5.1. ALM: Input files

Format of input files

Each input file should consist of entry fields. Available entry fields are

&general, &interaction, &cutoff, &cell, &position, and &optimize (&fitting).

Each entry field starts from the key label &field and ends at the terminate character “/”. (This is equivalent to Fortran namelist.)

For example, &general entry field of program alm should be given like

&general
  # Comment line
  PREFIX = prefix
  MODE = fitting
/

Multiple entries can be put in a single line. Also, characters put on the right of sharp (“#”) are neglected. Therefore, the above example is equivalent to the following:

&general
  PREFIX = prefix; MODE = fitting  # Comment line
/

Each variable must be given inside the appropriate entry field.

Description of input variables

“&general”-field

  • PREFIX-tag : Job prefix to be used for names of output files
Default:None
Type:String

  • MODE-tag = optimize | suggest | fitting
optimize (>= 1.1.0)
Estimate harmonic and anharmonic IFCs.
This mode requires an appropriate &optimize field.
fitting (deprecated)
An alias of MODE = optimize
suggest
Suggests the displacement patterns necessary
to estimate harmonic and anharmonic IFCS.
Default:None
Type:String

  • NAT-tag : Number of atoms in the supercell
Default:None
Type:Integer

  • NKD-tag : Number of atomic species
Default:None
Type:Integer

  • KD-tag = Name[1], … , Name[NKD]
Default:None
Type:Array of strings
Example:In the case of GaAs with NKD = 2, it should be KD = Ga As.

  • TOLERANCE-tag : Tolerance for finding symmetry operations
Default:1.0e-3
Type:Double

  • PRINTSYM-tag = 0 | 1
0 Symmetry operations won’t be saved in “SYMM_INFO”
1 Symmetry operations will be saved in “SYMM_INFO”
Default:0
type:Integer

  • PERIODIC-tag = PERIODIC[1], PERIODIC[2], PERIODIC[3]
0
Do not consider periodic boundary conditions when
searching for interacting atoms.
1
Consider periodic boundary conditions when
searching for interacting atoms.
Default:1 1 1
type:Array of integers
Description:This tag is useful for generating interacting atoms in low dimensional systems. When PERIODIC[i] is zero, periodic boundary condition is turned off along the direction of the lattice vector \(\boldsymbol{a}_{i}\).

  • HESSIAN-tag = 0 | 1
0 Do not save the Hessian matrix
1 Save the entire Hessian matrix of the supercell as PREFIX.hessian.
Default:0
type:Integer

“&interaction”-field

  • NORDER-tag : The order of force constants to be calculated. Anharmonic terms up to \((m+1)\)th order will be considered with NORDER = \(m\).
Default:None
Type:Integer
Example:NORDER = 1 for calculate harmonic terms only, NORDER = 2 to include cubic terms as well, and so on.

  • NBODY-tag : Entry for excluding multiple-body interactions from anharmonic force constants
Default:

NBODY = [2, 3, 4, …, NORDER + 1]

Type:

Array of integers

Description:

This tag may be useful for excluding multi-body clusters which are supposedly less important. For example, a set of fourth-order IFCs \(\{\Phi_{ijkl}\}\), where \(i, j, k\), and \(l\) label atoms in the supercell, can be categorized into four different subsets; on-site, two-body, three-body, and four-body terms. Neglecting the Cartesian coordinates of IFCs for simplicity, each subset contains the IFC elements shown as follows:

on-site
\(\{\Phi_{iiii}\}\)
two-body
\(\{\Phi_{iijj}\}\), \(\{\Phi_{iiij}\}\) (\(i\neq j\))
three-body
\(\{\Phi_{iijk}\}\) (\(i\neq j, i\neq k, j \neq k\))
four-body
\(\{\Phi_{ijkl}\}\) (all subscripts are different from each other)

Since the four-body clusters are expected to be less important than the three-body and less-body clusters, you may want to exclude the four-body terms from the Taylor expansion potential because the number of such terms is huge. This can be done by setting the NBODY tag as NBODY = 2 3 3 together with NORDER = 3.

