Descriptors
Two-Body
Below is a list of all two-body descriptors supported by Tadah:
D2_LJ
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class D2_LJ : public D2_Base
Standard Lennard - Jones descriptor.
\[ V_i = \sum_{j \neq i} 4 \epsilon \Bigg(\Big(\frac{\sigma}{r_{ij}}\Big)^{12} - \Big(\frac{\sigma}{r_{ij}}\Big)^6\Bigg) f_c(r_{ij}) \]or equivalently:
\[ V_i = \sum_{j \neq i} \frac{C_{12}}{r_{ij}^{12}} - \frac{C_6}{r_{ij}^6} f_c(r_{ij}) \]Note that machined learned coefficients \(C_6\) and \(C_{12}\) corresponds to \(\sigma\) and \(\epsilon\) through the following relation:
\[ \sigma = \Big(\frac{C_{12}}{C_6}\Big)^{1/6} \]\[ \epsilon = \frac{1}{4} \frac{C_6^2}{C_{12}} w(Z) \]where \(w(Z)\) is a species depended weight factor (default is an atomic number).The machine learned \(\sigma\) and \(\epsilon\) only make sense (say to compare with the literature ones) when BIAS false and NORM false and system in monatomic. It is ok thought to set them to true it’s just that numerical values will be different.
Required Config Key: INIT2B
D2_BP
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class D2_BP : public D2_Base
Behler-Parrinello two-body descriptor.
\[ V_i^{\eta,r_s} = \sum_{j \neq i} \exp{\Big(-\eta(r_{ij}-r_s)^2\Big)}f_c(r_{ij}) \]CGRID2B parameters control position \( r_s \) of the gaussian basis function.
SGRID2B parameters control width \( \eta \) of the gaussian basis function.
This is essentially a \( G^1_i \) descriptor from the below paper with an exception that it can use any cutoff function defined in Ta-dah!:
Behler, J., Parrinello, M. (2007). Generalized neural-network representation of high-dimensional potential-energy surfaces. Physical Review Letters, 98(14), 146401. https://doi.org/10.1103/PhysRevLett.98.146401
Required Config keys: INIT2B CGRID2B SGRID2B
D2_Blip
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class D2_Blip : public D2_Base
Blip two-body descriptor.
\[ V_i^{\eta,r_s} =\sum_{j \neq i} \mathcal{B}(\eta(r_{ij}-r_s))f_c(r_{ij}) \]where \( f_c \) is a cutoff function and \( \mathcal{B} \) is a blip basis function centered at \(r_s\) of width \(4/\eta\).
CGRID2B parameters control position \( r_s \) of blip centres.
SGRID2B parameters control width \( \eta \) of blips.
Blip basis function is built out of 3rd degree polynomials in the four intervals [-2,-1], [-1,0], [0,1], [1,2] and is defined as:
\[\begin{split} \begin{equation} \mathcal{B}(r) = \begin{cases} 1-\frac{3}{2}r^2+\frac{3}{4}|r|^3 & \text{if} \qquad 0<|r|<1\\ \frac{1}{4}(2-|r|)^3 & \text{if} \qquad 1<|r|<2\\ 0 & \text{if} \qquad |r|>2 \end{cases} \end{equation} \end{split}\]More details about the blip basis functions can be found in the following paper:
Hernández, E., Gillan, M., Goringe, C. (1997). Basis functions for linear-scaling first-principles calculations. Physical Review B - Condensed Matter and Materials Physics, 55(20), 13485–13493. https://doi.org/10.1103/PhysRevB.55.13485
Required keys: INIT2B CGRID2B SGRID2B
D2_EAM
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class D2_EAM : public D2_Base
Pair-wise part for the Embedded Atom Method descriptor.
\[ V_i = \frac{1}{2} \sum_{j \neq i} \psi(r_{ij}) \]This descriptor will load tabulated values for the two-body potential \( \phi \) from the provided SETFL file.
