Two-electron integral transformation

MP2 correlation energy calculations may increase the computational time because a two-electron integral transformation from atomic orbitals (AO s) to molecular orbitals (MO s) is required. HyperChem may also need additional main memory and/or extra disk space to store the two-electron integrals of the MO s. 

Until the advent of density functional theory (Chapter 13), thinking centred around means of circumventing the two-electron integral transformation, or at least partially circumventing it. The Mpller-Plesset method is one of immense historical importance, and you might like to read the original paper. 

N5 scaling of the two-electron integral transformation step. Of course, basis functions that describe the essence of the states to be studied are essential (e.g., Rydberg or anion states require diffuse functions, and strained rings require polarization functions). 

We thank Dr. Jeppe Olsen, Dr. Wesley Allen, and Matthew Leininger for helpful discussions, and Dr. Daniel Crawford and Dr. Justin Fermann for their work on the two-electron integral transformation algorithm. This work was sponsored by the U.S. National Science Foundation, grant no. CHE-9527468, and by NSF graduate and postdoctoral fellowships (grant no. CHE-9626094) to C.D.S. 

There are two variations on the above iterative process that have been proposed. In the case of small CSF expansion lengths it is appropriate to perform several c updates within an MCSCF iteration. This is because the expensive step of the MCSCF optimization process in this case is the two-electron integral transformation. The approximate transformations performed in step 3 of the micro-iterative procedure are less expensive than the exact transformations performed in step 1 of the MCSCF iteration. However, in the case of large CSF expansions, these updates should be avoided. This is because the exact transformation becomes a small part of the total iteration effort and it is preferable to perform the expensive CSF coefficient updates only with exact Hamiltonian operators. In this case step 5 and the cycle between steps 5 and 3 is not performed. Continuing this reasoning further, it may even be useful to perform several MCSCF iterations with the same CSF vector c to allow the orbitals to relax further for the exact Hamiltonian operators. 

Indeed, it is this convenient fact which enables all the SCF methods outlined so far to be implemented compactly the underlying two-electron integral transformation (four-index multiplications) has been contained into the formation of J and K matrices and some one-electron (two-index matrix multiplications) transformations. However, if we use any method which demands the existence of the MO-based repulsion integrals with either i j or k or both)... 

Although it is not obvious from Eq (2.89) that a two-electron integral transformation is not required to set up the Fock potential matrix for a general reference state, it becomes clear upon actually working out the matrix elements for a particular case. For example, for either a spin-unrestricted reference state or a closed-shell reference state, the Fock potentials of Eqs. (2.91) and (2.92), respectively, are seen to involve only a two-index transformation [e.g., sum over y in Eq. (2.92)J. 

Example 6.4 Data Distribution in the Two-Electron Integral Transformation... 

In the remainder of this chapter we will discuss different ways to parallelize the computation of the two-electron integrals with focus on how to achieve load balance. Two-electron integral computation in a quantum chemistry code is often performed in the context of a procedure such as Fock matrix formation or two-electron integral transformation, but we here consider the integral computation separately to specifically concentrate on strategies for distributing the work. Examples of parallel two-electron integral computation in quantum chemical methods are discussed in chapters 8,9, and 10. 

Hartree-Fock procedure, and is an atomic orbital. The two-electron integral transformation (Eq. 9.3) is usually performed as four separate quarter transformations... 

In the following we will discuss parallel implementation of the two dominant computational steps in the LMP2 procedure, namely the two-electron integral transformation and the computation of the residual. Parallelization of other computationally less demanding, but nonnegligible, steps is straightforward. ... 

The integral transformation itself is straightforward, but it must be broken into pieces because of the large and small components. For the Coulomb interaction, the two-electron integral transformation can be written... 

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