Setting the parameters for a bound-state calculation

Here we will demonstrate how to set the parameters for the determination of the bound states of the Ar...NO complex. Make the hibridon code Run this with a small basis set Setting bastst=t shows that you've selected a 6-state basis. Then, setting bastst=f and running the calculation reveals that the energies of the three lowest bound states are (in cm-1 units): We will change the integration parameters sequentially to obtain convergence, in the following order:

1. The parameter HSIMP controls the integration step size in the Simpson's rule integration to determine the matrix elements of the potential. This parameter is set to the value 0.1 in the input file. If we reduce the value of the step size to HSIMP=0.05, we get the same values of the energies. Thus we conclude that a step size of HSIMP=0.1 is sufficiently small.

2. Now we test the convergence with respect to the range of integration which extends from -DELR inside the first Gaussian function to +DELR beyond the last Gaussian. This parameter is set to DELR=0.8 in the input file. If we extend it to DELR=1.1, the calculated energies change to

with no further change if we extend DELR beyond 1.1 So, we will keep DELR=1.1.

3. Now let us test the convergence with respect to increasing the number of Gaussians. This is done by varying the parameter SPAC, which controls the SPACing between the centers of the Gaussians. (In fact, the input value of SPAC is just a suggestion. The program adjusts SPAC so that there are an integer number of Gaussians between R=R1 and R=R2).

Try SPAC=0.3 (instead of the value 0.4 in the original input file). The calculated energies of the lowest bound states change to

Lowering SPAC below 0.3 leads to no further change. So we will keep SPAC=0.3

4. Then, we will test convergence with respect to the overall range of integration. If we decrease R1 to 4.2, then there are no changes in the calculated energies. Thus, R1=4.5 is a good value. [Note that you can't decrease R1 too far, since the ab initio calculations on which the potential energy surface is based [M. H. Alexander, J. Chem. Phys. 111, 7426 (1999)] do not extend inside of R=4.5 bohr].

Since decreasing R1 from 4.5 to 4.2 gave no change, we can try increasing this parameter. Setting R1=5 gives no change in the calculated energies, so, we keep R1=5

5. Now, we try to increase R2, to R2=13. There is no change in the calculated energies of the first 12 levels, so we will keep R2=11. (Note, for higher bound-state levels one may need to increase R2 for convergence.)

6. The parameter C controls the width of the individual Gaussians. If this parameter is too small, then the overlap between the original Gaussians will be too large. For example, setting C=0.4 and running the calculation gives the error message

If you increase C to 0.6, there is no change in the calculated energies. So, keep C=0.5.

You have now set all the integration parameters with values

Now, we increase the size of the rotational basis, by increasing JMAX to obtain convergence. The results are shown in the following table:

JMAX E1 E2 E3 E4
1 -77.417 -75.062 -65.869 -64.031
2 -81.027 -79.022 -66.906 -66.149
3 -81.967 -81.362 -68.832 -67.317
4 -82.838 -82.244 -68.923 -67.879
5 -82.937 -82.791 -69.306 -68.076
6 -83.132 -82.853 -69.325 -68.171
7 -83.142 -82.926 -69.366 -68.192
8 -83.165 -82.930 -69.369 -68.202
9 -83.165 -82.936 -69.372 -68.203
10 -83.165 -82.937 -69.373 -68.204
11 -83.165 -82.937 -69.373 -68.204

Thus, to within 0.001 cm-1, the energies of the lowest four bound states are converged by JMAX=10, which corresponds to a total of 882 states (42 rotational states, each one multiplied by 21 Gaussian functions).
The representation of the R dependence of the bound-state wavefunction is by a distributed sum of Gaussians. The parameters SPAC, R1, R2, DELR, HSIMP control the accuracy of this description of the R dependence. When you add angular functions, you are increasing the accuracy of the description of the θ dependence. So, once you've determined the first set of parameters (those that control the accuracy of the description of the R dependence), you can keep these paramaters (SPAC, R1, R2, DELR, HSIMP) while increasing JMAX, which increases the accuracy of the description of the θ dependence.
The dependence on J (the total angular momentum) of the energies of the bound states can be assessed by increasing the paramter JTOT1. For the Ar–NO system you obtain

J Na timeb E1 E2 E3 E4
0.5 882 1 -83.167 -82.937 -69.373 -68.204
1.5 1722 8.8 -82.989 -82.700 -79.435 -79.014
2.5 2478 30.0 -82.679 -82.321 -79.090 -78.669
4.5 3738 117.6 -81.651 -81.146 -77.985 -77.565
6.5 4662 219 -80.077 -79.421 -76.328 -75.910
a. N is the total number of Gaussian functions used (here 21) multiplied by the number of rotational functions.
b. Relative cpu time (in s).

The energies of the bound states increase gradually with J, corresponding to a gradual increase in the centrifugal potential. Note that for J=1.5, two states appear with energy ~ 79 cm-1, which are not present for J=0.5. These correspond to two states where the nominal projection of the total angular momentum along the body-frame axis is K=1.5.

The dependence on J of the energies can be fit in the usual power series in J(J+1). Fitting the values in the above table to a quadratic in J(J+1) gives

and Thus, the Ar–NO rotational constants in the lowest and next-to-lowest bound states are 0.0619 and 0.0777 cm-1, respectively.

For a more in-depth discusion of the spectroscopy of complexes of NO with noble gas atoms, see B. Wen, Y. Kim, H. Meyer, J. Kłos, and M. H. Alexander, J. Phys. Chem. A 112, 9483 (2008).

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