Refinement

In this chapter details will be given of the refinement procedure implemented in TROVE. Refinement in this context means to adjust the ab intio potential energy surface by comparing computed rotational-vibrational energy levels to experimental values. Parameters of the PES are varied, energy levels re-computed and compared to experiment. This process is continued until acceptable agreement between the calculated and experimental energy levels is obtained. Usually there is relatively few experimental energies and so ab intio electronic energies are used to constrain the refinement to prevent over fitting. Although experimental data is usually at fairly low energies, it is often the case that correcting the lower energy region of the PES gives more accurate values at high energies also.

Theory

Details of the method of refinement implemented in TROVE have been published [11YuBaTe] and only a brief summary will be given here. Assuming a reasonable PES has already been obtained, a correction is added in terms of internal coordinates $\xi$

$\Delta V = \sum_{ijk...} \Delta f_{ijk...} \left(\xi_1^i \xi_2^j \xi_3^k ...\right)^A$

where $\left(\xi_1^i \xi_2^j \xi_3^k ... \right)^A$ corresponds to totally symmetric permutation of the internal coordinates so that all symmetry properties of the molecule are properly accounted for. $\Delta f_{ijk}$ are the expansion coefficients which are found by refinement. The Hamiltonian is now given as

$H = T + V + \Delta V = H_0 + \sum_{ijk...} \Delta f_{ijk...} \left(\xi_1^i \xi_2^j \xi_3^k \right)^A$

where $H_0$ is the initial Hamiltonian with ab initio PES.

If the eigenvalue problem for the initial Hamiltonian has been solved,

$H_0 \psi^{J,\Gamma}_{0,i} = E^{J,\Gamma}_{0,i} \psi^{J,\Gamma}_{0,i}$

where $J$ is the total angular momentum quantum number and $\Gamma$ is a symmetry label, then matrix elements of $H$ , using the $H_0$ solutions as a basis, are

$\langle \psi^{J,\Gamma}_{0,i} | H |\psi^{J,\Gamma}_{0,i'} \rangle = E^{J,\Gamma}_{0,i} + \sum_{ijk...} \Delta f_{ijk...} \Xi_{i,i'}^{J, \Gamma}$

where

$\Xi_{i,i'}^{J, \Gamma} = \langle \psi^{J,\Gamma}_{0,i} | \left(\xi_1^i \xi_2^j \xi_3^k ...\right)^A | \psi^{J,\Gamma}_{0,i'} \rangle.$

The derivatives of the energies with respect to adjustable parameters, which are required for least squares fitting, are given by the Hellman-Feynman theorem

$\frac{\partial E^{J,\Gamma}_{n} }{ \partial \Delta f_{ijk...} } = \langle \psi^{J,\Gamma}_{n} \left| \frac{\partial \Delta V}{\partial \Delta f_{ijk...} } \right |\psi^{J,\Gamma}_{n} \rangle = \langle \psi^{J,\Gamma}_{n} \left| \left(\xi_1^i \xi_2^j \xi_3^k ...\right)^A \right| \psi^{J,\Gamma}_{n} \rangle .$

where $E^{J,\Gamma}_{n}$ and $\psi^{J,\Gamma}_{n}$ are eigenvalues and eigenvectors of $H$ respectively. In the J=0 representation $\psi^{J,\Gamma}_{n}$ is given by

$\psi^{J,\Gamma}_{n} = \sum_i C_i^{J, \Gamma} \psi_{0,i}^{J, \gamma}$

Since TROVE typically uses linearised coordinates $\xi^{\rm lin}_i$ to represent PESs internally, i.e. in most cases different from the analytic representation of the ab initio PES, typically given in terms of some valence coordinates $r_\lambda$ , the TROVE eigenfunctions $\psi^{J,\Gamma}_{n}$ as well as basis functions $\psi^{J,\Gamma}_{0,n}$ are also solved and represented in terms of $\xi^{\rm lin}_i$ . Therefore, in order ro evaluate the integrals $\langle \psi^{J,\Gamma}_{0,i} | \left(\xi_1^i \xi_2^j \xi_3^k ...\right)^A | \psi^{J,\Gamma}_{0,i'} \rangle$ , each combination $\left(\xi_1^i \xi_2^j \xi_3^k ...\right)^A$ has to be Taylor re-expanded in terms of the linearised coordinates $\xi^{\rm lin}_i$ .

