J. Phys. Chem. 1993,97, 2443-2452
2443
Stereodirected Discrete Bases in Hindered Rotor Problems: Atom-Diatom and Pendular States Roger W. Anderson’ Chemistry Department, University of California, Santa Cruz, California 95604
Vincenzo Aquilanti, Simonetta Cavalli, and Gaia Grossi Dipartimento di Chimica dell’Universith, 06100 Perugia, Italy Received: August 26, 1992; In Final Form: October 20, 1992
The stereodirected discrete basis method is applied to the solution of hindered rotor problems. The assumptions of the method are presented along with its connections with finite basis representations (FBR), discrete variable representations (DVR), and finite difference (FD) approximations. The convergence of the energy eigenvalues is determined for four model atom-diatom problems and one pendular state problem. The method has an error that decreases as &-2 where Nb is the size of the discrete basis set. The discrete eigenvectors that result from the method have simple orthogonality relationships and use in matrix element calculations. The discrete (I&)) basis needs more functions for convergence than the helicity (bf’l)) basis, but the discrete basis calculations are easier because the Hamiltonian matrix is banded, no integrals over the potential are required, and only Nb scalar potential evaluations are needed. For coupled-channel atom-rotor problems the Hamiltonian matrix is heptadiagonal, and for the coupled-states approximation the Hamiltonian matrix is tridiagonal. Angular truncation can also be used to accelerate the discrete basis calculations.
1. Introduction
Hindered rotors appear in many problems in molecular physics. They play important roles in the spectroscopy and statistical mechanics of molecules, and they are also intimately involved in the scattering of atoms fromdiatomic molecules and in themotion of polar molecules in strong electric fields. Hindered rotor states can be characterized as rotations or librations. If the degree of hindrance is small, the rotor can still rotate (its wavefunction has significant amplitude in the same angular range sampled by a free rotor) and the state is characterized as a rotation. If the hindrance is great, the rotor motion is restricted to a small angular range and the state is vibrational-like and is characterized as a libration. The characterizationof rotor motion as rotational or librational has much importance in understanding the ranges of R, the distance between an atom and a diatom, that are most important in producing rotationally inelastic scattering. Traditionally the motion of diatom rotors in collisions with atoms or in triatomic complexes has been calculated in the hl), kQ), or IIA) representations. These representationsare analogous to Hund’s cases e, c, and d for the rotational analysis of electronic states of diatomics, and the atom-diatom representationshave been named cases e, y, and 6.’ Recently representations for atom-diatom scattering have been developedI-4 that are analogous to Hund’s cases a and b. These cases are called a and 8. In these cases an artificial projection quantum number (denoted by Y in the following) measures the relative directions of the approaching atom and the molecular axis. This quantum number is called the steric quantum number,2 and cases a and 8 can be called stereodirected representations. In these representations the potential energy matrix is diagonaland all coupling between states is found in the kinetic energy operator. In a recent paper2 we have transformed the S matrix for rotational excitation into these representations in order to display the quantum mechanical correlations between the precollision and postcollision angles between thediatom axis and the atom-diatom Jacobi vector. The S matrix in the stereodirected representations is easily obtained with orthogonal transformations from the S matrix for space fixed or body fixed representations. These cases are also interesting for the quantum mechanical treatment of open shell
atom collision^.^-^ As discussed elsewhere: the procedure can also be extended to the description of reactive systems. Recently the hindered rotation of polar molecules in electric fields has received considerable attention?-l4 There are two main motivations for this work. The first is that strong electric fields can be used to orient polar molecules with low rotational temperatures produced in supersonic expansions. Here the distinction between rotation and libration determines the nature of the orientation. The librational states may have a large degree of orientation with the dipole moment of the molecule aligned parallel to the electric field. However the rotational states have weak orientation antiparallel to the applied electric field except for If21 = j. The second motivation concerns the spectroscopy of polar molecules in electric fields. The line spacing and hence the appearance of the spectra depend on whether the polar molecule motion is rotational or librational. In this paper we develop and discuss the basic assumptionsand approximationsof the stereodirectedrepresentations. The paper also examines the application of the theory to obtain eigenvalues and eigenvectors for hindered rotor motion. The rate of convergence of energy eigenvalues is compared with that of the familiar body fixed bQ) basis, and the nature of the resulting wavefunctions is shown for several model problems. Section 2 considers the nature of hindered rotor Hamiltonians and r e p resentations. The bf2) representation is reviewed, and general limitations of stereodirected representations are presented. The section concludes with a derivation of stereodirected representations and discussion of an accurate method to calculate the orthogonal transformations that are required in the method. Connections between the stereodirectedrepresentationsand finite difference approximations, finite basis representations, and discrete variable representations are presented. Section 3 describes the model problems, and the convergence of the bQ) and IuQ) representations. Discussion of the results is given in section 4.
