Frames and vibrational coordinates¶

Here we introduce different ingredients available for triatomic molecules, including

Molecular frames $xyz$ ;
$3N-6$ ( $3N-5$ ) vibrational coordinates $\xi_n$ ;

For the linearised coordinates, the default frame is Eckart. The equilibrium structures, required for the definition of the linearised KEO and PEF, are chosen as the principal axis system (PAS). For the curvilinear KEOs, the frames are defined by the construction of the KEOs in their analytic representations.

Triatomics¶

XY₂ type molecules¶

A molecule type is defined by the keyword MolType. For the XY₂ example it is

MolType XY2

in the curvilinear KEO, it is common in TROVE to use the bisector frame for the XY₂ molecules, with the $x$ axis bisecting the bond angle and the $z$ in the plane of the molecule, but other embeddings are possible. The PAS frame coincides with the bisector frame at the equilibrium or non-rigid reference configuration (i.e. symmetric). In TROVE, the definition of the frame is combined with the definition of the internal coordinates via the keyword transform. In the following, these are described.

There are currently at least two exact, curvilinear KEO forms are provided for a quasi-linear XY₂ molecules, MLkinetic_xy2_bisect_EKE, MLkinetic_xy2_bisect_EKE_sinrho, see below.

`R-RHO-Z`¶

R-RHO-Z is used for (quasi-)linear molecules of the XY₂ type. it defined the curvilinear vibrational coordinates as the two bond angles $r_1$ and $r_2$ with the bending mode described by the angle $\rho = \pi - \alpha$ , where $\alpha$ is the interbond angle ( $\rho = 0 \ldots \rho_{\rm max}$ ). For the rigid reference frame (REFER-CONF RIGID), the actual internal coordinates are the displacements of $r_1$ , $r_2$ and $\rho$ from the corresponding equilibrium:

$\begin{split} \xi_1 &= r_1 - r_{\rm e}, \\ \xi_2 &= r_2 - r_{\rm e}, \\ \xi_3 &= \rho, \end{split}$

where $r_{\rm e}$ is the equilibrium bond length. If the non-rigid reference frame is used (REFER-CONF NON-RIGID), the bending mode is given on an equidistant grid, typically of 1000-2000 points, while the stretching modes are the displacements from the given $\rho$ point along the non-rigid reference frame, the latter is usually defined as the principal axes system with the bond length fixed to the equilibrium:

$\begin{split} \xi_1 &= r_1 - r_{\rm ref}, \\ \xi_2 &= r_2 - r_{\rm ref}, \\ \xi_3 &= \rho, \end{split}$

TROVE uses Z-matrix coordinates to build any user-defined coordinates. In this case, the Z=matrix is given by

ZMAT
    S   0  0  0  0  31.97207070
    H   1  0  0  0   1.00782505
    H   1  2  0  0   1.00782505
end

Alternatively, the reference value of the bond length $r_{\rm ref}$ can also vary with $\rho$ as e.g. in the minimum energy path (MEP) definition with $r_{\rm ref}$ being the optimised value at the given value of $\rho$ corresponding to the local energy minimum. In this case, the non-rigid frame must be defined using the MEP block (see the corresponding section).

For the linearised coordinates type (COORDS Linear), the actual internal coordinates are the linearised versions of $\xi_i$ above. More specifically, for the non-rigid reference configuration, the bending coordinate :math`rho` is kept curvilinear on a grid of :math`rho_k` points as before, while the stretching coordinates are defined by linearly expanding :math`r_1` and :math`r_2` in terms of the Cartesian displacement around the corresponding reference values $r_{\rm ref}$ . In the Rigid case, the bending coordinate is also linearised.

The advantage of the linearised coordinates is that the corresponding KEO can be constructed on the fly as part of the TROVE generalised procedure as a Taylor type expansion. The main disadvantage however is that the approximate linearised KEO operator is less accurate than the (exact) curvilinear EKO. Besides, the convergence of the variational solution is also poorer for the linearised case (see [15YaYu]).

