9.4 General Comments on MP2 Calculations, Practical Hints

Recommendations

• It is well-known, that perturbation theory yields reliable results only, if the perturbation is small. This is also valid for MP2, which means, that MP2 improves HF results only, if HF already provides a fairly good solution to the problem. If HF fails, e.g. in case of partially filled d-shells, MP2 usually will also fail and should not be used in this case.
• MP2 results are known to converge very slowly with increasing basis sets, in particular slowly with increasing l-quantum number of the basis set expansion. Thus for reliable results the use of TZVPP basis sets (or higher) is recommended. When using SVP basis sets at most a qualitative trend can be expected. Basis sets much larger than TZVPP usually do not significantly improve geometries of bonded systems, but still can improve the energetic description. For non–bonded systems larger basis sets (especially, with more diffuse functions) are needed.
• It is recommended to exclude all non-valence orbitals from MP2 calculations, as neither the TURBOMOLE standard basis sets SVP, TZVPP, and QZVPP nor the cc-pVXZ basis set families (with X=D,T,Q,5,6) are designed for correlation treatment of inner shells (for this purpose polarisation functions for the inner shells are needed). The default selection for frozen core orbitals in define (orbitals below -3 a.u. are frozen) provides a reasonable guess. If core orbitals are included in the correlation treatment, it is recommended to use basis sets with additional tight correlation functions as e.g. the cc-pwCVXZ and cc-pCVXZ basis set families.
• RI-MP2: We strongly recommend the use of auxiliary basis sets optimized for the corresponding orbital basis sets.

RI-MP2 calculations with the ricc2 program: All what is needed for a RI-MP2 gradient calculation with the ricc2 program is a $ricc2 data group with the entry geoopt model=mp2. If you want only the RI-MP2 energy for a single point use as option just mp2. To activate in MP2 energy calculations the evaluation of the D₁ diagnostic (for details see Sec. 10.1). use instead mp2 d1diag. (Note that the calculation of the D₁ diagnostic increases the costs compared to a MP2 energy evaluation by about a factor of three.)

Comments on the Output

• Most important output for ricc2, pnoccsd, and mpgrad are of course the MP2(+HF) energies (written to standard output and additionally to the file energy) and MP2(+HF) gradients (written to the file gradient).
• In case of MP2 gradient calculations the modules also calculate the MP2 dipole moment from the MP2 density matrix (note, that in case of mpgrad a frozen core orbital specification is ignored for gradient calculations and thus for MP2 dipole moments).

Further output contains indications of the suitability of the (HF+MP2) treatment.

• As discussed above, MP2 results are only reliable if the MP2 corrections to the Hartree-Fock results are small. One measure for size of MP2 corrections to the wavefunction are the doubles amplitudes, t_ab^ij, as is evident from the above equations. mpgrad by default prints the five largest amplitudes as well as the five largest norms of amplitude matrices t^ij for fixed i and j. The number of printed amplitudes can be changed by setting the data group $tplot n where n denotes the number of largest amplitudes to be plotted. It is up to the user to decide from these quantities, whether the HF+MP2 treatment is suited for the present problem or not. Unfortunately, it is not possible to define a threshold, which distinguishes a "good" and a "bad" MP2-case, since the value of individual amplitudes or amplitudes matrices t^ij are not orbital-invariant, but depend on the orbital basis and thereby under certain circumstances on the orientation, the point group, or the start guess for the MOs. Example: the largest norm of t-amplitudes for the Cu-atom (d¹⁰s¹, "good" MP2-case) amounts to ca. 0.06, that of the Ni-atom (d⁸s², "bad" MP2 case) is ca. 0.14.
• A more reliable criterion is obtained from the MP2 density matrix. Its eigenvalues reflect the changes in occupation numbers resulting from the MP2 treatment, compared to the Hartree-Fock level, where occupation numbers are either one (two for RHF) or zero. Small changes mean small corrections to HF and thus suitability of the MP2 method for the given problem. In case of gradient calculations ricc2 displays by default the largest eigenvalue of the MP2 density matrix, i.e. the largest change in occupation numbers (in %). If $cc2_natocc is set the full set of natural occupation numbers and orbitals will be save to the control file. For main group compounds largest changes in occupation numbers of ca. 5 % or less are typical, for d¹⁰ metal compounds somewhat higher values are tolerable.
• A similar idea is pursued by the D₂ and D₁ diagnostics [133,134] which is implemented in ricc2. D₂ is a diagnostic for strong interactions of the HF reference state with doubly excited determinants, while D₁ is a diagnostic for strong interactions with singly excited determinants.