In the fittings the derivatives of the energy levels with respect to the correction parameters are evaluated at each fitting iteration in line with the standard least squares fitting Newton procedures. This is all implemented in TROVE.

How to do refinement

Setting up Refinement

The specific inputs and checkpoint files required to carry out refinement of a PES using TROVE are discussed in this section.

Prior to refinement, TROVE requires checkpoint files and eigenfunctions for the basis set being used (see above). If a calculation of the rotational-vibrational levels using an unrefined PES has already been carried out, then all necessary files for refinement will have been generated. Refinement can be carried out in the $J=0$ basis.

As explained above, refinement in TROVE is represented as a correction $\Delta V(r)$ to the ab initio PES $V(r)$ an represented by refinement parameters $\Delta f_{ijk...}$ . In the current TROVE implementation, the refinement part $\Delta V(r)$ is required to have exactly the same analytic representation as $V(r)$ , i.e. the refined PES is represented by the expansion parameters $f'_{ijk...}$ given by

$f'_{ijk...} = f^{\rm ai}_{ijk...} + \Delta f_{ijk...}$

While the potential is defined in the POTEN block, the refined PES the external block on the TROVE input file. This is the same structure as used to define the dipole moment for intensity calculations and can assume a vector structure of dimension $D$ , for example in the case of DMS, the dimension is 3. For the refinement, each expansion term $\left(\xi_1^i \xi_2^j \xi_3^k ...\right)^A$ is treated as an independent function and thus the external field is represented as a vector of dimension $N$ , where $N$ is the number of expansion parameters $\Delta f_{ijk...}$ and each vector elements holds a combination $\left(\xi_1^i \xi_2^j \xi_3^k ...\right)^A$ .

Thus the structure of the external parameter section is just a repeat of the potential block.

..Note:: Only linear parameters like $\Delta f_{ijk...}$ can be fitted in TROVE. Non-linear parameters such as equilibrium positions, structural parameters currently cannot be refined in TROVE.

A typical fitting external section has the following form

external
dimension 102
Nparam  1
compact
TYPE  potential
COEFF   list  (powers or list)
dstep   0.005
Order   4
COORDS  morse morse linear
parameters
RE13            1.5144017558        fix
alphae          92.00507388         fix
a               0.127050746200E+01  fix
b1              0.500000000000E+06  fix
b2              0.500000000000E+05  fix
g1              0.150000000000E+02  fix
g2              0.100000000000E+02  fix
V0              0.0000000000000000
F_0_0_1     0.0              fit
F_1_0_0     0.0              fit
F_0_0_2    -0.173956405672E+05 fit
F_1_0_1     0.241119856834E+04 fit
F_1_1_0     0.0
....
end

Here

dimension is the number of all parameters $N$ of the PEF (number of lines between the cards parameters and end).

Nparam tells TROVE that each component of the $N$ dimensional external field ( $\left(\xi_1^i \xi_2^j \xi_3^k ...\right)^A$ ) is a single parameter object. Compare this with DIPOLE which can have 3 components each of which is an $N_i$ -dimensional analytic expansion ( $i=`1..3$ ) represented with $N_i$ parameters.

compact is the format without a fitting weight in the column before the parameter values (penultimate).

type in the case of the fitting must be potential (in the current implementation), which tells TROVE to refer to the functional type of the PEF (POT_TYPE), see Potential energy functions.

coeff is the card specifying whether that the parameters are given as a list with predefined order as implemented in the code or via list with powers-s (exponents). Here we use the list form. .. Note:: Although Coords in external does not have to coincide with that in POTEN, it is advised to used the same type in both fields for consistency.

dstep defines the derivation step size used to evaluate high order derivatives with finite differences.

Order defines the re-expansion order of each $\left(\xi_1^i \xi_2^j \xi_3^k ...\right)^A$ term.

Coords defines the types of the linearised coordinates used in the re-expansion.

parameters indicates the beginning of the section with parameters.

fix and fit are the keywords to distinguish the parameters to fix and parameters to fit. It is important that all structural parameters are marked with the fix card. This will insure that the derivatives and expansions of $\left(\xi_1^i \xi_2^j \xi_3^k ...\right)^A$ are evaluated correctly. fit needs to be set only to the parameters that will need t o be varied.

