Real space grid

Takahashi, H., Hori T., Hashimoto H. and Nitta T., A hybrid QM/MM method employing real space grids for QM water in the TIP4P water solvents. J.Comput.Chem. (2001) 22 1252-1261. 

Kohn-Sham DFT with Real-Space Grids... 

In the implementation of the QM/MM approach with the real-space method, the QM cell that contains the real-space grids is embedded in the MM cell. One should take care for the evaluation of the potential upc(r) defined as Eq. (17-20). When a point charge in MM region goes inside the QM cell, it makes a singularity in the effective Kohn-Sham Hamiltonian, which may give rise to a numerical instability. To circumvent the problem, we replace a point charge distribution... 

In Sections 17.2 and 17.3, we have reviewed the QM/MM approach based on the real-space grids [40,41,58,59,60,61,62] and the novel theory of solutions [14,15,16], respectively. As has been suggested, the theory of energy representation is readily applicable to a solute that is quantum chemically described. The present section is devoted to the details of the methodology, referred to as QM/MM-ER, developed by combining the QM/MM approach with the theory of energy representation [19]. The point of the method is to divide the total solvation free energy into the contributions due to the pairwise additive interaction between the solute and the solvent and the residual contribution due to the electron density fluctuation. A focus will be placed on the treatment of the many-body interaction inherent in the quantum chemical object. 

The computational details for the QM/MM simulation were almost common to those for the simulation described in Section 17.5.1. The major change was made in the setup of the real-space grids of the QM cell. The number of grid points for each axis was increased to 64 from 32 where the grid spacing h was set at h = 0.152 A. The thermodynamic condition of the MM water solvent was set at T = 300 K, p = 1.0g/cm3. The LJ parameters in AMBER95 force field were employed... 

A very different approach is the use of non-atom-centered basis functions such as plane waves. Due to their intrinsic periodic nature, they are mostly employed for electronic structure calculations of periodic solids [10]. A more recent development is the usage of real-space wavefunctions either by discretization on real-space grids or in a finite-element fashion [11], In a non-atom-centered basis, the basis set obviously does not depend on the atomic positions, which makes it ideally suited for ab initio molecular dynamics simulations, since the forces acting on the nuclei can be evaluated much more easily than in an atom-centered basis [10]. 

If Nx, Ny and N are products of small prime numbers, one can use Fast Fourier Transform (FFT) techniques which are very efficient to go from the real-space grid to the reciprocal-space grid and back. These algorithms indeed scale as N log N instead of N and are thus heavily used in plane wave codes [97-103,105,106,108]. 

The first term is calculated on the real-space grid defined by the plane wave expansion and the other two are efficiently and accurately calculated using atom centered meshes. 

The self-consistent field procedure in Kohn-Sham DFT is very similar to that of the conventional Hartree-Fock method [269]. The main difference is that the functional Exc[p] and matrix elements of Vxc(r) have to be evaluated in Kohn-Sham DFT numerically, whereas the Hartree-Fock method is entirely analytic. Efficient formulas for computing matrix elements of Vxc(r) in finite basis sets have been developed [270, 271], along with accurate numerical integration grids [272-277] and techniques for real-space grid integration [278,279]. 

The matrix elements of the first two terms involve only two-center integrals that are calculated in reciprocal space and tabulated as a function of interatomic distance. The remaining terms involve potentials that are calculated on a three-dimensional real-space grid. 

Time-dependent DET has the ability to calculate various physical and quantum quantities, and different techniques are sometimes favored for each type. For some purposes as, for example, if strong fields are present, it can be better to propagate forward in time the KS orbitals using either a real space grid or plane waves. For finite-order response, Fourier transforming to frequency space with localized basis functions may be preferable. We discuss in detail below how the latter approach works, emphasizing the importance of basis set convergence. 

The KS energy levels and KS orbitals for the LDA functional are shown in Table 7. The orbitals are calculated with two different numerical methods, the first is fully numerical basis set free (i.e., solved on a real space grid) while the other uses the Sadlej (52 orbitals) basis set [the OEP results for the EXX (KLI) approximation shown in Table 7 are also calculated basis set free]. Note that the eigenvalues for the higher unoccupied states are positive. This is due to the LDA potential being too shallow and not having the correct asymptotic... 

Commun., 140,315-322 (2001). Accurate Kinetic Energy Evaluation in Electronic Structure Calculations with Localized Functions on Real Space Grids. 

Embedded Divide-and-Conquer Algorithm on Hierarchical Real-Space Grids Parallel Molecular Dynamics Simulation Based on Linear-Scaling Density Functional Theory. 

A Hybrid QM/MM Method Employing Real Space Grids for QM Water in the TIP4P Water Solvents. 

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