Auxiliary Basis Set MP2 Methods
This feature employs standardized auxiliary basis expansions to model products of atomic orbitals. Effectively this replaces four-center two-electron integrals by three and two-center integrals, which are vastly fewer in number. This permits faster calculations, with results for relative energies that are very faithful to conventional calculations.
Auxiliary basis expansions will be available for MP2 energies, and also for local MP2 energies (using our highly accurate TRIM model). The speedup that is seen for 450 basis function 6-31G* calculation on the alanine tetrapeptide is approximately a factor of 7 for the MP2 part of the calculation, which then becomes cheaper than the Hartree-Fock calculation. Even larger speedups are obtainable with larger basis sets.
In addition to these very large speed-ups, the auxiliary basis MP2 implementation also offers greatly reduced memory and disk requirements. Memory demand is quadratic while disk storage requirements are cubic. This compares to the cubic memory requirements of the conventional MP2 code.
At this time, the auxiliary basis local TRIM-MP2 implementation is still in progress, but it promises to be about a factor of 1.5 times faster for the tetrapeptide. Since it scales one power of system size lower, this speedup will double for a doubling of system size -- thus it will be roughly a factor of 3 for an octapeptide. Relative to TRIM in our previous release, this auxiliary basis implementation offers speedups roughly similar to those obtained for conventional MP2. This feature is unique to Q-Chem.
These methods make large-scale MP2 energy calculations far faster and far more feasible. We are preparing the extensions to allow energy gradients for future release.
- type(s) of applications it is good for. Some features are more for a specific type of problem.
A great replacement for regular MP2 or the regular local TRIM-MP2 when they cost too much or require system resources that are too large.
- graphs or data. Graphics are better than data tables.
Alanine Tetrapeptide 6-31G*
Conventional MP2: 2085 seconds (2 GHz
Apple G5)
Aux. basis MP2: 275 seconds (2 GHz Apple G5)
(other timings will be available shortly using larger basis sets and for larger sytems)
- uniqueness. What's the innovation.
Auxiliary basis MP2 is available in rival codes, but the lower scaling auxilary basis local MP2 is unique. We believe our implementation of auxiliary basis MP2 is highly competitive.
- compared with competition. What do our competitors have.
See above.
- application limit. Practical limit on size.
Memory requirement: approximately 20 N^2 which is not much worse than for a DFT calculation. On a 1GB machine, one can do over 2000 basis functions.
- limit: derivatives, basis angular momentum, etc.
Set by integral code.
Y. Jung, R. Lochan, T. Dutoi and M. Head-Gordon, J. Chem. Phys. (in press)
TRIM work is still underway and not yet written up.
Variations of conventional MP2:
RI-MP2, SOS-MP2, RI-LMP2, MOS-MP2,
RI-MOS-MP2
(by Yousung Jung,
Roh Lochan, and Rob Distatio, 4/11/05)
Density fitting approximation is often used in electronic structure theories to reduce computational expense, in which the charge density (or individual product density) is expanded by a set of atom-centered one-electron auxiliary functions.
The fit coefficients C are typically determined by minimizing the self interaction of residual density. Finally the solution is given as:
In RI-MP2, product of occupied and virtual orbitals (ia pair) is expanded with auxiliary functions. In other words, one essentially approximates numerous 4-center 2-electron MO integrals, (ia|jb), by less numerous and more efficient (cheaper AO to MO transformations) 3-center 2-electron MO integrals (ia|K).
With optimized auxiliary basis functions, RI-MP2 has been shown to almost identically reproduce conventional MP2 results for relative energies, with an absolute energy (compared to conventional MP2) off by about 60 micro-hartree/atom. Speed-ups due to the advantages of RI explained above are about 5-20 times faster than conventional MP2. However, it should also be noted that, although much more efficient than MP2, RI-MP2 is still a 5-th order scaling correlation method, just like conventional MP2, and hence RI computation also becomes intractable if one goes to very large systems, for example, systems with ~200 heavy atoms and more than 2000 basis functions.
It is certainly desirable to explore enhancements to the basic MP2 method that permit increased accuracy as well as improved computational performance. Following this theme, recently, the idea of using separate scaling of the same-spin (SS) and opposite-spin (OS) correlation energies was proposed by Grimme and led to statistical improvement of MP2 results over a range of properties. This approach is called “spin-component scaled” (SCS) MP2. In MP2 theory, the correlation energy can be written as
E OSand E SS are the contributions of the opposite-spin ( a b) and same-spin ( a a and b b) components to the total MP2 correlation energy and are given in terms of canonical orbitals as:
The SCS-MP2 energy is defined as:
where Grimme’s recommended scaling factors are c OS = 1.2 and c SS = 0.3.
