the system to develop adiabatically. This makes it possible to introduce adiabatic labeling of the states in the combined fields, sort them out systematically, and define their directional characteristics, such as orientation and alignment cosines.

In Sec. III, we introduce an effective potential that makes it possible to regard the molecular dynamics in the combined fields in terms of a one-dimensional (1D) motion. The effective potential is an invaluable tool for making sense of some of the computational results obtained by solving the eigenproblem in question numerically.

In Sec. IV, we present the results proper. First, we construct correlation diagrams between the field free states of a symmetric top and the harmonic librator, which is obtained at high fields. Then, we turn to the combined fields and consider collinear fields and perpendicular fields in turn. It is where we discuss the details of the two major mechanisms responsible for the amplification of the orientation by the combined fields.

Section IV discusses the possibilities of applying the combined fields to a swatch of molecules (representing the symmetry combinations of the polarizability and inertia tensors) and of making use of the orientation achieved in a selection of applications. The main conclusions of the present work are summarized in Sec. VI.

We consider a symmetric-top molecule which is both polar and polarizable. The inertia tensor $I$ of a symmetric top molecule possesses a threefold or higher axis of rotation symmetry (the figure axis) and is said to be prolate or oblate, depending on whether the principal moment of inertia about the figure axis is smaller or larger than the remaining two principal moments (which are equal to one another). The principal axes $a$⁠, $b$⁠, $c$ of $I$ are defined such that the principal moments of inertia increase in the order $Ia<Ib<Ic$⁠.

The symmetry of the inertia tensor is reflected in the symmetry of the polarizability tensor $α$ in that the principal axes of the two tensors are collinear. However, a prolate or oblate inertia tensor does not necessarily imply a prolate or oblate polarizability tensor. Four combinations can be distinguished: (i) $I$ prolate and $α$ prolate, i.e., $Ia<Ib=Ic$ and $αa<αb=αc$ with $a$ the figure axis; (ii) $I$ prolate and $α$ oblate, i.e., $Ia<Ib=Ic$ and $αa>αb=αc$⁠, with $a$ the figure axis; (iii) $I$ oblate and $α$ prolate, i.e., $Ia=Ib<Ic$ and $αa=αb>αc$ with $c$ the figure axis; and (iv) $I$ oblate and $α$ oblate, i.e., $Ia=Ib<Ic$ and $αa=αb<αc$⁠, with $c$ the figure axis. The body-fixed permanent electric dipole moment $μ$ of a symmetric-top molecule is bound to lie along the molecule’s figure axis. The symmetry combinations of $I$ and $α$ are summarized in Table I in terms of the rotational constants

and polarizability components parallel, $α∥$⁠, and perpendicular, $α⊥$⁠, to the figure axis.

The rotational Hamiltonian $Hr$ of a symmetric top molecule is given by

with $J$ the angular momentum operator, $Jz$ its projection on the figure axis $z$ (⁠$z≡a$ for prolate and $z≡c$ for oblate top), and

The symmetric-top molecule is subject to a combination of a static electric field, $εS$⁠, with a nonresonant laser field, $εL$⁠. The fields $εS$ and $εL$ can be tilted with respect to one another by an angle, $β$⁠. We limit our consideration to a pulsed plane wave radiation of frequency $ν$ and time profile $g(t)$ such that

where $I$ is the peak laser intensity (in cgs units). We assume the oscillation frequency $ν$ to be far removed from any molecular resonance and much higher than either $τ−1$ (with $τ$ the pulse duration) or the rotational periods. The resulting effective Hamiltonian $H(t)$ is thus averaged over the rapid oscillations. This cancels the interaction between $μ$ and $εL$ (see also Refs. 23 and 24) and reduces the time dependence of $εL$ to that of the time profile,

Thus the Hamiltonian becomes

where the permanent, $Vμ$⁠, and induced, $Vα$⁠, dipole potentials are given by

with the dimensionless interaction parameters defined as follows:

The time-dependent Schrödinger equation corresponding to Hamiltonian (6) can be cast in a dimensionless form,

which clocks the time in units of $ℏ∕B$⁠, thus defining a “short” and a “long” time for any molecule and pulse duration. In what follows, we limit our consideration to the adiabatic regime, which arises for $τ⪢ℏ∕B$⁠. This is tantamount to $g(t)→1$⁠, in which case the Hamiltonian (6) can be written as

