Theoretical Performance Analysis

A common way to measure the performance of integral techniques is to count the number of floating point operations (flops) required. Table 5.1 lists the flop count for each part of the new algorithm. Only those flops in the O(N²) parts of the code are counted; for any large molecule the rest are insignificant. It is also assumed that ${\bf A}\ne {\bf B}$ and that the bra and www-theor are far apart (again this is true for the majority of shell-quartets in any large system).

  

Table 5.1: Flop costs for each part of the (0)^(m) construction

Task	When Called	Cost	n=10 cost
Geometric constants	Once	33	33
Form ${\bf H}$	Once per m value	5n2-9n-4+2m	406+2m
Generate ${\bf Hv}$	For each unique ${\bf Hv}$	n2	100
Generate ${\bf u}^{t}{\bf Hv}$	For each unique (0)(m)	2n+2	22
Copy duplicate (0)(m)s	For each duplicate	1	1

Note that most of the routines above have a cost depending on the length of the expansion, n. Decreasing this length will mean fewer shell-quartets pass the convergence tests, forcing more of the work to be done the traditional way. Thus there is a trade-off here and n=10 has proven to be a good compromise.

Using the above table, and how often each routine is called, a cost per class can be calculated. This cost is usually described in terms of the primitive, half-contracted and contracted contributions,
$$\begin{align}\mathrm{Cost} = xK^{4} + yK^{2} + z, \end{align}$$
where K is the degree of contraction of each of the four shells. Table 5.2 compares the contraction-first method (CO) described above with the fastest operator-first method (OC) of the COLD PRISM for the low momentum integrals.

  

Table 5.2: Comparison of CO and OC flop counts

	CO				OC
Class	x	y	z		x	y	z	Ktot CO win
(ss|ss)	0	0	561		8	0	0	12
(ps|ss)	0	0	1235		14	8	0	21
(ds|ss)	0	0	2121		22	24	0	31
(pp|ss)	0	0	2189		22	30	0	31
(ps|ps)	0	0	2368		26	28	0	32
(dp|ss)	0	0	3514		32	76	0	40
(ds|ps)	0	0	4019		42	76	0	41
(pp|ps)	0	0	4228		42	94	0	42
(dd|ss)	0	0	5554		44	176	0	48
(ds|ds)	0	0	7729		82	198	0	50
(dp|ps)	0	0	7035		62	224	0	50
(ds|pp)	0	0	8245		82	242	0	51
(pp|pp)	0	0	9493		94	296	0	52
(dp|dp)	0	0	32578		254	1482	0	63
(dd|dd)	0	0	109201		586	5962	0	67

In addition to the numbers presented in Table 5.2, the OC calculations require 1 square root and 2 divides as x-type work and CO needs 1 square root and 1 divide as z-type work. By making reasonable assumptions for the flop cost of these functions, the point at which CO becomes faster than OC can be estimated. These cross-over points are listed in the final column. The momentumless class shows the lowest CO/OC crossover. Here CO is faster whenever $K \ge 2$, making the new path extremely useful. The CO costs do grow rather fast with momentum though, indicating that it will only rarely be required for classes with a total momentum greater than 4. An exception may be calculations with transition metals and heavy main-group elements, where highly contracted d functions are used.

While flop-counts are a good guide to the performance of an algorithm, they are not the whole story. The number of memory operations, for example, is also an important consideration. Use of the computer's architecture is also relevant. For these reasons it is always important to perform a timings analysis.