Note

It is important to fix all structural parameters in the external section. For the example above, the potential function it is linked to has the pot_type POTEN_XY2_MORSE_COS and is defined as follows

POTEN
NPARAM  102
POT_TYPE  POTEN_XY2_MORSE_COS
compact
COEFF  list  (powers or list)
RE13              1.5144017558
alphae            92.00507388
a                 0.127050746200E+01
b1                0.500000000000E+06
b2                0.500000000000E+05
g1                0.150000000000E+02
g2                0.100000000000E+02
V0                0.000000000000E+00
F_0_0_1           0.000000000000E+00
F_1_0_0           0.000000000000E+00
F_0_0_2           0.173956405672E+05
F_1_0_1          -0.241119856834E+04
F_1_1_0           0.223873811001E+03
...
...
end

Here, the structural parameters are $r_{\rm e}$ , $\alpha_{\rm e}$ , Morse parameter $a$ as well as parameters $b_1$ , $b_2$ , $g_1$ , $g_2$ . The must be fixed to their values when doing the re-expansion of the external part.

Here is another example, where the potential function type poten_C3_R_theta was used:

POTEN
compact
POT_TYPE  poten_C3_R_theta
COEFF  powers  (powers or list)
RE12          0      0      0     1.29397
theta0        0      0      0     0.000000000000E+00
f000          0      0      0        0.00000000
f100          1      0      0        0.00000000
f200          2      0      0        0.33240693
f300          3      0      0       -0.35060064
f400          4      0      0        0.22690209
f500          5      0      0       -0.11822982
.....
.....
end

The external field is then given by

external
dimension 60
compact
NPARAM  1
compact
type potential
order 8
coords morse morse   linear
COEFF  powers  (powers or list)
parameters
RE12          1      0      0        1.29397   fix
theta0        0      0      0        0.0000000 fix
f000          0      0      0        0.00000000 fit
f100          1      0      0        0.00000000 fit
f200          2      0      0        0.00000000 fit
f300          3      0      0        0.00000000 fit
f400          4      0      0        0.00000000 fit
f500          5      0      0        0.00000000 fit
....
....
end

The fitted potential parameters in the external section can be assumed to be zero but never actually featured until step 4, so the actual values won’t matter at steps 1,2,3.

Calculation steps

At step 1 and 2, for the external field to be processed, the control block has to include the card external:

Step 1:

Control
Step 1
external
end

Step 2:

Control
Step 2
end

See TROVE Quick start.

Step 3 does not involve any operations with the external field and therefore should be processed as usual, e.g.

Control
Step 3
J 1
end

As discussed above, the refinement procedure requires matrix elements of the $H_0$ Hamiltonian and so eigenfunctions for each $J$ of interest must be computed.

After step 3, in the case of the refinements, in the control block we skip step 4 (intensities) and start step 5 (fitpot), at which matrix elements of the expansion terms $\left(\xi_1^i \xi_2^j \xi_3^k ...\right)^A$ are computed on the final ro-vibrational eigenfunctions obtained at step 3 for the ab initio model, for all values of $J$ and all symmetries considered. A typical Step 5 control block has the following structure:

Control
  Step 5  (FITPOT)
  external  3 60
  J  0, 1, 2, 3
  symmetries 1, 2, 3, 4
end

Here,

3-60 in the external is the range of the expansion terms (i.e. corresponding to the expansion parameters $f^{\rm ai}_{ijk...}$ ) to be processed.
J card lists all values of $J$ to be processed. Note that it is not a range but a list i.e. the parameters can appear in any combination or order. Alias Jrot.
symmetries card (alias gamma) is similar to the gamma card used in step 3. It gives a list of symmetries to be processed, again, in any combination or order.

An alias for step 5 is step fitpot. At this step, checkpoint files fitpot-J-Gamma-n.chk containing all matrix elements required for each $J$ , symmetry $\Gamma$ , for each expansion parameter $n$ are generated. Since a file is generated for each expansion parameter n, many files are generated in this step.

