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Sizes of Hamiltonian and Overlap matrices in OpenMX Date: 2022/09/05 13:49 Name: Wenfei   <liwenfei94@gmail.com> Dear developers of OpenMX, Greetings! This is Wenfei Li, a developer of a DFT software called ABACUS, and I'm reaching out for a question about the sizes of Hamiltonian matrix and Overlap matrix in OpenMX. The question arose during our recent collaboration with the developers of a method called DeepH , where they used machine learning to predict the Hamiltonian matrix from the Overlap matrix. In their original paper (https://www.nature.com/articles/s43588-022-00265-6), the matrices were obtained from OpenMX, and they got some pretty impressive results on twisted bilayer graphenes. During our collaboration, I got to know that the Hamiltonian and Overlap matrices extracted from OpenMX are the same in size, with non-zero elements coming from pairs of neighbouring atoms. This enabled them to have a consistent treatment in DeepH. But we encountered some difficulties in this regard on ABACUS. The question I have is related to the nonlocal pseudopotential contribution to the Hamiltonian. In ABACUS, the nonlocal pseudopotential part consists of terms such as <chi_mu | beta > <beta | chi_nu >, where the chi's are the atomic basis sets, and the beta’s are the projectors of the pseudopotentials. Suppose chi_mu, beta and chi_nu are located on atoms A,B, and C, respectively, then for this term to be nonzero, A and B are not necessarily neighbors with each other, as long as they share a common neighbor C. As a result, the Hamiltonian matrix from ABACUS contains more nonzero elements than the overlap matrix. For now, we resort to cutting those extra matrix elements from the Hamiltonian, so that its size will be consistent with the Overlap matrix. It seems that DeepH works pretty well with this treatment. Still, out of curiosity, I’m wondering if OpenMX is adopting a different formulation or some special treatments on the nonlocal pseudopotentials to allow the Hamiltonian and Overlap matrices to have the same size? I would be really grateful if I can get some elaboration on this question. Thanks in advance and best regards, Wenfei

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Re: Sizes of Hamiltonian and Overlap matrices in OpenMX ( No.1 ) Date: 2022/09/15 14:42 Name: T. Ozaki Hi, As you pointed out, the original matrix for the non-local projector has more non-zero elements than the overlap matrix. To make the sparse structure of the matrix for non-local projector equivalent to that of the overlap matrix, we introduce a damping function f(r) which is zero up to the second derivatives at r_{c}. Using the damping function, the matrix element is damped as f(r_{ij})<i\alpha| \hat{P} | j\beta>, where i and j are atomic indices, \alpha and \beta are orbital indices, and \hat{P} is the non-local operator. The treatment is important to ensure the smoothness of potential surface. Also, please note that the treatment does not violate the variational property of the total energy. The energy contribution of the non-local operator can be written as tr( \rho \times h^{(NL)}), where \rho is the density matrix, and h^{NL} is the matrix for the non-local operators. Thus, once the matrix elements are defined by f(r_{ij})<i\alpha| \hat{P} | j\beta>, the total energy can be variationally optimized w.r.t. both the LCAO coefficients and atomic coordinates during the geometry optimization. Regards, TO Re: Sizes of Hamiltonian and Overlap matrices in OpenMX ( No.2 ) Date: 2022/09/15 15:29 Name: Wenfei  <liwenfei94@gmail.com> Dear Professor Ozaki, Thank you very much! I appreciate your reply. Best, Wenfei Re: Sizes of Hamiltonian and Overlap matrices in OpenMX ( No.3 ) Date: 2022/09/15 15:55 Name: IK I'm sorry, I don't fully understand Prof. Ozaki's response, but is it correct that the HS.out Hamiltonian obtained by analysis_example.c includes the effect of non-local operator? Re: Sizes of Hamiltonian and Overlap matrices in OpenMX ( No.4 ) Date: 2022/09/15 16:38 Name: Wenfei  <liwenfei94@gmail.com> Dear Professor Ozaki, Thank you very much! I appreciate your reply. Best, Wenfei Re: Sizes of Hamiltonian and Overlap matrices in OpenMX ( No.5 ) Date: 2022/09/15 18:38 Name: T. Ozaki Hi, > I'm sorry, I don't fully understand Prof. Ozaki's response, but is it correct that the HS.out Hamiltonian > obtained by analysis_example.c includes the effect of non-local operator? Yes, it is. By diagonalizing the generalized eigenvalue problem with S and H obtained by analysis_example.c, one can reproduce the eigenvalue spectrum calculated by OpenMX. Regards, TO

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