Author Correction: Physics guided deep learning for generative design of crystal materials with symmetry constraints

Correction to: npj Computational Materials https://doi.org/10.1038/s41524-023-00987-9, published online 18 March 2023

The original version of this Article contained typos in both the PDF and the HTML versions.

In the fourth and fifth sentences of the first paragraph of the ‘Case study of three example stable materials’ section of the Results, the incorrect expression “β = β” has been replaced by “α = β”.

The third note of Equation 1 incorrectly read “(c) P = (a, b, c, β, β, γ) ∈ R are six lattice parameters that define three lengths and three angles of the unit cell;” The correct version states ‘(a, b, c, α, β, γ)’ in replace of (a, b, c, β, β, γ).

The Figure caption of Figure 5d incorrectly read “Inter- and intra-atom distance matrices (H_inter and H_inter) are calculated from three sets of base atom sites for both real and fake materials.” The correct version states ‘H_intra and H_inter’ in replace of ‘H_inter and H_inter’.

The first sentence after Equation (5) incorrectly read “where Linter constrains the distance in H_intra which describes inter-atom distance matrices.”. The correct version states ‘H_inter’ in replace of ‘H_intra’.

The third sentence after Equation (5) incorrectly read “S_intra are atom radius sum corresponding to each pair of atoms in H_intra and ${\phi }_{{\rm{inter}}}^{{\rm{upper}}}$ and ${\phi }_{{\rm{inter}}}^{{\rm{lower}}}$ are control weights for upper and lower bound of inter-atom distance, respectively.” In the correct version, ‘S_intra’and ‘H_intra’ have been replaced by ‘S_inter’and ‘H_inter’, respectively.

The first sentence of the fifth paragraph of “Implementation details” section incorrectly read “Atom Distance Matrices Given three sets of base atom sites (B⁰, B¹, B²), we calculate the atom distance matrices H_inter and H_inter as shown in sub-figure (d) of Fig. 5.” The correct version states ‘H_inter and H_intra’ in replace of ‘H_inter and H_inter’.

The third sentence of the fifth paragraph of “Implementation details” section incorrectly read “Then we select three atoms belonging to the same element to form a set of three atom sites for three elements and calculate pair-wise same atom distance matrix and again return only values in upper triangle of corresponding distance matrix termed by H_inter.” The correct version states ‘H_intra’ in replace of ‘H_inter’.

The last sentence of the fifth paragraph of “Implementation details” section incorrectly read “The final shape of H_inter and H_inter both is 3 × 3.” The correct version states ‘H_intra and H_inter’ in replace of ‘H_inter and H_inter’.

The sentence before Equation (8) incorrectly read “Fractional coordinates can be converted to Cartesian coordinates [x, y, y]^T .” The correct version states ‘[x, y, z]^T’ in replace of ‘[x, y, y]^T’.

The original version of this Article contained typographical errors in Equation (9).

The correct version of Equation (9) is:

$${\rm{A}}=\left[\begin{array}{ccc}a & {{b}}\,\cos {\boldsymbol{\gamma }} & {{c}}\,\cos {\boldsymbol{\beta }}\\ 0 & {{b}}\,\sin {\boldsymbol{\gamma }} & c\frac{\cos{\boldsymbol{\alpha }}-\cos{\boldsymbol{\beta }}\cos{\boldsymbol{\gamma }}}{\sin{\boldsymbol{\gamma }}}\\ 0 & 0 & \frac{V}{{{ab}} \sin{\boldsymbol{\gamma }}}\end{array}\right],$$

Where ${\rm{V}}={\rm{abc}}\sqrt{1-{\cos}^{2}{\boldsymbol{\alpha }}-{\cos}^{2}{\boldsymbol{\beta }}-{\cos}^{2}{\boldsymbol{\gamma }}+2\cos{\boldsymbol{\alpha }}\cos{\boldsymbol{\beta }}\cos{\boldsymbol{\gamma }}}$ is the volume of the unit cell.

Which replaces the previous incorrect version:

$${\rm{A}}=\left[\begin{array}{ccc}a & {{b}}\,\cos {\boldsymbol{\gamma }} & {{c}}\,\cos {\boldsymbol{\beta }}\\ 0 & {{b}}\,\sin {\boldsymbol{\gamma }} & c\frac{\cos{\boldsymbol{\beta }}-\cos{\boldsymbol{\beta }}\cos{\boldsymbol{\gamma }}}{\sin{\boldsymbol{\gamma }}}\\ 0 & 0 & \frac{V}{{{ab}} \sin{\boldsymbol{\gamma }}}\end{array}\right],$$

Where ${\rm{V}}={\rm{abc}}\sqrt{1-{\cos}^{2}{\boldsymbol{\beta }}-{\cos}^{2}\beta -{\cos}^{2}\gamma +2\cos{\boldsymbol{\beta }}\cos\beta \cos\gamma }$ is the volume of the unit cell.

The above changes have been made in the PDF version of the Article.