DF-MP2: Density-Fitted 2nd-Order Møller–Plesset Perturbation Theory

Code author: Robert M. Parrish

Section author: Robert M. Parrish

Module: Keywords, PSI Variables, DFMP2

Introduction

Second-order Møller–Plesset Perturbation Theory (MP2) occupies a unique role in quantum chemistry due to its small-prefactor \({\cal O}(N^5)\) treatment of dynamic electron correlation. This unusually cheap ab initio treatment of electron correlation may be made even more efficient by means of the Density-Fitting (DF) approximation (also known as Resolution-of-the-Identity or RI), wherein the quadratic \(ov\) products in the bra- and ket- of the \((ov|ov)\)-type Electron Repulsion Integrals (ERIs) appearing in MP2 are cast onto a linear-scaling auxiliary basis by least-squares fitting. Substitution of the DF factorization into the MP2 equations results in a formal scaling and prefactor reduction of MP2, and further speed gains are possible due to heavy utilization of matrix-multiplication kernels and minimal storage requirements in a DF approach. The method has been found to be quite robust and accurate, and it should be preferred unless extreme accuracy is required or a fitting basis is not defined for the primary basis and atom type encountered. In particular, we have found excellent efficiency and tractability gains when using DF-MP2 in concert with a DF-SCF reference. An efficient, threaded, disk-based DF-MP2 code is available in PSI4 for all single reference types available in the SCF module. MP2 defaults in PSI4 to the density-fitted code. See MP2_TYPE for performing a MP2 with conventional integrals.

An example utilization of the code is:

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molecule h2o {
0 1
O
H 1 1.0
H 1 1.0 2 104.5
}

set basis cc-pvdz
set scf_type df
set freeze_core True

energy('mp2')

The energy('mp2') call to energy() executes the predefined DF-MP2 procedure, first calling the SCF module with a default RHF reference and DF algorithm for the two-electron integrals. When the orbitals are converged, the DF-MP2 module is launched, which forms the density-fitted \((Q|ov)\) integrals and then builds the full \((ov|ov)\) tensor in blocks, evaluating the contributions to the MP2 energy as it goes. A RHF-MP2 wavefunction is selected automatically due to the RHF reference. In this example, we freeze the core, both for efficiency and because split-valence bases like cc-pVDZ do not contain core correlation functions. The result looks something like:

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-----------------------------------------------------------
 ==================> DF-MP2 Energies <====================
-----------------------------------------------------------
 Reference Energy          =     -76.0213974638823942 [Eh]
 Singles Energy            =      -0.0000000000000001 [Eh]
 Same-Spin Energy          =      -0.0512503270216563 [Eh]
 Opposite-Spin Energy      =      -0.1534098175176923 [Eh]
 Correlation Energy        =      -0.2046601445393486 [Eh]
 Total Energy              =     -76.2260576084217405 [Eh]
-----------------------------------------------------------
 ================> DF-SCS-MP2 Energies <==================
-----------------------------------------------------------
 SCS Same-Spin Scale       =       0.3333333333333333 [-]
 SCS Opposite-Spin Scale   =       1.2000000000000000 [-]
 SCS Same-Spin Energy      =      -0.0170834423405521 [Eh]
 SCS Opposite-Spin Energy  =      -0.1840917810212307 [Eh]
 SCS Correlation Energy    =      -0.2011752233617829 [Eh]
 SCS Total Energy          =     -76.2225726872441811 [Eh]
-----------------------------------------------------------

The theory, breakdown of results, and common keywords used in DF-MP2 are presented below.

Theory

Møller–Plesset Theory (MPn) or Many-Body Perturbation Theory (MBPT) through second order has the spin-orbital formula:

(1)\[E_{\mathrm{total}}^{(2)} = E_{\mathrm{Reference}} - \frac{f_{ia} f_{ia}}{\epsilon_a - \epsilon_i} - \frac{1}{4} \frac{\langle ij||ab\rangle^2}{\epsilon_a + \epsilon_b - \epsilon_i - \epsilon_j}\]

Here \(i\) and \(j\) are occupied spin orbitals, \(a\) and \(b\) are virtual spin orbitals, \(f_{ia}\) are the \(ov\) Fock Matrix elements, \(\epsilon\) are the orbital eigenvalues, and \(\langle ij||ab\rangle\) are the antisymmetrized physicist’s ERIs. For converged RHF and UHF references, the singles correction,

\[E_{\mathrm{MBPT}}^{(1)} = - \frac{f_{ia} f_{ia}}{\epsilon_a - \epsilon_i},\]

is zero due to the Brillioun Condition, and the first contribution to the perturbation series is at the second order:

\[E_{\mathrm{MBPT}}^{(2)} = - \frac{1}{4} \frac{\langle ij|ab\rangle^2}{\epsilon_a + \epsilon_b - \epsilon_i - \epsilon_j}.\]

