Download Semiclassical-DPF

A Semiclassical Direct Potential
Fitting Scheme for Diatomics
Joel Tellinghuisen
Department of Chemistry
Vanderbilt University
Nashville, TN 37235
Statement of Problem
Diatomic spectroscopy, traditional approach:
• Assigned lines  least-squares fit to energies E(u,J)
as sums of vibrational energy Gu , rotational energy
Bu k [k = J(J+1) for simplest states], and terms in k2,
k3, etc., to correct for centrifugal distortion.
• Gu and Bu  RKR potential curve  quantal
properties (centrifugal distortion constants, FCFs).
Alternative approach gaining momentum: Fit directly to
potential curve, computing E(u,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 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 Gu and Bu 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) + Ck/R2,
where the second term is the centrifugal potential (C a
constant). The SC eigenvalues are solutions to
(1)
1/ 2
h(u 1/2)  (8)

R2
[E
R1
- V (R,J)]1/ 2dR
for integer u, where h is Planck’s constant and  the
reduced mass. When this solution has been found, the
rotational constant Bu can be computed from r/t, where
these quantities are evaluated from similar integrals with
arguments proportional to [E - V(R,0)]-1/2 (t) and
R-2 [E - V(R,0)]-1/2 (r).
Solutions to Eq. (1) are obtained by successive approximation ( 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 (1-x2)1/2. Thus,
R2
1
2
1/ 2
[E -V (R,J)] dR   F(x)dx

R
-1
R
R
1
2
1
n
(2)


