Potential energy functions¶

TROVE provides a larger number of potential energy functions (PEFs) for different molecules already implemented. Most of these PEFs are in modules pot_* contained in file pot_*.f90.

pot_xy2.f90

pot_xy3.f90

pot_zxy2.f90

pot_abcd.f90

pot_xy4.f90

pot_zxy3.f90

These are a part of the standard TROVE compilation set. Alternatively, a user-defined PEF can be included into the TROVE compilation as a generic ‘user-defined’ module pot_user, see details below.

Potential Block¶

The Potential (Poten) block used to specify a PEF, has the following generic structure (using XY₂ as an example)

POTEN
NPARAM  102
POT_TYPE  POTEN_XY2_MORSE_COS
COEFF  list  (powers or list)
 RE13         1  0.15144017558000E+01
 ALPHAE       1  0.92005073880000E+02
 AA           1  0.12705074620000E+01
 B1           1  0.50000000000000E+06
 B2           1  0.50000000000000E+05
 G1           1  0.15000000000000E+02
 G2           1  0.10000000000000E+02
 V0           0  0.00000000000000E+00
 F_0_0_1      1 -0.11243403302598E+02
 F_1_0_0      1 -0.94842865087918E+01
 F_0_0_2      1  0.17366522840412E+05
 F_1_0_1      1 -0.25278354456474E+04
 F_1_1_0      1  0.20295521820240E+03
 F_2_0_0      1  0.38448640879698E+05
 F_0_0_3      1  0.27058767090918E+04
 F_1_0_2      0 -0.47718397149800E+04
...
end

For an example, see SiH2_XY2_MORSE_COS_step1.inp where this PES is used.

Here NPARAM is used to specify the number of parameters used to define the PES. POT_TYPE is the name of the potential energy surface being used which is defined in the pot_*.f90 file. The keywords COEFF indicates if the potential contains a list of parameter values (LIST) or values with the corresponding expansion powers (POWERS), e.g. (for H₂CO):

POTEN
NPARAM  114
POT_TYPE  poten_zxy2_mep_r_alpha_rho_powers
COEFF  powers
f_0_0      0 0 0 0 0 0   1  0.00000000000000E+00
f_0_1      1 0 0 0 0 0   1  0.00000000000000E+00
f_0_1      0 1 0 0 0 0   1  0.00000000000000E+00
f_0_2      0 0 0 1 0 0   1  0.00000000000000E+00
f_1_1      0 0 0 0 0 1   1  0.13239727881219E+05
f_2_1      0 0 0 0 0 2   1  0.46279621687684E+04
f_3_1      0 0 0 0 0 3   1  0.14394787420943E+04
f_4_1      0 0 0 0 0 4   1  0.10067584554678E+04
f_5_1      0 0 0 2 0 0   1  0.31402651686299E+05
.....
end

which is from the variational calculations of H₂CO, see the TROVE input 1H2-12C-16O__AYTY__TROVE.inp.

The potential parameters are listed after the keyword COEFF and terminated with the keyword END with exactly NPARAM. For the COEFF list option, the meaning of the columns is as follows:

Label

Index

Value

b1

0

0.80000000000000E+06

b2

0

0.80000000000000E+05

g1

0

0.13000000000000E+02

g2

0

0.55000000000000E+01

f000

0

0.00000000000000E+00

f001

1

0.25298724728304E+01

f100

1

0.76001446034650E+01

Here ‘Labels’ are the parameter names, used only for printing purposes and not in any calculations. The ‘Index’ field can be used to as a switch to indicate if the corresponding parameter was fitted or can be fitted. Otherwise it has no impact on any evaluations of the PEF values. ‘Values’ are the actual potential parameters, listed in the order implemented in the corresponding PEF POT_TYPE, for example

$\begin{split} V(r_1,r_2,\alpha) &= f_{000} + f_{001} y_3 + f_{100} [ y_1 + y_2 ] + f_{100} [ y_1 + y_2 ] + f_{002} y_3^2 + \ldots + \\ & + b_1 e^{-g_1 r_{\rm HH}} + b_2 e^{-g_2 r_{\rm HH}^2} \\ \end{split}$

where

$\begin{split} y_1 & = 1-e^{-a (r_1 - r_{\rm e})}, \\ y_2 & = 1-e^{-a (r_2 - r_{\rm e})}, \\ y_3 &= \cos\alpha-\cos\alpha_{\rm e}, \\ r_{HH}=\sqrt{r_1^2+r_2^2-2 r_1 r_2 \cos\alpha}.\\ \end{split}$