At step 6 (alias step refinement), the actual fits are taking place. At this step, the control block will have similar format as for step 5:

Control
  Step 6  (refinement)
  external  3 60
  J  0, 1, 2, 3
  symmetries 1, 2, 3, 4
end

or simply with Step refinement:

Control
  Step refinement
  external  3 60
  J  0, 1, 2, 3
  symmetries 1, 2, 3, 4
end

Fitting block

At step 6, additionally to the control block change, the user needs to include the fitting section. Here is an example of a fitting block used for SiH₂:

FITTING
itmax   0
fit_factor     1e4
geometries     poten.dat
output         f01
robust  0.0001
lock           100
target_rms     1e-18
fit_scale      0.25
thresh_obs-calc  10
OBS_ENERGIES
    0    1    1      0.00000  0  0   0   0   1.00
    0    1    3   1978.1533   0  0   0   2   1.00
    0    1    4   2005.469    0  1   0   0   1.00
    0    1    5   2952.7      0  0   0   3   1.00
    0    1    6   2998.6      0  1   0   1   1.00
    0    1    7   3907.4      0  1   0   2   1.00
.....
end

Here

itmax is the number of iterations of refining carried out. itmax 0 means no refinement and used for one straight-through calculation for checking purposes. TROVE will carry out refinement until the number of iterations specified is reached.

fit_factor is the relative weighting for the experimental data compared to ab initio energies $\omega$ in Eq. (1). The larger this is, the more importance will be given to the experimental energies. We initial value is usually of the order of 0.01 to 1, which is gradually increased to about 100000.

geometries is the name of the file which contains energies of an ab initio PES used in the constrained fit. This file should give geometries in the same valence coordinates as specified by the potential energy surface for the molecule of interest in TROVE followed by the ab initio energy (from MOLPRO for example) and a weighting. The format is explained below.

output is a string which specifies the pre-fix for auxiliary output file names, .en and .pot.

robust specifies whether Watson Robust Fit (WRF) is used, for 0.0 it is not, for 0.0001 it is. The main function of WRF is to control and remove outliers, but can be also used to adjust the weights according to the real uncertainty of the energy levels. The non-zero values also indicate how tight the robust weighting should distinguish between good and very good uncertainties. Currently, this is a trial-and-error parameter. A good staring value is about 0.0001.

lock (aka assignment) is the card specifying if the quantum numbers will be used to match the experimental and theoretical energies: zero means that assignment is not used. By default, the energies are matched using $J$ , symmetry $\Gamma$ and the running number $N$ . $J$ , $\Gamma$ and $N$ give a unique ID for all TROVE ro-vibrational energies. However experimental energies use quantum numbers as unique identifiers and thus need to be matched to the TROVE values, which must be done by manually checking the experimental and theoretical values stored in the auxiliary .en file. The disadvantage of the running state numbers as unique IDs $N$ is that they can change though the fit, which is a very common problem. If the lock value is not zero, TROVE will use an automatic matching using the TROVE quantum numbers and will “lock” its matching to the given state through the fit, regardless of if the running number will change. The lock value is in this case is used a threshold to match the quantum numbers. For example, lock 100 means that TROVE will attempt to find a QN match within 100 cm^-1 from the value associated with $N$ . $J$ , $\Gamma$ and $N$ .

target_rms is to value of the RMS error to terminate the fit when archived. In practice however, the desired RMS error is rarely achieved.

fit_scale is the parameter used to scale down the Newton-Raphson increment by this factor. fit_scale 1 means the full increment is used, while a smaller value should make the slower but more stable. It is especially useful when the parameters are strongly correlated and has the potential even to work with over-defined problems.

thresh_obs-calc is the threshold (cm^-1) to exclude accidental outliers from the fit. It is a common situation that in the middle of the fit, the state assignment of the calculated energies changes from the inial description, whether it is the running or the full set of quantum numbers are used, leading to a large residual and thus driving the fit to the wrong direction. The most reasonable approach is to exclude such an outlier from the current fit on the fly, let the process finish and then worry about the re-assignment later, before the next fit. For an almost converged fit, a typical thresh_obs-calc value is 2-5 cm^-1. For the initial stage, a recommended value is about 10-20 cm ^-1.

OBS_ENERGIES is the card indicating the beginning of the list with experimental (observed) energies.

Below is an example of a list of energies as an illustration of the format.

fitting
.......
.....
OBS_ENERGIES
  0    1    1      0.00000  0  0   0   0   1.00
  0    1    3   1978.1533   0  0   0   2   1.00
  0    1    4   2005.469    0  1   0   0   1.00
  0    1    5   2952.7      0  0   0   3   1.00
  0    1    6   2998.6      0  1   0   1   1.00
  0    1    7   3907.4      0  1   0   2   1.00
  0    1    8   3923.3      0  0   2   0   1.00
  0    1    9   3976.8      0  0   0   4   1.00
  0    1   10   3997.5      0  1   1   0   1.00
  0    4    1   1992.816    0  0   1   0   1.00
  1    2    1   11.801      2  0   0   0   1.00
  1    2    2   1010.64     2  0   0   1   1.00
  ......
  ......
end

The meaning of the columns is as follows.

