6.5 Local MP2 Methods

6.5.2 EPAO Evaluation Options

When a local MP2 job (requested by the LOCAL_MP2 option for CORRELATION) is performed, the first new step after the SCF calculation is converged is to extract a minimal basis of polarized atomic orbitals (EPAOs) that spans the occupied space. There are three valid choices for this basis, controlled by the PAO_METHOD and EPAO_ITERATE keywords described below.

  • Non-iterated EPAOs: The initial guess EPAOs are the default for local MP2 calculations, and are defined as follows. For each atom, the covariant density matrix (SPS) is diagonalized, giving eigenvalues which are approximate natural orbital occupancies, and eigenvectors which are corresponding atomic orbitals. The m eigenvectors with largest populations are retained (where m is the minimal basis dimension for the current atom). This non-orthogonal minimal basis is symmetrically orthogonalized, and then modified as discussed in Ref. 547 to ensure that these functions rigorously span the occupied space of the full SCF calculation that has just been performed. These orbitals may be denoted as EPAO(0) to indicate that no iterations have been performed after the guess. In general, the quality of the local MP2 results obtained with this option is very similar to the EPAO option below, but it is much faster and fully robust. For the example of the torsional barrier calculations discussed above,548 the TRIM RMS deviations of 0.03 kcal/mol from full MP2 calculations are increased to only 0.04 kcal/mol when EPAO(0) orbitals are employed rather than EPAOs.

  • EPAOs: EPAOs are defined by minimizing a localization functional as described in Ref. 547. These functions were designed to be suitable for local MP2 calculations, and have yielded excellent results in all tests performed so far. Unfortunately the functional is difficult to converge for large molecules, at least with the algorithms that have been developed to this stage. Therefore it is not the default, but is switched on by specifying a (large) value for EPAO_ITERATE, as discussed below.

  • PAO: If the SCF calculation is performed in terms of a molecule-optimized minimal basis, as described in Chapter 4, then the resulting PAO-SCF calculation can be corrected with either conventional or local MP2 for electron correlation. PAO-SCF calculations alter the SCF energy, and are therefore not the default. This can be enabled by specifying PAO_METHOD as PAO, in a job which also requests CORRELATION as LOCAL_MP2.

PAO_METHOD
       Controls the type of PAO calculations requested.
TYPE:
       STRING
DEFAULT:
       EPAO For local MP2, EPAOs are chosen by default.
OPTIONS:
       EPAO Find EPAOs by minimizing delocalization function. PAO Do SCF in a molecule-optimized minimal basis.
RECOMMENDATION:
       None

EPAO_ITERATE
       Controls iterations for EPAO calculations (see PAO_METHOD).
TYPE:
       INTEGER
DEFAULT:
       0 Use non-iterated EPAOs based on atomic blocks of SPS.
OPTIONS:
       n Optimize the EPAOs for up to n iterations.
RECOMMENDATION:
       Use the default. For molecules that are not too large, one can test the sensitivity of the results to the type of minimal functions by the use of optimized EPAOs in which case a value of n=500 is reasonable.

EPAO_WEIGHTS
       Controls algorithm and weights for EPAO calculations (see PAO_METHOD).
TYPE:
       INTEGER
DEFAULT:
       115 Standard weights, use 1st and 2nd order optimization
OPTIONS:
       15 Standard weights, with 1st order optimization only.
RECOMMENDATION:
       Use the default, unless convergence failure is encountered.

A local MP2 calculation (requested by the LOCAL_MP2 option for CORRELATION) consists of the following steps:

  • After the SCF is converged, a minimal basis of EPAOs are obtained.

  • The TRIM (and DIM) local MP2 energies are then evaluated (gradients are not yet available).

Details of the efficient implementation of the local MP2 method described above are reported in the recent thesis of Dr. Michael Lee.549 Here we simply summarize the capabilities of the program. The computational advantage associated with these local MP2 methods varies depending upon the size of molecule and the basis set. As a rough general estimate, TRIM MP2 calculations are feasible on molecule sizes about twice as large as those for which conventional MP2 calculations are feasible on a given computer, and this is their primary advantage. Our implementation is well suited for large basis set calculations. The AO basis two-electron integrals are evaluated four times. DIM MP2 calculations are performed as a by-product of TRIM MP2 but no separately optimized DIM algorithm has been implemented.

The resource requirements for local MP2 calculations are as follows:

  • Memory: The memory requirement for the integral transformation does not exceed OON, and is thresholded so that it asymptotically grows linearly with molecule size. Additional memory of approximately 32N2 is required to complete the local MP2 energy evaluation.

  • Disk: The disk space requirement is only about 8OVN, but is not governed by a threshold. This is a very large reduction from the case of a full MP2 calculation, where, in the case of four integral evaluations, OVN2/4 disk space is required. As the local MP2 disk space requirement is not adjustable.

The evaluation of the local MP2 energy does not require any further customization. An adequate amount of MEM_STATIC (80 to 160 MB) should be specified to permit efficient AO basis two-electron integral evaluation, but all large scratch arrays are allocated from MEM_TOTAL.

Example 6.2  A relative energy evaluation using the local TRIM model for MP2 with the 6-311G** basis set. The energy difference is the internal rotation barrier in propenal, with the first geometry being planar trans, and the second the transition structure.

$molecule
   0 1
   C
   C  1  1.32095
   C  2  1.47845  1  121.19
   O  3  1.18974  2  123.83  1   180.00
   H  1  1.07686  2  121.50  3     0.00
   H  1  1.07450  2  122.09  3   180.00
   H  2  1.07549  1  122.34  3   180.00
   H  3  1.09486  2  115.27  4   180.00
$end

$rem
   METHOD        local_mp2
   BASIS         6-311g**
$end

@@@

$molecule
   0 1
   C
   C  1  1.31656
   C  2  1.49838  1  123.44
   O  3  1.18747  2  123.81  1   92.28
   H  1  1.07631  2  122.03  3   -0.31
   H  1  1.07484  2  121.43  3  180.28
   H  2  1.07813  1  120.96  3  180.34
   H  3  1.09387  2  115.87  4  179.07
$end

$rem
   CORRELATION   local_mp2
   BASIS         6-311g**
$end