Descriptors

r_cut (real)

The cut-off of the descriptor (in Å).

Default r_cut=5.0

desc_forces (logical)

Compute or not the force descriptors. Also the writting of the force descriptors is desactivated. Active for all descriptors. In the case desc_forces=.false. only the positions are read.

Default desc_forces=.true.

descriptor_type (integer)

Type of descriptor to be used.

descriptor_type=1 G2
descriptor_type=2 G3
descriptor_type=3 Behler
descriptor_type=4 Angular Fourier Series (AFS)
descriptor_type=5 Smooth Overlap of Atomic Positions (SOAP)
descriptor_type=6 Power Spectrum SO3
descriptor_type=7 Bispectrum SO3
descriptor_type=8 Power Spectrum SO4
descriptor_type=9 bispectrum SO4
descriptor_type=14 hybrid G2 + AFS
descriptor_type=19 hybrid G2 + bispectrum SO4
descriptor_type=100 MTP\(^3\) (up to order \(M_{\mu,\nu}\), with \(\nu \le 3\))
descriptor_type=200 PiP permutationally invariant polynomials.

Default 1.

G2

To use G2, set descriptor_type=1.

Parameters

eta_max_g2 (integer)

The max value of \(\eta\). The grid will be taken between \(10^{-2}\) and this value.

Default eta_max_g2=0.8

n_g2_eta (integer)

The grid for \(\eta\) with the following generatig formulae:

\[\eta_p = 10^{-2} + \frac{p-1}{\textrm{n_g2_eta}}( \textrm{eta_max_g2}- 10^{-2})\]

where \(p\) range strats to 1 up to n_g2_eta.

Default 3

n_g2_rs (integer)

The grid for \(r_s\) with the following generating formulae:

\[r_s^p = p-1\]

where \(p\) range starts to 1 up to n_g2_rs.

Default 1

G3

To use G3, set descriptor_type=2.

Parameters

n_g3_eta (integer)

The grid for \(\eta\) with the following generatig formulae:

\[\eta_p = 10^{-2} + \frac{p-1}{\textrm{n_g2_eta}}(1 - 10^{-2})\]

where \(p\) range strats to 1 up to n_g2_eta.

Default 3

n_g3_lambda (integer)

the grid for \(\lambda\) with the following generating formulae:

\[\lambda_p = -1 + 2(p-1)\]

where \(p\) range starts to 1 up to n_g3_lambda. Is highly recommended the value n_g3_lambda\(=2\) i. e. \(\lambda\) will have the value \(1\) and \(-1\).

Default 2

n_g3_zeta (integer)

The grid for \(\zeta\) with the following generating formulae:

\[\zeta_p = 2^{p-1}\]

where \(p\) range starts to 1 up to n_g3_zeta.

Default 2

Behler

To use Behler, set descriptor_type=3.

Parameters

The parameters of the descriptors are controlled by the options of the case \(G2\) and \(G3\). Additional options are available:

strict_behler (logical)

All the parameters are set-up automatically following one of the Behler’s paper.

Default F

AFS

To use AFS, set descriptor_type=4.

Parameters

The parameters of the descriptors are controlled by the options of the number of radial channels and Tchbychev polynomials. There are two types of AFS descriptors introduced by the option afs_type.

afs_type (integer)

afs_type=1: integer this option activates the orginal AFS published in the PRB paper cite{bartok2013}:

\[\textrm{AFS}_{n,l}^j = \sum_{i,k} g_n(r_{ji}) g_n(r_{jk}) T_l (\cos{\theta_{ijk}})\]

and has the dimension equal to n_rbf \(\times\) (n_cheb + 1). The \(g_n\) and \(T_l\) are the radial channels and Tchebychev polynomials, respectivelly.
afs_type=2: integer this option activates a new AFS descriptor with strong coupling between the radial channels:

\[\textrm{AFS}_{n,n^\prime, l}^j = \sum_{i,k} g_n(r_{ji}) g_{n^{\prime}}(r_{jk}) T_l (\cos{\theta_{ijk}})\]

The dimension of this descriptor is equal to n_rbf\(^2\) \(\times\) (n_cheb + 1).

Default afs_type = 1

n_rbf (integer)

The number of radial channels.

Default n_rbf=4

n_cheb (integer)

The number of Tchebychev polynomials.

Default n_cheb=5

SOAP

To use SOAP, set descriptor_type=5.

Parameters

The parameters of the descriptor are controlled by the options of the number of Gaussian radial functions n_soap and maximum of the spherical harmonics l_max. If the nspecies_soap is set, the number of components are, if lsoap_diag=.true.: (l_max + 1) \(\times\) n_soap \(\cdot\) nspecies_soap\(\times\) (n_soap \(\cdot\) nspecies_soap +1)/2, whilst, if lsoap_diag=.false. the number of components is much less (l_max + 1) \(\times\) n_soap \(\times\) nspecies_soap\(\times\) (nspecies_soap +1)/2.

n_soap (integer)

The number of Gaussians.

Default n_soap=4

l_max (integer)

The max l of the spherical harmonics.