`R-RHO-Z-ECKART`¶

This Transform type is very similar to R-RHO-Z, but with the molecular frame define using the Eckart conditions.

`R-ALPHA-Z`¶

R-ALPHA-Z is very similar to R-RHO-Z with the difference in the bending coordinate, which in the interbond angle $\alpha$ in this case. In the ``Rigid reference configuration, it is a displacement from the equilibrium value $\alpha_{\rm e}$ :

$\begin{split} \xi_1 &= r_1 - r_{\rm e}, \\ \xi_2 &= r_2 - r_{\rm e},\\ \xi_3 &= \alpha-\alpha_{\rm e}. \end{split}$

In the Non-rigid reference configuration, $\alpha$ is given on a grid of points ranging from $\alpha_{\rm min}$ to $\alpha_{\rm max}$ and including the equilibrium value. In the linearised Rigid case, the bending coordinated is defined as a linear expansion of $\alpha$ at $\alpha_{\rm eq}$ in terms of the Cartesian displacements.

TROVE input example:

COORDS local (curvilinear coordinates) TRANSFORM r-rho-z (r1, r2, rho with the x parallel to the bisector) MOLTYPE XY2 REFER-CONF non-RIGID (Reference configuration)

Note

The text in brackets is used for comments.

R-RHO-Z-M2-M3¶

A ‘bisecting’ XY₂ frame used for isotopologies with slightly different masses of Y₁ and Y₂, for example O¹⁶CO¹⁷. Although this is an XYZ molecule, in this case it is formally treated as XY:sub:`2 but with non-symmetric masses and the Cs symmetry, e.g.:

TRANSFORM    R-RHO-Z-M2-M3
MOLTYPE      XY2
MOLECULE     CO2
REFER-CONF   non-RIGID

SYMGROUP Cs(M)

ZMAT
    C   0  0  0  0   11.996709
    O   1  0  0  0   16.995245
    O   1  2  0  0   15.9905256
end

XYZ type molecules¶

The main embedding here is the ‘bond’-embedding, with the $z$ axis placed parallel to the bond Y-Z with a heavier atom Z comparing to X (second bond). For molecules XYZ with comparable masses X and Z (e.g. in similar isotopologues), the bisector frames and associated TRANSFORM can be used.

`R1-Z-R2-RHO`¶

This is a ‘bond’-embedding with the same vibrational coordinates as in R-RHO-Z and $r_1$ along the $z$ axis. The coordinates are givem as above:

$\begin{split} \xi_1 &= r_1 - r_{\rm e}, \\ \xi_2 &= r_2 - r_{\rm e}, \\ \xi_3 &= \rho, \end{split}$

Here is an example of the Z-matrix for NNO.

ZMAT
    N   0  0  0  0   14.00307401
    N   1  0  0  0   14.00307401
    O   1  2  0  0   15.994915
end

`R1-Z-R2-ALPHA`¶

This is another ‘bond’-embedding with the same vibrational coordinates as in R-ALPHA-Z.

Tetratomics¶

XY₃ rigid molecules (PH₃ type)¶

Linearized KEOs use the Eckart frame with the PAS at the equilibrium configuration. The latter has the $z$ axis along the axis of symmetry $C_3$ with the $x$ axis chosen in plane containing the X-Y₁ bond and passing through $C_3$ .

`R-ALPHA`¶

For the rigid XY₃, like PH₃, the logical coordinate choice of the valence coordinates consists of three bond lengths $r_1$ , $r_2$ , $r_3$ , $\alpha_{23}$ , $\alpha_{13}$ and $\alpha_{12}$ . For the linearised KEO, these valence are used to form the linearised coordinates in the same way as before (1st order expansion in terms of the Cartesian displacement). For the curvilinear KEO (local), the vibrational coordinates are then defined as displacement from the corresponding equilibrium (or non-rigid reference) values:

$\begin{split} \xi_1 &= r_1 - r_{\rm e}, \\ \xi_2 &= r_2 - r_{\rm e}, \\ \xi_3 &= r_3 - r_{\rm e}, \\ \xi_4 &= \alpha_{23}-\alpha_{\rm e}, \\ \xi_5 &= \alpha_{13}-\alpha_{\rm e}, \\ \xi_6 &= \alpha_{12}-\alpha_{\rm e}. \end{split}$