   .......
   OBS_ENERGIES
   ---- ---- ---  ----------- -- -- --- ---- -----
     1    2    3       4       5  6   7   8    9
   ---- ---- ---  ----------- -- -- --- ---- -----
     0    1    1      0.00000  0  0   0   0   1.00
     0    1    3   1978.1533   0  0   0   2   1.00
     0    1    4   2005.469    0  1   0   0   1.00
     0    1    5   2952.7      0  0   0   3   1.00
     0    1    6   2998.6      0  1   0   1   1.00
     0    1    7   3907.4      0  1   0   2   1.00
     0    1    8   3923.3      0  0   2   0   1.00
     0    1    9   3976.8      0  0   0   4   1.00
     0    1   10   3997.5      0  1   1   0   1.00
     0    4    1   1992.816    0  0   1   0   1.00
     1    2    1   11.801      2  0   0   0   1.00
   ---- ---- ---  ----------- -- -- --- ---- -----


- col 1: Rotational angular momentum :math:`J` (rigourous QN);
- col 2: A symmetry count :math:`\Gamma`, e.g. for 1,2,3,4 for :math:`A_1`, :math:`A_2`, :math:`B_1` and :math:`B_2`, respectively in C :sub:`2v`(M);
- col 3: A block number, i.e. a state counting number of the states with the same :math:`J`, :math:`\Gamma`, sorted by energy.
- col 4: Experimental energy term values (cm\ :sup:`-1`) relative to ZPE.
- col 5: Rotational QN :math:`K` (non rigourous), assuming the TROVE assignment.
- col 6-8: Vibrational TROVE QNs :math:`v_1`, :math:`v_2`, :math:`v_3` etc. (non rigourous), assuming  the TROVE assignment.
- col 9: Fitting weight, which is usually inverse proportional to the experimental uncertainty of the state, but can be manipulated to influence the fit.

The format of the geometry file is as illustrated in the example below:

  -----  -------- --------------- --------- --------------
     2           3               4         5
  -----  -------- --------------- --------- --------------
520   1.520     1.570796327      0.0000     1.000000
520   1.520     1.649336143      0.3255     1.000000
520   1.500     1.649336143     13.7810     1.000000
500   1.520     1.649336143     13.7810     1.000000
520   1.500     1.570796327     18.2147     1.000000
500   1.520     1.570796327     18.2147     1.000000
520   1.520     1.675516082     47.3732     1.000000
500   1.520     1.675516082     59.5502     1.000000
.....

where

col 1-3: geometries in the input (usually valence) coordinates, the same as used to define the TROVE internal coordinates, in Angstrom for the bond lengths and radians for the angles for all $M=3N-6$ vibrational degrees of freedom.
col 4: Values of the reference “ab initio” PES for each geometry (cm^-1);
col 5: Fitting weights; usually estimated using the form suggested by Partridge and Schwenke, [97PaSc]

(2) $w_i=\left(\frac{\tanh\left[-0.0006\times(\tilde{E}_i {\rm cm} - 15{\,}000)\right]+1.002002002}{2.002002002}\right)\times\frac{1}{N \tilde{E}_i^{(w)} {\rm cm}}$

Refinement Output

The refinement procedure produces three output files. A regular .out file with a prefix the same as the .inp file and two auxiliary files .pot file and .en with prefixes as determined by the name given in the output keyword in the Fitting block.

