Frames and vibrational coordinates ********************************** Here we introduce different ingredients available for triatomic molecules, including - Molecular frames :math:`xyz`; - :math:`3N-6` (:math:`3N-5`) vibrational coordinates :math:`\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 ========== XY\ :sub:`2` type molecules --------------------------- .. sidebar:: .. figure:: img/XY2.jpg :alt: XY2 equilibrium structure An XY\ :sub:`2` type molecule and the bisector embedding. A molecule type is defined by the keyword ``MolType``. For the XY\ :sub:`2` example it is :: MolType XY2 in the curvilinear KEO, it is common in TROVE to use the bisector frame for the XY\ :sub:`2` molecules, with the :math:`x` axis bisecting the bond angle and the :math:`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 keywords ``transform`` or ``frame``. In the following, these are described. For quasi-linear triatomic molecules, it is also possible to use exact curvilinear KEO in implemented analytically, see below. ``R-RHO-Z`` ^^^^^^^^^^^ .. sidebar:: .. figure:: img/XY2-r-rho-z.jpg :alt: XY2 equilibrium structure The bisector embedding with the ``R-RHO-Z`` coordinates/frame type: :math:`r_1`, :math:`r_2`, and :math:`\rho`. - ``R-RHO-Z`` is used for (quasi-)linear molecules of the XY\ :sub:`2` type. it defined the curvilinear vibrational coordinates as the two bond angles :math:`r_1` and :math:`r_2` with the bending mode described by the angle :math:`\rho = \pi - \alpha`, where :math:`\alpha` is the interbond angle (:math:`\rho = 0 \ldots \rho_{\rm max}`). The exact orientation of the frame relative to the instantaneous position of the nuclei is critical for representing the dipole moment vectors. For the rigid reference frame (``REFER-CONF RIGID``), the actual internal coordinates are the displacements of :math:`r_1`, :math:`r_2` and :math:`\rho` from the corresponding equilibrium: .. math:: \begin{split} \xi_1 &= r_1 - r_{\rm e}, \\ \xi_2 &= r_2 - r_{\rm e}, \\ \xi_3 &= \rho, \end{split} where :math:`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 :math:`\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: .. math:: \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 :math:`r_{\rm ref}` can also vary with :math:`\rho` as e.g. in the minimum energy path (MEP) definition with :math:`r_{\rm ref}` being the optimised value at the given value of :math:`\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 :math:`\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 :math:`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`` (``frame``) 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 :math:`\alpha` in this case. In the ``Rigid`` reference configuration, it is a displacement from the equilibrium value :math:`\alpha_{\rm e}`: .. math:: \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, :math:`\alpha` is given on a grid of points ranging from :math:`\alpha_{\rm min}` to :math:`\alpha_{\rm max}` and including the equilibrium value. In the linearised ``Rigid`` case, the bending coordinated is defined as a linear expansion of :math:`\alpha` at :math:`\alpha_{\rm eq}` in terms of the Cartesian displacements. TROVE input example: :: COORDS local (curvilinear coordinates) frame 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' XY\ :sub:`2` frame used for isotopologies with slightly different masses of Y\ :sub:`1` and Y\ :sub:`2`, for example O\ :sup:`16`\ CO\ :sup:`17`. 