Stark effect of a rigid rotor
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1 J. Phys. B: At. Mol. Phys. 17 (1984) Printed in Great Britain Stark effect of a rigid rotor M Cohen, Tova Feldmann and S Kais Department of Physical Chemistry, The Hebrew University, Jerusalem , Israel Received 30 March 1984 Abstract. Energy levels of a rigid rotor in a uniform electric field of arbitrary strength have been computed using Rayleigh-Schrodinger perturbation theory. Rational fraction representations of the weak-field Taylor series, the strong-field asymptotic series, and of both series simultaneously are very accurate over limited ranges of the applied field strength. A scaled variational perturbation theory yields high accuracy over the entire range. 1. Introduction The influence of a static electric field on the energy levels of a diatomic polar molecule (which is described by a rigid rotor model) has been studied extensively by means of Rayleigh-Schrodinger ( RS) perturbation theory (IT). The most comprehensive treatment appears to be that of Propin (1978a, b), who performed separate calculations beginning with the field-free and infinite field limits. Propin s results, which correct and extend several earlier treatments cited in his work, are based on partial sums of the two RSFT power series. These series either display very limited regions of convergence or are asymptotically divergent. However, the computed coefficients may be used to construct approximants to the energy whose ranges of accuracy turn out to be much larger than those of the corresponding RSPT series. Nevertheless, approximants of various types, which may be obtained directly from a generalised RSFT procedure (Cohen and Feldmann 1981) or indirectly from the appropriate Taylor series and constructed from either the weak field or the strong-field RSFT series do not appear to converge for all field strengths. Furthermore, even two-point approximants (which require RSPT coefficients from both energy series) are not entirely satisfactory. In this paper, we first use Propin s (1978a, b) coefficients to calculate several different approximant forms and demonstrate their general superiority over the corresponding power series. We then present an alternative RSFT solution which is valid at all field strengths, and show that it yields good accuracy even in lowest order. 2. Exact solutions The Schrodinger equation for a rigid rotor with moment of inertia I and dipole moment p, placed in a uniform electric field of strength F in the z direction may be written /84/ $ The Institute of Physics 3535
2 3536 M Cohen, T Feldmann and S Kais (cf Peter and Strandberg 1957): 1 1 a sine- +--+hcose $(e,4)= [sin e a:( ai) sin2 e a42 1 w+(e,4). (1) Here, W =2EI/h2 where E is the energy, while A =2pFI/h2 is a natural RSPT parameter. A formal solution of equation (1) is obtained by writing in which the PF(cos 0) are the usual associated Legendre functions. Substituting (2) into (1) leads to a tridiagonal secular equation of injnite order but, in practice, we solve a series of truncated secular equations of steadily increasing order, N. This procedure is found to converge rapidly with N over a wide range of values of the parameter A. It yields eigenvalues which are essentially exact, and these serve as comparison data for the approximations we describe below. The form (2) is a consequence of the fact that m is a good quantum number for all A. Moreover, the energy depends only on m2, not on the sign of m, for any field strength. In the weak-field limit (A + 0), a single term of (2) provides an exact solution with, for example, aj(a) = 1 Uk(h)=O (k+ J) WO = J(J + 1). (3 1 Thus, the system is simply a rigid rotor with rotational quantum number J In the intense field limit (A +CO) an exact solution actually contains an injnite number of terms of (2). However, as shown by Peter and Strandberg (1957), asymptotically (W+A)-n(2A) 2 asa+oo (4) where, in the notation of Propin (1978a), n =2k+(ml+1 k=0, 1,2,.... (5) Thus k plays the role of a vibrational quantum number and the system behaves as a two-dimensional oscillator. At intermediate values of A, neither the rotational nor the vibrational description is strictly appropriate but, since there is no crossing of levels of fixed lml, we correlate successive states (J = Iml, Im/ + 1,... ) at A + 0 with successive states (k = 0, 1,... ) as A +=CO and, for convenience, label states according to their J values in the weak-field (A += 0) limit. 3. Approximants for the energy 3.1. Small A representations Standard RSPT methods yield a Taylor series, valid for small A: W(A)= WO+ W2A2+ W4A (6) The vanishing of all odd powers of A in (6) is a consequence of the odd parity of the perturbing field. Propin (1978b) has given analytical expressions for a few low-order coefficients W.(J, Iml) (is6) as well as numerical values for i up to 22 for several states; in principle, his calculations may be extended to higher order if desired.