The underlying Z-matrix coordinates are defined using the following Z-matrix:

ZMAT
    N   0  0  0  0  14.00307401
    H   1  0  0  0   1.00782505
    H   1  2  0  0   1.00782505
    H   1  2  3  1   1.00782505
end

This representation has been used for PH₃ [15SoAlTe], SbH₃ [10YuCaYa], AsH₃ [19CoYuKo], PF₃ [19MaChYa].

XY₃ non-rigid with umbrella motion (NH₃ type)¶

MolType XY3

Consider the Ammonia molecule NH3₃ with a relatively small barrier to the planarity. The three bending angles are not suitable in this case as they cannot distinguish the two opposite inversion configurations above and below the planarity. Instead, an umbrella mode has to be introduced as one of the bending modes. An example of an umbrella coordinate is an angle between the $C_3$ symmetry axis and the bond X-Y, see Figure. It is natural to use the non-rigid reference configuration along the umbrella, inversion motion and build the KEO as an expansion around it. For two other bending modes, in principle one can use two inter-bond angles, e.g. $\alpha_2$ and $\alpha_3$ , two dihedral angles $\phi_2$ and $\phi_3$ . However, for symmetry reasons, TROVE employs the symmetry-adapted bending pair $S_a$ and $S_b$ , defined as follows:

$S_a = \frac{1}{\sqrt{6}} (2 \alpha_{23}-\alpha_{13}-\alpha_{12}), \\ S_b = \frac{1}{\sqrt{2}} ( \alpha_{13}-\alpha_{12})$

or

$S_a = \frac{1}{\sqrt{6}} (2 \phi_{23}-\phi_{13}-\phi_{12}), \\ S_b = \frac{1}{\sqrt{2}} ( \phi_{13}-\phi_{12})$

The umbrella mode for any instantaneous configuration of the nuclei is defined in TROVE as the angle between a trisector

Linearized KEOs use the Eckart frame with the PAS at the equilibrium configuration. The latter has the $z$ axis along the axis of symmetry $C_3$ with the $x$ axis chosen in plane containing the X-Y₁ bond and passing through $C_3$ .

`R-S-DELTA`¶

For this TRANSFORM case, the following valence-based coordinates are used:

$\begin{split} \xi_1 &= r_1 - r_{\rm e}, \\ \xi_2 &= r_2 - r_{\rm e}, \\ \xi_3 &= r_3 - r_{\rm e}, \\ \xi_4 &= \frac{1}{\sqrt{6}} (2 \alpha_{23}-\alpha_{13}-\alpha_{12}), \\ \xi_5 &= \frac{1}{\sqrt{2}} ( \alpha_{13}-\alpha_{12}), \\ \xi_6 &= \delta. \end{split}$

The umbrella mode :math:\delta is defined as an angle between the trisector and any of the bonds X-Y. The other 5 coordinates are then used to construct the corresponding linearised vibrational coordinates (see above) for the linearised (linear) representation.

ZXY₂ (Formaldehyde type)¶

MolType ZXY2

The common valence coordinate choice for ZXY₂ includes three bond lengths , two bond angles and a dihedral angle $\tau$ . The latter can be treated as the reference for a non-rigid reference configuration in TROVE on a grid of $\tau_i$ ranging from :math`[-tau_{0}ldots tau_{0}]`, while other 5 modes are treated as displacement from their equilibrium values at each grid point $\tau_i$ . The reference configuration is always in the principle axis sysetm, i.e. for each value of the book angle $\tau$ , TROVE solve the PAS conditions to reorient the molecule.

