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# 4.6.3 Linear Scaling Exchange (LinK) Matrix Evaluation

(February 4, 2022)

Hartree-Fock calculations and the popular hybrid density functionals such as B3LYP also require two-electron integrals to evaluate the exchange energy associated with a single determinant. There is no useful multipole expansion for the exchange energy, because the bra and ket of the two-electron integral are coupled by the density matrix, which carries the effect of exchange. Fortunately, density matrix elements decay exponentially with distance for systems that have a HOMO/LUMO gap.1008 The better the insulator, the more localized the electronic structure, and the faster the rate of exponential decay. Therefore, for insulators, there are only a linear number of numerically significant contributions to the exchange energy. With intelligent numerical thresholding, it is possible to rigorously evaluate the exchange matrix in linear scaling effort. For this purpose, Q-Chem contains the linear scaling K (LinK) method833 to evaluate both exchange energies and their gradients834 in linear scaling effort (provided the density matrix is highly sparse). The LinK method essentially reduces to the conventional direct SCF method for exchange in the small molecule limit (by adding no significant overhead), while yielding large speedups for (very) large systems where the density matrix is indeed highly sparse. For full details, we refer the reader to the original papers.833, 834 LinK can be explicitly requested by the following option (although Q-Chem automatically switches it on when the program believes it is the preferable algorithm).

LIN_K

LIN_K
Controls whether linear scaling evaluation of exact exchange (LinK) is used.
TYPE:
LOGICAL
DEFAULT:
Program chooses, switching on LinK whenever CFMM is used.
OPTIONS:
$comment HF/3-21G single point calculation on a large molecule read in the molecular coordinates from file$end
$molecule read base_pair.inp$end
$rem METHOD HF Hartree-Fock BASIS sto-3g Basis set LIN_K TRUE Calculate K using LinK$end