1
(1- x 2 )1/ 2 Fw (x)dx 
-1
 Hi Fw (xi )
i1
R = (R1+R2)/2 + x (R2 - R1)/2, and
where x is defined2 by
Fw(x) = F(x)/(1-x )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].
Test of Methods — Rb2(X)
4000
R2,max
E (cm -1)
3000
V(J=0)
V(240)
2000
1000
R1
R2
0
3
4
5
6
7
R(
)
8
9
10
11
REQUIRED CONVERGENCE (RELATIVE) OF ACTION INTEGRALS =
WORKING POTENTIAL GENERATED OVER RANGE
2.800
REQUIRED CONVERGENCE IN V =
1.00E-07
TO
9.800
1.00E-05
ENERGIES AND EFFECTIVE BV VALUES FROM PHASE INTEGRALS FOR
V
E(TRIAL)
E(FOUND)
4
8
4
8
0
NQUAD =
NQUAD =
NQUAD =
NQUAD =
28.8596
4
8
4
8
10
NQUAD =
NQUAD =
NQUAD =
NQUAD =
591.0999
15
NQUAD =
NQUAD =
NQUAD =
NQUAD =
NQUAD =
NQUAD =
861.2656
R1
ACT
ACT
ACT
ACT
28.8588
=
=
=
=
ACT
ACT
ACT
ACT
591.0946
=
=
=
=
4
ACT
8
ACT
16
ACT
4
ACT
8
ACT
16
ACT
861.2610
=
=
=
=
=
=
J =
R2
.5000151
.5000151
.5000000
.5000000
4.0959921
4.3306829
10.5000966
10.5000971
10.4999995
10.5000000
3.7349882
4.8334555
15.5000831
15.5000865
15.5000865
15.4999966
15.5000000
15.5000000
3.6472901
4.9971049
0
4
ACT
8
ACT
16
ACT
4
ACT
8
ACT
16
ACT
2522.6533
=
=
=
=
=
=
50
NQUAD =
NQUAD =
NQUAD =
NQUAD =
NQUAD =
NQUAD =
2522.6609
4
ACT
8
ACT
16
ACT
32
ACT
4
ACT
8
ACT
16
ACT
32
ACT
3087.5089
=
=
=
=
=
=
=
=
65
NQUAD =
NQUAD =
NQUAD =
NQUAD =
NQUAD =
NQUAD =
NQUAD =
NQUAD =
3087.5205
4
ACT
8
ACT
16
ACT
32
ACT
4
ACT
8
ACT
16
ACT
32
ACT
3401.6658
=
=
=
=
=
=
=
=
75
NQUAD =
NQUAD =
NQUAD =
NQUAD =
NQUAD =
NQUAD =
NQUAD =
NQUAD =
3401.6815
50.4992890
50.5001847
50.5001854
50.4991036
50.4999992
50.5000000
3.3116691
6.0106878
65.4967910
65.5003310
65.5003387
65.5003387
65.4964524
65.4999923
65.5000000
65.5000000
3.2327590
6.5184197
75.4922973
75.5005223
75.5005479
75.5005479
75.4917497
75.4999743
75.5000000
75.5000000
3.1931709
6.9382654
ENERGIES AND EFFECTIVE BV VALUES FROM PHASE INTEGRALS FOR
Max E for potential =
NQUAD
NQUAD
NQUAD
NQUAD
NQUAD
NQUAD
4198.2110
=
=
=
=
=
=
at R =
4
8
16
32
64
128
ACT
ACT
ACT
ACT
ACT
ACT
J =
9.124113
=
=
=
=
=
=
79.6524183
79.6319452
79.6308426
79.6307567
79.6307506
79.6307501
EMAX =
4198.211
FOR E = EMAX, FOLLOWING RESULTS ARE OBTAINED:
E0 =
4198.2110
R1 =
3.3667956
R2 =
V
E(TRIAL)
0
1323.10746
5
1591.15984
10
1851.16183
15
2102.78165
20
2345.66380
25
2579.42934
30
2803.63160
35
3017.74220
40
3221.15559
45
3413.18638
50
3593.04952
55
3759.82864
60
3912.44560
65
4049.64081
70
4169.96107
75
4271.73723
NO SOLUTION; V =
E(FOUND)
R1
1279.13342
4.2404603
1545.78834
3.9954986
1804.27057
3.8755959
2054.25980
3.7886613
2295.38371
3.7192766
2527.20827
3.6613190
2749.22659
3.6116477
2960.84521
3.5684214
3161.36678
3.5304724
3349.96697
3.4970260
3525.66219
3.4675601
3687.26187
3.4417304
3833.29225
3.4193330
3961.86067
3.4002924
4070.36701
3.3846824
4154.65614
3.3728301
80.
EXCEEDS VMAX =
240
R2
9.1241135
NQUAD
4.4826019
4.8088244
5.0143559
5.1919133
5.3576869
5.5187621
5.6794973
5.8433007
6.0134023
6.1933698
6.3876448
6.6023355
6.8466797
7.1362810
7.5020638
8.0272369
79.1309
8
8
16
16
16
16
16
16
16
16
16
16
16
32
32
32
ITER
2
2
2
2
2
3
3
3
3
3
3
3
3
3
4
5
NV =
BV
2.237635E-02
2.208833E-02
2.178530E-02
2.146466E-02
2.112402E-02
2.076122E-02
2.037369E-02
1.995824E-02
1.951123E-02
1.902869E-02
1.850603E-02
1.793766E-02
1.731658E-02
1.663417E-02
1.588018E-02
1.504246E-02
79.1309431
BV(EFF)
2.087799E-02
2.054134E-02
2.018535E-02
1.980722E-02
1.940358E-02
1.897036E-02
1.850262E-02
1.799422E-02
1.743747E-02
1.682239E-02
1.613562E-02
1.535848E-02
1.446312E-02
1.340398E-02
1.209378E-02
1.029886E-02
DV(EFF)
1.29526E-08
1.33730E-08
1.38309E-08
1.43278E-08
1.48724E-08
1.54811E-08
1.61746E-08
1.69780E-08
1.79267E-08
1.90725E-08
2.04911E-08
2.22959E-08
2.46669E-08
2.79235E-08
3.27317E-08
4.10062E-08
Comparisons
0.015
CENDIS-Rb2X
0.010
E (cm-1)
0.005
semi-quant (J=0)
S-Q(100)
E,J=0 (Q - Dunham)
0.000
-0.005
-0.010
-0.015
-0.020
-0.025
0
10
20
30
u
40
50
60
70
80
With quantum defect (u = -0.000193) for SC
0.015
S-Q(0)
S-Q(100)
S-Q(150)
S-Q(190)
S-Q(220)
S-Q(240)
E (cm-1)
0.010
0.005
0.000
-0.005
0
20
40
u
60
80
100
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)
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
35
30
-400
E
(cm–1)
-800
20
0.016
16
0.014
13
0.012
10
8
-1200
Bu
6
0.008
4
0.006
2
0.004
0
0.002
-1600
3
4
5
R(Å)
6
NDE — Appadoo, et al.
(12 rotational parameters)
0.010
Mixed Representation Ñ JT
(6 NDE rotational parameters)
0.000
32
36
40
44
u
48
52
56
In spite of these efforts, including smoothing repulsive
branch above u = 30, this potential shows significant
Q-SC differences. Will these persist w/ DPF analysis?
0
0.050
E (Q-SC) (cm -1)
0.040
-4
0.030
-6
-8
0.020
-10
0.010
-12
0.000
0
10
20
u
30
40
50
105Bu (Q-constants)
-2
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 De). This to avoid anomalies outside R span of
data.
Results
• 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.)
• c2 values very close, and within 2% of best spectroscopic fit, but
20 MLJ parameters in model! (including Re & De)
1
E (Q-SC) (cm-1)
0
•
0.05
DPF-SC (w/ defect)
-1
-2
0.00
DPF-Q
-0.05
0
5
10
15
20
u
25
30
35
106Bu (Q-SC)
DPF-Q
• Convergence slow and
sensitve to starting potential, 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 c2 on number of parameters.
• Use of numerical derivatives makes it easy to test changes in the
model — like different parameters for the e/f W-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.