`R-THETA-TAU`¶

$\begin{split} \xi_1 &= r_1 - r_{\rm e}, \\ \xi_2 &= r_2 - r_{\rm e}, \\ \xi_3 &= r_3 - r_{\rm e}, \\ \xi_4 &= \theta_1, \\ \xi_5 &= \theta_2, \\ \xi_6 &= \tau. \end{split}$

Isotopologues of XY₃ as ZXY₂ type¶

The Z type can be used to define signle or double deturated isotopologues of an XY₃ molecule such as a rigid PH₃ or non-rigid NH₃. For PDH₂, we use R-THETA-TAU in combination with the Z-matrix given as follows:

ZMAT
    P   0  0  0  0  14.00307401
    D   1  0  0  0   2.01410178
    H   1  2  0  0   1.007825032
    H   1  2  3  2   1.007825032
end

Here, the equilibrium frame coinsides with the principle axis system with the $z$ axis in the plane conteining PD and bisetcing the angle betwen two PH bonds.

For a PH:sub:`2`D type isotopologue, the Z-matrix is given by

ZMAT
P 0 0 0 0 14.00307401 H 1 0 0 0 1.007825032 D 1 2 0 0 2.01410178 D 1 2 3 2 2.01410178

end

ZXY₃ (Methyl Chloride type)¶

MolType ZXY3

Similarilly, for the ZXY₃ type molecule we use valence coordinates consisting of four bond lengths $r_0$ , $r_i$ ( $i-1,2,3$ ), three bond angles $\beta_i$ and two symmetry adapted dihedral coordinates constructed from three dihedral angles $\tau_{12}, \tau_{23}, \tau_{13}$ , where $\tau_{12}+\tau_{23}+\tau_{13} = \pi$ . This is a rigid type, where all coordinates are treated as displacements from the corresponding equilibrium values. Currently, only the standard linearised KEO is available in TROVE.

`R-BETA-SYM`¶

$\begin{split} \xi_1 &= r_0 - r_{\rm e}^{(0)}, \\ \xi_2 &= r_1 - r_{\rm e}, \\ \xi_3 &= r_2 - r_{\rm e}, \\ \xi_4 &= r_3 - r_{\rm e}, \\ \xi_5 &= \beta_1-\beta_{\rm e}, \\ \xi_6 &= \beta_2-\beta_{\rm e}, \\ \xi_7 &= \beta_3-\beta_{\rm e}, \\ \xi_8 &= \frac{1}{\sqrt{6}} (2 \tau_{23}-\tau_{13}-\tau_{12}), \\ \xi_9 &= \frac{1}{\sqrt{2}} ( \tau_{13}-\tau_{12}). \\ \end{split}$

The Z-matrix coordinates (underlying basic TROVE coordinates) are as given by the Z-matrix rules:

ZMAT
    C   0  0  0  0  12.000000000
    Cl  1  0  0  0  34.968852721
    H   1  2  0  0   1.007825035
    H   1  2  3  0   1.007825035
    H   1  2  3  4   1.007825035
end

are as follows:

$r_0$
$r_1$ , $\beta_{1}$
$r_2$ , $\beta_{2}$ , $\alpha_{12}$
$r_3$ , $\beta_{3}$ , $\alpha_{13}$

where alpha_{12}` and $\alpha_{13}$ are interbond angles between the bonds X-Y_i. The Z-matrix coordinates are transformed to :math:`tau_{12}, tau_{23}, tau_{13} ` via the following trigonometric rules:

$\begin{split} \cos \tau_{12} &= \frac{\cos\alpha_{12}-\cos\beta_{1}\cos\beta_{2}}{\sin\beta_{1}\sin\beta_{2}}, \\ \cos \tau_{13} &= \frac{\cos\alpha_{13}-\cos\beta_{1}\cos\beta_{3}}{\sin\beta_{1}\sin\beta_{3}}, \\ \tau_{23} &= 2\pi - \tau_{12}-\tau_{13},\\ \cos \alpha_{23} &= \cos\beta_{2}\cos\beta_{3}+\cos(\tau_{12}+\tau_{13})\sin\beta_{2}\sin\beta_{3}.\\ \end{split}$

A chain ABCD type molecule (hydrogen peroxide type)¶

MolType ABCD

`R-ALPHA-TAU`¶

The six internal coordinates for the Transform R-ALPHA-TAU type consist of three stretching, two bending and one dihedral coordinates as given by

$\begin{split} \xi_1 &= R - R_{\rm e}, \\ \xi_2 &= r_1 - r_{\rm e}, \\ \xi_3 &= r_2 - r_{\rm e}, \\ \xi_4 &= \alpha_{123}-\alpha_{\rm e}, \\ \xi_5 &= \alpha_{234}-\alpha_{\rm e}, \\ \xi_6 &= \delta. \end{split}$

The non-rigid reference frame such that the $x$ axis bisects the dihedral angle.

