A Semiclassical Direct Potential

Fitting Scheme for Diatomics

Joel Tellinghuisen

Department of ChemistryVanderbilt UniversityNashville, TN 37235

Statement of ProblemDiatomic 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.

Reasons for optimism

• Results are exact for RKR curves, from common origin.

• RKR often shows quantum reliability to ~0.1 cm1.

• 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).

Theoretical Background

Consider effective potential V(R,J) = V(R,0) + C/R2, where the second term is the centrifugal potential (C a constant). The SC eigenvalues are solutions to

h( 1/2) (8)1/2 [E V (R,J)]1/2R1

R2 dR

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 R2 [E V(R,0)]1/2 ().

(1)

Solutions to Eq. (1) are obtained by successive approxi-mation ( e.g., Newton’s method). For each E , solve for turning points, R1 and R2, 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 (1x2)1/2. Thus,

2

R2 R1[E V (R,J)]1/2dR

R1

R2 F(x)dx 1

1

(1 x2)1/2 1

1 Fw (x)dx HiFw (xi )i1

n

where x is defined by R = (R1+R2)/2 + x (R2 R1)/2, and Fw(x) = F(x)/(1x2)1/2. For G-M quadrature, the pivots xi and weights Hi are obtained from simple trigonometric expressions [see, e.g. Z. Kopal, Numerical Analysis].

(2)

Test of Methods — Rb2(X)

0

1000

2000

3000

4000

3 4 5 6 7 8 9 10 11

V(J=0)V(240)E

(cm

1)

R()

R2,max

R2R1

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

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

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

-0.025

-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0 10 20 30 40 50 60 70 80

CENDIS-Rb2X

semi-quant (J=0)S-Q(100)E,J=0 (Q - Dunham)

E (

cm1

)

Comparisons

-0.005

0.000

0.005

0.010

0.015

0 20 40 60 80 100

S-Q(0)S-Q(100)S-Q(150)S-Q(190)S-Q(220)S-Q(240)

E

(cm

1)

With quantum defect ( = 0.000193) for SC

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: I2(A)

-1600

-1200

-800

-400

0

3 4 5 6

E(cm–1)

R(Å)

2

4

6

8

10

13

20

30

0

35

16

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

32 36 40 44 48 52 56

B

NDE — Appadoo, et al.(12 rotational parameters)

Mixed Representation Ñ JT(6 NDE rotational parameters)

Reasons for interest in this state:• Shallow, excited state

• Lots of data (9500 lines) extending to within 5% of De

• Requires lots of conventional or NDE parameters

0.000

0.010

0.020

0.030

0.040

0.050

-12

-10

-8

-6

-4

-2

0

0 10 20 30 40 50

E

(Q

-SC

) (c

m1

)

10

5B (Q

-con

stan

ts)

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

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 R15 small-R extension and Rp large-R (to De). This to avoid anomalies outside R span of data.

Results

-0.05

0.00

0.05

-2

-1

0

1

0 5 10 15 20 25 30 35

E (

Q-S

C)

(cm

1)

106B

(Q-S

C)

• DPF-SC (w/ defect)

DPF-Q

DPF-Q

• Q-SC differences remain, and are comparable for DPF-Q and DPF-SC analysis, even with the quantum defect (Y00) correction in the latter. (The potentials in each case adjust to the data, including differences in Te.)

• 2 values very close, and within 2% of best spectroscopic fit, but 20 MLJ parameters in model! (including Re & De)

• Convergence slow and sensitve to starting po-tential, but RKR works well; also, a point-wise model seems to solve this problem.

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.

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.