Default l_max=5

lsoap_diag (logical)

The SOAP descriptor is diagonal in radial functions.

Default lsoap_diag=.false.

lsoap_lnorm (logical)

The SOAP descriptor is normalized in each \(l\)-angular channel by a factor \(1/(2l+1)\).

Default lsoap_lnorm=.false.

lsoap_norm (logical)

The SOAP descriptor is normalized.

Default lsoap_norm=.false.

r_cut_width_soap (double precision)

The intermediate regime for the cutoff function.

Default r_cut_width_soap=0.5d0

Power Spectrum SO3

To use power spectrum SO3, set descriptor_type=6.

Parameters

The parameters of this descriptor are controlled by the number of radial functions, n_rbf, and the angular momentum of spherical harmonics l_max. This descriptor has two radial function choices given by radial_pow_so3 flag. The dimension of the descriptor will be: n_rbf \(\times\) (l_max + 1) if radial_pow_so3 = 1 and n_rbf + 1 \(\times\) (l_max + 1) if radial_pow_so3 = 2

radial_pow_so3 (integer)

The type of radial function. radial_pow_so3 = 1 is the default choice and give the original polynomial radial basis (the same as the default basis of AFS descritor) whilst radial_pow_so3 = 2 gives the radial basis based on Chebyshev polynomials described in refXXX. This option is common and acitve for others two descriptors: Power Spectrum SO3-3body and Bispectrum SO3 descriptors.

Default radial_pow_so3 = 1

n_rbf (integer)

The number of Gaussian (radial) functions.

Default n_rbf=4

l_max (integer)

The max values of the angular moment.

Default l_max=4

The \(n_{rbf} \times (1 + l_{\textrm{max}})\) or \((n_{rbf} + 1) \times (1 + l_{\textrm{max}})\) components of the power spectrum SO(3) descriptor of the \(j^{th}\) atom is written:

\[p_{nl}^j = \sum_{m=-l_{\textrm{max}}}^{l_{\textrm{max}}} c_{nlm}^{j*}c_{nlm}^j \nonumber\]

with \(n\) in the range \(0/1\) to \(n_{rbf}\), whilst \(l = 0, \ldots l_{max}\). The Wigner coefficients can be written:

\[c_{nlm}^j = w_j g_n(r=0) Y_{lm}(x,y,z \equiv \mathbf{0}) + \sum_{i \in v(j)} w_i g_n(r_{ij}) \cdotp Y_{lm}(x,y,z \equiv \mathbf{r}_{ij}) \nonumber\]

the functions \(g_n(r_{ij})\) are polynomial basis functions (the same as the radial function on which AFS is built) or Chebyshev polynomials from refXXX. \(Y_{lm}\) is the spherical function in the Cartesian form.

Power Spectrum SO3 3body

To use power spectrum SO3 3body, set descriptor_type=603. This descriptor follows the ideea of 3 body descriptor proposed by refXXXdeGironcoli.

Parameters

The parameters of this descriptor are controlled by the number of radial functions, n_rbf, and the angular momentum of spherical harmonics l_max. This descriptor has two radial function choices given by radial_pow_so3 flag. The dimension of the descriptor will be: \(n_{\textrm{rbf}}^2 \times (l_{\textrm{max}})\) if radial_pow_so3 = 1 or \((1 + n_{\textrm{rbf}})^2 \times (l_{\textrm{max}})\) if radial_pow_so3 = 2

radial_pow_so3 (integer)

The same utility as described before for Power Spectrum SO3 descriptor.

Default radial_pow_so3 = 1

n_rbf (integer)

The same utility as described before for Power Spectrum SO3 descriptor.

Default n_rbf=4

l_max (integer)

The max values of the angular moment.

Default l_max=4

The \(n_{rbf}^2 \times (1 + l_{\textrm{max}})\) or \((n_{rbf} + 1)^2 \times (1 + l_{\textrm{max}})\) components of the power spectrum SO(3) 3 body descriptor of the \(j^{th}\) atom is written:

\[p_{n_1 n_2 l}^j = \sum_{m=-l_{\textrm{max}}}^{l_{\textrm{max}}} c_{n_1 l -m}^{j*}c_{n_2 lm}^j \nonumber\]

with \(n_{1,é}\) are in the range of \(0/1\) to \(n_{rbf}\), whilst \(l = 0, \ldots l_{max}\). The Wigner coefficients are the same described for Power Spectrum SO3.

Bispectrum SO3

To use bispectrum SO3, set descriptor_type=7.

