For performing high-level ab initio computations (implying the usage of post-HF methods), three cornerstones must be taken into account: the level of theory for describing the electron correlation effects, the basis set size, and the level of theory for describing relativistic effects.

Currently, the coupled cluster-based techniques (CC) are very powerful ab initio methods for describing the electron correlation effects (for inorganic molecules). The hierarchy of these methods includes CCSD (with singles and doubles excitations), CCSDT (also triple excitations), CCSDTQ, etc. Within this series, the energy quickly converges to a hypothetical point of “full correlation” (for a specified basis set). The advantage of the coupled cluster theory, compared to the configuration interaction (CI) technique, is that with CC the energy converges more quickly upon increasing the excitation rank than with CI.

CCSD(T) is called the “golden standard” of quantum chemistry. Unlike the CCSDT method, in CCSD(T) the triples (triple excitations) are treated non-iteratively, in a perturbative fashion, which reduces computation cost quite significantly. CCSD does not give such good describing of electronic correlation, CCSDT is quite costly, while the CCSD(T) method is a good compromise; in addition, the success of the CCSD(T) approach stems from fortuitous error cancellation (however at Ref. [14] it is stated that CCSD(T) is worse than CCSDT for open shell systems, in contrast to closed shell systems).

Note that for all post-HF methods, upon increasing the basis set size the total energy converges relatively slowly; that’s why, it is a kind of naïve to expect “golden quality” results from the CCSD(T) method with the basis sets as small as DZ or even TZ (the DZ and TZ abbreviations stand for double-zeta and triple-zeta quality basis sets, respectively).

As we have written before, the so-called complete basis set (CBS) limit means that you first compute with cc-pVDZ, then cc-pVTZ, then cc-pVQZ, then cc-pV5Z, etc., and the energy should converge to a hypothetical “complete” basis set limit. At the same time, there are more than 10 extrapolation schemes which give nearly the same result after performing only 2-3 computations (however, these extrapolation schemes are empirical to some extent).

In the work of a quantum chemistry guru Kirk Peterson [1] the following is stated: “

*To directly achieve chemical accuracy with CCSD(T) without extrapolation would require correlation of both the valence and outer-core electrons with a basis set of at least aug-cc-pCV6Z or augcc-pCV7Z quality*”. The “chemical accuracy” term is usually interpreted in the thermochemistry literature as the error of 1 kcal/mol ( 4.184 kJ/mol) for atomization energies; Peterson offers the following arbitrary estimates for spectroscopic parameters: +-0.005 Å for bond lengths and +-15 cm-1 for vibrational frequencies. Without the extrapolation schemes, such computations are possible for very small molecules only; besides that, for heavy atoms (starting, e.g., from post-3d elements), the relativistic effects must be properly included to attain even semiquantitative accuracy [2, 3]. An efficient way of reducing the computation cost is the “composite approach”, which is based on the concept that smaller components are additive and that these terms can be most efficiently calculated at different levels of theory (but comparable levels of accuracy).

Previous composite methods like G1, G2, etc., or W1, W2 comprise computing chemical properties with molecular geometries obtained at HF/6-31G* or MP2/6-31G* level, and are much less reliable than fully ab-initio Feller–Peterson–Dixon method (see below) (unlike the FPD approach, these methods are a “black box”).

For very small molecules, the HEAT model (high-accuracy extrapolated ab initio thermochemistry) can be used; it produced a MAD of 0.09 kcal/mol relative to well-established experimental DHf(0 K) values (for a test set of 26 small molecules involving 5 atoms (H, C, N, O and F)) [5]. The maximum observed error was 0.33 kcal/mol (C2H2). The HEAT approach implies optimizing geometries at CCSD(T)(CV)//cc-pVQZ, then estimating the basis set limit by a 1/lmax 3 extrapolation [4] of aug-cc-pCVQZ and aug-cc-pCV5Z energies.

In the Feller–Peterson–Dixon (FPD) approach, the total energy of atomization of the molecule is usually computed as the sum of the following terms:

1) Estimated complete basis set (CBS) limit of the CCSD(T) energy;

2) Energy of correlation of outer-core electrons with themselves and with the valence electrons (core-valence (CV) term). This term is estimated at CBS limit too, if possible;

3) Scalar relativistic effects term (or, when these effects are already accounted for in the first two stages, the elimination of errors due to using the pseudopotential approximation);

Sometimes it is advised to use all-electron DK-contracted basis sets already for the first two stages if available relativistic sets are of the same size as non-relativistic ones.

4) Spin-orbit (SO) interaction term. This is a relativistic term too; more specifically, there are two levels of spin-orbit effects, the effects of second order are not always necessary to include. For closed shell species, only first order effects should be taken into account;

5) Corrections to the total atomization energy beyond the CCSD(T) level;

6) Zero-point energy (the energy of the first vibrational state). Usually this correction is computed in the harmonic approximation. For polyatomic molecules, this term may be reliably obtained from harmonic frequencies computed at lower levels of theory (DFT, MP2).