For this embedding, in order to be able to separate the vibrational and rotational bases into a product form, it is important to the an extended range for the dihedral angle $\delta = 0\ldots 720^\circ$ . Otherwise the eigenfunction is obtained double valued due to the $x$ axis appearing in the opposite direction to the two bonds after one $\delta = 360^\circ$ revolution.

H2O2 3 displays — Principal axis system with an extended torsional angle $\delta = 0\ldots 720^\circ$ for HOOH.¶

Principal axis system with an extended torsional angle $\delta = 0\ldots 720^\circ$ for HOOH.¶

A minimum energy path (MEP) as a non-rigid reference configuration¶

In MEP, the 5 internal coordinate displacements $\xi_i$ are defined around $\delta$ -dependent reference values. The latter are obtained as oprmised geometries by minimised molecule’s energy:

$\begin{split} \xi_1 &= R - R_{\rm ref}(\delta), \\ \xi_2 &= r_1 - r_{\rm ref}(\delta), \\ \xi_3 &= r_2 - r_{\rm ref}(\delta), \\ \xi_4 &= \alpha_{123}-\alpha_{\rm ref}(\delta), \\ \xi_5 &= \alpha_{234}-\alpha_{\rm ref}(\delta), \\ \xi_6 &= \delta, \end{split}$

where :math: the MEP values are given by a parameterised expansion, for example

$\zeta_i^{\rm ref} = \zeta_i^{\rm e} + \sum_{n} a_i^n (\cos\delta - \cos\delta_{\rm e})$

where ${\bf\zeta} = \{R,r_1,r_2,\alpha_{123},\alpha_{234}\}$ .

Fuive-atomic molecules¶

The XY₄ molecule (T_d) and the `XY4` type¶

MolType XY4

The frame for the tetrahedral molecule XY₄ spanning the T_d(M) symmetry group is chosen with the $xyz$ axes orthogonal to the faces of the box containing the molecule with the four atoms ${\rm Y}_i$ at its vertices, as shown in the figure, with the Cartesian coordinates at equilibrium given by

$\begin{split} H_{1x} &= -\frac{r_{\rm e}}{\sqrt{3}}, H_{1y} = \frac{r_{\rm e}}{\sqrt{3}}, H_{1z} = \frac{r_{\rm e}}{\sqrt{3}}, \\ H_{2x} &= -\frac{r_{\rm e}}{\sqrt{3}}, H_{2y} = -\frac{r_{\rm e}}{\sqrt{3}}, H_{2z} = -\frac{r_{\rm e}}{\sqrt{3}}, \\ H_{3x} &= \frac{r_{\rm e}}{\sqrt{3}}, H_{3y} = \frac{r_{\rm e}}{\sqrt{3}}, H_{3z} = -\frac{r_{\rm e}}{\sqrt{3}}, \\ H_{4x} &= \frac{r_{\rm e}}{\sqrt{3}}, H_{4y} = -\frac{r_{\rm e}}{\sqrt{3}}, H_{4z} = \frac{r_{\rm e}}{\sqrt{3}}. \\ \end{split}$

