Numerical Integration in DFT Patrick Tamukong The Kilina Group Chemistry & Biochemistry, NDSU

The Eigenvalue Problem in DFT  In DFT, we seek where the energy functional is and the electron density is subject to the constraints  Due to the analytical complexity of exchange and correlation energy formulas, integrations are performed numerically 2

Partitioning of the Integral  Express the integral as a sum over atomic centers where Partition or weight function Function to be integrated for any  The partition or weight function fulfills the conditions 3

Integral at Atomic Center  Each integral at atomic centers is approximated as a sum of shell integrals over a series of concentric spheres centered at the nucleus of the atom  The function to be integrated is 4 where integration over shell of radius Surface element in spherical coordinates

The Partition Function  The partition function or nuclear weight at a given point is  The are hyperbolic coordinates defined as 5 where is the unnormalized cell function of atom A, composed of independent pair contributions

The Cell Function  It must be close to unity near nucleus A and close to zero near other nuclei, thus the contribution between atoms A and B,, decreases monotonically as follows  is subject to the conditions 6 Gräfenstein, J.; Cremer, D. J. Chem. Phys. 2007, 127,

Becke’s Definition of The Cell Function  According to Becke  Becke found k = 3 to be the optimum value for a sufficiently well-behaved. Since, it follows that 7 Becke, A. D. J. Chem. Phys. 1988, 88, Where the polynormials are such that

Properties of the Hyperbolic Coordinates  Consider  Using the cosine rule 8 Tamukong, P. K. Extension and Applications of GVVPT2 to the Study of Transition Metals. Ph.D. Dissertation, University of North Dakota, Grand Forks, ND, 2014 (

Properties of the Hyperbolic Coordinates  Thus only within a sphere of radius  At a fixed radius of a given sphere around atom A, and are even functions of the angle 9 Tamukong, P. K. Extension and Applications of GVVPT2 to the Study of Transition Metals. Ph.D. Dissertation, University of North Dakota, Grand Forks, ND, 2014 ( that is iff  From

Properties of the Hyperbolic Coordinates  Thus has its maximum at and minimum at  Meanwhile has its maximum on the sphere 10 when and minimum when

Alternative Definition of Cell Function  Stratmann et al. alternatively define the cell function as 11 Where is a piece-wise odd function defined as Stratmann, R. E.; Scuseria, G. E.; Frisch, M. J. Chem. Phys. Lett. 1996, 257, 213.

Alternative Definition of Cell Function  Within the limits, the function is subject to the constraints 12  The function has zero second and third order derivatives at and leads to  The Stratmann et al. cell function satisfies reliably from the requirement that the derivatives of the Becke and Stratmann cell functions coincide at

Selection of Significant Functions  In performing integrations, advantage is taken of the fast decaying nature of Gaussian atomic orbitals such that for each grid point, only such functions that are numerically significant (according to a user-specified criterion, ) are considered 13  For grid point, a set of significant functions is chosen which satisfies radius of considered sphere ε

Selection of Significant Functions  To maximize computational efficiency, blocks of grid points are used, e.g., a sphere of grid points with a set of significant basis functions 14 if for some

Thank You 15