Contents: Table of Contents   CISD, CEPA, CCD, QCISD  Chapter 3  Direct MP2 calculations

Spin-projected MP2 calculations

As already mentioned UHF-MP2 calculations can suffer from significant spin-contamination. The best solution to this problem is not to use UHF at all and instead use the ROHF-MP2 techniques described in section 3.2. However, CADPAC also has an older technique based on applying a spin-projection operator to the UHF-MP2 expression. This method is called PUMP2. The following dataset provides an example. The implementation of PUMP2 is a conventional disk based scheme i.e. there is no 'direct' algorithm.

TITLE
SIC2H4  TRIPLET SCF and PUMP2 CALCULATION
NOPRINT OCCVECTORS
SYMMETRY
CNV 2
END
BASIS DZP
ATOMS
SI 14 0.0 0.0 -0.6615482
C1 6 2.8885198 0.0 -2.2904091
H1 1 4.7218001 0.0 -1.3803020
H2 1 2.9203532 0.0 -4.3435272
END
MULTIPLICITY 3

UHF
PUMP2
ENERGY
START

FINISH

This example is taken from a study of the Si + C₂H₄ surface - see J. Am. Chem. Soc. 112, 1499 (1990). This paper also contains references describing the origin of the PUMP2 method.

After the SCF output one finds the following,

 AFTER PROJECTION 0
 PROJECTED UHF ENERGY (PUHF)            -366.876041926
 PROJECTED SECOND ORDER ENERGY          -0.4050884096501
 TOTAL ENERGY (PUMP2) AFTER PROJECTION  -367.2811303357
 S-SQUARED AFTER PROJECTION             2.25464967077

 AFTER PROJECTION 1
 PROJECTED UHF ENERGY (PUHF)            -366.8861435227
 PROJECTED SECOND ORDER ENERGY          -0.404562841094
 TOTAL ENERGY (PUMP2) AFTER PROJECTION  -367.2907063638
 S-SQUARED AFTER PROJECTION             1.994429682714

 AFTER PROJECTION 2
 PROJECTED UHF ENERGY (PUHF)            -366.8856775658
 PROJECTED SECOND ORDER ENERGY          -0.4047783871538
 TOTAL ENERGY (PUMP2) AFTER PROJECTION  -367.290455953
 S-SQUARED AFTER PROJECTION             2.000038496833

The data under 'Projection 0' are the original UHF and UHF-MP2 results. It can be seen that the value of S² is significantly in error. The projection operator is then applied twice, both to the SCF energy and to the MP2 energy. After the second projection the value of S² is much better. The results after the first projection are best ignored. In most cases the total energies (and more particularly, energy differencies) are better after projection than the original UHF-MP2 energies. There are other spin-projection schemes in the literature. It should be noted that the PUMP2 scheme is not the same as the PMP2 scheme due to Schlegel - the PUMP2 scheme is more rigorous.