In the DFMP2 module, the first-order contribution, or “singles energy” is always evaluated. This term is a significant contributor to the total second-order energy if a ROHF reference is used. In this case, we have chosen to use the ROHF-MBPT(2) ansatz, in which the ROHF orbitals are semicanonicalized, the resultant nonzero Fock matrix elements \(f_{ia}\) are used to form the singles amplitudes, and then the second-order amplitudes are formed with the semicanonical spin orbitals via the same machinery as a UHF-MP2. Note that the singles energy should be very close to zero for RHF and UHF references; if it is not, there is a good chance your orbitals are not well converged. Tighten the SCF E_CONVERGENCE and/or D_CONVERGENCE keywords and try again.

To increase the efficiency of MP2 energy evaluation, spin integration and simplification is carried out. This also allows for the identification of Same-Spin (SS) and Opposite-Spin (OS) terms for use in Grimme’s Spin-Component Scaled (SCS) MP2. For RHF-MP2 (also labeled as RMP2), the spin-free equations are (note that the integrals are now chemist’s integrals over spatial orbitals)

\[E_{\mathrm{MBPT,OS}}^{(2)} = - \frac{(ia|jb)(ia|jb)}{\epsilon_a + \epsilon_b - \epsilon_i - \epsilon_j}\]

and

\[E_{\mathrm{MBPT,SS}}^{(2)} = - \frac{[(ia|jb)-(ib|ja)](ia|jb)}{\epsilon_a + \epsilon_b - \epsilon_i - \epsilon_j}.\]

For UHF-MP2 (also labeled as UMP2) and the second-order contribution to ROHF-MBPT(2) using semicanonical orbitals, the spin-free equations are

\[E_{\mathrm{MBPT,OS}}^{(2)} = - \frac{(ia^\alpha|jb^\beta)(ia^\alpha|jb^\beta)}{\epsilon_a + \epsilon_b - \epsilon_i - \epsilon_j}\]

and

\[E_{\mathrm{MBPT,SS}}^{(2)} = - \frac{1}{2}\frac{[(ia^\alpha|jb^\alpha)-(ib^\alpha|ja^\alpha)](ia^\alpha|jb^\alpha)} {\epsilon_a + \epsilon_b - \epsilon_i - \epsilon_j} - \frac{1}{2}\frac{[(ia^\beta|jb^\beta)-(ib^\beta|ja^\beta)](ia^\beta|jb^\beta)} {\epsilon_a + \epsilon_b - \epsilon_i - \epsilon_j}.\]

Note that the UHF-MP2 equations use three classes of integrals, while the RHF-MP2 equations use only one class. Because of this, a UHF-MP2 or ROHF-MBPT(2) energy should take roughly three times as long as an RHF-MP2 energy.

Recommendations

All-in-all, DFMP2 should be a simple module to use, with few keywords (fully documented in the Appendix DFMP2). Some basic recommendations are included below:

  • DFMP2 should be run with the \(ov\)-type RI or MP2FIT auxiliary basis sets, not the -JKFIT basis sets. The automatic basis selector should work fine for all of the Dunning and Pople bases (provided the auxiliary basis exists for the atom in question). If it does not, use the DF_BASIS_MP2 keyword to manually specify the basis.
  • DFMP2 likes memory. At a minimum, \(2Q^2\) doubles are required, where \(Q\) is the size of the auxiliary basis set. However, there is one disk transpose of the \((Q|ov)\) tensor in the RHF-MP2 algorithm [two for UHF-MP2 and ROHF-MBPT(2)], so more memory will reduce seek times. If you notice DFMP2 using more memory than allowed, it is possible that the threaded three-index ERI computers are using too much overhead memory. Set the DF_INTS_NUM_THREADS to a smaller number to prevent this in this section (does not affect threaded efficiency in the rest of the code).
  • DFMP2 likes disk. At a minimum, \(2Qov\) doubles are required for RHF-MP2, and \(4Qov\) doubles are required for UHF-MP2.
  • DFMP2 likes threads. Some of the formation of the \((Q|ov)\) tensor relies on threaded BLAS (such as MKL) for efficiency. The main \({\cal O}(N^5)\) step is done via small/medium-sized DGEMMs inside of OpenMP, so make sure to set the OMP_NESTED environment variable to FALSE to prevent thread thrash (or just as well, do not define OMP_NESTED at all).
  • Freezing core is good for both efficiency and correctness purposes. Freezing virtuals is not recommended. The DFMP2 module will remind you how many frozen/active orbitals it is using in a section just below the title.
  • ROHF-MBPT(2) may be preferred to UHF-MP2, as the latter can suffer from severe spin contamination in some cases.
  • MP2 is not suitable for systems with multireference character. The orbital energies will come together and an explosion will occur.