J . Phys. Chem. 1987, 91, 5462-5465
5462
picture proposed in Figure 1 and lead to quantitative predictions for the energy partitioning in the collision complex.
Conclusion CARS spectroscopy has been successfully applied to the study of energy partitioning and collision dynamics in the sodium-hydrogen system. For energy transfer to occur in the way observed,
C,, ( x ’ ) steric alignment of the colliding partners is necessary. A drastic change in the vibrational distribution of H2 was found upon red wing excitation. The population distribution appears to be sensitive to the presence of surface crossings. Modelling with ab initio potential energy surfaces is needed for a quantitative description of the results. Registry No. N a , 7440-23-5: H,, 1333-74-0.
Directed States of Molecules S . Kais and R. D. Levine* The Fritz Haber Research Center for Molecular Dynamics, The Hebrew University, Jerusalem 91 904, Israel (Received: April 13, 1987)
Stationary quantum mechanical states for diatomic and for symmetric top molecules whose geometrical axis is preferentially aligned in space are introduced and discussed.
Introduction Dynamical stereochemistry is very much concerned with the preferred direction of the approach or/and of the receding motion in a molecular collision. In classical mechanics there are no difficulties in considering a reagent molecule whose axis is oriented in space, and similarly for the final state. We have a clear image of what it means that, say, a diatomic molecules desorbs from a surface with its axis preferentially perpendicular to it.] In a computational study using classical trajectories one can select the spatial orientation of the reagents and/or classify that of the products. At first glance, such classical constructs do not have quantal analogues. Consider a diatomic molecule which at low levels of excitation can be regarded as a rigid rotor. The stationary states as given in textbooks,24 [j,m),are eigenstates of the angular momentum and its projection on the Z axis. In collision problems one can often choose the Z axis to a d ~ a n t a g e . ~However, ,~ for the intended applications in stereochemistry such states are not optimal since the axis of the diatomic molecule is “random” in the plane perpendicular to j. The object of this note is to introduce and examine quantum mechanical rotor (and symmetric top) states which, in the classical limit, correspond to a molecule with its axis preferentially aligned. A similar problem arises in providing a quantum mechanical e~planation’~ of the “directed” bonds of structural stereochemistry. To describe bonding using p (Le., j = 1) electrons, one does not use the three degenerate b,m) ( m = 1, 0, -1) states but three linear combinations which are referred to as px, py, and pz, respectively, and which are preferentially oriented along the three Cartesian axes. The corresponding states for d electrons are also familiar. Our problem here is analogous except that it is not the electrons but the axis of the molecule we want to “direct”. The states we (1) Luntz, A. C.; Kleyn, A. W.; Auerbach, D. J. Phys. Rev. B: Condens. Matter 1982, 25, 4273; Surf. Sci., in press. (2) Edmonds, A. R. Angular Momentum in Quantum Mechanics; Princeton University Press: Princeton, NJ, 1960. (3) Brink, D. M.; Satchler, G. R. Angular Momentum; Oxford University Press: Oxford, 1968. (4) Zare, R. N . Angular Momentum; Wiley: New York, 1987. (5) See, for example: Jellinek, J.; Kouri, D. J. In Theory of Chemical Reaction Dynamics; Baer, M., Ed.;CRC: Boca Raton, FL,1986. McCaffery, A . J.; Proctor, M. J.; Whitaker, B. J. Annu. Reu. Phys. Chem. 1986, 37, 223. (6) Levine, R. D.; Bernstein, R. B. Molecular Reacrion Dynamics and Chemical Reactivity; Oxford University Press: New York, 1987. (7) Pauling, L. Proc. Natl. Acad. Sci. U . S . A .1928, 14, 359 J . A m . Chem. SOC.1931, 53, 1367. (8) Slater, J. C . Phys. Reu. 1931, 37, 481. (9) Van Vleck, J . H. J . Chem. Phys. 1933, 1 , 177.