More examples:

NORDER = 2; NBODY = 2 2 includes harmonic and cubic IFCs but excludes three-body clusters from the cubic terms.

NORDER = 5; NBODY = 2 3 3 2 2 includes anharmonic terms up to the sixth-order, where the four-body clusters are excluded from the fourth-order IFCs, and the multi (\(\geq 3\))-body clusters are excluded from the fifth- and sixth-order IFCs.


“&cutoff”-field

In this entry field, one needs to specify cutoff radii of interaction for each order in units of Bohr. In the current implementation, cutoff radii should be defined for every possible pair of atomic elements. For example, the cutoff entry for a harmonic calculation (NORDER = 1) of Si (NKD = 1) should be like

&cutoff
 Si-Si 10.0
/

This means that the cutoff radius of 10 \(a_{0}\) is used for harmonic Si-Si terms. Please note that the first column should be two character strings, which are contained in the KD-tag, connected by a hyphen (’-’).

When one wants to consider cubic terms (NORDER = 2), please specify the cutoff radius for cubic terms in the third column as the following:

&cutoff
 Si-Si 10.0 5.6 # Pair r_{2} r_{3}
/

Instead of giving specific cutoff radii, one can write “None” as follows:

&cutoff
 Si-Si None 5.6
/

which means that all possible harmonic terms between Si-Si atoms will be included.

Caution

Setting ‘None’ for anharmonic terms can greatly increase the number of parameters and thereby increase the computational cost.

When there are more than two atomic elements, please specify the cutoff radii between every possible pair of atomic elements. In the case of MgO (NKD = 2), the cutoff entry should be like

&cutoff
 Mg-Mg 8.0
 O-O 8.0
 Mg-O 10.0
/

which can equivalently be written by using the wild card (’*’) as

&cutoff
 *-* 8.0
 Mg-O 10.0 # Overwrite the cutoff radius for Mg-O harmonic interactions
/

Important

Cutoff radii specified by an earlier entry are overwritten by a new entry that comes later.

Once the cutoff radii are properly given, harmonic force constants \(\Phi_{i,j}^{\mu,\nu}\) satisfying \(r_{ij} \le r_{c}^{\mathrm{KD}[i]-\mathrm{KD}[j]}\) will be searched.

In the case of cubic terms, force constants \(\Phi_{ijk}^{\mu\nu\lambda}\) satisfying \(r_{ij} \le r_{c}^{\mathrm{KD}[i]-\mathrm{KD}[j]}\), \(r_{ik} \le r_{c}^{\mathrm{KD}[i]-\mathrm{KD}[k]}\), and \(r_{jk} \le r_{c}^{\mathrm{KD}[j]-\mathrm{KD}[k]}\) will be searched and determined by fitting.


“&cell”-field

Please give the cell parameters in this entry in units of Bohr as the following:

&cell
 a
 a11 a12 a13
 a21 a22 a23
 a31 a32 a33
/

The cell parameters are then given by \(\vec{a}_{1} = a \times (a_{11}, a_{12}, a_{13})\), \(\vec{a}_{2} = a \times (a_{21}, a_{22}, a_{23})\), and \(\vec{a}_{3} = a \times (a_{31}, a_{32}, a_{33})\).


“&position”-field

In this field, one needs to specify the atomic element and fractional coordinate of atoms in the supercell. Each line should be

ikd xf[1] xf[2] xf[3]

where ikd is an integer specifying the atomic element (ikd = 1, …, NKD) and xf[i] is the fractional coordinate of an atom. There should be NAT such lines in the &position entry field.


“&optimize”-field (“&fitting”-field)

This field is necessary when MODE = optimize (or a deprecated option MODE = fitting).

  • LMODEL-tag : Choice of the linear model used for estimating force constants
“least-squares”, “LS”, “OLS”, 1 Ordinary least square
“elastic-net”, “enet”, 2 Elastic net
Default:

least-squares

Type:

String

Description:

When LMODEL = ols, the force constants are estimated from the displacement-force datasets via the ordinary least-squares (OLS), which is usually sufficient to calculate harmonic and third-order force constants.