This descriptor is usually used together with the many-body descriptor DM_EAM although this is not required and user can mix it with any other descriptors or use it on its own.
This descriptor will enforce cutoff distance as specified in a SETFL file. Set RCUT2B to the same value to suppress the warning message.
Required Config keys: INIT2B SETFL
D2_MIE
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class D2_MIE : public D2_Base
Mie descriptor.
\[ V_i = \sum_{j \neq i} C \epsilon \Bigg(\Big(\frac{\sigma}{r_{ij}}\Big)^{n} - \Big(\frac{\sigma}{r_{ij}}\Big)^m\Bigg) \]where
\[ C=\frac{n}{n-m}\Big( \frac{n}{m} \Big)^{\frac{m}{n-m}} \]Any cutoff can be used
Required Config Key: INIT2B TYPE2B
TYPE2B D2_MIE 12 6 ELEMENT1 ELEMENT2
will result in Lennard-Jones type descriptor
D2_ZBL
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class D2_ZBL : public D2_Base
ZBL Descriptor.
The ZBL (Ziegler-Biersack-Littmark) potential is an empirical potential used to model short-range interactions between atoms.
The constant term \( \frac{e^2}{4 \pi \varepsilon_0 } \) is set to 1 and will be fitted as needed.
The simplified expression for the ZBL potential is given by:
\[ V(r) = \frac{Z_1 Z_2}{r} \phi\left(\frac{r}{a}\right) \]where \( a \) is the screening length, expressed as:
\[ a = \frac{s_0 a_0}{Z_1^{p_0} + Z_2^{p_1}} \]Here, \( a_0 \), \( s_0 \), \( p_0 \), and \( p_1 \) are adjustable hyperparameters. Setting any of these to -1 uses the default values:
\( a_0 = 0.52917721067 \, \text{Å} \)
\( s_0 = 0.88534 \)
\( p_0 = 0.23 \)
\( p_1 = 0.23 \)
The screening function \( \phi \) is defined as:
\[ \phi(x) = 0.1818 e^{-3.2x} + 0.5099 e^{-0.9423x} + 0.2802 e^{-0.4029x} + 0.02817 e^{-0.2016x} \]Required Config Key: INIT2B TYPE2B
TYPE2B D2_ZBL \( a_0 \) \( s_0 \) \( p_0 \) \( p_1 \) ELEMENT1 ELEMENT2
Examples:
D2_Dummy
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class D2_Dummy : public D2_Base
Dummy two-body descriptor.
Use it to satisfy DescriptorsCalc requirements in case when two-body descriptor is not required.
D2_mJoin
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class D2_mJoin : public D2_Base, public D_mJoin
Meta two-body descriptor for combining multiple D2 descriptors.
This descriptor provides a convenient interface for concatenating multiple two-body descriptors. The resulting descriptor can then be used by Tadah! like any standard two-body descriptor.
Each descriptor must have a specified type in a configuration file, along with a cutoff function, cutoff distance, and optionally SGRID2B and CGRID2B values if applicable.
When listing descriptors under the TYPE2B key, you must include parameters relevant to this descriptor.
Here is an example of how to configure these descriptors:
TYPE2B D2_mJoin # <-- Meta descriptor for concatenating two-body descriptors TYPE2B D2_MIE 11 6 Ti Ti # <-- MIE exponents RCTYPE2B Cut_Cos RCUT2B 3.0 TYPE2B D2_Blip 6 6 Ti Nb Nb Nb # <-- grid sizes RCTYPE2B Cut_Tanh RCUT2B 7.5 SGRID2B -2 6 0.1 10 # Grid for D2_Blip, blips widths, auto generated CGRID2B 0 0 0 0 0 0 # Grid for D2_Blip, blip centers
Note: Grids can be specified on a single line, and the order of the grids is important.
There is no limit to the number of descriptors that can be concatenated.
Ensure the types and grids are correctly specified in the configuration file.