Dudley R. Herschbach’s enthusiasm and mastery of angular momentum theory have been a source of great inspiration for his students. His varied applicationsof the theory have captured the interest of his co-workers, and many of them have made significant applicationsin their independent work. This paper on aspects of angular momentum theory has been written on the occasion of
0022-3654/93/2091-2443504.~0/00 1993 American Chemical Society
Anderson et al.
2444 The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 Dudley's 60th birthday to celebrate some of his scientificinterests which includeangular momenta and the nature of pendular states.
2. Hindered Rotor Hamiltonians In this paper we consider two types of hindered rotor problems. The first is the hindered rotation of a linear rigid molecule interacting with an atom. The second problem is the hindered rotation of a polar linear molecule in a strong electric field (pendular states). Their Hamiltonian will be indicated as HJ and H, respectively. In either case the interaction potential can be expressed as V ( 9 ) . For the interaction between a structureless atom and a diatomic molecule at fixed R (distance between the atom and the diatom center of mass) 9 is the angle between R and the diatom interatomic distance vector r. The magnitude of the interaction depends on the specified value for R. In the case of pendular states the potential depends simply on the cosine of the angle between the dipole moment of the rotor and the applied electric field. The quantum mechanical theory of such systems except the high field Starkeffect has been extensivelystudied.15-'8 For the atom-diatom problem it is formulated with three angular momenta: j for the diatom, I for the atom-diatom system, and the total angular momentum J. Only j is needed for the pendular problem. The Hamiltonians, HJorH, for these hindered rotor problems can be written as the sum of three operators
H,+ H,+ V(9)
(1)
(H,),nJn, = bbjj.j ( j
The first term in eq 1 is the operator for the rotor energy,
H,= bj2
(2)
where for both problems the rotational constant, b, is given by
b = h2/21 (3) where I is the moment of inertia for the rotor. The second term is the operator for centrifugal motion energy,
H, = aI2
for this case. The rotational and centrifugal matrices of the Hamiltonian are diagonal for this case, and all coupling appears in the potential matrix. However for the present paper we are only interested in states with definite projections Q of j along R or an electric field. Hence we will ignore this space fixed frame and work with the body fixed frames corresponding to the other cases. 2.1. Body Fixed (BF) Frames: Coupling Cases y and 6. The coupling cases y and 6 are referred to representations IQ)and 11A),respectively.) The quantum number 52 is the component of j on the quantization axis: R in the atom-diatom case and E for the pendular case. For the atom-rotor case the projection of J along this body fixed axis is also D while the projection of 1is zero. If we choose r as quantization axis (case 6), the j component along this axis, j,, is zero, while the corresponding I and J projections, I, and J,, are equal to A. It is clear that we can use two alternative bases bQ) or IIA) to find the eigenvectors and eigenvaluesof H. Both have been considered in the l i t e r a t ~ r e l ~ - ~ ' but case y is much more familiar. In the bQ) representation the matrix of the molecular rotational interaction is diagonal, but couplings between Q' = Q f 1 are introduced in the centrifugal interaction matrix. The interaction potential matrix is diagonal in Q but has coupling between states with j # j'. Specificallythe rotational, centrifugal, and potential terms for this case are easily evaluated. The first two terms are diagonal in j and are given by the equations20.22
+ 1)
+ 1) + j ( j + 1) - 2Q2]u ~ ~ Y ~ ~ , , - , [ ( 1) J (-J Q(Q + + l))(j(j + 1) - Q(Q + 1))]Ii2 u~~~~,,,+,+ [ ( J1)(-J Q(Q - 1))(j(j + I) - Q(Q -
(HJj,gn, = aSj#&[J(J
(7)
The potential matrix is given as
(4)
For the atom-rotor problem
