Dipole moment functions ======================= TROVE provides a larger number of dipole moment functions (DMFs) 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 DMF can be included into the TROVE compilation as a generic 'user-defined' module ``pot_user``. Dipole (External) Block ------------------------ The DMFs are defined in the TROVE input file using the ``Dipole`` block, which is just an alias for the ``external`` input structure. A typical ``Dipole`` input is as follows: :: DIPOLE dimension 3 NPARAM 264 0 0 compact DMS_TYPE XY3_SYMMB COEFF list COORDS linear linear linear linear linear linear Order 6 6 6 dstep 0.005 parameters charge 0.00000000 nparamA 112.00000000 RhoE 90.00000000 RE14 1.01032000 beta 1.00000000 d0 4.56621083 f1a -9.36438932 f2a 32.96400671 ... end For an example, see :download:`14N-1H3__BYTe__TROVE__step1.inp <./input/14N-1H3__BYTe__TROVE__step1.inp>` where this DMF is used. - ``dimension`` (aliases ``rank``, ``dim``): This is the dimension of the external field. An "external" is treated in TROVE as a vector of dimension :math:`D`, which in the case of dipole can be up to :math:`D=3`, but will depend on the implementation. This parameter is to help structure the input dipole parameters according to the dipole components, if necessary. - ``NPARAM`` is used to specify the number of parameters to define the DMF and should contain :math:`D` input values. - ``compact``: is recently implemented card which switches to the "compact" format with no "fitting-indexes" column present. - ``DMS_TYPE`` (``TYPE``) is the name of the DMF as implemented in ``pot_*.f90 file`` and referenced in ``mlecules.f90``. - ``COEFF`` indicates if the DFM parameters are given as a list of parameter values (``LIST``) or values with the corresponding expansion powers (``POWERS``), see example below. - ``COORDS``: coordinate types used to re-expand the dipole field in terms of the internal TROVE coordinates. - ``Order``: The corresponding expansion order. - ``dstep``: finite difference step used in the re-expansion. The default value is 0.005 Ang. - ``parameters``: card indicating the section with the dipole parameters entries specific for the given ``DMS_TYPE``. The dipole moment parameters are listed after the keyword ``parameters`` and terminated with the keyword ``END``. The number of the entries should be equal exactly to the sum of ``NPARAM`` values. For the ``COEFF list`` option, the meaning of the columns is as follows: +---------+-----------------------+ | Label | Value | +---------+-----------------------+ | charge | 0.00000000 | +---------+-----------------------+ | nparamA | 112.00000000 | +---------+-----------------------+ | RhoE | 90.00000000 | +---------+-----------------------+ | RE14 | 1.01032000 | +---------+-----------------------+ | beta | 1.00000000 | +---------+-----------------------+ | d0 | 4.56621083 | +---------+-----------------------+ | f1a | -9.36438932 | +---------+-----------------------+ | f2a | 32.96400671 | +---------+-----------------------+ The first column with the name of the parameters, which is for clearness only. This field is only used for printing purposes and otherwise nor-referenced in the code in any way. The second column contains the actual value of the given parameter. The input is directly associated with the corresponding implementation and therefore the order is important. An alternative, legacy, format with no ``compact`` card assumes an additional column with the so-called "fitting-indexes" indicating if the parameter was varied in the fitting to the ab initio data. Here is an example: :: parameters charge 0 0.00000000 nparamA 1 112.00000000 RhoE 0 90.00000000 RE14 0 1.01032000 beta 0 1.00000000 d0 4 4.56621083 f1a 8 -9.36438932 f2a 8 32.96400671 f3a 7 -80.82339377 .... where column 2 contains the "fitting-indexes". These indexes are not used by TROVE. They are kept in order to simplify the interfacing