0022-3654/87/2091-5462.$01.50/0
shall discuss are those made up as linear combination of b , m ) states for a given j . As in the theory of hybridi~ation,~ one can also consider superpositions containing b , m ) states of different j ’ s . The extension is straightforward, but we shall not pursue it here as the states we consider already suffice to have the molecular axis aligned in any desired direction. The quantum mechanical states we consider are very much classical like and, as such, can be considered as generalized coherent states.lOlll However, they are not equivalent to the angular momentum (or spin) coherent states which have been previouslyI0-l4introduced. The latter are states where the angular momentum is directed while ours have the molecular axis directed. These will be shown to be stationary, minimal uncertainty states, which can be used as basis states in computational studies. A problem of pragmatic importance is how to prepare directed states in the laboratory. For polar symmetric tops, the states we consider can be prepared using inhomogeneous electric field^,'^-^' and such states will be oriented. It is indeed such molecules which have been used in the pioneering experimentsI5J6J8on the orientation dependence of the reactivity. For polar diatomics such states could be prepared by optical excitati~n.’~ We shall devote a separate study to this issue, our primary purpose in this note being to establish that diatomic molecules can also be discussed in terms of directed states.
Diatomic Molecules The direction of the molecular axis is specified, as usual, by the polar angles 6 and 4. In the directed state, In), the axis is preferentially (Le., uncertainty principle limited) aligned along the unit vector n. To introduce the directed state, we begin with ~
~~~
(10) Perelomov, A . Generalized Coherent States and Their Applications;
Springer-Verlag: West Berlin, 1986. (1 1) Klauder, J. R.; Skagerstam, B. S. Coherent States; World Scientific: Singapore, 1985. (12) Radcliffe, J. M . J . Phys. A : Gen. Phys. 1971, 4, 313. (13) Atkins, P. W.; Dobson, J. C. Proc. R . SOC.London, A 1971,321,321. (14) Arecchi, F. T.; Courtens, E.; Gilmore, R; Thomas, H . Phys. Rec. A 1972, 6,2211. (1 5) Bernstein, R. B. Chemical Dynamics via Molecular Beam and Laser Techniques; Oxford University Press: New York, 1982. (16) Stolte, S . Ber. Bunsenges. Phys. Chem. 1982, 86,413. (17) Bernstein, R. B.; Gandhi, S. R.; Xu, Q.; Curtiss, T. J. J . Phys. Chem., this issue. (18) Brooks, P. R. Science 1976, 193, 1 1 . (19) Zare, R. N. Ber. Bunsenges. Phys. Chem. 1982, 86,422.
0 1987 American Chemical Society
The Journal of Physical Chemistry, Vol. 91, No. 21, 1987 5463
Directed States of Molecules a semiclassical argument. For the familiar P,m) states of the rigid rotor, the spatial distribution is, in the semiclassical limit2-4~20
-
IY,m(0,+)12 I
I(0,41i,m)12
(1)
( 2 ~ ) - ~ ( s i na*- cos2 e)-1/2
+
Here cos2 a = m2/j(j 1) and the region cos2 0 > sin2 a is classically forbidden. For m = j , the molecule is fairly confined to the X-Yplane (0 = t / 2 ) and is uniformly distributed within that plane. For large j , such a state describes a rotor with an angular momentum vector aligned along the Z axis.2 This is not the state we are after. The molecular axis is not directed. Consider, however, the m = 0 state. Then sin2 a = 1 and the distribution (1) strongly favors 0 = 0 or T . Moreover, varying 4 leaves the molecular axis preferentially aligned along the Z axis. The state b,O) is directed, but along a particular direction, namely the Z axis. A state directed along an arbitrary axis n can be obtained by carrying out a rotation of coordinates such that the former Z axis is now along n. The unit vector n is specified by the polar and azimuthal angles (3 and a n = (sin P cos a , sin @ sin a , cos P )
(2)
and the axis of rotation is along the-unit vector b which is orthogonal to both n and the Z axis (Z = (O,O,l))
b = (sin a, - cos a, 0)
(3)
A rotation about b cannot change the magnitude of j. The directed state, obtained by such a rotation of V,O), is thus a linear combination of the 2 j 1 states b,m). The expansion coefficients are determined from the definition
+
In)
= R(a,P)li,O) = W,o(a,P)li,m) m
(4)
and the orthogonality condition (j,mu,m) = 6m,mt to be Dmo(a,P) Ci,mln) = (j,mlR(a,P)li,O)
The rotation about the b axis, which brings the Z axis (and hence the molecular axis) to be along n, carries a point specified by the angles 0 and q!~ into a point specified by 0’,4’ where
+ sin
sin 0 cos ( a - 4)
U(t)ln) = U ( t ) R(a,P)li,O) = R(a,P) U(Oli,O)
(10)
An explicit proof is to note that the eigenenergies are independent of m and hence from (9)
= exp(-iE/t/h)lC;o(O’,4?