For the COEFF Powers option, the meaning of the columns is as follows:

Label

n1

n2

n3

n4

n5

n6

Index

Value

f000000

0

0

0

0

0

0

1

0.00000000000000E+00

f100000

1

0

0

0

0

0

1

0.00000000000000E+00

f010000

0

1

0

0

0

0

1

0.00000000000000E+00

f000100

0

0

0

1

0

0

1

0.00000000000000E+00

f000001

0

0

0

0

0

1

1

0.13239727881219E+05

f000002

0

0

0

0

0

2

1

0.46279621687684E+04

f000003

0

0

0

0

0

3

1

0.14394787420943E+04

where

‘Labels’ are the parameter name, for printing purposes only;

‘n1’, ‘n2’, ‘n3’, … are the ‘powers’ of an expansion term, e.g. $V(r_1,r_2,r_3,r_4,r_5, r_6) = \sum_{n_1,n_2,n_3,n_4,n_5,n_1} f_{n_1,n_2,n_3,n_4,n_5,n_1} \xi_1^{n_1} \xi_2^{n_2} \xi_3^{n_3} \xi_4^{n_4} \xi_5^{n_5} \xi_6^{n_6}$

‘Index’ is a switch to indicate if the corresponding parameter was fitted or can be fitted, with no impact on any evaluations of the PEF values.

‘Values’ are the actual potential parameters. Their order is not important for this implementation as long as the corresponding powers are defined.

In case the definition of PEF requires also structural parameters, such as equilibrium bond lengths $r_{\rm e}$ , equilibrium inter-bond angles $\alpha_{\rm e}$ , Morse exponents $a$ etc., in the COEFF Powers form these parameters should be listed exactly in the order expected by the implemented of the PEF (similar to the COEFF LIST form), but with dummy “powers” columns so that their ‘values’ appear in the right column. For example:

POTEN
NPARAM  58
POT_TYPE  poten_C3_R_theta
COEFF  powers  (powers or list)
RE12          0      0      0      0     1.29397
theta0        0      0      0      0     0.000000000000E+00
f200          2      0      0      0        0.33240693
f300          3      0      0      0       -0.35060064
f400          4      0      0      0        0.22690209
f500          5      0      0      0       -0.11822982
.....

Here, RE12 and theta0 are two the equilibrium values and the three columns with 0 0 0 are given in order to parse their values using column 6.

Implemented PEFs¶

XY₂ type¶

There are several PEFs available for this molecule type.

`POTEN_XY2_MORBID`¶

This form is given by

$\begin{split} V(r_1,r_2,\alpha) &= f_{000} + f_{001} y_3 + f_{002} y_3^2 + f_{003} y_4 + \ldots \\ & + (f_{100} + f_{101} y_3 + f_{102} y_3^2 + f_{103} y_4 + \ldots) [y_1 + y_2] \\ & + (f_{200} + f_{201} y_3 + f_{202} y_3^2 + f_{203} y_4 + \ldots) [y_1^2 + y_2^2] \\ & + (f_{110} + f_{111} y_3 + f_{112} y_3^2 + f_{113} y_4 + \ldots) y_1y_2 \\ & + \ldots \\ & + b_1 e^{-g_1 r_{\rm HH}} + b_2 e^{-g_2 r_{\rm HH}^2} \\ \end{split}$

where

$\begin{split} y_1 & = 1-e^{-a (r_1 - r_{\rm e})}, \\ y_2 & = 1-e^{-a (r_2 - r_{\rm e})}, \\ y_3 &= \cos\alpha-\cos\alpha_{\rm e}, \\ r_{HH}=\sqrt{r_1^2+r_2^2-2 r_1 r_2 \cos\alpha}.\\ \end{split}$

For an example, see SO2_morbid_step1.inp where this PES is used.

`POTEN_XY2_MORSE_COS`¶

For description and example see ‘Potential Block’ above with the input file example for SiH₂ SiH2_XY2_MORSE_COS_step1.inp. This PES was used in [21ClYu}_ .

