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 nonlocal projector has more nonzero elements than the overlap matrix. To make the sparse structure of the matrix for nonlocal 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 nonlocal 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 nonlocal operator can be written as tr( \rho \times h^{(NL)}), where \rho is the density matrix, and h^{NL} is the matrix for the nonlocal 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 nonlocal 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 nonlocal 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