What follows is a study of the main properties of such states. The identification of $;o(O’,4? as the state directed along n was based on the semiclassical limit (1). We now verify that the state is not simply directed but that it is as directed as possible within the limitations of the uncertainty principle. For this purpose note that2-4 cos Obm) = [(j m l)(j - m 1)/(2j + 3)(2j + 1)]1/2b+ 1,m) + [(j m)(j - m)(2j 1)(2j - 1)]1/2b- 1,m) (12)
+ +
+
+
+
Hence
(jmlcos Obm) = 0
(1 3)
and (5)
The notation Omo for the expansion coefficients in (4) recognizes that they are elements of the rotation matrix. For the special case under consideration where one index is zero
cos 0’ = cos P cos 0
For practical applications there are two equivalent representations of the general directed state. One is (9) with cos 0’ given in terms of the coordinates O,$ in the laboratory-fixed system by (7). The direction n of the molecular axis is given by (2). The alternative form is given explicitly in terms of the laboratory system of coordinates by (8). That the coherent state In) is stationary when the diatomic molecule is unperturbed follows from the invariance of the molecular Hamiltonian under rotation. Hence, the time evolution operator U ( t ) commutes with the rotation operator R
(7)
The primed system of coordinates can be regarded as a molecule-fixed system with 2’along the molecular axis. The wave function (0’,4’ln) of the general directed state has the same value at the point O’,V that (0,4p,O) has at the point 0,4 that is carried into O’,@’by the rotation. Since the rotation carried the Z axis into the direction n, the directed state is indeed directed. One can also show this explicitly using (4), Le.
and (6) to give
(9)
where in the last line we have used the spherical harmonics addition t h e ~ r e m and ~ - ~P,(cos 0’) is the Legendre polynomial. (20) Brussaard, P.J.;Tolhoek, H. A. Physica (Amsterdam) 1957, 23, 955.
(jmlcos28bm) =
(j+ l - m ) ( j + 1 + m )
(2j + 1)(2j + 3)
+
(j-m)(j+m) (2j
+ 1)(2j - 1) (14)
Given a unit vector i along the molecular axis, its component i, along the Z axis is cos 0. The variance of the three components necessarily satisfies
(Q) + (iy2) +
(iz2) = 1
(15)
If we use (14), the variance ( P x 2 ) + ( i y 2is) minimal for m = 0. The directed state is aligned but is not oriented. Mathematically, the lack of distinction between the two ends of the molecule is reflected in the result (cf. (13)), (nlcos O’ln) = 0. When we come to symmetric tops, this will no longer be necessarily the case. Another manifestation of the directed nature of the state is that the angular momentum j has a vanishing component along n when the system is in the state In). To prove this we use the identity R(n)jZR-’(n) = n-j
(16)
Then, from the definition (4) mjln) = n.j R(n)b,O) = R(n)jzb,O) = OR(n)b,O) = Oln)
(17)
For any directed state Yio(O’,4’) it follows that the direction i of the molecular axis is as nearly as possible along n. The alignment of the state is uncertainty limited. That the directed state can be regarded as a generalized coherent follows directly from its definition, eq 2. The rotation operator is an element of the rotation group. In terms of the three Euler angles, we have the most general case
R(a,P,r)= exp(-Wz) exp(-iPjd ~ X P ( - W Z ) (18)
5464
Kais and Levine
The Journal of Physical Chemistry, Vol. 91, No. 21, 1987
The particular rotation we require is by an angle p about the b axis. The form of the rotation operator is then2-4*21
R = exp(
[email protected])
5.00
(19)
where
3.75
PCY Using (3) for b and the standard form for the operator components of j, one readily verifies that in terms of the standard form (1 8)
R = exp(-iajz) exp(-ipj,)
2.50
(21)
As is already implicit in the notation in (4), the rotation we require has y = 0. Only a and /3 are variable. States directed along different directions are not quite orthogonal. The overlap of two states can be computed from (4) and (6) (n’ln) = ZDmo(a,P) DL(a’,pI) m
(22)
1.25
0.00
0.00
45.0
= P,(n’.n)
+ sin /?sin p’ cos ( a - a’)
135.