The elastic net (LMODEL = enet) should be useful to calculate the fourth-order (and higher-order) force constants. When the elastic net is selected, the users have to set the following related tags: CV, L1_RATIO, L1_ALPHA, CV_MAXALPHA, CV_MINALPHA, CV_NALPHA, STANDARDIZE, ENET_DNORM, MAXITER, CONV_TOL, NWRITE, SOLUTION_PATH, DEBIAS_OLS


  • DFSET-tag: File name containing displacement-force datasets for training

New in version 1.1.0.

Default:None
Type:String
Description:The format of DFSET can be found here

  • DFILE-tag: File name containing atomic displacements in Cartesian coordinate

Deprecated since version 1.1.0: Use DFSET instead.

Default:None
Type:String
Description:The format of DFILE can be found here. This tag is deprecated and will be removed in a future major release. Please use DFSET instead.

  • FFILE-tag: File name containing atomic forces in Cartesian coordinate

Deprecated since version 1.1.0: Use DFSET instead.

Default:None
Type:String
Description:The format of FFILE can be found here. This tag is deprecated and will be removed in a future major release. Please use DFSET instead.

  • NDATA-tag : Number of displacement-force data sets
Default:None
Type:Integer
Description:If NDATA is not given, the code reads all lines of DFSET (excluding comment lines) and estimates NDATA by dividing the line number by NAT. If the number of lines is not divisible by NAT, an error is raised. DFSET should contain at least NDATA\(\times\) NAT lines.

  • NSTART, NEND-tags : Specifies the range of data to be used for fitting
Default:NSTART = 1, NEND = NDATA
Type:Integer
Example:To use the data in the range of [20:30] out of 50 entries, the tags should be NSTART = 20 and NEND = 30.

  • SKIP-tag : Specifies the range of data to be skipped for training
Default:None
Type:Two integers connected by a hyphen
Description:SKIP =\(i\)-\(j\) skips the data in the range of [\(i\):\(j\)]. The \(i\) and \(j\) must satisfy \(1\leq i \leq j \leq\) NDATA. This option may be useful when doing cross-validation manually (CV=-1).

  • ICONST-tag = 0 | 1 | 2 | 3 | 11
0 No constraints
1
Constraint for translational invariance is imposed between IFCs.
Available only when LMODEL = ols.
11
Same as ICONST = 1 but the constraint is imposed algebraically rather than numerically.
Select this option when LMODEL = enet.
2
In addition to ICONST = 1, constraints for rotational invariance will be
imposed up to (NORDER + 1)th order. Available only when LMODEL = ols.
3
In addition to ICONST = 2, constraints for rotational invariance between (NORDER + 1)th order
and (NORDER + 2)th order, which are zero, will be considered.
Available only when LMODEL = ols.
Default:11
Type:Integer
Description:See this page for the numerical formulae.

  • ROTAXIS-tag : Rotation axis used to estimate constraints for rotational invariance. This entry is necessary when ICONST = 2, 3.
Default:None
Type:String
Example:When one wants to consider the rotational invariance around the \(x\)-axis, one should give ROTAXIS = x. If one needs additional constraints for the rotation around the \(y\)-axis, ROTAXIS should be ROTAXIS = xy.

  • FC2XML-tag : XML file to which the harmonic terms are fixed upon fitting
Default:None
Type:String
Description:When FC2XML-tag is given, harmonic force constants are fixed to the values stored in the FC2XML file. This may be useful for optimizing cubic and higher-order terms without changing the harmonic terms. Please make sure that the number of harmonic terms in the new computational condition is the same as that in the FC2XML file.

  • FC3XML-tag : XML file to which the cubic terms are fixed upon fitting
Default:None
Type:String
Description:Same as the FC2XML-tag, but FC3XML is to fix cubic force constants.

