Questions & Answers about W1 and W2 theory

Q: In which paper have W1 and W2 theory been defined?
A: In JCP 111, 1843 (1999)

Q: Have any validation studies been carried out?
A: See particularly JCP 114, 6014 (2001)

Q: Are they ab initio methods?
A: W2 is a purely ab initio method in the sense that it is entirely free of empirical parameters. W1 theory does include an "empirical" parameter which is however derived from W2 calculations rather than experiment.

Q: How do they relate to G2, G3, CBS-Q, CBS-QB3 theory?
A: The above methods on average yield on average 1 kcal/mol accuracy, and are intended to be still usable on medium-sized systems. They all involve some degree of empirical parametrization. W1 and W2 theory yields more accurate results without empirical parametrization, at substantially increased computational expense. "There ain't no such thing as a free lunch."

Q: What is the largest system W1 and W2 theory can be applied to?
A: Depends on available hardware and symmetry of the molecule. We have successfully carried out a W1 calculation on benzene on a workstation; the largest W2 calculations carried out so far were SO₃ on a workstation, and SiF₄ on a Cray T90.

Q: Why are the SCF and valence correlation contributions extrapolated separately?
A: Because their convergence behaviors are fundamentally different -- the SCF energy appears to converge exponentially, while the valence correlation energy converges as an inverse-power series in the maximum angular momentum present in the basis set.

Q: How is the SCF energy extrapolated "old-style" and why?
A: By a 3-point geometric extrapolation A+B/Cn to 3 successive AVnZ (or AVnZ+2d1f for second-row elements) basis sets. By comparison with numerical Hartree-Fock results we know that agreement as close as 10 microhartree can be achieved.

Q: What is the difference between "old-style" and "new-style"? How is the SCF energy extrapolated "new-style"?
A: The difference is in the SCF extrapolations. In "new-style", the SCF part is extrapolated by a 2-point A+B/L⁵ formula. This gives essentially the same results as the exponential extrapolation in well-behaved cases, but is much more reliable in cases where the AVDZ basis set is particularly poor.

Q: How is the valence correlation energy extrapolated and why?
A: It has been known for some time (Schwartz, 1963; Hill, 1985) that the correlation energy in a helium-like atom follows an asymptotic series of the type A+B/L³+C/L⁴+... It has also been shown (Kutzelnigg and Morgan, 1992) that triplet-coupled pairs follow similar series A+B/L⁵+C/L⁶+... [In these expressions, L stands for the maximum angular momentum present in the basis set.] Our "leap of faith" was that since atomic as well as molecular correlation energies are dominated by pair correlation energies, perhaps the molecular correlation energy could be well extrapolated by an inverse-power series in L as well? In [Martin, Chem. Phys. Lett. 259, 669 (1996)] we found that this was indeed the case.
After extensive experimentation with various approximate expressions, we found that simply A+B/L³ works best if one can go up to L=5 (i.e. basis sets including h functions). This is the expression used in W2 theory. In W1 theory (which uses only up to L=4 inclusive), we use A+B/L^3.22, where the value 3.22 maximizes agreement with the W2 results.

Q: Why are the CCSD and (T) correlation energies extrapolated separately?
A: The (T) part is smaller in absolute magnitude, appears to converge more rapidly with the basis set, and --- most importantly --- requires a CPU time that scales as n³N⁴ rather than n²N⁴. (For systems with very many electrons correlated, the (T) term dominates the CCSD(T) calculation.) Hence we extrapolate it separately from smaller basis set results, such that only a CCSD calculation is required in the largest basis set.

Q: Why is the inner-shell correlation energy treated separately, and without extrapolations?
Particularly for second-row elements, the inner-shell correlation energy is a very large part of the total energy; however, the differential contribution to the total energy is quite small. In addition, the large numer of extra electrons correlated makes the cost of extrapolations prohibitive. Since we are interested mainly in BINDING, not in ABSOLUTE energies, we evaluate the differential contribution to the binding energy using a compact basis set optimized for inner-shell correlation (the so-called "Martin-Taylor small" basis set).

Q: Why should we bother with the scalar relativistic contribution?
A: Contrary to popular belief, they can make nontrivial contributions to the binding energy, particularly in systems involving several polar bonds to group VI or VII atoms. BF₃, SO₃, and SiF₄ are cases in point. These numbers are substantially larger than the residual error of W2 or even W1 theory, and the corrections take a fraction of the total CPU time, so we may as well do them. Neglect generally leads to an overestimate of the binding energy.

Q: Why are scalar relativistic effects treated at the ACPF/MTsmall level in W1/W2 theory?
A: First of all, for first-and second-row atoms, Darwin and mass-velocity (DMV) contributions should be sufficient. Secondly, neglect of electron correlation leads to a consistent overestimate of the relativistic contribution. Since the latter is most important for the innermost electrons, it seemed reasonable to use a basis set which is suitable for inner-shell correlation and to correlate all electrons. The ACPF (averaged coupled pair functional) correlation method is size-extensive and allows determination of the DMV contribution as a simple expectation value.