between the ab initio fitting and TROVE, but can be always omitted with the help of the card ``compact``. Here is an example of the input format using individual expansion "powers", ``COEFF powers`` (from CO\ :sub:`2`): :: DIPOLE rank 3 NPARAM 971 0 0 compact TYPE DIPOLE_AMES1 COEFF powers (powers or list) COORDS linear linear linear Orders 16 16 16 threshold 1e-8 Parameters re 0 0 0 0 1.15958d0 ae 0 0 0 0 180.00 d000 0 0 0 0 -0.4801402388843266D+00 d001 0 0 1 0 0.1203598337496481D+00 d002 0 0 2 0 -0.5662267278952241D-01 d003 0 0 3 0 -0.2529381009630170D-01 d004 0 0 4 0 -0.1271678002798687D+00 d005 0 0 5 0 0.3033049145401118D+01 d006 0 0 6 0 -0.1754036600894653D+02 ..... end See the TROVE input :download:`CO2_bisect_xyz_step1.inp <./input/CO2_bisect_xyz_step1.inp>`. Assuming the DMF form as an expansion .. math:: \mu_\alpha(\xi_1,\xi_2,\xi_3) = \sum_{k,j,k} d_{i,j,k} \xi_1^i \xi_2^j \xi_3^k, the input card has the following format +---------+-----------+----------+----------+-----+-------------------------+ | Label | :math:`i` | :math:`j`| :math:`k`|Index| Value :math:`_{i,j,k}` | +---------+-----------+----------+----------+-----+-------------------------+ | d000 | 0| 0 | 0 | 0| -0.4801402388843266D+00| +---------+-----------+----------+----------+-----+-------------------------+ | d001 | 1| 0 | 0 | 0| 0.1203598337496481D+00 | +---------+-----------+----------+----------+-----+-------------------------+ | d002 | 0| 1 | 0 | 0| -0.5662267278952241D-01| +---------+-----------+----------+----------+-----+-------------------------+ where - 'Labels' are the parameter name, for printing purposes only; - :math:`i`, :math:`j`, :math:`k` are the 'powers' of an expansion term; - '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. It is not present in the ``compact`` form. - 'Values' are the actual dipole parameters. For ``powers``, their order is not important. In case the definition of DMF requires also structural parameters, such as equilibrium bond lengths :math:`r_{\rm e}`, equilibrium inter-bond angles :math:`\alpha_{\rm e}`, 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, as in the example above, ``re`` and ``ae`` are two the equilibrium values and the three columns with ``0 0 0`` are given in order to parse their values using exactly column 6. Implemented DMFs ---------------- XY\ :sub:`2` type ----------------- See module pot_xy2.f90. There are several PEFs available for this molecule type. ``xy2_pq_coeff`` ^^^^^^^^^^^^^^^^ .. sidebar:: .. figure:: img/XY2_dm-pq.jpg :alt: PQ frame The :math:`pq` bisector frame ``xy2_pq_coeff`` used for for XY\ :sub:`2`. This is a bisector-frame DMF, given by two components, :math:`\mu^{(q)}` and :math:`\mu^{(p)}` with the :math:`q` axis being the bisector. The following expansions in terms of the coordinate displacements :math:`\Delta r_1 = r_{\rm 1} - r_{\rm e}`, :math:`\Delta r_2 = r_2 - r_{\rm e}`, and :math:`\cos\rho_{\rm e} - \cos\rho`, where :math:`\rho = \pi - \theta` are used, with :math:`\theta` is the bond angle, and :math:`r_1` and :math:`r_2` are the bond lengths: .. math:: :label: e-muQ-1 \begin{split} \mu^{(q)} (\Delta r_1, \Delta r_2, \Delta \alpha ) &= \sin\alpha \left[ \mu_0^{(q)}(\alpha) + \sum_{j} \mu_{j}^{(q)}(\alpha) \Delta r_j + \sum_{j\le k} \mu_{jk}^{(q)}(\alpha) \Delta r_j \Delta r_k \right. \\ & \left . + \sum_{j\le k \le m} \mu_{jkm}^{(q)}(\alpha) \Delta r_j \Delta r_k \Delta r_m + \sum_{j\le k \le m \le n} \mu_{jkmn}^{(q)}(\alpha) \Delta r_j \Delta r_k \Delta r_m \Delta r_n + \ldots \right], \\ \mu^{(p)} (\Delta r_1, \Delta r_2, \Delta \alpha ) &= \mu_0^{(p)}(\alpha) + \sum_{j}^{(p)} \mu_{j}^{(p)} (\alpha) \Delta r_j + \sum_{j\le k} \mu_{jk}^{(p)}(\alpha) \Delta r_j \Delta r_k \\ & + \sum_{j\le k \le m} \mu_{jkm}^{(p)}(\alpha) \Delta r_j \Delta r_k \Delta r_m + \sum_{j\le k \le m \le n} \mu_{jkmn}^{(p)}(\alpha) \Delta r_j \Delta r_k \Delta r_m \Delta r_n + \ldots , \end{split} where all