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Update potentials.rst
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Trovemaster committed Jun 7, 2023
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Expand Up @@ -94,28 +94,28 @@ The potential parameters are listed after the keyword ``COEFF`` and terminated w

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

.. math::
.. math::
\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

.. math::
.. math::
\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:

+---------+-----+--+--+--+--+--+-----+-----------------------+
Expand All @@ -136,17 +136,17 @@ For the ``COEFF Powers`` option, the meaning of the columns is as follows:
| f000003| 0| 0| 0| 0| 0| 3| 1| 0.14394787420943E+04 |
+---------+-----+--+--+--+--+--+-----+-----------------------+

where
where

- 'Labels' are the parameter name, for printing purposes only;
- 'n1', 'n2', 'n3', ... are the 'powers' of an expansion term, e.g.
- 'Labels' are the parameter name, for printing purposes only;
- 'n1', 'n2', 'n3', ... are the 'powers' of an expansion term, e.g.
:math:`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.
- '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 :math:`r_{\rm e}`\ , equilibrium inter-bond angles :math:`\alpha_{\rm e}`, Morse exponents :math:`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:
In case the definition of PEF requires also structural parameters, such as equilibrium bond lengths :math:`r_{\rm e}`\ , equilibrium inter-bond angles :math:`\alpha_{\rm e}`, Morse exponents :math:`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
Expand All @@ -160,9 +160,9 @@ In case the definition of PEF requires also structural parameters, such as equil
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.

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
================
Expand All @@ -177,19 +177,20 @@ There are several PEFs available for this molecule type.
``POTEN_XY2_MORBID``
^^^^^^^^^^^^^^^^^^^^

This form is given by
This form is given by

.. math::
.. math::
\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 \\
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

.. math::
Expand All @@ -198,6 +199,7 @@ where
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}
Expand Down Expand Up @@ -235,7 +237,7 @@ For description and example see 'Potential Block' above with the input file exam
end


Using the structural parameters in the ``POTEN`` section is important for PES refinements, see the corresponding section for details.
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 <https://raw.githubusercontent.com/Trovemaster/TROVE/develop/docs/source/input/h2s_step1.inp>`_ .


Expand All @@ -249,7 +251,7 @@ An empirical PES of CO\ :sub:`2` is from [17HuScFr]_. The input file example is



It is programmed using the ``powers`` format as follows:
It is programmed using the ``powers`` format as follows:
::


Expand Down Expand Up @@ -278,11 +280,11 @@ It is programmed using the ``powers`` format as follows:
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.

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



Expand All @@ -305,11 +307,11 @@ XYZ type

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

.. math::
.. math::
V = \sum_{ijk} f_{ijk} \xi_1^{i} \xi_2^{j} \xi_3^{k},
The vibrational coordinates are
The vibrational coordinates are

.. math::
Expand All @@ -318,8 +320,8 @@ The vibrational coordinates are
\xi_2 &= (r_2-r_2^{\rm eq})/r_2, \\
\xi_3 &= \alpha-\alpha_{\rm eq},
\end{split}
where the internal stretching coordinates :math:`r_1 = r_1^{\rm eq}`\ , :math:`r_2 = r_2^{\rm eq}`\ , the interbond angle :math:`\alpha = \angle({\rm e})`\ , and the equilibrium parameters are :math:`r_1^{\rm eq}`, :math:`r_2^{\rm eq}` and :math:`\alpha_{\rm eq}`. Note that the exponent :math:`k` associated with the bending coordinate :math:`\xi_3` assumes only even values because of the symmetry of the XYZ molecule.
where :math:`r_1 = r_1^{\rm eq}` and :math:`r_2 = r_2^{\rm eq}` are the internal stretching coordinates and :math:`\alpha` is the interbond angle, and the equilibrium parameters are :math:`r_1^{\rm eq}`, :math:`r_2^{\rm eq}` and :math:`\alpha_{\rm eq}`. Note that the exponent :math:`k` associated with the bending coordinate :math:`\xi_3` assumes only even values because of the symmetry of the XYZ molecule.


The input file example is `CaOH_Koput_step1.inp <https://raw.githubusercontent.com/Trovemaster/TROVE/develop/docs/source/input/CaOH_Koput_step1.inp>`_ where this PES is used.

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