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.: :: frame 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 ------------------ .. sidebar:: .. figure:: img/XYZ-r1.jpg :alt: XYZ equilibrium structure Frame ``R-RHO-Z-M2-M3``: An XYZ type molecule and the :math:`z` embedding along :math:`r_2` and :math:`r_3` with negative :math:`x`. The main embedding here is the 'bond'-embedding, with the :math:`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 ``frame`` can be used. ``R1-Z-R2-RHO`` ^^^^^^^^^^^^^^^ This is a 'bond'-embedding with the same vibrational coordinates as in ``R-RHO-Z`` and :math:`r_1` along the :math:`z` axis and :math:`r_2` in the negative direction of :math:`x`. .. figure:: img/XYZ_frame_R1-Z-R2-RHO.jpg :alt: XYZ equilibrium structure :width: 200 px :align: left Frame ``R1-Z-R2-RHO``: An XYZ type molecule and the :math:`z` embedding along :math:`r_2` and :math:`r_3` with negative :math:`x`. The coordinates are given as above: .. math:: \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``. ``R2-Z-R1-RHO`` ^^^^^^^^^^^^^^^ This is a 'bond'-embedding with the :math:`r_2` along the :math:`z` axis and :math:`r_1` in the positive direction of :math:`x`, which is illustrated in the figure. .. sidebar:: .. figure:: img/XYZ-r2.jpg :alt: XYZ equilibrium structure Frame ``R2-Z-R1-RHO``: An XYZ type molecule and the :math:`z` embedding along :math:`r_2` and :math:`r_3` with negative :math:`x`. Exact KEO frames for triatomic molecules ---------------------------------------- There several exact, curvilinear KEO forms are available in TROVE for quasi-linear triatomic molecules, XY\ :sub:`2` and XYZ. These KEOs are implemented in TROVE analytically, together with the corresponding matrix elements with the singularity resolution. These forms require a ``kinetic`` block in input with a reference to the specific frame. This is the difference with the linearised KEOs which use a general TROVE approach applicable for arbitrary molecules, except the linear ones. Exact KEO frames require that the ``COORDS`` card is set to ``LOCAL`` (aka ``CURVILINEAR``), which stands for the curvilinear coordinates. The associated kinetic expansion order ``KinOrder`` must be set to 2 in the following exact KEO. Here the expansion plays a formal role as this KEO i represented as a formal expansion of the 2nd order in terms of two stretches around the non-rigid reference configuration along the :math:`\rho` coordinate (see the ``rational`` expansion type in the basis set in the stretching subgroup 1). Each KEO presented case is constructed to be used with the specific basis set configuration and usually also for a specific frame. These must be always used together. ``KINETIC_XY2_EKE_BISECT`` ^^^^^^^^^^^^^^^^^^^^^^^^^^ This is a bisector frame for curvilinear coordinates of an XY\ :sub:`2` molecules with ``kinetic`` input block is given by :: KINETIC kinetic_type KINETIC_XY2_EKE_BISECT END It can be only used with the coordinates/frame type ``R-RHO-Z`` (see above), i.e. for the valence coordinates with :math:`\rho` as the bending angle (:math:`\rho=0` at the linear geometry), the basis set ``laguerre-k`` and with the ``NON-RIGID`` reference configuration. The ``laguerre-k`` basis functions are constructed using the Associated Laguerre polynomial with the factor :math:`\sqrt{\rho}` or :math:`\sqrt{\rho} \rho`, depending if :math:`K` (rotational quantum number) is zero or not, respectively. The associated kinetic expansion order ``KinOrder`` must be set to 2. Here is an input example for this case for the C\ :sub:`3` molecule: :: KinOrder 2 COORDS local frame r-rho-z MOLTYPE XY2 REFER-CONF NON-RIGID SYMGROUP C2v(M) ZMAT C 0 0 0 0 11.996709 