indices :math:`j, k, m`, and :math:`n` assume the values 1 or 2, .. math:: :label: e-muQ-exp \begin{split} \mu_{jk\ldots}^{(q)}(\alpha) =& \sum_{i=0}^{N} Q_{ij\ldots}^{(i)} (\cos\alpha_{\rm e} - \cos\alpha )^i, \\ \mu_{jk\ldots}^{(p)}(\alpha) =& \sum_{i=0}^{N} P_{ij\ldots}^{(i)} (\cos\alpha_{\rm e} - \cos\alpha )^i, \end{split} and the :math:`Q_{ij\ldots}^{(i)}` and :math:`P_{ij\ldots}^{(i)}` are molecular dipole parameters. The expansion coefficients in Eqs. :eq:`e-muQ-exp` are subject to the conditions that the functions :math:`\mu^{(q)}` are unchanged under the interchange of the identical protons, whereas the function :math:`\mu^{(p)}` is antisymmetric under this operation. There are 72 and 99 paramters :math:`Q_{ij\ldots}^{(i)}` and :math:`P_{ij\ldots}^{(i)}`, respectively. An example of ``xy2_pq_coeff`` is illustrated above and can be foound in :download:`H2S_EKE_basic-functions_step1.inp <./input/H2S_EKE_basic-functions_step1.inp>`. The implementation can be found in :code:`subroutine MLdms2pqr_xy2` from the module pot_xy2.f90. The transformation between the TROVE frame and the frame of the specifc dipole of the XY\ :sub:`2` is perfomed in the :code:`subroutine MLloc2pqr_xy2`, e.g.: .. code:: c ! select case(trim(molec%frame)) ! case('R-RHO-Z','R-RHO-Z-M2-M3','R-RHO-Z-M2-M3-BISECT','BISECT-Z') ! a0(2, 1) = -r(1) * cos(alpha_2) a0(2, 3) = -r(1) * sin(alpha_2) ! a0(3, 1) = -r(2) * cos(alpha_2) a0(3, 3) = r(2) * sin(alpha_2) case ... ``XY2_PQ_LINEAR`` ^^^^^^^^^^^^^^^^^ This is similar to ``xy2_pq_coeff``, but with the bending expansion in Eq. :eq:` e-muQ-exp` in terms of the displacement :math:`\alpha-\alpha_{\rm e}`: .. math:: :label: e-muQ-exp-2 \begin{split} \mu_{jk\ldots}^{(q)}(\alpha) =& \sum_{i=0}^{N} Q_{ij\ldots}^{(i)} (\alpha - \alpha_{\rm e} )^i, \\ \mu_{jk\ldots}^{(p)}(\alpha) =& \sum_{i=0}^{N} P_{ij\ldots}^{(i)} (\alpha - \alpha_{\rm e} )^i, \end{split} ``DIPOLE_AMES1`` ^^^^^^^^^^^^^^^^ This DMF is of the AMES1 type represented using the point-charge molecular bond frame [14HuScLe]_ given by projections on the molecular bond vectors :math:`\vec{r}_1` and :math:`\vec{r}_2`: .. math:: \vec{\mu} = \mu_x \vec{i} + \mu_z \vec{k} = \mu_1 \vec{r}_1 + \mu_2 \vec{r}_2 where :math:`\mu_x` and :math:`\mu_z` are the TROVE frame vectors and :math:`\mu_1` and :math:`\mu_2` are the *ab initio* dipoles in the molecular bond frame (the :math:`\mu_y` component is always zero). The two point-charge dipole moment components :math:`\mu_1` and :math:`\mu_2` are represented in terms of the vibrational coordinates as .. math:: :label: eq:coords_dms \begin{split} \zeta_1 &= r_1-r^{\rm ref}_1, \\ \zeta_2 &= r_2-r^{\rm ref}_2, \\ \zeta_3 &= \cos\alpha-1. \end{split} with the following analytic Taylor-type expansions used (see e.g. [14HuScLe]_): .. math:: \begin{split} \mu_1 &= \sum_{ijk} F^{(1)}_{ijk} \zeta_1^{i} \zeta_2^{j} \zeta_3^{k} , \\ \mu_2 &= \sum_{ijk} F^{(2)}_{ijk} \zeta_1^{j} \zeta_2^{i} \zeta_3^{k} , \end{split} As an example can be found of a system where this form was used, see :download:`CO2_bisect_xyz_step1.inp <./input/CO2_bisect_xyz_step1.inp>`. ``DIPOLE_SO2_AMES1`` ^^^^^^^^^^^^^^^^^^^^ This form is essentially the same as ``DIPOLE_AMES1`` but some specific characteristic used for the SO\ :sub:`2` molecule in [14HuScLe]_. ``XY2_C3_SCHROEDER`` -------------------- .. sidebar:: .. figure:: img/XY2_rot_from_Eckart.jpg :alt: Eckart frame Transfromation from tje Eckart to any other frames for XY\ :sub:`2` in the :math:`xz` plane. This DMF is based on the DMF form reported by Schroeder et al. [16ScSe]_ for C\ :sub:`3`. This DMF is in the Ecakrt frmame expressed in terms of two in-plane components, :math:`\mu^{\parallel}` and :math:`\mu^{\perp}`, as Taylor expansions around the equilibrium geometry: .. math:: \begin{split} \mu^{\parallel} &= D^{\parallel}_{ijk} \Delta r_1^i \,\Delta r_2^j \, \Delta \alpha^k , \\ \mu^{\perp} &= D^{\perp}_{ijk} \Delta r_1^i \,\Delta