The main output file for refinement is straightforward. The input is repeated as with other TROVE output files and then some information is given about the eigenfunctions which were read in, etc. After this TROVE prints the iteration number and then a list comparing the observed to calculated energies. For example

----------------------------------------------------------------------------------------------------
|## |  N |  J | sym|      Obs.    |    Calc.   | Obs.-Calc. |   Weight |  K   vib. quanta
----------------------------------------------------------------------------------------------------
  1    0  A1         0.0000       0.0000       0.0000   0.38E-05  (  0) (  0  0  0)
  3    0  A1      1978.1533    1981.4636      -3.3103   0.38E-05  (  0) (  0  0  2)
  4    0  A1      2005.4690    2008.9579      -3.4889   0.38E-05  (  0) (  1  0  0)
  5    0  A1      2952.7000    2956.2565      -3.5565   0.38E-05  (  0) (  0  0  3)
  6    0  A1      2998.6000    3003.6017      -5.0017   0.38E-05  (  0) (  1  0  1)
  7    0  A1      3907.4000    3914.4824      -7.0824   0.38E-05  (  0) (  0  2  0)*
  8    0  A1      3923.3000    3930.2323      -6.9323   0.38E-05  (  0) (  0  2  0)
  9    0  A1      3976.8000    3982.5964      -5.7964   0.38E-05  (  0) (  1  1  0)*
 10    0  A1      3997.5000    4003.2424      -5.7424   0.38E-05  (  0) (  1  1  0)
  1    0  B2      1992.8160    1996.2817      -3.4657   0.38E-05  (  0) (  0  1  0)

where: ## is the counting number of the experimental entries; N is the TROVE block number (counting number with $J$ and $\Gamma$ ); J is the rotational angular momentum $J$ ; sym is the irrep in the symmetry group of the molecule in question; Obs. is the experimental energy term value (cm^-1); Calc. is the calculated TROVE energy term value (cm^-1); Obs.-Calc. is the residual (cm^-1); Weight is the fitting weight value. This is modified from the input value, first by re-normalising all experimental weights to sum to 1, then scaling wih a fit_factor $\omega$ and then renormalising them together with the ab initio weights (see below), which iniially also normalised to 1. When the fit starts, these weights are also adjusted through the Robust Watson re-weighting procedure, which is then printed in this output at each iteration; K is the TROVE rotational QN; vib. quanta are the TROVE vibrational quantum numbers.

If an asterisk (*) is printed at the end of the row (as in the first row of this example) it means that TROVE has assigned the state differently to how it was labelled in the input in the Fitting block.

The energy output is followed by three blocks of the potential parameters.

The first block lists corrections $\Delta f_{ijk...}$ to the potential parameters is as follows:

Correction to potential parameters:
RE13          0.15144017558000E+01    fix
ALPHAE        0.92005073880000E+02    fix
AA            0.12705074620000E+01    fix
B1            0.50000000000000E+06    fix
B2            0.50000000000000E+05    fix
G1            0.15000000000000E+02    fix
G2            0.10000000000000E+02    fix
V0            0.00000000000000E+00
F_0_0_1       0.00000000000000E+00    fit
F_1_0_0       0.00000000000000E+00    fit
F_0_0_2      -0.17395640567200E+05    fit
F_1_0_1       0.24111985683400E+04    fit
F_1_1_0       0.00000000000000E+00
F_2_0_0       0.00000000000000E+00

The second block lists new values for the potential parameters $f'_{ijk...} = f_{ijk...} + \Delta f_{ijk...}$ at the current iteration:

Potential parameters:
RE13          0.15144017558000E+01
ALPHAE        0.92005073880000E+02
AA            0.12705074620000E+01
B1            0.50000000000000E+06
B2            0.50000000000000E+05
G1            0.15000000000000E+02
G2            0.10000000000000E+02
V0            0.00000000000000E+00
F_0_0_1       0.00000000000000E+00
F_1_0_0       0.00000000000000E+00
F_0_0_2      -0.77718868851662E-13
F_1_0_1      -0.16208272427320E-12
F_1_1_0       0.22387381100100E+03
F_2_0_0       0.38563857069600E+05

which is followed the corrections $\Delta f_{ijk...}$ again, but with their standard errors and also rounded according to their standard error:

Potential parameters rounded in accord. with their standard errors

RE13      -1              1.51440176
ALPHAE    -1             92.00507388
AA        -1              1.27050746
B1        -1         500000.00000000
B2        -1          50000.00000000
G1        -1             15.00000000
G2        -1             10.00000000
V0         0                    0.00
F_0_0_1    1                    -18.(            11)
F_1_0_0    1                   -109.(           242)
F_0_0_2    1                 -13218.(           390)
F_1_0_1    1                   1176.(          1047)
F_1_1_0    0                    0.00
F_2_0_0    0                    0.00
F_0_0_3    0                    0.00

The standard errors in the parentheses are understood as the last figures of the value shown, e.g. $-18.\pm 0.11$ . The standard errors are computed using the diagonal elements of the correlation matrix.