`POTEN_XY2_TYUTEREV`¶

POTEN_XY2_TYUTEREV is the essentially the same PES as POTEN_XY2_MORSE_COS, with the difference that the POTEN input part does not contain the structural parameters:

POTENTIAL
NPARAM  99
POT_TYPE  poten_xy2_tyuterev
COEFF  list
b1        0    0.80000000000000E+06
b2        0    0.80000000000000E+05
g1        0    0.13000000000000E+02
g2        0    0.55000000000000E+01
f000      0    0.00000000000000E+00
f001      1    0.25298724728304E+01
f100      1    0.76001446034650E+01
...
end

Using the structural parameters in the POTEN section is important for PES refinements, see the corresponding section for details. The input file example is h2s_step1.inp .

`poten_co2_ames1`¶

An empirical PES of CO₂ is from [17HuScFr]. The input file example is CO2_bisect_xyz_step1.inp where this PES is used.

It is programmed using the powers format as follows:

POTEN
NPARAM  234
POT_TYPE  poten_co2_ames1
COEFF  powers  (powers or list)
r12ref    0 0 0  1    1.16139973893
alpha2    0 0 0  1    1.0000000000
De1       0 0 0  1    155000.0000
De2       0 0 0  1    40000.0000
Ae1       0 0 0  1   40000.0000
Ae2       0 0 0  1   20000.0000
edamp2    0 0 0  1   -2.0000
edamp4    0 0 0  1   -4.0000
edamp5    0 0 0  1   -0.2500
edamp6    0 0 0  1   -0.5000
Emin      0 0 0  1  -0.000872131085d0
Rmin      0 0 0  1  0.1161287540428520D+01
rminbohr  0 0 0  1  0.2194515245360671D+01
alpha     0 0 0  1  1.00000000000000000000
rref      0 0 0  1  1.1613997389
f000    0    0    0  0    0.3219238090183E+01
f001    0    0    1  0   -0.2181831901567E+02
f002    0    0    2  0    0.7355163655139E+02
f003    0    0    3  0   -0.1531231748456E+03
f004    0    0    4  0    0.2090079612238E+03
f005    0    0    5  0   -0.1883325770080E+03
......
end

The first part contains some structural parameters with ‘powers’ indexes filled with dummy zeros to maintain the powers format.

Examples and References

XYZ type¶

`POTEN_XYZ_KOPUT`¶

The PEF is given by (see [22OwMiYu])

$V = \sum_{ijk} f_{ijk} \xi_1^{i} \xi_2^{j} \xi_3^{k},$

The vibrational coordinates are

$\begin{split} \xi_1 &= (r_1-r_1^{\rm eq})/r_1, \\ \xi_2 &= (r_2-r_2^{\rm eq})/r_2, \\ \xi_3 &= \alpha-\alpha_{\rm eq}, \end{split}$

where $r_1 = r_1^{\rm eq}$ and $r_2 = r_2^{\rm eq}$ are the internal stretching coordinates and $\alpha$ is the interbond angle, and the equilibrium parameters are $r_1^{\rm eq}$ , $r_2^{\rm eq}$ and $\alpha_{\rm eq}$ . Note that the exponent $k$ associated with the bending coordinate $\xi_3$ assumes only even values because of the symmetry of the XYZ molecule.

The input file example is CaOH_Koput_step1.inp where this PES is used.

ZXY₂ type¶

`poten_zxy2_morse_cos`¶

The PEF is expressed analytically as (see [15AlOvPo], [23MeOwTe])

$V = \sum_{ijklmn} a_{ijklmn} \xi_{1}^{i} \xi_{2}^{j} \xi_{3}^{k} \xi_{4}^{l} \xi_{5}^{m} \xi_6^{n},$

The vibrational coordinates are

$\begin{split} \xi_i &= 1 - \exp[-b_i(r_i-r_i^{\rm e})],\quad i={\rm 0, 1, 2}, \\ \xi_4 &= \alpha_1-\alpha_{\rm e}, \\ \xi_5 &= \alpha_2-\alpha_{\rm e}, \\ \xi_6 &= 1+\cos\tau. \end{split}$

where $a_{ijklmn}$ are the expansion parameters. Here, $r_0$ , $r_1$ and $r_2$ are the bond lengths, $\alpha_{1}$ and $\alpha_{2}$ are the bond angles, $\tau$ is the dihedral angle between two bond planes, and $b_i$ is a Morse-oscillator parameter.