Parameters

The parameters of the descriptor are controlled by the number of Gaussian radial functions, n_rbf, and maximal angular momentum of spherical harmonics, (l_max. The bispectrum SO3 descriptor components of the \(i^{th}\) atom are obtained from the power spectrum SO3 coefficients \(c_{nlm}^i\):

\[b_{n l l_1 l_2}^i = \sum_{m=-l}^{l}\sum_{m_1=-l_1}^{l_1} \sum_{m_2=-l_2}^{l_2} c_{ n l m}^{*i}C_{m m_1 m_2}^{l l_1 l_2} c_{n l_1 m_1}^i c_{n l_2 m_2}^i \nonumber\]

where \(C_{m m_1 m_2}^{l l_1 l_2}\) is the 3-dimensional Clebsch-Gordan coefficients. The dimension of this desriptor is difficult to know beforehand. The naive estimation of the dimension is n_rbf \(\times\) (l_max + 1)\(^3\). However, the selection rules of the Clebsch-Gordan coefficients reduce drastically this number e.g. for n_rbf=7 and l_max=5 the number of components is 140 (instead ). This number follow the GC/Karakala convention and take only the diagonal CG coefficients i.e. \(l_1=l_2\) in previous equation. This condition can be released using lbso3_diag=.false. (in the above mentioned the dimension becomes 483). For numerical reasons, is highly recommended to use ONLY the diagonal form.

n_rbf (integer)

The number of Gaussian (radial) functions.

Default n_rbf=4

l_max (integer)

The max values of the angular moment.

Default l_max=4

lbso3_diag (logical)

If are taken the full bispectrum coeffcient (overcomplete), .false. or only diagonal \(l_1=l_2\) .true.

Default lbso3_diag=.false.

Bispectrum SO4

To use bispectrum SO4, set descriptor_type=9.

Parameters

The parameters of the descriptors are controlled by maximum angular moment \(j_{max}\). The dimension is not easy to guess. Moreover, it depends on the choice on diagonal (\(j_1=j_2\)) or full the complete set of \(j_1, j_2\).

j_max (integer)

The maximum componenet of the spectral function.

Default j_max=1.5

inv_r0_input (real)

The value of the maximum projection at north pole in \(\pi\) units. The final value that will be used in code will be multiplied by \(\pi\). The value should be slightly lower that 1 but strictly positive. The default value is suggested by the brilliant paper of Bartok et al. If you do not have ideea about the \(SO(4)\) theory, trust the default value.

Default inv_r0_input= 1.d0 - 0.02/\(\pi\)

lbso4_diag (logical)

If .true. only the diagonal components are selected (as in GAP). If .false. is SNAP-like way and all the components are selected. It should be notted that the bi-spectrum is overcomplete descriptor and only diagonal components are mathematically justified. However, in the original SNAP potential of Thompsson is was used in full version.

Default lbso4_diag=.false.

MTP

To use MTP, set descriptor_type=100.

Parameters

The parameters of the descriptors are controlled by minimum and the maximum degree of the polynomials mtp_poly_min and mtp_poly_max. The degree of the generating radial function will be the internal parameter mtp_rad_order =mtp_poly_max - mtp_poly_min +1. The dimension of the descriptor i.e. number of basis function is given by mtp_dim=mtp_rad_order + 2*mtp_rad_order**2 + mtp_rad_order**3.

mtp_poly_min (integer)

The minimum degree of the radial function

Default mtp_poly_min=2

mtp_poly_max (integer)

The minimum degree of the radial function

Default mtp_poly_max=4

lambda_krr (real)

The regularization using \(L^2\) or \(L^1\) norm. It is active only for cases mld_fit_type=0 (for a fixed positive value of lambda_krr and mld_fit_type=10 (for a grid). For details see the correspoding documentation. For negative values this option is skipped and standard fit is performed.

Default lambda_krr=-1.d0

PiP

To use PiP (permutationally invariant polynomials), set descriptor_type=200.

Note

The current implementation of PiP treat only one single type of atoms

Parameters

This descriptors is based by a cluster expansion of the system energy:

\[E = \sum_i V_1 (r_i) + \frac{1}{2} \sum_{i,j} V_2(r_{i}, r_{j}) + \frac{1}{3!} \sum_{i,j,k} V_3(r_{i}, r_{j}, r_{k} ) + \ldots\]

The total energy is expressed as a sum of one-atom \(V_1\) , pair \(V_2\), threeatom (angle) \(V_3\) terms, and so forth. The order of expansion, or even a combination of expantion terms are driven by l_body_order. This version of Milady enables a developement up to the fourth order. The number of PiP terms for each term are driven by body_D_max, bond_dist_transform, bond_beta and bond_dist_ann.

l_body_order (logical vector)

This logical vertor of dimention 4 design the order of cluster expension. For example l_body_order(2)=.true., l_body_order(3)=.true., l_body_order(4)=.false. enable a cluster expension up to the third order.

Default l_body_order(:)=.true.

body_D_max (integer vector)

This integer vector fix the number of PiP terms for each term of expansion. body_D_max(2) for \(V_2\), body_D_max(3) for \(V_3\) and body_D_max(4) for \(V_4\). A reasonable choice can be 20, 9, 7. Try more less depending on th design that you made for your expension.

Default body_D_max(i)=i

bond_dist_transform (integer)

TODO: choose 3

Default: bond_dist_transform=3

bond_beta (real)

TODO: choose 2.0

Default: bond_beta=2.0

bond_dist_ann

TODO: choose bond_dist_ann=1.0

Default: bond_dist_ann=1.0