Hence, our task is limited to finding the eigenproperties of Hamiltonian (11). For $Δω=ω=0$⁠, the eigenproperties become those of a field-free rotor; the eigenfunctions then coincide with the symmetric-top wavefunctions $∣JKM⟩$ and the eigenvalues become $EJKM∕B$⁠, with $K$ and $M$ the projections of the rotational angular momentum $J$ on the figure and space-fixed axis, respectively. In the high-field limit, $Δω→±∞$ and/or $ω→∞$⁠, the range of the polar angle is confined near the quadratic potential minimum, and Eq. (11) reduces to that for a two-dimensional angular harmonic oscillator (harmonic librator), see Sec. IV A.

If the tilt angle $β$ between the field directions is nonzero, the relation

is employed in Hamiltonian (6), with $θ≡θL$ and $φ≡φL$⁠, see Fig. 1.

If the static and radiative fields are collinear (i.e., $β=0$ or $π$⁠), $M$ and $K$ remain good quantum numbers. Note that except when $K=0$⁠, all states are doubly degenerate. While the permanent dipole interaction $Vμ$ mixes states with $ΔJ=±1$ (which have opposite parities), the induced dipole interaction mixes states with $ΔJ=0∧±2$ (which have same parities) but, when $MK≠0$⁠, also states with $ΔJ=±1$ (which have opposite parities). Thus, either field has the ability to create oriented states of mixed parity.

If the static and radiative fields are not collinear (i.e., $β≠0$ or $π$⁠), the system no longer possesses cylindrical symmetry. The $cosφ$ operator mixes states which differ by $ΔM=±1$ and so $M$ ceases to be a good quantum number.

A schematic of the field configurations and molecular dipole moments, permanent and induced, is shown in Fig. 2.

The elements of the Hamiltonian matrix in the symmetric-top basis set $∣JKM⟩$ possess the following symmetries:

Since the Hamiltonian has the same diagonal elements for the two symmetry representations belonging to $+K$ and $−K$⁠, it follows that they are connected by a unitary transformation $U$⁠. On the other hand, complex conjugation $K$ of a symmetric top state yields

and so we see from Eqs. (13) and (14) that the $+K$ and $−K$ representations are related by the combined operation $UK$⁠, which amounts to time reversal. The two representations have the same eigenenergies and are said to be separable degenerate.²⁵ We note that the $+K$ and $−K$ representations are connected by time reversal both in the absence and presence of an electric field (see also Ref. 26); the time reversal of a given matrix element is effected by a multiplication by $(−1)M′−M$⁠. The symmetry properties, Eqs. (13) and (14), are taken advantage of when setting up the Hamiltonian matrix. In what follows, we concentrate on the case of collinear (i.e., $β=0$ or $π$⁠) and perpendicular fields (i.e., $β=π∕2$⁠).

We label the states in the field by the good quantum number $K$ and the nominal symbols $J̃$ and $M̃$ which designate the values of the quantum numbers $J$ and $M$ of the free-rotor state that adiabatically correlates with the state at $Δω≠0$ and/or $ω>0$⁠, $J→J̃$ and $M→M̃$⁠. We condense our notation by taking $K$ to be nonnegative, but keep in mind that each state with $K≠0$ is doubly degenerate, on account of the $+K$ and $−K$ symmetry representations. For collinear fields, $M̃=M$⁠.

In the tilted fields, the adiabatic label of a state depends on the order in which the parameters $ω$⁠, $Δω$⁠, and $β$ are turned on, see Sec. IV C. As a result, we distinguish among $∣J̃,K,M̃;ω,Δω,β⟩$⁠, $∣J̃,K,M̃;Δω,ω,β⟩$⁠, $∣J̃,K,M̃;ω,β,Δω⟩$⁠, and $∣J̃,K,M̃;Δω,β,ω⟩$ states, or $∣J̃,K,M̃;{p}⟩$ for short, with ${p}$ any one of the parameter sets $ω,Δω,β$ or $Δω,ω,β$ or $ω,β,Δω$ or $Δω,β,ω$⁠.

The $∣J̃,K,M̃;{p}⟩$ states are recognized as coherent linear superpositions of the field-free symmetric-top states,

with $aJMJ̃KM̃$ the expansion coefficients. For a given state, these depend solely on ${p}$⁠,

The orientation and the alignment are given by the direction-direction (two-vector) correlation²⁷ between the dipole moment (permanent or induced) and the field vector (⁠$εS$ or $εL$⁠). A direction-direction correlation is characterized by a single angle, here by the polar angle $θS$ (for the orientation of the permanent dipole with respect to $εS$⁠) or $θL$ (for the alignment of the induced dipole with respect to $εL$⁠). The distribution in either $θS$ or $θL$ can be described in terms of a series in Legendre polynomials and characterized by Legendre moments. The first odd Legendre moment of the distribution in $θS$ is related to the expectation value of $cosθS$⁠, the orientation cosine,

and the first even Legendre moment of the distribution in $θL$ to the expectation value of $cos2θL$⁠, the alignment cosine,

The states with same $∣K∣$ and same $MK$ have both the same energy,

and the same directional properties, as follows from the Hellmann-Feynman theorem,

and

The angular amplitudes of the permanent and induced dipoles are given, respectively, by

and

The elements of the Hamiltonian matrix, evaluated in the symmetric-top basis set, are listed in the Appendix. The dimension of the Hamiltonian matrix determines the accuracy of the eigenproperties computed by diagonalization. For the collinear case, the dimension of the matrix is given as $Jmax+1−max(∣K∣,∣M∣)$⁠, with $J$ ranging between $max(∣K∣,∣M∣)$ and $Jmax$⁠. For the states and field strengths considered here, a $12×12$ matrix yields an improvement of less than 0.1% of $E∕B$ over a $11×11$ matrix. However, the convergence depends strongly on the state considered. For the perpendicular case, the dimension of the matrix is $(Jmax+1)2−K2$⁠, with $J$ ranging between $∣K∣$ and $Jmax$ and $M$ between $−J$ and $J$⁠. Convergence within 0.1% can be achieved for the states considered with $Jmax=10$⁠.

The $∣J̃,K,M̃;{p}⟩$ states are not only labeled but also identified in our computations by way of their adiabatic correlation with the field-free states. The states obtained by the diagonalization procedure cannot be identified by sorting of the eigenvalues, especially for perpendicular fields when the states undergo numerous crossings, genuine as well as avoided. Therefore, we developed an identification algorithm based on a gradual perturbation of the field-free symmetric-top states. Instead of relying on the eigenvalues, we compare the wavefunctions, which are deemed to belong to the same state when their coefficients evolve continuously through a crossing as a function of the parameters ${p}$⁠. If $∣Ψ0⟩$ is the state to be tracked, one has to calculate its overlap $⟨Ψk∣Ψ0⟩$ with all the eigenvectors $∣Ψk⟩$ that pertain to the Hamiltonian matrix with the incremented parameters. The state with the largest overlap is then taken as the continuation of $Ψ0$⁠. This method has been used for the general problem of tilted fields.

In order to obtain the most probable spatial distribution (geometry), wavefunctions, such as

obtained by solving the Schrödinger equation in curvilinear coordinates, need to be properly spatially weighted by a nonunit Jacobian factor, here $sinθ$⁠. Alternatively, a wavefunction,

with a unit Jacobian can be constructed which gives the most probable geometry directly; such a wavefunction is an eigenfunction of a Hamiltonian

where $U$ is an effective potential.^20,21 Equation (27) shows that $Φ$ corresponds to the solution of a 1D Schrödinger equation for the curvilinear coordinate $θ$ and for the effective potential $U$⁠. Since $Φ$ can only take significant values within the range demarcated by $U$⁠, one can glean the geometry from the effective potential and the eigenenergy. In this way, one can gain insight into the qualitative features of the eigenproblem without the need to find the eigenfunctions explicitly. Conversely, one can use the concept of the effective potential to organize the solutions and to interpret them in geometrical terms.