Following this development, we recently suggested an even simpler variant of SCS-MP2 called “scaled opposite-spin” (SOS) MP2. We completely neglect the SS component of the correlation energy and scale only the OS part. The proposed scaling factors are now c OS = 1.3 and c SS = 0. This approach was shown to have a two-fold advantage in keeping with the theme of ‘improved accuracy and reduced scaling’. First, with only a single parameter, we were able to largely retain the statistical improvements obtained by the SCS-MP2 approach over a range of properties and second, we showed that SOS-MP2 energy can be evaluated with a fourth order scaling algorithm using a combination of auxiliary basis functions and a Laplace transform in contrast with the conventional fifth order scaling MP2 method.
Density fitting technique (again, essentially approximating 4-center ERIs with 3-center ERIs) can also be used to speed up other types of correlation methods as well, as in the recently implemented RI-assisted fast local MP2 method, namely RI-TRIM. TRIM is a 4-th order scaling algorithm, and again RI-TRIM does not alter the scaling of the existing method, but it reduces a pre-factor significantly. RI-TRIM provides overall 5-20 fold speed-ups compared to original TRIM (just like the efficiency difference between RI-MP2 and MP2). Errors introduced in RI-TRIM relative to TRIM are also very small – RI error seems to propagate consistently in RI-MP2 vs. MP2 and RI-TRIM vs. TRIM.
One can then think of combining the idea of scaled MP2 theory and RI-TRIM to gain further computational savings. It is based on the observation that exchange contribution in RI-TRIM needs not be computed if one is interested only in the opposite-spin component of correlation energy, as in SOS-MP2 method.
Despite statistically improved accuracy and low computational cost of SOS-MP2 theory, however, one undesirable feature of both the SCS-MP2 and SOS-MP2 methods, apart from their empirical parameter(s), is the incorrect physical description of the long-range correlation between two non-overlapping systems. In this long-range regime, if we assume that the two systems of interest are closed shell, the SS and OS components to the inter-system correlation energy should be exactly equal (since electrons in one monomer don’t know if the spin of electrons on the other monomer is up or down at that distance). Therefore the appropriate scaling factor contribute equally to the correlation energy, the scaling factor should approach 2. The scaling factors of SCS-MP2 and SOS-MP2 are around 1.5 and 1.3 respectively at this limit and thus tend to underestimate the MP2 correlation energy. Hence, these theories are not going to be very accurate for systems where long-range interactions are of critical importance. This assumes that MP2 theory itself is accurate for long-range dispersion interactions, which is usually (but not always) true.
In modified opposite-spin MP2 theory (MOS-MP2), we use distance-dependent scaling factor to correct the above weakness so that, at long-range limit, it gives the correct scaling factor which is 2. To this end, we define a long-range operator,
.
This operator was previously used in the context of separation
of the Coulomb operator (1/r) into a non-singular
but slowly and smoothly decaying long-range piece, 
and
a singular but rapidly decaying short-range part,
and
define a new “modified” two-electron operator, 
,
leading to a new set of “modified” integrals ( 
),
,
so that the “modified” opposite-spin (MOS)-MP2 energy is now given by
.
The variable c MOS above is easily fixed by the requirement
that 
as .
This new formalism MOS-MP2 is dependent on a single parameter w,
an optimal value for which can be determined empirically by performing
chemical tests. We recommend 0.6 for general purpose.
Keywords to specify:
Correlation = rimp2, sosmp2, rilmp2, mosmp2
Aux_basis = rimp2-VDZ, rimp2-cc-pVXZ, rimp2-aug-cc-pVXZ (X=D, T, Q)
Purecart = 1111
Supported basis sets:
rimp2-VDZ: H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br
rimp2-cc-pVXZ (X=D,T,Q): H He B C N O F Ne Al Si P S Cl Ar Ga Ge As Se Br Kr
Basically, rimp2-cc-pVXZ basis set is available for all atoms where the current QCHEM cc-pVXZ basis is available. But, I realize that QCHEM cc-pVXZ basis is not quite up to date at the moment (missing elements include Li, Be, Na, etc – these are available in pnl website).
With (very recent) additional developments of auxiliary basis sets for cc-pVXZ, rimp2-cc-pVXZ basis for these elements are now also available. So I think, as QCHEM updates the regular basis files (such as cc-pVXZ), I will add those to rimp2-cc-pVXZ files as well.