180.
135.
180.
e
where in obtaining the second line we have used the spherical harmonics addition t h e ~ r e m and ~-~ wn’ = cos /3 cos p’
90.0
5.00
(23)
The overlap (22) shows that the directed states are normalized (24)
3.75
and the overlap rapidly diminishes as n and n’ diverge. Since the directed state is aligned rather than oriented, the magnitude of the overlap begins to increase when n and n’ point out to different hemispheres. When n’ is antiparallel to n, l(n’ln)12 = 1. The directed states do form a complete basis despite the lack of orthogonality. This follows from the group property, Le., that two successive rotations are a rotation. Hence, the rotation and (4) can be inverted matrices are
Pf;“
(nln) = 1
2.50
1.25
where dn = d a d cos p and we have used 0.00
I 0 00
45.0
90.0
e where (np,m) is given by (5). Hence, any state which can be expanded as a linear combination of p,m) states with a given j
I+) = Zbmli,m)
Figure 1. The classical distribution Pcl(cos 8) vs. 8 for the rotor (j = 5 , m = 0) and the top ( J = 5 , K = 5 , M = 5 ) directed along the 2 axis. For any other axis n, the plot of Pcl(cos 0’) vs. 8’ (where 8’ is the polar angle with respect to n) will be the same.
m
can equally well be expanded in directed states
with
b(n)
[(2j + 1)/4*1 (nl+) = C b m [ ( 2 j+ 1)/4*lXo(a,@) m
(29)
Symmetric Tops We consider a rigid body with an axis of ~ y m m e t r y . ~ - ”The %~~ purpose again is to direct that axis. The new feature is that one can orient and not merely align the molecular axis. To emphasize this point, we show in Figure 1 the (semiclassical) probability distribution in the angle 0’ (between the molecular axis and the direction, n, of the state) for the two cases. For the symmetric tops it is also useful to examine Figure 4 of ref 24. As is the usual notation, let J be the total molecular angular momentum with projections M and K on a space-fixed Z axis and on the molecular symmetry axis, respectively. Resolving J into its components in a space-fixed system of coordinates, one finds that for a given M
It is important to note that, in the directed states for diatomic molecules, the molecular axis is aligned but is not oriented. In other words, in a heteroatomic molecule, either atom can be found in, say, the positive n direction. There is thus no distinction between the two “ends” of the molecule. The distinction is between ((AJX)’) ((AJy)’) = J(J 1) - M 2 (30) “on-axis” and “off-axis” directions. For stereochemical studies The minimal fluctuation is then for IMl = J . Similarly, for a this latter distinction can still be of considerable interest since it system of coordinates with the z axis as the axis of can delineate the angle dependence of the barrier to r e a c t i ~ n . ~ ~ . ~body-fixed ~ symmetry
+
(21) Biedenharn, L. C.; Louck, J. D. Angular Momentum in Quantum Mechanics; Addison-Wesley: Reading, MA, 1981, (22) Schechter, I.; Prisant, M. E.; Levine, R. D. J . Phys. Chem., this issue. (23) Janssen, M. H. M.; Stolte, S. J . Phys. Chem., this issue.
((AJJ2)
+ ((AJ,)’)
+
= J(J
+ 1) - @
(24) Choi. S. E.; Bernstein, R. B. J . Chem. Phys. 1985, 83, 4463.