r_2^j \, \Delta \alpha^k . \end{split} Since TROVE's frame is usually different from the DMF frame (e.g. bisector) in the ro-vibrational calculations, this dipole moments functions needs to be rotated. This is done using the rotation angle :math:`\phi` from an equilibrium bysector frame :math:`r_{n\alpha}^0` to the instantaneous frame :math:`r_{n\alpha}` (:math:`n=1,2,3` and :math:`\alpha=x,y,z`) in the in the :math:`xz` plane as given by .. math:: \tan\phi = \frac{m_{\rm X} ( r^0_{1,z} r_{1,x}-r^0_{1,x} r_{1,z} )+m_{{\rm Y}_1} (r^0_{2,z}r_{2,x}-r^0_{2,x}r_{2,z} )+m_{{\rm Y}_2}(r^0_{3,z}r_{3,x}-r^0_{3,x}r_{3,z}) }{ m_{\rm X} (r^0_{1,z} r_{1,z}+r^0_{1,x}r_{1,x})+m_{{\rm Y}_1}(r^0_{2,z}r_{2,z}+r^0_{2,x}r_{2,x})+m_{{\rm Y}_2}(r^0_{3,z}r_{3,z}+r^0_{3,x}r_{3,x}) } ``DIPOLE_PQR_XYZ_Z-FRAME`` ^^^^^^^^^^^^^^^^^^^^^^^^^^ .. sidebar:: .. figure:: img/XYZ-pqy_DM.jpg :alt: PQY frame The :math:`pqy` bisector frame ``DIPOLE_PQR_XYZ_Z-FRAME`` used for for XYZ. This is frame used to represent DMF of XYZ non-symmetri molecules with the :math:`z` (:math:`p`) axis along the vecror :math:`r_1` and other two axes defined using the following conditions: .. math:: \begin{split} \vec{p} &= \frac{\vec{r}_1}{r_1} \\ \vec{y} &= \frac{\vec{r}_1 \times \vec{r}_2 }{|\vec{r}_1 \times \vec{r}_2|} \\ \vec{q} &= \vec{y}\times \vec{p} \end{split} The corrsponding components :math:`\mu^{(q)}` and :math:`\mu^{(p)}` are expanded using the same form as in Eq. :eq:`e-muQ-1` but with no constraints on the permutations of the atoms. ``DIPOLE_PQR_XYZ_Z-FRAME_SINRHO`` ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The same as ``DIPOLE_PQR_XYZ_Z-FRAME`` but with :math:`\sin(\alpha_{\rm e}-\alpha)` as an expansion variable in Eq. :eq:`e-muQ-1` instead of :math:`\cos\alpha_{\rm e} - \cos\alpha`: .. math:: :label: e-muQ-exp-3 \begin{split} \mu_{jk\ldots}^{(q)}(\alpha) =& \sum_{i=0}^{N} Q_{ij\ldots}^{(i)} (\sin(\alpha_{\rm e}-\alpha))^i, \\ \mu_{jk\ldots}^{(p)}(\alpha) =& \sum_{i=0}^{N} P_{ij\ldots}^{(i)} (\sin(\alpha_{\rm e}-\alpha))^i, \end{split} When :math:`\alpha_{\rm e} = \pi` (linear molecules), :math:`\sin(\pi-\alpha) = \sin\rho`, which explanes the suffix ``_sinrho`` in the name of thi DMF, wich is aimed at linear molecules. DIPOLE_PQR_XYZ ^^^^^^^^^^^^^^ This a bisector dipole frame for the XYZ type molecules. It is defined by .. math:: \begin{split} \vec{q} &= \frac{\vec{r}_1+\vec{r}_2}{|\vec{r}_1+\vec{r}_2|} \\ \vec{y} &= \frac{\vec{r}_1 \times \vec{q}}{|\vec{r}_1 \times \vec{q}|} \\ \vec{p} &= \vec{q}\times \vec{y} \end{split} The exapnsion of the dipole moment components in terms of :math:`r_1`, :math:`r_2` and :math:`\alpha` as in Eq. :eq:`e-muQ-exp-2`. See :download:`7Li-16O-1H__OYT7__TROVE.model` for an example of a TROVE input. ``DIPOLE_PQR_XYZ_Z-BOND`` ^^^^^^^^^^^^^^^^^^^^^^^^^ Subroutine: ``MLdms2pqr_xyz_z_bond``. This is a generalisation of ``DIPOLE_PQR_XYZ_Z-FRAME``, which does not make any assumtion on the frame of the original dipole, only on its expansion form given as in Eq. :eq:`e-muQ-exp-2`. The role of ``DIPOLE_PQR_XYZ_Z-BOND`` is to transform it to the TROVE frame, which in this case is the with the $z$ axis oriented along the bond :math:`\vec{r}_1`: .. math:: \begin{split} \vec{x} &= \frac{\vec{n}_1+\vec{n}_2}{|\vec{n}_1+\vec{n}_2|} \\ \vec{y} &= \vec{z}\times \vec{x} \\ \vec{z} &= \frac{\vec{r}_{12}}{r_{12}} \end{split} ``DIPOLE_PQR_XYZ_BISECTING`` ^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Subroutine: ``MLdms2pqr_xyz_bisecting``. This is a generalisation of ``DIPOLE_PQR_XYZ``, which does not make any assumtion on the frame of the original dipole, only on its expansion form given as in Eq. :eq:`e-muQ-exp-2`. The role of ``DIPOLE_PQR_XYZ_BISECTING`` is to transform it to the TROVE frame, which in this case is the with the $x$ axis oriented along the bisector: .. math:: \begin{split} \vec{q} &= \frac{\vec{r}_1+\vec{r}_2}{|\vec{r}_1+\vec{r}_2|} \\ \vec{y} &= \frac{\vec{r}_1 \times \vec{q} }{|\vec{r}_1 \times \vec{q}|} \\ \vec{p} &= \vec{q}\times \vec{y} \end{split} ``DIPOLE_AMES1_XYZ`` ^^^^^^^^^^^^^^^^^^^^ This form is a modification of ``DIPOLE_AMES1`` for non-symmetric molecules. As an example can be found of a system where this form was used, see :download:`16O-12C-32S__OYT8__TROVE.model <./input/16O-12C-32S__OYT8__TROVE.model>` as well in `OYT8 spectroscopic model `__, where it was used to compute an ExoMol line list for OCS [24OwYuTe]_. ``XY2_SCHROEDER_XYZ_ECKART`` ^^^^^^^^^^^^^^^^^^^^^^^^^^^^ This is an XYZ version of the ``XY2_C3_SCHROEDER`` type. DIPOLE_H2O_LPT2011 ^^^^^^^^^^^^^^^^^^ DMF from [11LoTePo]_. It is included into :code:`subroutine MLdipole_h2o_lpt2011` in prop_xy2.f90. DIPOLE_PQR_XYZ_Z-BOND DIPOLE_PQR_XYZ_BISECTING` DIPOLE_BISECT_S1S2T_XYZ XY2_QMOM_SYM XY2_ALPHA_SYM XY2_QMOM_BISECT_FRAME TEST_XY2_QMOM_BISECT_FRAME XY2_SR-BISECT-NONLIN TEST_XY2_SR-BISECT-NONLIN XY3 --- See module pot_xy3.f90. XY3_MB ^^^^^^ .. sidebar:: .. figure:: img/XY3_DMF_frame_rigid.jpg :alt: XY3 frame Rigid XY\ :sub:`3` frame for ``XY3_MB``. This DMF is implemented as a subroutine :code:`MLdms2xyz_xy3_mb` and was reported in [06YuCaTh]_. This form was used to produce a line list for PH\ :sub:`3` in [13SoYuTe]_. An input example is in :download:`31P-1H3__SAlTY__TROVE.model` and can be also found at ``__. Consider the :math:`r_i` as the instantaneous value of the distance between X and Y\ :math:`_i`, where Y\ :math:`_i` is the nucleus labeled :math:`i` (= 1, 2, or 3); :math:`\alpha_i` denotes the bond angle :math:`\angle` (Y\ :math:`_j`\ XY\ :math:`_k`) where :math:`(i,j,k)` is a permutation of the numbers (1,2,3). We utilize the so-called Molecular-Bond (MB) representation to describe the :math:`r_i` and :math:`\alpha_j` dependence of the electronically averaged dipole moment vector :math:`\bar{\bf \mu}` for XY\ :sub:`3`. In the MB representation it is given by .. math:: :label: e-dm-bond-r \bar{\bf \mu} = \bar{\mu}_1^{\rm Bond}\, \vec{\bf e_1} + \bar{\mu}_2^{\rm Bond}\, \vec{\bf e_2} + \bar{\mu}_3^{\rm Bond}\, \vec{\bf e_3} where the three functions :math:`\bar{\mu}_i^{\rm Bond}`, :math:`i` :math:`=` 1, 2, 3, depend on the vibrational coordinates, and :math:`\vec{\bf e}_i` is the unit vector along bond :math:`i`, .. math:: \vec{\bf e}_i = \frac{\vec{\bf r}_i-\vec{\bf r}_4}{|\vec{\bf r}_i-\vec{\bf r}_4|} with :math:`\vec{\bf r}_i` (:math:`i` :math:`=` 1, 2, 3) as the position vector of proton :math:`i` and :math:`\vec{\bf r}_4` as the position vector of X. The representation of :math:`\bar{\bf \mu}` in Eq. :eq:`e-dm-bond-r` is "body-fixed" in the sense that it relates the dipole moment vector directly to the instantaneous positions of the nuclei (i.e., to the vectors :math:`\vec{\bf r}_i`). Following [06YuCaTh]_, we express the three functions :math:`\bar{\mu}_i^{\rm Bond}`, :math:`i` :math:`=` 1, 2, 3, as .. math:: :label: e-mu-bnd-mubond \bar{\mu}_i^{\rm Bond} = \sum_{j=1}^3 \left({\mathbf A}^{-1} \right)_{ij} \, \left( \bar{\bf \mu} \cdot \vec{\bf e}_j \right) where :math:`({\mathbf A}^{-1} )_{ij}` is an element of the non-orthogonal :math:`3 \times 3` matrix :math:`{\mathbf A}^{-1}` obtained as the inverse\footnote% {When the molecule is planar, the determinant :math:`\vert \mathbf A \vert` :math:`=` 0 and :math:`\mathbf A` cannot be inverted. In this case we set :math:`\bar{\mu}_3^{\rm Bond}` :math:`=` 0 in Eq. :eq:`e-dm-bond-r` and express :math:`\bar{\bf \mu}` in terms of :math:`\vec{\bf e}_1` and :math:`\vec{\bf e}_2` only, i.e., we determine :math:`\bar{\mu}_1^{\rm Bond}` and :math:`\bar{\mu}_2^{\rm Bond}` in terms of :math:`\bar{\bf \mu} \cdot \vec{\bf e}_1` and :math:`\bar{\bf \mu} \cdot \vec{\bf e}_2`.} of .. math:: {\mathbf A} = \left( \begin{array}{ccc} 1 & \cos\alpha_3 & \cos\alpha_2 \\ \cos\alpha_3 & 1 & \cos\alpha_1 \\ \cos\alpha_2 & \cos\alpha_1 & 1 \end{array} \right)\hbox{.