3 Stark eflect of a rigid rotor 3537 However, following Cohen and Feldmann (1981), we use the coefficients Wi to construct rational fractions W"'( A) having the general form where W(j'( A ) = N"'( A)/ D"'( A ) (7) and the coeflicients Ni, Di are related to the Wi through the equations No= WO i-l Ni = Di-IW, + W. ( l ~ i ~ j ). (9) f=o Equations (9) do not fix the pairs (Ni, Di) uniquely, and different possible solutions lead to Levin (1973) and Pad6 approximants (Baker 1975), as well as to the two-point approximants we describe below Large A representations Several different coordinate transformations (Peter and Strandberg 1957, Propin 1978a, Cohen and Feldmann 1982), lead to an asymptotic expansion of the energy, valid for A +a: where p = (2/A)"2, Analytical expressions for the lower w, (n, m) (i S 4) have been given by Cohen and Feldmann (1982), while numerical values for several states have been tabulated by Propin (1978a) for i up to 15. By analogy with (6) and (7), it seems natural to construct approximants w"'(p) having the form ~("(p) = -2/p2+ ~-l/p +n"'(p)/d"'(p) (12) where the coefficients (n,, d,) satisfy a set of equations analogous to (9) no= wo 1-1 ni = di-/wf + wi (1 s is j). I=O 3.3. Two-point approximants Our aim is to provide a single accurate representation of the energy function W(A), validfor all A. To this end, it is necessary for the 'small-a' approximant (7) to reproduce the leading behaviour of the asymptotic series (lo), or for the 'large-a' approximant (12) to reproduce the leading terms of the Taylor series (6). Now it is easily verified that (7) cannot yield (10) for any choice of Ni, Di whereas (12) can be made to yield (6). This simple observation possibly explains why the range of utility of the large-a approximant (12) extends to quite small values of A whereas the small-a approximant (7) has a much more limited useful range (see results below). It is therefore natural to consider a two-point approximant, having the fimctional form (12) with coefficients chosen so as to reproduce both the Taylor series (6) and
4 3538 M Cohen, T Feldmann and S Kais the asymptotic series (10). Thus, in addition to the set of equations (13), the coefficients (n,, d,) must also be chosen so that (writing Y in place of,l-') (::: ) r=o d,-$ + v' ( WO - w- Y + 2 v2 + 4 W, v W,v ) = 1 n,- I Y' + O( d). ( 14) The equations for (nl, d, ; i in matrix form: 1) which result from (13) and (14) are written conveniently 3.-[:I.-[$ +: n=ad where the vectors are simply d = -(B - A)-'w (15) and the matrices A, B are given explicitly as 0 4w2 0...I WO -W-l 2 (17) [WO ,.I WI WO B=l2 WI WO 0..I. In the following section, we discuss results of energy calculations based on the small-a approximants (7), the large-a approximants (12), and the two-point approximants with coefficients derived from (15). We shall see that whenever convergence occurs, it is so rapid that only a few leading series coefficients are required. 4. Accuracy of approximant energies We present results for the (J = 0, Iml= 0) and (J = 1, [mi= 1) states only; these seem to display all essential features of the calculations, without introducing unnecessary details. Moreover, although both Levin and diagonal Pad6 approximants were calculated from the separate small-a and large-a series, the numerical results are so similar that we present only one set for each state. Thus, table 1 contains diagonal Pad6 approximants for the (0,O) state, while table 2 contains Levin approximants for the (1, 1) state. Since the [nln] Pad6 approximant and the Levin approximant T(2"+l) both employ the same (2n + 1) leading series coefficients, our results for the two states are strictly comparable. The exact energies have been listed to five decimals only although they are actually obtained accurate to at least nine decimals. For these states, we list small-a approximant results for 1 6 A 6 5 and A = 10, and large-a approximant values for h = 5 and 10 s A < 50. (Note that A = 50 corresponds to a field strength of about 60000Vcm-' for the molecule CsF.) There is rapid
