Explicitly correlated PNO-MP2 and PNO-CCSD and its application to
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Supporting Information to: Explicitly correlated PNO-MP2 and PNO-CCSD and its application to the S66 set and large molecular systems Gunnar Schmitz1 , Christof H¨attig1 and David P. Tew2 1
Lehrstuhl f¨ ur Theoretische Chemie, Ruhr-Universit¨at Bochum, Universit¨atsstraße 150, D-44801 Bochum, Germany 2 Center for Computational Chemistry, University of Bristol, Bristol BS8 1TS
1
supplementary material
1.1
Integral screening for short range operators
For the integral screening we follow the work of Adler et al. [?] and implemented their introduced bounds. Additional we derived simpler and cheaper estimates by using only the most diffuse functions1 and setting the corresponding contraction coefficients Cµ etc. to one. In this way many quantities can be precomputed and with the additional approximation that the distance x of the Gaussians is set to the shortest distance x0 between the center of the auxiliary function and the line connecting the atoms at which, the two Gaussians are located (see Figure 1), we get always an upper bound to their estimate. To denote the we used the most diffuse functions we use a prime for the corresponding exponents α and β and basis functions µ, ν. The bounds are given as: atom a; basis func. i
2 − 3 ln π ≤ ln 1 + γx aµ 0 aν 0 2 3 r + ln (aµ0 ν 0 β 0 ) − γx0 + γ 2 / (4ξ 0 ) − 2 aµ0 ν 0 ab As usual Kµν is the preexponential factor of a Gaussian product center, αµν is the β sum of the exponents αµ αν and ξ is defined as ξ = ααµνµν+β . Moreover rab is the distance of two Gaussians centered at Atom A and B and Sµν an overlap. In our algorithm first the cheap estimate is used. If the matrix element is smaller than the threshold it is discarded, but if it is in tolerance range of +2 on a logarithmic scale it is checked again with the estimate involving the contracted GTOs. For large systems we expect a benefit of this approach. Nevertheless also the more evolved estimate is relatively cheap to compute.