a = h2/2rR2 (5) wherep is the reduced mass for the atom-diatom relative motion. For the pendular states, a = 0. For the pendular problem, the eigenvaluesand eigenvectorsof Hare the desired quantities, but the eigenvalues and eigenvectors of HJ for the atom-rotor problem are only part of the solution. These quantities are useful in solving a multichannel SchrMinger equation for atomdiatom scattering (continuum states) or for atom4iatom bound states. For a fixed value of the total angular momentum J and total energy E, the multichannel SchrMinger equation has the form
The eigenvectors and eigenvalues of the Hamiltonian defined by eq 1 can be obtained with different choices for basis functions. As suggested previously,j for hindered rotor problems we use Greek letters a,6, y, 6, and e to denote the five coupling cases. These basis functions correspond to different coupling schemes that are analogous to Hund's cases for rotating mole~ules.5-~ Different Hund's cases are used to describe open shell molecules depending on the relative magnitude of terms in the Hamiltonian. Similar arguments can be made for the hindered rotor case. For example for the very weak interaction between an atom and a diatomic molecule at large R, case e, is useful. In this case the states are labeled with bl). This indicates that free rotation of the rotor and of the atom about the rotor is a good starting point
For
V ( 9 ) = CVR.(COS 9 )
(9)
n
These equations are simple to apply if the potential is of the formgiven byeq9. However ifit isdifficult toexpand the potential in a series of Legendre polynomials, then the integrals in eq 8 must be evaluated with a numerical integration method of sufficient accuracy. This will involve evaluation of the potential and spherical harmonics at the quadrature points which is done in the finite basis representation (FBR) method.23 The FBR Hamiltonian is the same as that defined by eqs 7-10 but with numerical integrationofeq 8. Thediscretevariable representation (DVR) method23 uses a transformation matrix whose elements are the values of spherical harmonics weighted appropriately by the Gaussian integration weights to diagonalize the potential matrix and put all of the coupling into the kinetic energy terms. For rotor problems the DVR Hamiltonian matrix remains dense. We will see later that some of the effort required by the FBR and DVR methods can be avoided with a discrete stereodirected basis. In the IIA) representation the centrifugal operator motion is diagonal, and the rotational operator is diagonal in 1 but with coupling between A' = A f 1 . The interaction potential matrix
Stereodirected Discrete Bases in Hindered Rotor Problems is now diagonal in A and couples states with 1 # 1'. The bl),bQ), and IlA) representations are related one toeach other by orthogonal transformations. The matrix elements for this case (6) can be obtained from the ones for case y interchanging bothj and 1, and fl and A). However this case is less applicable to the hindered rotor systems of this paper. The convergence of the 6 representation is expected to be similar to that for the y case. 2.2. Toward Stereodirected (SD) Representations. We now examine representations for which the potential matrix is diagonal and all of the coupling is found in H,and &. Such representations as Hund's cases a and b in the open shell atom-atom problem have been recently introduced3for systems where the interaction is a function of the angle 9 between r and R or E. Our goal is to describe discrete bases of finite dimension that have diagonal potential energy, but we will first demonstrate such a discrete basis of infinite dimension. Consider the closure relationship (ref 24, page 143):
We see that the set of K,(9,9) for all 1 and m may be considered as an orthogonal transformation of the continuous variables, 9 and 9, to the discrete variables 1 and m. The transformation is stereodirected because it picks out a unique value for 9 and 9. Now if we use Kn1,-,tdefined by eq 8 and calculate the following sum
The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 2445 1 .o
