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A Semiclassical Direct Potential Fitting Scheme for Diatomics Joel Tellinghuisen Department of Chemistry Vanderbilt University Nashville, TN 37235
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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.
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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).
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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)
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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)
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Test of Methods — Rb 2 (X)
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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
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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
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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
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Comparisons
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With quantum defect ( = 0.000193) for SC
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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)
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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
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In spite of these efforts, including smoothing repulsive branch above = 30, this potential shows significant Q-SC differences. Will these persist w/ DPF analysis?
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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.
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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.
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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.
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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.
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