} For symmetry reasons, all three projections can be expressed in terms of a single function :math:`\mu_0(r_1,r_2,r_3,\alpha_1,\alpha_2,\alpha_3)`: .. math:: \begin{split} \bar{\bf \mu} \cdot \vec{\bf e}_1 &= \mu_0(r_1,r_2,r_3,\alpha_1,\alpha_2,\alpha_3) = \mu_0(r_1,r_3,r_2,\alpha_1,\alpha_3,\alpha_2), \\ \bar{\bf \mu} \cdot \vec{\bf e}_2 &= \mu_0(r_2,r_3,r_1,\alpha_2,\alpha_3,\alpha_1) = \mu_0(r_2,r_1,r_3,\alpha_2,\alpha_1,\alpha_3), \\ \bar{\bf \mu} \cdot \vec{\bf e}_3 &= \mu_0(r_3,r_1,r_2,\alpha_3,\alpha_1,\alpha_2) = \mu_0(r_3,r_2,r_1,\alpha_3,\alpha_2,\alpha_1), \end{split} and this function is chosen as an expansion .. math:: :label: e-mu-es-r \begin{split} \mu_0 = \mu_{0 }^{(0)} + \sum_{k} \mu_{k }^{(0)} \xi_k + \sum_{k,l} \mu_{k l }^{(0)} \xi_k \xi_l & + \sum_{k,l,m} \mu_{k l m }^{(0)} \xi_k \xi_l \xi_m \nonumber\\ & + \sum_{k,l,m,n} \mu_{k l m n}^{(0)} \xi_k \xi_l \xi_m \xi_n + \ldots \end{split} in the variables .. math:: \begin{split} \xi_k &= (r_k-r_{\rm e}) \exp \left( -\beta^2 \, (r_k-r_{\rm e})^2 \right), \;\; k=1,2,3, \\ \xi_l &= \cos \left(\alpha_{l-3}\right) - \cos \left(\alpha_{\rm e}\right) \hbox{,} \;\; l=4,5,6 \hbox{,} \end{split} where :math:`r_{\rm e}` and :math:`\alpha_{\rm e}` are the equilibrium values of the bond lengths and bond angles, respectively. We include the factor :math:`\exp \left( -\beta^2 (r_k-r_{\rm e})^2 \right)` in order to keep the expansion from diverging at large :math:`r_k`. The expansion coefficients :math:`\mu_{k,l,m,\ldots}^{(0)}` are pairwise equal. We have .. math:: \mu_{k',l',m', \ldots}^{(0)} = \mu_{k,l,m, \ldots}^{(0)} when the indices :math:`k',l',m', \ldots` are obtained from :math:`k,l,m, \ldots` by replacing all indices 2 by 3, all indices 3 by 2, all indices 5 by 6, and all indices 6 by 5. XY3_SYMMB ^^^^^^^^^ This form is a non-rigid analogy of ``XY3_MB`` allowing the pyrimidal molecule XY\ :sub:`3` to go through the planar configuration and was introduced in [09YuBaYa]_. It is implemented in :code:`subroutine MLdms2xyz_xy3_symmb` in pot_xy3.f90. This form has been used in several studies of non-rigid pyrimidal molecules including NH\ :sub:`3`, CH\ :sub:`3`, OH\ :sub:`3`\ :sup:`+`. A disadvantage of the ``XY3_MB`` representation is the ambiguity at and near planar geometries when the three vectors :math:`\vec{\bf e}_i` become linearly dependent, or nearly linearly dependent, and singularities appear in the determination of the :math:`\bar{\mu}_i^{\rm Bond}` functions. This is overcome by reformulating the :math:`\bar{\mu}_i^{\rm Bond}` functions in terms of symmetry-adapted combinations of the MB projections :math:`\left(\bar{\bf \mu}\cdot \vec{\bf e}_j \right)` .. math:: \begin{split} \mu_{A''_1}^{\rm SMB}&= \left( \bar{\bf \mu} \cdot \vec{\bf e}_{\rm N} \right) \\ \mu_{E'_a}^{\rm SMB} &= \frac{1}{\sqrt{6}} \left[ 2 \left( \bar{\bf \mu} \cdot \vec{\bf e}_1 \right) - \left( \bar{\bf \mu} \cdot \vec{\bf e}_2 \right) - \left( \bar{\bf \mu} \cdot \vec{\bf e}_3 \right) \right] \\ \mu_{E'_b}^{\rm SMB} &= \frac{1}{\sqrt{2}} \left[ \left( \bar{\bf \mu} \cdot \vec{\bf e}_2 \right) - \left( \bar{\bf \mu} \cdot \vec{\bf e}_3 \right) \right], \end{split} where an additional reference MB-vector :math:`\vec{\bf e}_{\rm N} = \vec{\bf q}_{\rm N}^{}/\vert \vec{\bf q}_{\rm N}^{}\vert` was introduced by means of the 'trisector' .. math:: \vec{\bf q}_{\rm N} = (\vec{\bf e}_1 \times \vec{\bf e}_2) + (\vec{\bf e}_2 \times \vec{\bf e}_3) + (\vec{\bf e}_3 \times \vec{\bf e}_1). This symmetrized molecular bond representation is denoted as "SMB". The subscripts of the :math:`\mu_{\Gamma}^{\rm SMB}` functions (:math:`\Gamma = A''_1,E'_a, E'_b`) refer to irreducible representations of D\ :sub:`3h`\ (M); :math:`\mu_{A''_1}^{\rm SMB}` has :math:`A''` symmetry in D\ :sub:`3h`\ (M), and :math:`(\mu_{E'_a}^{\rm SMB}, \mu_{E'_b}^{\rm SMB})` transform as the :math:`E'` irreducible representation. The symmetrized vectors .. math:: \begin{split} \vec{\bf