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

Q-Chem combines the Head-Gordon–Pople (HGP) method[Head-Gordon and Pople(1988)] and the COLD prism method[Adams et al.(1997)Adams, Adamson, and Gill] 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 II $^ x$ P term in the gradient evaluation, and the P II $^{xy}$ 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 II $^{x}$ 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 II $^{xy}$ 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.