B AOInts

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

Q-Chem combines the Head-Gordon–Pople (HGP) method358 and the COLD prism method24 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 IIx⁢y 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 IIx⁢y 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.