C 1 0 0 0 11.996709 C 1 2 0 0 11.996709 end BASIS 0,'JKtau', Jrot 0, krot 12 1,'numerov','rational', 'morse', range 0,30,r 8, weight 1.0, points 3000, borders -0.40,1.40 1,'numerov','rational', 'morse', range 0,30,r 8, weight 1.0, points 3000, borders -0.40,1.40 2,'laguerre-k','linear','linear', range 0,56, weight 1.0, points 10000, borders 0.,110.0 deg END KINETIC kinetic_type KINETIC_XY2_EKE_BISECT END KINETIC_XYZ_EKE_BOND_SINRHO ^^^^^^^^^^^^^^^^^^^^^^^^^^^ This is a bond frame KEO constructed to work with the basis set type ``sinrho-laguerre-k``. The associated frame is ``R1-Z-R2-RHO``, for example for :sup:`12`\ C\ :sup:`12`\ C\ :sup:`13`\ C, the corresponding TROVE input is as follows: :: COORDS local (curvilinear) TRANSFORM R-RHO-Z-M2-M3-BISECT (FRAME) MOLTYPE XY2 REFER-CONF NON-RIGID ZMAT C 0 0 0 0 12.000000 C 1 0 0 0 13.003355 C 1 2 0 0 12.000000 end KINETIC kinetic_type KINETIC_XYZ_EKE_bisect END The associated symmetry is either Cs(M) or C\ :sub:`ns`\ (M): :: symmetry Cs(M) ``KINETIC_XY2_EKE_BISECT_SINRHO`` ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ This a similar to the basis set ``KINETIC_XY2_EKE_BISECT``, which is introduced to work with the basis set ``sinrho-laguerre-k`` and only with this basis set. This basis set is constructed from the Associated Laguerre polynomial with the factor :math:`(\sin\rho)^{K+\frac{1}{2}}`. The associated TROVE configuration is as in the following input: :: KinOrder 2 COORDS local frame r-rho-z MOLTYPE XY2 REFER-CONF NON-RIGID BASIS 0,'JKtau', Jrot 0, krot 12 1,'numerov','rational', 'morse', range 0,30,r 8, resc 1.0, points 3000, borders -0.40,1.40 1,'numerov','rational', 'morse', range 0,30,r 8, resc 1.0, points 3000, borders -0.40,1.40 2,'sinrho-laguerre-k','linear','linear', range 0,56, resc 1.0, points 10000, borders 0.,110.0 deg END KINETIC kinetic_type KINETIC_XY2_EKE_BISECT_SINRHO END KINETIC_XYZ_EKE_BISECT ^^^^^^^^^^^^^^^^^^^^^^ For asymmetric triatomic molecules of type XYZ, there are several ways to orient the in-plane axes :math:`x` and :math:`z` at a general instantaneous geometry. The ``KINETIC_XYZ_EKE_BISECT`` KEO is constructed for the frame with the :math:`x` axis along the molecular bisector. The bisector XYZ frame is for asymmetric molecules XYZ with similar masses of Y and Z, i.e. when a bisector is a more natural description of the axis than a bond-frame. This KEO must be used with the correct XYZ-type bisector frames: ``R-RHO-Z-M2-M3-BISECT`` is used for general asymmetric molecules with similar masses of Y (M2) and Z (M3). In principle, the KEO should fully define the configuration of the problem to solve and the associated frame type should not matter for the solution of the Schroedniger equation. The point where the choice of the frame becomes critical is when the dipoles are involved, which need to be re-defined into the correct frame. The actual transformation of the dipole is performed in the subroutine ``MLloc2pqr_xyz``. For example for :sup:`12`\ C\ :sup:`12`\ C\ :sup:`13`\ C, the corresponding TROVE input is as follows: :: COORDS local (curvilinear) TRANSFORM R-RHO-Z-M2-M3-BISECT (FRAME) MOLTYPE XY2 REFER-CONF NON-RIGID ZMAT C 0 0 0 0 12.000000 C 1 0 0 0 13.003355 C 1 2 0 0 12.000000 end KINETIC kinetic_type KINETIC_XYZ_EKE_bisect END The associated symmetry is either Cs(M) or C\ :sub:`ns`\ (M): :: symmetry Cs(M) KINETIC_XYZ_EKE_BOND ^^^^^^^^^^^^^^^^^^^^ Is one of the bond-frames constructed for the XYZ type molecules (X is in the centre), with the :math:`z` axis along the instantaneous orientation of the bond :math:`r_1` (X-Y). Bond-frames are better suited for molecules with a light nucleus and the :math:`z` axis is assumed for the heavier nucleus. The associated ``frame`` is ``R1-Z-R2-RHO`` and the basis set type is ``laguerre-k``: :: KinOrder 2 COORDS local frame r-rho-z MOLTYPE XY2 REFER-CONF NON-RIGID BASIS 0,'JKtau', Jrot 0, krot 12 1,'numerov','rational', 'morse', range 0,30,r 8, resc 1.0, points 3000, borders -0.40,1.40 1,'numerov','rational', 'morse', range 0,30,r 8, resc 1.0, points 3000, borders -0.40,1.40 2,'laguerre-k','linear','linear', range 0,56, resc 1.0, points 10000, borders 0.,110.0 deg END KINETIC kinetic_type KINETIC_XYZ_EKE_BOND END KINETIC_XYZ_EKE_BOND-R2 ^^^^^^^^^^^^^^^^^^^^^^^ This is the case of the bond-frame with :math:`z` along the bong :math:`r_2` (Z nucleus). The associated frames and basis sets are ``R2-Z-R1-RHO`` and ``laguerre-k``, respectively. See the figure illustrating the ``R2-Z-R1-RHO`` frame above. KINETIC_XYZ_EKE_BOND_SINRHO ^^^^^^^^^^^^^^^^^^^^^^^^^^^ This is a bond frame KEO constructed to work with the basis set type ``sinrho-laguerre-k``. The associated frame is ``R1-Z-R2-RHO``, for example for HCN, the corresponding TROVE input is as follows: :: COORDS local (curvilinear) TRANSFORM R1-Z-R2-RHO (FRAME) MOLTYPE XY2 REFER-CONF NON-RIGID ZMAT C 0 0 0 0 12.00000000 N 1 0 0 0 14.00307401 H 1 2 0 0 1.00782503224 end KINETIC kinetic_type KINETIC_XYZ_EKE_BOND_SINRHO END The associated symmetry is either Cs(M) or C\ :sub:`ns`\ (M): :: symmetry Cs(M) Tetratomics =========== XY\ :sub:`3` rigid molecules (PH\ :sub:`3` type) ------------------------------------------------- Linearized KEOs use the Eckart frame with the PAS at the equilibrium configuration. The latter has the :math:`z` axis along the axis of symmetry :math:`C_3` with the :math:`x` axis chosen in plane containing the X-Y\ :sub:`1` bond and passing through :math:`C_3`. ``R-ALPHA`` ^^^^^^^^^^^ For the rigid XY\ :sub:`3`, like PH\ :sub:`3`, the logical coordinate choice of the valence coordinates consists of three bond lengths :math:`r_1`, :math:`r_2`, :math:`r_3`, :math:`\alpha_{23}`, :math:`\alpha_{13}` and :math:`\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: .. math:: \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} .. sidebar:: .. figure:: img/PH3.jpg :alt: PH3 equilibrium structure PH\ :sub:`3` equilibrium structure 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 PH\ :sub:`3` [15SoAlTe]_, SbH\ :sub:`3` [10YuCaYa]_, AsH\ :sub:`3` [19CoYuKo]_, PF\ :sub:`3` [19MaChYa]_. XY\ :sub:`3` non-rigid with umbrella motion (NH\ :sub:`3` type) --------------------------------------------------------------- :: MolType XY3 Consider the Ammonia molecule NH3\ :sub:`3` 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 :math:`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. :math:`\alpha_2` and :math:`\alpha_3`, two dihedral angles :math:`\phi_2` and :math:`\phi_3`. However, for symmetry reasons, TROVE employs the symmetry-adapted bending pair :math:`S_a` and :math:`S_b`, defined as follows: .. math:: S_a = \frac{1}{\sqrt{6}} (2 \alpha_{23}-\alpha_{13}-\alpha_{12}), \\ S_b = \frac{1}{\sqrt{2}} ( \alpha_{13}-\alpha_{12}) or .. math:: 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 .. sidebar:: .. figure:: img/umbrella.jpg :alt: Umbrella motions NH\ :sub:`3`: umbrella modes :math:`\rho` and :math:`\delta`. Linearized KEOs use the Eckart frame with the PAS at the equilibrium configuration. The latter has the :math:`z` axis along the axis of symmetry :math:`C_3` with the :math:`x` axis chosen in plane