The cutoff functions (RCTYPE2B) and distances (RCUT2B) must be defined for each descriptor.
Both SGRID2B and CGRID2B should be included if relevant, with their sizes matching the given descriptors.
Many-Body
This section lists all many-body type descriptors supported by Tadah! Some can calculate non-spherical charge distributions, thus overcoming the limitations of the Embedded Atom Method:
DM_Blip
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class DM_Blip : public DM_Base
Blip Many Body Descriptor.
\[ V_i^{L,\eta,r_s} = \sum_{l_x,l_y,l_z}^{l_x+l_y+l_z=L} \frac{L!}{l_x!l_y!l_z!} \Big( \rho_i^{\eta,r_s,l_x,l_y,l_z} \Big)^2 \]where density \( \rho \) is calculated using modified Gaussian Type Orbitals (expansion in the Blip basis instead of usual Gaussians):
\[ \rho_i^{\eta,r_s,l_x,l_y,l_z} = \sum_{j \neq i} x_{ij}^{l_x}y_{ij}^{l_y}z_{ij}^{l_z} \mathcal{B}{\Big(-\eta(r_{ij}-r_s)^2\Big)}f_c(r_{ij}) \]CGRIDMB parameters control position \( r_s \) of the gaussian basis function.
SGRIDMB parameters control width \( \eta \) of the gaussian basis function.
e.g. \(L_{max}=2\) will calculate descriptors with \( L=0,1,2 \) (s,p,d orbitals).
More information about this descriptor:
Zhang, Y., Hu, C.,Jiang, B. (2019). Embedded atom neural network potentials: efficient and accurate machine learning with a physically inspired representation. Journal of Physical Chemistry Letters, 10(17), 4962–4967. https://doi.org/10.1021/acs.jpclett.9b02037
Required Config keys: INITMB CGRIDMB SGRIDMB
DM_EAD
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class DM_EAD : public DM_Base
Embedded Atom Descriptor
\[ V_i^{L,\eta,r_s} = \sum_{l_x,l_y,l_z}^{l_x+l_y+l_z=L} \frac{L!}{l_x!l_y!l_z!} \Big( \rho_i^{\eta,r_s,l_x,l_y,l_z} \Big)^2 \]where density \( \rho \) is calculated using Gaussian Type Orbitals:
\[ \rho_i^{\eta,r_s,l_x,l_y,l_z} = \sum_{j \neq i} x_{ij}^{l_x}y_{ij}^{l_y}z_{ij}^{l_z} \exp{\Big(-\eta(r_{ij}-r_s)^2\Big)}f_c(r_{ij}) \]CGRIDMB parameters control position \( r_s \) of the gaussian basis function.
SGRIDMB parameters control width \( \eta \) of the gaussian basis function.
e.g. \(L_{max}=2\) will calculate descriptors with \( L=0,1,2 \) (s,p,d orbitals).
More information about this descriptor:
Zhang, Y., Hu, C.,Jiang, B. (2019). Embedded atom neural network potentials: efficient and accurate machine learning with a physically inspired representation. Journal of Physical Chemistry Letters, 10(17), 4962–4967. https://doi.org/10.1021/acs.jpclett.9b02037
Required Config keys: INITMB CGRIDMB SGRIDMB
DM_EAM
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class DM_EAM : public DM_Base
many-body part for the Embedded Atom Method descriptor.
\[ V_i = F\Bigg(\sum_{j \neq i} \rho(r_{ij}) \Bigg) \]This descriptor will load tabulated values for the density \( \rho \) and embedded energy \( F \) from the provided SETFL file.
This descriptor is usually used together with the two-body descriptor D2_EAM although this is not required and user can mix it with any other descriptors or use it on its own.
This descriptor will enforce cutoff distance as specified in a SETFL file. Set RCUTMB to the same value to suppress the warning message.