e}_{A'_1} &= \vec{\bf e}_{\rm N} \\ \vec{\bf e}_{E''_a} &= \frac{1}{\sqrt{6}} \left( 2 \vec{\bf e}_1 - \vec{\bf e}_2 - \vec{\bf e}_3 \right) \\ \vec{\bf e}_{E''_b} &= \frac{1}{\sqrt{2}} \left( \vec{\bf e}_2 - \vec{\bf e}_3 \right) \end{split} have :math:`A'_1` and :math:`E''` symmetry in the same manner. The dipole moment vector :math:`\bar{\bf \mu}` vanishes at symmetric, planar configurations of D\ :sub:`3h`\ geometrical symmetry. Also, the :math:`\mu_{A'_1}^{\rm SMB}` component is antisymmetric under the inversion operation :math:`E^*` and vanishes at planarity, which leaves only two independent components of :math:`\bar{\bf \mu}` at planarity. The advantage of having a DMS representation in terms of the projections :math:`\left( \bar{\bf \mu} \cdot \vec{\bf e}_j \right)` is that it is "body-fixed". It relates the dipole moment vector directly to the instantaneous positions of the nuclei (i.e., to the vectors :math:`\vec{\bf r}_i`). These projections are well suited to being represented as analytical functions of the vibrational coordinates. For intensity simulations, however, we require the Cartesian components :math:`\mu_{\alpha}`, :math:`\alpha=x,y,z` of the dipole moment along the molecule-fixed :math:`xyz` axes. These are obtained by inverting the linear equations .. math:: :label: e-dipol-car \mu_{\Gamma}^{\rm SMB} = \sum_{\alpha=x,y,z} A_{\Gamma,\alpha} \mu_{\alpha}, where :math:`A_{\Gamma,\alpha}`, is the :math:`\alpha`-coordinate (:math:`\alpha=x,y,z`) of the vector :math:`\vec{\bf e}_{\Gamma}` (:math:`\Gamma=A',E_a,E_b`). When the molecule is planar, :math:`\mu_{A''_1}^{\rm SMB}` is zero, as is the corresponding right-hand side in Eq. :eq:`e-dipol-car`. Thus, at planar configurations the system of linear equations in :math:`\mu_{\alpha}` contains two non-trivial equations only. At near-planar configurations :math:`\mu_{A''_1}^{\rm SMB}` is not exactly zero and cannot be neglected, and so Eq. :eq:`e-dipol-car` becomes near-linear-dependent. The symmetry-adapted representation of :math:`\mu^{\rm SMB}` appears to be well defined even for these geometries. The functions :math:`\mu_{\Gamma}^{\rm SMB}` (henceforth referred to as :math:`\mu_{\Gamma}`) are now represented as expansions: .. math:: \begin{split} \mu_{A''_1} & = \cos\rho \left[ \mu_{0}^{(A''_1)} + \sum_{k} \mu_{k}^{(A''_1)} \xi_k + \sum_{k,l} \mu_{k,l}^{(A''_1)} \xi_k \xi_l + \sum_{k,l,m} \mu_{k,l,m}^{(A''_1)} \xi_k \xi_l \xi_m + \cdots \right] \\ \mu_{E'_a} & = \mu_{0}^{(E'_a)} + \sum_{k} \mu_{k}^{(E'_a)} \xi_k + \sum_{k,l} \mu_{k,l}^{(E'_a)} \xi_k \xi_l + \sum_{k,l,m} \mu_{k,l,m}^{(E'_a)} \xi_k \xi_l \xi_m + \cdots \\ \mu_{E'_b} = & \mu_{0}^{(E'_b)} + \sum_{k} \mu_{k}^{(E'_b)} \xi_k + \sum_{k,l} \mu_{k,l}^{(E'_b)} \xi_k \xi_l + \sum_{k,l,m} \mu_{k,l,m}^{(E'_b)} \xi_k \xi_l \xi_m + \cdots \end{split} in terms of the variables .. math:: \begin{split} \xi_k &= (r_k - r_{\rm e}) \exp \left[ -\beta \, (r_k-r_{\rm e})^2 \right], \;\; k=1,2,3, \\ \xi_{4}&= \frac{1}{\sqrt{6}} \left( 2 \alpha_{1} - \alpha_{2} - \alpha_{3} \right)\hbox{,} \\ \xi_{5}&= \frac{1}{\sqrt{2}} \left( \alpha_{2} - \alpha_{3} \right)\hbox{,} \\ \xi_{6} &= \sin\rho_{\rm e}-\sin\rho \hbox{,} \end{split} Here .. math:: \sin {\rho} \, = \, \frac{2}{\sqrt{3}} \, \sin [ (\alpha_1 + \alpha_2 + \alpha_3) /6], and :math:`\sin(\rho_{\rm e})` is the equilibrium value of :math:`\sin(\rho)`. The factor :math:`\cos\rho = \pm \sqrt{1-\sin^2 {\rho}}` ensures that the dipole moment function :math:`\mu_{A''_1}` changes sign when :math:`\rho = 0\ldots \pi` is changed to :math:`\pi - \rho`. As in the rigid MB case, the factor :math:`\exp \left[ -\beta (r_k-r_{\rm e})^2 \right]` is used in order to keep the expansion from diverging at large :math:`r_i`. Chain type ABCD-type molecules ------------------------------ ``HOOH_MB`` ^^^^^^^^^^^ .. sidebar:: .. figure:: img/HOOH_frame_DMF.jpg :alt: Bisecting DMF frame HOOH Bisecting DMF HOOH frame ``HOOH_MB``. Subroutine: :code:`MLdms_hooh_MB`. The DMF frame :math:`xyz` is defined with the :math:`x` axis as a bisector between two planes, H\ :sub:`1`\ O\ :sub:`1`\ O\ :sub:`2` and O\ :sub:`1`\ O\ :sub:`2`\ H\ :sub:`1` and :math:`z` axis along the O\ :sub:`1`\ O\ :sub:`2` bond. Thus, the unit vector :math:`k` is defined as a normalised vector :math:`\vec{r}_{12}`, the unit vector :math:`i` is defined as a bisector between the normal to the planes 3-1-2 and 1-2-3 and the unit vector is defined via the rand-hand rule: .. math:: \begin{split} \vec{k} &= \frac{\vec{r}_{12}}{r_{12}} \\ \vec{i} &= \frac{\vec{n}_1+\vec{n}_2}{|\vec{n}_1+\vec{n}_2|} \\ \vec{j} &= \vec{j}\times \vec{i} \end{split} where the plane normals are defined as follows: .. math:: \begin{split} \vec{n}_1 &= \frac{\vec{r}_{21}\times \vec{r}_{12}}{|\vec{r}_{21}\times \vec{r}_{12}|} \\ \vec{n}_2 &= \frac{\vec{r}_{12}\times \vec{r}_{32}}{|\vec{r}_{12}\times \vec{r}_{32}|} \end{split} Here, the coordinate vectors :math:`\vec{r}_{21}`, :math:`\vec{r}_{31}` and :math:`\vec{r}_{42}` as well as :math:`\vec{i}`, :math:`\vec{j}` and :math:`\vec{k}` are given in the representation of the instantaneous TROVE frame :math:`xyz`. With this definition, the matrix constructed from these unit vecrtors :math:`\vec{i}`, :math:`\vec{j}` and :math:`\vec{k}` forms the rotation (unitary transformation) of the dipole vector between its original DMF frame and the instantaneous TROVE frame :math:`xyz`: .. math:: {\mathbf A} = \left( \vec{i}^T,\vec{j}^T,\vec{k}^T \right) and .. math:: \vec{\mu}^{\rm TROVE} = {\mathbf A}^T \vec{\mu}^{\rm DMF} The :math:`\mu_\alpha^{\rm DMF}` are expressed as Taylor-type expansion: .. math:: \begin{split} \mu_x^{\rm DMF} &= \cos\tau/2 \sum_{ijklmn} \mu_{ijklmn}^{(x)} \xi_1^i \xi_2^j \xi_3^k \xi_4^l \xi_5^m \xi_6^n \\ \mu_y^{\rm DMF} &= \sin\tau/2 \sum_{ijklmn} \mu_{ijklmn}^{(y)} \xi_1^i \xi_2^j \xi_3^k \xi_4^l \xi_5^m \xi_6^n \\ \mu_z^{\rm DMF} &= \mu_{ijklmn}^{(z)} \xi_1^i \xi_2^j \xi_3^k \xi_4^l \xi_5^m \xi_6^n \end{split} where .. math:: \begin{split} \xi_i & r_i - r_{i}^{\rm e} e^{-\beta_i (r_i - r_{i}^{\rm e}) }, t=1,2,3 \\ \xi_4 & = \alpha_1 - \alpha_{\rm e} \\ \xi_5 & = \alpha_1 - \alpha_{\rm e} \\ \xi_6 & = \cos\delta \end{split} The expansion paramters are the subjetc to the permutation constraint: .. math:: \begin{split} \mu_{i(jk)(lm)n}^{(x)} = \mu_{i(kj)(mn)n}^{(x)}, \quad {\rm for} j\ne k \quad l\ne m \\ \mu_{i(jk)(lm)n}^{(y)} = -\mu_{i(kj)(mn)n}^{(y)}, \quad {\rm for} j\ne k \quad l\ne m \\ \mu_{i(jk)(lm)n}^{(z)} = -\mu_{i(kj)(mn)n}^{(z)}, \quad {\rm for} j\ne k \quad l\ne m \end{split} This DMF form was used to produce the APTY line list for HOOH [15AlOvYu]_. The TROVE input file describing this spectroscopic model can be found in `APTY spectroscopic model `__. For this mode, the ``Dipole`` block is given by :: Dipole dimension 3 NPARAM 136 96 98 DMS_TYPE HOOH_MB COEFF powers COORDS linear Order 6 6 6 parameters re1 0 0 0 0 0 0 0 1.45538654 re2 0 0 0 0 0 0 0 0.96257063 alphae 0 0 0 0 0 0 0 101.08307909 beta1 0 0 0 0 0 0 0 1.00000000 beta2 0 0 0 0 0 0 0 1.00000000 xxxxx 0 0 0 0 0 0 0 0.00000000 mu000000 0 0 0 0 0 0 7 3.11323258 mu000001 0 0 0 0 0 1 6 -0.05725254 mu000002 0 0 0 0 0 2 5 -0.00374176 mu000003 0 0 0 0 0 3 4 -0.0048 .... where ``dimension`` is 3. ``HPPH_MB`` ^^^^^^^^^^^ Subroutine: MLdms_hpph_MB See :download:`./input/39K-16O-1H__OYT4_model_TROVE.inp` - ``HCCH_MB`` - ``HCCH_DMS_7D`` - ``HCCH_DMS_7D_7ORDER`` - ``HCCH_DMS_7D_7ORDER_LINEAR`` - ``HCCH_DMS_7D_LOCAL`` - ``HCCH_ALPHA_ISO_7D_LINEAR`` ZXY\ :sub:`2` ------------ - ``ZXY2_SYMADAP`` ZXY\ :sub:`3` ------------- ZXY3_SYM ^^^^^^^^ C\ :sub:`2`\ H\ :sub:`4` ----------------------- - ``DIPOLE_C2H4_4M`` Properties ---------- - ``XY2_SR-BISECT`` - ``XY2_SS_DIPOLE_YY`` ^^^^^^^^^^^^^^^^ - ``XY2_G-BISECT`` - ``XY2_G-ROT-ELEC`` - ``XY2_G-COR-ELEC`` - ``XY2_G-TENS-RANK3`` - ``XY2_G-TENS-NUC`` - ``XY3_NSS_MB`` - ``COORDINATES``