声子计算的几种方法：
Practical schemes for phonon calculations  （见castep说明）
A good review of the existing schemes can be found in Baroni et al. (2001). The theoretical study of phonon properties has to rely on one of the three available methods for determining the force constants matrix: analytical calculations, supercell calculations or linear response calculations. The analytical approach is only viable when the energy model is sufficiently simple to allow a direct evaluation of the second derivatives of the energy with respect to atomic displacements (e.g., empirical pair potential models). Therefore, it is unsuitable for first principles calculations. Further alternatives such as extracting vibrational properties from molecular dynamics runs (Arias et al. 1992) are less transparent and noticeably more expensive.

The supercell method involves perturbing the positions of the atoms slightly and calculating the reaction forces (Ackland et al. 1997). It is necessary to use supercells of the original cell when interatomic interaction in the system is long ranged. The main advantage of this method (and of the closely related frozen phonon technique) is that there is no need for a new formalism; any total energy scheme like CASTEP can be used to evaluate the forces at a number of carefully selected distorted configurations. The original frozen phonon scheme requires a displacement with the given wave vector and has been successfully used since the early 1980s (Yin and Cohen 1982, Ho et al. 1984). The force constants matrix evaluation in this formalism has been used to calculate interplanar force constants (Wei and Chou 1994) and thus phonon dispersion along high symmetry directions. More recent applications are based on the full reconstruction of the force constants matrix (Ackland et al. 1997, Parlinski et al. 1997, and references in Baroni et al. 2001).

Linear response calculations seek to evaluate the dynamical matrix directly for a set of q vectors. The starting point of the linear response approach is evaluation of the second-order change in the total energy induced by atomic displacements. The main advantage of the scheme is that there is no need to artificially increase the cell size in order to accommodate small values of the q vectors, as in the frozen phonon method, or to overcome the long range interaction problem (force constants matrix from supercell calculations). A more detailed description of the linear response method can be found in Baroni et al. 2001. The CASTEP implementation is described in the Linear Response topic.

   

第一性原理计算声子方法及常见程序：
一，直接法：
直接法，或称frozen-phonon方法，是通过在优化后的平衡结构中引入原子位移，计算作用在原子上的Hellmann-Feynman力，进而由动力学矩阵算出声子色散曲线。用该方法计算声子色散曲线最早开始于80年代初。由于计算简便，不需要特别编写的计算程序，很多小组都采用直接法计算材料性质。直接法的缺陷在于它要求声子波矢与原胞边界(super size)正交，或者原胞足够大使得Hellmann-Feynman力在原胞外可以忽略不计。这使得对于复杂系统，如对称性高的晶体、合金、超晶格等材料需要采用超原胞。超原胞的采用使计算量急剧增加，极大的限制了该方法的使用。这种方法不能很好的预言LO-TO splitting,只有在计算了Born effective charge和dielectric constant之后，进一步考虑了 non-analyticity term，才能计算出;但Direct Method本身并不能给出Born effective charge和dielectric constant.所以这也是它的一个缺陷.目前,vasp+phonon用的就是这种方法.
vasp+phonon（或者PHON或者fropho）
VASP能计算声子谱的都是采用一种直接的方法：构造超原胞，把原子移动一下，计算原胞中所有原子所受的力（这个根据体系的周期性，要多移动几个原子），然后根据这个力构造力常数矩阵。

1，PHONON Software
by Krzysztof PARLINSKI
Phonon is a software (see list of Publications) for calculating phonon dispersion curves, and phonon density spectra of crystals, crystals with defects, surfaces, adsorbed atoms on surfaces, etc. from either a set of force constants, or from a set of Hellmann-Feynman forces calculated within an ab initio program (not included). One can use VASP, Wien2k, MedeA of Materials Design , Siesta, or other ab initio code which is able to optimize a supercell and calculate the Hellmann-Feynman forces. Phonon builds a crystal structure, using one of the 230 crystallographic space groups, finds the force constant from the Hellmann-Feynman forces, builds the dynamical matrix, diagonalizes it, and calculates the phonon dispersion relations, and their intensities. Phonon finds the polarization vectors, and the irreducible representations (Gamma point) of phonon modes, and calculates the total and partial phonon density of states. It plots the internal energy, free energy, entropy, heat capacity and tensor of mean square displacements (Debey-Waller factor). Phonon finds the dynamical structure factor for the coherent inelastic neutron scattering and the incoherent doubly differential scattering cross section for a single crystal and polycrystal. For polar cystals the LO/TO mode splitting can be included.
Homepage：http://wolf.ifj.edu.pl/phonon/index.html

2，PHON
A program to calculate phonons using the small displacement method
This program calculates force constant matrices and phonon frequencies in crystals. From the frequencies it also calculates various thermodynamic quantities, like Helmholtz free energy, entropy, specific heat and internal energy of the harmonic crystal. The procedure similar to the one described in Ref. [1], i.e. is based on the small displacement method. It needs a code capable to calculate forces on the atoms of the crystal.
Homepage：http://chianti.geol.ucl.ac.uk/~dario/
E-mail: d.alfe@ucl.ac.uk
Telephone: +44 (0)20 7679 2361
Fax: +44 (0)20 7679 5166

3，fropho
is the open source implementation of the frozen phonon method.
Function:
Phonon band structure
Phonon DOS (Vibrational spectra)
Thermal properties
Mulliken notation assignment of vibration mode
fropho is the frozen phonon analyzer mainly for first principles (ab initio) calculation. Periodic boundary condition is assumed. fropho gives good combinations with VASP code or another codes which can derive Hellmann-Feynman forces.
Homepage：http://fropho.sourceforge.net/
Download: http://sourceforge.net/project/showfiles.php?group_id=161614
Contact: atz.togo@gmail.com
Authour: Atsushi Togo


二，DFPT方法：
1987年，Baroni、Giannozzi和Testa提出了一种新的晶格动力学性质计算方法--微扰密度泛函方法(Density Function Perturbation Theory)。DFPT通过计算系统能量对外场微扰的响应来求出晶格动力学性质。该方法最大的优势在于它不限定微扰的波矢与原胞边界(super size)正交，不需要超原胞也可以对任意波矢求解。因此可以应用到复杂材料性质的计算上。此外，能量对外场微扰的响应不仅可以推导出声子的晶体性质，还能求出弹性系数、声子展宽、拉曼散射截面等性质,这种方法本身就能算出Born effective charge dielectric constant,可以很好的预言LO-TO splitting甚至Kohn anomalies。这些优势使得DFPT一经提出就被广泛应用到了半导体、金属和合金、超导体等材料的计算上。比较常用的程序是pwscf和abinit，castep等采用的是一种linear response theory 的方法（或者称为  density perturbation functional theory，DFPT），直接计算出原子的移动而导致  的势场变化，再进一步构造出动力学矩阵。