0.5 P)
z
h
0
B
0.0 -0.5 1 .o 0.5
0.5
1
-1.0
-0.5
0.0
0.5
1 .o
Cos@ Figure 1. Partial closure (eq 17 in text) for m = 0 as a function of cos I9 for three values of cos 19' and three values for 1. The values of I are 5 , 10, and 40 for the top, middle, and bottom panels, respectively. The solid lines correspond to cos 8' = 0.0. The dashed lines are for cos.9' = 0.50 and the dotted lines for 0.9. 1 .o
0.5 P)
2
h
0
G
4
Id
0.0
-0.5 1 .o 0.5 0.0
.d
a2
4
-0.5 1.0 0.5
Y~n~(8,'P)rjfnf(92,(02) d(cos 9)d q we find that the sum is V(S,)S(Sl- &)6nnf. This transformation of infinite dimension produces a potential matrix that is diagonal in 9 and block diagonal in Q. Since Vis only a function of 0 , ~ ~ J isQ diagonal S in a. In view of this fact, it appears that the set of K,(9,q) for all 1 consistent with a given fixed m may be sufficient to produce a stereodirected basis that has unique values for 9.25 But infinite orthogonal transformations need infinite computer memories, so we now consider finite sums of the form /+m
We will call this sum: partial closure. As 1 becomes large for a given m we expect that this expression will more precisely pick out a localized value of 19. Figures 1 and 2 present numerical evaluation of eq 12. The graphs are constructed assuming that cos 9' = 0,0.5, and 0.9. For each value of cos Of, cos 9 is given values between-1 .Oand 1.O. The figures show clearly that indeed the finite dimension transformation does become more stereodirected as 1 is increased. For m = 0 (Figure 1) and m = 5 (Figure 2), the partial closureexpression has become substantially peaked at the values of cos 9 for 1 = 40. For this 1 value the angular halfwidth of the peaks is approximately I-' in agreement with uncertainty expressions for angles and angular momentum. The partial closure expression becomes much less localized for 1 = 10 and I = 5 . In these cases the peaks become broader and do not necessarily have their maxima at the values of cos 9. The maxima always agree for cos 9' = 0.0, because in this case only even 1 contribute for m = 0 and odd I for m = 5 . However the other values of co8 9' have peak positions that may be larger or smaller than the expected values. For example the 1 = 5 peak for cos 9' = 0.9 is at smaller cos I9 for m = 5 and larger for m
0.0
-0.5 -1.0
-0.5
0.0
0.5
1 .o
Cos@ Figure 2. Partial closure for m = 5 as a function of cos 6. Description as for Figure 1.
= 0. This is a general trend because the larger mvalues correspond to states that are constrained to motion nearer 9 = r / 2 . These figures demonstrate that stereodirected bases can be constructed from finite sets of spherical harmonics. However the bases are poorly stereodirected unless 1 is large. This is a fundamental restriction that arises from the uncertainty principle. This restriction also limits the stereodirectivity of the discrete bases for cases a and @ of the next section. The remaining problem with the bases is that a finite set of values for 9 must be specified as quadrature points for angular integrations. One way to do this is to use Gaussian integration abscissas to achieve the angular discretization as is done in the FBR and DVR a p p r ~ a c h e s .However ~~ we now describe stereodirected representations which result in diagonal potential energy matrices and banded kinetic energy matrices. 2.3. Stereodirected (SD) Representation: Coupling Casea a and 8. We define a case a and a case @ as the discrete representations correspondingrespectively to they and 6 coupling schemes. Our recipe3 starts by establishing a connection between cutting the angular range (0I9 Ir)in Nbslices and introducing an artificial angular momentum vector, A, where Nb = 2A 1. By defining N = Nb - 1 or A = N / 2 , and interpreting the integer or half integer Y which counts the slices (-N/2 IY IN / 2 ) as a projection quantum number, we can exploit angular momentum algebra at its full power. A consequence of our procedure is that the integrals for the matrix elements of V are approximated by summations.
+
2446
Anderson et a].
The Journal of Physical Chemistry, Vol. 97, No. IO, 1993 1 .o
0.0
i=Z,n=o -1.0
:
:
:
:
: . . :
a
1
. . . . ,
-1
.o
n
h
0"
CB
y
c
0.0
F -1
0.0
0.0
I
.o
."-1.0
j
=
0 , fl
-0.5
= 0 0.0
0.5
-1.0 -1
1 .o
cos0 Figure 3. Discrete and continuous spherical harmonics. The bottom panel shows YO 9,O) (solid line) and the corresponding discrete harmonic given by eq 13 with 9" given by eq 14 (plus symbols). The middle panel is for Y Icos 9,O) and the top panel is for Y2 9,O). In each case Nb = 10 ( N = 9 ) .
cos
cos
The basic tools of the method are the discrete analogs of spherical harmonics qn,which are essentially Clebsch-Gordan coefficients (see below). For large N, we have for integer j 0. Again the agreement improves as Nb increases and has the same dependence on lcos 91 as is seen in Figure 4. Figure 5 also illustrates a
.o
-0.5
0.0
1 .o
0.5
Cos@ Figure 4. Discrete and continuous spherical harmonics. All panels show Ylo 0,O) (solid line). Discrete harmonics (plus symbols) areshown in the bottom panel for Nb = 10, in the middle panel for Nb = 20, and in the top panel for Nb = 40.
cos
0.0
,."
4 ~.
. . . , . . .
.
, . . . .
I
.
.
"i
-1.0 [ E
cos 6, = -
-1.0
-0.5
0.0
0.5
1 .o
Cos6 Figure 5. Discrete and continuous spherical harmonics. All panels show YSCOS 9,O) (solid line). Discrete harmonics (plus symbols) are shown in the bottom panel for Nb = 10, in the middle panel for Nb = 20, and in the top panel for Nb = 40.
consequence of eq 14 that the range of cos 9 values is increased as Nb is increased. Larger Nb values allow more sampling of the nonclassical region for qn. Several general conclusions may be drawn from Figures 3-5 and other figures not shown here. The discrete harmonics start to approximate the continuous harmonics for Nb 1j. However satisfactory approximations are not found until Nb 1 2j, and good approximations are only found for Nb 1 4j. The discrete harmonics also do an excellent job of fitting the classical region of the U,n for 52 > 0. Now that the correspondence of the discrete harmonics to the U,, has been demonstrated, we can construct a basis for which the potential matrix is diagonal. An orthogonal transformation can be constructed from the discrete harmonics that will diagonalize the case y potential matrix in the limit of large N. Equation 8 can be integrated over cp to obtain the expression
This integral can be approximated with the midpoint rule3' to obtain
Stereodirected Discrete Bases in Hindered Rotor Problems where
The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 2447 v and 52
2v (17) N+Q+l and we assume that Y,R(9,0)is negligible in the nonclassical cos 9, = -
(v’Q’V(9)lvQ) =f(9,)6,,6,,t
(24)
For an eigenstate, li), of an operator we have
region where
Now using the discrete expressions for
qRfor 52 L 0 we obtain
There are obvious minor changes for negative 52. An orthogonal transformation, G, can be defined by specifying its elements to be proportional to the Y,n(S,,O) expressed in eq 13
Equation 19 shows that G N , define ~ ~ a representation that is similar to a finite basis representation (FBR). It is not strictly a FBR because the Clebscheordan coefficientsonlyapproximate the values of the spherical harmonics at the quadrature points. The GNvjnalso can be used to generate a DVR representation with a diagonal potential energy matrix but with a banded kinetic energy matrix. If an element of GVGT is calculated with ~ R J Q , given by eq 19
