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# Addivity And Basis-set Extrapolation Techniques

Additivity schemes and basis-set extrapolation for energies, geometries, and harmonic frequencies

Basis-set extrapolation

One possibility to reach basis-set convergence in quantum-chemical calculations is to use extrapolation techniques. Such techniques are nowadays well established for HF-SCF and CC calculations when using Dunning's correlation-consistent basis-sets. With X as the cardinal number of these basis sets, i.e. X=2 for cc-pVDZ, X=3 for cc-pVTZ, X=4 for cc-pVQZ, etc., the following exponential formula

{$E_{\infty}({\rm HF-SCF}) = E({\rm HF-SCF/cc-pVXZ}) + a \exp(-bX)$}

involving the three parameters {$a$},{$b$}, and {$E_{\infty}({\rm HF-SCF})$} is used at the HF-SCF level, while a two parameter expression of the form

{$\Delta E_{\infty}({\rm CC}) = \Delta E({\rm CC/cc-pVXZ}) - c/X^3$}

has been found most useful for the correlation-energy contribution.

The first equation means that the HF basis-set limit can be estimated using three HF-SCF calculations. A typical example might be that the required individual HF-SCF calculations are carried out with cc-pVTZ, cc-pVQZ, and cc-pV5Z and that from these calculations the basis-set limit {$E_{\infty}({\rm HF-SCF})$} is obtained using the given extrapolation formula. For the correlation energy, two calculations are sufficient to determine the parameter {$c$} and the basis set limit {$\Delta E_{\infty}({\rm CC})$}. This means that for, for example, calculations can be carried out at the CCSD(T)/cc-pV5Z and CCSD(T)/cc-pV6Z level in order to obtain the corresponding CCSD(T) basis-set limit.

Basis-set extrapolation schemes are often combined with additivity schemes for the various contributions. A common ansatz is here to write the total energy as

{$E = E_{HF} + \Delta E_{CCSD(T)} + \Delta E_{CCSDT} + \Delta E_{CCSDTQ} + ... + \Delta E_{core} + \Delta E_{relat} + ...$}

with the possibility that each of the given term can be treated as accurately as possible, i.e. by using appropriate basis sets and extrapolation schemes. Such additivity schemes combined with extrapolation techniques, for example, form the backbone of schemes such as Wn (n=1-4) or HEAT that are very popular for the accurate prediction of thermochemical energies.

CFOUR offers the possibility to perform calculations based on the described additivity schemes and/or basis-set extrapolations in an automatized manner for energies, geometry optimizations, and the calculation of harmonic frequencies.

Available addivity schemes and extrapolation techniques

The energy can be decomposed currently in the following contributions

{$E({\rm HF-SCF})$}
{$\Delta E_{\rm CCSD(T)} = E({\rm CCSD(T)})-E({\rm HF-SCF})$}
{$\Delta E_{\rm CCSDT} = E({\rm CCSDT})-E({\rm CCSD(T)})$}
{$\Delta E_{\rm CCSDTQ} = E({\rm CCSDTQ})-E({\rm CCSDT})$}
{$\Delta E_{\rm CCSDTQP} = E({\rm CCSDTQP})-E({\rm CCSDTQ})$}
{$\Delta E_{core} = E({\rm ae-CCSD(T)}) - E({\rm fc-CCSD(T)})$}
{$\Delta E_{relat} = E_{MVD1}({\rm CCSD(T)})$} or E_{DPT2}({\rm CCSD(T)}) or $\Delta E_{X2C-1e}({\rm CCSD(T)}) {$\Delta E_{DBOC}$} {$\Delta E_{ZPE}$} Note that a calculation does not need to include all contributions and a scheme should be selected which is suitable for the problem under consideration. b) HF-SCF extrapolation The HF-SCF energy can be extrapolated as described above. c) extrapolation of correlation-energy contributions Correlation energy contributions {$\Delta E_{CCSD(T)}$}, {$\Delta E_{CCSDT}$}, {$\Delta E_{CCSDTQ}$}, {$\Delta E_{CCSDTQP}$}, and/or {$\Delta E_{core}\$} can be extrapolated as described above.

Input description

The input file (ZMAT) consists of

a) normal standard geometry input (Z-matrix format or Cartesian coordinates)

b) a line specifying what property is to be calculated. Available options are here:

%energy request an single-point energy calculation
%geometry request a geometry optimization
%harmvib requests a calculation of harmonic frequencies

c) a section containing the keywords to be used in the individual calculations. The keywords for each type of calculations are given after a corresponding header

%scf_keyword
%ccsd(t)_keywords
%ccsdt_keywords
%ccsdtq_keywords
%ccsdtqp_keywords
%core_keywords
%mvd_keywords
%dpt_keywords
%x2c_keywords
%dboc_keywords
%zpe_keywords

and should be those needed for the corresponding calculations apart from the computational method (keyword CALC) and basis set (keyword BASIS). This means that the keywords specified here are those that give the convergence thresholds, the maximum of number of iterations, frozen-core options, memory requirements, AO-based algorithms, etc. (for further details, we refer to the provided examples)

d) A further section

%keyword

gives the options for the overall calculation in the case of a geometry optimization, i.e., the convergence thresholds for the forces and the maximum number of iterations. the convergence thresholds for the geometry optimization, etc.

e) The basis sets to be used are finally provided after the keywords

%scf_extrap
%ccsd(t)_extrap
%ccsdt_extrap
%ccsdtq_extrap
%ccsdtp_extrap
%core_extrap
%mvd_extrap
%dpt_extrap
%x2c_extrap
%dboc_extrap
%zpe_extrap

The input consists here of (i) the number of basis sets to be supplied (0 = no calculation, 1 = one calculation, 2 or 3 = basis-set extrapolation) followed by (ii) the to be used basis sets. Available are her

PVDZ, PVTZ, PVQZ, PV5Z, PV6Z
PCVDZ, PCVTZ, PCVQZ, PCV5Z, PCV6Z
and the corresponding augmented sets (AUG-PVDZ, etc. ..)

Note that all calculations employing computational levels higher than CCSD(T) should use the MRCC program by M. Kallay which has been interfaced to CFOUR (see Interface to MRCC). CFOUR is partially supported by the U.S. National Science Foundation.