If you've written your program so that the two-electron integrals are stored in a one-dimensional array, you may have recognized that your program spends quite a bit of its time calculating compound indices. Is there a way to speed up this process? Sure! Pre-compute the indices instead. Look again at the structure of a symmetric matrix, with the elements of the lower triangle numbered top-to-bottom/left-to-right:
We calculate these above values from the raw row and column indices, i and j, respectively using:
The expensive part of the above expression is the repeated evaluation of the i(i+1)/2 and j(j+1)/2 parts (which correspond to the first column of numbers in the above matrix). However, if we were to build an integer array (named “ioff”, for example) to store these values, then we could simplify the task to use array look-ups instead:
ij = (i>j) ? ioff[i] + j : ioff[j] + i;
We can build the ioff array using a recursion:
ioff = 0; for(int i=1; i < BIGNUM; i++) ioff[i] = ioff[i-1] + i;
where BIGNUM is some large number exceeding the maximum row/column dimension we expect to use in our calculations.