The approach provides flexibility in terms of the use of a particular level of theory for computing individual contributions. In some cases, a few of these components may be excluded without severe loss of accuracy. Below we present two sample formulas of computing the total energy in FPD approach, appropriate for a molecule comprising 2-5 atoms (the first formula demands lower hardware level than the latter):

Particularly, for the Q5 extrapolation algorithm:

N=4 for QZ, 5 for 5Z; here we have got 2 equations with two variables (Ecbs and A).

∆ESO includes atomic corrections taken from experimental data (NIST) and computed corrections for molecules.

The explanation of this formula is presented in Ref. [6]. We know that the ∆ESO can be computed using the Molpro program (via the state-interacting approach within the first-order perturbation theory) or the Dirac program (spin-orbit coupling is already included at the HF level of theory).

In the FPD approach, when high accuracy is necessary, the molecular geometry is usually optimized using CCSD(T) method with a sequence of the correlation-consistent basis sets in an effort to evaluate energy differences at or near the CBS limit geometries. Besides that, reasonable results can be usually obtained with lower level geometries (e.g. computed at DFT level).

The FPD approach currently seems to be the best solution for computing high-accuracy energies and other properties of relatively small molecules (usually inorganic). As an indication of the performance of the FPD procedure for atomization energies, a mean signed deviation of -0.04 kcal/mol, RMS = 0.28 kcal/mol and MAD = 0.17 kcal/mol is found when comparing against 121 molecules (small organic or inorganic molecules containing s and p elements, like CH4, C2N2, C2H2O, C4H6, SO3, HBr, H2SiO, I2, etc) whose experimental uncertainties are ±1 kcal/mol or less [7].

Now we should make an important statement. There is a persuasive hypothesis that the FPD approach never produces wrong results, if one reaches convergence in each of the computed contributions to the energy. And when the computation is performed by an experienced quantum chemist, he can always tell whether the convergence is reached in a particular component. For reliability of the FPD method it is required that the additivity approach works, and an experienced quantum chemist can test this approach too.

The term “wrong result” needs to be interpreted in more details. Any computation can be characterized by a specific accuracy. In chemistry, the concepts of “spectroscopic accuracy” and “chemical accuracy” are used. The spectroscopic accuracy (1 cm-1) is needed for the spectroscopy, the chemical accuracy (1 kcal/mole) is needed for thermochemistry, and for the majority of tasks probably even smaller accuracy is required. Using the FPD method on personal computers, one can reach chemical accuracy for the overwhelming majority of small stable inorganic molecules (diatomic and symmetrical polyatomic), or for small- to medium-sized organic molecules. It should be noted that for first and second row compounds the said accuracy can be achieved, using only the first and second step of the FPD procedure (elimination of basis set incompleteness and taking into account the core-valence correlation).

It has been shown [12], that for molecules containing d and f elements the experimental error exceeds 1 kcal/mol, and because of that, for these compounds it was proposed [13] to consider the chemical accuracy being 3 kcal/mol for d elements and 5 kcal/mol for f elements.

The most important point here is as follows: an experienced quantum chemist, if he can’t reach the required accuracy in the computation, is usually aware of that, because the convergence of any FPD component (contribution to the energy) can be estimated via its computation on a higher level of theory. The necessity to analyze individual components is both the disadvantage of the FPD approach (because it makes this method relatively laborious, demanding a high qualification of the chemist and the use of appropriate software), and its advantage (because the quantum chemist can decide which contributions should be calculated at a high level of theory, and which of them may be neglected). Because of that, the FPD method is definitely not a “black box”. And besides that, the error of the computation can be often estimated by comparing the results of the computations for similar molecules with the experimental data (mainly high-accuracy, particularly spectroscopic). For example, to estimate the computational errors for the NdI3 molecule, one can calculate the Nd3+ ion and compare the relative energies of its electronic states with the experiment.

At the moment, only a few groups are performing the FPD computations: groups of Peterson K.A., DeYonker N., Hill J. G., Solomonik V.G.

It should be mentioned that the abbreviation "FPD" should not be considered as some prescribed procedure; this term does not imply using any particular approaches when any energy contribution is computed. A researcher who applies this method must decide himself, what level of theory should be used in the computation of a particular energy contribution (depending on the task level and available hardware and software). So, a significant advantage of the FPD method is that it can immediately benefit from any newest developments of methods, basis sets, and hardware.