Q: What about the spin-orbit contribution? Is it an ab initio or an empirical term?
A: Most molecules (closed-shell systems, open-shell in nondegenerate electronic states) do not have first-order spin-orbit splitting. What will be affected are the constituent atomic energies. The corrections for the individual atoms can be obtained from experimental fine structures or from high-accuracy ab initio calculations (you only to do this once for each element, right?): to the accuracy relevant for our purposes, the result is the same. Once you have the corrections for each element, you need no more hardware than a pocket calculator.
Neglect of spin-orbit splitting usually results in underestimation of the molecular binding energy.
For molecular systems in degenerate ground states, one should also compute the molecular spin-orbit splitting if available. We found in the validation paper that results are acceptable even at the CISD/MTsmall level, particularly if the (2s,2p) electrons on second-row elements are correlated as well.

Q: What about zero-point energies?
A: For the utmost accuracy, particularly if any hydrogen atoms occur in the molecule, we recommend computing a quartic force field and obtaining a zero-point energy by vibrational second-order perturbation theory. Since this is rarely an option (unless one is also interested in the vibrational spectroscopy of the system and would want to do this type of calculation anyhow), we recommend scaled B3LYP/cc-pVTZ+1 harmonic frequencies as a second-best solution. The recommended scaling factor for zero-point energies (based on a comparison with exact values) is 0.985.

Q: What is all this business with "VTZ+1", "AVQZ+2d1f", etc.
A: It was shown some time ago [Bauschlicher and Partridge, CPL 240, 533 (1995); Martin, JCP 108, 2791 (1998)] that for second-row molecules with polar bonds, the inclusion of high-exponent d and f functions on the second-row atoms is essential for accurate properties. The most extreme example we have so far encountered is SO₃, where the inclusion of such functions makes a difference of between 10 and 40 kcal/mol to the SCF energy. This is a pure SCF level effect and has nothing to do with inner-shell correlation.
The notation "+2d1f" means that two d functions and one f function have been added to each second-row atom, with the exponents obtained by successively multiplying the highest one already present for that angular momentum by 2.5. The notation "+1" means that merely a single high-exponent d function has been added, with the same exponent as the highest one occurring in the cc-pV5Z basis set.

Q: What is the largest remaining source of potential error?
A: Imperfections in the CCSD(T) correlation method, which themselves can be decomposed into effects of connected quadruple excitations on the one hand, and imperfections in the treatment of connected triple excitations on the other hand. Upon taking these effects into account, our results for atomic electron affinities (where it is practical to do so) improve by an order of magnitude and on average agree with experiment to within 0.001 eV.

NEW Q: What is W3 theory?
A: In W3 theory, we attempt corrections for correlation effects that go beyond CCSD(T). The CCSDT - CCSD(T) difference is extrapolated from cc-pVDZ and cc-pVTZ basis sets; the CCSDTQ - CCSDT difference is computed with a cc-pVDZ basis set and scaled by 1.25. Additionally (but not very importantly for light-element systems), the scalar relativistic contributions are obtained from Douglas-Kroll CCSD(T) calculations with relativistically optimized basis set: this approach will be more robust for heavier-element systems.
Due to the extreme computational demains of the CCSDTQ calculation (scale as n4N6), W3 calculations are presently limited to systems about the size of ozone. However, W3 has no apparent trouble with molecules with significant nondynamical correlation effects, for which W2 may yield errors of a couple of kcal/mol.

NEWQ: How does the W1 implementation in Gaussian 03 differ from the standard?
A: Two ways. First of all, the scalar relativistic correction is obtained from a Douglas-Kroll CCSD(T) calculation (which should meet or exceed the reliability of the standard W1 relativistic correction). Secondly, CCSD(T) for open-shell systems is carried out from a UHF reference determinant. This may adversely affect accuracy for molecules that suffer from spin contamination at the UHF level. Needless to say, this is not an issue for closed-shell systems.

Q: What advantage is there in W2 theory over W1 theory?
A: Mostly rigor. The accuracy of W2 theory mainly seems to be limited by the performance of the CCSD(T) electron correlation method, and for well-behaved molecules, the results can be within as little as 0.17 kcal/mol, on average. Also, there is no bias towards first-row systems like W2 appears to have.

Q: What is the difference between W1 and W1' theory?
A: For first-row compounds, the two schemes are equivalent. For second-row compounds, W1' theory uses a basis set sequence in the extrapolations (AVDZ+2d, AVTZ+2d, AVQZ+2d1f) that appears to be somewhat better balanced than the standard W1 sequence (AVDZ+2d, AVTZ+2d1f, AVQZ+2d1f) and yields substantially improved results for second-row compounds at no additional expense. [J. M. L. Martin, Chem. Phys. Lett. 310, 271 (1999)]
In our W1/W2 validation paper, we found that much of the discrepancy is an artifact of the "old-style" SCF extrapolation: they largely disappear for the "new-style" extrapolation.

Q: Does it matter whether you extrapolate on the individual energies or on the computed reaction energy?
A: For "old-style" W1/W2 theory, yes (but the differences are generally tiny). For "new-style" W1/W2 theory, the results are independent of the order in which the extrapolations are carried out. (A nice by-product of getting rid of the 2-point exponential SCF extrapolation.)