  • SPARSE-tag = 0 | 1
0 Use a direct solver (SVD or QRD) to estimate force constants
1 Use a sparse solver to estimate force constants
Default:0
Type:Integer
Description:When you want to calculate force constants of a large system and generate training datasets by displacing only a few atoms from equilibrium positions, the resulting sensing matrix becomes large but sparse. For such matrices, a sparse solver is expected to be more efficient than SVD or QRD in terms of both memory usage and computational time. When SPARSE = 1 is set, the code uses a sparse solver implemented in Eigen3 library. You can change the solver type via SPARSESOLVER. Effective when LMODEL = ols.

  • SPARSESOLVER-tag : Type of the sparse solver to use
Default:SimplicialLDLT
Type:String
Description:Currently, only the sparse solvers of Eigen3 library can be used. Available options are SimplicialLDLT, SparseQR, ConjugateGradient, LeastSquaresConjugateGradient, and BiCGSTAB. When an iterative algorithm (conjugate gradient) is selected, a stopping criterion can be specified by the CONV_TOL and MAXITER tags. Effective when LMODEL = ols and SPARSE = 1.

See also

Eigen documentation page: Solving Sparse Linear Systems


  • MAXITER-tag : Number of maximum iterations in iterative algorithms
Default:10,000
Type:Integer
Description:Effective when an iterative solver is selected via SPARSESOLVER (when LMODEL = ols) or when LMODEL = enet.

  • CONV_TOL-tag : Convergence criterion of iterative algorithms
Default:1.0e-8
Type:Double
Description:When LMODEL = ols and an iterative solver is selected via SPARSESOLVER, CONV_TOL value is passed to the Eigen3 function via setTolerance(). When LMODEL = enet, the coordinate descent iteration stops at \(i\)th iteration if \(\sqrt{\frac{1}{N}|\boldsymbol{\Phi}_{i} - \boldsymbol{\Phi}_{i-1}|_{2}^{2}} <\) CONV_TOL is satisfied, where \(N\) is the length of the vector \(\boldsymbol{\Phi}\).

See also

Eigen documentation page: IterativeSolverBase


  • L1_RATIO-tag : The ratio of the L1 regularization term
Default:1.0 (LASSO)
Type:Double
Description:The L1_RATIO changes the regularization term as L1_ALPHA \(\times\) [L1_RATIO \(|\boldsymbol{\Phi}|_{1}\) + \(\frac{1}{2}\) (1-L1_RATIO) \(|\boldsymbol{\Phi}|_{2}^{2}\)]. Therefore, L1_RATIO = 1 corresponds to LASSO. L1_RATIO must be 0 < L1_ratio <= 1. Effective when LMODEL = enet. See also here.

  • L1_ALPHA-tag : The coefficient of the L1 regularization term
Default:0.0
Type:Double
Description:This tag is used only when LMODEL = enet and CV = 0. See also here.

  • CV-tag : Cross-validation mode for elastic net
0
Cross-validation mode is off.
The elastic net optimization is solved with the given L1_ALPHA value.
The force constants are written to PREFIX.fcs and PREFIX.xml.
>= 2
CV-fold cross-validation is performed automatically.
NDATA training datasets are divided into CV subsets, and CV different combinations of
training-validation datasets are created internally. For each combination, the elastic net
optimization is solved with the various L1_ALPHA values defined by the CV_MINALPHA,
CV_MAXALPHA, and CV_NALPHA tags. The result of each cross-validation is stored in
PREFIX.enet_cvset[1, …, CV], and their average and deviation are stored in PREFIX.cvscore.
-1
The cross-validation is performed manually.
The Taylor expansion potential is trained by using the training datasets in DFSET, and
the validation score is calculated by using the data in DFSET_CV for various L1_ALPHA values
defined the CV_MINALPHA, CV_MAXALPHA, and CV_NALPHA tags.
After the calculation, the fitting and validation errors are stored in PREFIX.enet_cv.
This option may be convenient for a large-scale problem since multiple optimization tasks with
different training-validation datasets can be done in parallel.
Default:0
Type:Integer
Description:This tag is used only when LMODEL = enet.

