Weizmann-1 theory

W1 (Weizmann-1) theory is a compound method aiming at kJ/mol accuracy in thermochemical calculations.
The determination of W1 energies involves extrapolated values of SCF and correlation energies based on
a series of calculations with increasingly large basis sets. First and second row elements are treated
somewhat differently at various stages of the procedure:

(1) Geometry optimization is performed at the Becke3LYP/cc-pVTZ+1 level of theory. The "cc-pVTZ+1"
basis set is identical to the standard cc-pVTZ basis set for first row elements and for hydrogen. For second row
elements a set of high-exponent d functions is added, taken from the cc-pV5Z basis set as the d-functions
with highest exponent.

(2) Zero-point vibrational energies and thermal corrections are obtained at the Becke3LYP/cc-pVTZ+1 level
of theory using a scale factor of 0.986.

(3) The Hartree-Fock infinite basis set limit (ESCF,INF) is derived from calculations with the augh-cc-pVDZ+2d,
augh-cc-pVTZ+2d1f and augh-cc-pVQZ+2d1f basis sets. These basis sets have different meaning for different
elements: for hydrogen the corresponding cc-pVnZ basis sets are used with n = D, T, Q. For first row elements
the aug-cc-pVnZ basis sets are used with n = D, T, Q. For second row elements the standard aug-cc-pVnZ basis
sets are augmented with high exponent d- and f-type basis functions necccesary to describe inner-shell correlation
effects correctly.

Two different extrapolation schemes have been tested for extrapolating to the SCF infinite basis set limit:

           (3a) Old style (three-point) extrapolation:
           ESCF,INF = ESCF,AVQZ - (ESCF,AVQZ - ESCF,AVTZ)2/ (ESCF,AVQZ - 2xESCF,AVTZ + ESCF,AVDZ)

           (3a) New style (two-point) extrapolation:
           ESCF,INF = ESCF,AVQZ + (ESCF,AVQZ - ESCF,AVTZ)/ ((4/3)5 - 1)

(4) The infinite basis set limit of the CCSD valence correlation energy (ECOR,CCSD,INF) is obtained from
frozen-core calculations with the augh-cc-pVDZ+2d, augh-cc-pVTZ+2d1f, and augh-cc-pVQZ+2d1f basis sets
using the following extrapolation scheme:

           ECOR,CCSD,INF = ECOR,CCSD,AVQZ + (ECOR,CCSD,AVQZ - ECOR,CCSD,AVTZ)/ ((4/3)3.22 - 1)

(5) The infinite basis set limit of the contributions of triple excitations to the CCSD(T) valence correlation
energy (ECOR,T,INF) is obtained from frozen-core calculations with the augh-cc-pVDZ+2d and augh-cc-pVTZ+2d1f
basis sets using the following extrapolation scheme:

           ECOR,T,INF = ECOR,T,AVTZ + (ECOR,T,AVTZ - ECOR,T,AVDZ)/ ((3/2)3.22 - 1)

(6) Relativistic effects and the effects of core-valence correlation are covered as the energy difference between
a frozen core CCSD(T) calculation and a scalar-relativistic calculation with the Douglas-Kroll-Hess model using
all electrons. In both cases the MTsmall basis set is used:

           ECV,DKH = Etot(CCSD(T,Full)/MTsmall/dkh) - Etot(CCSD(T,FC)/MTsmall)

The final total energy of the system is then simply the sum of the components:

           Etot(W1) = ESCF,INF + ECOR,CCSD,INF + ECOR,T,INF + ECV,DKH

Using ammonia (NH3) as an example, the following results are obtained:
 

basis set SCF
energy		 CCSD(FC)
energy		 CCSD(T,FC)
energy		 CCSD(T,Full)/dkh
energy
augh-cc-pVDZ -56.204899 -56.417907 -56.423109 -
augh-cc-pVTZ -56.220044 -56.471167 -56.479421 -
augh-cc-pVQZ -56.223777 -56.486415 - -
MTsmall - - -56.479725 -
MTsmall - - - -56.561568
            


Collecting the data from the above table the W1 total energy amounts to:

           Etot(W1) = -56.224998 + -0.270188 + -0.009389 + -0.081843 = -56.586418 au       (old-style extrapolation)

           Etot(W1) = -56.224938 + -0.270188 + -0.009389 + -0.081843 = -56.586358 au       (new-style extrapolation)

For this particular example the old-style and new-style extrapolation procedures yield very similar results differing by
only 0.000060 au (= 0.16 kJ/mol). Combination of the latter W1 total energy with zero point vibrational energies and
thermal corrections yield the energies at a given temperature:

E0 = -56.552637 au
H298 = -56.548825 au
G298 = -56.570666 au

The necessary energies can be calculated most efficiently in the following sequence:

The "augh-cc-pVnZ+2df" acronym can be used in Gaussian 03 to conveniently load the correct basis functions
for a given job step for first- and second row elements. The W1 method currently invoked in Gaussian 03
with the W1 keyword uses the "new-style" two-point extrapolation scheme for the SCF component.

Literature:

last changes: 09.03.2017, HZ
questions & comments to: zipse@cup.uni-muenchen.de