containing the X-Y\ :sub:`1` bond and passing through :math:`C_3`. ``R-S-DELTA`` ^^^^^^^^^^^^^ For this ``frame`` case, the following valence-based coordinates are used: .. math:: \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. ZXY\ :sub:`2` (Formaldehyde type) --------------------------------- :: MolType ZXY2 The common valence coordinate choice for ZXY\ :sub:`2` includes three bond lengths , two bond angles and a dihedral angle :math:`\tau`. The latter can be treated as the reference for a non-rigid reference configuration in TROVE on a grid of :math:`\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 :math:`\tau_i`. The reference configuration is always in the principle axis sysetm, i.e. for each value of the book angle :math:`\tau`, TROVE solve the PAS conditions to reorient the molecule. .. sidebar:: .. figure:: img/H2CO.jpg :alt: H2CO Valence coordinates and the bisector frame used for H\ :sub:`2`\ CO. Apart from the standard linearised KEO, a curvilinear exact KEO has been recently introduced into TROVE. This is exactly the ``R-THETA-TAU`` type, detailed below. ``R-THETA-TAU`` ^^^^^^^^^^^^^^^ .. math:: \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} Rigid isotopologues of XY\ :sub:`3` as ZXY\ :sub:`2` type ---------------------------------------------------------- The Z type can be used to define single or double deuterated isotopologues of an XY\ :sub:`3` molecule such as a rigid PH\ :sub:`3`. For PDH\ :sub:`2`, 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 :math:`z` axis in the plane contemning PD and bisecting the angle between 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 Non-rigid isotopologues of XY\ :sub:`3` as ZXY\ :sub:`2` type ------------------------------------------------------------- For non-rigid NH\ :sub:`2`\ D and NHD\ :sub:`2`, the choice of the non-rigid frame becomes important. During to the large amplitude motion, the frame can lead to a flip of the moments of inertia. In TROVE, the frame is chosen as the principle axis system (PAS), which, for most of the system, is straightforward to define. Therefore, frame and the internal coordinates are usually selected via a single keyword ``Transform``. For non-rigid systems, however, due to accidental degeneracies of the moments of inertia, the PAS along the non-rigid path must be carefully constructed to prevent such flips. Besides, for the ammonia isotopologues, despite the structure ZXY\ :sub:`2` being formally similar the structure of formaldehyde, the coordinates should be chosen as ammonia-like, not formaldehyde-like. Therefore, in this case we distinguish ``frame`` and ``transform``. For NHD\ :sub:`2`, the following setting is used: :: TRANSFORM R-S-DELTA frame R1-Z-R2-X-Y MOLTYPE XY3 REFER-CONF NON-RIGID The valence coordinates is defined using the ``Zmat`` card as follows: :: ZMAT N 0 0 0 0 14.00307401 D 1 0 0 0 2.01410178 H 1 2 0 0 1.007825032 H 1 2 3 1 1.007825032 end The coordinates type ``R-S-DELTA`` (card ``TRANSFORM``) defines the internal coordinates the same as in the case of XY\ :sub:`3`, while the frame ``R1-Z-R2-X-Y`` places the :math:`z` axis containing the vector :math:`\vec{r}_1` (ND), axis :math:`x` in the direction of the :math:`\vec{r}_1` (NH\ :sub:`1`) and the symmetry plane to be :math:`zy`, see the figure. This non-rigid frame of NH\ :sub:`2`\ D is illustrated in the side figure, where the evolution of the PAS is shown. At the planar configuration, the :math:`y` axis is normal to the plane with :math:`z` as the symmetry axis. .. sidebar:: .. figure:: img/NH2D.png :alt: NH2D