  • DFSET_CV-tag : File name containing displacement-force datasets used for manual cross-validation
Default:DFSET_CV = DFSET
Type:String
Description:This tag is used only when LMODEL = enet and CV = -1.

  • NDATA_CV-tag : Number of displacement-force validation datasets
Default:None
Type:Integer
Description:This tag is used only when LMODEL = enet and CV = -1.

  • NSTART_CV, NEND_CV-tags : Specifies the range of data to be used for validation
Default:NSTART_CV = 1, NEND_CV = NDATA_CV
Type:Integer
Example:This tag is used only when LMODEL = enet and CV = -1.

  • CV_MINALPHA, CV_MAXALPHA, CV_NALPHA-tags : Options to specify the L1_ALPHA values used in cross-validation
Default:CV_MINALPHA = 1.0e-4, CV_MAXALPHA = 1.0, CV_NALPHA = 1
Type:Double, Double, Integer
Description:CV_NALPHA values of L1_ALPHA are generated from CV_MINALPHA to CV_MAXALPHA in logarithmic scale. A recommended value of CV_MAXALPHA is printed out to the log file. This tag is used only when LMODEL = enet and the cross-validation mode is on (CV > 0 or CV = -1).

  • STANDARDIZE-tag = 0 | 1
0 Do not standardize the sensing matrix
1
Each column of the sensing matrix is standardized in such a way that its mean value
becomes 0 and standard deviation becomes 1.
Default:1
Type:Integer
Description:This option influences the optimal L1_ALPHA value. So, if you change the STANDARDIZE option, you have to rerun the cross-validation. Effective only when LMODEL = enet.

  • ENET_DNORM-tag : Normalization factor of atomic displacements
Default:1.0
Type:Double
Description:The normalization factor of atomic displacement \(u_{0}\) in units of Bohr. When \(u_{0} (\neq 1)\) is given, the displacement data are scaled as \(u_{i} \rightarrow u_{i}/u_{0}\) before constructing the sensing matrix. This option influences the optimal L1_ALPHA value. So, if you change the ENET_DNORM value, you will have to rerun the cross-validation. Effective only when LMODEL = enet and STANDARDIZE = 0.

  • SOLUTION_PATH-tag = 0 | 1
0 Do not save the solution path.
1 Save the solution path of each cross-validation combination in PREFIX.solution_path.
Default:0
Type:Integer
Description:Effective when LMODEL = enet and the cross-validation mode is on.

  • DEBIAS_OLS-tag = 0 | 1
0 Save the solution of the elastic net problem to PREFIX.fcs and PREFIX.xml.
1
After the solution of the elastic net optimization problem is obtained,
only non-zero coefficients are collected, and the ordinary least-squares fitting is
solved again with the non-zero coefficients before saving the results to PREFIX.fcs and
PREFIX.xml. This might be useful to reduce the bias of the elastic net solution.
Default:0
Type:Integer
Description:Effective when LMODEL = enet and CV = 0.

How to make a DFSET file

Format of DFILE and FFILE (deprecated)

Deprecated since version 1.1.0: Use DFSET instead.

The displacement-force data sets obtained by first-principles (or classical force-field) calculations have to be saved to DFILE and FFILE to estimate IFCs with MODE = fitting. In DFILE, please explicitly specify the atomic displacements \(u_{\alpha}(\ell\kappa)\) in units of Bohr as follows:

\begin{eqnarray*} u_{x}(1) & u_{y}(1) & u_{z}(1) \\ u_{x}(2) & u_{y}(2) & u_{z}(2) \\ & \vdots & \\ u_{x}(\mathrm{NAT}) & u_{y}(\mathrm{NAT}) & u_{z}(\mathrm{NAT}) \end{eqnarray*}

When there are NAT atoms in the supercell and NDATA data sets, there should be NAT \(\times\) NDATA lines in the DFILE without blank lines. In FFILE, please specify the corresponding atomic forces in units of Ryd/Bohr.