For collinear fields, the effective potential takes the form

Its first term, symmetric about $θ=π∕2$⁠, arises due to the centrifugal effects and, for $∣M∣>0$⁠, provides a repulsive contribution competing with the permanent (fourth term) and induced (fifth term) interactions. For $∣K∣>0$⁠, the second term just uniformly shifts the potential, either down when $ρ>0$ (prolate top) or up when $ρ<0$ (oblate top). The third term provides a contribution which is asymmetric with respect to $θ=π∕2$ for precessing states, i.e., states with $MK≠0$⁠. It is this term which is responsible for the first-order Stark effect in symmetric tops and for the inherent orientation their precessing states possess. The fourth term, due to the permanent dipole interaction, is asymmetric with respect to $θ=π∕2$ for any state and accounts for all higher-order Stark effects. The fifth, induced-dipole term, is symmetric about $θ=π∕2$⁠. However, it gives rise to a single well for $α$ prolate $(α∥<α⊥)$ and a double well for $α$ oblate $(α∥>α⊥)$⁠. This is of key importance in determining the energy level structure and the directionality of the states bound by the wells.

In the strong-field limit, a symmetric-top molecule becomes a harmonic librator whose eigenproperties can be obtained in closed form. The eigenenergies in the harmonic librator limit are listed in Tables II and III and used in constructing the correlation diagrams between the field-free and the harmonic librator limits, shown in Fig. 3 (for the permanent dipole interaction, $ω→∞$⁠) and Fig. 4 (for the prolate, $Δω→−∞$⁠, and oblate, $Δω→∞$⁠, induced-dipole interactions).

The correlation diagram for the permanent dipole interaction, Fig. 3, reveals that states with $K=0$ split into $J̃+1$ doublets, each with the same value of $∣M∣$⁠. The other states, on the other hand, split into $J̃+∣K∣$ at least doubly degenerate states, each of which is characterized by a value of $∣K+M∣$ for a given $J̃$⁠. For $∣K∣<J̃$⁠, some states are more than doubly degenerate. In the harmonic librator limit, the levels are infinitely degenerate and are separated by an energy difference of $(2ω)1∕2$⁠.

The correlation diagrams for the induced dipole-interaction reveal that states with $J̃<∣M∣+∣K∣$ [shown by black (full) lines in Fig. 4] for $α$ oblate and all states for $α$ prolate which have same $∣MK∣$ form degenerate doublets. In contradistinction, states with $J̃⩾∣M∣+∣K∣$ [shown by red (double-thick grey) lines and green (thick grey) lines in Fig. 4 for $K=0$ and $K≠0$⁠, respectively] bound by $Vα$ oblate occur as tunneling doublets. This behavior reflects a crucial difference between the $α$ prolate and $α$ oblate case, namely, that the induced-dipole potential $Vα$ is a double well potential for $α$ oblate and a single well potential for $α$ prolate.

The members of a given tunneling doublet have same values of $KM$ and $∣K∣$ but $J̃$’s that differ by $±1$⁠. The tunneling splitting between the members of a given tunneling doublet decreases with increasing $Δω$ as $exp(a−bΔω1∕2)$⁠, with $a,b⩾0$⁠, rendering a tunneling doublet quasidegenerate at a sufficiently large field strength. In the harmonic librator limit, the quasidegenerate members of a given tunneling doublet coincide with the $N+$ and $N−$ states, see Table III. The $N+$ and $N−$ states with $N+=N−$ for $J̃<∣M∣+∣K∣$ always pertain to the same $J̃$ but different $KM$ and so are precluded from forming tunneling doublets as they do not interact. In the $α$ prolate case, the formation of any tunneling doublets is barred by the absence of a double well. Note that in both the prolate and oblate cases, the harmonic librator levels are infinitely degenerate and their spacing is equal to $2∣Δω∣1∕2$⁠.

The correlation diagrams of Figs. 3 and 4 reveal another key difference between the permanent and induced dipole interactions, namely, the ordering of levels pertaining to the same $J̃$⁠. The energies of the levels due to $Vμ$ increase with increasing $∣K+M∣$⁠. For $Vα$ prolate, they decrease with increasing $∣M∣$ for levels with $∣M∣⩾∣K∣$ while states with $∣M∣⩽∣K∣$ have the same asymptote. The energy level pattern becomes even more complex for $Vα$ oblate, which gives rise to $∣K∣+1$ asymptotes. If $∣K∣=J̃$⁠, the energy decreases with increasing $∣M∣$⁠, while for $∣K∣<J̃$ it only decreases for levels with $∣M∣+∣K∣⩾J̃$⁠. All other levels connect alternately to the asymptotes with $N(±)=J̃$ or $J̃−1$⁠. This leads to a tangle of crossings, avoided or not, once the two interactions are combined, as will be exemplified below.