it is seen that G produces a diagonal potential matrix for the Iv52) basis. The diagonalization will become more accurate for large Nbecause then the discrete harmonics will more accurately agree with the continuous Y,Rand the midpoint rule will provide a more accurate quadrature. Since the errors in both the discrete harmonicappr~ximation~~ and the midpoint rulejl are proportional to N-2,we can anticipate that the error in the diagonalization will also scale as N-2. The previous derivation of the ( ~ 5 2 )stereodirected basis shows similarities with the FBR and DVR methods. However the stereodirectedapproach can also be demonstrated from a different approach. We have used the fact that Clebsch-Gordan coefficients can be discrete analogs for spherical harmonics. It can be shown that Clebsch-Gordan coefficientsare Hahn polynomial^^^ and that the same Hahn polynomials solve the finite difference equation that approximates to second order the differential equation that defines the associated Legendre polynomials. The discretevalues for 9,given above assure the second-order (- N-2) approximation. The connection between the Clebsch-Gordan coefficientsand the Hahn polynomialsassures that the difference equation is solved with the proper boundary conditions. The stereodirected basis can hence be considered as a second-order finite difference approximation to the spherical harmonics. The 1~52)basis has simple orthogonality and closure properties
and
CIvn)(vnl= 1
(23)
UR
In this basis matrix elements of functions of 9 are diagonal in
if the grids of 9, are commensurate for different 52. The amplitudesof the wavefunctionsare easily displayed at the discrete angles, 9,. It is straightforward to obtain the matrix elements for H and HJin the discrete 1~52)basis. The orthogonal matrix, G, is used to transform the 152)matrices for the molecular rotational and centrifugal operators. The matrix for the centrifugal operator has coupling between channels having different v and 52, while the molecular rotational interaction has coupling only between states with different v. Some simplified expressions have been presented for the calculation of these matrices4 coupling states with the same value of Q. Because of the relevance of these equationsfor the present workand to incorporatesmall corrections, these formulas are the following:
+ Q)(Q + 1) + 2(N2/4- v 2 ) )b6,,-,{(N/2 + v’)(N/2 + 1 - V ’ ) [ ( N + l)52 + Q2]+ (N/2 + lq2(N/2+ 1 - v’)2]1/2 - b6,,,+,{(N/2 - v’)(N/2 + 1 + V’)[(N+ l)52 + Q2] + (N/2 - v’)2(N/2 + 1 + v’)2)’/2 (Hc)vR,,,R = a6,,[J(J + 1) + ( N + 52)(52 + 1 ) + 2(N2/4v2) - 2Q2]- a6,,-,{(N/2 + v’)(N/2 + 1 - u ’ ) [ ( N + l)52 + Q2]+ (N/2 + v’)*(N/2 + 1 - Y’)~]’/~ - a6,,t+,{(N/2v’)(N/2 + 1 + V ’ ) [ ( N + l ) Q + Q2]+ (N/2 - v’)2(N/2+ 1 + v’)~]’/~(26) (Hr)vR,,,R = b6,,((N
Simplified formulas have also been found for the matrix elements when 52 is different from 52’ and lead to a heptadiagonal representation for the full coupled channel problem. However we have not exploited this yet. Of course the potential matrix is diagonal, and its elements are the discretized values of the potential function V ( 9 )computed at the angles 19”. Diagonalization of H will give the energy eigenvaluesand the components of the eigenvectors will directly give the amplitude of the wavefunctions at 9,. It is also possible to introduce a case 0 and a representation IvA), the role of 1 a n d j (and that of A and 52) being interchanged with respect to case a. The diagonal form for the potential is preserved. 2.4. Accurate Evaluation of G. It is very important to have accurate Clebsch-Gordan coefficients for the b52) to 1~52) transformations. The convergence of the eigenvalues and eigenvectors of H or HJ greatly suffer with inaccurate evaluation. Clebsch-Gordan coefficients are commonly calculated with explicit formulas that involve sums of ratios of factorials (ref 24, page 238). The large numerical range of the factorials is handled by working with the logarithms of the factorials, and taking the antilogs before ~ummati0n.j~We have found that this direct calculation of the Clebsch-Gordan coefficients starts to lose accuracy for N of the order of 30 with double precision (64 bit) arithmetic. This direct computation method is also slow even if the logarithms of the factorials are stored for future use. However accurate Clebsch-Gordan coefficientswith N as high as 127 are required in this work. The required accuracy can be easily and rapidly obtained by recursion. The elements of the
Anderson et al.
