1. A Semiclassical Direct Potential Fitting Scheme for Diatomics Joel Tellinghuisen Department of Chemistry Vanderbilt University Nashville, TN 37235

2. Statement of Problem Diatomic spectroscopy, traditional approach: Assigned lines  least-squares fit to energies E( ,J) as sums of vibrational energy G , rotational energy B   [  = J(J+1) for simplest states], and terms in  2,  3, etc., to correct for centrifugal distortion. G  and B   RKR potential curve  quantal properties (centrifugal distortion constants, FCFs). Alternative approach gaining momentum: Fit directly to potential curve, computing E( ,J) by numerically solving Schrödinger equation for each level (DPF methods). Question: Can DPF be implemented using semiclassical methods like those behind RKR method? Why? Perhaps 100 times faster.

3. Reasons for optimism Results are exact for RKR curves, from common origin. RKR often shows quantum reliability to ~0.1 cm  1. While this can greatly exceed spectroscopic precision, perhaps much of the error is “built in” at the start, RKR being an exact inversion of approximate G  and B  from fitting. Semiclassical (SC) and quantum (Q) agree exactly for several well-known potentials, like harmonic oscillator and Morse (for J = 0 only).

4. Theoretical Background Consider effective potential V(R,J) = V(R,0) + C  /R 2, where the second term is the centrifugal potential (C a constant). The SC eigenvalues are solutions to for integer , where h is Planck’s constant and  the reduced mass. When this solution has been found, the rotational constant B  can be computed from  / , where these quantities are evaluated from similar integrals with arguments proportional to [E  V(R,0)]  1/2 (  ) and R  2 [E  V(R,0)]  1/2 (  ). (1)

5. Solutions to Eq. (1) are obtained by successive approxi- mation ( e.g., Newton’s method). For each E, solve for turning points, R 1 and R 2, and then evaluate the integral. The latter computation can be done with remarkable accuracy using as few as 4 values of the integrand, and seldom requiring more than 16, using Gauss-Mehler quadrature, for the weight function (1  x 2 ) 1/2. Thus, where x is defined by R = (R 1 +R 2 )/2 + x (R 2  R 1 )/2, and F w (x) = F(x)/(1  x 2 ) 1/2. For G-M quadrature, the pivots x i and weights H i are obtained from simple trigonometric expressions [see, e.g. Z. Kopal, Numerical Analysis]. (2)

6. Test of Methods — Rb 2 (X)

7. REQUIRED CONVERGENCE (RELATIVE) OF ACTION INTEGRALS = 1.00E-07 WORKING POTENTIAL GENERATED OVER RANGE 2.800 TO 9.800 REQUIRED CONVERGENCE IN V = 1.00E-05 ENERGIES AND EFFECTIVE BV VALUES FROM PHASE INTEGRALS FOR J = 0 V E(TRIAL) E(FOUND) R1 R2 NQUAD = 4 ACT =.5000151 NQUAD = 8 ACT =.5000151 NQUAD = 4 ACT =.5000000 NQUAD = 8 ACT =.5000000 0 28.8596 28.8588 4.0959921 4.3306829 NQUAD = 4 ACT = 10.5000966 NQUAD = 8 ACT = 10.5000971 NQUAD = 4 ACT = 10.4999995 NQUAD = 8 ACT = 10.5000000 10 591.0999 591.0946 3.7349882 4.8334555 NQUAD = 4 ACT = 15.5000831 NQUAD = 8 ACT = 15.5000865 NQUAD = 16 ACT = 15.5000865 NQUAD = 4 ACT = 15.4999966 NQUAD = 8 ACT = 15.5000000 NQUAD = 16 ACT = 15.5000000 15 861.2656 861.2610 3.6472901 4.9971049

8. NQUAD = 4 ACT = 50.4992890 NQUAD = 8 ACT = 50.5001847 NQUAD = 16 ACT = 50.5001854 NQUAD = 4 ACT = 50.4991036 NQUAD = 8 ACT = 50.4999992 NQUAD = 16 ACT = 50.5000000 50 2522.6609 2522.6533 3.3116691 6.0106878 NQUAD = 4 ACT = 65.4967910 NQUAD = 8 ACT = 65.5003310 NQUAD = 16 ACT = 65.5003387 NQUAD = 32 ACT = 65.5003387 NQUAD = 4 ACT = 65.4964524 NQUAD = 8 ACT = 65.4999923 NQUAD = 16 ACT = 65.5000000 NQUAD = 32 ACT = 65.5000000 65 3087.5205 3087.5089 3.2327590 6.5184197 NQUAD = 4 ACT = 75.4922973 NQUAD = 8 ACT = 75.5005223 NQUAD = 16 ACT = 75.5005479 NQUAD = 32 ACT = 75.5005479 NQUAD = 4 ACT = 75.4917497 NQUAD = 8 ACT = 75.4999743 NQUAD = 16 ACT = 75.5000000 NQUAD = 32 ACT = 75.5000000 75 3401.6815 3401.6658 3.1931709 6.9382654