Required Config keys: INITMB SETFL
DM_mEAD
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template<typename F>
class DM_mEAD : public DM_Base Modified Embedded Atom Descriptor
REQUIRED KEYS: SGRIDMB, CGRIDMB, and KEYS OF THE EMBEDDING FUNCTION
This descriptor has a mathematical form very similar to DM_EAD but allows the usage of a custom-defined embedding function, \( \mathcal{F} \), in place of the default quadratic one.
\[ V_i^{L,\eta,r_s} = \sum_{l_x,l_y,l_z}^{l_x+l_y+l_z=L} \frac{L!}{l_x!l_y!l_z!} \mathcal{F}\Big( \rho_i^{\eta,r_s,l_x,l_y,l_z} \Big) \]where the density \( \rho \) is calculated using Gaussian Type Orbitals:
\[ \rho_i^{\eta,r_s,l_x,l_y,l_z} = \sum_{j \neq i} x_{ij}^{l_x} y_{ij}^{l_y} z_{ij}^{l_z} \exp{\Big(-\eta(r_{ij}-r_s)^2\Big)} f_c(r_{ij}) \]CGRIDMB parameters control the position \( r_s \) of the Gaussian basis function.
SGRIDMB parameters control the width \( \eta \) of the Gaussian basis function.
e.g., \(L_{max}=2\) will calculate descriptors with \( L=0,1,2 \) (s, p, d orbitals).
Required Config keys: INITMB CGRIDMB SGRIDMB
DM_mEAD functions
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class F_RLR : public F_Base
Implements an embedding function of the form: \( s x \log(c x) \).
This class supports embedding functions characterized by two main parameters:
SEMBFUNC: Controls the depth, \( s \), of the embedding function.
CEMBFUNC: Determines the x-intercept, with the x-intercept at \( 1/c \).
The number of keys for these parameters must match the entries in the mEAD descriptor.
Note
Ensure the configuration keys match the descriptor requirements.
DM_Dummy
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class DM_Dummy : public DM_Base
Dummy many-body descriptor.
Use it to satisfy DescriptorsCalc requirements in case when many-body descriptor is not required.
DM_mJoin
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class DM_mJoin : public DM_Base, public D_mJoin
Meta many-body descriptor for combining multiple DM descriptors.
This descriptor provides an interface for concatenating various many-body descriptors. The resulting descriptor can then be used by Tadah! like any standard many-body descriptor.
Each descriptor must have a specified type in a configuration file, along with a cutoff function, cutoff distance, and other optional keys that are typically expected for this descriptor, such as SGRIDMB and CGRIDMB.
When listing descriptors under the TYPEMB key, include parameters relevant to this descriptor.
Here is an example of configuring these descriptors:
TYPEMB DM_mJoin # Meta descriptor for concatenating many-body descriptors TYPEMB DM_EAD 1 5 5 * * # L number, cgrid, sgrid, list of element pairs RCTYPEMB Cut_Cos RCUTMB 3.0 CGRIDMB -1 5 0 3.0 # Grid for DM_EAD, blips centers, auto-generated SGRIDMB -2 5 1.0 10.0 # Grid for DM_EAD, blips widths, auto-generated TYPEMB DM_Blip 0 7 7 Ta Ta # L number, cgrid, sgrid, list of element pairs RCTYPEMB Cut_Tanh RCUTMB 7.5 SGRIDMB -2 7 0.1 10 # Grid for DM_Blip, blips widths, auto-generated CGRIDMB 0 0 0 0 0 0 0 # Grid for DM_Blip, blips centers
Note: Grids can be specified on a single line, and the order of the grids should match the order of descriptors.
There is no limit to the number of descriptors that can be concatenated.
Ensure the types and grids are correctly specified in the configuration file.
The cutoff functions (RCTYPEMB) and distances (RCUTMB) must be defined for each descriptor.
Both SGRIDMB and CGRIDMB should be included if relevant, with their sizes matching the given descriptors.