Figures 5 and 6 display the dependence of the eigenenergies, panels (a)–(c), orientation cosines, panels (d)–(f), and alignment cosines, panels (g)–(i), of the states with $0<J̃⩽3$⁠, $−1⩽MK⩽1$⁠, and $K=1$ on the dimensionless parameters $ω$ and $Δω$ that characterize the permanent and induced dipole interactions. These states were chosen as examples since they well represent the behavior of a symmetric top in the combined fields. The two figures show the dependence on $Δω$ for fixed values of $ω$ and vice versa. Note that negative values of $Δω$ correspond to $α$ prolate and positive values to $α$ oblate. The plots were constructed for $I$ prolate with $A∕B=2$ but, apart from a constant shift, the curves shown are identical with those for $I$ oblate. Thus Figs. 5 and 6 represent the entire spectrum of possibilities as classified in Table I.

The left panels of Figs. 5 and 6 show the eigenenergies and the orientation and alignment cosines for the cases of pure permanent and pure induced dipole interactions, respectively.

For an angle

such that $0<γ<π∕2$⁠, the pure permanent dipole interaction, $ω>0$ and $Δω=0$⁠, panels (a), (d), (g) of Fig. 5, produces states whose eigenenergies decrease with increasing field strength (i.e., the states are high-field seeking) at all values of $ω$ and their orientation cosines are positive, which signifies that the body-fixed dipole moment is oriented along $εS$ (right-way orientation). For states with $π∕2<γ<π$⁠, the eigenenergies first increase with $ω$ (i.e., the states are low-field seeking) and the body-fixed dipole is oriented oppositely with respect to the direction of the static field $εS$ (the so called wrong-way orientation). For states with $K=0$⁠, the angle $γ$ becomes the tilt angle of the angular momentum vector with respect to the field direction, which, for $J>0$⁠, is given by

At small $ω$⁠, states with $γ0>3−1∕2$ are low-field seeking and exhibit the wrong-way orientation, while states with $γ0<3−1∕2$ are high-field seeking and right-way oriented.

At large-enough values of $ω$⁠, all states, including those with $K=0$⁠, become high-field seeking and exhibit right-way orientation. In any case, the dipole has the lowest energy when oriented along the field. Since the asymmetric effective potential (28) disfavors angles near 180°, and increasingly so with increasing $KM$⁠, the eigenenergies for a given $J̃$ decrease with increasing $KM$⁠. The ordering of the levels pertaining to the same $J̃$ is then such that states with higher $KM$ are always lower in energy.

The eigenenergies and orientation and alignment cosines for a pure induced-dipole interaction, $ω=0$ and $∣Δω∣>0$⁠, are shown in panels (a), (d), and (g) of Fig. 6. The eigenenergies are given by

cf. Eqs. (2) and (11). However, Figs. 6(a)–6(c) and 12 only shows $E∕B+ω⊥≡λ$⁠, which increase with increasing laser intensity for $α$ prolate and decrease for $α$ oblate. Note that both prolate and oblate eigenenergies, $E∕B=λ−ω⊥$⁠, decrease with increasing laser intensity, and so all states created by a pure induced-dipole interaction are high-field seeking as a result. Note that

and thus the alignment cosines are given by the negative slopes of the curves shown in Fig. 6(a).