2448 The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 TABLE I: Accurate Eigenvalues for Hamiltonians
n
potential” VI VI v1 v2 v2 v2 v3 v3 v3 v4 v4 v4 VP VP VP VP
E2
Ei -79.084 015 83 23.771 705 58 39.914 699 40 -461.830 181 85 22.446 063 48 39.554 365 53 -65.676 198 04 -37.814 283 85 -14.536 066 20 -413.046 831 16 -381.525 830 37 -337.332 739 31 -86.362 538 53 -42.094 361 01 8.895 006 81 414.525 343 74
0 12 20 0 12 20 0 12 20 0 12 20 0 3 6 20
-23.062 634 77 74.907 8 I8 94 123.614 700 90 -348.105 61 1 68 66.236 828 IO 121.906 753 32 -13.175 944 78 33.076 830 04 76.068 623 26 -253.294 562 33 -209.968 8 13 73 -156.860 794 40 -59.1 17 403 00 -16.202 489 45 34.395 158 55 457.692 878 63
E3 1.338 285 72 129.267 524 18 211.019881 05 -216.588 391 54 105.420 539 65 206.448 041 34 2.999 390 05 93.839 474 66 167.474 960 12 -127.722 638 76 -69.430 196 20 -8.909 635 92 -32.967 761 11 8.576 889 32 59.185 166 50 502.609 158 05
E4 9.436 638 05 187.145 751 53 302.137 215 91 -89.814 322 57 151.846 144 60 293.110483 75 10.602 698 69 157.637 400 66 262.724 867 98 -41.494 157 86 34.561 034 07 110.268 535 00 -7.994 954 20 32.197 841 42 83.484 404 66 549.331 752 41
0 V1 has C = 100, xo = 1.0, and k = IO in eq 5 . V2 has C = 500, xo = 1.0, and k = 10 in eq 5. V3 has C = 100, xo = 0.3, and k = IO in eq 5 . V4 has C = 500, xo = 0.3, and k = IO in eq 5. VP has C = 100 in eq 6.
+ +
transformation matrix can be written as
Then the following recursion relation can be used (ref 24, page 256)30 ($+n-u-u N 2
/j - 2 0
)
=
10) -
O’-n)O’+n)(N+n-j+I)(N+n+J+l) 4(2j- 1)(2j
1
+ 1)
(f+f l - u 2~ u l j O ) ] (28) This equation is used in backward recurrence for j = N 1, N Q, ..., fl 2 with
+
NI
(;+R-u-u 2
+Q+
N+Q+lO)=O
(29)
and
The Nb ClebschGordan values for j ranging from to 52 + N are obtained by normalizing the relative values obtained by recursion to the actual values fori = Q given by the equation (ref 24, page 248) ( ; + n - u ~ u 2I n O )
=
;( + n - u ) ! ( f + n + u)!N(2n + l)! (-1)(NW
[
V ( X ) = -C exp(-k(x
- x0)’)
(32) where x = cos 0 . The parameters for the potentials Vi, V2, V3, and V4 are given in Table I. The fifth Hamiltonian is for motion of a polar rotor in a strong electric field. In this case a = 0, and b = 1. The pendular potential has the form
[-,(+ n - v %2l j -
+
with HJ= 12 j2 V(I9). For each of the four Hamiltonians, b and a in eqs 2 and 4 are set equal to 1. This choice is made for simplicity, but one consequence is that the results can be interpretedascalculationsin either theaorocases. The potential, V ( 9 )is chosen to have the form
(N+ 2 n + 1)!(f
+
U)!(f-
U)!(Q!)2
I/2
I
(31)
3. Convergence of the Discrete Bases for Eigenvalue C~~C~~OhS In this section we compare the convergence for calculations in the Ivn) basis with that for the VQ) basis. Five Hamiltonians (eq 1) are used in this work. Four are atom-diatom Hamiltonians
V ( x ) = -cx (33) where C can be interpreted as the quantity p E / b , where p is the rotor dipole moment and E is the magnitude of the electric field. This potential is named VP in Table I. The potentials for the atom-rotor problemsdiffer in the strength of the interaction and in the location of the minimum, XO. Potentials V1 and V2 have their minima at 19 = 0 and V3 and V4 have minima at 9 = 1.27 radians. V1 and V2 favor collinear configurations, while V3 and V4 favor triangular configurations. V2 and V4 have stronger interactions than V1 and V3. First the eigenvalues were obtained for the case y or IQ) representation for each of the five Hamiltonians. A simplification was made in the basis