Non-rigid reference frame for NH\ :sub:`2`\ D. The card ``MOLTYPE XY3`` means that all the associated transformation rules are implemented in the module mol_xy3.f90. .. sidebar:: .. figure:: img/NHD2.png :alt: NH2D Non-rigid reference frame for NH\ :sub:`2`\ D. The same non-rigid frame is used for NHD\ :sub:`2`, now with the :math:`z` axis containing the vector :math:`\vec{r}_1` (NH), axis :math:`x` in the direction of the :math:`\vec{r}_1` (ND\ :sub:`1`) and the symmetry plane to be :math:`zy`, see the figure. This non-rigid frame of NHD\ :sub:`2` and the evolution of PAS is illustrated in the side figure with the Z-matrix given by :: ZMAT N 0 0 0 0 14.00307401 D 1 0 0 0 2.01410178 H 1 2 0 0 1.007825032 H 1 2 3 1 1.007825032 end Although the definition of the frames is similar, the transformations of the corresponding PASs are very distinct. A chain ABCD type molecule (hydrogen peroxide type) ---------------------------------------------------- :: MolType ABCD ``R-ALPHA-TAU`` ^^^^^^^^^^^^^^ The six internal coordinates for the ``frame R-ALPHA-TAU`` type consist of three stretching, two bending and one dihedral coordinates as given by .. sidebar:: .. figure:: img/A2B2.jpg :alt: A2B2 Valence coordinates used for HOOH. .. math:: \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 :math:`x` axis bisects the dihedral angle. .. sidebar:: .. figure:: img/H2O2-bisector.jpg :alt: H2O2-bisector Molecular frame used for HOOH: the :math:`x` axis always bisecting the dihedral angle :math:`\delta` . 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 :math:`\delta = 0\ldots 720^\circ`. Otherwise the eigenfunction is obtained double valued due to the :math:`x` axis appearing in the opposite direction to the two bonds after one :math:`\delta = 360^\circ` revolution. .. figure:: img/H2O2_3_dysplays.jpg :alt: H2O2 3 displays Principal axis system with an extended torsional angle :math:`\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 :math:`\xi_i` are defined around :math:`\delta`-dependent reference values. The latter are obtained as oprmised geometries by minimised molecule's energy: .. math:: \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 .. math:: \zeta_i^{\rm ref} = \zeta_i^{\rm e} + \sum_{n} a_i^n (\cos\delta - \cos\delta_{\rm e}) where :math:`{\bf\zeta} = \{R,r_1,r_2,\alpha_{123},\alpha_{234}\}`. Five-atomic molecules ===================== The XY\ :sub:`4` molecule (T\ :sub:`d`\ ) and the ``XY4`` type -------------------------------------------------------------- :: MolType XY4 The frame for the tetrahedral molecule XY\ :sub:`4` spanning the T\ :sub:`d`\ (M) symmetry group is chosen with the :math:`xyz` axes orthogonal to the faces of the box containing the molecule with the four atoms :math:`{\rm Y}_i` at its vertices, as shown in the figure, with the Cartesian coordinates at equilibrium given by .. math:: \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} .. sidebar:: .. figure:: img/XY4.jpg :alt: XY4 The structure and molecular frame of the XY\ :sub:`4` molecule. ``TRANSFORM R-ALPHA`` ^^^^^^^^^^^^^^^^^^^^^ The tetrahedral five-atomic molecule XY\ :sub:`4` 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 :math:`r_i \equiv r_{{\rm C}-{\rm H}_i}` and six inter-bond angles :math:`\alpha_{{\rm H}_i-{\rm C}-{\rm H}_j} = \alpha_{ij}`. For the equilibrium value of the tetrahedral angle :math:`\alpha`, :math:`\cos(\alpha_{\rm e})` = :math:`-1/\sqrt{3}` which explains the factor :math:`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 :math:`\alpha_{ij}` is redundant as there should be only