The fitting iteration output is finished with a table which gives summary details on the fit for this iteration.

----------------------------------------------------------------------------------------
|  Iter  | Points | Params |    Deviat     |     ssq_ener  |    ssq_pot  | Convergence |
----------------------------------------------------------------------------------------
|      2 |   1924 |      4 |   0.46480E+02 |  0.30230E+01  |   0.289E+04 |  0.319E+00  |
----------------------------------------------------------------------------------------

This gives the statistics of the fit including both the experimental energies (ssq_ener) and the ab initio energies (ssq_pot) used to constrain the fit as well as the total weighted standard deviation (ab initio + experiment). The latter is usually less informative because of the weighted character and a large ab initio error contribution. The most informative number in this table is ssq_ener. The Obs-Calc table, tables with potential parameters and the fit statistics are then repeated for each iteration.

Auxiliary files

To help with the refinement process, two auxiliary files are created.

The .en file has a similar purpose as the Obs-Calc table in the output file but gives all calculated energies for all states calculated by TROVE. This file is very useful when matching the experimental energies to the calculated TROVE values. It is also useful for spotting and sorting out state swaps, i.e, when accidental assignment mismatches happen. It is relatively straightforward to identify which state a mismatched/replaced experimental value should be reassigned to in order to fix the match.

The .en printout generally repeats the format of the Obs-Calc table in the output:

 ----------------------------------------------------------------------------------------------------
|## |  N |  J | Sym|     Obs.    |    Calc.   | Obs.-Calc. |   Weight |    K    quanta   (Calc./Obs.)
 ----------------------------------------------------------------------------------------------------
    1    1    0  A1         0.0000       0.0000       0.0000   0.38E-05  ( A1 ;  0 ) ( A1 ;  0  0  0 )(  0  0  0  0)
    2    0    0  A1         0.0000     999.8872       0.0000   0.00E+00  ( A1 ;  0 ) ( A1 ;  0  0  1 )
    3    3    0  A1      1978.1533    1981.4636      -3.3103   0.38E-05  ( A1 ;  0 ) ( A1 ;  0  0  2 )(  0  0  0  2)
    4    4    0  A1      2005.4690    2008.9579      -3.4889   0.38E-05  ( A1 ;  0 ) ( A1 ;  1  0  0 )(  0  1  0  0)
    5    5    0  A1      2952.7000    2956.2565      -3.5565   0.38E-05  ( A1 ;  0 ) ( A1 ;  2  0  1 )(  0  0  0  3)*

with additional QNs after in the last columns. They show the “experimental” QNs, i.e. QNs from the input file. This is to help with (re-)assignment and (re)-matching. In the case the QNs do not match, am asterisk (*) is added. It is also added if the residual obs-calc is too large.

The .pot file is a ab initio counterpart of the .en file. It list ab initio PES energies with the corresponding geometries from the geometry file used for constraining the fit. The calculated PEF values are compared to the ab initio once and the differnes are printed (cm^-1), together with the fitting ab initio weights.

Here is an example of a .pot file:

                                                       Ref.         Calc.        Ref.-calc     weight
520000000         1.520000000         1.570796327   0.0000       27.487        -27.48678   0.5661E-03
520000000         1.520000000         1.649336143  0.32550       26.223        -25.89790   0.5661E-03
520000000         1.500000000         1.649336143   13.781       40.492        -26.71074   0.5661E-03
500000000         1.520000000         1.649336143   13.781       40.492        -26.71074   0.5661E-03
520000000         1.500000000         1.570796327   18.215       46.703        -28.48839   0.5661E-03
500000000         1.520000000         1.570796327   18.215       46.703        -28.48839   0.5661E-03
520000000         1.520000000         1.675516082   47.373       72.300        -24.92634   0.5661E-03
500000000         1.520000000         1.675516082   59.550       85.239        -25.68921   0.5661E-03

where the first $M$ columns (number of the vibrational degrees of freedom) list the valance coordinates as defined in the geometry file, followed the ab initio or “reference” (Ref.) PES values, calculated (Calc.) values, the differences Ref.-Calc. and the fitting weights (after being normalised within the ab initio set, combined with the experimental weights and re-normalised to 1 again).

Watson Robust fitting

In progress