B AOInts

B.11 More Efficient Hartree–Fock Gradient and Hessian Evaluations

(May 16, 2021)

Q-Chem combines the Head-Gordon–Pople (HGP) method 428 Head-Gordon M., Pople J. A.
J. Chem. Phys.
(1988), 89, pp. 5777.
and the COLD prism method 23 Adams T. R., Adamson R. D., Gill P. M. W.
J. Chem. Phys.
(1997), 107, pp. 124.
for Hartree-Fock gradient and Hessian evaluations. All two-electron four-center integrals are classified according to their angular momentum types and degrees of contraction. For each type of integrals, the program chooses one with a lower cost. In practice, the HGP method is chosen for most integral classes in a gradient or Hessian calculation, and thus it dominates the total CPU time.

Recently the HGP codes within Q-Chem were completely rewritten for the evaluation of the P IIx P term in the gradient evaluation, and the P IIxy P term in the Hessian evaluation. Our emphasis is to improve code efficiency by reducing cache misses rather than by reducing FLOP counts. Some timing results from a Hartree-Fock calculation on azt are shown below.

Basis Set AIX Linux
Gradient Evaluation: P IIx P Term
Old New New/Old Old New New/Old
3-21G 34 s 20 s 0.58 25 s 14 s 0.56
6-31G** 259 s 147 s 0.57 212 s 120 s 0.57
DZ 128 s 118 s 0.92 72 s 62 s 0.86
cc-pVDZ 398 s 274 s 0.69 308 s 185 s 0.60
Hessian Evaluation: P IIxy P term
Old New New/Old Old New New/Old
3-21G 294 s 136 s 0.46 238 s 100 s 0.42
6-31G** 2520 s 976 s 0.39 2065 s 828 s 0.40
DZ 631 s 332 s 0.53 600 s 230 s 0.38
cc-pVDZ 3202 s 1192 s 0.37 2715 s 866 s 0.32
Table B.1: AIX timings were obtained on an IBM RS/6000 workstation with AIX4 operating system, and Linux timings on an Opteron cluster where the Q-Chem executable was compiled with an Intel 32-bit compiler.