five independent bending vibrations, with the following redundancy condition: .. math:: :label: e-redund \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 . XY\ :sub:`4` belongs to the T\ :sub:`d`\ (M) molecular symmetry group, which consists of five irreducible representations, :math:`A_1`, :math:`A_2`, :math:`E`, :math:`F_1` and :math:`F_2`. One way to define independent bending modes is to reduce the six inter-bond angles :math:`\alpha_{ij}` to five symmetry-adapted irreducible combinations, which, together with four bond lengths :math:`r_i` form nine independent vibrational modes :math:`\xi_i` as follows: four stretches .. math:: :label: e-vects-i \xi_i =r_i, \;\; i = 1,2,3,4, two :math:`E`-symmetry bends .. math:: :label: e-vects-5-6 \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 :math:`F`-symmetry bends .. math:: :label: e-vects-7-9 \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 :math:`r_i` can also be in principle combined into symmetry-adapted coordinates in T\ :sub:`d`\ (M): .. math:: :label: e-CH4-xi1=4 \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} ZXY\ :sub:`3` (Methyl Chloride type) ------------------------------------ :: MolType ZXY3 For the ZXY\ :sub:`3` type molecule we use valence coordinates consisting of four bond lengths :math:`r_0`, :math:`r_i` (:math:`i-1,2,3`), three bond angles :math:`\beta_i` and two symmetry adapted dihedral coordinates constructed from three dihedral angles :math:`\tau_{12}, \tau_{23}, \tau_{13}`, where :math:`\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. .. sidebar:: .. figure:: img/CH3Cl.jpg :alt: CH3Cl Valence coordinates and the bisector frame used for CH\ :sub:`3`\ Cl. ``TRANSFORM R-BETA-SYM`` ^^^^^^^^^^^^^^^^^^^^^^^^^ .. math:: \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: - :math:`r_0` - :math:`r_1`, :math:`\beta_{1}` - :math:`r_2`, :math:`\beta_{2}`, :math:`\alpha_{12}` - :math:`r_3`, :math:`\beta_{3}`, :math:`\alpha_{13}` where \alpha_{12}` and :math:`\alpha_{13}` are interbond angles between the bonds X-Y\ :sub:`i`. The Z-matrix coordinates are transformed to :math:`\tau_{12}, \tau_{23}, \tau_{13} ` via the following trigonometric rules: .. math:: \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} The CH\ :sub:`3`\ D molecule (C\ :sub:`3v`\ ) of the ``ZXY3`` type ------------------------------------------------------------------ ``TRANSFORM R-BETA-SYM`` ^^^^^^^^^^^^^^^^^^^^^^^^^ This is a similar to the Methyl Chloride molecule type (``MOLTYPE ZXY3``) in terms of the symmetry properties, although being a methane-deuterated product. The same coordinate ``Transform`` type ``TRANSFORM R-BETA-SYM`` as for fZXY\ :sub:`3` can be used for CH\ :sub:`3`\ D with the following setting: :: TRANSFROM R-BETA-SYM MOLTYPE ZXY3 but with the PES of methane (see **Potential energy functions**). Six-atomic molecules ==================== The C\ :sub:`2`\ H\ :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: .. sidebar:: .. figure:: img/C2H4.jpg :alt: C2H4 The structure and molecular frame of the C\ :sub:`2`\ H\ :sub:`4` molecule. .. math:: \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 .. math:: \tau = \left\{ \begin{array}{cc} \delta, & \delta <\pi, \\ \delta - 2\pi, & \delta >\pi, \\ \end{array} \right. .. sidebar:: .. figure:: img/C2H4_coords.jpg :alt: C2H4 coordinates The vibrational coordinates of the ``C2H4_2BETA_1TAU`` type used for the C\ :sub:`2`\ H\ :sub:`4` molecule. This type can be used both for rigid and non-rigid molecule types. The non-rigid coordinate is :math:`\xi_{12}` in the latter case.