`R-ALPHA`¶

The tetrahedral five-atomic molecule XY₄ has 9 vibrational degrees of freedom. For a semi-rigid molecule (i.e. ignoring any isomerisation that can occur at higher energies), they can be characterised by four bond lengths $r_i \equiv r_{{\rm C}-{\rm H}_i}$ and six inter-bond angles $\alpha_{{\rm H}_i-{\rm C}-{\rm H}_j} = \alpha_{ij}$ . For the equilibrium value of the tetrahedral angle $\alpha$ , $\cos(\alpha_{\rm e})$ = $-1/\sqrt{3}$ which explains the factor $1/\sqrt{3}$ in the definition of the Cartesian coordinates. There should, however, be only 9 independent vibrational degrees of freedom in a 5 atomic molecule. One of the inter-bond angles $\alpha_{ij}$ is redundant as there should be only five independent bending vibrations, with the following redundancy condition:

(1)¶ $\left| \begin{array}{cccc} 1 & \cos\alpha_{12} & \cos\alpha_{13} & \cos\alpha_{14} \\ \cos\alpha_{12} & 1 & \cos\alpha_{23} & \cos\alpha_{24} \\ \cos\alpha_{13} & \cos\alpha_{23} & 1 & \cos\alpha_{34} \\ \cos\alpha_{14} & \cos\alpha_{24} & \cos\alpha_{34} & 1 \end{array} \right| = 0 .$

XY₄ belongs to the T_d(M) molecular symmetry group, which consists of five irreducible representations, $A_1$ , $A_2$ , $E$ , $F_1$ and $F_2$ . One way to define independent bending modes is to reduce the six inter-bond angles $\alpha_{ij}$ to five symmetry-adapted irreducible combinations, which, together with four bond lengths $r_i$ form nine independent vibrational modes $\xi_i$ as follows: four stretches

(2)¶ $\xi_i =r_i, \;\; i = 1,2,3,4,$

two $E$ -symmetry bends

(3)¶ $\begin{split} \xi_5^{E_a} &= \frac{1}{\sqrt{12}} (2 \alpha_{12} - \alpha_{13} - \alpha_{14} - \alpha_{23} - \alpha_{24} + 2 \alpha_{34} ), \\ \xi_6^{E_b} &= \frac{1}{2} (\alpha_{13} - \alpha_{14} - \alpha_{23} + \alpha_{24} ), \end{split}$

and three $F$ -symmetry bends

(4)¶ $\begin{split} \xi_7^{F_{2x}} &= \frac{1}{\sqrt{2}} ( \alpha_{24} - \alpha_{13} ), \\ \xi_8^{F_{2y}} &= \frac{1}{\sqrt{2}} ( \alpha_{23} - \alpha_{14} ), \\ \xi_9^{F_{2z}} &= \frac{1}{\sqrt{2}} ( \alpha_{34} - \alpha_{12} ), \end{split}$

where the corresponding symmetries of the bending modes are indicated.

The stretching modes $r_i$ can also be in principle combined into symmetry-adapted coordinates in T_d(M):

(5)¶ $\begin{split} \xi_1^{A_1} &= \frac{1}{2} \left( r_1 + r_2 + r_3 + r_4\right), \\ \xi_2^{F_{2x}} &= \frac{1}{2} \left( r_1 - r_2 + r_3 - r_4\right), \\ \xi_3^{F_{2y}} &= \frac{1}{2} \left( r_1 - r_2 - r_3 + r_4\right), \\ \xi_4^{F_{2z}} &= \frac{1}{2} \left( r_1 + r_2 - r_3 - r_4\right). \end{split}$

Six-atomic molecules¶

The C₂H:sub:4 molecule and the `C2H4` type¶

MolType C2H4

`C2H4_2BETA_1TAU`¶

The internal coordinates are defined using the following 12 valence coordinates: 5 stretching (molecular bond) coordinates, 4 bending (inter-bond angles) and 3 dihedral coordinates, with the last mode as a book angle describing the relative motion of two moieties:

$\begin{split} \xi_1 &= r_0 - r_{\rm e}^{(0)}, \\ \xi_2 &= r_1 - r_{\rm e}, \\ \xi_3 &= r_2 - r_{\rm e}, \\ \xi_4 &= r_3 - r_{\rm e}, \\ \xi_5 &= r_4 - r_{\rm e}, \\ \xi_6 &= \beta_1-\beta_{\rm e}, \\ \xi_7 &= \beta_2-\beta_{\rm e}, \\ \xi_8 &= \beta_3-\beta_{\rm e}, \\ \xi_9 &= \beta_4-\beta_{\rm e}, \\ \xi_{10} &= \theta_1 - \pi, \\ \xi_{11} &= \theta_2 - \pi, \\ \xi_{12} & = \theta_1 + \theta_2 - 2\tau, \end{split}$

where

$\tau = \left\{ \begin{array}{cc} \delta, & \delta <\pi, \\ \delta - 2\pi, & \delta >\pi, \\ \end{array} \right.$

This type can be used both for rigid and non-rigid molecule types. The non-rigid coordinate is $\xi_{12}$ in the latter case.

Frames and vibrational coordinates¶

Triatomics¶

XY₂ type molecules¶

`R-RHO-Z`¶

`R-RHO-Z-ECKART`¶

`R-ALPHA-Z`¶

R-RHO-Z-M2-M3¶

XYZ type molecules¶

`R1-Z-R2-RHO`¶

`R1-Z-R2-ALPHA`¶

Tetratomics¶

XY₃ rigid molecules (PH₃ type)¶

`R-ALPHA`¶

XY₃ non-rigid with umbrella motion (NH₃ type)¶

`R-S-DELTA`¶

ZXY₂ (Formaldehyde type)¶

`R-THETA-TAU`¶

Isotopologues of XY₃ as ZXY₂ type¶

ZXY₃ (Methyl Chloride type)¶

`R-BETA-SYM`¶

A chain ABCD type molecule (hydrogen peroxide type)¶

`R-ALPHA-TAU`¶

A minimum energy path (MEP) as a non-rigid reference configuration¶

Fuive-atomic molecules¶

The XY₄ molecule (T_d) and the `XY4` type¶

`R-ALPHA`¶

Six-atomic molecules¶

The C₂H:sub:4 molecule and the `C2H4` type¶

`C2H4_2BETA_1TAU`¶

Table of Contents

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Frames and vibrational coordinates¶

Triatomics¶

XY2 type molecules¶

R-RHO-Z¶

R-RHO-Z-ECKART¶

R-ALPHA-Z¶

R-RHO-Z-M2-M3¶

XYZ type molecules¶

R1-Z-R2-RHO¶

R1-Z-R2-ALPHA¶

Tetratomics¶

XY3 rigid molecules (PH3 type)¶

R-ALPHA¶

XY3 non-rigid with umbrella motion (NH3 type)¶

R-S-DELTA¶

ZXY2 (Formaldehyde type)¶

R-THETA-TAU¶

Isotopologues of XY3 as ZXY2 type¶

ZXY3 (Methyl Chloride type)¶

R-BETA-SYM¶

A chain ABCD type molecule (hydrogen peroxide type)¶

R-ALPHA-TAU¶

A minimum energy path (MEP) as a non-rigid reference configuration¶

Fuive-atomic molecules¶

The XY4 molecule (Td) and the XY4 type¶

R-ALPHA¶

Six-atomic molecules¶

The C2H:sub:4 molecule and the C2H4 type¶

C2H4_2BETA_1TAU¶

XY₂ type molecules¶

`R-RHO-Z`¶

`R-RHO-Z-ECKART`¶

`R-ALPHA-Z`¶

`R1-Z-R2-RHO`¶

`R1-Z-R2-ALPHA`¶

XY₃ rigid molecules (PH₃ type)¶

`R-ALPHA`¶

XY₃ non-rigid with umbrella motion (NH₃ type)¶

`R-S-DELTA`¶

ZXY₂ (Formaldehyde type)¶

`R-THETA-TAU`¶

Isotopologues of XY₃ as ZXY₂ type¶

ZXY₃ (Methyl Chloride type)¶

`R-BETA-SYM`¶

`R-ALPHA-TAU`¶

The XY₄ molecule (T_d) and the `XY4` type¶

`R-ALPHA`¶

The C₂H:sub:4 molecule and the `C2H4` type¶

`C2H4_2BETA_1TAU`¶