Update manual

doc/reference.rst

@@ -4,13 +4,30 @@ References
 Method used in phonopy
 -----------------------
 
+.. _reference_force_constants:
+
+Generation of force constants
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+In phonopy, force constants are generated based on finite displacement
+method. Crystal symmetry is used to reduce the calculation cost and
+numerical noise of the force constants. Firstly a symmetry reduced set
+of atomic displacements is generated. After the atomic force
+calculations, the set of atomic displacements are expanded using the
+symmetry and then all the elements of force constans between atoms in
+a primitive cell and the supercell are fit to the symmetry expanded
+forces of atoms in supercells using Moore–Penrose pseudoinverse. This
+procedure may considered as a variant of :ref:`reference_plk`. Some of
+the details are found in the appendix of the following paper:
+
+- \L. Chaput, A. Togo, I. Tanaka, and G. Hug, Phys. Rev. B, 84,
+  094302 (2011)
+
 .. _reference_plk:
 
 Parlinski-Li-Kawazoe method
 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
 
-- \K. Parlinski, Z. Q. Li, and Y. Kawazoe, Phys. Rev. Lett. 78, 4063 (1997)
-
 Parlinski-Li-Kawazoe method is based on the supercell approach with
 the finite displacement method. 
 
@@ -21,15 +38,39 @@ pseudoinverse is easy to handle arbitrary number of displacements
 amplitudes and directions, and can rely on the exisiting library,
 e.g., LAPACK.
 
-In phonopy, the symmetry reduced elements of force constants are not
-prepared. Simply all the elements of force constans between atoms in
-a primitive cell and the supercell are fit. 
+- \K. Parlinski, Z. Q. Li, and Y. Kawazoe, Phys. Rev. Lett. 78, 4063 (1997)
+
+
+.. _reference_thermal_expansion:
+
+Thermal expansion using quasi-harmonic approximation
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+In :ref:`phonopy-qha <phonopy_qha>`, thermal properties at constant
+pressure is obtained from the thermodynamic definition.  To achieve
+Legendre transformation, volume-energy function is generated from a
+set of Helmholtz free energies and *pV* terms at volumes by fitting to
+a smooth function for which equations of states are prepared in phonopy-qha.
+
+The volume dependence of the Helmholtz free energy is included from
+quasi-harmonicity. When using DFT-GGA (-LDA), often we should have
+some amount of error in the absolute value since phonon frequencies
+are underestimated (overestimated). However the value of some ratio
+like thermal expansion coefficient is often very well estimated. An
+example is shown in the following paper:
+
+- \A. Togo, L. Chaput, I. Tanaka, G. Hug, Phys. Rev. B, 81, 174301-1-6 (2010)
 
 .. _reference_NAC:
 
 Non-analytical term correction
 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
 
+Non-metallic crystals are polarized by atomic displacements and the
+generated macroscopic field changes force constants near
+:math:`\Gamma` point. This contribution is included through
+non-analytical term correction.
+
 - \R. M. Pick, M. H. Cohen, and R. M. Martin, Phys. Rev. B 1, 910, (1970)
 - \P. Giannozzi, S. Degironcoli, P. Pavone, and S. Baroni,  Phys. Rev. B 43, 7231 (1991)
 - \X. Gonze, and C. Lee, Phys. Rev. B 55, 10355 (1997)
@@ -39,17 +80,17 @@ Non-analytical term correction
 Interpolation scheme at general *q*-points with non-analytical term correction
 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
 
+The above non-anarytical term correction can be applied only at
+:math:`\mathbf{q}\rightarrow \mathbf{0}`. To connect smoothly to
+general **q** points, the following interpolation scheme is employed
+in phonopy (:ref:`nac_tag`).
+
 - \Y Wang , J J Wang , W Y Wang , Z G Mei , S L Shang , L Q Chen and
   Z K Liu, J. Phys.: Condens. Matter. 22, 202201 (2010)
 
-.. - \K. Parlinski, Z. Q. Li, and Y. Kawazoe, Phys. Rev. Lett. 81, 3298 (1998)
 
-Interpolation scheme at getenral *q*-points with non-analytical term
-correction is implemented according to Wang *et al* (``--nac``
-option).
-
-Other methods for calculating force constants
------------------------------------------------
+Other methods and software for calculating force constants
+-----------------------------------------------------------
 
 .. _reference_small_displacement: