5.5 DFT Numerical Quadrature

5.5.4 Multi-resolution Exchange-Correlation (MRXC) Method

The multi-resolution exchange-correlation (MRXC) method is a new approach, courtesy of the Q-Chem development team,487, 805, 168 for accelerating computation of the exchange-correlation (XC) energy and matrix for any given density functional. As explained in Section 4.6.5, XC functionals are sufficiently complicated integration of them is usually performed by numerical quadrature. There are two basic types of quadrature. One is the atom-centered grid (ACG), a superposition of atomic quadrature described in Section 4.6.5. The ACG has high density of points near the nucleus to handle the compact core density and low density of points in the valence and non-bonding region where the electron density is smooth. The other type is even-spaced cubic grid (ESCG), which is typically used together with pseudopotentials and plane-wave basis functions where only the valence and non-bonded electron density is assumed smooth. In quantum chemistry, an ACG is more often used as it can handle accurately all-electron calculations of molecules. MRXC combines those two integration schemes seamlessly to achieve an optimal computational efficiency by placing the calculation of the smooth part of the density and XC matrix onto the ESCG. The computation associated with the smooth fraction of the electron density is the major bottleneck of the XC part of a DFT calculation and can be done at a much faster rate on the ESCG due to its low resolution. Fast Fourier transform and B-spline interpolation are employed for the accurate transformation between the two types of grids such that the final results remain the same as they would be on the ACG alone, yet a speedup of several times is achieved for the XC matrix. The smooth part of the calculation with MRXC can also be combined with FTC (see Section 4.6.5) to achieve a further gain in efficiency.

MRXC
       Controls the use of MRXC.
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       0 Do not use MRXC 1 Use MRXC in the evaluation of the XC part
RECOMMENDATION:
       MRXC is very efficient for medium and large molecules, especially when medium and large basis sets are used.

The following two keywords control the smoothness precision. The default value is carefully selected to maintain high accuracy.

MRXC_CLASS_THRESH_MULT
       Controls the of smoothness precision
TYPE:
       INTEGER
DEFAULT:
       1
OPTIONS:
       i⁢m An integer
RECOMMENDATION:
       A prefactor in the threshold for MRXC error control: i⁢m×10-i⁢o

MRXC_CLASS_THRESH_ORDER
       Controls the of smoothness precision
TYPE:
       INTEGER
DEFAULT:
       6
OPTIONS:
       i⁢o An integer
RECOMMENDATION:
       The exponent in the threshold of the MRXC error control: i⁢m×10-i⁢o

The next keyword controls the order of the B-spline interpolation:

LOCAL_INTERP_ORDER
       Controls the order of the B-spline
TYPE:
       INTEGER
DEFAULT:
       6
OPTIONS:
       n An integer
RECOMMENDATION:
       The default value is sufficiently accurate