5 Stark effect of a rigid rotor 3539 Table 1. Diagonal Pad6 approximant energies -[n/n] for the (0,O) state Accurate Small-A approximant (7) Large-h approximant (12) t t " t t Converged values; there is no change with increasing n. Table 2. Levin approximant energies T(2n+1) for the (1, 1) state. A Accurate Small-A approximant (7) t t t t Large-A approximant (12) t t f Converged values. convergence towards the exact energies from either the small-a or the large-a approximants, except for the range 3<A <5 for the (0,O) state. However, (7) is apparently converging at A = 3, whereas Propin's (1978b) estimate of the radius of convergence of the Taylor series is only For the (1, 1) state, (7) is more rapidly convergent than (12) even at A = 10, although the estimated radius of convergence of the small-a series is 5.56! The economy of these approximant representations is noteworthy; thus, we obtain five figure accuracy at A = 2 for the (0,O) state with nine
6 3 540 M Cohen, T Feldmann and S Kais coefficients and at A = 5 for the (1, 1) state with only seven coefficients, whereas the corresponding Taylor series yield no more than three figure accuracy even when all computed coefficients are used! At intermediate values of A (a region which is evidently different for each state) it does not seem possible to predict whether either the small-a approximant (7) or the large-a approximant (12) will prove satisfactory and it was anticipated that the two-point approximants based on (15) would resolve the uncertainty. Tables 3 and 4 contain our results for the two states and it appears that the two-point approximants do not converge for small A (even though they may eventually yield accurate results for very small A.) This is probably due to the fact that the large-a series (10) is asymptotic, so that including progressively more coefficients can make the results poorer at small A. At sufficiently large A, the two-point approximants (15) are converging only slightly less rapidly than the large-a approximants (12), but since the regions of useful accuracy are so similar, there seems little advantage in using (15) rather than (12) in this problem. Table 3. Two-point approximant energies for the (0,O) state. A Exact I IO t Converged values t t t Table 4. Two-point approximant energies for the (1, 1) state. A n Exact IO t 'r T t t Converged values.
7 5. An alternative solution Stark efect of a rigid rotor 3541 To obtain energies of high accuracy for all values of A, it is desirable to include the leading effect of the field appropriately in lowest order. This may be achieved conveniently by introducing into the RSPT zero-order solution a variational scaling parameter which is field dependent (cf Cohen and Kais 1984). First, we rewrite the Schrodinger equation (l), making the substitutions COS e = I - 2 ~ $( e,+) = e"+[x( 1 - x)]"/'f(x) (18) so that F(x) satisfies the equation where 2F = EF (Osxsl) (19) 2=-[x(l-x)d2/dx2+(m+1)(1-2x)d/dx-2Ax] E =[ W- m(m +1) +A]. Equation (19) has been studied previously using standard RSPT by Cohen and Feldmann (1982), who observed that the change of coordinate scale x + ay leads to a convenient decomposition of 2 provided that CY'= 1/8A. This asymptotic choice of CY yields the energy series (lo), which is obtained by replacing the range of the scaled variable 0 s y s 1/ CY by 0 s y <CO; this is justified only when A In the present work, we choose a variationally for each value of A and it is more convenient to work with the unscaled variable x over a fixed range 06 x s 1, and to introduce the scaling parameter a into the zero-order solution. In the asymptotic limit, the eigenfunctions and eigenvalues (for given m) are given by (Cohen and Feldmann 1982) (K = 0, 1,2,...) FKmb) = N K e-y'22f(~) ckm = i(2k + m + 1) (21) where -%':(U) is the associated Laguerre function. Here, we consider only the (0,O) and (1, 1) states for both of which K = 0 and, for such states, we choose (20) Fo( x) = N = ;( m + 1). (22) with a still to be determined. We now decompose the operator 2 according to where 2=20+21 (23) Z0= -[x( 1 -x) d2/dx2 +(m + 1)(1-2x) d/dx] +(m + 1)ax +ia2x( 1 -x) which yields Hermitean operators, and is thus to be preferred to the alternative (non-hermitean) choice of Cohen and Feldmann (1982). The first-order energy correction is now calculated from (24) E, = [2A - (m + 1). -~a2](x)o+~a2(x2)o (25) where the integrals include a weight function xm( 1 - x)" (cf equation (18) above) and are given explicitly by (x"), = 47rN' jol e-axx"+m( 1 - x)" dx.