for the four atom-rotor cases by using only one value of Q for each of the j in the basis. This is equivalent to the coupled states or j , conserving approximation. In this approximation the total J is not a good quantum number, and since it only contributes to a diagonal centrifugal barrier (eq 26), it is assumed to be zero in the following. This assumption of a unique uncoupled Q is not an approximation for the pendular Hamiltonian. For all five Hamiltonians, the values of j were chosen to be 0, 1, 2, ...,(Nb - l), with Nb = 4,8, 16,32,64. The four Gaussian potentials were expanded in a series of Legendre polynomials (eq 9). Values of n from 0 to 30 were used and resultedinfittingthepotentialsVl,V2,V3,andV4toanabsolute error of less than 2ClC9 where C is given at the bottom of the table. The matrix elements for the rotational, centrifugal, and potential terms in the Hamiltonians were evaluated with eqs 7 and 10. The V Q ) calculations converged rapidly as Nb was increased. The eigenvalues of the four lowest energy states for potentials V1, V2, and V3 always agree to more than 9 digits for Nb = 32 and Nb = 64 calculations. For values of 0 that result only in ‘rotor”-like states (E > 0), this convergence was generally found with Nb = 16. The pendular problem had convergence to more than 9 digits for Nb 1 16 for all values of 0. Only for the v 4 potential was the convergence less impressive. For Q = 0 the
Stereodirected Discrete Bases in Hindered Rotor Problems
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The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 2449
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fourth energy agreed to only 5 digits for the Nb = 32 and Nb = 64 calculations. The s2 = 12 calculation has 8-digit agreement, and the s2 = 20 results show 11-digit agreement. Because of this "snapin* convergence, the l a ) calculations with Nb = 64 were taken as the exact eigenvalues. The four lowest energy eigenvalues are listed in Table I for all the Hamiltonians for different a. Figures 6-10 show the convergenceof the vs2) calculations for relatively smallvaluesof Nb. For atom-rotor systemswith bound states (E < 0) convergence to 0 only functions that vanish at the end points can be
represented. However this restriction does not exclude the cases for which we use such bases. (26) Smirnov, Y. F.; Suslov, S.K.; Shirokov, A. M. J . Phys. A 1984,17,
2157. (27) Alder, K.; Bohr, A.; Huus, T.; Mottelson, B.; Winther, A. Reu. Mod. Phys. 1956, 28,432. (28) Brussaard, P. J.; Tolhoek, H. A. Physica 1957, 23, 955. (29) Regge, T.; Ponzano, G. Spectroscopicandgroup theoreticalmethods inphysics; Bloch, F., Eds.; North Holland Publishing Co.: Amsterdam, 1968. (30) Schulten, K.; Gordon, R. G. J . Math. Phys. 1975, 16, 1971. (31) Davis, P. J.; Rabinowitz, P. Methods of Numerical Integration; Academic Press: Orlando, FL, 1984. (32) Nikiforov, A. F.; Suslov, S.K.; Uvarov, V. B. Classical Orthogonal Polynomials of a Discrefe Variable; Springer Verlag: Berlin, 1991. (33) Zare, R. N. Angular Momentum; Wiley & Sons: New York, 1988. (34) Brink, D. M.; Satchler, G. R. Angular Momentum; Clarendon: Oxford, 1968. (35) Wigner, E. P. Group theory and its application to the quantum mechanics of atomic spectra; Academic Press: New York. 1959. (36) Parlett, B. N. The Symmetric Eigerwalue Problem; Prentice-Hall: Englewood Cliffs, NJ, 1980. (37) Light, J. C.; Whitnell, R. M.;Park, T. J.; Choi, S.E.Supercomputer
Algorithms for Reactiuity, Dynamics and Kinetics of Small Molecules; Lagana, A., Ed.; Kluwer: Dordrecht, 1989.
