@@ -35,7 +35,9 @@ Caesar Anharmonic Calculation Library

Purpose of Module

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The :ref:`Caesar <Caesar>` anharmonic calculation library aims to provide an efficient method for calculating vibrational properties beyond the harmonic approximation; under the vibrational self-consistent harmonic approximation (VSCHA) [cite] or using vibrational self-consistent field theory (VSCF) [cite].

The :ref:`Caesar <Caesar>` anharmonic calculation library aims to provide an efficient method for calculating vibrational properties beyond the harmonic approximation; under the vibrational self-consistent harmonic approximation (VSCHA) [Errea_ea]_ or using vibrational self-consistent field theory (VSCF) [Christiansen]_.

.. [Christiansen] Vibrational structure theory: new vibrational wave function methods for calculation of anharmonic vibrational energies and vibrational contributions to molecular properties. https://doi.org/10.1039/B618764A

Theory

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@@ -43,11 +45,11 @@ ______

Fitting the Potential Energy Surface

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:ref:`Caesar <Caesar>` models the nuclear potential energy surface (PES) using a truncated Taylor expansion in normal-mode co-ordinates. Constructing and fitting this model happens over several steps

:ref:`Caesar <Caesar>` models the nuclear potential energy surface (PES) using a truncated Taylor expansion in normal-mode coordinates. Constructing and fitting this model happens over several steps:

- Firstly, a set of symmetry-invariant basis functions are generated, using the crystal symmetries as calculated by `spglib <https://github.com/spglib>`_.

- Secondly, a set of nuclear co-ordinates :math:`\mathbf{r}` are generated at which the PES will be sampled.

- Thirdly, electronic structure calculations are performed at each co-ordinate, using the :ref:`Caesar_Electronic_Interface`.

- Secondly, a set of nuclear coordinates :math:`\mathbf{r}` are generated at which the PES will be sampled.

- Thirdly, electronic structure calculations are performed at each coordinate, using the :ref:`Caesar_Electronic_Interface`.

- Finally, the results of the electronic structure calculations, including calculated energies, forces and other information, are used to calculate the basis function coefficients.

As with the harmonic calculation, the anharmonic calculation uses the non-diagonal supercell method [Lloyd-Williams_Monserrat]_ to reduce the total computational cost of the electronic structure calculations where possible.

@@ -34,7 +34,7 @@ Caesar electronic structure interface

Purpose of Module

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Calculating vibrational properties requires a mapping of the nuclear potential energy surface (PES) :math:`V(\mathbf{r})`, where :math:`\mathbf{r}` is the collective co-ordinate describing the locations of the nuclei. :ref:`Caesar <Caesar>` maps the PES by sampling it at a number of nuclear configurations :math:`\mathbf{r}_i`.

Calculating vibrational properties requires a mapping of the nuclear potential energy surface (PES) :math:`V(\mathbf{r})`, where :math:`\mathbf{r}` is the collective coordinate describing the locations of the nuclei. :ref:`Caesar <Caesar>` maps the PES by sampling it at a number of nuclear configurations :math:`\mathbf{r}_i`.

In software terms, each PES sample represents a single electronic structure calculation, where the electronic structure code is given the nuclear configuration :math:`\mathbf{r}_i`, and calculates the value of the PES at that configuration, :math:`V(\mathbf{r}_i)`, optionally along with other quantities such as the forces :math:`\mathbf{f}(\mathbf{r}_i)=-\frac{\partial}{\partial \mathbf{r}}V|_{\mathbf{r}_i}` and the Hessian matrix :math:`H(\mathbf{r}_i) = \frac{\partial}{\partial \mathbf{r}}\frac{\partial}{\partial \mathbf{r}}V|_{\mathbf{r}_i}`.

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@@ -58,7 +58,7 @@ where:

- :math:`L'` is the reciprocal supercell lattice matrix of the supercell, defined as :math:`L'=L^{-T}`.

- :math:`S` is the supercell matrix, which relates the supercell lattice matrix :math:`L` to the primitive cell lattice matrix :math:`L_p` as :math:`L=SL_p`.

- :math:`S'` is the reciprocal supercell matrix, defined as :math:`S'=S^{-T}`.

- :math:`z_i` and :math:`\mathbf{r}_i` are the species label and cartesian co-ordinate of the :math:`i`'th atom.

- :math:`z_i` and :math:`\mathbf{r}_i` are the species label and cartesian coordinate of the :math:`i`'th atom.

- :math:`\{R_i\}` are the R-vectors of the primitive cell which are contained within the supercell.

- :math:`\{G_i\}` are the G-vectors of the reciprocal supercell which are contained within the reciprocal primitive cell.

- Subscripts :math:`x`, :math:`y` and :math:`z` denote cartesian components.

The nuclear potential energy surface :math:`V` is a function of the :math:`3n`-dimensional nuclear co-ordinate :math:`\mathbf{r}`. Under the harmonic approximation [Hoja_ea]_, this function is approximated as quadratic in the difference between :math:`\mathbf{r}` and the value of :math:`\mathbf{r}` for the undisplaced structure, :math:`\mathbf{r}^{(0)}`. Formally, this is

The nuclear potential energy surface :math:`V` is a function of the :math:`3n`-dimensional nuclear coordinate :math:`\mathbf{r}`. Under the harmonic approximation [Hoja_ea]_, this function is approximated as quadratic in the difference between :math:`\mathbf{r}` and the value of :math:`\mathbf{r}` for the undisplaced structure, :math:`\mathbf{r}^{(0)}`. Formally, this is

.. math::

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@@ -51,7 +51,7 @@ The nuclear potential energy surface :math:`V` is a function of the :math:`3n`-d

where :math:`H` is the Hessian matrix.

By making a co-ordinate transform from :math:`\mathbf{r}=\sum_ir_i\hat{\mathbf{r}}_i` to :math:`\mathbf{r}=\sum_ju_j\hat{\mathbf{u}}_j`, the Hessian matrix can be diagonalised, to give

By making a coordinate transform from :math:`\mathbf{r}=\sum_ir_i\hat{\mathbf{r}}_i` to :math:`\mathbf{r}=\sum_ju_j\hat{\mathbf{u}}_j`, the Hessian matrix can be diagonalised, to give