8 3542 M Cohen, T Feldmann and S Kais From equations (20), (22) and (25) we obtain upper bounds to the energy validfor all A ; explicitly we find for the (0,O) state and for the (1, 1) state W(O,O)=-A +(a/4+2a/a)-;-(4h -a)/[2(e"- l)] (27) W(l,1)=2-A+a CY (2A - CY)CY' 2-CY 3-a (y ;) 22-CY (a-2)*ea+(a2-4)' Variation of these expressions with respect to CY may be carried through analytically when A and CY are very large or very small, and we find and W(0,O)- -A W(1, 1)--A +V~-~+O(A-'/~) +2fi-~+O(A-'/') corresponding to the large-a value CY = 2&, w(o, 0) - -ih2 +o(a~) while and (30) w(1, ~)-~-&,A*+o(A~) corresponding to the small-a values, CY = 2A/(m + 1). Thus, equations (27) and (28) yield the correct leading terms of both the small-a and the large-a series expansions, a consequence of our choice of F,,(x) which approaches the correct functional forms both as A +O and as A +CO. At intermediate values of A, the variation must be performed numerically, and leads to the parameter values and first-order energy bounds presented in table 5. The maximum error of this first-order energy calculation does not exceed 0.82% for any A for either state; moreover, greater accuracy may easily be obtained by calculating the RSPT first-order correction F, (x). This yields second- and third-order energy Table 5. Upper bound energies for the (0,O) and (1,O) states. (0,O) state (1, 1) state h a First-order Accurate a First-order Accurate I
9 Stark efect of a rigid rotor 3543 corrections directly, and Fo(x) +~F,(x) may be used as a variational approximation to obtain a third-order upper bound to the total energy (Dalgarno and Stewart 1961). We have obtained the first-order solution for the (0,O) state, using the variational method of Hylleraas (1930). We present in table 6 both rigorous upper bounds and the [ 1/ 11 Pad6 approximant based on the present RSPT energy coefficients. In this case, the [ 1/ 11 Pad6 approximants are practically indistinguishable from the third-order energy bounds, while both of these sets reproduce the accurate values to four or five figures. Table 6. Energies - W for the (0,O) state. h (3) (4) i4.i~ (1) First-order upper bound. (2) Third-order upper bound. (3) [1/1] Pad6 approximant. (4) Accurate values. 6. Discussion and conclusions Both the successes and the failures of the various approximants have been noted above. Our present uniform treatment, carried through here only in lowest-order RSPT, owes something of its numerical accuracy to the fact that our simple choice of F,(x) (equation (22) above) yields an exact solution in both the large-a and the small-a limits. Excited states can be treated in a similar way, but will require more elaborate Fo(x). For example, the (m + 1, m) states require two field dependent variational parameters, so that F~( x) = N e-ax 2( 1 - px) (31) if F(x) is to become exact in both limits (a + 0, p + 2 as A + 0; a = p + 2J2h as A + a), and more highly excited states will involve additional parameters. However, unrestricted variation of the energy through first order may no longer yield upper bounds to the exact energy for such excited states at some intermediate values of A. Greater accuracy can be obtained quite easily by calculating higher-order RSFT corrections, but we have not felt this necessary here. It seems clear that the scaling procedure has the effect of maximising the zero-order terms and simultaneously minimising the first-order terms in the complete Hamiltonian, thus making RSFT a particularly effective approximation, even in lowest order.
10 M Cohen, T Feldmann and S Kais References Baker G A Jr 1975 Essentials of Pad6 Approximants (New York: Academic) Cohen M and Feldmann T 1981 J. Pbys. B: At. Mol. Pbys J. Pbys. B: At. Mol. Pbys Cohen M and Kais S 1984 Cbem. Pbys. Lett Dalgarno A and Stewart A L 1961 Proc. Phys. Soc. I Hylleraas E A 1930 Z. Pbys Levin D 1973 rnt. J. Comput. Math Peter M and Strandberg M W P 1957 J. Cbem. Phys Propin R 1978a J. Pbys. B: At. Mol. Pbys b J. Pbys. E?: At. Mol. Pbys
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