While modeling spectra, it is desirable to ensure that the Born-Oppenheimer approach is suitable for the molecule studied. By the term “an approach is non-suitable” we mean that the accuracy of the computation within this approach is worse than required. The Born-Oppenheimer approach can yield large errors, if there are low-lying electronic states in the system: in this case, the reciprocal influence of the PESs of different electronic states will be observed, and this will lead to significant discrepancies between the computed and experimental spectra. When the Born-Oppenheimer approach fails, the electronic and vibrational states interact, and the electronic and vibrational (nuclear) solutions cannot be separated [8]. Ab initio methods, in fact, allow one to deal with the electronic part, while the Born-Oppenheimer approach is related to the nuclear one.

For very heavy elements, other relativistic elements can play a role, in particular the effects of quantum electrodynamics (first of all, the Lamb shift). Kirk Peterson writes:

*While generally neglected in most composite methodologies, it has been shown previously by Dyall et al. that the contribution of the Lamb shift to the atomization energy can approach 3–5% of the scalar relativistic contribution. In cases where the scalar relativistic correction is large, for example when the valence s orbital occupation strongly changes, the Lamb shift can contribute ~0.1 to 0.2 kcal/mol to the atomization energy for even relatively light systems like AlF3 and GaF3. *
The FPD method is usually applied for inorganic molecules. It should be mentioned, that for molecules with complicated electronic structure, even small, the convergence of all contributions for achieving the chemical accuracy still can’t be reached.

Nevertheless, at the moment the properties of the majority of molecules with up to 4 atoms can be computed by the FPD method with accuracy, in practice not worse than the experimental one.

The FPD method is based on the assumption of the additivity of the contributions, which are enlisted above. In some cases, when the researcher needs a very high (spectroscopic) accuracy, the standard approach implying separate accounting for various contributions may fail to yield such accuracy. This can occur when individual contributions (for example, spin-orbit) are relatively high. In such situations several contributions should be taken into account simultaneously.

For molecules, which are to big to be computed via the “traditional” FPD, the following approach becomes righteous: the geometry is optimized at DFT level with triple-zeta basis set, and then a single point FPD computation is performed (for example, in the Peterson’s work [9]).

It should be noted that for the majority of small first and second row compounds, as it can be seen, the chemical accuracy can be reached at the DFT level (not systematically however). In Ref. [10] the computations of 211 molecules were performed using the PBE and PBE0 functionals and quintuple zeta-quality basis sets (aug-cc-pV5Z), and the authors obtained MAEs of lower than 1 kcal/mol and MaxAEs of 2-5 kcal/mol. See this blog [11] for more information.

Refs:

[1] Theor Chem Acc (2012) 131:1079

[2] Methods in Computational Chemistry; Wilson , S., Ed.Vol. 2; Plenum Press: New York, 1988.

[3] Hess, B. A.; Dolg, M. Relativistic Quantum Chemistry with Pseudopotentials and transformed Hamiltonians. Wiley Series in Theoretical Chemistry; John Wiley & Sons: Chichester , 2002; Vol. 57.

[4] Helgaker T, Klopper W, Koch H, Noga J (1997) J Chem Phys. 106:9639

[5] Chemical accuracy in ab initio thermochemistry and spectroscopy: current strategies and future challenges. Kirk A. Peterson • David Feller • David A. Dixon. Theor Chem Acc (2012) 131:1079 DOI 10.1007/s00214-011-1079-5

[6] David A. Dixon, David Feller, Kirk A. Peterson. Annual Reports in Computational Chemistry. Volume 8, 2012, Pages 1–28

[7] D. Feller, K.A. Peterson, and D.A. Dixon, "A survey of factors contributing to accurate theoretical predictions of atomization energies and molecular structures", J. Chem. Phys. 129, 204105 (2008).

[8] https://en.wikipedia.org/wiki/Born–Oppenheimer_approximation

[9] R. Craciun, D. Picone, R.T. Long, S. Li, D.A. Dixon, K.A. Peterson, and K.O. Christe, “Third row transition metal hexafluorides, extraordinary oxidizers and Lewis acids: Electron affinities, fluoride affinities, and heats of formation of WF6, ReF6, OsF6, IrF6, PtF6, and AuF6”, Inorg. Chem. 49, 1056 (2010).

[10] Jensen, Stig Rune; Saha, Santanu; Flores-Livas, José Abdenago; Huhn, William; Blum, Volker; Goedecker, Stefan; Frediani, Luca, 2017, "GGA-PBE and hybrid-PBE0 energies and dipole moments with MRChem, FHI-aims, NWChem and ELK", doi:10.18710/0EM0EL, UiT Open Research Data Dataverse, V3

[11] http://www.compchemhighlights.org/2017/03/the-elephant-in-room-of-density.html

[12] DeYonker et al. J. Phys. Chem. A, Vol. 111, No. 44, 2007

[13] Stephanie Grimmel, George Schoendorff, and Angela K Wilson. J. Chem. Theory Comput., 05 Feb 2016, DOI: 10.1021/acs.jctc.5b01193.

[14] http://www.compchemhighlights.org/2016/01/assessment-of-accuracy-of-coupled.html