9. ENERGIES AND EFFECTIVE BV VALUES FROM PHASE INTEGRALS FOR J = 240 Max E for potential = 4198.2110 at R = 9.124113 NQUAD = 4 ACT = 79.6524183 NQUAD = 8 ACT = 79.6319452 NQUAD = 16 ACT = 79.6308426 NQUAD = 32 ACT = 79.6307567 NQUAD = 64 ACT = 79.6307506 NQUAD = 128 ACT = 79.6307501 EMAX = 4198.211 FOR E = EMAX, FOLLOWING RESULTS ARE OBTAINED: E0 = 4198.2110 R1 = 3.3667956 R2 = 9.1241135 NV = 79.1309431 V E(TRIAL) E(FOUND) R1 R2 NQUAD ITER BV BV(EFF) DV(EFF) 0 1323.10746 1279.13342 4.2404603 4.4826019 8 2 2.237635E-02 2.087799E-02 1.29526E-08 5 1591.15984 1545.78834 3.9954986 4.8088244 8 2 2.208833E-02 2.054134E-02 1.33730E-08 10 1851.16183 1804.27057 3.8755959 5.0143559 16 2 2.178530E-02 2.018535E-02 1.38309E-08 15 2102.78165 2054.25980 3.7886613 5.1919133 16 2 2.146466E-02 1.980722E-02 1.43278E-08 20 2345.66380 2295.38371 3.7192766 5.3576869 16 2 2.112402E-02 1.940358E-02 1.48724E-08 25 2579.42934 2527.20827 3.6613190 5.5187621 16 3 2.076122E-02 1.897036E-02 1.54811E-08 30 2803.63160 2749.22659 3.6116477 5.6794973 16 3 2.037369E-02 1.850262E-02 1.61746E-08 35 3017.74220 2960.84521 3.5684214 5.8433007 16 3 1.995824E-02 1.799422E-02 1.69780E-08 40 3221.15559 3161.36678 3.5304724 6.0134023 16 3 1.951123E-02 1.743747E-02 1.79267E-08 45 3413.18638 3349.96697 3.4970260 6.1933698 16 3 1.902869E-02 1.682239E-02 1.90725E-08 50 3593.04952 3525.66219 3.4675601 6.3876448 16 3 1.850603E-02 1.613562E-02 2.04911E-08 55 3759.82864 3687.26187 3.4417304 6.6023355 16 3 1.793766E-02 1.535848E-02 2.22959E-08 60 3912.44560 3833.29225 3.4193330 6.8466797 16 3 1.731658E-02 1.446312E-02 2.46669E-08 65 4049.64081 3961.86067 3.4002924 7.1362810 32 3 1.663417E-02 1.340398E-02 2.79235E-08 70 4169.96107 4070.36701 3.3846824 7.5020638 32 4 1.588018E-02 1.209378E-02 3.27317E-08 75 4271.73723 4154.65614 3.3728301 8.0272369 32 5 1.504246E-02 1.029886E-02 4.10062E-08 NO SOLUTION; V = 80. EXCEEDS VMAX = 79.1309

10. Comparisons

11. With quantum defect (  =  0.000193) for SC

12. From these results, it appears that SC DPF analysis of these data would indeed yield a potential requiring little further adjustment to achieve quantum reliability. Next step: Develop DPF codes and compare their performance. Example: I 2 (A)

13. Reasons for interest in this state: Shallow, excited state Lots of data (9500 lines) extending to within 5% of D e Requires lots of conventional or NDE parameters

14. In spite of these efforts, including smoothing repulsive branch above  = 30, this potential shows significant Q-SC differences. Will these persist w/ DPF analysis?

15. Computational Approach: Code for both SC and Q analysis, for performance comparisons. Numerical derivatives — both centered and one-sided. Employ modified Lennard-Jones (MLJ) potentials, as in much previous work. For now, use R  15 small-R extension and R  p large-R (to D e ). This to avoid anomalies outside R span of data.

16. Results Q-SC differences remain, and are comparable for DPF-Q and DPF-SC analysis, even with the quantum defect (Y 00 ) correction in the latter. (The potentials in each case adjust to the data, including differences in T e.)  2 values very close, and within 2% of best spectroscopic fit, but 20 MLJ parameters in model! (including R e & D e ) Convergence slow and sensitve to starting po- tential, but RKR works well; also, a point-wise model seems to solve this problem.

17. Performance With dR = 0.001 Å, Q fitting is about 100 times slower than SC; however, dR can be increased to 0.004 Å for preliminary work and 0.002-0.0025 Å for final, dropping this concomitantly. Potential from DPF-SC analysis is not significantly better than RKR for starting DPF-Q. But DPF-SC analysis provides valid information about models, including dependence of  2 on number of parameters. Use of numerical derivatives makes it easy to test changes in the model — like different parameters for the e/f  -doubling (warranted). One-sided derivatives are as good as centered. In the DPF-SC analysis, with 9500 lines and 24 adjustable parameters, one iterative cycle takes about 25 s on an inexpensive PC.

18. And now, the real “pony in the manure” … Because the semiclassical energies track the quantal so closely, the partial derivatives needed for the nonlinear least-squares fitting can all be computed semiclassically, rendering the DPF-Q method only a factor of ~2 more time-demanding than DPF-SC.