The expansion has a symmetry adapted form, which is invariant upon interchanging the two equivalent atoms Y₁ and Y_2`, i.e. for $r_1 \leftrightarrow r_2$ simultaneously with $\alpha_1 \leftrightarrow \alpha_2$ .

The TROVE function for this PEF is MLpoten_zxy2_morse_cos, which can be found in the module mol_zxy2.f09.

poten_zxy2_morse_cos uses the powers form for the potential parameters. An input file example for H₂CS is H2CS_zxy2_morse_cos_step1.inp where this PES is used.

XY₃ type (pyramidal)¶

`POTEN_XY3_MORBID_10`¶

This PEF is for ammonia (non-rigid) molecules with an umbrella coordinate representing the 1D inversion motion and other 5 are displacements from their equilibrium values, but it can be equally used for rigid molecules like phosphine as well. As all PEFs in TROVE, POTEN_XY3_MORBID_10 is a symmetry adapted to be fully symmetric for all operations (permutations) of the D:sub: 3h(M) (and also C_3v(M)) molecular group symmetry. It uses the following coordinates

$\begin{split} \xi_i &= 1 - \exp[-a(r_i-r_i^{\rm e})],\quad i={\rm 1, 2, 3}, \\ \xi_4 &= \frac{\sqrt{2}}{\sqrt{3}} (2\alpha_1-\alpha_2-\alpha_3), \\ \xi_5 &= \frac{\sqrt{2}}{2} (\alpha_2-\alpha_3), \\ \xi_6 &= \sin\rho_{\rm e}-\sin\rho, \end{split}$

where $r_i$ ( $i=1,2,3,$ ) are three bond lengths, $\alpha_i$ ( $i=1,2,3,$ ) are three bond angles with $\alpha_i$ opposite to $r_i$ and $\rho$ is an umbrella coordinate defined as an ‘average’ angle between three bonds and an average symmetry axis as follows:

$\sin\rho = \frac{2}{\sqrt{3}}\sin\left(\frac{\bar\alpha}{2}\right)$

and

$\bar\alpha = \frac{1}{3} (\alpha_1+\alpha_2+\alpha_3).$

The corresponding Fortran function is MLpoten_xy3_morbid_10, which can found in mol_xy3.f90. It uses the Coeffs LIST form. A TROVE input example for NH₃ is NH3_BYTe_step1.inp , see [09YuBaYa] where this empirical PES was used to compute the BYTe line list for ammonia.

`POTEN_XY3_MORBID_11`¶

This is a similar PES to POTEN_XY3_MORBID_10, but with the structural parameters ( $r_{\rm e}$ , $\alpha_{\rm e}$ and $a$ (Morse parameter)) included into the POTENTIAL block:

POTEN
NPARAM   307
POT_TYPE  poten_xy3_morbid_11
COEFF  list
 re           0  0.97580369000000E+00
 alphae       0  0.11195131680000E+03
 beta         0  0.21500000000000E+01
 VE           0  0.00000000000000E+00
 FA1          1  0.19897404278336E+02
 FA2          1  0.34574086467373E+06
 FA3          1 -0.45576605893993E+06
 FA4          1  0.21457903182324E+07
 .....
end

The Fortran function is MLpoten_xy3_morbid_11, which can found in mol_xy3.f90. A TROVE input example for H₃O⁺ is 1H3-16O_p__eXeL__model-TROVE.inp from [20YuTeMi] , where it was used to compute an ExoMol line list for this molecule.

A chain molecule of HOOH type¶

`POTEN_H2O2_KOPUT_UNIQUE`¶

The HOOH PES is given by the expansion

$V = \sum_{i_1,i_2,i_3,i_4,i_5,i_6} f_{i_1,i_2,i_3,i_4,i_5,i_6} \xi_1^{i_1} \xi_2^{i_2} \xi_3^{i_3} \xi_4^{i_4} \xi_5^{i_5} \cos(i_6 \delta)$

in terms of the following coordinates:

$\begin{split} \xi_1 &= \frac{R - R_{\rm e}}{R}, \\ \xi_2 &= \frac{r_1 - r_{\rm e}}{r_1}, \\ \xi_3 &= \frac{r_2 - r_{\rm e}}{r_2}, \\ \xi_4 &= \alpha_1-\alpha_{\rm e}, \\ \xi_5 &= \alpha_2-\alpha_{\rm e}, \\ \xi_6 &= \delta, \end{split}$

where $R, r_1, r_2$ are three bond lengths, $\alpha_i$ ( $i=1,2,$ ) are two bond angles with $\alpha_i$ and $\delta$ is a dihedral angle.