In panels (d) and (g), one can see that only the precessing states are oriented, and that their orientation shows a dependence on the $Δω$ parameter which qualitatively differs for $α$ oblate and $α$ prolate: For $Δω>0$⁠, the orientation is enhanced for states with $J̃<∣M∣+∣K∣$ and suppressed for states with $J̃⩾∣M∣+∣K∣$⁠, while for $Δω<0$ it tends to be suppressed by the radiative field for all states. This behavior follows readily from the form of the effective potential, Eq. (28), as shown in Fig. 7. For the prolate case $(Δω<0)$⁠, the potential becomes increasingly centered at $θ→π∕2$ with increasing field strength and therefore tends to force the body-fixed electric dipole (and thus the figure axis) into a direction perpendicular to the field. On the other hand, for the oblate case $(Δω>0)$⁠, the effective potential provides, respectively, a forward $(θ→0)$ and a backward $(θ→π)$ well for the $N+$ and $N−$ states of a tunneling doublet (for $J̃⩾∣M∣+∣K∣$⁠) or of a degenerate doublet (for $J̃<∣M∣+∣K∣$⁠). However, only for the degenerate-doublet states does the increasing field strength result in an enhanced orientation at $εS=0$⁠. This distinction is captured by the effective potential, Fig. 7, whose asymmetric forward (for $MK>0$⁠) or backward $(MK<0)$ well lures in the $J̃<∣M∣+∣K∣$ states. The $J̃⩾∣M∣+∣K∣$ states become significantly bound by the $Vα$ oblate potential at $Δω$ values large enough to make them feel the double well, which makes the two opposite orientations nearly equiprobable.

Figure 8 shows the dependence of the $J$-state parity mixing on the $Δω$ parameter at $ω=0$ [panel (a)] and $ω=10$ [panel (b)]. The mixing is captured by a parameter

with $n$ an integer. In the absence of even-odd $J$ mixing, $ξ=0$ for $J̃$ even and $ξ=1$ for $J̃$ odd; for a “perfect” $J$-parity mixing, $ξ=12$ for either $J̃$ even or odd. We see that for the sampling of states shown, the nonprecessing states become parity mixed only when $ω>0$⁠. However, all precessing states are $J$-parity mixed as long as $Δω≠0$⁠.

The eigenenergies as well as eigenfunctions in the harmonic librator limit for both the prolate and oblate polarizability interactions have been found previously by Kim and Felker,²² and we made use of the former in constructing the correlation diagram in Fig. 4. We note that in the prolate case, the eigenenergies and alignment cosines, as calculated from Kim–Felker’s eigenfunctions, agree with our numerical calculations for the states considered within 4% already at $Δω=−50$⁠. The prolate harmonic librator eigenfunctions render, however, the orientation cosines as equal to zero, which they are generally not at any finite value of $Δω$⁠.

For the $α$ oblate case, the eigenenergies in the harmonic librator limit agree with those obtained numerically for the states considered within 5% at $Δω=50$⁠. For $Δω≈350$⁠, the difference between the alignment cosines obtained numerically and from the analytic eigenfunctions is less then 3%. We note that for $J̃<∣M∣+∣K∣$⁠, only one eigenfunction exists, pertaining either to $N+$ for $KM>0$ or to $N−$ for $KM<0$⁠, cf. Table III. This eigenfunction is strongly directional, lending a nearly perfect right-way orientation to an $N+$ state and a nearly perfect wrong-way orientation to an $N−$ state. These eigenfunctions pertain to the degenerate doublets. The orientation cosines can be obtained from the analytic wavefunctions within 0.01% for $Δω=100$⁠. For $J̃⩾∣M∣+∣K∣$ (including the cases when either $K=0$ or $M=0$⁠), both the $N+$ and $N−$ analytic solutions exist, cf. Table III, and pertain to the tunneling doublets. The degeneracy of the doublets in the $Δω→+∞$ limit precludes using the analytic eigenfunctions in calculating the orientation cosines. In contrast to the numerical results, the analytic solution predicts a strong orientation, which, in fact, is not present, as shown in Fig. 7. The linear combination of the analytic eigenfunctions $f+$ and $f−$⁠, which is also a solution, given as $f1,2=1∕2(f+±f−)$⁠, does not exhibit any orientation, just alignment.

The above behavior of symmetric-top molecules as a function of $Δω$ at $ω=0$ sets the stage for what happens once the static field is turned on and so $ω>0$⁠. For $Δω>0$⁠, the combined electrostatic and radiative fields act synergistically, making all states sharply oriented. For $Δω<0$⁠, the orientation either remains nearly zero (for states with $KM=0$⁠) or tends to vanish (for states with $KM≠0$⁠) with increasing $∣Δω∣$⁠.

The synergistic action of the combined fields arises in two different ways, depending on whether $J̃<∣M∣+∣K∣$ or $J̃⩾∣M∣+∣K∣$⁠.