(31)
Directed States of Molecules
The Journal of Physical Chemistry, Vol, 91, No. 21, 1987 5465
For a linear rotor K 0 and one cannot minimize this uncertainty. J is always perpendicular to the plane of rotation. Here, however, for llyl = J one can have the molecular axis directed along the space-fixed 2 axis. Moreover, by selecting the sign of K, one can have the molecular oriented; Le., one particular end of the molecule is pointing in the positive 2 direction. This also follows from the result24 KM ( JKM~cos6’1JKM) = J ( J + 1) Here, as in section 2, 0 is the angle beiween the molecular axis and the space-fixed Z axis, cos 6’ = 2.Z. For K = 0 we recover (13), but for K = M = AJ, the state is not only directed but oriented. The result analogous to (14) is ( JKMlcos2 6lJKM) =
[3@
+ 1)/8?~~]’/’D$J~,b’,x)
(34)
To have a state directed along another axis n, we repeat the same procedure as for the linear rotor
In) = R(a,P)IJ,-J,-J) (35) With respect to the new axis, the wave function of the directed state has the same value as that of (J,-J,-J) with respect to the fixed 2 axis. Hence, here too one can write the wave function for the directed state either as +JJJ(~’,~’,x’)= [(2J
+ ~ ) / ~ ? F ~ I ’ ~ ~ D $ J ( ~ ’ , ~ ’ (36) ,X’)
using the coordinates of the new system or, explicitly, in terms of the original coordinates and the rotation matrix. The relation2-4.21 D$J#J’J’,x’) = CD$M(LY,P) oJ,,(4,0,x) M
(37)
between the rotation matrices provides the required connection so that the analogue of (8) is
+JJJ(6”,4’9X’) = CD$M((Y,P) +JMJ(4AX) M
(38)
The rotation matrix which appears in (38) is here, too, a special case D$M(a,P)
=
exp (iJ a )
+ sin a sin b cos y
(40)
+
+
= [(2J
Concluding Remarks The directed nature of the states considered is readily visualized in the semiclassical20 limit. For both the rotor and the top, the probability density of the molecular axis is uniformly distributed in the angle y, where25 y is the azimuthal angle specifying the rotation of the molecule-fixed z axis with respect to the total angular momentum, Pc,(r) = (2?~)-l.To direct the state, one needs to have the angle 6’ (the polar angle which specifies the molecular axis relative to the space-fixed Z axis) confined to a limited range as y varies. Now, in general cos 0 = cos a cos b
- J ( J + 1)][3P - J ( J + l)] J ( J + 1)(2J - 1)(2J + 3)
showing that the variance (ix2) (i:) (= 1 - ( i z 2 )is)minimal 4= 1 4 = J. for 1 A state of a symmetric top directed (and oriented) along the space-fixed Z axis is thus IJ, -J, -J) or, explicitly in terms of wave f~nctions~-~ (PIJ,-J,-J)
Note that the value of K is unchanged since its the projection of J upon the body-fixed z axis which is unchanged by the rotation. For K (which is the rightmost index of +) equal zero, (38) reduces to the result (8) for the linear rotor.
(J+M)!(J-M)! (39)
where cos2 a = @ / J ( J 1) and cos2 b = P / J ( J + 1) (or 0 for a linear rotor). It follows that cos 6’ spans the range (cos (a b ) , cos (a + b ) ) and that the classical distribution in 6’ is P~I(COS 0) I Pcl(y)/ld COS o/dyl = i / ? F [ ( ~ o0s - COS emin)(COSemax- COS 0)11/2 (41) which for K = 0 (cos2 0 < sin2 a) reduces to (1). For a symmetric top, Pclneed not be even in cos 0 and hence the top can be oriented and not only aligned. For the top, b is the angle between the molecular axis and J. The choice K = J makes cos2 b maximal and hence confines the precession of the molecule about the J axis. For K = J , it is optimal to choose M = J so that the range of 6’ is also very restricted and the top is not only directed but also oriented. For the linear rotor K 0 and the molecule rotates in a plane perpendicular to j. The choice m = j confines 8, but the molecular axis is not directed due to the precession. Rather, the choice m = 0 does direct the molecular axis (Figure 1). An analogous state for the top is M = 0, K = 0. Here the rotor is directed but not oriented. The results (40) and (41) remain valid after rotation of the coordinates except that now 0 is O’, the polar angle measured in the new system. That is the one essential (albeit, straightforward) point. The distribution of molecular axis with respect to the new system of coordinates at the angle 0’ is equal to the distribution before rotation at that angle 6 which is carried into 6” by the rotation. The general directed state can be obtained by a rotation from the state directed along the Z axis. Acknowledgment. We thank Professors R. B. Bernstein, S. Stolte, and R. N. Zare for discussions. This work was supported by the US. Air Force Grant AFOSR-86-0011 and by the Stiftung Volkswagenwerk. The Fritz Haber Research Center is supported by the Minerva Gesellschaft fur die Forschung, mbH, Munich, BDR. (25) y is the dihedral angle between the planes containing Z and J (or I) and containing J (or I) and z . See also ref 4 and 24.