This a powers type:

POTEN
NPARAM  166
POT_TYPE  POTEN_H2O2_KOPUT_UNIQUE
COEFF  powers  (powers or list)  (oo oh1  oh2 h1oo h2oo rho)
f000000       0  0  0  0  0  0  1  0.14557772800000E+01
f000000       0  0  0  0  0  0  1  0.96253006000000E+00
f000000       0  0  0  0  0  0  1  0.10108194717000E+03
f000000       0  0  0  0  0  0  1  0.11246000000000E+03
f000000       0  0  0  0  0  0  1  0.39611300000000E-02
f000001       0  0  0  0  0  1  1  0.48279121261621E-02
f000002       0  0  0  0  0  2  1  0.31552076592194E-02
f000003       0  0  0  0  0  3  1  0.27380895575892E-03
f000004       0  0  0  0  0  4  1  0.53200000000000E-04

The Fortran function is MLpoten_h2o2_koput_unique, which can found in mol_abcd.f90. A TROVE input example for HOOH is HOOH_step1.inp from [15AlOvYu] , where it was used to compute an ExoMol line list for this molecule.

Potential energy functions¶

Potential Block¶

Implemented PEFs¶

XY₂ type¶

`POTEN_XY2_MORBID`¶

`POTEN_XY2_MORSE_COS`¶

`POTEN_XY2_TYUTEREV`¶

`poten_co2_ames1`¶

XYZ type¶

`POTEN_XYZ_KOPUT`¶

ZXY₂ type¶

`poten_zxy2_morse_cos`¶

XY₃ type (pyramidal)¶

`POTEN_XY3_MORBID_10`¶

`POTEN_XY3_MORBID_11`¶

A chain molecule of HOOH type¶

`POTEN_H2O2_KOPUT_UNIQUE`¶

Table of Contents

Previous topic

Next topic

This Page

Label	Index	Value
b1	0	0.80000000000000E+06
b2	0	0.80000000000000E+05
g1	0	0.13000000000000E+02
g2	0	0.55000000000000E+01
f000	0	0.00000000000000E+00
f001	1	0.25298724728304E+01
f100	1	0.76001446034650E+01

Label	n1	n2	n3	n4	n5	n6	Index	Value
f000000	0	0	0	0	0	0	1	0.00000000000000E+00
f100000	1	0	0	0	0	0	1	0.00000000000000E+00
f010000	0	1	0	0	0	0	1	0.00000000000000E+00
f000100	0	0	0	1	0	0	1	0.00000000000000E+00
f000001	0	0	0	0	0	1	1	0.13239727881219E+05
f000002	0	0	0	0	0	2	1	0.46279621687684E+04
f000003	0	0	0	0	0	3	1	0.14394787420943E+04

Potential energy functions¶

Potential Block¶

Implemented PEFs¶

XY2 type¶

POTEN_XY2_MORBID¶

POTEN_XY2_MORSE_COS¶

POTEN_XY2_TYUTEREV¶

poten_co2_ames1¶

XYZ type¶

POTEN_XYZ_KOPUT¶

ZXY2 type¶

poten_zxy2_morse_cos¶

XY3 type (pyramidal)¶

POTEN_XY3_MORBID_10¶

POTEN_XY3_MORBID_11¶

A chain molecule of HOOH type¶

POTEN_H2O2_KOPUT_UNIQUE¶

XY₂ type¶

`POTEN_XY2_MORBID`¶

`POTEN_XY2_MORSE_COS`¶

`POTEN_XY2_TYUTEREV`¶

`poten_co2_ames1`¶

`POTEN_XYZ_KOPUT`¶

ZXY₂ type¶

`poten_zxy2_morse_cos`¶

XY₃ type (pyramidal)¶

`POTEN_XY3_MORBID_10`¶

`POTEN_XY3_MORBID_11`¶

`POTEN_H2O2_KOPUT_UNIQUE`¶